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Selected Topics on Constrained and Nonlinear Control, Workbook

1. I ur gt m h1 MATLAB gt L w H 012 rozsah Function h2 gt Saturation Q gt U Saturation1 REAL h2 a betat2 sart u 1 u 2 E dr 4 beta2 sqrt u 1 posun 4 3j m mare Step Mux w hi Step1 T Step2 w Fig 7 7 Control structure for nonliner PD1 implemented in Matlab Simulink Matlab function block call PD2 u 1 u 2 u 3 u 4 global alfal alfa2 Qimin Qimax mic m2c betal beta12 beta2 posun 4 4 S1 alfa1 alfa2 closed loop poles Qimin Qimax limit values for the pump input mic m2c waighting coefficients for specifying operating points for linearization mic m2c 0 corresponds to exact linearization i e linearization around actual state leading to overshooting thanks to nonmodelled dynamics mic m2c 1 corresponds to linearization around reference state too slow closed loop dynamics recommended to work with mic m2c gt 0 0 1 0 2 betal betai2 beta2 valve coefficients just betal2 7 beta2 required posun offset of the output valve difference between tank bottom and valve output orifice S1 tank area load basics calibParams load basics valvel load basics valve2 load basic
2. A A Gain2 Transfer Fcn2 Light i intensity 1 5 mea Fig 9 18 FIo controller Simulink model The experimental results can be plot using following commands figure stairs yl 1 yl 2 k hold on stairs yl 1 yl 4 k xlabel t s ylabel light intensity legend system output setpoint Using the recommended three process gains in the experiment you should obtain similar results to Fig 9 19 9 20 Fig 9 21 9 22 and Fig 9 23 9 24 For the plant used in this example the desired control quality was achieved when the process gain K 17 was used The control results can be seen in Fig 9 23 306 Peter Tap k and Mikul Huba Light channel output 70 T T T System output 60 Setpoint 4 50r J Light intensity o T 105 0 10 20 30 40 50 60 time s Fig 9 19 Experimental results for K 3 From the experience with the previous experiments one can assume that a higher value of the process gain leads to lower overshoot with slower tran sients and vice versa After getting some experience with the controller let us practice robust controller tuning At first put dow
3. 0 i L i L 0 20 40 60 80 100 time s Fig 9 52 FSP for filtered optical channel system output 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 333 T T 4 f 1 5m 4H 4 x l 4 SU X am S Td Mar S i aae Tnt eel 227 E 3 a 1 b 4 l i 0 20 40 60 80 100 time s T T 6 r 1 A ae Bulb voltage V wo EE r vragen I 2l if Mian deg if I I ir i 1 i 0 L L I L 0 20 40 60 80 100 time s T T 8 r 1 Bulb voltage V 0 20 40 60 80 100 time s Fig 9 53 FSP for filtered optical channel control signal 334 Peter Tap k and Mikul Huba Comments Remarks 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 335 Comments Remarks 336 Peter Tap k and Mikul Huba Comments Remarks 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 337 Comments Remarks
4. 255 7 4 Exact Feedback Linearization lt 257 T Do o E aso eire Does Sorten t O UE 260 p EE d 0 0000 su sed t d E hh ed See Me Nee deen ees 269 eferemdes oi cece dns hid bere Wand aera tan a dee dad kbc aed aie enh PU WESEL 269 Contents xi 8 Remote Laboratory Software Module for Thermo Optical P ord eS cl O atas ses odeurs oe ene 275 Pavol Bistak ME S oco oe oe ote ea ed ek Rn be dee ba BOB 275 8 2 Technical Requirements eee eee eee 276 PORE 1E 276 6 2 2 Client Computer cicius cases ges iod eia ees 276 23 GRA OVA coe ok ck hawt tet A ee EEERME de banana des 276 8 3 1 Server Installation 5 osse 276 8 3 2 Client lastallatian cose e RR eds 277 8 4 Running the Client Server Application 277 8 5 Client User Interf c 0 ous aie seb Ras 279 Shl Senge TRECE 280 8 5 2 Server IP Address and Control Buttons 281 8 6 Running the Experiment 0 eee en 283 8 7 Rules for Creation of Models in Simulink 284 S Condis u c ls eee qu RD RE eer RE rds 286 Constrained PID control Tasks for Controlling the Thermo Optical Plant 2 291 Peter Tap k and Mikul Huba 9 1 Thermo optical Plant uDAQ28 LT Quick Start 292 9 1 1 Installation in Windows Operating System 292 DX Light Channel Control occ ced dell eee gushed 298 9 2 1 Feedforward Control 299 O22 Ig C
5. The above expressions give the loss for the given d and n but the issue is the find the magnitude of the loss for the set bounded as d Here magnitude can be defined in different ways and the worst case loss Halvorsen et al 2003 and average loss Kariwala et al 2008 for a given H are given by lt 1 2 17 2 Luc 55 My 2 18 L L IMI 2 19 79 ny na d l where M H J HG HY 2 20 Here 6 M denotes the singular value induced 2 norm of the matrix M H and M z denotes the Frobenius norm normal 2 norm of the matrix M Use of the norm of M to analyze the loss is known as the exact local method 2 4 4 Loss Method for Finding Optimal H The optimal H can be found by minimizing either the worst case loss 2 18 or the average loss 2 19 Fortunately Kariwala et al 2008 prove that the H that minimizes the average loss in equation 2 19 is super optimal in the sense that the same H minimizes the worst case loss in 2 18 Hence only minimization of the Frobenius norm in 2 19 is considered in the rest of the paper The scaling factor ES does not have any effect on the solution 2 Measurements for Control Optimization and Estimation 61 of the equation 2 19 and hence it is omitted in the problem formulation Similarly the square does not effect the optimal solution and can be omitted In summary the problem is to find the combination matrix H that mini mizes the
6. 55r TVOmean 5 0295e 005 Fig 9 30 Performance portrait FPrlo controller e Compare the control quality with the Ig controller using a real time ex periment In practice it is often not efficient and sufficient to compensate large time constant s influence just by restricting the closed loop bandwidth Active compensation of dominant loop time constant leads to control structures such as PIo controller Fig 9 33 9 34 The output of filtered optical channel will be used to practice FPIp controller tuning An analogue first order filter is used for non filtered light channel filtering The process can so be approximated as K e Tas lum Tis t1 9 17 The time constant T represents a analogue filter time constant dead time Ty is used to approximate the lag between a bulb voltage step and the corresponding change in the light intensity Obtain the parameters of filtered light channel The exnum command with experiment no 5 can be used For the plant used in this example it gives 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 315 70 lt Setpoint ee FPrl 60r Flo a o T S o T Light intesity o T 10r 0 50 100 150 200 250 300 350 400 450 time s Fig 9 31 FPrlo controller vs FIo controller under 5s transport delay plant output K 8 8728 19 8939 T 17 0096 25 8687 Ta 0 0 6 The dead
7. 1 Problems in Anti Windup and Controller Performance Monitoring 19 1 Til s 1 Kpt Transfer Fen 1 Int Gain Outi In3 Ob wa Transfer Fen1 In2 1 Gain1 Ti2 s 1 Fig 1 19 Implementation of two PI controllers with simple anti windup technique in Simulink 1 1 7 4 Conclusion on Simulation Results The simulations show that all three controller types are able to achieve good control of the plant For all three controllers the performance is significantly degraded in the presence of saturation although to different degrees for the different controllers 1 1 8 Anti windup In this section anti windup is implemented for all three controllers and the simulations re run 1 1 8 1 PI Controllers The simple PI anti windup scheme in Fig 1 1 is used This simple anti windup scheme is implemented in Simulink as illustrated in Fig 1 19 The figure shows the implementation of both PI controllers where input 1 is the reference input 2 is the measurement and input 3 is the actual measured values of the manipulated variables The results when using this anti windup technique are shown in Fig 1 20 We see that the performance is markedly improved due to anti windup although output 2 is not able to return to its setpoint due to saturation 1 1 8 2 Decoupler Although the inverse decoupler reduces the problems of input saturation com pared to ordinary decoupling anti windup is still ne
8. Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh 4 White noise disturbances 4 and measurement noise j hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh kw 0 001 Wvar kw ones 2 1 Disturbance variance QXU eye 7 kv 0 0004 Vvar kv ones 2 1 Measurementn noise variance 1 1 9 2 Simple PI Controller without Anti windup Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh PI controller Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh s tf s Kp1 0 5 Kp2 0 5 Signs are decided through dcgain A B C D which results in 88 3573 86 8074 108 0808 107 9375 Til 2 Ti2 2 ki Kp1i Tii s 1 Ti1 s kinum kiden tfdata k1 kinum kinum i kiden kiden 1 k2 Kp2 Ti2 s 1 Ti2 s k2num k2den tfdata k2 k2num k2num 1 k2den k2den 1 1 1 9 3 Decoupling Transfer function matrix G s defined in file model m 4 which must be run before this file 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 Problems in Anti Windup and Controller Performance Monitoring 25 y Similarly the individual PI controllers used with the decouplers are 4 in this case identical to the controllers used for ordinary PI control The file 4 PIcont m should therefore also run before this file Ahhhhhhhhhhhhhhhhhhhhhhhh
9. So why is least squares not optimal 80 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke The problem objectives function for normal least squares is to get the best match of the available data ie minimize Y H X and it does not consider how the estimate y Hr is going to be used in the future So why is the loss approach expected to be better In the loss approach the problem is Given that y Ha find an optimal H such that the average magnitude of y ll is minimized for the expected future ds and n s which are assumed 2 norm bounded Here we use the data to obtain Y Y opr G and G step 1 This step may possibly be improved but it seems reasonable The main advantage is that in step 2 we obtain the estimate y Hy that will work best on average for the expected disturbances and measurement noise as are indirectly given by the data in Yop that is we consider the future use of the estimator and not just fitting of the data 2 10 Discussion 2 10 1 Gradient Information How can we use the proposed approach in practice for example to find the optimal policy for a marathon runner We need to be able to distinguish between optimal data Y and nonoptimal data to find G and G1 A simple approach is to set y4 Ju that is we want to estimate the gradient Then we know that optimal data corresponds to y Ju 0 and we can do exactly as above Data
10. There are a field for setting the server IP address and following control buttons in the bottom part of the client interface e Server IP the field for setting the server IP address in the form of four three digit numbers separated by points The IP address should be fixed and the user should know it For testing purposed it is possible to start the client application on the server computer In this case it is necessary to enter the value 127 0 0 1 or the string localhost e Connect this button realize the connection to the server A user is in formed about the result of the connection in the field Output When the connection is successful the message is Connected to server In the op posite case wrong IP setting e g the message is Could not connect to 282 Pavol Bist k remote host some time it takes a longer period In the case when some one is already connected to the server the connection with another client will be not realized because in one time only one user can operate a real hardware This time the message is Server busy disconnecting Fig 8 7 Connection e Disconnect the button that cancels the connection with the server and thus the server is free to accept connection from another user There is a message Disconnected in the field Output when the client is disconnected If a client is not active for longer period or because of network failure she he could be disconnected from the server automatical
11. 1 4 4 1 B 1 1 C 1 0 D 0 syst ss A B C D Ts 0 02 create a system structure by discretizing the continous time model sysStruct mpt_sys syst Ts define system constraints sysStruct ymax 10 sysStruct ymin 10 sysStruct umax 1 sysStruct umin 1 Now we can define a set of initial conditions XO and a set of admissible inputs UO as polytope objects 4 set of initial states XO polytope 0 9 0 1 0 9 0 1 1 1 0 1 1 1 0 1 set of admissible inputs UO unitbox 1 0 1 inputs should be such that ul lt O 1 Finally we can compute the reachable sets N 50 R mpt_reachSets sysStruct XO UO N plot the results plot XO r R g The reachable sets green as well as the set of initial conditions red are depicted in Figure 4 24 Fig 4 24 Reachable sets for Example 4 1 4 Multi Parametric Toolbox 143 To compute reachable sets for linear or hybrid systems whose inputs are driven by an explicit control law the following syntax can be used R mpt reachSets ctrl X0 N where ctrl is the controller object as generated by mpt control XO is a poly tope which defines a set of initial conditions z 0 Xo and N is an integer which specifies for how many steps should the reachable set be computed The resulting reachable sets R are again returned as polytope array Example 4 2 In this example we illustrate the reachability computation
12. 91 942 Gar 004568 9 ga 00 3 55918 Gy 97 3 942 8 or 32 noisy data set generated from ideal data G G 4 data sets with 2 or 8 different noise sets to get a total of 8 data sets We here generate the data by adding noise to data from a model calibration set and the validation set is the noise free data To center the data I used opposite sign when the data was repeated The noise was generated using randn command in Matlab It was found that when tested on a single example then almost any of the methods could be the winner To avoid this effect the comparison was run many times with different random noise Table 2 5 shows the average value of Y HX p for 8 data sets after running 250 times The best method is PCR val 0 4931 PLS with 4 com ponents val 0 5137 is the best PLS This is not surprising since we know that the data contains 4 directions The new loss method is not doing so well in this case val 1 055 which is not so surprising since with only 8 data sets there is limited information about the noise Note that it is even worse than least squares val 0 9985 As expected the improvement by adding noise was significant val 0 5850 but it is still not quite as good as PCR and PLS Surprisingly smoothening of y had absolutely no effect in this case even when I added noise on the y data results not shown 2 Measurements for Control Optimization and Estimation 79 Tab
13. Exercises 93 It is assumed that the unmeasured disturbances are k and kz and all other variables are known except of cg which is assumed difficult to measure The unmeasured variables are summarized in Table 3 1 and all measurements Symbol Description ky Reaction constant for reaction 1 k2 Reaction constant for reaction 2 cg Concentration of desired product Table 3 1 Unmeasured variables and known parameters are shown in Table 3 2 The task is to find a con Symbol Description Type Value Unit q Feed flow rate Known input varying m min cA Outlet concentration A Measurement varying kmol m cc Outlet concentration C Measurement varying kmol m V Tank volume Known parameter 0 9 m car Feed concentration A Known parameter 10 0 kmol m cpr Feed concentration B Known parameter 0 0 kmol m ccr Feed concentration C Known parameter 0 0 kmol m Table 3 2 Known variables Inputs measurements and parameters trolled variable which can be controlled using the total flow rate and which maximizes the desired concentration Subtasks Set up the steady state component balances Set up the optimization problem Write the optimality conditions Calculate the reduced gradient Eliminate the unknown Variables Cup poop 3 4 Solution 3 4 1 Component Balance We do a The steady state component balances for the system read g ar qcA kica V 0 92 qCBF QCB kicaV kaegV 0 3 3 92 Wor 9c kocgV
14. One can analyze the loss with alternatives choices for c using the exact local method so why do we need the maximum gain rule The motivation for using the maximum gain rule is at least threefold 1 It is simpler to compute 2 It given insight in particular that we want to control sensitive variables with a large scaled gain Select variables c where the optimal variation c Copt from disturbances and including measurement noise is small compared to the achievable variation c Gu from inputs 3 The scaled gain matrix is Gs S1GS2 Here the gain matrix G is ob tained by linearizing in a single operating point To find the scaling matrices S and 52 we need to reoptimize for the disturbances to find C Copt needed for 4 and to find the second derivative with respect to the inputs to find 5 J zi 2 which can be rather involved calculations If this information missing then one may often get good results by esti mating the optimal variations to find S for example based on operating data and by setting S5 I one should in this case scale the inputs so that their expected effect on the cost is similar This maximum gain rule has also proven to work surprisingly well on many applications Nevertheless if one has data for the optimal sensitivity F then our recommendation is to use the exact local method instead of the maximum gain rule This is because one can analyze alternative ch
15. e Use the dointersect function to detect hit of the target remember Ex ercise 5 e You can plot the whole polytope array R in one shot by plot R Exercise 4 9 Given is a discrete time LTI system xy441 Ax Bux sub ject to constraints tmin S k S Tmax Umin S Uk S Umax lake the following data 11 1 5 5 A l i B los gt Tmin je gt Tmax H gt Umin l Umax 1 Let s stabilize the system by an LQR feedback uz K zp with K dlqr A B eye 2 1 which gives K 0 52 0 94 Obviously such a controller doesn t take constraints into account However there are some states for which all constraints are satisfied e g x 0 1 0 1 clearly satisfies state constraints u K x 0 146 clearly satisfies input constraints and some for which they are violated e g x 1 1 4 satisfies state constraints u K x 1 46 but violates input constraints 4 Multi Parametric Toolbox 121 Now comes the question what is the set of states for which all constraints hold Can it be represented by a polytope As you might have guessed the answer can be found using the MPT s poly tope library In fact the set of states for which all state and input constraints are satisfied can be represented by Ds x Emin lt Tk S Tmax Umin Uk Umax 4 18 Using the fact that uj Kv leads to V z min S Tk Cmax Umin Kik Umax 4 19 which can be rewritten using a compact m
16. h OP 6a n l n 1 Gn 2 p 3 Gna n5 bhi Uno OONO O RUN 0 5558 0 3492 0 2938 0 2399 0 2534 0 1782 0 2118 0 1633 0 1364 F Ch 0 1253 0 0095 0 0235 0 0204 0 5164 7 6000 72 2300 0 0299 0 0104 0 0119 0 0024 0 6424 6 0450 37 8500 0 0698 0 0331 0 0251 0 0295 0 5599 5 5910 0 1069 0 0288 0 0148 0 0188 0 0175 2 8360 7 6770 109 9813 0 0522 0 0292 0 0274 0 0266 2 3090 7 7400 98 4255 0 0245 0 0054 0 0032 0 0023 2 6220 5 4040 60 9862 0 0221 0 0126 0 0247 0 0188 2 2820 6 8110 98 1203 0 0186 0 0175 0 0146 0 0220 3 1980 5 4450 98 2250 0 0486 0 0383 0 0167 0 0464 3 2560 6 0385 89 0348 Table 5 3 Fuel path model parameters h OP ah n i p amp Q 00 1 O C 4 C 2 0 2516 0 2132 0 2470 0 1571 0 1888 0 1555 0 1217 0 1715 0 1935 ano ang Gna no bhi bn al 0 0827 0 0771 0 0636 0 0431 0 1463 0 5987 0 9737 0 0728 0 0636 0 0441 0 0373 0 1474 0 6242 0 7749 0 0602 0 0607 0 0377 0 0448 0 1467 0 5805 0 8746 0 0402 0 0835 0 0181 0 0579 0 3897 0 5257 0 4549 0 0475 0 0865 0 0203 0 0601 0 3509 0 4987 0 9676 0 0299 0 0667 0 0183 0 0428 0 4020 0 4820 0 7991 0 0218 0 0471 0 0077 0 0168 0 5273 0 3537 0 7585 0 0025 0 0562 0 0081 0 0227 0 5931 0 3097 0 0154 0 0083 0 0478 0 0136 0 0041 0 6168 0 2477 0 0373 tr ma e mg v ot rpm 18 20 22 19 21 23 20 22 24 6 658 8 166 9 296 5 668 7 275 8 625 4 987 6 437 7 663 122 5 141 5 157 5 107 5
17. has been performed at the same time These changes simulate varying working regimes of an engine which is adapting its run to a daily traffic Changes in t and nen quantities are determining the engine load at the same time ensuring that the engine passes through several work ing points during its operation As mentioned in Section 5 3 3 2 the engine revolutions are not included among explicit variables of local models but they build together with a delayed throttle valve position a vector of an working point k The quality of control is sufficient Fig 5 23 variable A with exceptional acceptable overshoots in both directions These overshoots of the controlled variable A have been caused by smaller model precision due to its distance from the working point at which the system identification has been per formed This effect is caused by the approximation of a particular model 5 Implementation of MPC Techniques to Real Mechatronic Systems 221 22 sU LILI LI LILI LL gt 217 J 20 40 60 80 100 120 140 160 2600 T T T i T T T S 2300 I c 2000 real l 7 7 z desired jo 60 80 100 120 140 160 1 2 T T T T T T T L 1 R 0 8 I I l I 1 20 40 60 80 100 120 140 160 PEE 10 A 9 T T T T T T T E 5 L 1 L L L L L 20 40 60 80 100 120 140 160 ts Fig 5 23 Results of a real time control of the SI engine from the other working points
18. models The overshoots were also diminished by an introduction of a penalization of a difference from the terminal state This means that into cost function is added a member comprising the difference between the instantaneous and the desired terminal state of a state vector As it can be seen skipping this member in a cost function reduces the guality of control significantly Fig 5 22 The corresponding control action computed by the controller is shown in Fig 5 23 variable Inj time The initial engine warm up to 80 C eliminated model plant mismatch caused by temperature dependent behavior of the engine The control has been performed by choosing the penalization r 0 1 Utilizing the member pl N f N 3 of a cost function by setting p 1 0 allowed us to shorten the control horizon to Np 20 what significantly unloaded the computational unit and stabilized the controlled output of the engine on this shortened horizon as well The best control has been achieved in the neighborhood of working points what is logically connected to the most precise engine model at those points In other working points the control is still good enough with small deviations from the stoichiometric mixture 222 Gergely Takacs et al 5 3 6 Conclusion Considering the preliminary results from the real time experiments at the engine it can be concluded that the idea of the AFR model predictive con trol based on local ARX models is
19. sysStruct A 0 0315 0 0 0315 0 0315 sysStruct B 0 0769 0 sysStruct C 0 1 sysStruct D 0 must be specified even if it is zero constraints sysStruct xmin 21 21 sysStruct xmax 3 5 3 5 sysStruct umin 17 sysStruct umax 3 sysStruct ymin 21 sysStruct ymax 3 5 symbolic labels sysStruct StateName level 1 level 2 sysStruct InputName inflow sysStruct OutputName level 2 verify the setup mpt_verifySysStruct sysStruct simulate the evolution for 5 steps x 5 10 4 initial state u 2 ones 10 1 inputs to use in the simulation X x 4 store of the simulated states for k 1 5 x mpt_simSys sysStruct x u k X X x X X23 end plot the state trajectories t O size X 1 1 plot t X Solution 4 12 for Exercise 4 12 probStruct N 6 4 Multi Parametric Toolbox 163 probStruct Q eye 2 probStruct R 1 probStruct norm 2 mpt_verifyProbStruct probStruct Solution 4 13 for Exercise 4 13 1 What is the optimal control action for z 0 755 we assume sysStruct and probStruct have been defined ctrl mpt_control sysStruct probStruct online 4 optimal u associated to x0 5 10 u ctrl 5 10 u 0 0121 2 Which input corresponds to z 0 1 5 Can you give the answer even before running the code in MATLAB hint take a look at state constraints in sysSt
20. 1 on line controller controller online mpt control sysStruct probStruct online explicit controller controller explicit mpt control sysStruct probStruct explicit initial state for the closed loop simulations x0 3 0 timing of the on line controller tic X sim controller online x0 4 Multi Parametric Toolbox 165 t toc t size X 1 timing of the explicit controller tic X sim controller explicit x0 t toc t size X 1 Solution 4 16 for Exercise 4 16 run ex 2 to load all necessary data close all plot the controller regions plot controller explicit plot the two initial conditions of interest x1 4 1 x2 4 2 hold on plot xi 1 x1 2 kx x2 1 x2 2 ko markersize 12 optimal control action associated to x1 since x1 is contained in one of the regions we expect a feasible answer u controller explicit x1 check that x1 is indeed in one of the regions isinside controller explicit Pn x1 optimal control action associated to x2 since x2 is outside of the colored area there should be no control action associated to this state in which case u u controller explicit x2 check that x2 is indeed outside of the regions isinside controller explicit Pn x2 now plot the feedback laws close all plotu controller explicit rotate the graph manually to inspect it 166 Mich
21. 1 KO Is Tf s 1 Fig 9 34 Plo controller Simulink Model to get FPIo controller you need to add pre filter to the reference setpoint input Gains corresponding to approximation of step responses produced by incremental input chang 20 a o o o 00 o 18r o A 16r L 2 o 14 o o Measured gains S 12r 99 o Mean gain D B 10r id 8 o a 8 6 E 4r 2 I oO L L L 1 J 0 1 2 3 4 5 Bulb voltage Fig 9 35 Process gain in several operation points as function of the plant input T Tio 0 8 0 6 0 06 9 20 T Tio 0 8 and T Tio 0 6 should give transients of the undelayed system output with up to 2 overshooting and the delayed system output should not overshoot T T10 0 06 corresponds to controller tuning where Ty e max T4 1 6310 9 21 318 Peter Tap k and Mikul Huba Process time constants 30r 25 o o o a o o 96 O O 9 00 20 o o o o9 o o o o o Q Measured time constants Mean time constant Process time cosntant a T o T oO L L L L J 0 1 2 3 4 5 Bulb voltage Fig 9 36 Process time constant in several operation points as function of the plant input This tuning can yield to transients with approximately 10 overshooting of the undelayed system output and the delayed system output should not overshoot see the performance portrait in Fig 9 37 Compare the results with the FIo controller Fig 9 17 Use the foll
22. 127 5 145 0 95 7 114 1 130 3 Ch u ms m y mg v 8 2735 9 7026 10 7074 7 3588 8 7353 9 9395 6 5651 7 7898 8 8919 1500 1500 1500 2000 2000 2000 2500 2500 2500 ot rpm 1500 1500 1500 2000 2000 2000 2500 2500 2500 VIZ Te 39 SYLL e 5 Implementation of MPC Techniques to Real Mechatronic Systems 215 amp Comb engine T gt u f z real process Nady identification Ms a Air path 1 Len gt V Au Fuel path ms f Fig 5 19 Scheme of the identification process 5 3 3 4 Weighting Functions The full working range of the engine has been covered by discrete amount of local linear models LLMs identified at particular working points Due to a request of a smooth and continuous global engine model design of weighting functions defining validity of each local model according to an instantaneous working point of the engine was crucial There were designed particular in terpolation functions for every LLM assigning it 10096 validity exactly at the belonging working point with decreasing trend depending on the change of the throttle valve opening t or the engine revolutions nen The Gaussian functions were used as the local weighting functions due to their suitable shape fulfilling approximation properties Each one is defined as Pr k z 0 exp Anen k At k a EN Peal 5 24 The choice of tuning
23. 2009 Normey Rico and Camacho 2009 considers compensation of an output disturbance by correction of the reference value whereby the disturbance is reconstructed by using the PPM However despite to the proclaimed unification it separately presents solutions corresponding to stable integral and unstable plants Use FSP structure from Fig 9 51 with filter 322 Peter Tap k and Mikul Huba Controller output FPI T T 9 0 8 iati FPI T T 9 0 6 FPI T T 9 0 06 Flo controller 2 5 Bulb Voltage 0 L L ir 1 i J 0 200 400 600 800 1000 1200 1400 time s Fig 9 40 Experimental results control signal 1 Bus F s 9 28 nd TT 9 28 Use FOPDT approximation 9 18 whereby Ks K T and a 1 T Choose K o max K 0 max Ta Tio max Ti ao min a Set up the experiments in the same way as in the previous exercise to be able to compare the results For controller tuning use following rules The P action should be set to Kp 1 T ao K o 9 29 Filter parameter 11 set to Bu Tio 1 1 Te Tio 1 T Tyo e 9 To 9 30 Try multiple T and T settings e g T T10 2 4 8 16 Tp T 2 4 8 In this example various T settings were used The filter time constant was set to Ty T 4 Experimental results are shown in Figs 9 52 9 53 Questions e Was there any overshooting in the experiments e How did the increasing of parameter T affe
24. 2009 Bidirectional branch and bound for controlled vaiable selection part ii Exact local method for self optimizing control Computers and Chemical Engineering 33 1402 1414 Kariwala V Cao Y Janardhanan S 2008 Local self optimizing control with average loss minimization Ind Eng Chem Res 47 1150 1158 Yelchuru R Skogestad S 2010 Miqp formulation for optimal controlled variable selection in self optimizing control In PSE Asia 2010 July 25 28 Singapore 2 Measurements for Control Optimization and Estimation Comments Remarks 87 88 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Comments Remarks 2 Measurements for Control Optimization and Estimation Comments Remarks 89 90 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Comments Remarks Chapter 3 Measurement polynomials as controlled variables Exercises Johannes J schke and Sigurd Skogestad Abstract In this chapter we present two exercises for finding controlled variables which are polynomials in the measurements Detailed solutions and maple source code are included so that the reader can easily follow the procedure 3 1 Introduction To illustrate concepts from described in the textbook we present some small problem and go through the solution step by step The reader is encouraged to experiment on his own to understand the ideas better For solving the CSTR case study the multires package is require
25. 3 plot I y hold on plot Q r Q1 g Q2 b Q3 m opts hold off Solution 4 4 for Exercise 4 4 Plot the the set difference between Q and Q3 D Q Q3 plot D g hold on plot Q r Q3 m opts hold off 4 Multi Parametric Toolbox 125 Fig 4 13 Intersection of polytopes Q Q2 and Q3 How many polytopes define the difference length D ans Solution 4 5 for Exercise 4 5 unitbox 2 unitbox 2 0 1 P1 P2 05r 0 5 215 Fig 4 14 Set difference between Q and Q3 126 Michal Kvasnica 4 enumerate extremal vertices Vi extreme P1 V2 extreme P2 nvi size V1 1 number of vertices of P1 4 initialize the store of vertices v for each vertex of Vl add v1 V2 to the store for i l size V1 1 vi Vi V V repmat vi nvi 1 V2 end compute the minkowski addition as convex hull sum polytope V is the output equal to MPT s native implementation sum P1 P2 ans Solution 4 6 for Exercise 4 6 Here is a general solution P1 polytope 1 1 1 1 1 1 1 1 P2 polytope 0 1 0 1 0 1 0 1 0 0 1 A1 bi double P1 A2 b2 double P2 At zeros size A1 1 size A2 2 A1 A2 A2 bt bi b2 Pt polytope At bt sum projection Pt 1 size A2 2 is the sum equal to the output of P1 P2 sum P1 P2 ans Solution
26. GI Ga F G JIi Jua G3 G G1 Ga G4 In addition we need to handle setpoint changes for the primary variable Y s which requires changes in the setpoint for c Note that y only affects the cost function and may be viewed as a price variable p so from 2 22 the required change in the setpoint is Ac HF Ay s where Fp may be obtained from the model exercise derive the expression 2 8 2 Indirect Control of y Based on Estimator Note that we still have not used the available degrees of freedom in H To simplify the setpoint adjustment we may use the degrees of freedom in H to make HF I or equivalently c y This means that c should be an estimate of y Note that y G4u and c H Giu setting d 0 for simplicity These two gains need to be identical so we use the extra degrees of freedom in H to make HG G 2 34 It is then easy to include changes in setpoint we just control c at y s Some comments on this estimator e What kind of estimator is this If we look at the problem formulation then we see that it is be the optimal estimator in the following sense When we control y at the estimate using the unconstrained degrees of freedom then this minimizes the deviation from the given value setpoint for the expected range of disturbances and measurement noise e For practical purposes when obtaining the model it may be smart to let the primary outputs be the degrees of freedom u y that
27. It consists of the estimation of the flow coefficients of valves and of the pump s parameters according to the above mentioned process 6 2 3 Automatic Calibration and Identification The above mentioned procedure of calibration and identification is obviously time consuming and also not easy to calculate manually To perform the such long time step by step measurements of the pump s characteristics plus to save the measured values subsequently after each measurement and evaluate them afterwards we have to sacrifice an enormous quantity of time That s all is the reason why we developed the software package for automatic iden tification of system s individual parameters and for identification of all valve and pump parameters The control is very easy only one click is needed to get the required identification in the menu Menu s environment can be seen in Fig 6 5 As the processes were described in details in the previous chapter the individual functionalities will be mentioned now just shortly At the identifi cation of the valve 1 2 or 3 the tank is fill up to the height of approximately 24 5 cm then we wait till the liquid s level is stabilized and the tank is af terwards emptied to 0 8 cm At almost zero level the valve s behaviour is strictly non linear and as we are not planning to regulate the level under 1 cm we don t consider the lowest boundary The measured values are saved to gether with all used constants into the ma
28. Plant model Calculated measuremehts State estimates Fig 1 3 Illustration of anti windup for controllers based on static state feedback combined with state estimation In many applications it is desired to have offset free control at steady state This requires the use of integral action This is often incorporated in a state estimator state feedback control design as illustrated in Fig 1 4 The state estimator only estimates actual plant states whereas the state feedback is designed for a model where integrators which integrate the con trol offset are appended to the plant model When implementing the con troller the integrators are a part of the controller in the control system The values of the integrators are thus directly available in the control system and clearly there is no need to estimate these states 8 Morten Hovd and Selvanathan Sivalingam References Model for state feedback Controller design Plant r i State inputs icd feedback gt Plant Tee r Integrator s controller I State estimator State estimates Fig 1 4 State estimator and static state feedback augmented with integral action However when integration is incorporated in this way the integrating states may wind up even if the actual input values are sent to the state estimator Fig 1 5 illustrates how the anti windup signal to the inte
29. time s FPI T T 970 6 38 T T T T 36r J 34r E Light intensity 1060 1080 1100 1120 1040 1000 1020 1140 time s PP T T 970 06 38r J gt o 36r J E 34 4 z o 32r J 1000 1020 1040 1060 1080 1100 1120 1140 time s Fl controller 38r J D o 36 E 34 o 32 30 1000 1020 1040 1060 1080 1100 1120 1140 time s Fig 9 43 Experimental results detail delayed system output 325 disturbance step 326 Peter Tap k and Mikul Huba FPI T T 970 8 38 36 34 32 Light intensity ggob 0 1000 1020 1040 1060 1080 time s FPI T T 0 6 1100 1120 1140 38 36 34 Light intensity 32 30b 77 1000 1020 1040 1060 1080 time s FPI T T 4070 06 1100 1120 1140 38 36 34 Light intensity 32 30 1000 1020 1040 1060 1080 1100 1120 1140 time s Fly controller o Dv Light intensity Co R c2 o 1000 1020 1040 1060 1080 1100 1120 1140 time s Fig 9 44 Experimental results detail undelayed system output disturbance step 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 327 Fig 9 45 PI controller FF Pa Ks gt Att me gt P gt j gt A i Temperature 1 Thermo optical Repeating P asction Saturation 3 Seguence v 0 gt 16 fo interface Gain
30. 0 94 Johannes J schke and Sigurd Skogestad 3 4 2 Optimization Problem The objective function is J cp 3 4 which we want to minimize subject to the process model min J s t g 0 3 5 g2 0 ga 0 3 4 3 Optimality Conditions We write the Lagrangian L J z ATg z 3 6 where z c4 cp co q T The first order optimality conditions are then VZzJ z Vzg z 0 aia es 3 7 Next we calculate the null space of the constraints N n1 n2 n3 na with ni car ca q kV 3 8 qcpr CBrkiV qep t cgkaV k1Vcar kica V Pee UN eee a k2V q la V n3 copd corqk V copkaVq cork2V7ky coq coqkiV cck2Vq cckoV ki kaVqcpr koV cgrki koVqcp koV cpki kaV kicAr kaV kica q kaV q kV a 3 10 n4 1 3 11 Eliminate Lagrangian multipliers using the null space we write the opti mality conditions c N z TV J z 0 3 12 3 Measurement polynomials as controlled variables Exercises 95 Since we control c to zero we need to consider only the numerator which is Num c qcepr CBrkiV geB eglaV k1Vcapr kicaV 3 13 This expression cannot be used for control yet because it contains unknown variables These have to be eliminated in the next step 3 4 4 Eliminating Unknowns k k2 and cp We use the package multires to construct the matrix for the toric resultant qcBF q cBFV VcAF cAV V 0
31. 0 0 4 5 6 1 Import of Models from External Sources MPT can design control laws for discrete time constrained linear switched linear and hybrid systems Hybrid systems can be described in Piecewise Affine PWA or Mixed Logical Dynamical MLD representations and an efficient algorithm is provided to switch from one representation to the other form and vice versa To increase user s comfort models of dynamical systems can be imported from various sources Models of hybrid systems generated by the HYSDEL language MLD structures generated by the function mpt_pwa2mld Nonlinear models defined by mpt_nonlinfcn template State space and transfer function objects of the Control toolbox System identification toolbox objects MPC toolbox objects In order to import a dynamical system one has to call sysStruct mpt sys object Ts where object can be either a string in which case the model is imported from a corresponding HYSDEL source files or it can be a variable of one of the above mentioned object types The second input parameter Ts denotes sampling time and can be omitted in which case Ts 1 is assumed Example 4 6 The following code will first define a continuous time state space object which is then imported to MPT sampling time Ts 1 continuous time model as state space object di ss 1 1 0 1 1 0 5 1 0 0 1 0 0 import the model and discretize it sysStruct mpt sys di Ts Note 4 1 If the
32. 0 0 qcBF q cAV 0 0 Vo qcAF qcA 0 cAV 0 0 0 0 M 0 0 qcBF g cAV 0 V 3 14 0 0 qcAF qcA 0 cAV 0 0 qcCF qcC 0 0 0 0 vo L 0 0 qcC F qcC 0 0 0 V 3 4 5 The Determinant The factorized determinant of M is c q cAV cAFCA CAFCOF CAFCC c 3 15 We see that the pre factor q cAV is nonzero under operation so the condi tion for optimal operation is only the last factor CAFCA Carcor CAFCO CA 0 3 16 Now the we have ended up with a controlled variable combination which con tains only known variables It may be confirmed by the reader that controlling c to zero leads to the same solution as solving the optimization problem 3 5 oma rt O U K WN F W W o o o o W o Oo wn YN NNN NN NN E FB FE E FE F FE FE HH O 340 0 R ON R O 4 AAR WN HF O O NJN O a A O N F O 96 Johannes J schke and Sigurd Skogestad 3 4 6 Maple Code The maple code for the CSTR example is given below Maple code for the CSTR example SAHARA HAHAHA HHH BHP HHH HH This is the file mapleCSTR mpl Simple CSTR Johannes Jaeschke Nov 2010 THEIBIBBHHHBHBRHHHBHHBHHHHHHEHHHBHHBHERHHHBHHBHBHEHBHHHEHHBHHBHBHHHEHHEHHBBHE with LinearAlgebra with VectorCalculus Define the cost to minimize J cB Setting up the constraints gi q cAF q cA ki cA V g2 q cBF g cB ki cA V k2 cB V g3 q cCF q cC k2 cB V g 7 g1 g2 g3 Derive to obtain first or
33. 1 1 F M ni rec Transfer Fcn 1 WEN T il s 1 Step Light v intensity 1 f sta O us Tis Fig 9 46 PI controller Simulink model e Which controller performed better in comparison with PI controller References Astr m K and H gglund T 2005 Advanced PID control In ISA The Instru mentation Systems and Automation Society Research Triangle Park NC Normey Rico J E Bordons C and Camacho E F 1997 Improving the robustness of dead time compensating pi controllers Control Engineering Practice 5 801 810 Normey Rico J E and Camacho E F 2009 Unified approach for robust dead time compensator design J Process Control 19 38 47 Normey Rico J E Guzman J Dormido S Berenguel M and Camacho E F 2009 An unified approach for dtc design using interactive tools Control Engi neering Practice 17 1234 1244 Visioli A 2006 Practical PID Control Springer London 328 Peter Tap k and Mikul Huba Pl controller output of filtered light channel K 19 70 r T i e delayed output lo undelayed output 60 setpoint J i l so i D o M J NE Try E 5 0 i f L L 1 0 20 40 60 80 100 time s
34. 1 Introduction This subsection introduces the use of multi parametric programming based MPC in active vibration attenuation The Multi Parametric Toolbox MPT is a freely available and distributed Matlab toolbox for the design analysis and rapid deployment of PWA con trollers Kvasnica et al 2004 2006 The multi parametric control laws cre ated via the MPT toolbox are not only usable in Matlab but it is possible to deploy them onto rapid software prototyping platforms using the Real Time Workshop More details about the MPT toolbox can be found in Chapter 4 This workbook assumes the current release Version 2 6 2 of the toolbox available online Matlab is assumed to be used for multi parametric con troller calculation and simulations while the real time code is assumed to be implemented using the xPC Target protocol and Simulink 5 2 5 2 Off line Controller Computation The control objective assumed in this example is to regulate toward origin since the beam equilibrium is located at the origin of the state space Let the cost function to be set as a quadratic 2 norm Penalization and constraints are identical to the NRMPC and QPMPC case input penalty R 10E 4 state penalty matrix was set to O CT C Input constraints are set to 120 V and output or state constraints are not engaged We begin with loading the system model load n4s2A mat sysStruct A n4s2 A sysStruct B n4s2 B 1 Software package and extended doc
35. 102 4 0 Computational Geometry in MPT 0 103 A2l POliSDeE sooo cer athens ia pet era ma 103 4 22 Polytope Arraya 5 sessanta 106 4 2 3 X Operations on Polytopes 107 42 4 Functions Overview 2 0 0 0 isle reu 113 qu BXOFDBAB olores A herein eee HE Parier E Ete ee 115 T JSODDIOBE lt c fo 8 bane cao ed PE eee d UE dece E 122 4 5 Model Predictive Control in MPT suus 131 ADL BeoscuUsBg vo cube cable k dek eae 133 4 5 2 Closed loop Simulations 0 eee eee 134 4 5 3 Code Generation and Deployment 135 4 5 4 Advanced MPC using MPT and YALMIP 136 455 ABS 234 eda vnc ark he eon eke Ronee bran 141 4 5 6 System Structure sysStruct 147 4 5 7 Problem Structure probStruct 151 ONEMS ALS TORT Ec a Pe ee ee ee Pe 156 E ins ER 162 SICH oened ek ee Mew hdd ad eee eden ees 166 Contents Implementation of MPC Techniques to Real Mechatronic SVOTE 25 ieee OR L ee O eda eed e p debe s 171 Gergely Takacs and Tom s Pol ni and Boris Rohal Ilkiv and Peter imon i and Marek Honek and Mat Kopa ka and Jozef Csamb l and Slavom r Wojnar VNMEME CI ID nn PERSEQUERIS METER 172 5 2 MPC Methods for Vibration Control 173 Boll OCI oe ce ok no dea zbo len EC edes 173 2 2 RWS ode eed cde enh abe eaten bd 174 5 2 3 Quadratic Programming based MPC 177 5 24 New
36. 2 ein an vale 12 VO Interface raw signal from sensor3 in EU pf calibP aramsB valves calibP aramst5 I3 m valve 2 gt offset 3 Gain4 valves 4 M tank3 valve 2 3 valves 3 4 Terminator valve 3 Fig 6 4 Basic diagram for calibration and configuration Under manual calibration we will partially fill the calibrated tank and empty it subsequently Raw integer number from each output is sensor offset Tiempty This is specified in engineering units EU Afterwards we fill up the tank to the maximum height of liquid s level and we define the output s value Ziempty in EU We deduct once again the value of sensor s outputs The conversion to liquid s level in meters is defined by the relation itu Tiempty Ax Tlactual Viempt g Tisetual Flempty 6 5 sensorGaini sensorGaini 236 Vladim r ilka and Mikul Huba Here T1actual is the actual value of liquid s level in EU X1empty is the output in EU at the zero liquid level and 7154 is the value of the output in EU at maximum liquid level and Az is the height of the maximum liquid level For example if we know that at the zero level 1empty 800 EU at the maximum level Az 0 25 m and output 3300 EU for the output 2300 EU it holds 3300 800 sensorGaim 795 10000 EUm 2300 800 r 10000 0 15m After the calibration it is suitable to perform identification of system s pa rameters
37. 2 w dim 1 3 w dim 1 5 size Q 1 Matrix square roots of the penalization variables are computed These are required in the construction of the invariance condition The closed loop linear quadratic gain is calculated as well and it makes use of the prediction model and penalties introduced earlier sqrtR sqrtm R sqrtQ sqrtm Q K dlqr An B Q sqrtR sqrtR Phi0 An B K 5 2 4 3 Off line Variables and Constraints Four optimization variables are declared according to the dimensionality of the problem Matrix N is fully parametrized and sguare while the rest of the optimization variables are real valued and symmetric Xg sdpvar w dim 1 1 w dim 1 1 Yg sdpvar w dim 1 1 w dim 1 1 N sdpvar w dim 1 1 w dim 1 1 fu11 M sdpvar w dim 1 2 w dim 1 1 The LMI constrain the semi definite programming problem The set com mand instructs the parser YALMIP to construct a constraint in an LMI form Invi Yq Xq Xq Xq Inv3 PhiO Yq B M Phi0 Xg N Phi0 Yg B M PhiO Xq if gamma lt 1 tolerance gInv gamma eye w dim 1 5 w dim 1 2 zInv zeros w dim 1 5 w dim 1 2 2 w dim 1 1 Inv2 blkdiag sqrtQ sqrtR Yq Xq K YqtM K Xq F set gInv zInv Inv2 zInv Invi Inv3 Inv2 Inv3 Invi gt 0 else F set Invi Inv3 Inv3 Invi gt 0 end The if construct checks whether there is an input constraint defined or not If yes the feasibility condition is translated to the proper LMI and added
38. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 1 Problems in Anti Windup and Controller Performance Monitoring Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Reference signals Parameters in signal generator blocks for references Ahhhhhhkhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh ylamp 1 yifreq 0 005 y2amp 0 y2freq 0 005 Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh System parameters 4 Distillation column model 4 hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhlhhh lhl A 0 005131 0 0 0 0 i 0 0 07366 0 0 0 0 0 0 1829 0 0 0 0 0 0 4620 0 9895 0 0 0 0 9895 0 4620 B 0 629 0 624 0 055 0 172 0 030 0 108 0 186 0 139 1 230 0 056 0 7223 0 5170 0 3386 0 1633 0 1121 a ll Output 1 1 5 T l l 1 L l L L 50 100 150 200 250 300 350 400 Output 2 I 450 500 2 T ib 2 L L l L L L l L 0 50 100 150 200 250 300 350 400 Fig 1 24 Responses with LQG controller with integral action and anti windup L 450 500 23 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 10 24 Morten Hovd and Selvanathan Sivalingam 0 8913 0 4728 0 9876 0 8425 0 2186 Bd 0 062 0 067 0 131 0 040 0 022 0 106 0 188 0 027 0 045 0 014 D 00 00 sys ss A B C D G tf sys Used in decoupler
39. 6 7 Pump implementation in simulink For removing sediments from the walls of the tanks use a soft brash appro priate for bottle cleaning Pump out the water containing such sediments and flush the system several times by clean water After that you need to pump out also the standard water used for cleaning and again fill the container by the distilled one For eliminating the evaporation fill in the water into tanks that are covered and have smaller evaporation area After a longer period without use it may happen that some valve does not open or close In such a case it is necessary to repeat open close procedure several times even when it does not function and then slightly hit the metal screws in the valve centre This may help to release the valve When you know that the system will not be used for a longer time pump out all water When you keep all these rules the system does not require any mainte nance Only in extreme situations when the system was not used for longer time period without pumping out all water the sediments may not only cover the walls of containers but even tubes pressure sensors and valves When facing problems with pressure sensor after emptying the containers by a gen tle pull remove the sensor from the tube and clean the tube by a thin wooden tool When the sediments clog up the valve it is necessary to unscrew fit tings from both sides and to flush the valve under pressure e g by a rubber ball filled b
40. 8 Sos p a 2 E a 6 206 a 04 4 0 2 0 2 4 6 8 10 12 0 02 04 06 08 1 12 14 16 18 pump voltage V pump 1 output m s x10 a IO characteristic b Inverse IO characteristic The characteristic of the flow coefficient of the valve o5 Outflow characteristic fer teal system an model tenet 0 038 identification 2cm steps Ll nea Set 1 o identification steady states 0036 approximation 2 0 084 2 S 0 032 amp 0 15 0 03 5 E 0 028 2 0 026 0 1 8 5 0 024 2 0 022 0 05 0 02 0 018 0 i i i i i n i o 0 05 0 0 2 0 25 o 5 10 15 2 30 35 40 45 1 0 15 0 25 level m t s c Determining c1 by the first method d Determining c1 by the second method Fig 6 6 Measured Input Output pump characteristic and its approximation by the 4th degree polynomial and inverse input output pump characteristic and its approx imation by the 4th degree polynomial above Determination of the valve coefficient c with two approaches comparing the measurement and approximative data for tank 1 below The measurement of pump s characteristics is more time consuming In order to achieve the required accuracy we continue with increasing the pump input by the step 396 of the maximal range 10096 up to the moment where the pump starts slowly to draw the liquid In our case it will be around 19 21 96 At lower inputs the pump is not working At the beginning of measurement we empty the tank and
41. Az t Bult 4 23 or a Piecewise Affine PWA system with f a t u t Aix t Byu t f if Wl aD 4 24 u t where the state update matrices change according to the position of the state input vector i e the tuple A Bi fi is valid if Bel Dj where D i 1 D are polytopes of the state input space Furthermore it is assumed that the system variables are subject to constraints zx t ce X ult EU vt 20 4 25 where T and U are polytopic sets In MPC optimal control actions are calculated by formulating and solving a suitable optimization problem which usually takes the following form N 1 min Pyanlp X 10x Ruzllp 4 26a k 0 s t zo x t 4 26b k 1 f k uk k 0 N 1 4 26c EX k 0 N 4 26d RAM We oco 4 266 where x and ux denote respectively the state and input predictions at time instants t k initialized by the measurements of the current state z t and Py Q R are penalty matrices The prediction is carried out over a finite prediction horizon N In addition Pz p with p 2 stands for the square of the weighted 2 norm of z i e z7 Pz Similarly p 1 and p oo denote the 1 and co norms of some vector The aim of MPC is to find the vector Uy ud uT uT T of optimal control inputs which minimizes the cost function 4 26a If f in 4 26c 132 Michal Kvasnica is linear as in 4 23 the MPC formulation can be rewritten as a Quadra
42. FW4W ns The objective is to find the nonsquare ne x ny matrix H note that nu ne We will here consider some special cases of this problem which historically were developed before the convex and analytical solutions presented above 2 7 1 No Measurement Noise Nullspace Method full H For the special case with no measurement noise Wy 0 and with more independent measurements than independent inputs and disturbances Ny Nu na it is possible to find H such that HF 0 2 28 that is the loss is zero This is called the nullspace method Alstad and Skogestad 2007 because H is in the nullspace of F In this case Gy and W a do not matter for finding the optimal H The nullspace method is very simple and has been found to be very useful in applications Since the nullspace method neglects the effect of measurement error it is important to use preselect a subset of the measurements that are expected to be insensitive to measurement errors Also one cannot include too many disturbances because otherwise one cannot satisfy the requirement ny gt ny ng One limitation with the analytical formulas in 2 24 and 2 25 is that they do not give the nullspace method as a special case This is because Y FWa Wn at most has rank ng when W ny 0 Thus the ny x ny matrix YY at most has rank nq and is not invertible because this would require the rank to be ny However the convex optimization problems in Theorem
43. M2 1 nc 1 nx nc 1 nx nx R eye nc F F1 M2 1 nc 1 nx nc 1 nx nx Qe M1 1 nc 1 nx nc 1 nxtnx G G1 M1 1 nc 1 nx nc 1 nx nx Qe M1 1 mc 1 nx nc 1 nx nx To ensure feasibility and stability beyond the prediction horizon the con straint checking horizon is calculated as well This process is started up by an initialization procedure 180 Gergely Takacs et al Ki K Ki 2 K A B K i 1 Nc 0 u uh 1 The length of the constraint checking horizon is computed in the following loop while u gt uh Ki i 2 2K A B K i 1 f Ki i 2 Am Ki 1 i 1 Ki 1 i 1 b uh ones i 1 1 ul ones i 1 1 xO linprog f Am b u Ki i 2 x0 Nc Nc 1 i i 1 end This can be followed by defining the constraints and re formulating them to be useful for direct guadratic programming solution This formulation assumes symmetric input constraints Aci eye nc bO uh ones nc Nc 1 ul ones nc Nc 1 Bxi zeros nc 1 nx for i 0 Nc Aci nc i Ki i 1 M21 Bxi nc i Ki i 1 M11 end Ac Ac1 Ac1 Bx Bxi Bx1 5 2 3 4 Re formulating for the Simulink Interface This sub subsection introduces a way to reformulate the prediction matrices so they can be used directly with the Simulink interface of qpOASES First the cost prediction matrix H is re formulated so it is suitable to pass on to the qpOASES problem Hap for i 1
44. Meeting on Cyberntetics and Systems Research Vienna Austria 270 Vladim r ilka and Mikul Huba Huba M 1999 Dynamical classes in the minimum time pole assignment control In Computing Anticipatory Systems CASYS 98 Woodbury American Institute of Physics pp 499 512 Huba M 2001 Constrained control of the two tank system In 9th Mediterranean Conference on Control and Automation Dubrovnik pp 93 98 Huba M 2003 Gain scheduled PI level control of a tank with variable cross section In 2nd IFAC Conference on Control Systems Design Bratislava Slovakia Huba M 2005 P und pd polvorgaberegler fiir regelstrecken mit begrenzter stell gr sse Automatisierungstechnik AT 53 6 273 283 Huba M 2006 Constrained pole assignment control In Current Trends in Nonlinear Systems and Control L Menini L Zaccarian Ch T Abdallah Edts Birkh user Boston pp 163 183 Huba M 2010 Designing robust controller tuning for dead time systems In IFAC Int Conf System Structure and Control Ancona Italy Huba M 2011a Basic notions of robust constrained pid control In Selected topics on constrained and nonlinear control M Huba S Skogestad M Fikar M Hovd T A Johansen B Roha Ilkiv Editors STU Bratislava NTNU Trondheim Huba M 2011b Constrained pole assignment controller for sopdt plant complex poles In NIL Workshop on constrained and nonlinear control M Huba S Sko gestad M Fikar M Hovd T A Johansen
45. N and M are converted into the standard double precision matrix format from the YALMIP optimization variable no tation info solvesdp F geomean blkdiag Yq Xq ops Yq double Yq Xq double Xq N double N M double M 5 2 4 5 Factoring Storing and Preparing Parameters for the On line NRMPC Run After the optimization variables are available the parameters used in the on line NRMPC run have to be factored out and stored 5 Implementation of MPC Techniques to Real Mechatronic Systems 187 V XiU lu eye w dim 1 1 Yq Xq XiU XiU Qzi inv Xq XiU XiU V Yq XiU Qz Yq V V Xq XiU W Code segments V Yq XiU and Xq XiU V actually implement mathe matical operations V Y X U and X4X U V The full optimized shift matrices Ag and Co are calculated according to AO Xq XiU N V CO M V Qi1 inv Xq Q12 XiU Q21 XiU Q22 V Yq XiU Kt K CO where code segment Xq XiU N V is equivalent to the operation Xa XiU KVT A matrix right division is used in the segment M V to implement the operation M VT 1 After this respective partitions of Q are stored in vari ables for the needs of the on line NRMPC code It is true that partitions Q12 Q are related in symmetry 5 2 4 6 Cost Transformation The following code segment is related to cost transformation and the result ing conversion of augmented states The cost to be minimized in the on l
46. P P2 described in Sec tion 4 2 3 6 is easy to implement of the polytopes P and P2 are given in their V representation ie P convh x1 m P2 convh y1 yp Then Pi P convh zi yj Vi Vj 4 14 Now assume that P and P2 are as follows gt gt P1 unitbox 2 gt gt P2 unitbox 2 0 1 0 8 0 6 04 02 i i fi i Ll i L i i 0 8 0 6 04 02 0 0 2 04 0 6 0 8 1 i 04 1 Fig 4 10 Set difference between Q3 and Q 118 Michal Kvasnica i e hypercubes in 2D centered at the origin The task is to implement Minkowski addition via 4 14 Hint use the extreme function to enumerate vertices of P and P2 Compare your result to the output of MPT s internal implementation i e to P1 P2 Exercise 4 6 Recall that Minkowski addition of two polytopes is given by P B Pa x y x Pi y Po Assume that the polytopes are given by their H representation i e P1 x Aix lt bi Pa y Aoy bo Define a new variable z x y Then we have P Pa y x Pi y E Po 4 15a x y Aix lt b A2y bo 4 15b z z a y Aix lt by A2y lt b2 4 15c dete A KED 4 15e z1 l j lt 8 4 158 z iz st Z lt b 4 15g Note that in 4 15d we have eliminated the variable y by replacing it with z x The last line 4 15g states that the Minkowski difference can be in fact computed by projecting the poly
47. Reduction MPT also addresses the issue of complexity reduction of the resulting explicit control laws As explained in more detail in Section 4 5 the on line evalua tion of explicit control laws involves checking which region of the controller contains a given measured state Although such an effort is usually small it can become prohibitive for complex controllers with several thousands or even more regions Therefore MPT allows to reduce this complexity by sim plifying the controller partitions over which the control law is defined This simplification is performed by merging regions which contain the same ex pression of the control law By doing so the number of regions may be greatly reduced while maintaining the same performance as the original controller The results of the merging procedure for a sample explicit controller of a hybrid system is depicted in Figure 4 26 10 3 10 10 10 10 10 8 10 s 6 m 2 0 2 4 D 8 6 A 0 4 6 a Regions of an explicit controller before b Regions of an explicit controller after simplification 252 regions simplification 39 regions Fig 4 26 Region merging results To simplify the representation of a given explicit controller by merging regions which contain the same control law one has to call ctrl simple mpt simplify ctrl 4 Multi Parametric Toolbox 147 If the function is called as indicated above a heuristic merging will be used It is al
48. T L L L 0 50 100 150 200 250 300 350 400 450 500 Output 2 2 T T T T jt X a 0 gg S n Mc necs aL 4 2 l l l l l l l l l 0 50 100 150 200 250 300 350 400 450 500 Fig 1 16 Responses with the inverted decoupler with saturation in the inputs 1 1 7 3 LQG Controller with Integral Action The responses in the outputs when using the LQG controller are shown in Fig 1 17 We observe that the performance is good and there is very little interaction between the outputs The corresponding responses when the input is constrained between are shown in Fig 1 18 Note that the Kalman filter is here supplied 0 5 the 18 Morten Hovd and Selvanathan Sivalingam Output 1 15 1 L L L L L L L L 50 100 150 200 250 300 350 400 450 500 Output 2 2 T TE 4 0 AL 4 A L L L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 17 Responses with the LOG controller with integral action in the absence of saturation Output 1 L L L L L fi 0 50 100 150 200 250 300 350 400 450 500 Output 2 an n L L L fi 0 50 100 150 200 250 300 350 400 450 500 Fig 1 18 Responses with the LQG controller with integral action with saturation in the inputs actual constrained saturated input values but there is no anti windup for the augmented integrators The performance is significantly poorer than for the case without saturation
49. T We would like to recall as well the special case of two tanks connection It differs from 6 1 in the way that the outflow from the first tank is at zero valve 1 is closed the second pump is turned off and the liquid s height in the second tank is controller by controlling the inflow into the first tank We can write down the system equations as it follows 260 Vladim r ilka and Mikul Huba 1 t As Ci2 V X1 T2 1 t2 12V 1 T2 Co t2 y T2 7 8 The above mentioned system has the relative degree r 2 That s why we have to proceed when looking for the feedback from the equation v so u1 can be defined as 2A T1 uj um T2 1 12V 1 12 2A102 T2 A 12 2 Different from the previous examples by application of this feedback we obtain from the original system 7 8 linear system y v with transfer function Thus we would have to apply a different approach to continuous gainschedul ing of the controller paramters based on 7 9 whereby the new input vi should be computed in a way respecting all basic performance limitations nonmodelled dynamics model uncertainty meausrement and quantization noise and the contorl signal constraints 7 5 PD controller In reality the control signal is always constrained what can be expressed as Ur U1 Us Ui 0 Uz 7 9 So despite to the fact that the control signal is usually generated by linear contr
50. The liguid from the bottom tank can not flow into the upper one plus the upper tank s inflow impact does not depend on the liguid s level in the lower tank Due to the available eguipment in the practical part of this chapter we will consider just coupling in series with mutual interaction of tanks In order to be able to demonstrate theoretical aims of the control algorithm design and verify real processes by simulation using their models we need to know the process model and its parameters Sometimes the simplest possible model is reguired but for some tasks the most accurate model possible is wellcommed In case of inaccurate mathematical model the accociated con troller if not sufficiently robust would function only in simulations That s why we will be dealing from now with the analytical description of the real scheme necessary for construction of the model with the identification of the proposed model parameters as well as with confrontation of the model with the real process and with control processes associated with control design based on considered models 6 2 Coupled Tanks Hydraulic Plant Coupled tanks represent one of the most common examples of non linear system The task is usually related to basic manipulation reguired for dealing with the available physical system System considered in our publication is shown in Fig 6 1 This hydraulic system is composed of 2 pumps and 3 coupled tanks each of them with the own dra
51. a scalar factor a 0 gives R ax x P alpha 0 6 R alpha P e Affine transformation of P by an invertible matrix M gives R Mx x PY M 1 5 0 O 1 R M P Now practice on the triangle defined in Exercise 1 1 Let Q1 denote the shift of Q in the direction of the vector 0 5 0 2 2 Let Q2 be Q1 scaled by a factor of 0 8 3 Let Q3 represent Q2 rotated by 45 4 Plot Q in red Q1 in green Q2 in blue and Q3 in magenta Moreover plot all polytopes in the same graph using wireframe Hints e Rotation is achieved by applying the following matrix transformation with 0 being the angle in radians 0 sin 0 M so cos 0 ids e To plot several polytopes in the same graph do it in a standard MATLAB fashion i e plot P1 r P2 b P3 g e To plot using wireframe use additional options opts struct wire 1 linewidth 3 plot Pi r P2 b P3 g opts 116 Michal Kvasnica Exercise 4 3 Now concentrate only on Q and Q3 from the previous exercise It is obvious from the plot that these two polytopes overlap i e they have a non empty intersection To check this use dointersect dointersect Q3 Q ans 1 Since the intersection of any two polytopes is again a polytope the intersec tion can be computed explicitly by calling the overloaded amp operator Q4 Q3 amp Q To visualize the intersection we plot Q4 in solid yellow and Q3 and Q using wiref
52. all practical control problems the range of actuation for the control input is limited Whenever the input reaches the end of its range of actuation the control input is saturated the feedback path is broken If the controller has been designed and implemented without regard for this problem the controller will continue operating as if the inputs have unlimited range of actuation but further increases in the controller output will not be implemented on the plant The result may be that there is a large discrepancy between the internal states of the controller and the input actually applied to the plant This problem often persists even after the controlled variable has been brought back near its reference value and controllers that would work fine with unlimited inputs or with small disturbances may show very poor performance once saturation is encountered The problem described is typically most severe when the controller has slow dynamics integral action is particularly at risk since a pure integration corresponds to a time constant of infinity An alternative term for integral action is reset action since the integral action resets the controlled variable to its reference value at steady state When the input saturates while there remains an offset in the controlled variable the integral term will just continue growing it winds up The problem described above is therefore often termed reset windup and remedial action is corr
53. and we have the Johannes 1 2 L u d J u d Jope d 5 a i 2 11 From these expressions we see that minimizing the loss L is equivalent to minimizing z which is equivalent to minimizing z 1 2 u Uopt ES Thus we have the important conclusion that minimizing the loss is equiv alent to minimizing the weighted 2 norm of the gradient J with the weight being given by the matrix J m However for the normal case when there are no restrictions like fixing some elements to zero on the matrix H we will show below that the weight J a does not have any effect on the optimal H or 2 2 3 4 Optimal Sensitivities Note from 2 6 that we can write Uop Fud where F o More generally we can write yt Fd 2 12 where F is the optimal sensitivity of the outputs measurements with respect to the disturbances Here F can be obtained using 2 2 and 2 6 y G uopi Gid G J qu Jua G d that is F G J Jua GU ede However Juu can be difficult to obtain especially if one relies on numerical methods and also taking the difference can introduce numerical inaccuracy Thus for practical use it is often better to obtain F from its definition F dy dd by numerically reoptimizing the model for the disturbances 2 Measurements for Control Optimization and Estimation 59 2 4 The Loss Method Now we are finally ready to derive the main results 2 4 1 The Loss Var
54. approach 1 or 2 However note that this means that we need some non optimal data where we know the value of J 2 10 2 Relationship to NCO tracking Finally some ideas related to NCO tracking The NCO idea is to set the Gradient 0 An important difference com pared to the proposed loss approach is that in NCO tracking one tries to find an expression for u open loop implementation similar to deadbeat control On the other hand in the loss method self optimizing control we stop when we have the expression for the gradient c J the implementation to find u that gives J 0 is by feedback control From 2 10 and 2 11 we see that we want to minimize u Uopr or Ju but weighted by Juu Recall here that in the full H case Juu is not needed Still it remains unclear if this means that we can just minimize Ju 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 2 Measurements for Control Optimization and Estimation 81 Another problem with NCO idea to Control gradient to zero is that this is not really possible since gradient can not be measured Thus it needs to be estimated For the case with no noise the estimate J is same as nullspace method so c Hy Ju For noisy case not so clear but may as well use c Hy 2 11 Appendix This is file matlab test cases m Load data from C Documents and Se
55. bench is a combustion engine itself rigidly connected to the eddy current brake As an interface and a communication part is utilized a rapid control prototyping system based on dSpace hardware This allows to execute the designed controllers of the air fuel ratio on the engine instead of the original electronic control unit ECU 5 3 2 1 Combustion Engine For the experiments and measurements has been used a spark ignition engine from the Volkswagen Polo 1 4 16V Fig 5 10 It is a four cylinder in line engine with a DOHC valve train Engine characteristics Engine code AUA Volume 1390ccm Number of cylinders 4 Number of valves 16 Bore x stroke 76 5 x 75 6mm Compression ratio 10 5 Power 55kW 75k at 5000rpm Torque 126Nm at 3800rpm Injection system multipoint sequential 1 injector per valve Ignition electronic independent injectors Fuel Natural 95 Emissions processing three way catalys The original Bosch ECU has been completely replaced by a rapid proto typing system It is dedicated for the control of power stages as injectors and ignitors as described in the Subsection 5 3 2 3 202 Gergely Takacs et al Fig 5 10 Spark ignition engine VW Polo 1 4 with a brake 5 3 2 2 Engine Brake Dynamometer For the sake of identification experiments resulting in acquisition of the local ARX models of the air and fuel path it was necessary to keep the engine at predefined working points This aim has been reach
56. close all 7 Constrained PID Control Tasks for Coupled Tanks Control 263 PD2 controller levels 1 2 0 25 0 2 E 0 15 gt 0 1 0 05 0 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 time s x 10 PD2 controller output 1 5 pump power m s 150 200 250 300 350 time s 0 50 100 Fig 7 5 Typical setpoint steps in the loop with PD controller with tuning 7 21 may contain up to two step at saturation limits after larger setpoint steps at its output but for stable systems the 2nd interval at saturation corresponding to braking the transient may disappear or not reach the saturation 264 Vladim r ilka and Mikul Huba PD2 controller levels 1 2 0 25 0 2 E 0 15 gt 2 0 1 0 05 0 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 time s x 10 PD2 controller output 1 5 s l o o o E 2 20 5 0 1 1 1 1 1 0 50 100 150 200 250 300 350 time s Fig 7 6 Typical setpoint steps in the loop with PD controller with tuning 7 22 derived by the performance portrait method Huba 2011b gives a bit tighter control and faster dynamics of transient responses than the traditionally used tuning 7 21 what may be e g observed by longer period at the maximal level of h in the first tank and yet more by higher noise amplification 7 Constrained PID Control Tasks for Coupled Tanks Control 265
57. condition MPT toolbox is shipped with several freely available LP QP MILP MIQP solvers to perform this task without further effort from the users Clearly as solving the optimization problem is more difficult compared to a simple table lookup execution of the u 7 ctrl x took 50 miliseconds 4 5 2 Closed loop Simulations MPT provides several functions to perform closed loop simulations of MPC based controllers The most general approach is to use the specially crafted Simulink block which allows MPC controllers to be used for virtually any type of simulations A screenshot of such a block is shown in Figure 4 19 where the controller is connected in a feedback fashion The Simulink block supports both explicit as well as on line MPC controllers Another option is to use command line interfaces which perform closed loop simulations according to the receding horizon principle X U X U sim ctrl simmodel xO Tsim simplot ctrl simmodel x0 Tsim The sim function takes a controller object representing either an explicit or an on line MPC controller the simulation model the initial condition and the number of simulation steps as the inputs and provides closed loop evolution of system states and control actions as the output The simplot function behaves in a similar fashion but in addition generates a graphical 2 4 GHz CPU 2GB RAM MATLAB 7 4 MPT 2 6 2 4 Multi Parametric Toolbox 135 Dynamical Sy
58. counteract interactions due to the inputs that are actually implemented on the plant is to implement the decoupler as illustrated in Fig 1 6 Unsaturated Saturated inputs inputs M E Eo 5e Controller 912 loop 1 911 i Controller 921 lap loop 2 922 T2 k2 1 s y2 Plant Fig 1 6 Implementation of decoupler in order to reduce the effect of input satura tion The decoupler will only attempt to counteract interactions due to inputs that are actually implemented on the plant The implementation in Fig 1 6 is easily extended to systems of dimension higher than 2 x 2 When the inputs are unsaturated the Decoupler with saturation in Fig 1 1 corresponds to the decoupling compensator W s G s G s where G s denotes the diagonal matrix with the same diagonal 10 Morten Hovd and Selvanathan Sivalingam elements as G s The precompensated plant therefore becomes GW G i e we are nominally left only with the diagonal elements of the plant Note that if the individual loop controllers k s contain slow dynamics which is usually the case PI controllers are often used they will still need anti windup In this case the anti windup signal to the controller should not be the saturated input but the saturated input with the contribution from the decoupling removed i e the decoupling means that
59. effects of noise and therefore does not indicate the presence of stiction Again the routine bicoherence m returns a negative value for NGI and a positive value for NLI and therefore nonlinearity has not been detected for this PV time series Similarly the routine g1stat m finds no clear indication of PV being non Gaussian the probability of false alarm being estimated to 1 0 None of these routines therefore indicate stiction in the loop 44 Morten Hovd and Selvanathan Sivalingam o L L L L L L L L 500 400 300 200 100 0 100 200 300 400 500 Fig 1 46 The OP to PV cross correlation function for data set 4 0 6 0 2 PV 0 2 0 2 0 15 0 1 0 05 0 0 05 0 1 0 15 Fig 1 47 OP PV plot for data set 4 1 3 3 Conclusions 1 Data set 1 is a traditional stiction example as seen from Fig 1 34 The oscillations are here at a significantly lower frequency than the dominant dynamics of the plant 2 Data set 2 is obtained with the same stiction model parameters but with a higher proportional gain in the controller The oscillations are therefore 1 Problems in Anti Windup and Controller Performance Monitoring 45 faster in a similar frequency range as the dominant dynamics of the plant 3 Data set 3 is also obtained with the same stiction model parameters How ever here integral action is used in the valve positioner This results in rather fast oscillations so fast that the oscil
60. for Coupled Tanks Control 271 Comments Remarks 272 Vladim r ilka and Mikul Huba Comments Remarks 7 Constrained PID Control Tasks for Coupled Tanks Control 273 Comments Remarks 274 Vladim r ilka and Mikul Huba Comments Remarks Chapter 8 Remote Laboratory Software Module for Thermo Optical Plant Pavol Bistak Abstract Virtual and remote laboratories play an important role in the education process of engineers Their expansion is closely connected to the growth of Internet This contribution describes how it is possible to install and use the remote laboratory for the thermo optical plant It explains the process of installation both the server and the client applications Then the user interface of the client application is discussed in details An administrator of the remote laboratory can modify existing control structures therefore there are special rules included in one section how to create new Simulink block diagrams in order to be compatible with the remote laboratory software Thus one can set up the remote laboratory easily and it can be modified to her his needs quickly 8 1 Introduction The remote laboratory software module for the uDAQ28 LT thermo optical plant is represented by the client server application which main task is to enable remote experiments via Internet Although the software has been de signed to co operate with the uDAQ28 LT real plant its usage is not limited only
61. ia v ike v av PVO B bounding_box P Q range P A f Q domain P A f R projection P dim Check if two polytopes are equal P Q Check if two polytopes are not equal P O Check if P D O Check if P C Q Check if PD O Check if P c Q Intersection of two polytopes PN Q Union of two polytopes PU Minkowski sum P O cf 4 10 Pontryagin difference P Q cf 4 11 Set difference operator cf 4 12 Minimal hyper rectangle containing a polytope P Affine transformation of a polytope Q Ar f x eP Compute polytope that is mapped to P O x Ax f EP Orthogonal projection of P onto coordinates given in dim cf 4 7 4 Multi Parametric Toolbox 115 4 3 Exercises Exercise 4 1 Perform following tasks 1 Create a polytope P as the 2D unit box centered at the origin i e P f x x2 1 lt rz lt 1 1 lt r2 lt 1 2 Create an another polytope Q as the convex hull of points v 1 0 Uo T 0 7 U3 0 i 3 Plot P in red and Q in blue don t forget to use figure to open a new plot 4 Is P a subset of Q Check this using vertices of Q 5 Is 0 as subset of P Check this using vertices of P Exercise 4 2 MPT offers an easy way to perform affine transformations of polytopes These include e Shifting the polytope P in the direction given by a vector v leads a new polytope R x v x P It can by obtained by v 0 8 1 4 R P v e Scaling P by
62. identical with the Matlab workspace data format mat file and therefore it possible to read data into the Matlab environment later on and process the saved data more carefully Fig 8 10 Saving data into the file 8 6 Running the Experiment The sequence of running the experiment is following 1 on The basic assumption is that the server is running If it does not it is necessary to run it as it was written in Section 8 4 Start the client application according to the guide in Section 8 4 Set the IP address of the server into the Server IP field Press the Connect button and check if the message Connected to server appeared in the Output field Fig 8 7 Set the the parameters in the lower left sector Settings e Select the block diagram and initial parameters by entering the names of corresponding files Model Scheme Model Parameters e Set the desired value sampling period and duration of the experiment Press the Run button to start the operation of the selected block diagram This will cause running the remote experiment During the period of the 284 Pavol Bist k experiment running specified in Running Time it is not possible to in teract with the experiment and it is necessary to wait until it finishes to take over the control of the application again Depending on the network traffic and the load of the server it could happened that the experiment does not run correctly or the data are not transferred
63. is to use 72 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke closed loop data this may seem strange but it is OK Then we have G I 2 9 Estimator for y Based on Data The idea is to use the same approach as for the previous problem but using data instead of a model The objective is to estimate y based on measure ments y which we from now on will call to follow statistics notation That is we want to find y Hr where y The available information is given by the data Yau Y X Note that the data must first be centered To use our method we first need to know the expected optimal variation Y Here optimal means that y is constant In addition we also need to obtain G and G from the data This means that the data must contain non optimal variations in u and not only contain optimal data where u Uopt d Comment The setup is the same as for the previous problem expect that it is not clear how noise in y can be included It is a bit similar to imple mentation error which has been neglected since we assumed integral action 2 9 1 Data Approach 1 Here we assume that Y 4 is obtained from two different sources of data 1 Optimal data with constant y X Y opt This is closed loop data for y with y constant for various disturbances d and also with noise It should be representative data for the expected operation This directly gives the matrix Y Y
64. it really contributes to the enhancement of Slovak Norwegian coopera tion and to improvement of the educational framework at both participating universities Thereby I would like to thank all colleagues participating in the project activities at both participating universities and especially pro fessor Sigurd Skogestad coordinator of activities at the NTNU Trondheim associate professor Katarina Z kov for managing the project activities and professor Miroslav Fikar for the patient and voluminous work with collect ing all contribution and compilation of all main three project publications textbook workbook and workshop preprints Bratislava Mikul Huba 2 1 2011 Project coordinator iii Acknowledgements The authors and editors are pleased to acknowledge the financial support the grant No NIL I 007 d from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism This book is also co financed from the state budget of the Slovak Republic Contents Problems in Anti Windup and Controller Performance IM GWIEOPINE 2cicias ce a ob eee DRE REP ea Ed Ea Morten Hovd and Selvanathan Sivalingam 1 1 Anti Windup Control of a Distillation Column with Input oo ocn NEED L4 1 BNOUNBIOB o ese oem e e bedded 1 1 2 Some Background Material on Anti windup 1 1 3 Decoupling and Input Constraints 1 1 4 The Plant Model used in the Assignment LIS ASCO adde
65. nc Hagpt H i 5 Implementation of MPC Techniques to Real Mechatronic Systems 181 Hqp Hqp Hqpt end Passing on F is possible with the original formulation The cost matrix Ac needs to be transformed likewise AcQPh Ac 1 nc Nc AcQP for i 1 nc Nc AcQPt AcQPh i AcQP AcQP AcQPt end where the matrices need to be divided in the usual C programming style along with constraint matrices Bx and b0 BxOP Bx 1 nc Nc bOQP b0 1 nc Nc H Hqp Cost H COx F gt Matrix Cost F A ultiply Matrix Multiply BxOP gt Matrix b ultiply u pred Bx R Matrix ultiply b00P Auh comp H ACOP Bx x Matrix constraint PK Ac Gainl x status ub qpOASES nWSR Simple symmetric tp bounds gt Lo gt Pie ubA Hi Fig 5 3 Parsing the MPC problem to the qpOASES interface 182 Gergely Takacs et al 5 2 3 5 Off line Simulation In case an off line simulation is necessary for results verification or other purposes we may use the Matlab default quadratic solver named quadprog To do this one needs to launch a cycle with one iteration for each sampling instant and supply the optimization problem to the solver in the following fashion for k 1 run U1 k f statusi 1 k output quadprog H F X1 k Ac bO Bx X1C
66. ny Gy Yalli ny 1 r1 1 ny Xopt Yalli ny 1 r1 ny 1 c1 step 2 AHopt pinv Xopt Xopt Gy analytical expression Hopt loss soc_avg Gy Xopt D Hopt Gy inv G1 Hopt4 inv D Hopt resb Hopt4 X resbO Hopt4 X0 a41 norm resb Y1 a42 norm resb0 Y10 method4 loss NEW modified loss method for case with noise on Y redo step 1 New estimate of Y1 Yismooth Hopt4 X then redo step 1 and 2 163 164 165 166 167 168 169 170 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 2 Measurements for Control Optimization and Estimation 85 Yallsmooth Yismooth X u s v svd Y1smooth Yallsmooth Yall v Gismooth Yallsmooth 1 ny 1 ny Gysmooth Yallsmooth ny 1 r1 1 ny Xsmooth Yallsmooth ny 1 r1i ny 1 c1 step 2 Hopt loss soc_avg Gysmooth Xsmooth D Hopt Gysmooth inv Gismooth HoptiO inv D Hopt resa Hopt10 X resa0O Hopt10 X0 a101 norm resa Y1 a102 norm resa0 Y10 method10 uulossuw smooth loss method add artificial noise weights noisemag 2 e 4 noisemag tol Noise Alt 1 just additive noise noise noisemag eye rx Xoptnoise Xopt noise Noise Alt2 Add noise proportional to variation in each output 4 rx cx size Xopt Amag 4for
67. of the input the windup problem goes away This is because desired changes that violate the actuation constraints simply will not have any effect Note that the actuator should implement new desired value present value Aux If the previous desired value is used instead of present value above the velocity form of the controller will not remove windup problems The velocity form can also be found for more complex controllers in par ticular for PID controllers However derivative action is normally rather fast and the effects thereof quickly die out It is therefore often not considered necessary to account for the derivative action in anti windup of PID con trollers 1 1 2 3 Anti windup in Cascaded Control Systems For ordinary plant input it is usually simple to determine the range of actu ation For instance a valve opening is constrained to be within 0 and 100 maximum and minimum operating speeds for pumps are often well known etc In the case of cascaded control loops the plant input seen by the outer loop is actually the reference signal to the inner loop and the control is typically based on the assumption that the inner loop is able to follow the reference changes set by the outer loop In such cases the available range of 1 Problems in Anti Windup and Controller Performance Monitoring 5 actuation for the outer loop may be harder to determine and may depend on operating conditions An example of this problem ma
68. on the double integrator example load system and problem parameters Double Integrator compute explicit controller ctrl mpt control sysStruct probStruct 4 define the set of initial conditions XO unitbox 2 1 3 0 compute the 5 Steps reachable set N 5 R mpt_reachSets ctrl XO N plot results plot ctrl Pn y XO r R g The reachable sets green as well as the set of initial conditions red are depicted on top of the controller regions yellow in Figure 4 25 Reachable sets green set of initial conditions X0 red controller regions yellow Fig 4 25 Reachable sets for Example 4 2 144 Michal Kvasnica 4 5 5 2 Verification Reachability computation can be directly extended to answer the following question Do the states of a dynamical system whose inputs either belong to some set of admissible inputs or whose inputs are driven by an explicit control law enter some set of unsafe states in a given number of steps Example 4 8 In this example we show how to answer the verification ques tion for the first case i e system inputs belong to some set of admissible inputs u k Uo Although we use a linear system here exactly the same procedure applies to hybrid systems in PWA representation as well define matrices of the state space object A 1 4 4 1 B 1 1 C 10 D 0 syst ss A B C D Ts 0 02 create a system structur
69. one inflow and one outflow up to two level system linearized by generalized input output linearization and controlled by the PD controller 7 2 Basic P and PI controllers At the beginning it would be the best to introduce the simplest case of hydraulic system s configuration In doing so we will consider one level system which has all the valves closed The pump feeding the liquid into this tank will represent the actuator Under control such system behaves as simple integrator Considering 6 1 we will define differential equations of such system in the following form EE dj A yi T 7 1 Obviously we are dealing with linear system with the transfer function _ i s _ 1 U s Ais Fis ie with integrator having the integral time constant A gain 1 A1 From theory we know that to stabilize such a system 7 1 P controller with a gain P would be sufficient The closed loop transfer function will be P F s u P G s m 1 PES m Ais Pi If we want to control this system we can request it to behave as the first order linear system with the transfer function 1 sS Tis 1 In such a case the P controller s gain results as A1 P 1 Ti It is clear that by choosing the time constant T we are able to define the dynamics of the closed loop that might also be expressed by its closed loop pole a 1 T By decreasing this time constant when the closed loop pole is being shifted to oo we
70. op including the weights needed in 2 26 2 Non optimal data with constant d This is data for and y collected with varying u From this data we directly obtain G and G1 By selecting u y one may also here used closed loop data but with y varying in this case G1 I 3 Find optimal H using 2 26 with HG I 2 9 2 Data Approach 2 Loss Regression More generally we do not have separate optimal and non optimal data Instead we have combined data Yai where y and x vary simultaneously 2 Measurements for Control Optimization and Estimation 73 Note that we here use the notation from statistics chemometrics and call the measurements y for a We can then do a two step procedure In the first step we split up the data Y 4 data to find G1 G and Y opr and in step 2 we proceed as normal to find the optimal H Step 1A We rearrange the data Yq such that the y values are in the first rows and the y measurements are in the rest called X Yau Y1 X Step 1B We now want to separate the data into optimal and nonop timal data The data can generally be transformed by multiplying by a real unitary matrix V because HYau HYauV for the 2 norm Thus we can use the SVD of Y U S v to transform the data to split up the data G 0 YauVi 7 G X opt Now we have G1 G and Y X and can proceed as normal using our derived fo
71. or 7 19 and by using 7 9 the real control signal is achieved When substituting for 71 72 in the above transformations directly the actual level values we use standard approach of the exact linearization Due to the non modeled dynamics it is however better to work with generalized operating points for x1 x2 defined somewhere between the actual and the final state as i w mi d 1 m h m 0 1 i 1 2 7 20 In this way we are using a method that represents combination of the exact linearization and of linearization around fixed operation point Example of transient responses are in Fig 7 5 and Fig 7 6 The tradition ally used tuning derived by the conditions of the triple real pole Q1 2 0 321 Ty 7 21 used in older works the closed loop poles were now chosen according to Huba 2011b as 01 2 0 321 T4 7 22 where Ty 0 8s is the identified loop dead time was now confronted with newer one Huba 2011b derived by the performance portrait method The dynamics of the control signal changes is obviously from the dynamical class 2 But due to the plant character the second pulse in the control transients is not so dominant as the first one what is a typical feature of all stable systems The controller structure in Simulink is depicted in Fig 7 7 Control pro gram which works with results of automatic calibration and indentification is listed below 7 5 0 2 Program for Matlab Simulink cle clear all
72. parameters o 250 and 01 2 0 8 used in the weighting functions has been chosen experimentally awaiting continuous and 216 Gergely Takacs et al smooth output of the modeled system At the same time it has been found out that there can be used identical weighting functions for weighting of an air and fuel path parameters as well All the weighting functions were at the end normalized 5 25 so the sum of values of all weighting functions belonging to a particular working point Fig 5 20 equals exactly one or 55 pn O k 1 PR P R ENACO 2 rel weight 1500 2000 24 2500 t deg Don rpm Fig 5 20 Relative weighting Gaussian functions 5 3 4 Predictive Control The strategy of an exceeding oxygen amount control using a predictive controller is based on a prediction of a controlled quantity A and subsequent minimization of a chosen cost function on the horizon Np expressed in a standard quadratic form The value of A is predicted by utilization of partially linear models of the air and fuel path Through the independent air path model the proper amount of fuel is predicted and enters the cost function J Hence the target of the cost function minimization is to determine such a control law that the measured system output A is stoichiometric The second modeled subsystem the fuel path is an explicit component of the objective function where the amount of the fuel is the function of optimized contr
73. quality compared to DCO controllers e What was the most obvious difference in control signal shape compared to DCO controllers e Was there any overshooting in the experiments 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 321 System output Comparison of the undelayed system output 60 a FPI T T 9 0 8 v oio PP T T 9 0 6 E FP T T 0 06 lo controller 40 pene setpoint Light intensity wo o 20 600 800 1000 1200 1400 time s 0 L L 0 200 400 Fig 9 39 Experimental results undelayed system output 9 2 6 Filtered Smith Predictor FSP Tasks e Control the filtered light channel output by FSP e Analyze the control quality for various controller tunings e Compare results with control performance of PI controller The FSP was originally proposed in Normey Rico et al 1997 for stable FOPDT processes to improve robustness of the traditional SP Later the disturbance filter F s has been also proposed to decouple the reference setpoint and the disturbance response and to stabilize the controller loop in case of unstable and integral plants Normey Rico and Camacho 2009 It may be interpreted as a structure with the dynamical feedforward control and the reference plant model Astr m and Hagglund 2005 Visioli 2006 or the 2DOF IMC structure The unified approach to designing FSPs for the FOPDT plants introduced in Normey Rico et al
74. range considered there may be several constrained regions and the procedure of finding H needs to be repeated in each constrained region Switching be tween the regions will then be needed and we will show that monitoring the controlled variables c Hy in neighboring regions can be used for switching Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Techno ogy in Trondheim Norway e mail skoge chemeng ntnu no Ramprasad Yelchuru Department of Chemical Engineering Norwegian University of Science and Techno ogy in Trondheim Norway e mail ramprasad yelchuru chemeng ntnu no Johannes J schke Department of Chemical Engineering Norwegian University of Science and Techno ogy in Trondheim Norway e mail jaschkeGchemeng ntnu no 53 54 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke What we here call the loss method is the same as what is called the exact local method in these papers Halvorsen et al 2003 Alstad et al 2009 The new material in this summary paper is mainly related to using data Y X as the basis and for example to use the loss method for regression see section 5 data approach 1 and 2 2 2 Problem Formulation 2 2 1 Classification of Variables u inputs degrees of freedom for optimization and control it does not actually matter what they are as long as they form an independent set e d disturbances includ
75. state space object is already in discrete time domain it is not necessary to provide the sampling time parameter Ts to mpt sys Af ter importing a model using mpt sys it is still necessary to define system constraints as described previously 4 Multi Parametric Toolbox 149 MPT allows to define following types of constraints Min Max constraints on system outputs Min Max constraints on system states Min Max constraints on manipulated variables Min Max constraints on slew rate of manipulated variables 4 5 6 2 Constraints on System Outputs Output equation is in general driven by the following relation for PWA sys tems y k Ci k Drulk g 4 30 and by y k Ca k Du k 4 31 for LTI systems It is therefore clear that by choice of C I one can use these constraints to restrict system states as well Min Max output constraints have to be given in the following fields of the system structure sysStruct ymax outmax sysStruct ymin outmin where outmax and outmin are ny X 1 vectors 4 5 6 3 Constraints on System States Constraints on system states are optional and can be defined by sysStruct xmax xmax sysStruct xmin xmin where xmax and xmin are n X 1 vectors 4 5 6 4 Constraints on Manipulated Variables Goal of each control technigue is to design a controller which chooses a proper value of the manipulated variable in order to achieve the given goal usually to guarantee stability but other a
76. the set plot PSI compute extremal vertices of PSI 4 Multi Parametric Toolbox 129 E extreme PSI for i 1 size E 1 x E u K x if isinside unitbox 1 u amp amp isinside X x fprintf Vertex 7 d 0K n i else fprintf Constraints violated for vertex Ai Nn i end end A Fig 4 16 Target hi 4p 3 2 0 1 2 3 4 n Fig 4 17 Set of states for which the LQR controller satisfies all constraints 130 Michal Kvasnica Solution 4 10 for Exercise 4 10 the sets X and U were defined in the previous example PSI X amp U k 1 the algorithm has no guarantee of finite time convergence kmax 10 while k lt kmax PSI k 1 inv A B K PSI k amp PSI k k k 1 if PSI k PSI k 1 break end end PHI PSI end plot all sets plot PSI 4 or a one liner PHI mpt_infset A B K PSI 1 kmax Fig 4 18 The invariant subset the innermost region 4 Multi Parametric Toolbox 131 4 5 Model Predictive Control in MPT We consider the class of discrete time time invariant systems given by the state space representation c t 1 f x t u t 4 22 where x t is the state vector at time t x t 1 is the successor state u t denotes the control input Moreover f is the state update function which depends on the type of the model which can either describe a linear time invariant LTI system with f a t u t
77. then rcx D 0m Thus for case 3 we do not have a convex problem formulation that is we need to solve the nonconvex problem in 2 21 with additional constraints on the structure of H This is not surprising as decentralized control is generally a nonconvex problem Nevertheless Theorems 2 1 and 2 4 with additional constraints on the structure of H give convex optimization problems that provide upper bounds on the optimal H for case 3 In particular in Theorem 2 4 4 we may make use of the extra degree of freedom provided by the matrix Q Yelchuru and Skogestad 2010 Also note that as opposed to cases 1 and 2 Juu is needed to find the optimal solution for case 3 This may seem a bit surprising Case 4 Decentralized control using single measurements that is nys ne where nys is the number of selected measurements Theorems 2 1 2 4 hold also in this case This is a special case of case 3 where we use the fewest number of measure ments This case is different from case 2 in that H is a diagonal matrix The reason why Theorems 2 1 2 4 hold in this case is that we can still keep D full because the non zero part of H is square and we can to change it to anything so we can treat it a special case of full H 2 7 Some Special Cases Nullspace Method and Maximum Gain Rule The general optimization problem is 2 Measurements for Control Optimization and Estimation 67 1 2 y 1 min Ji HG HY 2 27 F where Y
78. these constraints time varying one can exploit the fact that the symbolic decision variables V x and V u are stored as cell arrays indexed by the prediction index k gt gt G C 5 lt V x 1 lt 5 gt gt G C 6 lt V x 2 lt 6 gt gt C C 1 lt V x 3 lt 1 Such a construction will force the predicted states to stay within of given time varying bounds 4 Multi Parametric Toolbox 139 4 5 4 3 Move Blocking Move blocking is a popular technique frequently used to reduce complexity of MPC optimization problems by eliminating some degrees of freedom from the sequence Uy ud u1 ul T In practice usually three types of move blocking are used 1 Keeping first Ne moves uo uw free and fixing the remained uy 41 UN 2 uyx 1 0 This can be achieved by gt gt C C V u Nc 1 N 1 0 2 First Ne moves are free while the remaining control moves are driven by a state feedback u Ka ie un 4 n Kap k 2 1 N Ne 1 for k Nc 1 N 1 C C V u k K x k end 3 Introduce blocks of control moves which are to stay fixed e g ug u1 us and ua u4 gt gt C C V uf 1 V u 2 V u 2 V u 3 gt gt C C V uf 4 V u 5 The optimal values of ug and ua obtained by solving the corresponding optimization problems then pre determine the values of u1 u2 and u4 4 5 4 4 Contraction Constraints Another way of achieving closed loop stability is to i
79. time Tq is relatively small comparing to the process time constant T we have so called lag dominant plant thus it can be neglected in further calculations Figs 9 35 and 9 36 show how the process gain and time constant vary through the operational range independence on the plant input Again the controller tuning can be done using the performance portrait method Fig 9 37 It is best to choose Tio max T1 Kio max K 9 18 which for this plants is Tio 25 8687 Ko 19 8939 9 19 Try multiple disturbance observer filter time constants Tp In this example following T T 9 ratios were used 316 Peter Tap k and Mikul Huba FP 4 5 F A o oa 99 Control signal no n oa E ol 0 5 0 50 100 150 200 250 300 350 400 450 time s Fig 9 32 FPrlo controller vs FIo controller under 5s transport delay control signal Fig 9 33 FPIo controller 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 317 Gain3 u Step Temperature 1 VO Interface 4 Light intensity 1 Ps Pit KO Thermo eptical Plant Saturation 3 gt 0 M Fan Gain1 1 Naandiee Fond 0 ni rec ERE ransfer Fcn Tf s 1 LED 4 Gain2 Transfer Fcn2 X T10 s 1
80. value will be overwritten by the value set in the Sample T ime field of the client application The desired value is given by the global variable denoted as w the Final value parameter of the Step block in the supplied block diagram e g This denotation must be kept everywhere If a variable with this name is used in the initial parameter file m file or somewhere in the block diagram mdl file its value will be overwritten by the value set in the Input Desired Value field of the client application Signals that should be displayed in the client application interface have to be gathered using the multiplexer block into the only one Scope block whose name is identical with the name Scope and its settings coincide with those of Fig 8 13 N 9v Scope Fig 8 12 Gathering signals into the Scope block When creating new block diagrams it is recommended to start with the supplied block diagram Fig 8 11 and modify its structure while keeping the blocks drawn by the orange color unchanged Thus it is possible to add an arbitrary controller to the thermo optical plant block that can be re alized in the Simulink The block diagram in the Fig 8 11 and its initial parameter file have been developed by a simple modification of the original files PI1 windupless mdl a init PI1 windupless m that are supplied on the installation CD of the uDAQ28 LT thermo optical plant Warning Time responses from the supplied block diagram and the p
81. with the reference changes and noise variances prescribed above Plot the results and compare to the performance without anti windup make comparable plots with and without anti windup implemented Comment on the results 12 Morten Hovd and Selvanathan Sivalingam 1 1 6 Solution 1 1 6 1 Model Implementation The model is implemented in Simulink as illustrated in Fig 1 7 Note that the state space model for the distillation column has been implemented with the disturbance as the second input in accordance with 1 10 A manual switch is used to include exclude saturation of the manipulated variable and clearly this switch will have to be set in accordance with the case to be simulated The Matlab code to define the model is listed in Section 1 1 9 1 1 1 6 2 Controller Designs PI Controllers The PI controllers without anti windup are implemented using transfer func tion blocks in Simulink cf Fig 1 8 Alternatively the PID Controller block in the Continuous library can be used but one should be aware that the tuning parameters in that block does not correspond to the standard pro portional gain and integral time parameters consult Simulink documentation for details The Matlab code used to define the controller numerator and denomina tor polynomials are given in subsection 1 1 9 2 Note that the sign of the proportional gain is adjusted to ensure negative feedback Dynamic Decoupler The dynamic decoupler is
82. 0 measurements and the validation data set 50 measurements specified in files I received The results are shown in Table 2 4 The loss method is the best validation error 2 9 compared to 3 1 for the traditional methods but the differences are small The difference between the methods is small for this test case and in this case the loss method gives better validation 2 9033 than least squares 3 1092 in spite of the fact that the calibration fit is perfect in both cases This is a case where one would expect it to help to add artificial noise to the loss method The reason is that we have 701 as but only 50 data sets for calibration so the data would not be expected to contain sufficient informa tion about the expected noise However the numerical results do not confirm this and the fit gets worse when we add noise 78 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Y contains both the two y variables water and protein presumably better fit can be obtained by fitting one at a time Table showing the fit Y HX p to calibration and validation data Table 2 4 Wheat Spectra Calibration and Validation data 0 0000 3 1092 least squares PCR 50 0 3136 3 0871 PCR 47 tol 1 1 e 3 Again smoothening of y has no effect which again is not surprising since the fit was perfect 2 9 7 Test 3 Our Own Example The data contains 2 yi 7 x 2 d G E Ga E bl gy 0 2034563 gy 00 213489 G
83. 1 to 4 to USB Serial Port 296 Peter Tap k and Mikul Huba 9 1 1 2 Driver and Software Package Installation in Matlab To install driver and software package in Matlab follow these instructions e Before installation it is good to make sure that any unterminated process matlab exe is not in the memory if it is terminate it tab Processes in Task Manager activated by keys CTRL ALT DEL e Run Matlab e Change the working directory to the directory from the package of sup porting software where installation file udaq_setup p is located e g by typing command cd e dirname e Run file udaq setup p by typing command udaq setup After successful completion of installation there is directory matlabroot u dag udag28LT and particular subdirectories and files copied on the local hard drive matlabroot represents the name of the directory displayed after typing command matlabroot in Matlab command window Two mdl files are open after installation in Matlab udaq28LT iov2 mdl located in matlabroot u dag udag28LT examples and the library containing two blocks drivers rep resented by mdl file matlabroot udaq udaq28LT blks udaq28L T lib mdl You can create your own simulation mdl file that communicates with thermo optical plant by copying the driver block from the library or from other functioning mdl file into your own mdl file 9 1 1 3 Thermo optical Plant Communication Interface Thermo optical plant communication interfac
84. 1073 1 14 48 Morten Hovd and Selvanathan Sivalingam With this modified benchmark variance we find a modified Harris index of HI nog 1 49 or NHImoa 0 327 Thus with the chosen closed loop time constant as basis for comparison the motivation for retuning the controller is significantly reduced as the observed variance is not much larger than the acceptable variance Acknowledgements The authors are pleased to acknowledge the financial support by a grant No NIL I 007 d from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism The contributions of Anders Fougner and Anders Willersrud in preparing the anti windup assignment are also gratefully acknowledged References Hanus R Kinnaert M Henrotte JL 1987 Conditioning technique a general anti windup and bumpless transfer method Automatica 23 6 729 739 Hovd M Sivalingam S 2011 A short introduction to anti windup In Huba M Skogestad S Fikar M Hovd M Johansen TA Rohal Ilkiv B eds Selected Top ics on Constrained and Nonlinear Control Textbook STU Bratislava NTNU Trondheim Sivalingam S Hovd M 2011 Controller performance monitoring and assessment In Huba M Skogestad S Fikar M Hovd M Johansen TA Rohal Ilkiv B eds Selected Topics on Constrained and Nonlinear Control Textbook STU Bratislava NTNU Trondheim Skogestad S Postlethwaite I 2005 Multivariable Feedback Control Analysis and De
85. 4 7 for Exercise 4 7 The implementation literally follows the five steps listed in Exercise 4 7 4 Multi Parametric Toolbox 127 input data C1 polytope 0 2 02 24 2 4 4 2 4 2 C2 polytope 4 2 4 2 2 4 2 4 0 2 0 2 C C1 C2 Q 5 unitbox 2 UJ the algorithm H hull C D H B E H C F G E B DNF is the output equal to C B G C B ans Solution 4 8 for Exercise 4 8 Hit or miss in 10 steps input data Xi unitbox 2 3 0 Xf O 5 unitbox 2 A 0 5 0 75 0 75 0 5 N 10 initialize the data storage S Xi for k 1 N compute the set Xk1 A Xk r AxS end add Xki at the end of S s S r end plot the sets remember that Xi is part of S plot Xf k S hit or miss if dointersect Xf S disp Target hit else disp Target missed 128 Michal Kvasnica end Fig 4 15 Reachable sets for N 10 show a miss If a miss how large must N be for a hit N 0 S Xi while dointersect S Xf N N 1 A S end S r r S end plot X k S N Solution 4 9 for Exercise 4 9 input data A 1 1 O 1 B 1 0 5 K dlqr A B eye 2 1 xmax 5 5 xmin 5 5 umax 1 umin 1 set of states which satisfy state and input constraints X polytope eye 2 eye 2 xmax xmin U polytope K K umax umin PSI X amp U plot
86. 9 22 Experimental results for K 10 e Make a disturbance by a LED voltage step during the experiments e Compare the control quality with the FIo controller using a real experi ment Tuning of the closed loop systems involving dead time still represents a challenging domain of control research an increase of the dead time values with respect to the dominant plant time constant leads in the loops with PID controllers to rapid performance deterioration Therefore filtered predictive Io FPrIo controller will be used in this exercise Under the FPrIo controller controller we will understand the static feedforward control with the gain 1 Ko extended by the input disturbance reconstruction and compensation Fig 9 27 with the disturbance observer filter time constant T and by the pre filter with the time constant Tp Ty Robust tuning of the FPrlIo controller may again be done by the perfor mance portrait The information on plant parameters is needed For the plant used in this exercises the performance portrait for undelayed plant output is in Fig 9 29 the delayed plant output performance was analyzed in Fig 9 30 Compare the performance of filtered predictive Ip controller vs FIo controller Use additional transport delay when controlling a non filtered optical chan nel In the following example 5s transport delay was added to the non filtered light channel output so the transport delay of the plant ranges from 5 4 to 310 Peter Tap k and
87. Assign symbolic labels level 1 level 2 to the two state variables in flow to the input and level 2 to the output variable 3 Verify that sysStruct contains valid entries by running gt gt mpt_verifySysStruct sysStruct If sysStruct is mis configured the verification function will output an error 4 Simulate the evolution of the system for 5 steps starting from the initial condition z 0 p and using the control inputs u t 2 t 1 10 hint use the mpt_simSys function see help mpt_simSys for more details Exercise 4 12 For the two tanks system defined in the previous exercise we would like to synthesize an MPC controller which would minimize the following performance objective N min DETO u Ru 4 32 k 0 Assume N 6 Q 9 and R 1 Create a problem structure probStruct which will reflect this settings do not forget to set prob Struct norm 2 to indicate that a quadratic performance objective should be used Verify consistency of the generated structure by running gt gt mpt_verifyProbStruct probStruct 4 Multi Parametric Toolbox 157 Exercise 4 13 Consider again the two tanks setup from the previous two exercises Finally we will synthesize an MPC controller To do that run gt gt ctrl mpt_control sysStruct probStruct online The online flag indicates that we will use an on line MPC controller After the controller object is constructed we can obtain the optimal con
88. B Roha Ilkiv Editors STU Bratislava NTNU Trondheim Huba M 2011c Robust controller tuning for constrained double integrator In NIL Workshop on constrained and nonlinear control M Huba S Skogestad M Fikar M Hovd T A Johansen B Roha Ilkiv Editors STU Bratislava NTNU Trond heim Huba M Bist k P 1999 Dynamic classes in the pid control In Proceedings of the 1999 American Control Conference San Diego AACC Huba M Sovi ov D Oravec I 1999 Invariant sets based concept of the pole assign ment control In European Control Conference ECC 99 Duesseldorf VDI VDE Pan H Wong H Kapila V de Queiroz MS 2005 Experimental validation of a non linear backstepping liquid level controller for a state coupled two tank system Control Engineering Practice 13 1 27 40 Zilka V Hal s M 2010 Disturbance decoupling of coupled tanks From theory to practice In IFAC Symposium on System Structure and Control CSSS 2010 Ancona Italy Zilka V Hal s M 2011 Noninteracting control of coupled tanks From theory to practice In 13th International Conference on Computer Aided Systems Theory EUROCAST 2011 Las Palmas Gran Canaria Zilka V Hal s M Huba M 2009 Nonlinear controllers for a fluid tank system In R Moreno D az F Pichler A Quesada Arencibia Eds Computer Aided Systems Theory EUROCAST 2009 Springer Berlin Germany Lecture Notes in Computer Science pp 618 625 7 Constrained PID Control Tasks
89. Estimation 75 Wn is a tuning parameter but a systematic approach is the following Plot the singular values of Xp and select wn such that the singular values of X extended follow the same trend 2 9 3 Modification Smoothening of Data The loss regression method is based on the loss method where it is assumed that a model is available When we are using data then the model has to be extracted first step 1 and this step of the method is not based on a rigorous approach to the final objective which is to use the model for future predictions Therefore it is likely that this step may be improved One possible modi fications is suggested next although it seems from the later numerical tests that it actually may have no effect 2 9 3 1 Smoothening of y1 The above procedure including step 1 assumes that all the noise is in the x y One way of dealing noise in y is to run through the procedure twice First we go through the procedure steps 1 and 2 and find the optimal Ho Next we go through the procedure again but with a modified step 1A where use Ho to estimate the smoothened fitted values of y that corre spond to the measured X yoon HX and then we use this smoothened data in the other steps Y ui s X Yismooth Hoptloss X then redo step 1 and 2 Yallsmooth Yismooth X u s v svd Y1smooth 4 etc However from the numerical examples this smoothening has no effect on the results which in
90. Fig 9 47 PI controller filtered light channel control for K 19 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 329 Pl controller control signal K 19 5 T T T A T L 9 oi T L A T fi Bulb voltage V no c1 N T 1 5r 0 5r 0 i L L 0 20 40 60 80 100 time s Fig 9 48 PI controller filtered light channel control for K 19 330 Peter Tap k and Mikul Huba Pl controller output of filtered light channel K 9 70 T T T T DUE delayed output undelayed output 60r setpoint J Light intensity 0 20 40 60 80 100 time s Fig 9 49 PI controller filtered light channel control for K 9 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 331 Pl controller control signal K 9 5 T T T A T L i ol T L A T fi Bulb voltage V no c1 N T 0 5F 0 i L L 0 20 40 60 80 100 time s Fig 9 50 PI controller filtered light channel control for K 9 IISINININISISIIEREEEREEERERERERERERERERERERERERTRTRRRRRRIRI Disturbance Filter Disturbance Observer Fig 9 51 Modified P FSP with the primary loop using 2DOF P controller with the disturbance filters 332 Peter Tap k and Mikul Huba Filtered light channel output 45 T T T T 40 c1 o N o 20 Light intensity
91. Figure 4 8 X X 1 a Polytopes Pi and 75 b Set difference O P P2 Fig 4 8 Illustration of the set difference operation 4 2 4 Functions Overview A compact overview of the functions and operators described above is given in Tables 4 1 and 4 2 For more information see the output of help mpt help polytope and references therein Table 4 1 Computational geometry functions P polytope A b Constructor for the polytope P x R Az b P polytope V Constructor for creating a V represented polytope double P Access the H representation e g A b double P extreme P Access the V representation e g V extreme P display P Displays details about the polytope P nx dimension P Returns dimension of a given polytope P nc nconstr P For a polytope P x Ax b returns number of constraints of the A matrix i e number of rows Lag Horizontal concatenation of polytopes into an array e g PA P1 P2 P3 3 Subscripting operator for polytope arrays e g PA i returns the i th polytope in PA length PA Returns number of elements in a polytope array P4 end Returns the final element of an array c r chebyball P Center c and radius r of the Chebychev ball inside P bool isfulldim P Checks if polytope 7 is full dimensional bool isinside P x Checks if r P Works also for polytope arrays 114 Michal Kvasnica Table 4 2 Operators defined for polytope objects vu as ia s iko s has
92. Frobenius norm of M in 2 19 that is min 72076 av 2 21 H F where Y FW 4 W ns We call this the minimum loss method for finding optimal linear measurement combinations c H y The objective in 2 21 is to find the nonsquare n xn matrix H note that Ny ne In most cases it may be recast as a convex optimization problem as given in 2 23 below The exception is if H has a specified structure for example H is a selection matrix which is discussed in Section 2 6 Further Comments 1 Using the norm of M to analyze the loss is known as the exact local method and finding the optimal H is the exact local method optimization problem However in this paper we simply call it the loss method 2 To include changes in the weights in the cost function p prices we need to find the optimal sensitivity to price changes y Fpp The corrected setpoint for the variables c Hy is then cs Hy HF p 2 22 3 The effect transfer function from c to z is My J1 HG and from c to u is G HG Since there are extra degrees of freedom in H which are not set by the optimization problem either of these M or G can be selected freely see below for details Exercise 2 1 Consider a scalar case nu ne 1 with no disturbances F 0 and assume that the measurements y have been scaled such that Wry I noise of equal magnitude on all outputs For the scalar case Jua does not m
93. I bic f max bicoherence pv gives NGI 0 0014 and NLI 0 4394 Since both these indices are re 38 Morten Hovd and Selvanathan Sivalingam 05r PV o T 0 5 15 1 1 l 1 L 1 5 1 0 5 0 0 5 1 15 Fig 1 36 OP PV plot for data set 1 guired to be positive in order to indicate significant non linearity no stiction is indicated In contrast the command sg sl glstat pv 0 51 128 shows that the signal is non Gaussian the Pfa probability of false alarm is zero However the estimated inter quartile range R is quite close to the theoretical value for a linear system and thus glstat does not find any clear sign of non linearity 1 3 2 2 Data Set 2 The time plots of OP and PV for data set 2 are shown in Fig 1 37 The OP resembles the sawtooth shape but here PV also resembles a sawtooth shape and stiction is therefore not readily identifiable from the time plots The cross correlation function for data set 2 is shown in Fig 1 38 This is nearly an even valued function and this test therefore does not indicate the presence of stiction The OP PV plot for data set 2 is shown in Fig 1 39 The plot looks like an ellipsoid seen from the side but due to the angle it is difficult to assess whether there are any sharp corners in the plot In Fig 1 40 the PV time series is therefore shifted by 30 samples The plot is a little messy since the time shifting destroys the correspondence between the effects
94. M Huba S Skogestad M Fikar M Hovd T A Johansen B Rohal Ilkiv Editors Selected Topics on Constrained and Nonlinear Control Workbook STU Bratislava NTNU Trondheim Copyright 2011 authors Compilation Miroslav Fikar Cover Tatiana Hubova Printed and bounded in Slovakia by Miloslav Roubal ROSA Dolny Kubin and Tla iare Vr bel Doln Kubin ISBN 978 80 968627 3 3 Preface This workbook was created within the NIL I 007 d Project Enhancing NO SK Cooperation in Automatic Control ECAC carried out in 2009 2011 by university teams from the Slovak University of Technology in Bratislava and from the Norwegian University of Science and Technology in Trondheim As it is already given by the project title its primary aim was enhancing co operation in academic research in the automatic control area in the partner institutions This was achieved by supporting broad spectrum of activities ranging from student mobilities at the MSc and PhD level staff mobilities organization of multilateral workshop and conferences joint development of teaching materials and publishing scientific publications With respect to the original project proposal the period for carrying out the foreseen activi ties was reasonably shortened and that made management of the all work much more demanding Despite of this the project has reached practically all planned outputs this workbook represents one of them and we believe that
95. Manager Ele Action view Help eH mepsu x e J MATOKAM BB Acronis Devices Wl Batteries H Computer ee Disk drives a Display adapters DVD CD ROM drives 3 IDE ATA ATAPI controllers 4 gt Keyboards 77 Mice ard other pcinting devices be Modems ES a Monitors HB Network adapters PCMCIA adapters Forts COM amp LPT JZ Communications Part COMA 7 ECP Printer Port LPT1 JZ gt Processors SCSI and RAD controllers 9 Sound video and game controllers Fig 9 5 Device manager USB serial port Advanced Settings for COM3 COM Pert Number coma CIMT nuse USB Transter 5ges 00M2 m Select lower setting CIMA N fonce problems at low baud rates Select highe settings lor laste performance Receive Byles 4056 Tranem Bytes 4036 Select lower settings to coriec response problems Serial Enurreictor Serial Printer Latency Timer mzec 1 v Cancel li Power Off Timecuts Evert On Surprise Removal Set ATS On Chee Minimum Asad Timeout mses 0 Disable Modem CtlAt Startup Minimum Wrke Timeout msect n l BM Options Miscellaneous Dprions F r r F r r 295 Fig 9 6 Assignment COM port number to USB Serial Port Latency Timer setting other applications or programs first assign change com port number bigger than 4 to one of those programs and then assign available number from
96. Mikul Huba Light channel output 45 T T T T System output 40r s Setpoint f DD O1 eo c1 T n e Light intensity E oa o T L 0 L L L L L 0 10 20 30 40 50 60 time s Fig 9 23 Experimental results for K 17 5 7 s The process gain still range in interval 3 17 For 1 overshooting tol erance it yields to filtered predictive Ip controller parameters Tao 6 48 9 11 Ty 4 28 9 12 Ko 14 15 9 13 For FIo controller it gives Tao 5 7 9 14 Ty eTao 5 16 9 15 Ko 17 9 16 The performance of both controllers is compared in Fig 9 31 9 32 Feel free to make this experiment with larger dead time e g 10s Questions e What type of plant is FPrlg controller suitable to control for e Was there any overshooting in output and input transients during the experiments e Which controller performed better in disturbance rejection 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 311 Control signal 3 5 T T T Bulb voltage 0 54 4 0 L L L L L 0 10 20 30 40 50 60 time s Fig 9 24 Experimental results for K 17 k Step Response NonOvershooting 8 Monotonic Control 0 8 0 6 0 4 0 2 04 K K mi Fig 9 25 Performance portrait for FIo controller 312 70 60 R 91 o o o Light intensity 10 Peter Tap k and Mikul Huba Setpoint c
97. Operating temperatures range Power supply Communication interface 0 70 C 12V 2A DC external adapter USB virtual serial port 298 Peter Tap k and Mikul Huba Block Parameters s uncioni m uDAQ28 LT Communication Interface mask link c 2006 Martin Kamensky Serial port USB serial port number assigned to uDAQ28 LT system Sampling time sec the lowest working value in Windows is about 0 04 0 05 s Sample delayed number of events percentually when simulink loop can take more than time determined by sample time Read timeout msec timeout for reading data from the port the lowest working value is about 25 ms Matlab priority NORMAL PRIORITY priority of common Windows application ABOVE NORMAL PRIORITY recommended higher priority than common Windows application priority HIGH PRIORITY recommended high priority REALTIME PRIORITY NOT recommended takes almost all system resources Parameters Serial pot z Sample time sec perme me eee M Sampling delayed 1 Read timeout msec lower than sample time Ts 2 1000 Matlab priority ABOVE NORMAL PRIORITY E jv Warming if delayed error if unchecked Detailed timing and timeout printout Cancel Help Apply Fig 9 8 User dialog window of communication interface Data transfer speed 250 kbit s 9 2 Light Channel Control The non filtered and filtered light cha
98. Static state feedbacks can be found by solving equation yf Ui The proof technical details and additional references can be found in Conte et al 2007 For the system 6 1 the static state feedback that solves the linearization problem can be found by solving the equation Yi v for u1 that is ui A101 Ac y x1 7 7 7 Constrained PID Control Tasks for Coupled Tanks Control 259 When we apply this the feedback we get from the original system 6 1 linear system v with transfer function Fils lt Now it is enough to design controller which will provide the required loop behavior It will be necessary for it to have opened loop transfer function Tr When we talk about simple first degree integrator in this case it is enough use simple P controller Its gain will be 1 Pras 1 Ti For the two level system 6 2 we will do the same By requiring j v j2 v2 the solution will be given equations for u and u2 uj Avl A1 12 sign x z2 y x x2 A1c1 71 uo As Agcy sign zi x2 V x z l A 2C24 T2 Similarly to previous case after application of these feedbacks we obtain two independent decoupled systems j1 v1 jo v2 having the transfer functions Fi s F s wSBlRewle We can continue again in the same way as with the one level system As we now have two independent liear systems for each level we can design separate P controller 1 R 1 Ti 1 Ps 2
99. Ywheat Y 2 y s 701 x s 100 data set X Xwheat idxcal Y1 Ywheat idxcal 50 calibration sets XO Xwheat idxval Y10 Ywheat idxval 50 validation sets X Xwheat idxval Y1 Ywheat idxval 50 calibration sets switched XO Xwheat idxcal Y10 Ywheat idxcal 50 validation sets switched Jump to MATLAB code starts here 4 SS a a n a eS Se ee eae 4 Test case 3 Own example G1 1 0 0 1 Gdi 0 0 0 0 gy1 0 20 3 4 5 6 3 gy2 0 0 2 1 3 4 9 gdi 0 0 4 5 6 8 9 gd2 0 0 3 5 5 9 18 Gyd gd1 gd2 Y10 G1 Gd1 XO Gy Gyd Y100 Y10 na 0 aa a 0 Run repeatedly from here for several cases Y10 Y100 8 data sets Noise O 5 randn 7 8 X XO X0 Noise 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 2 Measurements for Control Optimization and Estimation 83 yi Y10 Y10 use Y10 and Y10 to get centered data 4432 data sets 4X XO X0 Noise 0 5 randn 7 32 4X X X X X Noise Y1 Y10 Y10 use Y10 and Y10 to get centered data 4Y1 Y1 Y1 Y1 Y1 A4 NoiseY 0 0 randn 2 32 with 2 yl s Y1 Y1 Yi Y1 Y1 NoiseY 4 100 times where I take out data each time 4 MATLAB code starts here Least squares PCR with default tol
100. a rameter file should not be the optimal ones First it is necessary to identify 286 Pavol Bist k J Scope parameters General Data history Tip try right clicking on axes Limit data points to last sai jv Save data to workspace Variable name ScopeData Format Janay DK Cancel Help Apply Fig 8 13 Scope block parameter settings each individual piece of the uDAQ28 LT thermo optical plant and conse quently to modify parameters of the controller in the initial parameter file 8 8 Conclusion Remote laboratory software module extends possibilities how to exploit the thermo optical plant in that way that thank to the Internet it is accessible for greater number of interested persons If the safety conditions are kept it can run 24 hours per day Thus the time and distance barrier are decreased and the effectiveness of the real plant usage is enhanced To increase the accessibility of the remote laboratory it is recommended to include the starting Web page of the client application into a Learning Management System LMS or into other system for Web pages administra tion and provide the detailed description of the remote experiments that are available introduce the file names of block diagrams and initial parameters that can be chosen It is also advantageous to provide users with the value of the server IP address if its value is not predefined in the client application The client application is univer
101. a device Click Next to continue Fig 9 1 Found new hardware wizard Found New Hardware Wizard This wizard helps you install software for DIGICON uDAQ28 LT If your hardware came with an installation CD QE or floppy disk insert it now What do you want the wizard to do O Install the software automatically Recommended Click Next to continue Fig 9 2 Install driver from specific location selection 293 294 Peter Tap k and Mikul Huba Found New Hardware Wizard Please choose pour search and installation options Seach for the best driver in these locations Use the check boxes below to limit or expand the delault search which includes local paths and removable media The best driver found will be installed v Search removable media lloopy CD ROM V Include thes location in the search E Vdriver v Browse O Dont search will choose the driver to install Choose this option to select the device driver from a kst Windows does not guarantee that the driver you choose wil be the best match for your hardware Fig 9 3 Driver location selection Found New Hardware Wizard Completing the Found New Hardware Wizard The wizard has finished installing the software for Q USB Serial Converter Click Finish to close the wizard Cancel Fig 9 4 Successful completion of installation 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant Device
102. ables temperature and its filtered value ambient temperature light in tensity its filtered value and its derivative the ventilator speed of rotation and its motor current The temperature and the light intensity control chan nels are interconnected by three manipulated variables the bulb voltage the heat and light source the light diode voltage the light source and the ven tilator voltage the system cooling The plant can be easily connected to standard computers via USB when it enables to work with the sampling pe riods 40 50 ms and larger Within the Matlab Simulink scheme the plant is represented as a single block limiting use of costly and complicated software package for the real time control 9 1 1 Installation in Windows Operating System 9 1 1 1 Device Driver Installation New hardware is detected automatically by operating system and Found New Hardware Wizard will start after plugging device into electrical power network and connecting it to PC by USB cable Fig 9 1 Choose driver installation from specific location Fig 9 2 Directory driver from the package of supporting software is needed to select as device driver location path Fig 9 3 Complete installation requires two cycles of adding new hardware USB Serial Converter and USB Serial Port is installed into operating system Found New Hardware Wizard starts automatically second time The user is informed on successful completion of installation see the windo
103. ack all inequalities into a matrix A 1 0 10 01 O 1 b t 1 1 1 polytope A b 1 2 Create an another polytope Q as the convex hull of points v 1 0 7 12 1 0 U3 0 1 vi 0 5 0 v2 0 5 0 v3 0 0 5 Q polytope vi v2 v3 don t forget the transpose 3 Plot P in red and Q in blue don t forget to use figure to open a new plot figure plot P r figure plot Q b 4 Is P a subset of Q Check this using vertices of Q Or a one liner P lt Q ans 0 5 Is Q as subset of P Check this using vertices of P Or a one liner Q lt P ans 1 Solution 4 2 for Exercise 4 2 1 Let Q1 denote the shift of Q in the direction of the vector 0 5 0 2 01 Q 0 5 0 2 2 Let Q2 be Q1 scaled by a factor of 0 8 Q2 0 8 Q1 3 Let Q3 represent Q2 rotated by 45 th Q3 4 Plot Q in red Q1 in green Q2 in blue and Q3 in magenta Moreover plot all polytopes in the same graph using wireframe 45 180 pi M cos th sin th sin th cos th M Q2 124 Michal Kvasnica opts struct wire 1 linewidth 3 plot Q r Q1 g Q2 b Q3 m opts axis equal Fig 4 12 Resulting plot for Exercise 4 2 Solution 4 3 for Exercise 4 3 Plot the intersection of Q Q2 and Q3 Q amp Q2 amp Q3 H s M I Q amp Q2 I I amp Q3 opts struct wire 1 linewidth
104. akacs stuba sk Peter Tap k Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail peter tapak stuba sk Slavomir Wojnar Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail slawomir wojnar stuba sk Ramprasad Yelchuru Department of Chemical Engineering Norwegian Univer sity of Science and Technology Trondheim Norway e mail ramprasad yelchuru chemeng ntnu no Vladimir Zilka Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail vladimir zilkaAstuba sk Chapter 1 Problems in Anti Windup and Controller Performance Monitoring Morten Hovd and Selvanathan Sivalingam Abstract This chapter provides assignments for control engineering students on the topics of anti windup and controller performance monitoring The as signments are provided with detailed problem setups and solution manuals Windup has been recognized for decades as a serious problem in control applications and knowledge of remedies for this problem i e anti windup techniques is essential knowledge for control engineers The first problem in this chapter will allow students to acquire and apply such knowledge The subsequent problems focus on different as
105. al Ilkiv B Johansen TA 2007 Multiple ARX model based air fuel ratio predictive control for SI engines In IFAC Workshop on advanced fuzzy and neural control Valenciennes France conference paper MO5 3 Pol ni T Johansen TA Rohal Ilkiv B 2008 Identification and modeling of air fuel ratio dynamics of a gasoline combustion engine with weighted arx model network Transaction of the ASME Journal of Dynamic Systems Measurement and Control 130 6 061009 Preumont A 2002 Vibration Control of Active Structures 2nd edn Kluwer Aca demic Publishers Rossiter JA 2003 Model based predictive control a practical approach 1st edn CRC Press LCC Song G Qiao PZ Bibienda WK Zhou GP 2002 Active vibration damping of com posite beam using smart sensors and actuators Journal of Aerospace Engineering 15 3 97 103 Sturm JF 1999 Using SeDuMi 1 02 a MATLAB toolbox for optimization over symmetric cones Optimization Methods and Software Special issue on Interior Point Methods 11 12 625 653 Takacs G Rohal Ilkiv B 2009 MPC with guaranteed stability and constraint feasi bility on flexible vibrating active structures a comparative study In Proceedings of The eleventh IASTED International Conference on Control and Applications Cambridge United Kingdom Wills AG Bates D Fleming AJ Ninness B Moheimani SOR 2008 Model predictive control applied to constraint handling in active noise and vibration control IEEE Transactions on Contr
106. al Kvasnica finally look at it from the top view 2 Acknowledgements The authors are pleased to acknowledge the financial support of the Scientific Grant Agency of the Slovak Republic under the grants 1 0071 09 and 1 0537 10 and of the Slovak Research and Development Agency under the contracts No VV 0029 07 and No LPP 0092 07 It is also supported by a grant No NIL I 007 d from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism References Borrelli F 2003 Constrained Optimal Control Of Linear And Hybrid Systems Lec ture Notes in Control and Information Sciences vol 290 Springer Geyer T 2005 Low Complexity Model Predictive Control in Power Electronics and Power Systems Dr sc thesis ETH Zurich Zurich Switzerland available from http control ee ethz ch index cgi page publications action details id 2124 Kvasnica M Fikar M Cirka L Herceg M 2011a Complexity reduction in explicit model predictive control In Huba M Skogestad S Fikar M Hovd M Johansen TA Rohal Ilkiv B eds Selected Topics on Constrained and Nonlinear Control Textbook STU Bratislava NTNU Trondheim pp 241 288 Kvasnica M Rauov I Fikar M 2011b Separation functions used in simplification of explicit mpc feedback laws In Huba M Skogestad S Fikar M Hovd M Jo hansen TA Rohal Ilkiv B eds Preprints of the NIL workshop Selected Topics on Constrained and Nonlinear Control STU Brati
107. al Plant 279 8 5 Client User Interface The client application serves as the control panel for remote experiments Its interface must provide remote users with tools that will enable to exploit all functions of the server side Generally this interface allows its users to connect to the server set parameters of experiments control the run of experiments visualize experiments in the form of animations graphs numerical data and finally to save data for later processing T T T T 200 300 400 500 Fig 8 4 Graphical user interface of client application The graphical user interface of the client application can be divided to four quadrants he two upper quadrants cover a graphical visualization of data from the running experiment In the left upper quadrant the experiment is animated by the series of pictures selected from a database according to the current state values of the experiment In the upper right quadrant there are experiment data displayed in the form of graphs There are graphs of input desired output and control val ues These responses give an overview how the experiments evolves More detailed responses can be acquired after saving the numerical data and dis playing them using different software for graphs Matlab e g In the dis played graphs the scale of the axis is continuously modified according to the actual values of the displayed data Below the graphs there is a Legend for
108. al level of the liquid before opening the valve and T1final its final level when we finish experiment by closing the valve Finally At represents the duration of the experiment The second way of determination c does not require to know the exact inflow from the pump which can leads to higher accuracy This way is also 234 Vladim r ilka and Mikul Huba faster Time reguired to prepare and perform the measurement takes just about minute In both cases it is convenient to perform several measurements at different level heights x and to define the resulting value c as a mean value It is caused mainly by the fact that the solenoid valves do not have aperture of a simple circular shape which together with the relatively low pressure and low level height may lead uncertain and to slightly fluctuating values of the valve constant ci Indetermining the valve constant c12 describing interconnection between the tanks we can use similar approach The difference would only be in the fact that during the experiment the outflow valves from tank 1 and 2 will be closed and before its beginning one tank will be filled to the maximum level and the other one made empty We stop the measurement just before the complete alignment of both levels The resulting relation defined from 6 2 for c19 will be Ino hao y hu hai iC 12 Atiz With hio hao as the initial liquid s levels in the first or second tank h11 h21 as the f
109. alue from the certain interval of values e Sample Time sampling period Model state variables are measured and changed only in the regular instants characterized by this value The min imal sampling period is given by physical possibilities of the system 8 Remote Laboratory Software Module for Thermo Optical Plant 281 e Running Time duration of the experiment It limits the time how long a selected block diagram with specified parameters will be running During this time the model state variables are visualized by the help of graphs and animations Also the numerical values of the model state variables are displayed in the field Output e Save to File the name of the file placed on the user computer to which time sequences of the model state variables will be saved after pressing the button Save It is necessary to enter the whole path C output mat e g If the file does not exist it will be created ModelScheme PL Model Parameters initPI1 Input Desired Value 15 000 Sample Time 0 20 Running Time 50 00 Save to File cloutputmat Fig 8 5 Settings Warning When entering any parameter the corresponding field is high lighted by yellow color that indicates data editing The new parameter is valid only if the editing is finished by pressing the Enter key 8 5 2 Server IP Address and Control Buttons Connect Disconnect me q 9 9 i seem uns Fig 8 6 Control buttons
110. anical Engi neering Slovak University of Technology in Bratislava e mail marek honekAstuba sk Mat Kopa ka Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail matus kopacka stuba sk Jozef Csamb l Institute of Measurement Automation and Informatics Faculty of Mechanical Engi neering Slovak University of Technology in Bratislava e mail jozef csambal stuba sk Slavomir Wojnar Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail slawomir wojnar stuba sk 171 172 Gergely Takacs et al wide application possibilities of MPC and the numerous examples of mecha tronic systems two real life laboratory examples have been selected First an active vibration attenuation system is discussed with numerous hardware implementation details This is then followed by a detailed discussion of the implementation of three very different stabilizing MPC approaches tradi tional quadratic programming based MPC pre computed explicit MPC and a sub optimal approach called Newton Raphson s MPC Details are given on software and code implementation using coding examples in the Matlab and C programming environment The next section of this chapter deals with a very different practical example a spark ignition engine example This section will introduce the details o
111. anteed stability Automatica 34 10 1205 1217 Dong X Meng G Peng J 2006 Vibration control of piezoelectric smart structures based on system identification technique Numerical simulation and experimental study Journal of Sound and Vibration 273 680 693 Ferreau H 2006 An online active set strategy for fast solution of parametric quadratic programs with applications to predictive engine control Master s thesis University of Heidelberg Ferreau H Bock H Diehl M 2008 An online active set strategy to overcome the limitations of explicit mpc International Journal of Robust and Nonlinear Control 18 8 816 830 dSPACE GmbH 2009 HelpDesk Application Hou Z 2007 Air fuel ratio control for gasoline engine using neural network multi step predictive model 3rd international conference on intelligent computing Qingdao China Kouvaritakis B Rossiter J Schuurmans J 2000 Efficient robust predictive control IEEE Transactions on Automatic Control 45 8 1545 1549 Kouvaritakis B Cannon M Rossiter J 2002 Who needs QP for linear MPC anyway Automatica 38 879 884 5 Implementation of MPC Techniques to Real Mechatronic Systems 223 Kvasnica M Grieder P Baoti M 2004 Multi Parametric Toolbox MPT Online available http control ee ethz ch Kvasnica M Grieder P Baotic M Christophersen FJ 2006 Multi Parametric Tool box MPT Extended documentation Lara A Burch JC Schloss JM Sadek IS Adali S 2003 Piezo patch sensor act
112. ape but here PV also resembles a sawtooth shape and stiction is therefore not readily identifiable from the time plots The oscillations are of modest amplitude around an order of magnitude larger than the measurement noise 1 Problems in Anti Windup and Controller Performance Monitoring 41 1 05 0 95 4 0 9 L L i L L 7000 7050 7100 7150 7200 7250 7300 Time 0 8 L L L L L 7000 7050 7100 7150 7200 7250 7300 Time Fig 1 41 OP and PV for data set 3 30 20 10 0 10 20 30 Fig 1 42 The OP to PV cross correlation function for data set 3 The cross correlation function for data set 3 is shown in Fig 1 42 This is an almost perfectly even valued function and this test therefore does not indicate the presence of stiction The OP PV plot for data set 3 is shown in Fig 1 43 The plot looks like an ellipsoid seen from the side but due to the angle it is difficult to assess whether there are any sharp corners in the plot In Fig 1 44 the PV time series is therefore shifted by 5 samples Again we find that the plot is a 42 Morten Hovd and Selvanathan Sivalingam L L L 0 04 0 03 0 02 0 01 Fig 1 43 OP PV plot for data set 3 0 25 fi L L 0 0 01 0 02 0 03 0 04 PV k 5 L L L 0 04 0 03 0 02 0 01 L L 0 0 01 0 02 0 03 0 04 Fig 1 44 OP PV plot for data set 3 with the PV data series shifted 5 samples little messy since the tim
113. ariables without anti windup 1 2 4 4 Simulation with Anti windup The simple anti windup technique for PI controllers described in Hovd and Sivalingam 2011 is used and is implemented as illustrated in Fig 1 29 Note that if we want to account for saturation in the valve in addition to the effect of the selector this could easily be done in the simulation by putting a saturation element in series after the selector before sending the anti windup signal to each controller In a real life implementation it would be preferable to use for anti windup a direct measurement of the valve position The simulated responses in the controlled variables when including anti windup are shown in Fig 1 30 and the corresponding responses in the con troller outputs are shown in Fig 1 31 We see that only small deviations of I I Flow controller output Pressure controller output Input applied fi L fi fi L fi L n 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Fig 1 28 Responses in controller outputs without anti windup 1 Problems in Anti Windup and Controller Performance Monitoring 33 kf u E m 0 1 Gaint Flow setpoint deviation variable Tif s 1 Transfer Font f Flow uf deviation variable 1 Flow controller output 20s 1 5 Flow disturbance 10s 1 Flow W min ur MinM
114. ase where H may be any full matrix In terms of selecting controlled variables c Hy this means that any combination of measurements are allowed However in practice there may be constraints on H for example one wants to use a subset of the measurements or one want to use a decentralized structure for H We will consider the following special cases Case 1 No restrictions on H This is the case already considered where Theorems 2 1 2 4 hold 66 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Note that key for deriving Theorems 2 1 2 4 was that if H is a solution then so is H DH where D is any non singular matrix Case 2 H contains a subset of the measurements but is otherwise full Theorems 2 1 2 4 hold also in this case The reason is that H DH will have the same structure as H for any nonsingular D This is because if H has columns equal to zero then these columns will remain zero in DH even if D is full For example if H i ea then we can still allow a full D z where is 0rx0zc Tc any number and keep the structure of H in H DH Case 3 H contains measurements from disjoint set so H has a block diagonal decentralized structure Theorems 2 1 2 4 do not hold in this case The reason is that for H DH to have the same structure D must have a structure similar to H For example let H eee then 00022 zx 0 LLLLL D la o where x is any number and if H k 00x a
115. atrix formulation as cesis jes Ies 4 20 which clearly is an intersection of two polytopes in the x space hence a poly tope itself Can you compute and plot the set V using the numerical data above Verify constraint satisfaction by checking the extremal vertices of W Exercise 4 10 We still consider the LQR case from the previous example We have seen that for any state r V the LQR feedback u K respects constraints But for how long Let s investigate one of the corner points of V E extreme PSI x E 1 u K x u 1 0000 What is now the successor state tp41 Av Buy xn Axx Bru xn 2 1713 3 3287 But this point is outside of W isinside PSI xn ans 0 which means that for this new state some constraints would be violated u K xn u 2 0002 122 Michal Kvasnica In other words V does not possess the set invariance property In simple words a set P is invariant if for all starting points xg inside such set all future trajectories x remain in the set Vk gt 0 Set invariance is crucial for MPC in order to guarantee stability of the closed loop system Now that we checked that W is not invariant can we compute its invariant subset Following definition might help Py k k E Y A BK zy V Vk gt 0 4 21 Following iterative algorithm can be used to compute Py Step 1 k 1 Wp Y Step 2 repeat Step 3 V 2 E Wg A BK z Wk St
116. atter for the optimization problem which becomes ming HG H ie and we want to find the optimal H a Consider the case with 2 measurements so G column vector gi g and H h hg row vector and solve the problem analytically Also compute the optimal norm j HG H b Derive the result more generally for the scalar case with any number of measurements by making use of the definition of the induced 2 norm sin zT G h gular value which for a vector gives G VG G maxn 2 62 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke note that for a vector the Frobenius norm p and 2 norm 3 are the same c Use the analytical formula presented below to derive the general result for the multivariable case nu ne gt 1 Solution 2 1 a Two measurements HG hiha H h g haga scalar P HGY H h h2 higi h292 1 27 g1 292 where x ha hi Set d j dx 0 to find the optimal x After a lit tle simplification x 1 x go g1 xg 0 which gives x go gi Conclusion ho hi 92 91 that is we prefer to control the measurements corresponding to large el ements in G Also find 7 1 4 g g2 which we note is equal to 1 IG l b Any number of measurements Let h H be a column vector G is already a column vector Since HG hT G is a scalar it is equal to its transpose and we have tha
117. aw k2a k2b inv k2d k2c k2bw zeros size k2b k2b inv k2d uz size k2b inv k2d k2cw k2c k2dw k2d zeros uz 1 1 9 4 LOG Controller Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh LQG controller with integral action Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Augment plant with 2 integrators ny size C 1 nx nu size B n 5 m 2 Zxx zeros nx nx Zxu zeros nx nu Zux zeros nu nx Zuu zeros nu nu Aint A Zxu C Zuu Augment plant with integrators Bint B D Augment plant with integrators at plant output states 5 use old system gt old number of states Kalman filter only for true states not needed for integrators Q kw eye 2 R kv eye 2 sys d ss A B Bd C D D 4 D is zero anyway kestold KFL P M Z kalman sys_d Q R LQR control states Q 5xeye states R 1 eye 2 N 0 lqrK S e lqr Aint Bint Q R Shows that we only need the A and B matrices for LQR control larKs lqrK 1 nx Plant state feedback part lqrKint larK nx 1 nx ny 4 Integrator state feedback part 36 37 38 39 40 41 42 43 44 45 46 AT 48 49 50 51 52 53 1 Problems in Anti Windup and Controller Performance Monitoring 27 lqrKii inv lqrKint Aki zeros 2 2 State space representation of two integrators in parallel Bki eye 2 Cki eye 2 Dki 1
118. ax Input applied deviation variable 10 100s 1 Pressure E 20s 1 Transfer Fen gt p P Pressure deviation variable kp u x3 Gain 0 Pressure setpoint deviation variable Pressure controller output 1 a x i Tip s 1 Transfer Fcn2 Fig 1 29 Simulink implementation of model and control system with anti windup the controlled variables above their setpoints occur T here is no appreciable windup effect and active control quickly switches to the other controller upon reversal of the sign of the disturbance For the inactive control loop there are still large deviations below the setpoint i e in the safe direction This is clearly unavoidable when controlling only one variable at the time Comparing to the simulations without anti windup we observe much bet ter control of deviations in the controlled variables above the loop setpoints Anti windup is therefore very important when the value of a manipulated variable is determined by selecting between the outputs of several controllers especially if good control in one direction is required for safety The simulations also illustrates the error in the widely held belief that the active control switches to the variable whose value is above setpoint In Fig 1 27 we can easily find periods where both controlled variables are above the setpoint and in Fig 1 30 we see that the active co
119. ble represents a valid controller object which can be further post processed or deployed as real time executable code as illustrated in the previous section In the sequel we present several examples illustrating how the concept of adding new constraints can be used to achieve typical control tasks 4 5 4 1 Terminal Constraints One way of ensuring that the MPC controller will provide closed loop stability is to add a terminal state constraint xy 0 This requirement can be easily added to the set of constraints by gt gt C C V xfend 0 138 Michal Kvasnica Evolution of states Evolution of outputs 1 1 X Y 0 H 0 JYX H a 1 2 1 amp on 2 Ven o 2 3 3 4 4 0 10 20 30 0 10 20 30 Sampling Instances Sampling Instances Evolution of control moves Active dynamics 2 1 5 8 3 E a o 1 s a 0 5 1 0 0 10 20 30 0 10 20 30 Sampling Instances Sampling Instances Fig 4 21 Closed loop simulation using the simplot function where V xfend represents the final predicted state zw Another option is to use terminal set constraints i e to require that xw 7 for some polytope T x Hx K This type of constraints can also be introduced easily by gt gt C C H V xfend lt K 4 5 4 2 Time varying Constraints In the optimization problem 4 26 constraints 4 26d and 4 26e are time invariant To make
120. box provides easy to use access to various tasks ranging from modeling of dynamical systems through control synthesis up to verification and code deployment The purpose of this chapter is twofold First the toolbox is introduced and its functionality is explained by means of simple illuminating examples Then a set of advanced exercises is provided for the reader to practice and master their skills 4 1 Multi Parametric Toolbox Optimal control of constrained linear and piecewise affine systems has gar nered great interest in the research community due to the ease with which complex problems can be stated and solved The aim of the Multi Parametric Toolbox MPT is to provide efficient computational means to obtain feed back controllers for these types of constrained optimal control problems in a MATLAB programming environment As the name of the tool hints it is mainly focused on calculation of feedback laws in the parametric fashion in which the feedback law takes a form of a PWA look up table But the toolbox is also able to formulate and solve MPC problems on line in the receding hori zon fashion i e by solving the optimization problem for a particular value of the initial condition at each time step Michal Kvasnica Faculty of Chemical and Food Technology Slovak University of Technology in Bratislava e mail michal kvasnica stuba sk 101 102 Michal Kvasnica In short the Multi Parametric Toolbox can be described as being a
121. by evaluating f Rv 5 13 where f is the perturbation vector R is an input from the off line optimiza tion and v is a vector product re calculated at each sampling interval 5 Implementation of MPC Techniques to Real Mechatronic Systems 193 if W2 gt tol dgemv_ chn amp order korder amp mone R amp order Vec amp onei amp zero f amp onei The other option in the else construct is to evaluate for the unknown A using the Newton Raphson procedure This conditional statement launches a while loop which cycles through the NR procedure until the floating point absolute value of error is larger than the pre set tolerance The first part of this code segment serves only to evaluate the matrices used in the NR loop These simplifications increase computational speed and are based on the assumptions about function A and its i th derivatives The second part of the following code segment is the Newton Raphson algorithm itself Here the first step is to evaluate the value of A and its derivative The ratio of the function value and its derivative is the error which is subtracted from the result for A from the previous step elsei while fabs err to1 1 descal korder klamN Sm m dediv_ amp order W1 m W1d dediv amp order Wid m Widd fval ddot korder vec konei Widd konei W2 fderval 2 ddot amp order Wid amp onei Widd amp onei err fval fderval lamN lamN err Since the value of A has been acq
122. cal implementation details while sug gesting the simplest possible solutions Therefore no customized on line quadratic programming solvers or algorithms of multi parametric program ming are discussed here All methods will be implemented in Matlab Simulink if possible using off the shelf solutions for off and on line solvers Typically the problem setup will be carried out in Matlab while the on line problem is running on an xPC Target platform system We will begin our discussion with introducing the experimental system used as an example 5 2 2 Brief Description of the Experimental Hardware A clamped cantilever beam may represent many real life under damped flex ible structures such as helicopter rotor beams antenna masts manipulating arms and solar panels on space structures The experimental setup considered as an example to demonstrate different MPC implementations may represent the vibration attenuation of any of the aforementioned physical systems The aim of the MPC based vibration control system is to minimize beam tip vibrations while respecting constraints set by the maximal allowable volt age levels on the piezoelectric patches Since piezoelectric materials are prone to depolarization maximal voltage input levels are set Additional require ments placed on the control system are the need for guaranteed stability and constraint feasibility 5 Implementation of MPC Techniques to Real Mechatronic Systems 175 5 2 2 1 Hardwa
123. can be than computed as K ys us In this example it gives kom Us 2 5 Modify the Simulink model to use the inverse process gain to control the plant Fig 9 11 Do not forget to add the input saturation in the model 8 9 1 300 C udag28LT_iov2 Elle Edit View Simulation Format Tools Help DOBA BmBe 551 gt Temperature Optical Thermal Plant P VO Interface 3 inienerfy Fan current Fan rpm Peter Tap k and Mikul Huba Light intensity SB 0 9 9h 02 Time offset 0 Fig 9 10 Basic Simulink model for the udag28 LT plant because the bulb voltage is limited from 0 V to 5 V Add the setpoint signal to the light intensity scope Set the simulation time to infinity Ele Edt View Semuahon Format Took Help Optical Thermal Plant VO Interface Fig 9 11 Static feedforward control Light intensity 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 301 Make multiple setpoint steps in a wide operational range It can be done while the experiment is running Light intensity Fig 9 12 Experimental results Light intensity scope You should observe a steady state error in several working points The smallest steady state error can be seen around the point where the process gain was measured It is not difficult to conclude that the process parameters vary throu
124. can speed up the transients processes And contrary by increasing 71 when the pole is shifted towards the origin we 7 Constrained PID Control Tasks for Coupled Tanks Control 249 can slow down them In increasing the P controller gain we have to be careful and speed up the transients only until certain limit 0 8 0 6 a 1 T 1 2 100 a 1 T 1 2 100 0 4 J 0 4 0 2 J 0 2 0 0 0 1 2 3 4 5 0 1 2 3 4 5 Fig 7 1 Impact of the P controller tuning on the tranient responses The limitations are mainly given by factors as nonmodelled dynamics plant uncertainty measurement noise and control signal saturation For shorter time constant 71 saturation of the control signal typically occur and for T1 0 the control signal tends to rectangular pulse of the minimum time control Fig 7 1 P controller for integrator plant represents solution of the dynamical class 0 If we increase the amplification of controller too much the sensitivity of the circuit to the measurement and quantisation noise will increase and this can lead to system s destabilization increased wear of active elements or damage in this case by switching the pump on and off over a too long period to maximum or zero voltage On the other hand too small controller gain can slow down the transient processes or in an extreme case the pump will even not start to move In order to get some insight which controller gain may be appli
125. ce and feasibility conditions LMI are parsed using YALMIP introduced by L ofberg 2004 to the SeDuMi optimization software suite as formulated by Sturm 1999 The first portion of the off line code initializes the algorithm Amongst others simulation stop time is defined along with the desired deflection which in the case of the vibration attenuation example is always zero A linear time invariant state space model is loaded from a file Sampling time is also defined as T s Tstop 60 yd 0 load m2ss mat A m2ss A B m2ss B C m2ss C D m2ss D Ts 0 01 Other types of settings and tuning parameters are also declared at the beginning of the script file Symmetric input constraints are stated as uh State penalties are set as Q CT C which includes the output deflection in the computed cost Input weighting is declared as the variable R uh 120 Q C C R 1e 4 Prediction cost performance bound y is stated which is necessary to be limited in order to preserve numerical stability of the process A tolerance limit is also set which is used to modify the behavior of YALMIP regarding the handling of strict inegualities tolerance 1e 8 5 Implementation of MPC Techniques to Real Mechatronic Systems 185 gamma 1e6 Dimensionality of the problem is determined to allow the use of different state space models for generating predictions dim size An dim dim 1 1 w dim size B w dim 1 3 w dim 1 1 w dim 1 4 w dim 1
126. cessary if integral action 20 Morten Hovd and Selvanathan Sivalingam Output 1 15 1 L fi fi L L L L L 50 100 150 200 250 300 350 400 450 500 Output 2 2 T f L L L i L L i 0 50 100 150 200 250 300 350 400 450 500 Fig 1 20 Response with PI controllers with anti windup E P gi2num s Transfer Fen giinum s yes i Transfer Fent n mP Ax Bu e yref ne Jaa ei y Cx Du 1 k1 Hanus 0 01s 1 Fast lag yi to break algebraic loop1 y u u measured pa qe 1 y v gt X Ax Bu a y Cx Du 0 01s 1 k2 Hanus Fast lag g21num s to break algebraic loop a g22num s 7 y Fig 1 21 Simulink implementation of inverse decoupler with anti windup for outer PI controllers is used in the controllers for the individual loops that result after decoupling compensation This is the common case which also is used in this example However even if simple PI controllers are used in the individual loops we may can no longer use the simple anti windup scheme in Fig 1 1 Rather we have to calculate the range of manipulated input variable movement that is available after accounting for the action of the decoupling elements This is is illustrated by the Simulink implementation in Fig 1 21 Note that here the anti windup of the PI controllers has been achieved using the so called Ha
127. ch depending on which region R x t Hix t Ki i 1 Nreg contains the state x t Theorem 4 1 suggests that the RHMPC feedback Un a t can be con structed off line and stored as a lookup table which consists of the regions R and the associated feedback gains F G The advantage of such an approach is that value of Uw x t for a particular value of x t can be obtained by simply evaluating the table as captured by Algorithm 1 The algorithm tra Algorithm 1 Sequential table traversal INPUT Regions R feedback laws F Gi number of regions Nreg state measure ment x t OUTPUT Optimal RHMPC control input uo z t 1 for r 1 Nreg do 2 if H z t lt K then 3 Un x t Fpa t Gr 4 uo a t I 0 0 Uv lt t 5 return uo z t 6 end if 7 end for verses through the regions sequentially stopping once it finds a region which 4 Multi Parametric Toolbox 133 contains x t In such case the optimal control action is calculated by evalu ating the corresponding control law and returned back Clearly in the worst case the algorithm has to search through all regions But since the operations performed in Steps 2 4 only consist of simple matrix multiplications and ad ditions for a large class of problems running Algorithm 1 is faster compared to obtaining uo x t by solving 4 26 on line as an LP QP MILP MIQP using off the shelf solvers The second benefit is that the explicit representa tion of the f
128. controller_explicit is not contained in the regions of inv_controller To see that we compute the set difference between the regions of the two controllers gt gt NI controller_explicit Pn inv_controller Pn and we plot it plot regions of the invariant subset in yellow 4 Multi Parametric Toolbox 161 4 and the states which are not invariant in red gt gt plot inv controller Pn y NI r You should see the figure similar to the one below Fig 4 28 Set of invariant states yellow and those which are not invariant the two small red patches in the upper left and lower right corners With the invariant controller available in inv controller we can return back to stability analysis gt gt L mpt lyapunov inv controller pwq This time the Lyapunov function is found mpt_getPWQLyapFct SUCCESS Found Piecewise Quadratic Lyapunov function which shows that the explicit controller inv_controller will definitely ren der the closed loop system stable You can check this fact by running a bunch of closed loop simulations using the point and click interface gt gt simplot inv_controller Since the controller is invariant and guarantees stability do you expect to find a starting point for which the simulation would fail 162 Michal Kvasnica 4 7 Solutions Solution 4 11 for Exercise 4 11 it s always wise to wipe the sysStruct variable clear sysStruct plant dynamics
129. created in the form of a Java applet inserted into a Web page For the testing purposes the installation is not needed because it can be run directly from the installation CD by clicking the client html file placed in the Client Application folder But the client application is aimed for a Web publication when it can be accessible for all potential users In this case it is necessary to upload the content of the Client Application folder to the desired Web server Then after setting the correct Internet address the client application is downloaded into the Web browser window that enables running of Java applets 8 4 Running the Client Server Application The run of a client server application must be synchronized First it is necessary to run the server part of the application After the server has been successfully installed it is possible to run it by a double click to the RemLabServer exe file that is after the correct installation placed in the CA RemLab Workspace folder Depending on the server computer safety set tings it can happen that the system will ask for confirmation of a trust to the application 278 Pavol Bist k cx C RemLabServer exe connection connection t connection connection waits for client connection Fig 8 2 Server application waiting for client connection It can be noticed that the run of the server application invokes the run of Matlab application automatically After starting the server is held i
130. ct that the air mass flow is measured as an absolute value it was necessary to integrate this amount during the particular time and express the air and fuel quantity as relative mass densities grams cylinder Hence the air fuel ratio is defined as Malk Liymg k Where ma k and my k are relative mass amounts of air and fuel in a cylinder and Lin 14 64 is the theoretical amount of air necessary for the ideal combustion of a unit amount of fuel Considering the A k modeling the engine has been divided into two subsys tems with independent inputs namely into A k 5 15 air path with the air throttle position as the disturbance input and fuel path with the input of fuel injector opening time 5 Implementation of MPC Techniques to Real Mechatronic Systems 207 ORAS 4 Fig 5 15 Case with converters and security circuits Another disturbance like acting quantity were engine revolutions implicitly included in the engine model particularly for each working point The output ratio of both paths is the value of A 208 Gergely Takacs et al 5 3 3 2 SI Engine Modeling using ARX Models The engine modeling is based on the weighted linear local model with single input single output SISO structure Pol ni et al 2008 The parameters of local linear ARX models with weighted validity Murray Smith and Jo hanssen 1997 are identified to model AFR nonlinear dynamics The prin ciple of this nonlinear modeling techniqu
131. ct control quality 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 323 FPI T T 570 8 Light intensity 80 100 120 140 160 180 200 220 240 260 280 time s FPI T T 970 6 52r 4 Light intensity 80 100 120 140 160 180 200 220 240 260 280 time s PP T T 970 06 52r 4 Light intensity 80 100 120 140 160 180 200 220 240 260 280 time s Fl controller RK c Cc 00 o N T 1 Light intensity BA o 100 120 140 160 180 200 220 240 260 280 time s co o Fig 9 41 Experimental results detail delayed system output setpoint step 324 Peter Tap k and Mikul Huba FPI T T 0 8 52r 4 D D o E o 60 80 100 120 140 160 180 200 220 240 260 280 time s PP T T 0 6 52r 4 D Ez 2 E E o 60 80 100 120 140 160 180 200 220 240 260 280 time s FPI T T 0 06 52r J D 5 o mnie erue darpa iaa E 48r J e 46r 1 60 80 100 120 140 160 180 200 220 240 260 280 time s controller 0 52 5 50 E 48 o 46 60 80 100 120 140 160 180 200 220 240 260 280 time s Fig 9 42 Experimental results detail undelayed system output setpoint step 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant PPI T T 9 0 8 38 3 Light intensity 1100 1040 1060 1080 1000 1020 1120 1140
132. ctric actuators As in most real systems direct control feed through is not considered therefore matrix D is set to equal zero The experimental procedure involved measurements in the frequency do main The system has been excited by an amplified chirp signal entering the actuators PZT1 and PZT2 The amplitude of this signal has been set to reach maximal allowable levels of 120 V in order to minimize vibration caused by the surroundings Since the first eigenfrequency of the beam is 8 127 Hz the chirp signal frequency started from 0 1Hz up to 20Hz The measured signal has been sampled 5000 Hz and subsequently re sampled for different models The predictions are generated by the following state space system sampled by 100 Hz 5 Implementation of MPC Techniques to Real Mechatronic Systems 177 an 0 9981 1 2233 B 6 697 E 9 10 0021 0 9985 8 001E 6 C 0 5774 0 7069 5 2 5 2 8 Quadratic Programming based MPC 5 2 3 1 Introduction This subsection will elaborate on the practical implementation of a dual mode quadratic programming based MPC controller with stability and fea sibility guarantees The on line quadratic programming solver considered in this application is called qpOASES and it has been introduced by Ferreau et al 2008 and subsequently Ferreau 2006 This solution has been chosen for its easy implementation properties and the fact that unlike generic QP solvers it has been fine tuned for MPC application
133. d 3 2 Simple excercise Exercise 3 1 Cox 1992 Check whether the two polynomials f 2x 3x 1 and fo 7a x 3 have a common root in C Solution 3 1 The resultant is the determinant of the Sylvester matrix Johannes J schke Department of Chemical Engineering NTNU Trondheim Norway e mail jaschke chemeng ntnu no Sigurd Skogestad Department of Chemical Engineering e mail skoge chemeng ntnu no 1 The software can be downloaded at www sop inria fr galaad software multires 91 a AB U NR 92 Johannes J schke and Sigurd Skogestad 2070 3217 1331 0103 Res fi fa det 153 0 3 1 There exist no common root since the resultant is nonzero Maple code for the Sylvester matrix example with LinearAlgebra f1 2 xx 243 x4 1 f2 T x 2 x 3 Syl SylvesterMatrix f1 f2 Res Determinant Syl 3 3 Isothermal CSTR Case Study Consider a CSTR as in Figure 3 1 with a feed stream containing component A and with two first order chemical reactions A B n kc d 3 2 B C ra kocp Of the products formed B is the desired procuct while C is an undesired side product The manipulated variable is the feed stream q which can be adjusted to achieve profitable performance The operational objective is to maximize the concentration of the desired product CAF 2 L LY A CO CA CB CO q Fig 3 1 Isothermal CSTR 3 Measurement polynomials as controlled variables
134. d PV for data set 1 are shown in Fig 1 34 The OP resembles the sawtooth shape and PV resembles the square wave shape that together are held to be typical of stiction The cross correlation function for data set 1 is shown in Fig 1 35 This is nearly an odd valued function and this test therefore indicates the presence of stiction The OP PV plot for data set 1 is shown in Fig 1 36 for approximately two oscillation periods Despite the imperfect trend removal the plot clearly shows the signals oscillating around an open area and sharp corners in the plots T his test therefore also indicates the presence of stiction 1 Problems in Anti Windup and Controller Performance Monitoring 37 OP I L L 1 i L L L L 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time PV L L L L L L L L L 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time Fig 1 34 OP and PV for data set 1 1500 T 1000 F al 500 F 4 1000r 1500 L L L L L L L i 1000 800 600 400 200 0 200 400 600 800 1000 Fig 1 35 The OP to PV cross correlation function for data set 1 The routines bicoherence m and glstat m are both run to detect the presence of nonlinearity in the output Default parameters are used for both routines Elements 5001 9096 of the PV data series are used a data series of length a power of 2 is used since these routines are based on the FFT The command bic waxis NGI NL
135. d dele RP bo data Ena LAS TONTOR ollm ncn elie k nhs See add aera LELY Sarlations 2 es data E sated 11 8 JAutiwindup i reikia bee eee LLI Matlab Gaede wn liest cette 1 2 Anti windup with PI Controllers and Selectors 1 2 1 Introduction to Selective Control 1 2 3 The Control Problem in the Assignment 1 2 9 PSO cocus E LT O E ER Re 12 4 OH 213 teat sooo ide lek e REISCRLSA US hoo Deed cae ein sae RAE TREE dU RE Rd Lal ASAIO 2 2122 iessteeee RR OR EP P Reed Lee OIM DIE aereum RR Re Rex FRE REL 1450 MOS o hogtied ee ed 1 4 Controller Performance Monitoring using the Harris Index 1 4 1 Assignment eccesso eem eee 14 2 BOA 2216 eee Rene R ne bee a ed PETR onus ie blbondce Su equ Pe eode de dob de ies eed a B viil Contents Optimal Use of Measurements for Control Optimization and Estimation using the Loss Method Summary of Existing Results and Some New uussu 53 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke 2 1 LubcducboH lt pei ee eked bebe Dia EEEE E 53 22 Problem Formulabiol csset Rede 54 2 2 1 Classification of Variables 54 22 Cost PBC cz ce ccs odd on herr See dou A Red des 54 2 2 3 Measurement Model 0 0000 cece eee 55 22 4 Assumptions 2 6 i000 ncaon denna b Re gees eek 55 2 2 5 Expected Set of Disturbances and Noise 55 2 2960 Probl le hm RE ER RR 56 2 2 7 Examples of this Prob
136. d only if 1 z t 1 Fig 4 23 Visualization of the lookup table for the control setup with the login equivalence constraint 1 lt z t 1 0 5 uo lt 0 5 4 5 5 Analysis The toolbox offers broad functionality for analysis of hybrid systems and verification of safety and liveliness properties of explicit control laws In ad dition stability of closed loop systems can be verified using different types of Lyapunov functions 4 5 5 1 Reachability Analysis MPT can compute forward N steps reachable sets for linear and hybrid sys tems assuming the system input either belongs to some bounded set of inputs or when the input is driven by some given explicit control law To compute the set of states which are reachable from a given set of initial conditions XO in N steps assuming system input u k Uo one has to call R mpt reachSets sysStruct XO UO N where sysStruct is the system structure XO is a polytope which defines the set of initial conditions r 0 A UO is a polytope which defines the set of admissible inputs and N is an integer which specifies for how many steps should the reachable set be computed The resulting reachable sets R are returned as a polytope array We illustrate the computation on the following example Example 4 1 First we define the dynamical system for which we want to compute reachable sets 142 Michal Kvasnica define matrices of the state space object A
137. den s Pl2 Fig 1 8 Implementation of PI controllers without anti windup 14 Morten Hovd and Selvanathan Sivalingam e2 gi2num s gtinum s kinum s Transfer Fcn1 a kiden s 1 Transfer Fcn3 0 01941 Fast lag to break algebraic loop u u measured je k2num s 1 k2den s 0 01s 1 Transfer Fcn2 Fast lag g21num s to break algebraic loop g22num s Transfer Fen Fig 1 9 Simulink implementation of inverted decoupler without anti windup of PI controllers Int a O lqrK In2 7 1 C 3 u gt C u s In3 B Integrator1 C L Au A Fig 1 10 Simulink implementation of LOG controller Input 1 are the values of the augmented integrating states which are available to the controller whereas input 2 is the measurement vector nnn 00 Square 0 Constant P yref gt l 1 E gt 5 Integrator Measurement gt y nt om jeasuremen P in2 Outt int ig Out gt u LQGint LV w saturation To Workspace Measured input Measurement Fig 1 11 Overall simulation setup for LQG controller with integrating states no anti windup is provided for the integrating states of the augmented LQ regulator 1 Problems in Anti Windup and Controller Performance Monitor
138. der optimality constraints Jac Jacobian g cA cB cC q gradT Jacobian J cA cB cC q Calculate the null space of the constraints N NullSpace Jac N N 1 Transpose and mutlitpy G Transpose gradT NT Transpose N The reduced gradient is then Cv VectorMatrixMultiply NT G cv simplify numer Cv 1 pick only numerator Unknown variables to be eliminated read multires mpl varlist k2 k1 cB unknown variables polylist cv gi g2 g3 optimality conditions BigMat spresultant polylist varlist Construct resultant matrix LargeMat det BigMat Calculate determinant c factor LargeMat factorize the CV save Jac grad N Cv cv c invariant save results 3 Measurement polynomials as controlled variables Exercises Comments Remarks 97 98 Johannes J schke and Sigurd Skogestad Comments Remarks 3 Measurement polynomials as controlled variables Exercises Comments Remarks 99 100 Johannes J schke and Sigurd Skogestad Comments Remarks Chapter 4 Multi Parametric Toolbox Michal Kvasnica Abstract This chapter introduces the Multi Parametric Toolbox MPT for MATLAB which is a freely available tool for Model Predictive Control and computational geometry MPT simplifies and automates many tasks a control engineer has to go through when designing and validating optimal control laws based on the MPC principle The tool
139. e 3 eye 2 A with a small non singular D term to make the Hanus form implementable Ahki Aki Bki inv Dki Cki Two integrators in parallel in Hanus form Bhki zeros ny ny Bki inv Dki Chki Cki Dhki Dki zeros ny nu hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh tlh h Fast low pass dynamics to break algebraic loop in Simulink Af 1000 eye 2 Bf eye 2 Cf 1000 eye 2 Df zeros 2 2 1 2 Anti windup with PI Controllers and Selectors Selectors and overrides are often used in plants when operational objectives change with operating conditions Clearly selectors makes the control system non linear and mistakes are easily made when designing such control systems leading to sustained oscillations or even instability It may reasonably be argued that more advanced control techniques specifically MPC can be a good alternative to regulatory control systems in which standard single loop controllers i e PI are combined with selectors since changes in the set of active constraints are handled relatively easily with MPC This assignment will address a particular problem of windup when us ing selectors This windup is not caused by constraints in the manipulated variable but occurs for inactive deselected controllers due to the absence of feedback when the controller is not selected In this assignment a simple remedy to this problem is tested 1 2 1 Introduction to Selective Control Selective contro
140. e Controller Computation MPT Toolbox features rapid code deployment functionality The controller stored as a Matlab multi field variable and can be exported as a stand alone C code by using the command mpt_ezportc ctrl where ctrl is the controller name This C code can be then integrated into a given application The MPMPC controller has been integrated into a custom S Function block The function code built through the S Function Builder takes the state vector as its input and has a single output the direct controller voltage The core code is very simple and involves calling the routine supplied by the MPT Toolbox in the form double region region mpt getInput x0 u 5 Implementation of MPC Techniques to Real Mechatronic Systems 197 Fig 5 7 MPMPC controller regions plotted in two dimensional state space where x is the state vector obtained through an observer block u is the con troller output An additional variable region returns the index number of the polyhedral region corresponding to the acquired current state measurement For the correct functionality of the S Function block the sampling rate is defined so is the external function declaration of the MPMPC routine extern double mpt getInput double double and a declaration of the C source file as well mpt_getInput c The process of associating a state with a region and the corresponding PWL function is repeated at each sampling interval and is impleme
141. e by discretizing the continous time model sysStruct mpt_sys syst Ts define system constraints sysStruct ymax 10 sysStruct ymin 10 sysStruct umax 1 sysStruct umin 1 define the set of initial condintions as a polytope object XO polytope 0 9 0 1 0 9 0 1 1 1 0 1 1 1 0 1 set of admissible inputs as a polytope object UO unitbox 1 0 1 inputs should be such that ul lt 0 1 4 set of final states the unsafe states Xf unitbox 2 0 1 0 2 0 2 number of steps N 50 perform verification canreach Nf mpt verify sysStruct XO Xf N UO If the system states can reach the set X canreach will be rue otherwise the function will return false In case Xf can be reached the optional second output argument Nf will return the number of steps in which Xf can be reached from X0 Example 4 4 It is also possible to answer the verification question if the sys tem inputs are driven by an explicit control law 4 Multi Parametric Toolbox 145 load dynamical system Double_Integrator compute explicit controller expc mpt control sysStruct probStruct define set of initial condintions as a polytope object XO unitbox 2 1 3 0 4 set of final states the unsafe states Xf unitbox 2 0 1 0 2 0 2 number of steps N 10 perform verification canreach Nf mpt verify expc XO Xf1 N 4 5 5 3 Invariant Set Computat
142. e calculation of the con troller regions The main calling function takes the system and the problem structure as an argument and outputs the multi parametric MPC problem 196 Gergely Takacs et al ctrlex mpt_control sysStruct probStruct Regions of the controller can be determined by regions length ctrlex The controller can be saved in its original structure so it can be later loaded into the workspace or used in off line simulations via the standard Matlab interface save ctrlex mat ctrlex The export of the controller into a C code is very simple and straightfor ward and it can be carried out using the following command mpt_exportc ctrlex One may need such data as the volume of the region of attraction This for example may be used to compare different controller designs or models The total volume of the region of attraction is the sum of the individual volumes and can be simply calculated by result areareach sum volume ctrlex Pn The maximal absolute deflection of the beam can be calculated by creating a convex hull around the region of attraction transforming this into a vertex representation and by multiplying the individual edges of the region of at traction with the output matrix C we may get the direct output equivalents The maximum of this is the maximal possible deflection at the beam tip P Pn hull ctrlex Pn result V extreme P result maxdef max abs sysStruct C result V 5 2 5 3 On lin
143. e case of the fuel path Malk T mp rlaus k DIB 33 pl r N 5 N IS The chosen MPC approach utilizes the state space representation and its structure uses a control deviation for the correction of the prediction Due to a disturbance d k the steady state values of u and x have to be adapted so that the assumption J 0 is valid This problem solves an explicit inclusion of the disturbance into the model The fuel injectors are controlled by a fuel pulse width what is at the same time the control uy The optimal injection time can be computed by minimization of a cost function 5 32 which has after expansion by the fuel path prediction equation form Ja Ji BE reis Op u 1 Lly k m s 9 Aur D ple ap I 2 gt 5 33 r An analytical solution of 4 0 of 5 33 without constraints leads to a 4 definition determining the change of fuel injector opening time in a step k 5 Implementation of MPC Techniques to Real Mechatronic Systems 219 as Au QTR Ir pOT Qn GT hu k P803 LG M pO ANE pote 9 Hence the absolute value of the control action in a step k is given by a sum of a newly computed increment in a control 5 34 and an absolute value of the control in a step k 1 uf k uj k 1 Zur k 5 35 5 9 5 Results of a Real Time Application of a Predictive Control The ability to control the mixture concen
144. e computational unit necessary for the real time implementation of the MPC control is based on a powerful and freely programmable control system based on dSpace and RapidPro units or Rapid Control Prototyping System RCP Fig 5 13 dSPACE GmbH 2009 It is built up on the processor board ds1005 and hardware in loop platform 204 Gergely Takacs et al d52202 HIL The main components are Fig 5 12 e DS1005 PPC Board processor module e DS2202 HIL I O Board module of input output interface e DS4121 ECU Interface Board interface for the communication with the ECU DS1005 PPC Board DS1005 PPC Board is a core processor module of the dSpace system It has a huge computational power for the real time applications and creates an interface between the input output circuits and the host computer DS1005 is suitable for the systems with fast dynamics with a high computational load Basic features e processor PowerPC 750GX 1 GHz e fully programmable from Simulink environment e fast communication between dSpace modules over PHS bus Module DS2202 HIL I O Board The DS2202 HIL I O Board module has been designed for the hardware in the loop simulations in automobile industry applications It supports mea surements and and control of various peripheral devices typical for different automobile systems Module features e 20 D A channels a 16 A D channels e 38 digital inputs and 16 digital outputs e supports 12 to 42V voltag
145. e eR eR dee 72 2 9 2 Data Approach 2 Loss Regression 72 2 9 83 Modification Smoothening of Data 75 294 Numerical Teste 2222222 eee RR 76 2 9 5 Test 1 Gluten Test Example from Harald Martens 76 2 9 6 Test 2 Wheat Test Example from Bjorn Alsberg Malivas 1800 sccsectecsehea cir kt pr ERE 77 Contents ix 2 9 7 Test 3 Our Own Example 78 2 9 8 Comparison with Normal Least Squares 79 ol BOUSS co oid gee eee Aes eta he on eed aa 80 210 1 Gradient Information eaae dis 80 2 10 2 Relationship to NCO tracking 80 211 Appendix cece Eee gee ede ae a bade RI CD die 81 Referentes DES 86 3 Measurement polynomials as controlled variables Exercises 222 0622 ue Phe b RR EA COR ERR Pe a ee 91 Johannes J schke and Sigurd Skogestad 34 Datrod cti t coetus Re eee 91 2 MPR exrcation rnaen daia iad Cena CIE dS 91 3 8 Isothermal CSTR Case Study cee eee 92 Oe E MED ELT 93 241 Component Balance ouem c tr ERR edn 93 512 Optimization Problem oce 94 3 4 3 Optimality Conditions 94 3 4 4 Eliminating Unknowns k k2 and ep 95 2445 The Determinant uec kika REID RE 95 BAG Maple Cade eccl Rad eu E Er eds 96 4 Multi Parametric Toolbox esse 101 Michal Kvasnica 4 1 Mlulti Parametrie Toolbox suele yt 101 4 1 1 Download and Installation
146. e inner loop being faster than the outer loop otherwise performance may suffer when the plant input in the inner loop is not in saturation There is usually a clear timescale separation between cascaded loops 1 1 2 4 Hanus Self conditioned Form Hanus self conditioned form Hanus et al 1987 Skogestad and Postleth waite 2005 is a quite general way of preventing windup in controllers Assume a linear controller is used with state space realization Apu T Bre 1 4 Cgu Dre 1 5 where v are the controller states e are the ordinary controller inputs and is the calculated output from the controller desired plant input The corresponding controller transfer function may be expressed as K s esse Ck sI Ak 1 Bg Dk 1 6 6 Morten Hovd and Selvanathan Sivalingam The corresponding implementation of the same controller in Hanus self conditioned form is illustrated in Fig 1 2 where K s given by sol ji s BP i 1Ck 0 BrDy Dg 0 1 7 Saturation Fig 1 2 Illustration of anti windup with the controller K s implemented in its self conditioned form K s From 1 7 we see that when the plant input u is not saturated i e when u the controller dynamics are given by 1 5 When the plant input is saturated the steady state controller output will be y Ck Ax BKD Cg u De 1 8 If Be Dy Cp gt Ax we get unxut Dre 1 9 and thus the plant input will sta
147. e is in partitioning of the engine s working range into smaller working points A net of local ARX models weighted for a particular working point is defined as Y pn OK An 9 isi Zm Pn 6 E Bu a ulk Yt Oen e O defined by polynomials Aj and Bp An g 2 1c aniq b Gh nd wl Bn g banata g 0 bh nupa go dh 5 17 where symbolics q denotes a sample delay e x q y k y k i ah and bh j ap are parameters of h th local function and dp is its delay Pa rameter nj represents the number of local models The p denotes a weighting function for a particular ARX model see Sec 5 3 3 4 and e k is a stochastic term with a white noise properties The engine working point itself is defined by engine revolutions nen and the throttle valve position tr hence 6 k nes k tr k The absolute term Cp of the equation is computed from the steady state values of the system output Ye n and the system input wep as ny nu h Veh Yen hi Ue gt bhy 5 18 The model output is computed from the eguation ys k 35s Palok R 5 19 ye n iq ys k x 10 b tan 2 u k n Introducing the estimated parameter vector 6p and the regression vector vlk equation 5 19 becomes k V pr o k 9n V 7 pn lk en 5 20 h 1 h 1 5 Implementation of MPC Techniques to Real Mechatronic Systems 209 5 3 3 3 Model Identification Parameters of the local ARX model
148. e is represented in Matlab by one of the blocks udaq28LT_v1R13 Fig 9 7 or udaq28LT_v2R13 located in the library matlabroot udaq udaq28LT blks udaq28LT lib mdl Double clicking on the udaq28LT_v1R13 block brings up the block parameters menu Fig 9 8 9 1 1 4 Measurement and Communication System The inputs and outputs of the communication interface refer to these signals Inputs Bulb 0 5 V to 0 20W of light output Fan 0 5 V to 0 6000 fan rpm LED 0 5 V to 0 100 of LED light output T D microprocessor inputs for the purpose of calculation of the first light light channel derivative sample period the microprocessor samples light channel with minimal possible value is 1 ms and coefficient of actual sample for the 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 297 Fan1 LED1 Ts_deriv1 D filtt udag28LT v2R13 Derivative of light intensity1 Fan current Fan rpm1 Fig 9 7 Communication interface block in Simulink Outputs Ambient temperature Temperature Filtered temperature Light intensity Filtered light intensity discrete filter of the first order with accuracy of 3 decimal positions sensor PT100 range 0 100 C accuracy better than 9996 1st order filter with time constant cca 20s 1st order filter with time constant cca 20s Filtered derivative of the first light intensity channel Current consumption by fan 0 50 mA Fan revolutions 0 6000 rpm
149. e shifting destroys the correspondence between the effects of measurement noise in the OP and PV time series Furthermore since the measurement noise is here of significant magnitude compared to the amplitude of the oscillations it is also in this plot not possible to identify any sharp corners in the plot The routine bicoherence m returns a negative value for NGI and a posi tive value for NLI and therefore nonlinearity has not been detected for this 1 Problems in Anti Windup and Controller Performance Monitoring 43 I L L 1 i L L L L 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time L L L L L L L L L 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Time Fig 1 45 OP and PV for data set 4 PV time series either Similarly the routine glstat m finds no clear indica tion of PV being non Gaussian the probability of false alarm being estimated to 1 0 None of these routines can therefore detect stiction in the loop 1 3 2 4 Data Set 4 The time plots of OP and PV for data set 4 are shown in Fig 1 45 Here both OP and PV look like noisy sinusoidal signals without any clear indication of stiction The cross correlation function for data set 4 is shown in Fig 1 46 This is nearly an even valued function and this test therefore does not indicate the presence of stiction The OP PV plot for data set 3 is shown in Fig 1 43 The plot looks like a smooth ellipsoid when disregarding the obvious
150. e systems Modul DS4121 Module DS4121 is an interface between the ECU or RapidPro unit and the dSpace modular system It has a support for two independent ECUs The flexibility of this solution allows to control the power stages of the engine gearbox or the valve train for a up to 12 cylinder engine 5 Implementation of MPC Techniques to Real Mechatronic Systems 205 Module features e supports the communication with 8 16 and 32 bit architecture of micro controllers e has 2 LVDS channels for fast communication e includes interfaces for MATLAB Simulink Fig 5 12 Modules DS1005 DS2202 DS4121 The RCP ensures sufficient headroom for the real time execution of com plex algorithms Arsie et al 2008 and lets all engine tasks to be controlled directly Also the customized variants of the controller can be performed immediately Typical RCP system consists of e A math modeling program prepared in Simulink Symbolic input output blocks e A real time target computer embedded computer with an analog and digital I O e A host PC with communication links to target computer e A graphical user interface GUI which enables to control the real time process The RCP system enables to use a support in the form of embedded func tions which make the preparation of algorithms easy and fast It is a great help because one can then concentrate on significant problems development and debugging of algorithms without the spending
151. ear transformation Fz to polytope P can also be achieved by multiplying the polytope by a square matrix F gt gt Q F P Needless to say all these operators can be combined together i e 4 Multi Parametric Toolbox 109 gt gt Q alpha F P t The inverse image of P under the affine map Fx g with F not necessarily non singular is given by Q x Fr geP 4 5 If P is given in its H representation then Q z A Fr 9 lt b z AF z lt b Ag 46 S A b The command to compute the inverse image in MPT is called domain Q domain P F g 4 2 3 3 Orthogonal Projection Given a polytope P x R y R A 7 b C R the orthogonal projection of P onto the z space is defined as proj P x R dy R st A lt b 4 7 In MPT the orthogonal projection of a polytope P on some of its coordinates is achieved by calling the projection method gt gt Q projection P coordinates to project on As an example take a unit box in 3D centered at the origin and project it on its 2nd and 3rd coordinate P unitbox 3 gt gt Q projection P 2 3 It should be noted that computing projections is considered a hard problem in general MPT implements several projection methods and automatically chooses the best depending on the dimension of P An example of projecting a 3D polytope onto a 2D plane is shown in Figure 4 3 4 2 3 4 Intersection The int
152. easy to change in a case of failure they don t require any maintenance they don t corrode and water sediments don t attach to them On the other hand the measurement depends on the surrounding pressure thus it is recommended to perform at least at the beginning of the measurement day the calibration of these sensors It consists of the definition of offset and sensor s gain values what should be done for each liquid s level separately The dependency is linear The basic scheme of the conversion of raw signals on liquid s levels in meters can be found in Fig 6 4 All the necessary parameters for signal adjustment lay in one array denoted as calibParams Array pumps contains the pump s entrances performance in from 0 to 100 and field valves contains the valves configuration 0 close 1 open for each valve raw signal from sensor 1 in EU loalibP arams 2 gt 3 calibP arams 1 pumpstt gt gt gt offset 1 Gain 1 pumpi Saturationg ain amp tank 1 M m um IEEE pumps gt gt raw signal from sensor 2 in EU Gain3 le pump2 Saturation3 Gain iba FP calibP arams 3 valves t __ tj offset 2 Gain2 Het Two Pumps Hydraulic UWZ valves
153. ed by utilizing a Schenck dynamometer Fig 5 11 loading the engine and so keeping it at the desired revolutions The operation of a used brake is based on an eddy current principle The braking torque is excited by the electromagnetic induction of a coil which excites eddy currents in the brake s rotor The brake supports predefined braking profiles and also manual control of the braking moment The brak ing moment is measured by a tensometric force sensor fixed to an arm Basic brake features e maximal revolutions 10 000 rpm covered by a voltage output 0 10V e maximal braking torque 750 Nm covered by a voltage output 0 10V 5 Implementation of MPC Techniques to Real Mechatronic Systems 203 The control unit of a brake is an independent device allowing the manual brake control by analog voltage signals It includes the following circuits e thyristor phase control circuit braking moment e revolutions measurement circuit e braking moment measurement circuit The thyristor phase control circuit is based on the integrated circuit IC TCA785 Function of this IC is a phase control of a brake s power stage The control voltage span is 0 10 V corresponding to the braking moment The braking moment is used for the aim of e regulation to constant revolutions based on a discrete PI controller e regulation to a particular braking moment Fig 5 11 Engine brake 5 3 2 3 Rapid Control Prototyping System Th
154. ed in the cost function and the controller will regulate the output s to the given references usually zero or provided by probStruct yref probStruct tracking 0 no tracking resulting controller is a state regu lator which drives all system states or outputs if probStruct Qy is given towards origin 1 tracking with Au formulation The controller will drive the system states or outputs if probStruct Qy is given to a given reference The optimization is performed over the difference of manipulated variables u k u k 1 which involves an extension of the state vector by nu additional states where nu is the number of system inputs 2 tracking with out Au formulation The same as probStruct tracking 1 with the excep tion that the optimization is performed over u k i e no Au formulation is used and no state vector extension is needed Note however that offset free tracking cannot be guaranteed with this setting Default setting is prob Struct tracking O probStruct yref instead of driving a state to zero it is possible to refor mulate the control problem and rather force the output to zero To ensure this task define probStruct Qy which penalizes the difference of the actual output and the given reference probStruct P_N weight on the terminal state If not specified it is assumed to be zero for quadratic cost objectives or Py Q for linear cost probStruct Nc the control horizon Specifies the number of
155. ed without lead ing to parasitic oscillations it is necessary to approximate the non modelled loop dynamics that can be characterized by equivalent dead time Ty This can be identified as the time between a control signal step and beginning of the reaction of the measured output Then it is recommended not to use shorter time constants faster poles as those given by T eTa o 1 e T4 7 2 For the given hydraulic system notice that the pump is reacting only to positive values of the input voltage So its control is only possible in one way to increase the liquid s level in tank That s why it is necessary to empty the tank before the experiment or after it 250 Vladim r ilka and Mikul Huba 7 2 1 PI controller Now let s consider situation with opened outflow valve characterized with the flow coefficient c1 In this case the one level system will be in the form 6 1 This system does no longer behave as pure integrator Let s try to control it at the beginning with P controller designed in the previous part By examining few operating point we will realize that P controller can t control this system without the permanent control error It is no surprise if we take into account that the opened outflow requires permanent inflow but the zero control error leads to zero inflow With zero control error generates the P controller zero action but considering the fact that the liquid outflows from the tank
156. eedback law allows to further analyze and post process the MPC controller For instance as will be shown in the next section lookup table styled controllers allow stability of the closed loop system to be rigorously analyzed by searching for a certain types of Lyapunov functions 4 5 1 Basic Usage MPT is a MATLAB based toolbox which allows for user friendly design sim ulation analysis and deployment of MPC based controllers Control design is divided into two intuitive steps At first the model of the plant is entered either in the LTI form 4 23 or as a PWA system 4 24 We will illustrate the procedure on the double integrator example given by the state space rep resentation x t 1 4 1 a t 9 s u t which is subject to constraints 5 lt a t 5 1 u t lt 1 The plant model can then be entered using the command line interface sysStruct A 1 1 0 1 sysStruct B 1 0 5 sysStruct C eye 2 sysStruct D zeros 2 1 sysStruct xmin 5 5 sysStruct xmax 5 5 sysStruct umin 1 sysStruct umax 1 With the prediction model at hand the user then defines parameters of the MPC optimization problem 4 26 the prediction horizon N penalty matrices Py Q R and the norm p probStruct N 3 probStruct P_N 1 0 0 1 probStruct Q 1 0 O 1 probStruct R probStruct norm 1 can be 1 2 or Inf ll m Finally the MPC controller is synthesized To solve 4 26 explicitly f
157. egrator in the controller and the controller state goes to Umar with a time constant Tz Thus the integrating state does not wind up Note also that when the controller state has reached its stationary value of Umax the 4 Morten Hovd and Selvanathan Sivalingam controller output will stay at its maximum value until the measurement y has crossed the reference value rej This anti windup scheme is straight forward and simple to implement pro vided any actuator dynamics is fast compared to the PI controller time con stant Tr 1 1 2 2 Velocity Form of PI Controllers The PI controller in 1 1 is in position form i e the controller output cor responds to the desired position value of the plant input Alternatively the controller output may give the desired change in the plant input Whereas the equations for PI controllers in position form are often ex pressed in continuous time even though the final implementation in a plant computer will be in discrete time the velocity form of the PI controller is most often expressed in discrete time Let the subscript denote the discrete time index and ej Yref Yk be the control offset at time k The discrete time equivalent of 1 1 may then be expressed as Aug Uk UR Tea K ex k 1 1 3 where T is the sample interval Here Aux represents the change in the plant input at time k If this change is sent to the actuator for the plant input instead of the desired position
158. eld of active vibration attenuation Due to the nature of the problem very short sampling times are required moreover large actuator asymmetry can render the MPC problem quite formidable If the control process is augmented with guaranteed stability and feasibility requirements the useful states are limited inside a portion of the otherwise infinitely large state space This portion of the state space is called the region of attraction If the expected deformations of under damped systems belong to a wide scale region we have to ensure a matching region of attraction A good example of such and similar vibrational systems are for example helicopter rotor beams or solar panels in a microgravity environment This chapter introduced a simple laboratory system which emulated the physical properties and dynamic behaviour of the aforementioned class of engineering 5 Implementation of MPC Techniques to Real Mechatronic Systems 199 Y Fig 5 9 MPMPC cost function J z problems Details have been given on the hardware implementation of this laboratory test bench so the if interested one may replicate the setup This has been followed by the introduction of three different MPC imple mentations traditional dual mode infinite horizon quadratic programming based MPC pre computed explicit multi parametric programming based MPC and finally a sub optimal method called Newton Raphson s MPC AII of these methods are computationally efficient in s
159. employment of neural networks while Mao et al 2009 uses for engine modeling CARIMA models In the engine control itself became popular fuzzy logic Hou 2007 neural network control Arsie et al 2008 and model predictive control MPC ap proaches Lorini et al 2006 and Muske and Jones 2006 General topics on an issue of stability and robustness in MPC can be found in Mayne et al 2000b or Zeman and Rohal Ilkiv 2003 Our approach introduced in Pol ni et al 2007 is utilizing an analytical model predictive controller with a penalization of a terminal state It uses a multi model approach using a weighted net sugeno type fuzzy logic of 5 Implementation of MPC Techniques to Real Mechatronic Systems 201 autoregressive models ARX as a system model The ARX models were identified in the particular working points of the engine as black box mod els This method of engine modeling offers an easy way of global nonlinear system model acquisition with subsequent utilization in the model based sys tem control The preliminary real time predictive control results presented in this paper indicate that the proposed controller could be suitable alternative toward the air fuel ratio control through the look up tables 5 3 2 Hardware Description The engine test bench consists of several components building up together a flexible and freely programmable system perfectly suitable for engine control and research One part of the test
160. en prior to running the command Now we move to plotting the associated feedback laws by 4 Multi Parametric Toolbox 159 close all plotu controller_explicit Again such a graphical information is valuable to control engineers because it gives us an insights into the internals of the optimal solution to a given MPC optimization problem Specifically we can see for which range of initial conditions the controller responds by a saturated control action To see this in an even more clear way run the view 2 command to watch the result from the top You will see dark blue regions which represent all initial conditions for which the optimal control action is saturated at u umin 1 dark red ones where u umax 1 and the rainbow stripe in between with 1 lt u lt 1 Exercise 4 17 Visualization is not the only way to analyze explicit MPC controllers MPT provides a wide range of analysis functions For instance one can try to compute a Piecewise Quadratic Lyapunov function as a cer tificate that the controller provides closed loop stability guarantees gt gt L mpt lyapunov controller explicit pwq When you run this command you should see the following output mpt_getPWQLyapFct Partition is not invariant and therefore it cannot be asymptotically stable The reason for this is that the controller is not invariant We have already seen what invariance means in Example 6 of the MPT Polytope Library part
161. en them but as more practical it seems to schedule the controller parameters continuously This is usually done by the input output feedback linearization In general it is supposed a non linear system in the following form i f x u y g x where the state x R the input u R the output y RP Find if possible a static state feedback u e z 258 Vladim r ilka and Mikul Huba in such a way that the system would be from point of view of new input v and output y linear For the solution to exist it has to be given that the system s relative degree r is natural number The system s relative degree ri is a natural number that defines the number of output s derivations y we have to perform in order to obtain the direct dependence on the output u In case of non existence of such number we say that the output has the infinite relative degree ays ri min k N Du 0 More information can be found for example in Conte et al 2007 For system 6 1 we may say 1 Y 5 v 7 6 1 that it has the relative degree 1 But also the system 6 2 has the relative degree 1 because y gu c2 sign z 22 z1 3 C121 ja zu c2sign z 23 z1 z3 Co 22 the basic condition is satisfied The problem of the static state feedback lin earization is solvable if r1 re o Yi es Yp u rank p where r are the relative degrees of outputs y for i 1 p
162. enotes the iterations of the Newton Raphson procedure One may take advantage of expressing the matrix Q f as the an eigenval ue eigenvector decomposition Qi RART 5 6 where R A is defined by the eigenvalue eigenvector decomposition of Q f and i is the i th power or inverse Using the decomposition 5 6 we may denote M as M Rdiag 1 1 AS RT 5 7 where diag denotes a diagonalization operation 9 is a vector of eigen values gained from A and denotes the piecewise division operation This substitutes the inversion of the full matrix expression I Q f in the on line algorithm This expression occurs not only in evaluating the perturbation vector f but also in the function A and its first derivative Let my 1 AS and m diag 1 m then B A cf Wim Wor Wa 5 8 d X a 2z1 Wam Wore 5 9 5 Implementation of MPC Techniques to Real Mechatronic Systems 189 The equation yielding the perturbation vector f will successively transform to f Atm Wiz 5 10 Matrices W1 W2 Ws can be calculated offline therefore saving some time avoiding unnecessary multiplications at every NR iteration and sample time Matrix Wy can be calculated before the NR process for the actual sample time initiates Wi Qas R Ws WF Ws WA W x1 W3 W2 z1Q r 1 5 2 4 8 Real time Code Implementation to C The on line part of the NRMPC controller is simple The algorithm does no
163. ents it was necessary to wisely propose a PRBS so that the air fuel mixture is always ignitable The local ARX models can be subsequently determined from the measured values of instantaneous A k and A k belonging to the air path and fuel path utilizing relative air and fuel mass densities as Malk Ma e Aa k 5 21 mf myl DERE 5 22 The final formula describing the aif fuel ratio dynamics is built up of local linear ARX models of the air and fuel paths is in the form ji As k ty VZ 5 SRA Pan K b ant Dp Ay Pan 009 Ean 5 23 VF k ney Pan lOl OF n Der Pg n 0 R 6g A Where y is the regression vector of system inputs and outputs na isthe amount of working points p is the interpolation function 9 is the vector of a working point 210 Gergely Takacs et al 0 is the vector of ARX parameters c is the absolute term of an ARX model In accordance with the general model structure presented the key variables used for the air and fuel path are defined in the Table 5 1 and coupled with the general variables used in the presented formulas Table 5 1 Symbol connection between general expression and the model general air path fuel path operating symbol model model point y k ma k mj k u k tr k uj k Vk Malk 14 k On Oah O p n pn 6 pan 9 5 py 06 Ch Ca h Cf h sO nd The schematics of the engine identification described above in this Section is depicted in Fig 5 19 The
164. ep 4 k k 1 Step 5 until V Z V 1 Step 6 4 V The only non trivial stage is Step 3 Remember the forward reach sets of Example 4 Take a look again at 4 16 and you should spot some similarities In fact the set computed in Step 3 is the one step backwards reachable set i e the subset of Uz which remains in Y for one time step Since forward reach sets were computed by Xx41 AX we can compute the backward set by X44 A 1 X5 intersected with Xj itself i e PSI k 1 PSI k amp inv A B K PSI k The rest of the solution should be clear You will however need to replace repeat until by a while end statement and negate the stopping condition accordingly Now it s your turn compute the invariant subset of PSI and plot it Note that many theoretical subtleties were untold in this story One of them being that the presented algorithm has no guarantee of finite time conver gence there you should place an upper limit on the number of iterations 10 is a safe choice 4 4 Solutions Solution 4 1 for Exercise 4 1 1 Create a polytope P as the 2D unit box centered at the origin i e P z1 z2 1 lt 1 1 1 x2 X L1 4 Multi Parametric Toolbox 123 convert all inequalities into the form al a2 x1 x2 lt b x1 lt 1 i e 1 0 x1 x2 lt 1 x d lt 1 i e 1 O x1 x2 lt 1 y xo lt f i e 0 1 x1 x2 lt 1 Kk x_2 lt 1 i e 0 1 x1 x2 lt 1 now st
165. er using inverse or parallel plant model Since the hydraulic plant 6 1 252 Vladim r ilka and Mikul Huba is stable inversion of the plant dynamics may be directly derived also for the nonlinear model as U M 01 Ciy Y 7 3 The input disturbance v may then be reconstructed by comparing the recon structed plant input 41 and the controller output according to j Uu 7 4 n 175 v and then compensated at the P controller output up Such a compensation may be treated as a special case of nonlinear disturbance observer in Fig 7 2 Besides of the P controller gain the new tuning parameter Ty may be used to increase system robustness against uncertainties and non modelled dynamics and to filter measurement noise Disturbance Observer Fig 7 2 Rejection of the input disturbances v by correction of the P controller output in the nonlinear PI controller for system dy dt g y u v f y You may test your integral controller by filling to different levels also the second tank and then opening or closing the inteconnecting valve among the tanks The controller structure in Simulink is depicted in Fig 7 4 Control pro gram which work with results of automatic calibration and indentification is listed below 7 2 2 1 Programs for Matlab Simulink cle clear all close all 7 Constrained PID Control Tasks for Coupled Tanks Control Pl1 controller system outpu
166. erating i e with the selector Comment on the results of the simulation 4 Simulation with anti windup Implement the PI controllers with anti windup and redo the simulation Comment on the results and compare to the simulation results without anti windup 1 2 4 Solution 1 2 4 1 Model Implementation The Simulink implementation of the plant model with the disturbance and including PI controllers without anti windup and selector is shown in Fig 1 26 2 n addition the variables are scaled so not too much emphasis should be placed on the magnitude of the variables 1 Problems in Anti Windup and Controller Performance Monitoring 31 1 2 4 2 Controller Tuning Not much effort has been put into the controller tuning For each of the controllers the integral time T has been set equal to the dominant time constant of the open loop and a proportional gain of 1 is used These tuning parameters are found to result in reasonable responses to setpoint changes for each individual loop 1 2 4 3 Simulation without Anti windup The system has been simulated with the disturbance active The response in the controlled outputs are shown in Fig 1 27 We observe large variations in the controlled variables including variations where the measurement is larger than the setpoint 0 for both loops The corresponding controller outputs are shown in Fig 1 28 We observe that the pressure controller winds up so severely that it needs a long t
167. ersection of two polytopes P x R Aja by and Pa x R Aga lt ba of the same dimension is the polytope Q el Alr lt 3 4 8 This operation is implemented in MPT using the overloaded amp operator 110 Michal Kvasnica Fig 4 3 Illustration of orthogonal projection gt gt Q P1 amp P2 Note that MPT only computes full dimensional intersections Full dimen sionality of the result Q can be checked by isfulldim Q An example of the intersection operation is shown in Figure 4 4 Fig 4 4 Full dimensional intersection of two polytopes 4 2 3 5 Convex Hull The convex hull of two or more polytopes P i 1 m is the smallest convex set which contains them i e convh vert P1 vert Pm 4 9 4 Multi Parametric Toolbox 111 where vert P denotes the extremal vertices of the i th polytope Given an array of polytopes their convex hull can be computed as follows gt gt Q hull P1 P2 P3 P4 An example of the convex hull operation is depicted in Figure 4 5 i i i i i i i 25 i i i 2 1 o 1 2 4 5 D 1 2 4 b Convex hull of the polytopes yel low a 3 random 2D polytopes Fig 4 5 Illustration of convex hull 4 2 3 6 Minkowski Addition The Minkowski addition also called set addition between two polytopes P and P is defined as Pi P2 zi cz x E Pi T2 Pa 4 10 and is implemented in MPT using the overl
168. es on an experimental laboratory model This device is basically a simple lightly damped mechanical structure The algorithms con sidered here include the traditional infinite horizon dual mode quadratic pro gramming based MPC QPMPC pre computed optimal multi parametric MPC MPMPC and the efficient albeit sub optimal Newton Raphson s MPC NRMPC The control of automotive internal combustion engines is one of most com plex control problems due to nonlinearities and variable time delays This is reflected in the time variability of the model parameters in different regions 5 Implementation of MPC Techniques to Real Mechatronic Systems 173 of the engine operating space The section 5 3 analyzes the practical aspects of real time implementation of a multi model MPC technique to cope with the problem of nonlinearities and variable time delays in controlling of the SI engine air fuel ratio In both sections the main practical software and hardware aspects of the suggested real time solutions are discussed in details 5 2 Implementation of Various MPC Methods for the Vibration Control of Lightly Damped Cantilevers 5 2 1 Introduction The algorithmic support of active vibration suppression applications is usu ally limited to simple positive position feedback PPF or for example strain rate feedback SRF control Song et al 2002 Other essential strategies include PID or linear quadratic LQ control Preumont 2002 Because of the si
169. espondingly termed anti reset windup or simply anti windup 1 Problems in Anti Windup and Controller Performance Monitoring 3 Anti windup techniques remain an active research area and no attempt is made here to give a comprehensive review of this research field The aim is rather to present some important and useful techniques that should be known to practising control engineers 1 1 2 1 Simple PI Control Anti windup A simple PI controller with limited actuation range for the control inputs i e controller outputs may be implemented as illustrated in Fig 1 1 Here the actual input implemented on the plant is feed back to the controller through the low pass filter 1 rrs 1 If the actual plant input is not measured it suffices to know the range of actuation for the input The actual input can then easily be calculated Saturation Umax Yref Y u Kp Umin ct i Tis 1 Fig 1 1 Simple anti windup scheme for a PI controller From Fig 1 1 it is easy to see that when the plant input is not saturated when u we get TIS H 1 TIS u Ky Yrej v 1 1 That is we get the normal behaviour of a PI controller On the other hand consider the case when the input is in saturation at its upper limit Umar 1 maxr 1 2 TIS 1 2 K yref y The internal feedback path in the controller is now broken there is no open int
170. f z for any allowed c lt 1 is equal to c Ji G W and we want this as small as possible From singular value properties we have that the A I 1 o A that is we want to maximize smin Wz GJ 1 We have then derived the maximum gain rule Under the assumption that c X 1 the worst case loss is given by Lmax SoG where G 1G8 2 33 and S W diag 1 c Copt il S2 J i Note that S includes the sum of the optimal variation as given by the F matrix and the expected measurement error Thus to minimize the loss we should select c Gu with a large minimum singular value of the scaled gain matrix Gs The only non exact step in deriving this rule comes from the assumption that all c lt 1 are allowed which means that we neglect some of the variations in c Copt that are correlated Nevertheless since the presence of measurement noise means that there is always some uncorrelated variation at least if we consider individual measurements c y this implies that we can safely exclude candidate c s with a small gain that is with a small value of e G 70 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Note that Juu enters into the maximum gain rule whereas it is actually not required when H is the optimal full matrix see 2 26 However in general Juu must be included see case 3 in the discussion following 2 26 e Do we need the maximum gain rule
171. f Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail matus kopacka stuba sk Michal Kvasnica xiii xiv List of Contributors Institute of Information Engineering Automation and Mathematics Faculty of Chemical and Food Technology Slovak University of Technology in Bratislava e mail michal kvasnicaQstuba sk Tomas Pol ni Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail tomas poloni stuba sk Boris Rohal Ikiv Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail boris rohal ilkiv stuba sk Peter imon i Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail peter simoncic stuba sk Selvanathan Sivalingam Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim Norway e mail selvanathan sivalingam itk ntnu no Sigurd Skogestad Department of Chemical Engineering Norwegian University of Science and Technology Trondheim Norway e mail skogeQ chemeng ntnu no Gergely Tak cs Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail gergely t
172. f a full laboratory setup hardware implementation details and practical issues regarding the use of MPC for the real time control of air fuel ratio in automotive engines 5 1 Introduction The merits of well designed control algorithms have been long recognized by the industry where a performance improvement of a fraction point leads to significantly increased revenue or decreased production cost Performance is not the only criterion to judge the qualities of a control strategy Every real actuator and process has inherent physical limits which are to be constrained in order to preserve safety further influence production output or to safeguard equipment As claimed by Rossiter 2003 the only control method capable of handling process constraints on an algorithmic level is model predictive control MPC MPC has been used for decades in applications where sampling rates are in the order of several minutes or even hours for example in the petrochemical industry Now the situation turns to applications with very short sampling periods in orders of milliseconds mainly in mechatronic systems This chapter contributes to experimental implementation and real time verification of chosen efficient MPC techniques applied for control of fast mechatronic systems namely laboratory vibration system subsection 5 2 and internal combustion engine subsection 5 3 The section 5 2 discusses the implementation properties of three differ ent MPC approach
173. for finding the optimal H it would be required for finding a numerical value for the loss The analytical solutions are useful in particular for their insights they yield but for practical calculations it is usually faster and more robust to compute the optimal H by solving the convex quadratic optimization prob lems In addition the convex optimization problems do not need the require ment that YY is non singular Based on the insight in Corollary 2 1 the quadratic optimization in Theorem 2 1 Alstad et al 2009 can be further reformulated to a more general form Yelchuru and Skogestad 2010 Theorem 2 4 Generalized convex formulation An optimal H for the problem in 2 23 is min HY H 2 26 st HG Q where Q is any non singular ne x Ne matrix for example I but Q must be fixed while minimizing HF p Proof The result follows from Corollary 2 1 but can more generally be de rived as follows The problem in 2 23 is to minimize J1 HGY HY S P X F 2 Measurements for Control Optimization and Estimation 65 The reason why we can omit the ne x n matrix X is that if H is an opti mal solution then so is H DH where D is any nonsingular n x n see proof of Theorem 2 1 However note that the matrix X or equivalently the matrix Q must be fixed during the optimization so it needs to be added as a constraint O The fact that Q can be chosen freely Theorem 2 4 can be useful fo
174. formance of previous controllers The PI controller structure and Simulink model are in Figs 9 45 9 46 PI may also be used for controlling unstable plant To cover all stable and unstable plants by one transfer function it is necessary to use the pole zero form instead of the time constant one and to express the plant as K G s e Te 9 23 sca For robust controller tuning the performance portrait method could again be used For simple nominal tuning use the following rules based on the notion of the so called equivalent poles a of the proportional controller and ae r of 320 Peter Tap k and Mikul Huba System output Comparison of the delayed system output 60 FPI T T 0 8 aeui e FPI T T 70 6 FPI T T 70 06 Fl controller OST E setpoint Light intensity wo o 20 0 L L 2 1 i J 0 200 400 600 800 1000 1200 1400 time s Fig 9 38 Experimental results delayed system output the controller with disturbance compensation I action ae 1 aTy 4T4 9 24 Qer Ae 1 3 9 25 P aer a Ks 9 26 Ty eTa 9 27 Plant parameters from FOPTD approximation 9 18 can be used whereby K K T and a 1 T The experimental results for the plant used in this chapter are in Figs 9 47 9 48 9 49 9 49 Questions e Which process gain used for controller tuning gave better control quality What was the most obvious difference in control
175. free MATLAB toolbox for design analysis and deployment of MPC based con trol laws for constrained linear nonlinear and hybrid systems Efficiency of the code is guaranteed by the extensive library of algorithms from the field of computational geometry and multi parametric optimization The toolbox offers a broad spectrum of algorithms compiled in a user friendly and ac cessible format starting from modeling systems which combine continuous dynamics with discrete logic hybrid systems through design of control laws based on different performance objectives linear quadratic minimum time to the handling of systems with persistent additive and polytopic uncertain ties Users can add custom constraints such as polytopic contraction or collision avoidance constraints or create custom objective functions Result ing optimal control laws can either be embedded into control applications in the form of a C code or deployed to target platforms using Real Time Workshop MPT can also be viewed as a unifying repository of hybrid systems de sign tools from international experts utilizing state of the art optimization packages The list of included software packages includes packages for linear programming CDD GLPK quadratic programming CLP mixed integer linear programming GLPK and semi definite programming SeDuMi In addition MPT ships with a dedicated solver for computing projections of convex polytopes called ESP a boolean optimizati
176. free control moves in the optimization problem probStruct Tset a polytope object describing the terminal set If not pro vided and probStruct norm 2 the invariant LQR set around the origin will be computed automatically to guarantee stability properties Since MPT 2 6 it is possible to denote certain constraints as soft This means that the respective constraint can be violated but such a violation is penalized To soften certain constraints it is necessary to define the penalty on violation of such constraints 4 Multi Parametric Toolbox 155 e probStruct Sx if given as a nx x nx matrix all state constraints will be treated as soft constraints and violation will be penalized by the value of this field e probStruct Su if given as a nu x nu matrix all input constraints will be treated as soft constraints and violation will be penalized by the value of this field e probStruct Sy if given as a ny x ny matrix all output constraints will be treated as soft constraints and violation will be penalized by the value of this field In addition one can also specify the maximum value by which a given constraint can be exceeded e probStruct sxmax must be given as a nx x 1 vector where each element defines the maximum admissible violation of each state constraints e probStruct sumax must be given as a nu x 1 vector where each element defines the maximum admissible violation of each input constraints e probStr
177. gh the operational range in other words the I O characteristics of the non filtered light channel is not linear The exnum command can be used to measure I O characteristics of the plant and to obtain the process parameters in multiple working points Choose the experiment no 2 for I O characteristics measurement The following fig ures will give you the information on the I O characteristic the process gain and dead time through the operational range Short delay in a light intensity change can be observed after a bulb voltage step The uDAQ28 LT device converts bulb voltage steps into a steep ramp to prevent undesired distur bances in the plant Let us approximate this delay as a dead time After running the I O characteristics measurement plot the step responses from 302 Peter Tap k and Mikul Huba which the I O characteristics of the light channel was obtained by the fol lowing commands stairs yl 1 yl 2 xlabel t s ylabel light intensity title Step responses of non filtered light channel Input output characteristic 70 T T T T 60 1 30r 1 20 J 10 J Fig 9 13 I O characteristics measurement results 9 2 2 Io Controller Tasks e Add a disturbance observer to the feedforward control to compensate steady state error e Use the I O characteristics measurement results to tune the controller by hand Tune the controller using the performance portrait method Make a step d
178. grators must represent the range of movement available for the integrating states i e with the contribution from the actual state feedback removed Controller C tice 7 Reference i Integrator s I K I z w anti windup re Actual Measure i y input ments i ro Pa Plant gt l H Kstate l i Ka m Y State gt estimator Fig 1 5 Implementation of anti windup for state estimator and static state feedback augmented with integral action 1 Problems in Anti Windup and Controller Performance Monitoring 9 Remark Note that if Hanus self conditioned form is used for the anti windup this requires a non singular D matrix resulting in a PI block instead of a purely integrating block The size of this D matrix may affect controller performance depending on how and whether it is accounted for in the state feedback control design 1 1 3 Decoupling and Input Constraints Decouplers are particularly prone to performance problems due to input con straints This is not easily handled by standard anti windup because much of the input usage can be related to counteracting interactions Therefore if an output is saturated but other outputs are adjusted to counteract the effects of the unsaturated output severe performance problems may be expected One way of ensuring that the decoupler only tries to
179. gt Gain3 u P a gt Thermo optical Step Saturation3 Eve VO Interface Fan Gain1 1 o p ni rec meer TransferFent s LED Light j intensity1 3 IMs E t Til s 1 Fig 8 11 Supplied block diagram a The block diagram depicted in the Fig 8 11 can be found in the CA RemLab Workspace folder after successful installation The core of this block diagram 8 Remote Laboratory Software Module for Thermo Optical Plant 285 is created by the Thermo optical Plant I O Interface block that represents the physical model of the uDAQ28 LT thermo optical plant The blocks drawn by the orange color are important for the correct operation of the remote laboratory software module These are input output blocks of this block di agram and the client application is programmed in that way that it could exchange the data with these blocks The following three rules have to be kept where the first two are related to input parameters and the third one to input output parameters of the block diagram 1 The sampling period of an individual block as well as the whole block diagram is given by the global variable denoted as T s This denotation must be kept everywhere If a variable with this name is used in the initial parameter file m file or somewhere in the block diagram mdl file its
180. hat by designing the controller we proceed from the system 7 5 which works with deviations from steady state not with the absolute varaibles That s why the controller is using them as well i e it holds u where AU1 s and AE1 s are Laplace transforms of the deviations of the controller output Au ui u1 and of the control error Ae e1 1 from steady states u1 a e1 Of course for control error in steady state is 1 0 In this case Ae ei but at the controller output one still has to consider deviation signal Au The system 6 1 which we are in fact controlling has as its input directly u So we need to arrange the controller output by adding the stabilized steady state value u1 and thus obtaining directly u1 uy Au Ulo The value u1 represents the stabilized value of the pump s voltage at which the liquid s level or liquid s volume in the tank has the stabilized value x1 The numerical value u1 is off course needed at the implementation of the controller We ll define it based on system s 6 1 In a steady state all time derivatives are zero thus 34 0 After substituting into 6 1 we ll obtain U1 defined as Ulo A161 X15 7 4 Exact Feedback Linearization In this task we would like to control the liquid s level for different operating points By appropriate gain scheduling it is possible to have various operating points and the associated controllers and then to switch betwe
181. he cable terminal and finally the measuring card The LASER system settings are controlled via the host computer using a proprietary settings software via USB pro tocol The control and measuring chain is demonstrated by the simplified schematics in Fig 5 1 Note that the capacitive sensor is not utilized as a feedback signal in these examples a direct readout using the laser triangulation system is considered 176 Gergely Takacs et al a Laboratory test bench b Piezoelectric actuators Fig 5 2 A photo of the laboratory test bench with the piezoelectric actuators bonded on the beam A photo of the laboratory test bench with the piezoelectric actuators bonded on the beam is visible In Fig 5 2 5 2 2 2 System Identification To predict future system behaviour a second order linear time invariant state space mathematical model of the vibrating system is considered To obtain this model Lara et al 2003 used a direct phenomenological model of the system or for example it is possible to utilize the results of finite element modeling FEM harmonic analysis to acquire the state space representation as well Dong et al 2006 An experimental identification procedure will be utilized to obtain a dis crete linear time invariant state space system in the form Tk41 Ary Buk Yk Cay Duk 5 1 where yx represents the beam tip deflection in millimeters and uz is the high level voltage signal directly fed to the piezoele
182. here implemented in two different ways both as a standard decoupler W s G s G s where G s is the diagonal matrix corresponding to the diagonal elements of G s and using the inverted decou pler structure shown in Fig 1 6 Note that G s here represents the transfer function matrix from manipulated variables only to the measurements i e C4 Int X Ax Bu Saturation 6 y Cx Du Manual Switch Out1 State Space Disturbance Y Measurement noise Fig 1 7 Implementation of distillation column model in Simulink 1 Problems in Anti Windup and Controller Performance Monitoring 13 without the disturbance inputs Note also that in order to invert G s a full rank D matrix is necessary Small diagonal elements are therefore added to the D matrix and it will be necessary to check that these do not intro duce right half plane zeros which would lead to unstable poles after system inversion The Matlab Control Systems Toolbox allows using the command inv G to find the inverse of G s Alternatively a state space formula for the inverse can be found in Skogestad and Postlethwaite 2005 The standard decoupler is simulated by simply putting the PI controllers and the decoupler W s represented by a state space model block in series The inverted decoupler with PI controllers is shown in Fig 1 9 Note that anti windup of the PI controllers is not inc
183. hhhhhhhhhhhhhhhhhhhh Dynamic decoupling controller Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Gdiag G 1 1 0 0 G 2 2 4 We need a non singular D matrix to calculate inv G Find an approximate G using a new D matrix that does not introduce poles in the right half plane dd 1e 3 0 0 1e 3 Gapp ss A B C dd tzero Gapp Check for zeros W inv Gapp Gdiag W minreal W Wa Wb Wc Wd ssdata W Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Anti windup for dynamic decoupling Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Note that all elements share the same denominator glinum giiden tfdata sys 1 1 gilden giiden 1 giinum giinum 1 g12num gi2den tfdata sys 1 2 gi2den gi2den 1 gi2num gi2numi1 g21num g21den tfdata sys 2 1 g2iden g2iden 1 g21num g2inum 1 g22num g22den tfdata sys 2 2 g22den g22den 1 g22num g22num 1 Using Hanus self conditioned form for anti windup of PI controllers k2a k2b k2c k2d ssdata k2 kla kib kic kid ssdata k1 klaw kia kib inv kid kic kibw zeros size kib kib inv k1id uz size kib inv kid kicw klic kidw kid zeros uz 46 AT 48 49 50 51 10 11 12 13 14 15 16 IT 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 26 Morten Hovd and Selvanathan Sivalingam k2
184. i 1 rx Amag mag norm Xopt i the magnitudes are about 0 8 end noise noisemag diag mag 4Xoptnoise Xopt noise step 2 with artificial noise semilogy svd X semilogy svd X X0 Hopt loss soc_avg Gy Xoptnoise D Hopt Gy inv G1 Hopt9 inv D Hopt resb Hopt9 X resbO Hopt9 X0 a91 norm resb Y1 a92 norm resb0 Y10 method9 loss w noise Summary of results 207 208 86 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke methods methodi method5 method6 method7 method4 method10 method9 a all a12 a51 a52 a61 a62 a71 a72 a41 a42 a101 a102 a91 a92 For repeated case 1 and case 3 na na 1 aa aata aaa aa na For repeated case 1 end Acknowledgements The authors are pleased to acknowledge the financial support from the NIL I 007 d project and the Norwegian Research Council References Alstad V Skogestad S 2007 Null space method for selecting optimal measurement combinations as controlled variables Ind Eng Chem Res 46 846 853 Alstad V Skogestad S Hori E 2009 Optimal measurement combinations as con trolled variables Journal of Process Control 19 1 138 148 Halvorsen IJ Skogestad S Morud JC Alstad V 2003 Optimal selection of controlled variables Ind Eng Chem Res 42 Hori ES Skogestad S Alstad V 2005 Perfect steady state indirect control Industrial amp Engineering Chemistry Research 44 4 863 867 Kariwala V Cao Y
185. iable z as a Function of Disturbances and Noise We start from the loss expression in 2 8 with izlli where z J U Uopr We want to write z as a function of d and n The first step is to write U Uopr as a function of c Copt We have c Hy so c Hy HG u HG d Cop Hy HG uop HG d Thus c Copp HG u opt or u m Uopt HG c Copt where G HG is the transfer function from u to c The next step is to express c Copt as a function of d and n From 2 4 we have that H y n cs constant or c Hy Hn c Here c 0 since we assume the nominal point is optimal From 2 12 we have that Cop H Fd Since the signs for n and d do not matter for the expressions we derive below we can have both positive and negative changes we derive C Copt H Fd n H FW d W nun HHFWa Wry Fl Note that W4 and W 4 are usually diagonal matrices representing the mag nitude of the disturbances and measurement noises respectively 2 4 2 Loss for Given H Disturbance and Noise Analysis In summary we have derived that for the given normalized disturbances d and for the given normalized measurement noises n the loss is given by 60 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Le pa 2 14 where i d z JV u top Ji HG HY sv 2 15 M H Y FW4 Wn 2 16 2 4 3 Worst case and Average Loss for Given H Analysis
186. ics using low cost hardware To push into this direction the MPT toolbox can automatically generate real time executable C code version of Algorithm 1 The export can be initiated by calling gt gt mpt_exportC ctrl target_filename The function generates the files target filename cand target filename h which will contain respectively the table evaluation code and the table data the regions R and the feedback gains K and L cf 4 27 The code can be subsequently linked to any application written in the C language Another option is to use the Real Time Workshop RTW which is a de facto standard code generation tool supporting different types of digital signal processors and other CPUs MPT provides a RTW compatible C code implementation of the Algorithm 1 which makes code generation and de ployment a single click operation To use this option the controller is first embedded into the Simulink environment using a provided block as shown in Section 4 5 2 The block is subsequently connected to signals from A D and D A converters to close the control loop Then one simply clicks on the respective icon in the Simulink window to initiate RTW which compiles the overall control scheme including the lookup table and then automatically downloads it to any supported target CPU 4 5 4 Advanced MPC using MPT and YALMIP In the previous section we have shown how to formulate solve analyze and deploy MPC controllers using the MPT to
187. ime to wind down again and therefore only is selected active for a short period during each disturbance cycle This explains the long periods where the pressure measurement is above the setpoint There are also large deviations in the flow measurement but since the flow controller is selected most of the time the deviations in the flow measurement are quite quickly removed Kf Tif 1 5 Tits Transfer Font Flow setpoint deviation variable 0 f Flow gt uf deviation variable 1 Flow controller output 20s 1 5 Flow disturbance 10s 1 Flow M oo00 min gt ur 00 MinMax Input applied Disturbance deviation variable 10 d gt 100s 1 Pressure Ce 2 le Save disturbance 20s 1 Transfer Fen gt up P 4 Pressure deviation variable kp Tip 1I la n Tip s Pressure controller Pressure setpoint deviation variable Pressure controller output Fig 1 26 Simulink implementation of model and control system without anti windup 32 Morten Hovd and Selvanathan Sivalingam J M ab i li tl ood AN L L L L fi f f 0 200 400 600 800 1000 1200 1400 1600 1800 2000 I Flow Pressure Disturbance jj Fig 1 27 Responses in controlled v
188. in It was designed according to ilka 2007 a Front view b Back view Fig 6 1 Coupled tanks uDAQ28 3H2 Front and back view 6 Laboratory Model of Coupled Tanks 231 The system is easily connectable to the PC via USB interface There is no need of additional data acquisition cards hence the controlling is per formed directly from the Matlab SIMULINK or Scilab Scicos environment By switching the valves between the tanks and between the tanks and the lower reservoir it enables to create different configurations e g the one level system two levels system with one inflow and one drain or the multi input multi output MIMO system with several inflows and outflows and the mu tual tanks coupling At the beginning we will start with the easiest model we will imagine only one tank where the level of liquid is influenced only by inflow from the pump and drain from the valve This valve is only two positions open close This kind of connection s model can be seen in Fig 6 2 12 a One tank system b Coupled Tanks Fig 6 2 Models of tanks The liquid s level is denoted as x with the liquid s inflow u1 tank s cross sections A1 and the valve s outflow coefficient c This tank model can be described by various differential equations acquired through the application of the law of substance preservation Thus it applies for the given tank that the change in volume V is defined by the difference betwee
189. inal liquid s levels after finishing the experiment and 15 as the duration of the measured interval It is important to mention that in case with the tank s outflow orifice not being exactly at the same altitude as the valve s drain orifice but lower we have to consider this difference Then it is to consider that the real liquid s level in the tank producing the resulting pressure at the valve orifice is T1 Litank T1oflset where zitank represents the hight of the water collumn in the first tank and offset denotes the offset between tank orifice and valve outflow The pump identification seems to be easier It may be based on determining time periods in which the tank with closed valves will be completely filled under consideration of different pump s input voltage A121 At q It is vital to choose various working points in whole pump s working scale Finally since the pump is a reasonably nonlinear element we approximate the measured pump characteristic by the curve fitting methods e g with the polynomials of third or fourth degree The input output characteristic of the first pump the inverse input output characteristic of the pump and the characteristic of the first valve are depicted in Fig 6 6 6 Laboratory Model of Coupled Tanks 235 6 2 2 Sensors Calibration To measure the height of liquid s level in the plant in Fig 6 1 the pressure sensors are used These have several advantages they are
190. individual responses When the rules for creation of block diagrams Sec tion 8 7 will be kept the blue curve will represent the input signal the red curve will be the control value and the green one is for the output value of the real system The two lower quadrants can be characterized as alphanumerical ones be cause they serve for input and display of textual and numerical data of ex 280 Pavol Bist k periments Beside this there is a group of buttons placed in the bottom part of the interface that enables to control the whole client application The lower left sector Settings assures the setting of the following experi ment parameters the name of the controlled model model parameters input desired value sampling period duration of experiment and the name of a data file to which the state variables of the running experiment can be saved The lower right sector Output displays the experiment outputs in the nu merical form If the user does not want to exploit the possibility to save data into the data file it is possible to copy data from the field Output through the clipboard to arbitrary program for further processing Besides this the field Output displays also the application state messages whether the connection has been successful if no then why the state of saving data to the external file etc Further it shows warnings and error messages in the case they are generated by the system The buttons for the control of run
191. ine NRMPC run has been expressed as Jyrupo fT f which can be only true in the case the augmented states are transformed to make the cost equivalent with the original MPC formulation Mx Mc dlyap PhiO Q K R K dlyap A0 CO R B Mx B CO In order to minimize an equivalent transformed cost and still having the same simple function the augmented states z have to be transformed in the on line optimization task V D eig Mc d sqrt max diag D tolerance invT V diag 1 d V invTT blkdiag eye w dim 1 1 invT Select parameters are passed onto the on line formulation while some mi nor practical operations are performed in the final code segment Pt invTT Qzi invTT 188 Gergely Takacs et al R S eig Pt dim 1 2 dim dim 1 2 dim Sm diag S 021 Pt dim 1 2 dim 1 dim 5 2 4 7 The Newton Raphson Root Search Algorithm The first problem in the on line formulation is to find A For this one needs to use the Newton Raphson procedure The underlying concept is very simple Anstee 2006 and in relation to the NRMPC procedure it can be stated as d X 0 D X Ec ke 95 The procedure itself is also trivial and is represented by the following algo rithm At each sampling instant initialize with A 0 and perform an iteration which calculates P X dB A aX Anti n 5 5 lt until the change in A is smaller than the pre determined tolerance where subscript n d
192. inearization we talk about approximation of the non linear system 256 Vladim r ilka and Mikul Huba i f zi ui yi g zi by a linear system Ax AA B u Ay C Az where Ax T1 zi Au1 U1 U1 Ay Y1 yi and T19 Ulo yi iS our fixed operating point It is to stress that the linearization is considered around a steady state state input and output are replaced by deviations from it as Ax Au and Ay and the matrices A B and C are in general given as of of Og 6 1 2719 U1 U19 In case on one level hydraulic system 6 1 we get by such a linearization s E Aj 2 1o AY Az We would like to point that 7 5 is not valid for 71 0 Taking into account that our system 7 5 is linear we can consider its velocity transfer function mo 250 A 1 AU s ost ONT Now for this system we propose PI controller for the first order linear sys tem Transfer function of the controller should again be made in such a way to yield an open transfer function 1 T1s what may e g be achieved by choosing R 1 1 s 2 A1 Alc s YY o Tis Fi s STs T 2T T1 5 where A p l Ti A f 101 2T Tio are gains of P a I1 action of the controller Note that the time constant T can be easily obtained from step response of the system We should not 7 Constrained PID Control Tasks for Coupled Tanks Control 257 omit the fact t
193. ing state space realization 11 1 Teti o 4 Te g 5 Ue Yk 1 0 Tk which is subject to constraints 1 lt uj lt 1 and 5 sas First step for you is to create the corresponding sysStruct structure for such a system make sure to run clear to wipe the workspace When you have that create the probStruct structure assuming the following assign ment 158 Michal Kvasnica e prediction horizon 5 e quadratic type of the performance objective e Q I and R 1 Finally create two controller objects one called controller online and representing an on line MPC controller and the other one controller explicit where the solution will be pre calculated Hint use either online or explicit string flags as the third input to mpt control Once you have both controllers verify that they are indeed equivalent Which control action does controller online give for x0 3 0 And what is the value of controller_explicit for the same state Of course the two values should be identical since the two controllers represent the same optimization problem The only difference is in the evaluation speed To see it in a more prominent fashion fire up closed loop simulations for both controllers using the sim command and with xO 3 0 as the initial state In both cases measure the execution time using tic and toc functions What do you see keep in mind that this will measure the execution time of the whole cl
194. ing 15 1 1 7 Simulations 1 1 7 1 PI Controller The responses in the controlled outputs when no saturation is present are shown in Fig 1 12 We observe that the setpoint changes are followed quickly However there is some interaction between the two loops causing offset in loop 2 following setpoint changes in loop 1 These interactions are counter acted relatively quickly by the controllers The corresponding responses when the input is constrained between 0 5 are shown in Fig 1 13 It is obvious that the saturation severely degrades control performance 1 1 7 2 Decoupler The responses in the controlled outputs when no saturation is present are shown in Fig 1 14 The observed performance is good and could probably have been further improved by more careful design of the PI controllers and the decoupling is practically perfect with no observable interaction between the two outputs Note that these results do not depend on whether a standard or an inverted decoupler implementation is used The response with the standard decoupler when the input is constrained between 0 5 are shown in Fig 1 15 Clearly performance is horrible and the interactions between the outputs extreme Note however that the system in Output 1 1 5 T L L L L L L L L 50 100 150 200 250 300 350 400 450 500 Output 2 2 T L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 12 Responses with PI controlle
195. ing parameter changes e y all available measurements will later call a subset of these for in ac cordance with statistics notation The manipulated variables MVs often the same as the inputs u are generally included in the measurement set y This will allow for example for simple control policies where the inputs are kept constant Of course the set y also includes measured disturbances dm a subset of d e n measurement noise error for y Ym y n e p prices weights that enter into cost function do not affect y 2 2 2 Cost Function The objective is to choose the input u to minimize the quadratic cost function J u d u Q u 4 d Q d amp u Q4d 2 1 Note that for simplicity we have not included linear terms in the cost func tion Any linear term in u can be removed by shifting the zero point for u to be at the optimal point On the other hand a linear term in d cannot be counteracted by choosing the input u so excluding it does not change the solution The same applies to any constant term in the cost If we compare 2 1 with a second order Taylor series expansion of the cost around a nominal point u d then we have that 1 1 O 37 Q Jia Qs 5 ad 2 Measurements for Control Optimization and Estimation 55 and u represents deviation from the optimal point u d 0 0 at which J 0 2 2 8 Measurement Model A linear model is assumed for the effect on u and d on measurements
196. ion For controllers for which no feasibility guarantee can be given a priori the function mpt_invariantSet can compute an invariant subset of a controller such that constraints satisfaction is guaranteed for all time ctrl inv mpt invariantSet ctrl 4 5 5 4 Lyapunov type Stability Analysis In terms of stability analysis MPT offers functions which aim at identifying quadratic sum of squares piecewise quadratic piecewise affine or piecewise polynomial Lyapunov functions If such a function is found it can be used to show stability of the closed loop systems even in cases where no such guarantee can be given a priori based on the design procedure To compute a Lyapunov function one has to call ctrl lyap mpt lyapunov ctrl lyaptype where ctrl is an explicit controller and lyaptype is a string parameter which defines the type of a Lyapunov function to compute Allowed values of the second parameter are summarized in Table 4 3 Parameters of the Lyapunov function if one exists will be stored in lyapfunction ctrl lyap details lyapunov 146 Michal Kvasnica lyaptype Type of Lyapunov function quad Common quadratic Lyapunov function gt sos Common sum of squares Lyapunov function pwa Piecewise affine Lyapunov function gt pwq Piecewise quadratic Lyapunov function gt pwp Piecewise polynomial Lyapunov function Table 4 3 Allowed values of the functiontype parameter in mpt_lyapunov 4 5 5 5 Complexity
197. ired 4 posun offset of the output valve difference between tank bottom and 4 valve output orifice 4 RRS 1enals transformations kkkokokokokok h2m yr y yr v 4s hifting reference state to the origin wi beta2 betal2 2 wtposun w 4 steady state level in tank 1 4 corresponding to setpoint reference level w in tank 2 posun valve orifice offset hi mic wit 1 mic him h2 m2c wt 1 m2c h2m 4 hi h2 operating points for linearization of tank 1 and 2 4 mic m2c waighting coefficients for specifying 4 operating points for linearization 4 mic m2c 0 correspond to exact linearization i e linearization around actual state leading 4 to overshooting thanks to nonmodelled dynamics 4 mic m2c 0 correspond to linearization around reference state 4 too slow closed loop dynamics 4 recommended to work with m c m2c gt 0 0 1 0 2 h12 abs h1 h2 z sign hi h2 a betal2 2 S1 sqrt h12 b betal12 2 1 2 2 beta2 h2 beta2 posuntbeta2 h1 beta12 h1 h2 1 2 h2 posun 1 2 1 2 beta2 2 4 transformation to the normal form for output y h2 real control ur of pump with constraints Qimin Qimax 4 replaced by fictive input of a double integrator 4 with limits Q12 and Q11 268 Q12 Vladim r ilka and Mikul Huba a Qimax b 011 a Qimin b limit values of fictive input signal of double integrator xk kkkkkkkkkkcontrol algorithm kkkkkkkkk contro
198. isturbance by a LED voltage step during the experiments 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 303 Gain of open loop 18 T T T T T 16r 14 12 0 10 20 30 40 50 60 70 Fig 9 14 Process gain in multiple operating points as function of the output variable e Compare the results with various controller tuning The disturbance observer can be added to the static feedforward control to compensate the steady state error To obtain a structure equivalent to I controller the pre filter with time constant equal to the observer time con stant has to be added as well Let us denote the controller as Io controller Fig 9 17 and the controller with pre filter as FIo Tuning of the Ip and FIg controller requires information on a process gain and approximation of the non modeled dynamics usually by the dead time The approximation of the delay in this example is Tg 0 4 The filter time constant of the disturbance observer is restricted by the parasitic time delays of the non modeled dynamics in the process Use following formula to set up the disturbance observer filter time constant Tri eTa 9 2 Try to use multiple process gains for controller tuning to improve con trol quality Use the lowest the average and the maximum process gain from Fig 9 13 Choose experiment no 3 with the exnum command to control non filtered optical channel by FIo controller You will be prompted for
199. it decreases the liquid s level below the desired values To compensate the outgoing liquid we have to use more advanced controllers The easiest way to compensate the outgoing water is to add parallel signal to the output of the controller Such constant signal set usually manually was originally called as reset offset Later this was replaced by automatic reset denoted today as the integral action and produced directly by the con troller By this modification we will obtain the PI controller The goal of the integral action is to integrate the control error values multiplicated by a suitable constant what will continuously increase the controller output until the complete removal of the control error The transfer function of such PI controller is I R s P s where P represents the gain of the proportional action of the controller and I is the gain of the integral action An easy experimental way to set up the PI controller s parameters may be based on measured step responses of the system The step responses of the system contain information about the basic dynamic attributes of the system and this is why it enables us to set up the controller s parameters better than by the frequently used trial and error method At the same time we have to realize that the step response method was developed for the responses of linear system to a unit step Our system 6 1 is however a nonlinear one We can t thus talk about the system s step resp
200. k DH D DH D D options X1 k 1 A X1 k B U1 1 k Yi k C X1 k end where X1 is the matrix containing the states and Y 1 is the vector containing the deflection data The last two lines assume that there is no need for state observation this is to make the simulation simpler If the cost is needed as well one needs to include either one of the following lines in the code J k U1C EJ H U1 K 2 X10 k F U1C K X1 C k2 G X1 k J2 k X1 k 0 X1 k U1 1 k R U1 1 k The optimization procedure can be fine tuned by options optimset LargeScale off Display off TolFun 1e 12 It is also possible to substitute the Matlab built in quadprog function with the qpOASES Matlab interface After compilation the sequential qpOASES solver can be simply called by using the following code for k 1 run 0bj0A U2 k yd status2 1 k nWSRout2 1 k gp0ASES seguence i H F X2 k Ac bO Bx X2 k 10 X2 k 1 A X2 k B U2 1 k Y2 k C X2 k end 5 2 3 6 Controller Loop in Simulink The QPMPC controller loop is featured in Fig 5 3 The controller loop con tains means for A D and D A conversion state observation data logging and others The block containing the QPMPC controller takes the actual ob server state as an input and outputs controller voltage It may be required to 5 Implementation of MPC Techniques to Real Mechatronic Sy
201. l algorithm for constrained control of double integrators with real closed loop poles alfai alfa2 if Q11 Q12 gt 0 if him h2m gt w w1 u Qimin else u Q1imax end ur u else if y gt 0 Q1j else Q1j Q11 end Q12 di 011 alfa1 d2 012 alfa1 if dr gt d2 amp dr d1 u alfal alfa2x y alfail alfa2 dr linear pole assignment else u 1 alfa2 y 0 5 dr dr Q1j Q1j alfai alfal dr 01j nonlinear pole assignment end constraining output if u gt 012 u Q12 end if u lt Q11 u Q11 end 4 kkkkkkkkkkinverse transformation KKKKK 4 inverse transformation from the fictive 7 Constrained PID Control Tasks for Coupled Tanks Control 269 double integrator input u to real input ur ur utb a end out ur 7 6 Conclusion We have described the basic possibilities of control of available hydraulic sys tem established by different configuration of one or two containers with one or two pumps It is important to mention that in case of control of real sys tem with the mentioned controllers we will not obtain the ideally supposed results Certainly due to modelling errors we will register some deviations between supposed and real transients The control performance will depend on the accuracy of identification and robustness of controller As we will never be able to define the system s parameters with 100 precision the system wild never behave as the perfect integ
202. l is sometimes used when there are more than one candidate controlled variable for a manipulated variable For each of the candidate controlled variables a separate controller is the used and the value of the 28 Morten Hovd and Selvanathan Sivalingam manipulated variable that is implemented is selected among the controller outputs A simple example of selective control with pressure control on one side and flow control on the other side of a valve is shown in Fig 1 25 Normally one selects simply the highest or lowest value A few points should be made about this control structure e Clearly a single manipulated variable can control only one controlled vari able at the time i e the only variable that is controlled at any instant is the variable for which the corresponding controller output is implemented It might appear strange to point out such a triviality but discussions with several otherwise sensible engineers show that many have difficulty com prehending this Thus one should consider with some care how such a control structure will work e The selection of the active controller is usually based on the controller outputs not the controller inputs Nevertheless the local operators and engineers often believe that the selection is based on the controller inputs or that the control switches when the a measurement passes its setpoint In principle the selection of the active controller may also be based on the controller input
203. l loop Since the setpoint is 0 the mea surement and control offset are equivalent 2 Calculate the Harris Index or Normalized Harris Index for the control loop Is there scope for significant improvements in control performance 3 Select a reasonable value for the desired closed loop dominant time con stant Instead of the minimum variance benchmark used in the Harris Index use a modified benchmark reflecting a desired closed loop time con stant to assess control performance The calculations involved are relatively simple and appropriate Matlab routines may be programmed by the student Alternatively the Matlab rou tines ar and impulse may be useful 1 4 2 Solution First the mean of the data series OP is removed with the command opd detrend op constant Next a 10th order auto regressive model from the unmeasured disturbance to the controlled variable is identified using the command mod ar opd 10 The routine ar finds parameters for the model y k ony k 1 agy k 2 a 10 y k 10 a k 1 12 A q y k d k where y k is the measurement at time k and a k is the unmeasured distur bance Note that the leading coefficient of the polynomial A q7 1 a 0 1 The coefficients of the polynomial A q are found in mod a and the es timated variance of a k is found in mod NoiseVariance The AR model is converted to an impulse response model by simple polynomial long divi 1 Proble
204. l plants control This may be just partially achieved by computer simulations and the feedback received in such a way is never complete Therefore it is important to acquire already during the university studies some practice and experience related competences in designing and testing of controllers on real systems Due to the simple manipulation easily understandable and observable pro cesses and their time constants one of many systems used at universities fre quently as pedagogical instruments in control education the system of coupled tanks has to be mentioned The theoretical design of controllers included in this publication comprehends two basic configurations control of one or of two coupled tanks In considering coupled tanks one of possible configura tions is the coupling in series with interaction when the liquid s level in one of the tanks influences the other tank s liquid s level as the liquid may flow Vladimir Zilka Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail vladimir zilkaQ stuba sk Mikul Huba Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail mikulas hubaQstuba sk 229 230 Vladim r ilka and Mikul Huba between the tanks in both direction The other two tank consideration is the cascade coupling with one way influence where the tanks are placed one un der another
205. labFcn following equation was made c_ 12 sign u 1 u 2 sqrt abs u 1 u 2 6 2 1 Identification For simulating or controlling the real system it is indispensable to know all parameters of its mathematical model While the measurement of the level surfaces in tanks does not represent a serious problem to determine the values of the valve s flow coeficient is usually a bit more difficult By taking the more detailed view on 6 1 or 6 2 we will realize that we can define at least two approaches for determining parameters ci co and c12 One is 6 Laboratory Model of Coupled Tanks 233 Sum level4 Integrator 1 c1 sqri u outflow from 1 tank level2 Integrator 2 c2 sqritu outflow Fig 6 3 Coupled tanks Simulink model based on measurement with constant inflow the other on experiment with zero inflow tank s emptying In the first approach based on measuring steady state under constant in flow the resulting state equation reduces to 6 1 is 1 A Ciy Ti U1 Ai mi The second approach is based on evaluating transients under zero inflow i e during tanks emptying The flow coefficient c of the valve can be easily obtained from 6 1 If the pump is off the state eguation 6 1 reduces to t aVT1 y C 6 3 T In this case by integrating above equation c can be get as u 2 Tlinit 2y T1 final Ad 6 4 C1 where T1init represents the initi
206. lating the unconstrained linear quadratic optimal gain along with the terminal weighting matrix K S e dlqr A B Q R K K Qe dlyap A B K Q K R K 5 Implementation of MPC Techniques to Real Mechatronic Systems 179 The forced and free state prediction matrices are calculated through a set of nested loops according to the following script M2 zeros nc nx for n 1 nc M1 n nx nx 1 n nx A7n for na 0 nx nc nx na O0 nx nc nx m2 nx n 1 na 1 nx n 1 na nx n A na nx B end end M2 m2 1 nx nc where several other possible solutions may exist These solutions can be equiv alently good and while their runtime may differ this should not be an issue in an off line problem setup process The last n rows of the matrices M1 and M2 are also selected M11 M1 nx nc nx 1 nx nc M21 M2 nx nc nx 1 nx nc The next step is to create the cost prediction matrices One has to begin with initialization H1 0 F1 0 G1 A70 Q This is then followed by creating the cost prediction matrices H F and G first by running the following loop to get partial results for i 0 nc 2 H1it M2 1 i nx i nxtnx Q M2 1 i nx i nxtnx H1 H1 Hit F1it M2 1 i nx i nxtnx Q M1 1 i nx i nxtnx F1 F1 Fit G1it M1 1 i nx i nxtnx Q M1 1 i nx i nxtnx G1 G1 G1t end And finally assembling cost prediction matrices H F and G H H1 M2 1 nc 1 nx nc 1 nx nx Qe
207. lations in the output are small due to the low plant gain at high frequencies Note however that the oscillations in the valve position due to the stick jump are nevertheless significant in magnitude and may result in premature wear on the valve 4 Data set 4 is obtained with a sinusoidal external disturbance and the control loop itself is linear We find that the cross correlation method therefore perform well on data sets 1 and 4 and the OP PV plot at least when allowing for time shifting between the OP and PV data series perform well on data sets 1 2 and 4 The methods using only the PV bicoherence m and glstat m failed to detect significant nonlinearity for sets 1 2 and 3 while correctly finding no indication of nonlinearity for data set 4 One should note however that e t is not really fair to compare a stiction detection method using only PV to methods using both OP and PV e Both bicoherence m and glstat m have tunable parameters that may help in detecting nonlinearity in some or all of these cases However in the absence of any clear recommendations on how to select these parame ters we have chosen to use default values throughout None of the methods tested were able to detect stiction in data set 3 Logging the actual valve position which must be measured at least locally in order to implement a valve positioner would be very helpful for detection of stiction when the stiction in combination with the val
208. le 2 5 Our own example 8 data sets in each run average of 500 runs waoe 0 2934 0 9985 LS PCR 8 0 5496 0 4931 PCR tol 1 most cases PCR 5 Now to show that the loss method does better when there is more data Table 2 6 shows the average value of Y1 HX for 32 data sets after running 300 times Table 2 6 Our own example 32 data sets in each run average of 300 runs 1 7700 0 2690 loss method no noise 1 7700 0 2690 loss method smooth y1 1 7703 0 2687 loss method noise 0 5 The loss method is the winner val 0 269 in spite of the fact that it has no tuning parameters Here there is little effect of adding artificial noise with the loss method presumable because we have enough data 2 9 8 Comparison with Normal Least Squares Normal least square solution Problem Find H such that the magnitude of Y HY is minimized for the given set of data for Y and Y Solution H Y pinv Y This minimizes Y HY and for cases where this JH is not unique minimizes H This is the same as finding H to minimize H Y au which we know is not the optimal solution Proof The data matrix is Yau Y Y Assume that we seek Hau to minimize Han Y ai We have degrees of freedom in Hau so we set Hau I H Then we want to minimize Hau Y 1 Y Y 1 HY the best solution is given by the pseudo inverse H Y ipinv Y which is the usual least square solution O
209. lem 56 2 2 8 Comments on the Problem 56 2 3 Solution to Problem Preliminaries 57 2 3 1 Expression fex told oreco betrar tas tee ei 57 2 3 2 Expression for J around topla 57 2 3 8 Expression for J around Moving uopr d 58 2 3 4 Optimal Sensitivities 06 0026 eee ee 58 24 The Loss Method et bud etn eee pta RED Ba td eee dna 59 2 4 1 The Loss Variable z as a Function of Disturbances and Noele oues pricei tinie DER deal ed bial acts doe 59 2 4 2 Loss for Given H Disturbance and Noise Analysis 59 2 4 8 Worst case and Average Loss for Given H Analysis 60 2 4 4 Loss Method for Finding Optimal H 60 2 5 Reformulation of Loss Method to Convex Problem and Explicit Solution cerise 22 meme eet eme 6 aeree 63 2 6 Structural Constraints on H ee eee 65 2 7 Some Special Cases Nullspace Method and Maximum Gu Nod cies ke P can eva ae Ex A ER iE ERR 66 2 7 1 No Measurement Noise Nullspace Method full H 67 2 Na Dhsburbaneem vesci sed rt eR Ree 68 2 7 9 An Approximate Analysis Method for the General Case Maximum Gam Rule eee ee ae 68 2 8 Indirect Control and Estimation of Primary Variable 70 2 8 1 Indirect Control of Yy ooo met 71 2 8 2 Indirect Control of y Based on Estimator 71 2 9 Estimator for y Based on Data 0 00 cee ee 72 280 Data Approach I c neu
210. load them and have a model that will use them correctly The basic program would after wards look as follows load calibParams offsets and gains of pressure sensors load valvel parameters of drain valve from tank 1 load valve2 parameters of drain valve from tank 1 load valve12 parameters of valve between tank 1 amp 2 load pumpi pump 1 parameters load pump2 pump 2 parameters A 0 001 4 tank area T 0 25 sampling period open model mdl open simulink model Pump s subblock is depicted in Fig 6 7 where function block U QI1 con tains PU1 1 u 1 4 PU1 2 u 1 34PU1 3 u 1 2 PU1 4 u 1 PU1 5 This 4 degree polynomial enables linearization of the pump The input for this function is inflow from controller in m s Function convert this input to Volts The Gain4 changes Volts to 96 because the pump input in matlab block is in 96 There is also treshold For too small input smaller than 10 9m s push to output 0 This feaure respects the fact that the pump can not process smaller inputs 6 2 4 Some Recommendation for Users At the end we would like to add some recommendations that may help you in keeping your plant in optimal work Use always distilled water even in the case you need to add just small amount of it In this way you may reasonably prolonged intervals between clearings 240 Vladim r ilka and Mikul Huba Constant2 Switch Fig
211. lows extern double ddot_ int double int double int extern void dediv_ int double double double Element wise division of two vectors is performed by the following func tion void dediv_ int n double x double y double z int i 0 for i i lt n i z il x il ylil 5 Implementation of MPC Techniques to Real Mechatronic Systems 191 where vectors x and y have a common dimension of n Their elements are indexed with the locally declared variable 7 The function takes the dimension n and vectors x and y as its input and places the result in vector z After the unknown A is calculated by the algorithm each element of vector x is scaled by it This simple scaling operation is extended by an additional step for computational efficiency Each scaled element of x is subtracted from 1 and the result is placed in the vector y void descal int n double lamN double x double y 1 int i 0 for i i lt n i y Ci 1 lamN x i The function takes scalar dimension n vector x and scalar A as its input The output is placed in y where the dimensions of both vectors are n An inside loop performs the formerly described simple operation where the elements are indexed with the locally defined variables i 5 2 4 10 The On line NRMPC Code in C language A pre requisite for the correct functionality of this code is the correct call ing of external BLAS functions and the
212. luded in this figure Note also that fast low pass filters are introduced in order to break so called algebraic loops in the Simulink simulation Such algebraic loops are a problem only in simulation not in a physical implementation If the low pass filters are significantly faster than the rest of the dynamics in the system they should have little effect on simulation results However very fast low pass filters could lead to a stiff simulation problem causing long simulation times or requiring the use of special purpose integration routines LQG Controller The Simulink setup for the LQG controller is shown in Fig 1 10 The Kalman filter part of the LQG controller is designed with the physical model states and the specified noise covariances The LQ regulator part of the LQG con troller is designed with a model that is augmented with integrators in order to include integral action The weights used in the LQ design are found by trial and error in order to approximately achieve the specified closed loop band width The overall simulation setup including integrating controller states LQG controller and distillation column model is shown in Fig 1 11 Note that provided the Kalman filter is provided with the saturated input values it will provide good state estimates also when the inputs are saturated However kinum s k1den s A O A m Reference k2num s Measurement Controller output k2
213. ly She he is informed by the message Connection timeout or network error occurred Disconnected from server Fig 8 8 Automatic disconnection e Run it serves for running the block diagram specified in the Model Scheme parameter First all parameters are sent to the server where also the parameters from the file given in the Model Parameters field are read into the Matlab and consequently the experiment starts with continuous data transfer and visualization that takes until the time set in the Running Time field is over In the Output field the message Simulation starts will appear and the transfer data characterizing the experiment states begin to be displayed The button is available only after a successful connection to the server e Clear its task is to clear display fields of the client application namely the Output and Graphs sectors The application will be then prepared for the next run within the existing connection to the server e Save this button is available after the duration Running Time of the experiment is over When it is activated the time sequences of the model 8 Remote Laboratory Software Module for Thermo Optical Plant 283 Fig 8 9 Experiment run state variables coming from the just finished experiment begin to save into the file specified by the Save to File parameter In the Output field the user is informed about the process and success of saving data The format of the saved file is
214. mand A b double TV A 0 7071 0 7071 0 7071 0 7071 106 Michal Kvasnica O 1 0000 b 0 7071 0 7071 0 Notice that in this output all rows of A b have been automatically normalized such that the 2 norm of each row is equal to one to improve numerics 4 2 2 Polytope Arrays Instances of the polytope object can be concatenated into arrays Currently only one dimensional arrays are supported by MPT and it does not matter if the elements are stored row wise or column wise An array of polytopes is created using standard MATLAB concatenation operators e g A B C D It does not matter whether the concatenated elements are single poly topes or polyarrays To illustrate this assume that we have defined polytopes P1 P2 P3 P4 P5 and polyarrays A P1 P2 and B P3 P4 P5 Then the following polyarrays M and N are equivalent M A B N P1 P2 P3 P4 P5 Individual elements of a polyarray can be obtained using the standard refer encing i operator i e P M 2 will return the second element of the polyarray M which is equal to P2 in this case More complicated expressions can be used for referencing Q M 1 3 5 will return a polyarray Q which contains first third fourth and fifth element of polyarray M If the user wants to remove some element from a polyarray he she can use the referencing command as follows M 2 which will remove the second element from the polyarray M Again m
215. may also be successfully applied to any stable plant but in such situations it is no more possible to speed up the control signal transient up to the step function just to keep it monotonic that guarantees that all such controllers may again be successfully treated by the linear theory as well In the second part basic structures of the DC1 will be introduced that may already typically have one constrained period in their control signal step responses Here control structures with integral action based on disturbance observers will be introduced that do not exhibit windup phenomenon and so enable simpler one step tuning that in the case of traditional linear controllers extended by the anti windup circuitry Peter Tap k Institute of Control and Industrial Informatics Faculty of Electrical Engineering and IT Slovak University of Technology Ilkovi ova 3 812 19 Bratislava Slovakia e mail peter tapakAstuba sk Mikul Huba Institute of Control and Industrial Informatics Faculty of Electrical Engineering and IT Slovak University of Technology Ilkovi ova 3 812 19 Bratislava Slovakia e mail mikulas huba stuba sk 291 292 Peter Tap k and Mikul Huba 9 1 Thermo optical Plant uDAQ28 LT Quick Start This section gives a short guide to uDAQ28 LT plant Get familiar with thermo optical plant interface For more information on the device please re fer to the user s manual This device offers measurement of eighth process vari
216. mentioned file with date and time that way we are able to see the history at any time We apply the similar procedure at the other measurements as well Basically in the same way we identify the valve between the tanks After the measurement is launched the first tank is filled and the second emptied the interconecting valve is opened afterwards and the measurement is per formed until the leveling of two liquid s levels Then we perform the second measurement the second tank is filled up and the first emptied We iden tify again the flow coefficient and we use those two measurements to define the mean coefficient c12 We save the measured values following the above mentioned procedure just this time the mat file will be named valvel2 mat for the actual value and valvel2day time mat for archiving We ll use the identified coefficient for the simulation and we draw the comparison of model s course and the real into the graph In case of lesser compliance we repeat the procedure 238 Vladim r ilka and Mikul Huba The inverse i o characteristic of the pump 1 The Input output characteristic of the pump and approximation with 4rd degree polynomial x 10 and approximation with 4rd degree polynomial 14 18 r Real i o characteristic Real i o characteristic Approximation 16H Approximation 12 14 Biz zs S i a 9
217. methodi 2 LS Hls Yl pinv X res Hls X resO Hls X0 aii norm res Y1 a12 norm res0 Y10 PCR vary tol PCR To find cut off to find no of components semilogy svd X method6 PCR tol 1 e 4 Hpcr Yi pinv X to1 npcr rank pinv X tol no of components used in PCR res Hpcr X resO Hpcr XO0 a61 norm res Y1 a62 norm res0 Y10 Weighted least squares to get relative noise assumption rather than additive noise method5 weightedLS rx cx size X mag for i 1isrx mag mag norm X i the magnitudes are about 0 8 end Xs inv diag mag X Hlss Yi pinv Xs 128 129 130 131 132 133 134 135 136 137 138 139 141 142 143 144 148 149 150 151 152 153 154 156 157 158 159 160 161 162 84 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Hlss Hlssi inv diag mag ress Hlss X ressO Hlss X0 abi norm ress Y1 ab2 norm ress0 Y10 pls npls 35 4npls npcr XL y1 XS YS beta PCTVAR plsregress X Y1 npls Note that PLS has an additional bias centering parameter yfit ones size X 1 1 X beta yfitO ones size X0 1 1 X0 beta a7i norm yfit Y1 a72 norm yfit0 Y10 method7 11 PLS Loss method step 1 Yall Y1 X u s v svd Y1 Yalli Yall v r1 ci size Yall r2 c2 2size Y1 ny r2 G1 Yalli 1 ny i1
218. milar transients using both controller tunings Fig 9 26 The 2 V LED voltage step was made at time 105s Following table summarizes the robust FIo controller tuning for various overshooting tolerances Questions 308 Peter Tap k and Mikul Huba Light channel output 45 T T T k System output 40r be Setpoint N O1 o c1 T l n o Light intensity E oa o T L 0 L L L L L 0 10 20 30 40 50 60 time s Fig 9 21 Experimental results for K 10 Table 9 1 Controller tuning ey 96 10 5 4 2 1 0 1 0 01 0 001 0 0 T K Q 1 724 1 951 2 0 2 162 2 268 2 481 2 571 2 625 2 703 2 718 q 2 amp 0 58 0 515 0 5 0 465 0 441 0 403 0 389 0 381 0 37 0 368 What type of plant is FIg controller suitable to control for Which plant s parameter restricts the controller s dynamics Was there any overshooting in the experiments Why the performance of different controller tunings in Fig 9 26 is almost the same 9 2 3 Filtered Predictive Ig Controller Tasks e Add a dead time to the non filtered light channel output e Modify the Ip controller to compensate the delay e Tune the controller using a performance portrait 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 309 Control signal 3 5 T T T P M ol T L N T L Bulb voltage oa T fi 0 54 4 0 L L L L L 0 10 20 30 40 50 60 time s Fig
219. mplicity of these approaches no issues regarding computational imple mentability arise Even though these control schemes are often sufficient in certain applications the advantages of MPC cannot be overlooked in vibra tion damping Unfortunately the fast dynamics of vibration attenuation systems require very short sampling times which limit the use of computationally intensive on line calculations usually associated with MPC Traditional quadratic pro gramming based MPC QPMPC has been successfully implemented by Wills et al 2008 where the problems raised by the fast sampling were tackled by implementing a machine code optimized quadratic programming solver on a specialized hardware While this algorithm handles process constraints it fails to address the important consequences of stability and constraint feasi bility The stability and constraint feasibility of model predictive control algo rithms may be guaranteed through the deployment of terminal constraints Maciejowski 2002 This unfortunately reduces the range in which the con trol algorithm may operate by assigning a region of attraction of all feasible initial conditions in the state space Amongst others the size of this region is dependent on the prediction horizon Clamped cantilever beams actuated by piezoelectric strips are a very spe cial case in the eye of the control engineer Model predictive control with guaranteed constraint stability and feasibility applied on s
220. ms in Anti Windup and Controller Performance Monitoring 47 0 5 Impulse response coefficient 0 2 01r 0 L L L t 100 50 0 50 100 150 200 250 300 350 Time Fig 1 48 Calculated impulse response sion or using the command h t impulse mod The resulting impulse response is plotted in Fig 1 48 It can be seen that the model is as expected semi proper There is no time delay from manipulated variable to the controlled variable so the minimum variance can be found from OZ h W o 1 13 We thus find o 2 489 10 The actual variance of the output is found to be o 1 95 10 We thus find a Harris index of HI 31 94 or a normalized Harris Index of NHI 0 969 We conclude that there is a significant scope for improved control performance A frequently quoted rule of thumb is that the closed loop time constant should be 2 3 time faster than the open loop time constant However this clearly cannot apply to integrating processes with infinite time constant and in practice will depend on the available range and speed of actuation for the manipulated variable Nevertheless the rule of thumb would indicate that a closed loop time constant of 40 is reasonable considering the open loop dominant time constant of 100 This corresponds to a desired closed loop discrete time pole of u e 1 49 0 9753 The modified benchmark variance therefore becomes m 1 4 Omod 1 z a2 5 103
221. n contains the integration from net force acting on the valve to the valve velocity as well as the calculation of the friction 1 Problems in Anti Windup and Controller Performance Monitoring 35 force based on the simple model described in Sivalingam and Hovd 2011 The submodel is shown in Fig 1 33 For simulations without stiction the submodel Intstiction is replaced by the simple integration from net force to velocity with a positive kg in Fig 1 32 used to simulate viscous linear friction 1 3 1 Assignment Prepare by downloading required software if not already installed and the data series used in the assignment 1 Download the Higher Order Spectral Analysis hosa toolbox from Matlab Central and or the file bicoherence m from Dr MAA Shoukat Choud hury s website teacher buet ac bd shoukat 2 Download the Matlab data files Setidata mat Set4data mat from http www itk ntnu no ansatte Hovd_Morten NIL Each of the Matlab data files contain three arrays of variables i OP the controller output ii PV the process measured variable and iii tp the corresponding time index For each of the data sets do the following 1 Plot both OP and PV versus time Do these plots indicate oscillations caused by stiction 2 Use the cross correlation method to detect possible stiction Comment on the result 3 Use the OP PV plot to detect stiction If the plot is hard to interpret try time shifting the data se
222. n the state of waiting for the connection from the client After the client connects to the server it starts to fulfill client s demands The server usually runs without breaking if the safety issues are covered In the case it is necessary to finish the server application it can be done by closing the RemLabServer exe application window and closing manually the Matlab application If the server starts successfully it is possible to connect to it from the client application To run the client application means simply to open the starting Web page client html placed in the Client Application folder Usually the client application is installed on a Web server Then it is necessary to set the corresponding Web page address into the Web browser Depending on the Web browser safety settings it could be required to confirm a trust in running the Java applets http www kar elf stuba sk pbistak appletKE client html Windows Internet Explorer EEK G E http www kar elf stuba sk pbistakfappletKEjclient htmi X lel File Edit View Favorites Tools Help Google VM search der Bie CO sidewk S Orvo We Be hpiiwewkar elf stubasklopbistaklappletkE dlent 5t deh Page C Tools Model Parameters Input Desired Value Connect Server IP 127001 j e Internet Fig 8 3 Start of the client application after downloading Java applets 8 Remote Laboratory Software Module for Thermo Optic
223. n the inflow and outflow If we know the inflow u4 and we define the outflow as uout then AV ui Uout V 1144 dx ld U1 Uout To determine uou we can use the Toricelli formula which defines the speed of the liquid outflowing the tank s aperture Vout With the outflow we have to take into account the co actor as well u for the water 0 63 Plus if we already know the cross sections the valve s aperture Avo we can put down the following 232 Vladim r ilka and Mikul Huba Vout V 29gX1 dai Ar uy H vo y 2gz1 Now by applying the substitution and defining the flow coefficient of the valve C1 a y M The resulting differential eguation for the one level system would be 1 Li A CIV TI y 11 6 1 In this way we can derive the differential equations for coupled tanks System is then composed of two mutually interacting tanks with two possible pumps Each tank would have a separate inflow u1 a ug separate outflow depending on x and x2 and the flow between the tanks specified by difference of the levels and the constant c19 Such tank model can be then described by differential equations 1 ty au c2 sign z z2 4 x z9 ei 21 1 1 to Te cig sign x z2 y 1 x2 eo 2 2 yi T Y2 T2 6 2 The liquid s levels in the first and second tank are denoted as T1 r9 and the level surfaces as A1 Ag For Simulink model see Fig 6 3 In block Mat
224. n the intervals in which the process gain and the dead time range The data from Fig 9 14 and Fig 9 15 can be used In this example the dead time 74 ranges in interval 0 4 0 7 and the process gain ranges in interval 3 17 Now determine the interval in which normalized variable amp ranges amp represents normalized variable where Ko stands for the process gain used in the controller K corresponds to real process gain which varies through the operational range The goal is to fit the uncertainty box into area of performance portrait Fig 9 25 where the controller gives monotonic transients Try to use two limit values of process gain for Ko in this example it gives which leads to amp 3 17 1 For the maximal value of Ko 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 307 Control signal 5 T T T 3 51 M 2 57 1 Bulb voltage 0 10 20 30 40 50 60 time s Fig 9 20 Experimental results for K 3 Ko Kman 1T 9 5 it gives x 1 17 3 Q Ta Ts 9 6 Calculate filter time constant T for both selections of Ko For Ko 9 4 it yields NQ 0 0649 9 7 T Ta 0 7 0 0649 10 7825 9 8 For Ko 9 5 it yields Q 0 3679 9 9 Ty Ta 2 0 7 0 3679 0 7e 1 9028 9 10 Verify controller tuning by real experiment for various setpoint steps Add a disturbance using a LED to verify input disturbance compensation You should observe si
225. ncorporate a contrac tion constraint v 1 p lt a xx p which requires the controller to push the predicted states towards the origin with some fixed decay rate 0 a lt 1 for k 1 N 1 C C norm V xf k 1 1 lt a norm V x k 1 end Notice however that norm constraints are nonconvex and the resulting op timization problem will contain binary variables having negative impact on the time needed to solve the optimization probStruct 4 5 4 5 Logic Constraints Constraints involving logic implication and equivalence can be also formu lated and automatically translated to a corresponding mixed integer rep resentation by YALMIP Take again the double integrator example of the 140 Michal Kvasnica previous section In order to make the controller less aggressive one can re quire the first optimal control move to satisfy 0 5 lt ug lt 0 5 but this constraint should only be imposed if the state is contained in a small box around the origin If the state is outside of this interval the control action is to respect 1 lt ug lt 1 which is the default constraint already contained in the sysStruct structure To formulate such a control goal with MPT one can make use of the implies operator YALMIP provides gt gt C C implies H V x 1 lt K 0 5 lt V u 1 lt 0 5 where H eye 2 eye 2 and K 1 1 1 1 represent the area of inter est ie the box 71 lt a t lt 1 When such a constraint i
226. ng library to make it possible to choose whether the saturation should be turned on or off for the simulations The disturbance input is modelled as white noise independently normally distributed ran dom variables with variance 0 001 The measurement noise is white noise with variance 0 0004 2 Controller design Design three different controllers for the model a A controller based on two PI PID loops b A controller based on dynamic decoupling c A controller based on multi variable controller synthesis LQG H Hoc All the controllers should be designed to give zero stationary deviation The dominant time constant for the closed loop should be around one minute PI PID controller design should be well known The multi variable con troller design method chosen is also assumed known all the three con troller synthesis methods proposed above are of the state estimator state feedback type Decoupling is also a relatively common controller design see Wade 1997 for a thorough description of decoupling for 2 x 2 systems if necessary 3 Simulation with disturbances Simulate each of the three controllers in Simulink Use the following reference signals Yi ref Square pulse amplitude 1 frequency 0 005Hz U2 ref 0 Simulate each controller design both without and with saturation in the inputs Discuss the results of the simulation 4 Anti windup and simulation Implement anti windup for all three controllers Simulate
227. nguish two cases 1 All inputs are discrete 2 Some inputs are discrete the rest are continuous 4 5 6 7 Purely Discrete Inputs Typical application of discrete valued inputs are various on off switches gears selectors etc All these can be modelled in MPT and taken into account in controller design Defining discrete inputs is fairly easy all that needs to be done is to fill out sysStruct Uset Uset where Uset is a cell array which defines all possible values for every control input If the system has for instance 2 control inputs and the first one is just an on off switch i e u1 0 1 and the second one can take values from set 5 0 5 it can be defined as follows sysStruct Uset 1 0 1 sysStruct Uset 2 5 0 5 4 Multi Parametric Toolbox 151 where the first line corresponds to u and the second to us If the system to be controlled has only one manipulated variable the cell operator can be omitted i e one could write sysStruct Uset 0 1 The set of inputs doesn t have to be ordered 4 5 6 8 Mixed Inputs Mixed discrete and continuous inputs can be modelled by appropriate choice of sysStruct Uset For each continuous input it is necessary to set the corre sponding entry to Inf Inf indicating to MPT that this particular input variable should be treated as a continuous input For a system with two ma nipulated variables where the first one takes values from a set 2 5 0 3 5 and the
228. ning the application are placed at the bottom part of the interface and play the role of a connection establishment or cancellation running the experiment saving the experiment data clearing the interface before running a new experiment Besides the control buttons there is also the textual field in the bottom part to which it is necessary to enter the IP address before the connection is established 8 5 1 Settings The following parameters can be set up using the lower left sector Settings e Model Scheme the name of the Simulink file mdl that represents the block diagram model of the experiment This file must be placed on the server computer in the specified folder CA RemLab Workspace that was created during the server installation The block diagram must keep the rules written in Section 8 7 The user should be informed about the existing names of mdl files that are available e Model Parameters the name of the parameter file that is represented by Matlab m file Also this file must be placed on the server computer in the folder CA RemLab Workspace Usually it contains the values of constants and calculations of initial parameters necessary for running the experiment block diagram model e Input Desired Value input value of the block diagram that in the case of controller represents a desired value of the output value Because the real system has limits included in the block diagram it has only sense to set the desired v
229. nlinear model in a fixed operating point by the input output linearization in the actual state exact linearization and around a generalized operating point are developed and verified for the one level and two level system 7 1 Introduction A plant to be controlled may be modelled and identified in many ways It is e g possible to derive mathematical models of the plant that for a given data show nice matching but when used for the controller design the resulting performance will be pure In case of inaccurate or inappropriate mathemati cal model the resulting regulators would function only in simulations That s why we will be dealing from now with the description of the real plant estima tion of its model identification of its parameters as well as with confrontation of results achieved in control of the plant with the performance reguired in the controller design In the next paragraph we will show some examples of simple P PI and PD controllers for different tanks configurations from the simplest one tank Vladim r ilka Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail vladimir zilka stuba sk Mikul Huba Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail mikulas hubaQstuba sk 247 248 Vladim r ilka and Mikul Huba system with closed output valves through one level systems with
230. nnels of uDAQ 28LT plant are going to be controlled in this section Simple alternatives to linear I controllers will be practiced 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 299 Fig 9 9 Basic electrical diagram of thermo optical plant uDAQ28 LT 9 2 1 Feedforward Control Tasks Identify non filtered light channel parameters Control non filtered light channel using inverse process gain Analyze the steady state error for various setpoint changes Measure the I O characteristics of the non filtered light channel Let us start with getting to know the light channel characteristics The non filtered light channel represents a very fast process which can be approx imated as memoryless plant In an ideal case static feedforward control with inverse process gain should be sufficient for such process Measure one point of the I O characteristic to obtain the process gain The basic I O Simulink model matlabroot udaq udaq28LT examples udaq28LT_iov2 mdl can be used As other alternative use the eznum com mand and choose experiment no 1 which opens up basic I O Simulink model of the plant Set the bulb voltage to us 2 5 V and run the experiment for 2s Put down the steady state value of the light intensity By default it is represented by the yellow transient in the light intensity scope Fig 9 10 The steady state value of the light intensity in this example is approxi mately ys 20 The process gain
231. nsidered e Worst case loss Lwc e Average loss Lavg 2 2 7 Examples of this Problem 1 Identify controlled variables c Hy squaring down Then use feed back control to adjust u such that cm Hym Cs 2 Find invariants for quadratic optimization problems 3 Obtain estimate of primary variables c 4 Hy Problem Given that 4 Hy find optimal H such that magnitude of y Y1 is minimized for the expected d s and n s 2 2 8 Comments on the Problem 1 The controlled variables are c Hy and the objective is to find the non square n x n matrix JH note that n nu In general H is a full combination matrix However it may also be interesting to consider control of individual measurements in which case H is a selection matrix with n number of columns with single 1 and the rest of columns are zero mathematically HH I 2 Minimizing the magnitude of the loss L is close to but not quite the same as minimizing the cost J In some cases they give identical results in terms of the optimal H for example if we consider the average loss or cost for given disturbances because then the same cost function is subtracted So it seems it is the same for the 2 norm Frobenius of M see below However there will be some difference if we consider the worst case loss or cost 2 Measurements for Control Optimization and Estimation 57 2 3 Solution to Problem Preliminaries The objective is to de
232. nted in the source file mpt_getInput c a part of the MPT Toolbox The controller look up tables are included in a separate file referenced in this work as mpt_getInput_h and declared within the function source itself The block scheme implementing the MPMPC controller loop along with data acquisition D A A D conversion and other functional parts is featured in Fig 5 6 Fig 5 7 shows multi parametric MPC controller regions plotted in a two dimensional state space Controller partitions shown on the image belong to a controller based on the second order model of the experimental device and conforming to the example presented here Prediction horizon here is 70 steps The region of attraction is divided into 10099 polyhedral regions with associated control laws The figure shows partitions without additional optimization or merging 198 Gergely Takacs et al Fig 5 8 MPMPC controller input in volts MPMPC controller action expressed in Volts is plotted against the state space in two dimensions in Fig 5 8 Note how the control law is essentially reduced to a switching behaviour In fact many real and simple vibration control systems utilize this highly simplified saturated effect Fig 5 9 shows the plot of the quadratic cost function value J related to the polytopic controller regions in state space 5 2 6 Conclusion This section has been dealing with the implementation of model predictive control algorithms in the engineering fi
233. nter of the Chebychev Ball is not unique in general i e there can be multiple solutions e g for rectangles Given a polytope object P the center and radius of its Chebyshev s ball can be computed by calling the chebyball function gt gt xc r chebyball P A sample Chebyshev s ball inscribed in a 2D polytope is shown in Figure 4 2 4 2 3 2 Affine Transformations Consider a polytope P x R Ax lt b and an affine function f x Fx g with F R and g R F non singular Then the image of the 108 Michal Kvasnica xi Fig 4 2 Illustration of the Chebychev s ball polytope P under the function f x is the polytope Q Q Fr g Ax lt b 4 3 To see that Q is indeed a polytope define y Fx g For invertible F we get x Foty F 1g Substituting this expression into y Ax lt b we get Q y AF y lt b AF g 4 4 Y A b which is again a polytope in the H representation y Ay lt b Note that dimensions of r and y are the same since F is assumed to be a square non singular matrix This direct affine transformation of a polytope can by obtained by MPT using its range function gt gt Q range P F g Two special cases of affine transformations are scalar scaling and translation Scaling a polytope P with a scalar a R is achieved by gt gt Q alpha P Translation of P by a vector t R is obtained by gt O P t Applying the lin
234. ntrol actually switches when both controlled variables are below their setpoint T his is of course due to the selection of the active controller being determined by the values of the controller outputs rather than the values of the measurements or setpoints goog 00 Disturbance d Save disturbance 34 Morten Hovd and Selvanathan Sivalingam I Flow Pressure SIE Disturbance A JH l fi L L fi L f 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Fig 1 30 Responses in controlled variables with anti windup I T Flow controller output Pressure controller output Input applied H fi fi Ll fi i f f f L 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Fig 1 31 Responses in controller outputs with anti windup 1 3 Stiction Detection In process control valves are commonly used as manipulated variables Stic tion in control valves is a common cause for poor control performance often causing continuous oscillations in the loop This assignment will study the detection of valve stiction using a few of the available stiction detection meth ods Background on stiction detection methods can be found in Sivalingam and Hovd 2011 A Simulink model is used to simulate a simple control loop with a possi bly sticking valve A printout of the Simulink model is shown in Fig 1 32 The submodel Intstictio
235. ntroller design may now be used as well When the open loop transfer function was in the case of the P controller design given as 1 T1s the closed loop had the transfer 1 Tis 1 Now we will design controller to yield the same open loop transfer function as previously i e 1 T1s what yields 1 1 Tins 1 Ti 1 R v Z Z 1 5 Tis Fi s KiTis KiTi T KiTis Thus the P and J actions will be T P gt Kin 1 n Kin After calculating and testing this tuning we recommend you to experiment a bit and to track the influence of the changes of P and J on the control performance You may observe that the achieved dynamics is differnt from the one possible pulse at saturation that was typical for the P controller design This soultion represents basic part of the PI controller of the dynamical class 0 it should yet be completed by a prefilter with the time constant Ti then the control signal after a setpoint step exponentially increases monotonically to its new steady state value Attempts to achieve faster dynamics may be connected with strong overshooting of the output signal 7 2 2 PI controller Improved dynamics for larger setpoint steps without producing windup effect is possible to achieve by the so called PI controller that typically has one interval of the control signal at the saturation limit This controller is based on reconstruction of the equivalent input or output disturbances by appropriate observ
236. nus form This could equivalently have been done using the scheme in Fig 1 1 The resulting responses when using the inverted decoupler with anti windup are shown in Fig 1 22 Comparing to Fig 1 16 we see that the 1 Problems in Anti Windup and Controller Performance Monitoring 21 Output 1 1 5 T L L L L L fi fi fi 50 100 150 200 250 300 350 400 450 500 Output 2 2 T o B Ue Ab 4 L L L L i L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 22 Responses with inverted decoupler with anti windup responses are significantly improved although far from as good as in the unconstrained case in Fig 1 14 1 1 8 3 LQG Controller with Integral Action The LQG controller with integral action and anti windup is implemented as illustrated in Fig 1 5 The Simulink implementation is shown in Fig 1 23 Note the fast low pass filter used to break the algebraic loop in the anti windup loop of the controller Due to the order of execution in digital control systems this is not required in practice but significantly simplifies numerical integration in continuous time simulations The time constant of these low pass filters should be significantly shorter than any other dynamics in the system but very short time constants may lead to a numerically stiff system When anti windup is included in the LQG controller with integral action the results in Fig 1 24 are obtained The resul
237. oaded plus operator gt gt Q Pi P2 Illustration of Minkowski addition is given in Figure 4 6 4 2 3 7 Pontryagin Difference The Pontryagin difference also called set erosion or set dilation between two polytopes P and P is given by Pi O Po a1 21 22 Pi V2 P3 4 11 In MPT this operation is implemented by means of the overloaded minus operator 112 Michal Kvasnica i i i i i i 15 i Ais a 05 o 05 1 15 Ais 1 a Polytopes Pi and P2 b Minkowski addition Pi Po Fig 4 6 Illustration of Minkowski addition gt gt Q P1 P2 Important to notice is that in general the Pontryagin difference and the Minkowski sum operations are not dual to each other i e P1 P2 P2 P Graphical interpretation of Pontryagin difference is shown in Figure 4 7 EM i i i i i a i s 1 5 D 05 1 15 s 1 a Polytopes P and P2 5 o 05 1 15 b Pontryagin difference Pi O P Fig 4 7 Illustration of Pontryagin difference 4 2 3 8 Set Difference The set difference between two polytopes P and 3 is the set P P2 x x P1 P2 4 12 and is implemented by the overloaded backslash operator gt gt Q P1 P2 4 Multi Parametric Toolbox 113 Note that the set difference of two convex sets can in general be non convex In such a case the output will be an array of polytopes An example of the set difference operation is shown in
238. ocess Wo is a by product resulting the logical simplification of matrix operations The first code line creates a temporary vector a result of the matrix vector operation Q11xo while the second finishes the task by evaluating the rest dsymv chu amp order amp one Q11 amp order X0 amp onei amp zero tempv2 amp onei WO ddot amp order xO amp onei tempv2 amp onei 1 In case the resulting vector will be Wo lt 0 the following code portion calculates the next by product a vector re used in later code portions v RT Qai20 S 5 12 where v denotes the vector result of this operation and is an element wise division This operation is carried out in two steps First a general matrix matrix multiplication saves the result of RT s into a temporary matrix Then the expression v is calculated by multiplying the result with the negative of the current state measurement and its elements divided by vector S if W0 0 1 dgemm cht chn amp order amp order amp order amp one R amp order Q21 amp order amp zero tempm amp order dgemv chn amp order amp order amp mone tempm amp order X0 amp onei amp zero W1 amp onei dediv korder W1 Sm vec Another partial result is calculated by evaluating a dot product of two vectors and adding Wo to the result W2 ddot amp order vec amp onei W1 amp onei WO where in case W2 gt 0 the perturbation vector can be directly calculated
239. od for a given H If we want to compare analyze alternative choices for H for example al ternative choices of individual measurements then the maximum gain rule is effective and gives considerable insight The maximum gain rule has also been used to find optimal H especially for the case where one wants to control individual measurements and Yu and Kariwala have devised efficient branch and bound methods for solving this problem 2 Measurements for Control Optimization and Estimation 69 In the maximum gain rule one considers the scaled gain matrix G HG from u to c To derive the maximum gain rule we return to the loss expression 1 J 27 a where z Ju us JI2G c cop c Cop H Fd n G HG Here c Copt may be viewed as the optimal or expected variation in the selected variables c Hy caused by disturbances and measurement noise The magnitude of c Cop HFd Hn is obtained by adding the magnitude of the contributions from JH F d and Hn and we assume in the following that c Copp W C where W is a diagonal matrix for the expected optimal variation optimal span in c and we assume that all Ile lt 1 are allowed e Copt translates into changes in the inputs u opt by the transformation u G c and to a loss through the matrix J Ma We want u Uopt small so we want the norm of G small More specifically the largest worst case value o
240. oe IE min K max 20 40 60 80 100 120 140 time s Fig 9 26 Real experimental results for FIo controller Prefilter Disturbance Observer Fig 9 27 FPrlo controller 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 313 hh ro Taah ee PY w ly Temperature 1 Ri ti Saturati DA Transport EM Plant a v F R Gain1 an 1O Interface Do rri 1 Transfer Font gt gu le Tisi Step Bici Y Gain2 Transfer Fcn2 Light P a intensity C3 x Mt i Wy K Tisi Fig 9 28 FPrlo controller Simulink model Pri MO Yo Setp e 0 01 Tf 2 3516 K0 17 4675 Td0 5 8065 IAEOmean 22 5849 6 T T T T T 55 4 5 TVOmean 0 00042767 Fig 9 29 Performance portrait FPrlo controller 9 2 4 Plo and FPIg Controllers Tasks Identify the parameters of filtered light channel Use PIo controller to compensate the delay of the filtered light channel Tune the controller using a performance portrait Make a disturbance by a LED voltage step during the experiments with the filtered light channel 314 Peter Tap k and Mikul Huba Pri MO yi Setp 20 01 Tf 4 2811 K0 14 1525 Td0 6 4865 IAE1mean 29 9608 6 T r T T
241. of measurement noise in the OP and PV time series However the sharp corners in the plot are now Clearly identifiable indicating the presence of stiction 1 Problems in Anti Windup and Controller Performance Monitoring 39 OP f i L L L i L L L L 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 Time 0 L L L L L L L L L 7000 7100 7200 7300 7400 7500 7600 7700 7800 7900 8000 Time Fig 1 37 OP and PV for data set 2 800 T 600 400 200 200 400 600 800 L L L L L L L L L 100 80 60 40 20 0 20 40 60 80 100 Fig 1 38 The OP to PV cross correlation function for data set 2 The routine bicoherence m returns a negative value for NGI and a posi tive value for NLI and therefore nonlinearity has not been detected for this PV time series either The routine g1stat m finds no clear indication of PV being non Gaussian the probability of false alarm being estimated to 0 8665 None of these routines can therefore detect stiction in the loop 40 Morten Hovd and Selvanathan Sivalingam 0 8 0 6 1 PV 0 2 0 4 4 0 6 1 1 l 1 L 1 5 1 0 5 0 0 5 1 15 Fig 1 39 OP PV plot for data set 2 0 8 0 6 0 4 0 2 PV k 30 Fig 1 40 OP PV plot for data set 2 with the PV data series shifted 30 samples 1 3 2 3 Data Set 3 The time plots of OP and PV for data set 3 are shown in Fig 1 41 The OP resembles the sawtooth sh
242. of this workshop In simple terms the problem is that there are some initial states for which there exists a feasible control move at the beginning e g gt gt x 5 4 gt gt u controller_explicit x ed but then the successor state Ax Bu becomes infeasible gt gt xp model A x model B u xp 2 0000 3 5000 gt gt u controller explicit xp MPT GETINPUT NO REGION FOUND FOR STATE x 2 3 5 160 Michal Kvasnica To see this graphically run gt gt x 5 4 gt gt simplot controller explicit struct x0 x You will see that the closed loop trajectory leaves the colored area which means that constraint satisfaction is not guaranteed for all time for this initial point On the other hand the state x 5 0 belongs to the invariant subset as shown by the existence of the following closed loop simulation also shown on the figure below gt gt x 5 0 gt gt simplot controller explicit struct x0 x Closed Loop Trajectory for initial state 5 0 Fig 4 27 Closed loop simulation illustrating that the state x 5 0 belongs to the invariant subset To isolate the invariant subset of the given explicit controller run gt gt inv_controller mpt_invariantSet controller_explicit Ok but what is the difference between the two controllers Specifically which part of controller_explicit was NOT invariant i e which sub set of regions of
243. oices for c and H more exactly by computing the norm Frobenius norm or max singular value of M J rn HY Kariwala and Cao 2009 have derived efficient branch and bound algorithms for finding the measurement choice optimal structured H that minimize either the norm of M as well as the scaled gain c G Although the computation times for minimizing the latter are somewhat lower the benefit is not sufficiently large to justify using the maximum gain rule provided we have the necessary information available for the first method 2 8 Indirect Control and Estimation of Primary Variable These two problems are very similar and can be written as a special case of the loss method involving the same matrices 2 Measurements for Control Optimization and Estimation 71 2 8 1 Indirect Control of y The objective is to keep the primary output y close to its setpoint so the cost function is J Yi Vis However y is not measured but we have to use some other measurements y Thus we want to achieve indirect control of y by keeping c Hy at a given setpoint To find the optimal full H using the loss method we need four pieces of information F G W a Wnv In our case the optimal sensitivity is F dy P dd dy dd y It may be obtained by simulations where we keep y constant for the various disturbances Instead of using simulations we may write y Gu Gad and then Hori et al 2005 Ju GiGi Jua
244. ol action Pol ni et al 2008 5 Implementation of MPC Techniques to Real Mechatronic Systems 217 5 3 4 1 Predictive Model The applied control strategy is based on the knowledge of the internal model IM of air path predicting the change of air flow through the exhaust pipe and consequently setting the profile of desired values of the objective function on the control horizon In this case we will consider the state space SS formulation of the system therefore it is necessary to express linear local ARX models in parameter varying realigned SS model z a py k 1 Aca p 6 a p E Bea py OUa py Kk Ths a f K Cla f t a f 5 26 The weighted parameters of a multi ARX models are displayed in matrices Aa f and Ba f for both subsystems This is a non minimal SS representation whose advantage is that no state observer is needed The fuel pulse width control is tracking the air mass changing on a prediction horizon from IM of the air path by changing the amount of injected fuel mass Due to tracking offset elimination the SS model of the fuel path 5 26 index f with its state space vector xf is written in augmented SS model form to incorporate the integral action amp g k 1 Ag Q ig k By b Aus k 5 27 rp k 1 _ Ash BrP x5 k uj k zi 0 1 Ph k Auy k ms p k y y k DyAug k 5 28 ms r k Cg Dg amp k Dp Aus k The prediction of the air mass ma on the p
245. ol Systems Technology 16 1 3 12 Zeman J Rohal Ilkiv B 2003 Robust min max model predictive control of linear systems with constraints IEEE International Conference on Industrial Technology pp 930 935 224 Gergely Takacs et al Zhai YJ Yu DL 2009 Neural network model based automotive engine air fuel ratio control and robustness evaluation Engineering Applications of Artificial Intelli gence 22 171U180 Zhai YJ Ding WenYu Hong YuGuo DLYu 2010 Robust air fuel ratio control with adaptive DRNN model and AD tuning Engineering Applications of Artificial In telligence 23 283 289 5 Implementation of MPC Techniques to Real Mechatronic Systems 225 Comments Remarks 226 Gergely Takacs et al Comments Remarks 5 Implementation of MPC Techniques to Real Mechatronic Systems 227 Comments Remarks 228 Gergely Takacs et al Comments Remarks Chapter 6 Laboratory Model of Coupled Tanks Vladimir Zilka and Mikul s Huba Abstract This chapter gives introduction into work with frequently used hydro system of coupled tanks The chapter explains analytical modelling for one and two level systems of coupled tanks including the experimental identification of system s parameters and sensor s calibration 6 1 Introduction To be able to use their knowledge in solving practical tasks for graduates it is important to develope skills in applying control theory related to linear and non linear systems to rea
246. olbox The control design was based on two main input objects the sysStruct structure which contains infor mation about the system dynamics and the probStruct structure which defined parameters of the MPC optimization problem 4 26 Although intuitive and easy to use the command line interface outlined in Section 4 5 1 only allows for basic MPC design In particular the MPC problem to be formulated and solved can only be adjusted by changing the prediction horizon or the penalty matrices in 4 26a In this section we de scribe a new addition to the Multi Parametric Toolbox which allows more advanced MPC setups to be formulated easily and with low human effort The new framework for advanced model predictive control utilizes the YALMIP package which is a versatile modeling language for construction of generic optimization problems in MATLAB The design procedure goes as 4 Multi Parametric Toolbox 137 follows First given the model and problem structures MPT formulates the basic MPC problem 4 26 using YALMIP symbolic decision variables by a single function call gt gt C obj V mpt_ownmpc sysStruct proStruct which returns constraints 4 26c 4 26e in the variable C and the objective function 4 26a in obj formulated using symbolic variables V x and V u These variables represent the prediction of systems states and inputs through the finite horizon N stored as cell arrays i e V x k denotes x 1 and V u k stand
247. ollers the input to the plant u is modified by a nonlinear function that can be denoted as the saturation function Uz u gt U2 gt 0 u t sat u t sat u qu U lt u lt U 7 10 U u lt U lt 0 In the classical period of control up to late 60s in the 20th century the constraints in the control were treated by a huge amount of papers Then in the subsequent decades in the main stream of the control theory 7 Constrained PID Control Tasks for Coupled Tanks Control 261 the problem of constraints practically disappeared Just a limited number of authors dealing with the anti windup design continued to investigate this important feature Today the problem of constraints is again in the focus of the research activities and it is hardly possible to give here just a brief overview of different approaches For the sake of brevity we will show here briefly one possible approach to controlling the coupled tanks combining constrained pole assignment con trol Huba and Bist k 1999 Huba 1999 Huba et al 1999 Huba 2001 2005 2006 2010 2011a c with extended exact linearization method that is based on transformation of the existing nonlinear system by 7 9 into the double integrator one and in controlling such double integrator system by the PD controller that fully considers the existing control saturation limits Let us start with considering the double integrator system dyt dt For y y w w being the reference inpu
248. ome way or other either by design or by implementation details A thorough detail is given on all three methods Although their imple mentation is connected with the vibration detail this section may be used as a guide to create a functional MPC controller for any application Not only the on line part of the algorithms is discussed in detail but the off line part as well The approach utilized in this chapter is down to Earth and simple lacking any sort of over complicated theoretical discussions Moreover espe cially in the case of the QP and MP based implementations if possible off the shelf and freely available solutions have been preferred 200 Gergely Takacs et al 5 3 Implementation of Predictive Control of Air Fuel Ratio in Spark Ignition Engine 5 3 1 Introduction A run of a spark ignition engine SI is highly dependent on the mixture of the sucked air and injected fuel present in the cylinder waiting to be ignited by the spark Incorrect ratio of this two components may lead to the poor engine power ineffective functionality of the catalytic converter resulting in higher level of emissions polluting the environment and in the extreme case this can lead to the engine stoppage Due to this reason it is crucial to keep the air fuel ratio AFR at the stoichiometric level which means that both the air and the fuel are completely combusted Due to above mentioned rea sons and all the time tightening emission standards the car p
249. on package ESPRESSO as well as with the HYSDEL modeling language The main factor which distinguishes this toolbox from other alternatives is the big emphasis on efficient formulation of the problems which are being solved This means that the toolbox provides implementation of novel control design and analysis algorithms but also offers the user an easy way to use them without the need to be an expert in the respective fields MPT aims at providing tools which can be used in the whole chain of the process of successful control design It allows users not only to design optimization based controllers but also to formally verify that they behave as desired investigate the behavior of the closed loop system and to post process the resulting feedback laws in order to simplify them without loosing prescribed design properties 4 1 1 Download and Installation MPT is freely available for download from the project web site at http control ee ethz ch mpt After download follow the installation instructions and add all MPT s sub directories to MATLAB path To start off check your MPT installation by running the following command gt gt mpt init 4 Multi Parametric Toolbox 103 which should provide the output similar to this one looking for available solvers MPT toolbox 2 6 3 initialized LP solver CDD Criss Cross QP solver quadprog MILP solver GLPKCC MIQP solver YALMIP Vertex enumeration CDD Any questions sugges
250. one should note that there is little additional complexity involved in also having limited range of actuation of the valve e g to the range 1 A simple model of the plant is given as E TF u s ku d s 1 11 p s 100s F1 20s 1 30 Morten Hovd and Selvanathan Sivalingam where f is the flowrate p is the pressure u is the valve position and d is the disturbance Note that these variables as always when using transfer function models are expressed in deviation variables and thus both negative valve positions pressures and flowrates do make sense 1 2 3 Assignment 1 Model implementation Implement the a model of the plant in Simulink The two controllers may both be given the setpoint zero The disturbance should be modelled as a square wave with period 500 2 Controller tuning Tune each of the PI controllers without taking saturation or the selector into account Any tuning methodology could be used this should not be very difficult anyway since the plant in each of the loops is Strictly Positive Real However completely unrealistic tunings should be avoided i e for asymptotically stable plants like in this example the closed loop time constant should not be orders of magnitude faster than the open loop time constant 3 Simulation without anti windup Implement the resulting PI controllers in Simulink without accounting for windup and simulate with the disturbance active and both controllers op
251. onses for the whole scope of liquid s levels The obtained information will depend on the choice of concrete operating point in which the step will be measured For the PI controller design having the best attributes in the most ample areas it may seem to be useful to choose the operating point approximately in the middle of the working area In the surroundings of chosen operating point we will approximate our non linear system 6 1 by the first order linear model This will be obtained by the step response method from step responses measured firstly by setting up by appropriate input pump voltage the liquid s level approximatelly to the the middle of the tank hight and here to stabilize it for some time optimally few centimeters under the middle When the liquid s level will be stabilized 7 Constrained PID Control Tasks for Coupled Tanks Control 251 we will switch the pump input to higher voltage by approximately 1V step and wait again for the liquid s level to stabilize it should stabilize few cen timeters over the middle From the measured step response we will find out by constructing tangent at the origin the approximate value of time constant Ti and the value of the approximative plant gain K1 Ay Au So in a close surroundings of the operating point corresponding to the input voltage about uj 4V the system may be approximated by linear transfer function K F s Tepi Then in the previous paragraph mentioned P co
252. ontroler cessent REI ESCRITO R e dhs 302 9 2 3 Filtered Predictive Ig Controller 308 9 2 44 PIg and FPIp Controllers 0 0 eee 313 9025 Ply ecoucollir cos REDIMERE SERE 319 9 2 6 Filtered Smith Predictor FSP 321 eferem ed 12lseseedeem quee O ree RR MER C RA PPP PPPSIS 327 List of Contributors Pavol Bistak Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail pavol bistak stuba sk Jozef Csambal Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail jozef csambal stuba sk Marek Honek Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail marek honek stuba sk Morten Hovd Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim Norway e mail morten hovd itk ntnu no Mikul s Huba Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava e mail mikulas hubaAstuba sk Johannes J schke Department of Chemical Engineering Norwegian University of Science and Technology Trondheim Norway e mail jaschke chemeng ntnu no Matus Kopacka Institute o
253. or all possible initial conditions one would call gt gt ctrl mpt control sysStruct probStruct explicit For the double integrator example and the MPC setup as above the MPT toolbox constructed the explicit representation of Uy as a function of x t 134 Michal Kvasnica cf 4 27 in just 0 9 seconds generating 50 regions If instead the aim is to design an on line MPC controller the calling syntax is gt gt ctrl mpt_control sysStruct probStruct online In both cases the result is stored in the ctrl variable which represents an MPC controller object The object oriented philosophy of the Multi Parametric Toolbox then allows to process the ctrl object in a number of ways The basic one is to ask the controller to provide the optimal value of the control action uo for a given initial condition x t In MPT the controller object behaves as a function which allows for the following intuitive syntax gt gt x 4 1 measured state gt gt u ctrl x evaluation of the controller If the ctrl object denotes an explicit MPC controller the optimal control action u is calculated by traversing the lookup table For the example above the MPT implementation of Algorithm 1 only takes 2 miliseconds to execute If on the other hand the object represents an on line MPC controller the optimal control input is obtained by solving 4 26 as an optimization problem in this case an LP with x being its initial
254. osed loop simulation More interesting is the average execution time of one simulation step Exercise 4 16 Probably the best way to understand an explicit MPC con troller is to visualize it We just quickly remind that such an controller consists of several polytopic regions with associated affine feedback laws of the form u Kjx Li We start by inspecting the controller s regions close all plot controller_explicit This figure tells us two important properties of the underlying MPC formu lation First we see how many regions there are in total Quite intuitively the higher number of regions the more complex the MPC problem is and the more computationally demanding is its implementation in real time But even more important is the portion of the colored area compared to the white space surrounding it Any point from within of the colored patches is a feasible initial point i e one for which there exists a solution satisfying constraints for the whole prediction horizon N On the other hand any point from the white areas is infeasible Again this information is quite important for real time implementation as you can easily reject bad initial conditions Let s verify these statements by running u controller_explicit x0 once for x0 4 1 and the other time for x0 4 2 Do the re sults support our claim about feasibility What would happen if we use u controller online x0 instead Can you give an answer ev
255. owing FlIo controller tuning Ko max K Ty e max T1 9 22 Make several setpoint steps make a LED voltage step as well when the system output is settled Do not forget to keep more time between setpoint steps when using Io controller It is useful to make a simulation first to deter mine setpoint step time interval sufficient to settle the system output between them The experiments results are shown inf Figs 9 38 9 39 9 40 Compare the overshooting for setpoint step and disturbance step The detailed view on these transients is shown in Figs 9 41 9 42 9 43 and 9 44 Questions e What type of plant are PI and FPIp controllers suitable to control for e Analyze the amount of overshooting in the experiments e Which controller performed better in disturbance rejection 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 319 PPI u TV VSM PPI TV 2 TI 2 T 1 8r 1 8 1 67 1 6 1 4 1 47 o 1 o 1 2 1 2 i E FE n l 0 8 0 8 t 1 0 6 0 67 al 0 4 0 47 Q4 4 S 0 2 ANN I N 2 A 0 5 1 1 5 2 0 5 1 1 5 2 gt WT WT Fig 9 37 Performance portrait of the FPIo controller e Was there any advantage of using FIo over FPIg e g in noise sensitivity 9 2 5 PI controller Tasks e Control the filtered light channel output by PI controller e Analyze the control quality for various controller tunings e Compare results with control per
256. particular identification results of the air and fuel paths are shown in Fig 5 17 and Fig 5 18 respectively The figures show static characteristics of both engine paths defining the relative amount of air and fuel flowing through the engine as a function of throttle valve opening angle or the injector opening time in the case of fuel injectors at specific engine revolutions The step responses of both engine subsystems for various engine revolutions are depicted as well The identified model parameters are situated in Tab 5 2 and 5 3 These are at the same time the final results of the identification and they used for the prediction of future states of the system in the controller 5 Implementation of MPC Techniques to Real Mechatronic Systems 211 21 5 T 21 20 57 Throttle valve position mN o T 18 5 l 200 205 210 Time s a Excitation PRBS signal 215 Lambda measurement model fit 74 61 200 205 210 Time s 215 b Response curve of the simulated system to the PRBS excitation Fig 5 16 ARX identification at a particular working point air path 212 Gergely Takacs et al Relative amount of air mg cylinder 1500 rpm 100 vy 2000 rpm O 2500 rpm 80 i i 18 21 22 23 24 Throttle valve position deg a Static characteristics of the air
257. path 15 T T T T 4 TL Mear aP ieme CES NE SEM lei nn LL ee UE 10 mi g d A p lI EJ B a 3 B 3 i OPI 13 44mg cylinder 1500rpm OP2 09 84mg cylinder 1500rpm P3 07 12mg cylinder 1500rpm OP4 12 52mg cylinder 2000rpm OP5 11 39mg cylinder 2000rpm OP6 09 41mg cylinder 2000rpm OP7 10 50mg cylinder 2500rpm OP8 10 41mg cylinder 2500rpm P9 09 17mg cylinder 2500rpm 0 1 1 n n 1 1 0 0 5 1 5 2 2 5 3 3 5 Time s Fig 5 17 Results of the air path identification b Response curve of the air path in the operating points 5 Implementation of MPC Techniques to Real Mechatronic Systems O 1500 rpm D 2000 rpm O 2500 rpm Relative amount of fuel mg cylinder 5 5 5 6 6 5 v 75 8 8 5 9 Injector opening time ms a Static characteristics of the fuel path 1 6 T T T T T 14r 12 1 k 0 8L OP1 1 54g s 1500rpm l OP2 1 36g s 1500rpm 0 6 OP3 1 32g s 1500rpm i OP4 1 42g s 2000rpm OP5 1 42g s 2000rpm 0 4 OP6 1 29g s 2000rpm an OP7 1 12g s 2500rpm 0 2 OP8 1 21g s 2500rpm 1 OP9 1 15g s 2500rpm 0 L L 1 l L 0 1 2 3 4 5 Time s b Response curve of the fuel path in the operating points Fig 5 18 Results of the fuel path identification 213 Table 5 2 Air path model parameters
258. pects of Controller Performance Monitoring CPM an area that has seen rapid developments over the last two decades CPM techniques are important in particular for engineers work ing with large scale plants with a large number of control loops Such plants are often found in the chemical processing industries 1 1 Anti Windup Control of a Distillation Column with Input Constraints This assignment lets the student apply three different controller design meth ods to the control of a 2 x 2 distillation column model Subsequently the con troller implementations should be modified to account for input constraints Morten Hovd Department of Engineering Cybernetics Norwegian University of Science and Tech nology e mail morten hovd itk ntnu no Selvanathan Sivalingam Department of Engineering Cybernetics Norwegian University of Science and Tech nology e mail morten hovd itk ntnu no 2 Morten Hovd and Selvanathan Sivalingam 1 1 1 Notation Consider a linear continuous time state space model given by t Ax Bu y Cz Du where z is the state vector u is the input vector and y is the output vector and A B C D are matrices of appropriate dimension The corresponding transfer function model is given by G s C sI A B D The following equivalent shorthand notation is adopted from Skogestad and Postlethwaite 2005 and will be used when convenient eo 1 1 2 Some Background Material on Anti windup In virtually
259. posuni betal ci beta12 0 beta2 0 Qmin 0 Qmax q_max1 7 Constrained PID Control Tasks for Coupled Tanks Control 255 w11 0 10 setpoint values for tank 1 level 3 steps w12 0 20 w13 0 10 Td 0 996 4 dead time Ks 1 81 plant gain h10 w11 operating point choise a betai 2 sqrt hi0 posun linearization around operating point alfae 0 2 directly given closed loop pole alfae exp 1 a Td Td a closed loop pole based on dead time estimate alfae 1 a Td exp 1 ax Td Td exp 1 a Td Kr a alfae Ks P controller gain Tfil 1 alfae 4 obsever filter time constant ci beta1 2x sgrt h10 posun tf Ks 1 c1 inputdelay Td plant model around the operating point c a Gp sim C PIireal2 zilka mdl str datum num2str floor clock str datum strrep str datum 7 yuu file name sprintf reg_PI1 s str datum eval save file name 7 3 Linearization around a fixed operating point Analytical linearization around a fixed operating point shows one of the sim plest ways of designing controller for the one level system Linearized model may be determined directly from the system s equation 6 1 For a chosen working point and its closest environment the non linear system will behave as linear and therefore we can use this as the basic possibility for deriving linear controllers e g the PI one While making this l
260. prediction model and possibly an initial state for Kalman filtering or simulation purposes In this example the model is loaded from a saved system identification file load n4s2A A n4s2 A B n4s2 B C n4s2 C nx length A X1 1 zeros 1 nx Penalization for the inputs and states needs to be stated as well The input penalization can be determined by direct experimentation with the algorithm or simply evaluating different linear guadratic controllers in simulation and determining a good balance between controller performance and aggressive ness In this case the input penalty has been found by using an LO controller with the settings R le 4 and Q C x C while balancing the input somewhat above the constraints R 1e 4 Q C C 5 2 3 3 Prediction Matrices It is possible to pass the information to a stand alone function This custom function uses the system model penalty matrices the constraints a hori zon and possibly a system order information Its output are the properly formulated prediction matrices and possibly the re formulated constraints H F G Ac bO Bx Ki Nc predmodelqp A B C R Q umax nc nx Now let us begin with examining what such a function may do in order to generate the prediction matrices and cost function prediction matrices for the on line run As in most vibration damping applications this implementation assumes a symmetric constraint on the input ul uh This is followed by calcu
261. put for the current process measure ment 1 Provided appropriate scaling of variables is used the auctioneering control structure may be a better alternative to using selective control with the selection based on controller inputs 1 Problems in Anti Windup and Controller Performance Monitoring 29 Flow Pressure controller Low select controller mE C WW Disturbance Flow Pressure transmitter transmitter X Valve Fig 1 25 Schematic of simple plant with two controllers and a selector Note that the applied control signal is fed back to the controllers For PI controllers the simple PI anti windup scheme in Fig 1 1 may be used 1 2 2 The Control Problem in the Assignment Consider the small plant depicted in Fig 1 25 In normal operation the flow controller should be active but the pressure controller should take over if the downstream pressure becomes too high This is achieved by selecting the lower of the two controller outputs and applying that as the valve opening A downstream disturbance affects both pressure and flow and can cause the pressure control to take priority over the flow control In Fig 1 25 solid arrows indicate process flows in pipelines dashed lines indicate signal transmission and the dash dotted lines indicate feedback from the signal actually applied to the valve back to the controllers for avoiding windup In this assignment we will only study the effects of the selector but
262. r numerical reasons or finding improved bounds for cases with constraints on H see below Once we have found an optimal H using any of the Theorems above we can use the non uniqueness of optimal H to find another H1 DH with desired property or structure For example one can select D such that G HG I Alternatively one can specify selected elements in H1 for example H K In the latter case write H H H and assume H is full rank then H K DH DH and we find dy Hy and K Hj 1H Required information To find the optimal full H using the loss method we need four pieces of in formation First for the measurements we need the optimal disturbance sen sitivity F and input sensitivity GY These are obtained from the model Next we must specify the disturbance magnitudes W4 and the noise magni tudes Wx The matrix Juu is not needed except when there are structural constraints as discussed in the next section Note that changes disturbances in the prices parameters in the cost function do not change the optimal H based on the assumption that we still have a quadratic optimization problem with constant weights However as given in 2 22 the setpoint for c needs to be adjusted c HF pp and for this we need for the measurements the optimal price sensitivity Fp which can be obtained from the model 2 6 Structural Constraints on H In the previous section we considered the normal c
263. rames a combined solid and wireframe plot cannot be achieved by a single command therefore we split it in two steps opts struct wire 1 linewidth 3 plot Q4 y hold on plot Q r Q3 m opts hold off i fi Ll fi i i i 1 0 8 0 6 04 0 2 0 0 2 04 0 6 0 8 1 04 L y Fig 4 9 Intersection of polytopes Q3 and Q Now practice Plot the intersection of Q Q2 and Q3 Hint you can use amp in a chain i e Q amp Q2 amp Q3 Exercise 4 4 The set difference operation between polytopes P and Q is defined as PVO x x P x In MPT this is achieved by using the V operator D Q3 Q plot D g hold on plot Q r Q3 m opts hold off 4 Multi Parametric Toolbox 117 Notice that the set difference between two convex sets is not necessarily con vex In fact the example above illustrates such case and we have D represented as an array of two polytopes D D Polytope array 2 polytopes in 2D The number of elements of D can be determined by length D and its com ponents can be accessed using standard MATLAB indexing e g Di D 1 D2 D 2 D3 D end same as D 2 You can also create a polytope array on your own e g polyarray D 1 Q D 2 Q3 Now practice Plot the set difference between Q and Q3 notice that ordering makes a difference How many polytopes define the difference Exercise 4 5 The Minkowski addition operation
264. rator and P controller will not assure the necessary quality of control That s why we propose to design the robust controllers having sufficient robustness against influence of disturbances un certainties and measurement noise At the same time these controllers should also respect the available system s limits Such proposals for one level system can be found for example Huba 2003 Pan et al 2005 Hal s 2006 Almu tairi and Zribi 2006 Zilka et al 2009 Further development of the above mentioned procedures as e g the disturbance decoupling problem of coupled tanks was studied in Zilka and Halds 2010 and non interacting control of coupled tanks in Zilka and Hal s 2011 Acknowledgements The work has been partially supported by the Slovak Grant Agency grants No VG 1 0656 09 and VG 1 0369 10 It was also supported by a grant No NIL I 007 d from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism This project is also co financed from the state budget of the Slovak Republic References Almutairi NB Zribi M 2006 Sliding mode control of coupled tanks Mechatronics 16 7 427 441 Conte G Moog C Perdon A 2007 Algebraic Methods for Nonlinear Control Systems Theory and Applications 2nd edn Communications and Control Engineering Springer Verlag London Hal s M 2006 Quotients of noncommutative polynomials in nonlinear control sys tems In 18th European
265. re Description The beam is made of aluminium marked by the designation EN AW 1050A Its dimensions are 550x40x3 mm The four piezoelectric patches bonded to the beam surface are identical of the make MIDE QP16 Wafers marked with PZT1 2 in Fig 5 1 are connected counter phase and are used in actuator mode for control The rest of the patches are short circuited and are not utilized in the upcoming implementation examples Capacitive LASER CPU LASER Mounting proximity sensor and filtering Triangulation base FEAR and clamp Control PZT 1 amp 2 xPC Host Fig 5 1 Simplified schematics of the control and measuring chain applied to the vibrating clamped cantilever beam Piezoelectric actuators are connected to 20V V gain MID EL 1225 power amplifiers which in turn receive low level analogue voltage input from a cable terminal fed to a high speed National Instruments PCI 6030E measuring card This measuring card is a part of a computer running Matlab xPC Target real time control environment Development of the control algorithms and measuring tasks is carried out using a host computer running Matlab Simulink and is connected to the target computer via Ethernet Beam tip deflection is measured using a KEYENCE LK G 82 industrial LASER triangulation system indicated as Laser triangulation on the figure Digital low band pass filtering and signal processing is realized in a propri etary unit which outputs an analogue voltage signal to t
266. rediction horizon JV is solely dependent on the throttle position tr and is computed as Malk Tuld ra k 246 1 k 1 5 29 where the x denotes the state space vector of the air path Due to the unprecise modeling IM strategy biased predictions of the air mass future trajectory and consequently biased fuel mass might occur This error can be compensated by the term L mp k ms k in the fuel mass prediction equation m 218 Gergely Takacs et al my p k THOE Oz Ask 1 E L my k ms s k The matrices of free response Ia I s and forced response Na Qf are computed from models 5 26 and 5 27 respectively Maciejowski 2000 Since there is only A k measurable in equation 5 15 the value of ma k needs to be substituted using IM of the air path then l Msalk iri Um XE 5 31 The estimate p k is used to compensate for possible bias errors of pre dicted m p k in 5 30 5 3 4 2 Cost Function The main part of a cost function is a sum of deviations of predicted outputs from a set point i e a weight of a future control deviations Its another elements are a penalization of control increments r and a p penalizing a deviation between a predicted and desired end state To eliminate a the steady state control error the criterion 5 32 is defined through the control increments That guarantees in a steady state that Jx 0 also for the system with no integration properties what is th
267. ries relative to each other Comment on the result Outi i d Vary P s Vonka gt ues Integrator To Workspace Intstiction procoum 4 34 To Workspaced onou 00 Sawtooth signal PID j O contout 1 To Workspace rence Fig 1 32 Simulink model for simulating simple loop with sticky valve 36 Morten Hovd and Selvanathan Sivalingam intinput Out To Workspace2 1 C1 CO gt c P velocity Ini Integrator To Workspace p Sign MATLAB Function Fs MATLAB Fen Fs Fe Fc P posinput dout To Workspace3 To Workspace1 Fig 1 33 Simulink submodel for calculating friction force and valve velocity 4 Use glstat m from the hosa toolbox and or bicoherence m to detect stiction Comment on the result 1 3 2 Solution The data series contain an initial setpoint change from 0 to 1 It may therefore be advisable to select parts of the data series after the PV has come close to the setpoint The length of the selected data series should contain multiple oscillations You may also use the Matlab function detrend to remove any linear trends in the data 1 3 2 1 Data Set 1 The time plots of OP an
268. rive the solution to the above problem It has been previously been known as the exact local method but we will here call it the loss method However first we need some preliminaries 2 3 1 Expression for Uopt d We want to find the optimal input u for a given disturbance d Expanding the gradient J around the nominal point u d 0 0 gives Ju I Jinu t Jiad M 4 J53 H Ww where J J u d 0 because the nominal point is assumed to be optimal To remain optimal u Uop d we must have J 0 and we derive topt J aI uad 2 6 where we have dropped the superscript either because we consider small deviations or because we assume that the problem is truly quadratic 2 3 2 Expression for J around Uop d Consider a given disturbance d Then expanding the cost J around a moving Uopt d gives 1 J u d J uopt d d Ju u opt zlu uopt J uu u uopt 2 7 Jopt d 0 Here J 0 since we are expanding around an optimal point so we get the following expression for the loss 1 1 L u d J u d Jon d 5272 5 elle 2 8 where z JY u uopr d 2 9 58 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke 2 3 3 Expression for J around Moving Uop d A similar expansion but now of the gradient gives Ju Ju opt 4 Juu u Uopt Juu u Uopt 2 10 0 Combining this with 2 9 gives z J expression for the loss 2J u
269. rmulas see earlier Step 2 Find the optimal H by solving the convex optimization in 2 26 with Y X opt and the constraint HG G1 4 Loss method step 1A Yall Y1 X step 1B u s vl svd Y1 Yalli Yall v r1 ci size Yall r2 c2 size Y1 ny r2 G1 Yalli 1 ny 1 ny Gy Yalli ny i ri 1 ny Xopt Yalli ny ti ri ny i c1 step 2 4Hopt pinv Xopt Xopt Gy Zanalytical expression Hopt loss soc_avg Gy Xopt D Hopt Gy inv G1 Hoptloss inv D Hopt 74 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Comments e Alternatively in step 2 provided Y Y has full rank we may use the analytical expression H7 Y Y 7 1G in 2 25 and then rescale H to get a new H DH which satisfies H1G G1 which gives H Gi HG H If Y Y does not have full tank one may use some pseudo inverse similar to PCR This adds degrees of freedom to the method It has not been tested out but some preliminary results are promising e The method seems a bit similar to PLS in that we use the data for y to affect the x data we get Xo from X by using the SVD of Y i and also use G when minimizing H X opt 2 9 2 1 Modification for Case with Too Few Experiments e g Spectroscopic Data If we start with a model then the data matrix Y FW 4 W ns is a fat matrix this is clear since the noise magnitude matrix W ny is a square matrix usually diagonal Thus there exis
270. roducers are improving the control of the air fuel ratio Traditional control of air fuel ratio is based on a feedforward control using predefined tables determining how much fuel has to be injected into a cylin der based on the information from the mass air flow meter This fuel amount is subsequently corrected using the information from the lambda probe so the stoichiometric mixture can be reached Due to a lambda probe position at the engine exhaust a delay arises causing an improper feedback cor rection at the unstable engine regimes like acceleration or deceleration On the other side this kind of control guarantees stability and robustness at all conditions and therefore is still preferred by car producers despite its disad vantages in control The academic field have started to publish other kinds of air fuel control mostly model based ones The model based approaches are bringing good quality of control but are also more sensitive to the model precision and is sues with stability and robustness appear A survey through popular mean value engine modeling is described in Bengtsson et al 2007 This analytical way of engine modeling is very clear but requires exact knowledge of the sys tem and the model error has to be taken into account explicitly Other ways of a model acquisition are based on the experimental identification black box modeling Works of Zhai et al 2010 Zhai and Yu 2009 and Hou 2007 are specialized in
271. rs in the absence of saturation 16 Morten Hovd and Selvanathan Sivalingam Output 1 2 T L L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Output 2 2 T och P LL 4 d k map 2 L L L i L i L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 13 Responses with PI controllers with saturation in the inputs Output 1 2 T L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Output 2 2 T L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 14 Responses with decoupler in the absence of saturation this case remains stable although the responses leave the window displayed in the figure The corresponding response with the inverted decoupler are shown in Fig 1 16 We see that the performance is much better than with the stan dard decoupler although there is still significant interactions between the outputs These responses still suffer from the absence of anti windup for the PI controllers used for the individual loops 1 Problems in Anti Windup and Controller Performance Monitoring 17 Output 1 2 T T T T 0 2 P el 8 L L L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Output 2 2 T T T T aL O m 4 ims ac 2l A L L L L L L L L L 0 50 100 150 200 250 300 350 400 450 500 Fig 1 15 Responses with the standard decoupler with saturation in the inputs Output 1 2
272. ruct xmax u ctrl 0 12 5 u NaN Here the NaN output indicates that the MPC problem was infeasible for a given initial condition In fact the second element of z 0 violates the second state constraint 0 12 5 lt sysStruct umax 3 Plot the closed loop evolution of system states and control inputs for 10 steps starting from the initial condition z 0 7 closed loop simulation x 5 10 initial state X xX store for closed loop states U store for closed loop inputs for k 1 5 u ctrl x obtain optimal control action x mpt simSys sysStruct x u 4 simulate the system X X x WU ul X xX plot the results 164 Michal Kvasnica tx O size X 1 1 figure plot tx X tu O size U 1 1 figure stairs tu U Solution 4 14 for Exercise 4 14 By enlarging the input penalty the regulation process is slowed down i e it takes longer for the states to reach the zero levels Conversely by enlarging the probStruct Q penalty the controller will react more aggressively Solution 4 15 for Exercise 4 15 clear prediction sysStruct for the double integrator sysStruct A 1 1 0 1 sysStruct B 1 0 5 sysStruct C 1 0 sysStruct D 0 4 constraints sysStruct umin 1 sysStruct umax 1 sysStruct xmin 5 5 sysStruct xmax 5 5 objective function probStruct N 5 probStruct norm 2 probStruct Q eye 2 probStruct R
273. ructor returns a polytope object which can be processed in multiple ways Let us start by simple plots figure plot TH r figure plot TV g Fig 4 1 Plot of polytopes TH left and TV right The output of these commands is shown in Figure 4 1 It appears that the two polytopes are equal But how to check this in an automated fashion Easily use the standard equivalence MATLAB operator ans 1 See help polytope for more information 4 Multi Parametric Toolbox 105 1 Other supported operators include P lt Q which checks whether the poly tope P is a subset of Q and P gt Q which checks whether P is a superset of Q Another frequent operation is to check whether a given point is contained in the polytope The isinside function provides a YES NO answer to this question x 0 5 0 5 is this point contained in our first triangle isinside TH x ans 1 of course the function also works for V polytopes isinside TV x ans now try a point outside the triangle x 10 10 isinside TV x ans 0 Regardless of whether the polytope was constructed using inequalities or vertices it is always possible to extract both the H as well as the V representation To compute the extremal vertices of a given polytope use the extreme function see also help polytope extreme E extreme TH ans 0 1 1 0 1 0 To extract the H representation use the double com
274. s The C program is contained within this S Function which has been created using the S Function Builder In addition to the specified parameters includ ing their data type input and output ports are declared Here the input is the estimated state and output the scalar controller variable Discrete sampling 190 Gergely Takacs et al Estimated PCI 6030E Defl stately og Kalman eflection lNational Instr Filter Analog Input Kalman Filter LASER PCI 6030E AD Deflection gm File Scope Id 2 TET xPC Target Data Logginc TET 1 PCI 6030E lNational Instr I x0 enrmpc z nput Analog Output iterp Unit Delay amplifier PCI 6030E DA Compensation NRMPC Colntroller Fig 5 5 Block scheme of the NRMPC controller in Simulink for the active vibration cancellation application time depends on the application The NRMPC block provides voltage output directly therefore have to be compensated by the amplifier gain 5 2 4 9 Custom C functions In addition to the main C code and BLAS functions there are two additional custom C functions One performs element wise division of two vectors while the other one is an algorithm specific operation scaling the elements of a vector by a scalar value and subtracting it from 1 Both of these functions have to be declared similarly to the BLAS routines as externals Function declarations are as fol
275. s Some type of scaling will then often be necessary in order to compare different types of physical quantities e g comparing flowrates and pressures e If the controllers contain integral action a severe problem that is similar to reset windup can occur unless special precautions are taken The con trollers that are not selected should be reset for normal PID controller this is done by adjusting the value of the controller integral such that for the present controller measurement the presently selected manipulated variable value is obtained Commonly used terms for this type of function ality are putting the inactive controllers in tracking mode or using a feedback relay This functionality should be implemented with some care as faulty implementations which permanently lock the inactive controllers are known to have been used On a digital control system the controllers should do the following for each sample interval 1 Read in the process measurement 2 Calculate new controller output 3 The selector now selects the controller output to be implemented on the manipulated variable 4 The controllers read in the implemented manipulated variable value 5 If the implemented manipulated variable value is different from the con troller output the internal variables in the controller typically the integral value should be adjusted to obtain the currently implemented manipu lated variable value as controller out
276. s The problem setup creating prediction and cost prediction matrices is carried out in the Mat lab scripting environment The final problem parameters are passed onto the Simulink interface of the qpOASES solver Stability in the traditional QPMPC formulation is guaranteed through suitably formulated state feedback and terminal cost function matrices The deployment of dual mode predictions is the part of the formulation as well the first mode considers n free control moves while the second mode assumes the LQ control law Chen and Allg ver 1998 Mayne et al 2000a Feasibility of process constraints is ensured beyond the prediction horizon by the inclusion of a constraint checking horizon First let us begin with the setup of the problem For this the code shall be implemented in the Matlab m script language 5 2 3 2 Setup First it is required to specify certain parameters for the QP controller Some of these parameters are the sampling period T s if it is an off line simulation a stop time is required as well and it is also essential to state the prediction horizon nc In case symmetric bounds on the input are assumed the con straints are set using umaz This is basically the polarization voltage of the piezoelectric transducers Ts 0 01 T 0 5 nc 70 run T Ts 178 Gergely Takacs et al umax 120 where run is the runtime in case an off line simulation is needed In the next step it is required to load and specify a
277. s 2 1 and 2 4 do give the nullspace method as a special case Comment In general with measurement noise included or with few mea surements so ny lt Nu Na it is not possible to make HY zero Explicit Expression for H for Nullspace Method The following explicit expression applies for H Alstad and Skogestad 2007 H J uu Jua G 2 29 Proof Here is a proof which is much simpler than that given in Alstad and Skogestad 2007 Want to find c Hy with zero loss 68 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke 1 Measurement relationship y eu H Inverting this H G ly 2 30 2 Optimality condition NCO Ju 0 2 31 3 First order expansion of gradient Ju Fut Teste Had Fou Fe where we use J 0 4 2 30 and 2 31 then give Juu JualG y 0 and it follows that H J Jual G O 2 7 2 No Disturbances The case with no disturbances has limited practical significance but is nev ertheless and interesting limiting cases We assume there are no disturbances W 4 0 and we scale the measure ments y so that they all have unity measurement noise W y I From the analytical expression 2 25 we then have that an optimal solution is HT G 2 32 This gives the important insight that we prefer sensitive measurements 2 7 3 An Approximate Analysis Method for the General Case Maximum Gain Rule The maximum gain rule is an approximate analysis meth
278. s added to the control setup an explicit MPC controller can be calculated by ctrl mpt ownmpc sysStruct probStruct C obj V as a lookup table Re gions of the table and a PWA representation of the optimal feedback law are depicted in Figure 4 22 Notice that the implies operator only works one way i e if x t is in the box delimited by Ha t K then 0 5 uo 0 5 will hold To restrict the control action to stay within of these limits if and only if x t is in the box the iff operator can be used gt gt C C iff H V x 1 lt K 0 5 lt V uf 1 lt 0 5 The lookup table corresponding to such a constraint modification is visualized in Figure 4 23 Controller partition with 111 regions Value of the control action U over 111 regions 5 lt A 3 2 1 0 1 2 3 4 5 a Regions of the lookup table con b PWA function uo x t Notice sisting of 111 regions that the control action stays within the interval 0 5 0 5 if 1 lt x t lt 1 Fig 4 22 Visualization of the lookup table for the control setup with the one way logic implication constraint 1 lt a t lt 1 gt 0 5 u lt 0 5 4 Multi Parametric Toolbox 141 Controller partition with 115 regions Value of the control action U over 115 regions 1 2 3 4 5 a Regions of the lookup table con b PWA function uo x t Notice sisting of 115 regions that the control action stays within the interval 0 5 0 5 if an
279. s for u for some index k Subsequently the user can modify the MPC problem by adding or custom constraints and or by changing the objective function To illustrate this con cept consider again the double integrator example of the previous section The closed loop profile of control actions shown in the lower left of Figure 4 20 reveals an abrupt change of the control signal from u 0 8 to u 0 at time instant t 2 Such a large change can be mitigated for instance by adding a constraint which only allows the control signal to change by a certain quantity say by 0 5 from step to step This can be achieved by adding the constraints 0 5 uk ug441 lt 0 5 to the MPC setup 4 26 In MPT this is achieved by using a standard YALMIP syntax gt gt C C 0 5 lt V u k V ufk 1 lt 0 5 with k 1 N More examples of constraint modification are presented in the seguel Once completed modified constraints and performance objective are passed back to MPT which converts them into a valid controller object gt gt ctrl mpt ownmpc sysStruct probStruct C obj V type If the object should represent an on line MPC controller then type online If type explicit then an explicit representation of the solution to the optimization problem represented by constraints C and objective obj is pre computed for all possible initial conditions and stored as a lookup table In both cases the ctr1 varia
280. s have been estimated from the data acquired from the exhaust gas oxygen sensor and an air flow sensor The identification has been designed so that the dynamics of the air path and fuel path stayed uncoupled Dynamics of both paths were measured indi rectly The first experiment started at the stoichiometric value of A4 in the opera tion point To excite the air path dynamics the throttle valve position was oscillating around its steady position according to a pseudo random binary signal PRBS Fig 5 16 while the fuel injectors were delivering constant fuel mass Mf e The change in Aa value has been recorded During the experiment the engine had been braked at constant revolutions The PRBS signal is a signal changing its value binary that means from one level to another with a pseudo randomly changing period Fig 16 a This period has to be chosen with respect to the identified system so that for even smallest change in PRBS a clear system response can be noticed This condition has to be ensured to excite the entire system dynamics and con sequently to perform a relevant system identification The Fig 16 b shows an example of a correctly identified system with a good fit compared to the original data system output The identification of the fuel path dynamics has been done similarly but with the fixed throttle valve delivering a constant air mass Ma e The PRBS is vary ing the fuel injectors opening time In both experim
281. s valve12 266 Vladim r ilka and Mikul Huba load basics pumpl T 0 25 sampling period S1 0 001 4 tank area posun posun2 offset 2 mic 0 12 m2c 0 12 beta1 0 beta12 b12 beta2 c2 valves flow coeff Qimin 0 Qimax q maxi w11 0 05 w12 0 10 w13 0 05 setpoint values for tank 2 level 3 steps D 0 996 considered dead time c 1 alfae 0 321 c D closed loop pole choice c gt 1 is used for PID2 controller tuning Tfil 1 alfae Tfd Tfi1 10 4 Time constants for Disturbance Observer filter and derivative filter alfal alfae alfa2 alfal tuning for controller with real poles sim PD2_REAL_zilka mdl str datum num2str floor clock str datum strrep str datum j file name sprintf reg_PD2 s str datum eval save file name Finally also PD2 algorithm PD2 m function out PD2 dr yr w him 4 constrained pole assignment controller for double integrator plant dr dh2 dt estimated according to the first state equation 7 Constrained PID Control Tasks for Coupled Tanks Control 267 yr h2 measured level in tank 2 4 w reference value for tank 2 4 him hi measured level in tank 1 global alfai alfa2 Qimin Qimax mic m2c betal beta12 beta2 posun S1 alfai alfa2 closed loop poles Qimin Qimax limit values for the pump input 4 betal betal2 beta2 valve coefficients just betal beta12 requ
282. sal so it can be easily modified to create an user interface for remote control of other real plants This has been suc cessfully tested for hydraulic plant magnetic levitation system and rotational pendulum 8 Remote Laboratory Software Module for Thermo Optical Plant 287 Comments Remarks 288 Pavol Bist k Comments Remarks 8 Remote Laboratory Software Module for Thermo Optical Plant 289 Comments Remarks 290 Pavol Bist k Comments Remarks Chapter 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant Peter Tap k and Mikul s Huba Abstract Tuning of simple controllers respecting input constraints will be considered in this chapter and verified by controlling simple physical device of thermo optical plant The chapter starts with short user s manual and installation guide to the uDAQ28 LT device which will be used as a real plant to apply the control on In the introduction several fundamental con trollers of the Dynamical class 0 DCO will be considered that avoid control saturation by guaranteeing monotonic transients among steady states at the controller output Processes of the DCO are typically met in situations where the dynamics of transients may be neglected i e it is not connected with a reasonable energy accumulation In such cases the ideal control response following a setpoint step may also converge to step function not having a saturation phase Controllers of the DCO
283. se it to derive a simplified analytical solution 64 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Theorem 2 3 Simplified analytical solution Under the assumption that Y Y is full rank another analytical solution for the problem in 2 23 is H VY G 2 25 Proof This follows trivially from Theorem 2 2 since if HT is a solution then a so is HT H7 D and we simply select D GY YY7 1G 1JV Ji Gv YYT 1G which is a ne x ne matrix D Note that the analytical expressions in Theorems 2 2 and 2 3 require YY to be full rank This implies that they generally do not apply to the case with no measurement error Wny 0 but otherwise they apply for any number of measurements One exception but not so common in practice when the analytical expressions for H do apply also for WY 0 is when ny lt na because YY then remains full rank Corollary 2 1 Important insight Theorem 2 3 gives the very important insight that Juu is not needed for finding the optimal H provided we have the normal case where H can be any ne x ny matrix This means that in 2 21 we can replace I by any non singular matriz and still get an optimal H This can greatly simplify practical calculations because Juu may be difficult to obtain numerically because it involves the second derivative On the other hand we found that F which enters in Y is relatively straightforward to obtain numerically Although Juu is not needed
284. second one is continuous one would set sysStruct Uset 1 sysStruct Uset 2 2 5 0 3 5 Inf Inf 4 5 6 9 Text Labels State input and output variables can be assigned a text label which overrides the default axis labels in trajectory and partition plotting i u and yi respectively To assign a text label set the following fields of the system structure e g as follows sysStruct xlabels position speed sysStruct ulabels force sysStruct ylabels position speed Each field is an array of strings corresponding to a given variable If the user does not define any or some labels they will be replaced by default strings xi u and yi The strings are used once polyhedral partition of the explicit controller or closed loop open loop trajectories are visualized 4 5 7 Problem Structure probStruct Problem structure probStruct is a structure which states an optimization problem to be solved by MPT 152 Michal Kvasnica 4 5 7 1 Quadratic Cost Problems In case of a performance index based on quadratic forms the optimal control problem takes the following form N 1 min z N T Pyz N u k Ru k z k T Ox k k 0 z k 1 fayn x k u k w k Umin lt u k lt Umax Amin S ulk u k 1 Aumax Umin lt Gayn x k u k lt Ymax x N Tia If the problem is formulated for a fixed prediction horizon N we refer to it as to Constrained Finite Time Op
285. sets Exercise 4 8 Given an autonomous linear discrete time system x k 1 Ax k the set of initial conditions z 0 X and a target region X the following question is of imminent importance when investigating safety will the system states hit the target region in N steps We will employ the MPT polytope library to give a yes no answer to such a question by computing forward reachable sets The set of states x k 1 which are reachable in one time step from x k Xy is given by Xk 1 Ax rc Xx 4 16 When X is an H polytope given by X x Ma lt L Xy41 is represented Xp x MA lt L 4 17 In MPT given a polytope Xk the one step reachable set Xk1 is easily com puted by applying the affine matrix transformation Xk1 A Xk 120 Michal Kvasnica Assume the following input data A 0 5 0 75 0 75 0 5 Xi unitbox 2 3 0 unit box centered at 3 0 Xf O 5 unitbox 2 4 box with sides of 0 5 4 centered at 0 0 N 10 Now answer the following questions 1 Will the system states starting from Xi hit the target region in N steps 2 If no by how much would N need to be increased 3 Plot the initial set the terminal region and the reachable sets Hints e Apply 4 17 in a loop for k 1 N starting with X1 X e Use a growing array to store the intermediate results e g R polytope for k 1 N compute polytope P R R P add P at the end of the array end
286. sign John Wiley amp Sons Ltd Chichester England Wade HL 1997 Inverted decoupling a neglected technique ISA Transactions 36 3 10 1 Problems in Anti Windup and Controller Performance Monitoring Comments Remarks 49 50 Morten Hovd and Selvanathan Sivalingam Comments Remarks 1 Problems in Anti Windup and Controller Performance Monitoring Comments Remarks 51 52 Morten Hovd and Selvanathan Sivalingam Comments Remarks Chapter 2 Optimal Use of Measurements for Control Optimization and Estimation using the Loss Method Summary of Existing Results and Some New Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Abstract The objective of this chapter is to study the optimal use of mea surements and measurements combinations c Hy in optimization and estimation based on the loss method 2 1 Introduction In this paper we consider a steady state unconstrained quadratic optimiza tion problem with linear measurement relationships The main objective is to find a linear measurement combination c Hy such that control of these indirectly leads to close to optimal operation with a small loss L in spite of unknown disturbances d and measurement noise error n If the original optimization problem is constrained then we assume that any ac tive constraints are kept constant controlled and we consider the lower dimensional unconstrained subspace Depending on the disturbance
287. slava NTNU Trondheim pp 48 53 4 Multi Parametric Toolbox 167 Comments Remarks 168 Michal Kvasnica Comments Remarks 4 Multi Parametric Toolbox 169 Comments Remarks 170 Michal Kvasnica Comments Remarks Chapter 5 Implementation of MPC Techniques to Real Mechatronic Systems Gergely Takacs and Tom s Pol ni and Boris Rohal Ikiv and Peter Simon i and Marek Honek and Mat Kopa ka and Jozef Csamb l and Slavom r Wojnar Abstract This chapter is focused on the implementation details of model pre dictive control for mechatronic systems with fast dynamics From the very Gergely Tak cs Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail gergely takacsAstuba sk Tom Pol ni Institute of Measurement Automation and Informatics Faculty of Mechanical Engi neering Slovak University of Technology in Bratislava e mail tomas poloniAstuba sk Boris Rohal Ilkiv Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail boris rohal ilkiv stuba sk Peter imon i Institute of Measurement Automation and Informatics Faculty of Me chanical Engineering Slovak University of Technology in Bratislava e mail peter simoncicQstuba sk Marek Honek Institute of Measurement Automation and Informatics Faculty of Mech
288. so possible to use optimal merging based on boolean minimization ctrl simple mpt simplify ctrl optimal Note however that the optimal merging can be prohibitive for dimensions above 2 due to an exponential complexity of the merging procedure Geyer 2005 See clipping method Kvasnica et al 2011a in Textbook and ap proach based on separation functions Kvasnica et al 2011b in Preprints for alternative ways with better convergence properties and larger reduction ratio 4 5 6 System Structure sysStruct LTI dynamics can be captured by the following linear relations x k 1 Aa k Bulk 4 28a y k Ca k Du k 4 28b where x k R is the state vector at time instance k z k 1 denotes the state vector at time k 1 u k R and y k R v are values of the control input and system output respectively A B C and D are matrices of appropriate dimensions i e A is a nz x n matrix dimension of B is n x Nu C isa ny x nz and Dany x n matrix Dynamical matrices are stored in the following fields of the system struc ture sysStruct A sysStruct B sysStruct C sysStruct D ool Jawe Example 4 5 Assume a double integrator dynamics sampled at 1 second z k 1 i l l bos u k 4 292 P k i a k H ui 4 29b In MPT the above described system can be defined as follows 148 Michal Kvasnica sysStruct A 1 1 0 1 sysStruct B 1 0 5 sysStruct C 1 0 0 1 sysStruct D
289. software prototyping interface The computationally efficient NRMPC formulation with guaranteed stabil ity and feasibility of constraints has been first introduced by Kouvaritakis et al 2000 In this approach the on line optimization task is performed through the Newton Raphson root search algorithm Although this is a sub optimal approach its application to vibration suppression systems is attractive 184 Gergely Takacs et al Optimality levels may be enhanced by extending NRMPC by a simple explicit additional optimization as introduced by Kouvaritakis et al 2002 The further development of NRMPC introduced by Cannon and Kouvari takis 2005 proves to be essential for systems with a significant asymmetry in actuator capabilities and deformation range The optimization of predic tion dynamics formulated by Cannon and Kouvaritakis 2005 recovers the maximal possible ellipsoidal region of attraction The algorithm described by Kouvaritakis et al 2000 Kouvaritakis et al 2002 and Cannon and Kouvaritakis 2005 serves as a basis for the practi cal implementation applied to the active vibrating system For more details on the theory behind the NRMPC approach the reader shall refer to the aforementioned publications Let us begin with initializing the off line part of the NRMPC code 5 2 4 2 Initialization of the Off line Code The controller structure is obtained by evaluating linear matrix inequalities LMI defined by the invarian
290. some sense is good but on the other hand it does not offer any extra degrees of freedom 76 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke 2 9 4 Numerical Tests The Matlab files for these tests can be found at the home page of S Skogestad and the commands are also listed in the Appendix The objective is to find H called beta in PLS Matlab such that y Ha Note that PLS in all cases has an extra degree of freedom because it fits y Hx ho where ho is nonzero This may help for the fitting but not for validation In any case all the data is centered so we can assume ho is close to 0 2 9 5 Test 1 Gluten Test Example from Harald Martens Data GlutenStarchNIR 1 y gluten 100 x NIR absorbents 100 data set This data has no noise on yj We first chose to use the 50 first data set for calibration and the 50 last for validation Table 2 1 shows the fit Y HX p to calibration and val idation data The best result for the traditional methods is PCR with 33 Table 2 1 Spectroscopic Example Calibration and Validation data Val 00000 3471 POR AI tol Le4 T0 60 09470 PLS 6 principal components corresponding to a tolerance of 2e 4 when taking the pseudo inverse which gives a validation fit of 0 3142 PLS can use much fewer components but the fit is not as good validation error is 0 3445 as PCR As expected the minimum loss method given perfect fit to the data and gives re
291. spects like optimality may also be consid ered at this point In most real plants values of manipulated variables are restricted and these constraints have to be taken into account in controller design procedure These limitations are usually saturation constraints and can be captured by min max bounds In MPT constraints on control input are given in sysStruct umax inpmax sysStruct umin inpmin 150 Michal Kvasnica where inpmax and inpmin are n x 1 vectors 4 5 6 5 Constraints on Slew Rate of Manipulated Variables Another important type of constraints are rate constraints These limitations restrict the variation of two consecutive control inputs du u k u k 1 to be within of prescribed bounds One can use slew rate constraints when a smooth control action is required e g when controlling a gas pedal in a car to prevent the car from jumping due to sudden changes of the controller action Min max bounds on slew rate can be given in sysStruct dumax slewmax sysStruct dumin slewmin where slewmax and slewmin are n X 1 vectors Note 4 2 This is an optional argument and does not have to be defined If it is not given bounds are assumed to be oo 4 5 6 6 Systems with Discrete Valued Inputs MPT allows to define system with discrete valued control inputs This is especially important in a framework of hybrid systems where control inputs are often required to belong to certain set of values We disti
292. stem MPT Controller Fig 4 19 MPT controller block in Simulink plot of the closed loop trajectories as can be seen in Figure 4 20 Note that the simulation model simmodel can be different from the prediction model used to design the MPC controller This allows to verify the controller s performance in situations with model plant mismatch Evolution of states Evolution of outputs 1 1 X 0 X M 0 Yo Isl a 1 2 1 g amp on gt 92 o 2 3 3 4 4 0 10 20 30 0 10 20 30 Sampling Instances Sampling Instances Evolution of control moves Active dynamics 8 a o 1 s a 0 5 0 0 10 20 30 0 10 20 30 Sampling Instances Sampling Instances Fig 4 20 Closed loop simulation using the simplot function 4 5 8 Code Generation and Deployment As discussed in Section 4 5 if the MPC problem 4 26 is solved off line and the explicit representation of the feedback law Uw x t is obtained as a 136 Michal Kvasnica lookup table the optimal control action can be obtained at each sampling instant by Algorithm 1 As the algorithm only performs trivial arithmetic operations it can be implemented using any high or low level programming language Simplicity of the individual operations also allows the table to be traversed using low computational power This makes the approach of high practical relevance when applying MPC to systems with fast dynam
293. stems 183 output such variables as iteration data or cost function but that is reserved for diagnostic purposes The state observation in this example is carried out via a product default Kalman filter The subsystem marked as QPMPC in Fig 5 3 is featured in its full detail in Fig 5 4 Here it is visible how the predicited cost matrices calculated in the problem setup stage are parsed onto the qpOASES interface One shall refer to the qpOASES documentation for details Estimated i PCI 6030E Defl state X est Kalman 7 eflection 5 41 National Instr Filter Analog Input Kalman Filter Laser Setting PCI 6030E AD Deflection Input File Scope g Irdi 2 TET x u xPC Target Data Logging 2 TET QPMPC control Unit Delay PCI 6030E 1 National Instr I aput Analoq Output Amplifier Compensation PCI 6030E DA Fig 5 4 Block scheme of the QPMPC controller in Simulink for the active vibration cancellation application 5 2 4 Newton Raphson s Suboptimal MPC 5 2 4 1 Introduction This subsection will introduce the practical implementation of the Newton Raphson MPC algorithm into the vibration attenuation of lightly damped structures As with the other cases problem setup will be carried out using the Matlab script language while the on line part of the algorithm will be implemented in the C language combined with the xPC Target
294. structural constraints it can be reformulated as a constrained quadratic programming problem Alstad et al 2009 min HY 2 23 st HG Jl Proof From the original problem in eguation 2 21 we have that the optimal solution H is non unique because if H is a solution then H DH is also a solution for any non singular matrix D of size ne x ne This follows because J H G HY JI HG D DHY J HG HY One implication is that we can freely choose G HG which is a ne x ne matrix representing the effect of u on c c Gu Thus in 2 21 we may use the non uniqueness of H to set the first part of the expression equal to the identity matrix which is equivalent to setting HG J 1 2 This identity must then be added as a constraint in the optimization as shown in 2 23 IN The reason for the non uniqueness is that since n gt ne H is fat ne x n matrix with more columns than rows Theorem 2 2 Analytical solution Under the assumption that Y Y T is full rank an analytical solution for the problem in 2 23 is Alstad et al 2009 HT YYT G G Y YT Gv i2 2 24 Proof The result is proved in Alstad et al 2009 and is based on first vector izing the problem and then using standard results from constrained quadratic optimization The analytical solution in Theorem 2 2 results in a H satisfying HG J y2 However recall that the optimal solution H is non unique and we may u
295. suitable and applicable for the SI engine control Proposed flexible design of a predictive controller offers easy tuning possibilities extension of the global engine model to other working regimes of the engine and implementation of constraints This fact also indicates the next project step which is the overshoot elimination in the A control by the identification of wider net of local engine models Acknowledgements The authors gratefully acknowledge the financial support granted by the Slovak Research and Development Agency under the contracts LPP 0096 07 LPP 0075 09 and LPP 0118 09 This research is also supported by the grant from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism This grant is co financed from the state budget of the Slovak Republic References Anstee R 2006 The Newton Rhapson Method Lecture notes Arsie I Iorio SD Noschese G Pianese C Sorrentino M 2008 Optimal air fuel ratio dSpace Magazine 1 20 23 Bengtsson J Strandh P Johansson R Tunestal P Johansson B 2007 Hybrid model ing of homogenous charge compression ignition HCCI engine dynamics a survey International journal of control 80 11 1814 1847 Cannon M Kouvaritakis B 2005 Optimizing Prediction Dynamics for Robust MPC IEEE Transactions on Automatic Control 50 11 1892 1597 Chen H Allg ver F 1998 A quasi infinite horizon nonlinear model predictive control scheme with guar
296. sults identical to least squares Thus noise has to be added and a noise level of 2 e 4 gives an error to the validation data 0 3115 which is even better than the best PCR Use of smoothened y data has no effect which is expected as the fit to the calibration data is perfect 2 Measurements for Control Optimization and Estimation 77 Table 2 2 Spectroscopic Example Calibration and Validation data with sets inter Cal Val 0000 1687 73 PCR 12 tol e changed o n o PLS 7 MUS xe min loss w noise 5 e 4 Table 2 3 Spectroscopic Example Calibration 99 measurements and Validation data 1 rotating measurement Cal ove 0 0000 3 0681 Least squares ls min loss 0 2689 0 3609 PCR tol 5 e 4 0 2769 0 3129 PLS 7 0 2471 0 3556 min loss w noise 5 e 4 However if we interchange the calibration and validation data set then the results are very different see Table 2 2 The best is now PLS 7 val 0 2357 whereas minimum loss with noise is at 0 3798 Finally one data set was excluded at a time and used for validation The average norm of the calibration fit and the norm of the validation fit for the 100 runs are given in Table 2 3 Again the PLS method is best val 0 3129 whereas the minimum loss method with noise is the second best val 0 2556 2 9 6 Test 2 Wheat Test Example from Bjorn Alsberg Kalivas 19977 Data wheat spectra 2 y 701 x 100 data set The calibration data contains 5
297. t 0 25 0 2 E 0 15 gt 2 0 1 0 05 0 1 1 1 1 1 1 1 0 50 100 150 200 250 300 350 time s x10 Pl1 controller output 1 5 e s o o o E 2 20 5 0 1 1 1 1 1 1 0 50 100 150 200 250 300 350 time s 253 Fig 7 3 Typical setpoint and disturbance steps in the loop with PI controller con tain one step at saturation after setpoint steps ur rozsah w H f u e Pu m gt Constant Saturation Q gt U Saturation1 hi Subsystem 1 Tfil s 1 oe 1 sta Ks Tfil s 1 S1 beta1 sqrt u 1 posun ur rozsah 2 y w i Kr eco f u gt 7 100 12 U m Constant Saturatign Y Q gt U Saturation1 hi 1 Subsyst gt ystem Tfil s 1 w z y LP S1 betat sqrt u 1 posun 4 n 1 a s Integrator Fig 7 4 Control structure for linear and nonliner PI1 implemented in Matlab Simulink load basics calibParams b XL E 7 load load load load basics valvel basics valve2 basics valve12 basics pump1 T 0 25 S1 0 001 posun
298. t contain full matrix inversions only essential piecewise inversions Matrix and vector operations within the NRMPC on line algorithm are performed using the Basic Linear Algebra Subprograms BLAS package The Simulink scheme implementing the NRMPC controller gains its feed back signal directly from a block reading the analogue input of the measure ment card The scheme is visible in Fig 5 5 The output from this block is scaled according to the current settings of the LASER head so its output is given directly in millimetres The File Scope block ensures real time data logging onto the xPC Target PC hard drive which can be later re used for measurement processing Feedback measurement passes through a product default Kalman filter a part of the Signal Processing Blockset toolbox MathWorks 2007 The Kalman filter is always enabled and the number of filters is set to one Initial condition is assumed to be zero The estimated state conforming to the dimensions of model considered when creating the NRMPC controller is passed onto a custom S Function block implementing the on line NRMPC code in C language This block takes parameters R T Sm Q21 Q11 K and Co as its set up and starting param eters All of these parameters are the result of running the Matlab script responsible for initialization and off line NRMPC algorithm evaluation The S Function block has the core C program and two custom functions See 5 2 4 9 specified as module
299. t x t y t d t d dy dt the system state and K t 7 11 u K t U K U i 1 2 7 12 being the normalized control it can be described in the state space as TO a t butt A hoo b 1 cam The linear pole assignment PD controller is given as wens r ro r u Ks 7 14 The PD controller is composed from the linear pole assignment controller used for the relatively low velocities and given as at aol A ro Qa1Q2 T Q1 Q2 7 16 For d d dg the corresponding constrained pole assignment controller is again given by 7 15 7 16 and with limiter 7 10 For higher velocities d the nonlinear algorithm is used that moves the representative point in the state space towards the so called Reference Braking Trajectory and guaran teeing braking with dynamics specified by one of the closed loop poles and without overshooting The controller for the double integrator is then given as 262 Vladim r ilka and Mikul Huba u d U 7 17 u sat u K 7 18 j 3 sign y 2 7 19 When controlling the nonlinear system 7 9 it is at first necessary to transform by means of inverse equation to 7 9 the existing saturation limits e g 0 and 5 V into the limits valid for the fictive controller input v1 Then using information about the plant state and the new saturation limits con sidered in 7 19 the fictive control signal v is computed by means of 7 15
300. t HG G h Our optimization problem then becomes er 1 max 2 1 G X All g min j min H h Gn We have here made use of the induced 2 norm and the fact that both the Frobenius and 2 norm are the same for a vector Thus the optimal j is the inverse of the 2 norm of G which generalizes the solution found for the case with two measurements The optimal h cG where c is any scalar since only the relative magnitudes matter that is HT cG which generalizes the result above c Multivariable case c is no longer required to be a scalar From 2 25 we derive with Y I F 0 and measurement noise of magnitude 1 for all outputs that an optimal solution is HT G which generalizes the results above Thus for the case where only mea surement noise is a concern and all the measurements have the same noise 2 Measurements for Control Optimization and Estimation 63 magnitude the optimal is to select the measurements according the gain matrix G This means that sensitive measurements with large elements in G are preferred for control 2 5 Reformulation of Loss Method to Convex Problem and Explicit Solution We consider here the normal case where H is a full matrix with no structural constraints Theorem 2 1 Reformulation as a convex problem The problem in equation 2 21 may seem non convex but for the normal case where H is a full matrix with no
301. t file in the following format valveValvenumber year month day hour minute second mat For example valvei 2009 12 20 16 26 43 mat and so on We don t consider the first and the last measurement sample from the other values with the step approximately 1 5cm we use each sixth sample define the value c as well as the possible offset value of the valve of f seti At the end we define the mean value of offset and of flow coefficient and we save those two values together with the time course of emptying into the valvel valve2 and valve3 mat We apply them directly to the proposed model we simulate the emptying and we draw the comparison of simulation 6 Laboratory Model of Coupled Tanks 237 MENU ES Your choice Calibration test Calibration Valve 1 Valve 2 Valve 1 2 Valve 3 Valve 2 3 All valves Pump 1 Pump 2 Both pumps All params End of identification Fig 6 5 Menu of the software module for calibration and identification and real data into the graph of emptying characteristics such as Fig 6 6 This procedure allows us to check visually quality of the valve s identification on the graph The registration to the actual mat file allows us to read variables in the program according to our choice at the direction of the model or at the work with the real system The previous identification may sometimes look better than the new one that s why the measurements are stored in the
302. t lev level of optimality Table 4 4 Mandatory fields of the problem structure probStruct 4 5 7 4 Level of Optimality MPT can handle different setups of control problems Specifically 1 The cost optimal solution that leads a control law which minimizes a given performance index This strategy is enforced by probStruct subopt lev 0 The cost optimal solution for PWA systems is currently supported only for linear performance index i e probStruct norm 1 or probStruct norm Inf 2 Another possibility is to use the minimum time setup i e the control law will push a given state to an invariant set around the origin as fast as possible This strategy usually leads to simpler control laws i e fewer controller regions are generated This approach is enforced by probStruct subopt lev 1 154 Michal Kvasnica 3 The last option is to use a low complexity control scheme This approach aims at constructing a one step solution and subsequently a PWQ or PWA Lyapunov function computation is performed to verify stability properties The approach generally results in a small number of regions and asymp totic stability as well as closed loop constraint satisfaction is guaranteed If one wants to use this kind of solution he she should set probStruct subopt_lev 2 4 5 7 5 Optional Fields Optional fields are summarized next probStruct Qy used for output regulation If provided the additional term llQ y Yref p is introduc
303. the pro cess gain value and the delay time constant In Fig 9 18 there is the Simulink model which should pop up when the correct experiment starts The exper 304 Peter Tap k and Mikul Huba Dead time 0 75 T T T T I T l 0 7 0 65 0 6 Td 0 55 0 5 0 45 5 Fig 9 15 Dead time in multiple operating points as function of the input variable Step responses of nor filtered light channel Step res pons es of ncn filtered light channel T m T T T 19 45 aop 1 el 35 BO 4 2 7 ai 1 i 16 Bal b 15k 4 15 0L 4 sl u 0 n 1 1 A i n 1 i o 5 10 15 20 25 30 35 15 155 18 16 5 T7 17 5 ts tis Fig 9 16 Step responses of the non filtered optical channel overview and a detail of one step response These are the responses to 0 5V bulb voltage steps made in 3s intervals 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant 305 Prefilter Disturbance Observer Fig 9 17 FIo controller structure equivalent for Tp Ty to I controller iment will run with multiple setpoint steps Feel free to modify the model to make your own setpoint steps sequence The goal is to achieve quick non overshooting transients m gt R u del gt Tied p gt L L Temperature1 Repeating Saturationt Beg Transfer Fen3 P regi i Terapia Ly lan Gaint 1 o Fan VO Interface ni rec m TransterFont ji Tfil s 1 TED
304. the saturation limitations for the individual loop controllers k s are time variant 1 1 4 The Plant Model used in the Assignment The model used is the detailed LV model of a distillation column given in 13 19 of Skogestad and Postlethwaite 2005 A B Ba s vl wear d 1 10 0 005131 0 0 0 0 07366 0 A 0 0 0 1829 0 0 0 0 T 0 dis 0 0 0 0 9895 0 4620 0 629 0 624 0 055 0 172 B 0 030 0 108 0 186 0 139 1 230 0 056 c 70 7223 0 5170 0 3386 0 1633 0 2 ho sit 0 8913 0 4728 0 9876 0 8425 0 xe 00 0 062 0 067 0 131 0 040 Ba 0 022 0 106 Dj D 0 188 0 027 0 045 0 014 where the two measurements are the top and bottom product concentrations the first input is the reflux flowrate at the top of the distillation column the second input is the boilup rate at the bottom of the distillation column the first disturbance is the feed flowrate and the second disturbance is the feed composition The time scale in the model is in minutes For further informa tion about the model and the scalings applied please refer to Skogestad and Postlethwaite 2005 1 Problems in Anti Windup and Controller Performance Monitoring THE 1 1 5 Assignment 1 Model implementation Implement the LV model in Simulink Use saturation blocks found in the discontinuities library of 0 5 for the inputs to the LV model You may use manual switch blocks found in the signal routi
305. the www javasoft com and install it 8 Remote Laboratory Software Module for Thermo Optical Plant 277 The proper installation of the uDAQ28 LT thermo optical plant is another preposition The installation should be carried out according to the guide on the installation CD To check if the control of the real plant works locally it is possible to run one of the predefined experiments from the installation CD In the case the above mentioned requirements are fulfilled the server ap plication could be installed Open the Server folder on the installation CD and double click the server install bat file During the process of installation there will be a new folder called RemLab Workspace created in the root di rectory of the C drive Necessary Matlab files will be copied into this folder Moreover the server application RemLabServer exe will be copied into the RemhLab Workspace folder Before running the server application it is recom mended to check the setting of the serial port in supplied block diagrams that are placed in the directory CA RemLab Workspace Warning Do not delete or move the folder CA RemLab Workspace This folder is fixed working directory of the server application The RemLab Server eze application file can be moved from the CA RemLab Workspace folder to any different place and it is also possible to create a shortcut placed at the desktop in order to run it quickly 8 3 2 Client Installation The client application is
306. then fill it up by given input up to the fullest state or during 60 seconds That s because at the lowest levels of input the filling up would take too much time After the last measurement for full input we define the maximum inflow of the pump as well We put the given inflows and power tensions into the field and save them at the end of experiment by same way as valves into the mat files The first pump s name will obviously start by pumpl and the second one by pump2 6 Laboratory Model of Coupled Tanks 239 During the individual steps we include into the measurement the artificial break of 40s to prevent the changes in pump s parameters its heating or to partially isolate the previous measurement from the actual one We draw the measured IO or inverse IO characteristics into the graph and compare them with the approximations acquired thanks to curve fitting functions As the buttons names indicate by the above described procedures we can identify all valves both pumps and all system s parameters To clean the older data in order to delete the measured and saved val ues more easily we can use the script cleaning which will preserve only the measured values and will delete all the mat files that contain the date in the name That s why one has to be very careful when using it We should use it only if we are sure that we don t want to archive the older values After we get all the required parameters we only have to
307. tic Program QP if p 2 or as a Linear Program LP if p 1 00 Finally if the predictions in 4 26c are governed by a PWA system the optimization problem can be stated as a mixed integer QP MIQP for p 2 or as a mixed integer LP MILP for p 1 or p oo see e g Borrelli 2003 MPC is usually implemented in the receding horizon RHMPC fashion At each sampling instant t measurements or estimates of x t are obtained first Then the optimal control sequence Un x t is calculated by assuming x t as the initial condition for problem 4 26 Subsequently only the first el ement of the sequence is extracted i e ug z t 1 0 0 Ux and uo x t is fed back to the plant The whole procedure then repeats at subsequent time instants ad infinitum This introduces feedback into the scheme As explained in the introduction numerical values of Uy can be obtained by solving 4 26 as a corresponding LP QP MILP MIQP problem on line at each sampling time for a particular value of x t An alternative is to pre calculate the optimal solution to all possible values of the initial condition x t Theorem 4 1 Borrelli 2003 Consider the MPC optimization prob lem 4 26 with f in 4 26c being either linear as in 4 23 or piecewise affine as in 4 24 Then the optimizer Un x t is a piecewise affine function of x t i e Un a t Fiz t Gi Va t Ri 4 27 where different feedback gains F G swit
308. timal Control CFTOC problem If NV is infinity the Constrained Infinite Time Optimal Control CITOC problem is formulated Objective of the optimization is to choose the manipulated variables such that the performance index is minimized 4 5 7 2 One and Infinity Norm Problems The optimal control problem with a linear performance index is given by p NSB Met Ye a Mp lQx F lIp a k 1 fayn a k u k w k Umin lt u k lt Umax Amin S u k u k 1 Aumar Umin lt Jayn a k u k lt Umax z N Ty 4 Multi Parametric Toolbox 153 where u vector of optimized control inputs N prediction horizon p norm indicator can be 1 2 or Inf Q weighting matrix on the states R weighting matrix on the manipulated variables Py weight imposed on the terminal state Umin Umax constraints on the manipulated variable s Aumin dUmax constraints on slew rate of the manipulated variable s Umin Ymax Constraints on the system outputs Tet terminal set the function fayn x k u k w k is the state update function and is dif ferent for LTI and for PWA systems see Section 4 5 6 for more details 4 5 7 3 Mandatory Fields In order to specify which problem the user wants to solve mandatory fields of the problem structure probStruct are listed in Table 4 4 probStruct N prediction horizon probStruct Q weights on the states probStruct R weights on the inputs probStruct norm 1 2 or Inf norms in the cost probStruct subop
309. time on not so important tasks how to handle features of RCP system at low level programming The RCP is equipped with many input and output ports implemented on the terminal board Fig 5 14 managing the whole communication between the main computational unit and the engine To the RapidPro is directly connected the knock sensor and the broadband lambda probe The crankshaft and camshaft sensors are attached through a converter case developed during the project lifetime It is utilized by air temperature and pressure sensor and the cooling water and oil temperature sensor as well Among other functions it indicates correct functionality of injectors igniters and oil pressure sensor Actual battery voltage is displayed The security circuits protect the engine by switching off the power stages and the fuel pump if an error occurs 206 Gergely Takacs et al Ateos Ea RapidPro Power Unit Master Unit C ion B RapidPro His us RCP system SC Unit Unit Connection Bus UCB Host PC RapidPro with ConfigurationDesk Control Unit KJ CJ 90 ve IVDS link USB for configuration ASPACE Link Board DS8 1 9 or slot CPU Fig 5 13 Rapid control prototyping scheme 5 3 38 AFR Model Design 5 3 3 1 Air Fuel Ratio The model of the air fuel ratio dynamics A of a spark ignition engine is based on the mixture defined as mass ratio of the air and fuel in a time step k Due to the fa
310. tions and or bug reports should be communicated to mpt control ee ethz ch 4 2 Computational Geometry in MPT 4 2 1 Polytopes First we introduce the computational geometry features of MPT MPT con tains a rich library for manipulating convex geometrical objects known as polytopes Definition 4 1 Convex Set A set C C R is convex if the line segment connecting any two points in C lies entirely in C i e 021 1 0 x9 C C for all zj z2 C and0 R 0 0 1 Definition 4 2 Polyhedron Convex set represented as the intersection of a finite number of closed half spaces aZ x lt bj i e P x R Ar lt b with di bi A b 4 1 is called a polyhedron Definition 4 3 Polytope Polytope is a bounded polyhedron In general polytopes can be described in two possible representations 1 The H representation i e P x Ax b where the input data A and b are matrices representing the defining half spaces ala lt bi or 2 the V representation i e P convh z1 m where convh denotes the convex hull operator i e convh z1 m x v Aui Amtim 0 A L 35A 1 Here the input data are the vertices 71 Zm of the polytope 104 Michal Kvasnica In MPT polytopes are constructed using the polytope constructor A 1 1 11 0 1 bib 1 0 TH polytope A b xi 1 0 x2 1 0 x3 0 1 TV polytope x1 x2 x3 Calling the const
311. to 186 Gergely Takacs et al the set of constraints defining the SDP problem Input constraints are defined by if isempty uh F F set uh 2 K Yq M K Xq K Yg M K Xq Invi gt 0 end 5 2 4 4 Solver Setup and Off line Solution Initiation Options are passed to the LMI parser and also to the solver in this case SeDuMi Strict inequality constraints are relaxed and perturbed by the shift setting ops sdpsettings verbose 0 ops sdpsettings ops shift 10 tolerance ops sdpsettings ops solver sedumi sedumi eps 0 Solution of the above defined SDP problem is initiated by the solvesdp YALMIP command The LMI defining constraints are passed onto the solver as the variable F options are contained in the ops parameter The aim of this optimization problem is to maximize the volume of the ellipsoids defining the region of attraction and target set This can be carried out by utilizing the fact that the volume of an ellipsoid is proportional to its determinant maxe detP 5 3 where P is the optimization parameter in general and m is the dimension of P In this case optimization objectives and parameters are Yq and Xg defining the projection and intersection of the augmented ellipsoid into x space It is desirable to maximize the volumes of ellipsoids defined by Y q and Xq at the same time by including them in a block diagonal construct Optimization parameters Y q X q
312. to the client In this case the button Run must be pressed repeatedly 7 When it is required that data from the just finished experiment have to be saved set the name of the data file into the Save to File field and press the Save button It is necessary to wait for the end of the saving process that is announced in the Output field Fig 8 10 Then it is possible to process the data saved in the specified file using another application 8 In the case of repeating the experiment with new parameters clear the Output and Graphs fields by pressing the Clear button and repeat this procedure starting from the point 5 9 Press the button Disconnect when it is not desired to continue working with remote experiments This will free resources of the remote laboratory for other users To finish the client application close the corresponding window of the Internet browser 8 7 Rules for Creation of Models in Simulink Remote laboratory administrators can create new block diagrams and placed them together with initial parameter files on the server in the fixed given folder C RemLab Workspace Thus remote users can have more possibilities to select the type of the experiment when choosing the corresponding names of files mdl and m files in the Model Scheme and Model Parameters fields To assure the application is working correctly several simple rules must be kept when creating these new block diagrams
313. to this real system The server part of the software is created by the program application written in the Java programming language The server application should be installed on the computer that directly controls the real plant The client part of the software is realized in the form of a Web page with embedded Java applet This provides a user friendly environment to carry out remote experiments Pavol Bist k Institute of Control and Industrial Informatics Slovak University of Technology in Bratislava e mail pavol bistak stuba sk 275 276 Pavol Bist k REAL PLANT i SERVER o f A i a i i ites een ees ee een M Remote Control Fig 8 1 Remote laboratory overview 8 2 Technical Requirements 8 2 1 Server Windows operating system XP Vista Seven Java Virtual Machine Matlab with Simulink fixed IP address free port No 23 8 2 2 Client Computer e Internet browser with Java applet support 8 3 Installation 8 3 1 Server Installation First it is necessary to check whether a Java Run Time JRE environment is installed on the server computer By default Java is installed into the Program Files folder To check the Java installation you can also open the Command Window Start Run cmd and type java command to see whether the Java installation exists or no If there is no Java installed on the server computer you should download the installation file from
314. ton Raphson s Suboptimal MPC 183 5 25 Multi Parametric MPC 0005 194 Bee COONODBSIOR Loose RR E Rp i bodn drt SS 198 Gu AFR trl 15e O eee haan ieee 200 Dal TONG e eni Eun pL duc petes 200 5 3 2 Hardware Description 0 020 4 ceca eee dis 201 5 8 8 AFR Model Design 0 0 eee eee 206 5 3 4 Predictive Control 5 212 socius 216 5 8 5 Results of a Real Time Application of a Predictive OMRON C Beda kes aged ane ey au odd uza ea 219 5 2300 Copclu ion coss deme RR RR Rae ges ks 222 BieferemdeH csooocacde el bed ood ahead a ib qe per dde und 222 Laboratory Model of Coupled Tanks 229 Vladim r Zilka and Mikul s Huba 6 1 Dabduelii vec IDEE dace dla aie ble ade DERE ERRORS 229 6 2 Coupled Tanks Hydraulic Plant 230 Dal Identification osuere m erem 232 6 2 2 Sensors Calibration 2 22 22 2g dol li leri 235 6 2 3 Automatic Calibration and Identification 236 6 2 4 Some Recommendation for Users 239 ilg RD 241 Constrained PID Control Tasks for Coupled Tanks COUPON Lolo ila eor nace ER D UR ne ur a a e UR RES 24T Vladim r ilka and Mikul Huba Tok atrodom 22 eg eee bet ovl eens ba van k OR RS 247 2 Basie Pand Pl controllers z occ ccc ven Land Rl eis 248 Gal PilEeanuolek llli1zlzkiencig3 vue ie bade aah 250 pc Ply controlat PRPRRCOTOTRTOITERPTENMPRUE 251 7 3 Linearization around a fixed operating point
315. tope P A res Fa onto the z space cf 4 7 Your task is to implement this projection based procedure and compare its output to MPT s native algorithm available by running P1 P2 Assume the following input data gt gt P1 polytope 1 1 1 1 1 1 1 1 gt gt P2 polytope 0 1 0 1 0 1 0 1 0 0 1 Exercise 4 7 The Pontryagin difference operation discussed in Section 4 2 3 7 was so far defined under the assumption that C and B in 4 11 are single polytopes However the operation is also well defined when C is a polytope array i e C Uj C In such a case the difference G C O B can be obtained by the following procedure 1 H convh C 2 D HOB 3 H C 4 F E B G DNF The algorithm is explained graphically in Figure 4 11 Assume the following input data gt gt C1 gt gt C2 polytope 0 2 0 2 2 4 2 4 4 2 4 2 polytope 4 2 4 2 2 4 2 4 0 2 0 2 4 Multi Parametric Toolbox 119 gt C C1 C2 gt gt B 0 5xunitbox 2 and implement the procedure above using MPT s basic operators summarized in Table 4 2 Compare the output to the default implementation of C B a Polygone C U C b Convex hull H c Set D HOB and polytope B convh C wu F X R 5 B 1 sE 1 al i 4 5 s a ro 1 2 3 4 ro 1 2 3 4 d Set E C e Set F 2E 8 B Pontryagin Difference G D F Fig 4 11 Pontryagin difference for non convex
316. tration at stoichiometric level us ing MPC is demonstrated through the real time SI engine control Fig 5 21 This has been performed using the AFR predictive control strategy described Combustion Engine MPC Controller Fig 5 21 Control scheme in the previous section designed in Matlab Simulink environment and com piled as a real time application for a dSpace platform It has been applied to the VW Polo engine Fig 5 10 1390 cm with 55kW 5000rpm not equipped with a turbocharger or an exhaust gas recirculation system The 220 Gergely Takacs et al control period was 0 2s The result of an identification are 9 local linear mod els LLM for each air and fuel path dependent on a throttle valve opening and engine revolutions 2500 E a real desired 2000 I i i i i po N 20 40 60 80 100 120 140 160 180 Inj time s o 1 i 1 i i 1 i 1 20 40 60 80 100 120 140 160 180 Time s Fig 5 22 Results of a real time control of the SI engine without desired final state penalization The primary target of a control Fig 5 23 was to hold the air fuel ratio in a stoichiometric region A 1 in the worst case to keep the mixture ignitable 0 7 lt A lt 1 2 During the experiment the change in throttle valve opening between 21 and 22 degrees Fig 5 23 variable t and the change of engine revolutions Fig 5 23 variable nen
317. trol input associated to a given initial state by running gt gt uopt ctrl x Now answer following questions 1 What is the optimal control action for z 0 755 2 Which input corresponds to z 0 2 Can you give the answer even before running the code in MATLAB hint take a look at state constraints in sysStruct xmax 3 Plot the closed loop evolution of system states and control inputs for 10 steps starting from the initial condition z 0 75 Exercise 4 14 Assume again the setup from the previous exercise but change the probStruct R penalty to 10 Run again the closed loop simu lation and explain what has changed We also have a good news for you MPT provides a simple command to visualize a closed loop simulation simplot ctrl x0 number of simulation steps If you only want to compute the numerical data use the sim command X U Y sim ctrl x0 number of simulation steps Exercise 4 15 Now we will practice with explicit MPC controllers In the explicit approach the MPC problem is pre solved and the optimal control action is computed for all possible initial conditions The solution then takes a form of a look up table which consists of polyhedral regions with associ ated affine feedback laws Evaluation of such controllers then reduces to a mere table lookup which can be performed much faster compared to on line controllers We consider control of the double integrator described by the follow
318. ts are drastically improved compared to the case without anti windup and are quite similar to what was obtained for the decoupler with anti windup above 10 11 22 Morten Hovd and Selvanathan Sivalingam 1 1 8 4 Conclusions on Anti windup The results obtained clearly illustrate the benefits of anti windup when the manipulated variables saturate for all three controller types studied in this assignment 1 1 9 Matlab Code 1 1 9 1 Definition of Model and External Signals Ahhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh Assignment Controller design and anti windup Solution Model and signal definitions hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh clc clear all close all P yet nnn 00 Square X Ax Bu y Cx Du 9 Parallel integrators Constant in Hanus form x Ax Bu lorkii y CDu R Fast low pass Gain to break algebraic loop lgrkKs x Ku A Measurement t gt y Outt m wins LO Measureme In ous Py int md aput Out u Out2 Ing LV w saturation To Workspace LQGint states Measured input Fig 1 23 Simulink implementation of LV distillation column with LOG controller The LQG controller has integral action shown separately from the rest of the LOC controller and anti windup 12 13 14 15 16 17 18 19
319. ts no matrix H such that HY 0 However if we start with data and have many measurements e g spec troscopic data then Y Xop is likely a thin matrix and there will exist an or actually infinitely many H such that HY 0 Since the experimental data Y contains measurement noise this means that H is fitting the noise The proposed method will then be very similar to least squares although the constraint H G G can make it different as seen from the numerical examples below Extra step 1C To fix this up one may add artificial measurement noise to get a better representation of the actual measurement noise Since there will always be some independent noise for each measurement it is suggested to add a diagonal matrix W 4v to the original data X extended X opt Wa where X opt was found in Step 1B above The problem is now to choose Wy One approach is to estimate it from the data using a preprocessing step For example one may do some prelimi nary regression and from this obtain an estimate of the noise A very simple approach which is tested on some applications below is to assume that the measurements have been scaled for example by the norm of the variation in each measurement such that they have similar expected magnitudes Thus we use W ny w4 Y where wn is a scalar noise wn eye rx Xoptnoise Xopt noise Hopt loss soc_avg Gy Xoptnoise 2 Measurements for Control Optimization and
320. ttings skoge My Documents MATLAB NN E Test case 1 Martens data load GlutenStarchNIR X Variablene 13 112 er NIR absorbanser log 1 T fra 850 1048 nm yi Variabel 4 er kvantitiv analytt konsentrasjon gluten Matrix 100 rows 112 cols regular MATLAB matrix VarLabels 112 rows 17 cols MATLAB character string 4 ObjLabels 100 rows 7 cols MATLAB character string Yall Matrix XX Yall 13 112 YY Ya11 4 X XX 1 50 X0 XX 51 100 Y1 YY 1 50 Y10 YY 51 100 4 repeated Martens 4 100 times where I take out data each time for nsis 1 98 4X XX 1 nsis nsis 2 100 XO XX nsist1 Y1 YY 1 nsis nsis 2 100 Y10 YY nsis 1 4 Two end cases are handled separately 4nsis 0 4X XX Insis 2 100 XO XXC nsis 1 4Y1 YYC nsis 2 100 Y10 YY nsis 1 4nsis 99 4X XX 1 nsis XO XX nsis 1 4Y1 YYC 1 nsis Y10 YY nsis 1 34 35 36 37 38 39 40 41 42 43 44 45 46 AT 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 82 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke Jump to MATLAB code starts here E o ee 4 Test case 2 Bjgrn Alsberg data epost 15 11 2010 load wheat spectra Your variables are 4X Y idxcal idxval moist protein Xwheat X
321. uator control of the vibrations of a cantilever under axial load Composite Structures 62 423 428 L ofberg J 2004 YALMIP A toolbox for modeling and optimization in MATLAB In Proceedings of the CACSD Conference Taipei Taiwan Lorini G Miotti A Scattolini R 2006 Modeling simulation and predictive control of a spark ignition engine In Predimot ed Predictive control of combustion engines TRAUNER Druck GmbH amp CoKG pp 39 55 Maciejowski J 2002 Predictive Control with Constraints lst edn Prentice Hall Maciejowski JM 2000 Predictive control with constraints University of Cambridge Mao X Wang D Xiao W Liu Z Wang J Tang H 2009 Lean limit and emissions improvement for a spark ignited natural gas engine using a generalized predictive control GPC based air fuel ratio controller Energy amp Fuels 23 6026 6032 MathWorks T 2007 Matlab signal processing blockset v6 6 r2007b Software Mayne DQ Rawlings JB Rao CV Scokaert POM 2000a Constrained model pre dictive control Stability and optimality Automatica 36 789 814 Mayne DQ Rawlings JB Rao CV Scokaert POM 2000b Constrained model pre dictive control Stability and optimality Automatica 36 789 814 Murray Smith R Johanssen TA 1997 Multiple model approaches to modelling and control Taylor amp Francis Muske KR Jones JCP 2006 A model based SI engine air fuel ratio controller Amer ican Control Conference Minneapolis USA Pol ni T Roh
322. uch and similar systems raises not only questions associated with algorithm speed but also with minimal useful prediction horizon length Due to the stability require ment the useful range the stable MPC controller can operate is severely lim 174 Gergely Takacs et al ited calling for extremely long prediction horizons Takacs and Rohal IIkiv 2009 This is partly caused by the large relative deflections compared to the max imal possible control effort by the piezoelectric actuators and short sampling times Optimization of prediction dynamics introduced by Cannon and Kou varitakis 2005 offers a remedy to this problem as it maximizes the region of attraction which defines the useful controller range however this idea is executed for for a computationally efficient albeit sub optimal MPC approach called Newton Raphson s MPC This subsection describes the practical implementation of different MPC methods onto systems of active vibration control with under damped dynam ics Three MPC methods are introduced here e traditional dual mode quadratic programming based MPC QPMPC with stability and feasibility guarantees e a computationally efficient sub optimal control strategy introduced by Kouvaritakis et al 2000 Kouvaritakis et al 2002 and Cannon and Kou varitakis 2005 called Newton Raphson MPC NRMPC e and finally an optimal multi parametric programming based MPC ap proach MPMPC This section places focus on practi
323. uct symax must be given as a ny x 1 vector where each element defines the maximum admissible violation of each output constraints The aforementioned fields also allow to specify that only a subset of state input or output constraint should be treated as soft constraints while the rest of them remain hard Say for instance that we have a system with 2 states and we want to soften only the second state constraint Then we would write probStruct Sx diag 1 1000 probStruct sxmax 0 10 Here probStruct sxmax 1 0 tells MPT that the first constraint should be treated as a hard constraint while we are allowed to exceed the second con straints by at most 10 and every such violation will be penalized by the factor of 1000 156 Michal Kvasnica 4 6 Exercises Exercise 4 11 Consider a model of two liquid tanks given by the following discrete time state space model x t 1 y t 0 1 x t 0 0315 0 0 0769 0 0315 e a 0 ew where the state vector z t Bal represents the deviations of liquid levels from some steady state levels and u t is the deviation of the liquid inflow from the steady state value The state input and output variables are as sumed to be constrained as follows e z2 lt 2 lt 33 e 17 lt u t lt 3 e 21 lt y t lt 3 5 Perform following tasks 1 Create a sysStruct representation of such a system by filling out the structure fields as described in Section 4 5 6 2
324. uired in the previous step the only task left is to evaluate for the perturbation vector f which in this case can be also stated as f TRWia 5 14 This single mathematical operation is divided into three parts for the C code First the value of A is negated then a temporary vector is created from the product of viemp ARWia The final step is to calculate f by multiplying this temporary vector by T from the left f T vtemp lamN lamN dgemv_ chn amp order amp order amp lamN R amp order Wid amp onei amp zero amp tempv amp onei dsymv chu amp order amp one T amp order tempv amp onei amp zero f amp onei With the perturbation value calculated in the previous step the final task is only to evaluate the current control move according to u Kao Cof This is performed in the C code by summing up results of two vector dot operations u 0 ddot amp order K amp onei xO amp onei 194 Gergely Takacs et al ddot amp order CO amp onei f amp onei The other option implies that the loop is already optimal thus the pertur bation f 0 There is no need for optimization this is part of an if than else decision In this case the fixed feedback matrix is used to calculate the control move from the observed state by evaluating u Kao This is again a simple vector dot product elsef u 0 ddot amp order K amp onei xO amp onei 5 2 5 Multi Parametric MPC 5 2 5
325. ultiple indices can be specified e g MC 1 3 0 will erase first and third element of the given polyarray If some element of a polyarray is deleted the remaining elements are shifted towards the start of the polyarray This means that assuming N P1 P2 P3 P4 P5 after 4 Multi Parametric Toolbox 107 MG 31 2 the polyarray N P2 P4 P5 and the length of the array is 3 No empty positions in a polyarray are allowed Similarly empty polytopes are not being added to a polyarray A polyarray is still a polytope object hence all functions which work on polytopes support also polyarrays This is an important feature mainly in the geometric functions Length of a given polyarray is obtained by 1 length N A polyarray can be flipped by the following command Nf fliplr N ie ifN P1 P2 P3 P4 P5 then Nf P5 P4 P3 P2 P1 4 2 3 Operations on Polytopes Next we review some standard operations on polytopes 4 2 3 1 Chebyshev Ball The Chebychev s ball of a polytope P x R Ax lt b is the largest hyperball inscribed in P i e B e x R xe r such that B P The center and radius of such a hyperball can be easily found by solving the following LP max r 4 2a subj to Aize r Ai o lt bi 4 2b where A denotes the i th row of A The polytope P is empty if and only if the LP 4 2 is infeasible If r 0 then the polytope is lower dimensional Note that the ce
326. umentation is available at http control ee ethz ch mpt 5 Implementation of MPC Techniques to Real Mechatronic Systems 195 n4s2 C 0 sysStruct C sysStruct D The process is followed by naming the state variables and setting con straints on the inputs Output constraints are set to infinity therefore prac tically they are neglected sysStruct StateName 1 sysStruct StateName 2 sysStruct umin 120 sysStruct umax 120 sysStruct ymin inf sysStruct ymax inf x1 5 x2 Sub optimality level is zero this means that the software generates an optimal controller with a quadratic cost function the norm is set to 2 Input and state penalties are defined as well along with the controller horizon probStruct norm 2 probStruct subopt lev 0 probStruct Q sysStruct C sysStruct C probStruct R 1e 4 probStruct N 70 Estimated T PCI 6030E State X est c Zi Position 1 National Instr Analog Input I LASER Kalman Filter Gain PCI 6030E AD PCI 6030E K gt National Instr Analog Output Input Amplifier PCI 6030E DA E M 1 Compensatio d Lplco mpmpc z Input pFile Scop rp Unit Delay Id 2 Data Logging MPMPC Colntroller xPC Target TET Fig 5 6 Block scheme of the MPMPC controller in Simulink for the active vibration cancellation application The next step is the most important of all that is th
327. ve positioner results in such fast oscillations 1 4 Controller Performance Monitoring using the Harris Index In large scale chemical plants the number of control loops is simply too high for operators or engineers to continuously monitor the performance of each loop Studies show that disappointingly many control loops perform poorly The need for automated tools to assess control performance is therefore clear In this assignment one of the most popular performance monitoring methods the Harris Index will be illustrated More background on the Harris Index and its modifications can be found in Sivalingam and Hovd 2011 In this assignment the Harris Index is used to assess the performance of a simple control loop The sampling interval of the loop is 1 time unit and the open loop dominant time constant is estimated to be around 100 the 46 Morten Hovd and Selvanathan Sivalingam knowledge of the dominant time constant is not known for calculating the Harris Index but is used in one of the more common modifications of the index The plant is strictly proper but there is otherwise no time delay from input to output i e h 0 0 h 1 4 0 where h k is the impulse response coefficient for lag k The plant is open loop stable and has no zeros outside the unit disk 1 4 1 Assignment 1 Download the Matlab data file cpmsim mat from http www itk ntnu no ansatte Hovd_Morten NIL The file contains the measurement OP of a contro
328. w on the Fig 9 4 When the device has been installed virtual serial port is available on the list of hardware devices labeled as USB Serial Port One can get to its settings through Device Manager Start Settings Control Panel System Hardware Device Manager if Ports COM amp LPT USB Serial Port is selected Fig 9 5 The port has automatically assigned by the system one of com port numbers which is not in use If the assigned number is bigger than 4 it is necessary to change it to different com port number from the range 1 4 which is not in use at the moment It is possible to perform on the tab Port Settings Advanced COM Port Number Fig 9 6 The last action after installation to be done is setting parameter Latency Timer msec on value 1 which minimizes data loss during data transfer Parameter Latency Timer msec is available on the same tab as COM Port Number Fig 9 6 In the case that all com port numbers from the range 1 4 are already in use by 9 Constrained PID control Tasks for Controlling the Thermo Optical Plant Found New Hardware Wizard Welcome to the Found New Hardware Wizard Windows will search for current and updated software by looking on your computer on the hardware installation CD or on the Windows Update Web site with your permission Read our privacy policy Can Windows connect to Windows Update to search for software O Yes this time only O Yes now and every time connect
329. xPC optimized BLAS code library The two external custom functions described in 5 2 4 9 are also needed to be present and properly declared at compilation time At the on line control process the following real time NRMPC algorithm is called on and evaluated at each sampling interval Local variables are declared at the beginning of the code The BLAS func tions require character variables where for example transposition of matrices is controlled by N and T as in not to transpose and transpose Some of these functions also require to mark whether the upper or lower triangular portion of a symmetric matrix is to be read in The order of the system is declared just as some common values as zero one or minus one The value of A is set to zero at starting time tolerance and error thresholds are also stated Finally local matrix and vector variables are declared as well char chn N cht T chu U chl L int onei 1 order 2 double one 1 0 mone 1 0 zero 0 0 lamN 0 tol 1e 5 err 2e 5 192 Gergely Takacs et al double WO W2 fval fderval double tempv 2 tempm 4 vec 2 tempv2 2 W1 2 Wid 2 Widd 2 m 2 2 After the local variable declarations are expressed the following mathe matical operation is performed in two steps Wo x Quo 1 5 11 where xp marks the current observed state and Qj is a partition of the matrix defining the invariant ellipsoid as calculated in the off line pr
330. y deviation variables y G u Gid G 2 2 2 2 4 Assumptions e No constraints u spans unconstrained subspace e We want to find as many controlled variables as there are degrees of free dom ne dim c dim u nu Then HG is a square n x n matrix e We use at least as many measurements as there are degrees of freedom Ny Ny Ne 2 2 5 Expected Set of Disturbances and Noise We write d W4 d where W is a diagonal matrix giving the expected magnitude of each disturbance and d is a normalization vector of unit mag nitude Similarly n Wn ny where W 4 is a diagonal matrix with the mag nitude of the noise for each measurement and the vector n is of unit mag nitude More precisely the combined normalization vectors for disturbances and measurement noise are assumed to have 2 norm less than 1 lt 1 2 3 d al 1 nY The choice of the 2 norm rather than for example the vector infinity norm is discussed in the Appendix of Halvorsen et al 2003 56 Sigurd Skogestad and Ramprasad Yelchuru and Johannes J schke 2 2 6 Problem Given that H y n c constant 0 nominally 2 4 Um find the optimal H such that magnitude of the loss L J u d Jop d 2 5 is minimized for the expected d and n The expected set of the disturbances and noise is defined above The magnitude of the loss still needs to be defined Two possibilities are co
331. y at its limit until the corresponding element of Dre changes sign Clearly the use of this anti windup methodology requires Dx to be invert ible and hence also of full rank Thus the controller must be semi proper The rate at which the controller states converge towards the steady state so lution when in saturation is given by the eigenvalues of Ax Bk Dy Cx This matrix obviously has to be stable A small but non singular Dx will generally make the convergence fast In Hanus et al 1987 self conditioning is presented in a more general setting potentially accounting also for time varying or non linear controllers However only in the case of linear time invariant controllers do the resulting controller equations come out in a relatively simple form 1 Problems in Anti Windup and Controller Performance Monitoring ri 1 1 2 5 Anti windup in Observer based Controllers Many advanced controllers are or may be implemented as a combination of static state feedback controllers and a state observer estimator This is the case for LQG Hz optimal controllers as well as H optimal controllers For such controllers anti windup is achieved by ensuring that the state observer estimator receives the actual plant input that is implemented on the plant This is illustrated in Fig 1 3 State feedback Saturation Umax Measurements y ci c K Plant State estimator observer
332. y be a temperature control system where the temperature control loop is the outer loop and the inner loop is a cooling water flow control loop with the valve opening as the plant input In such an example the maximum achievable flowrate may depend on up and downstream pressures which may depend on cooling water demand elsewhere in the system Possible ways of handling anti windup of the outer loop in such a situation include e Using conservative estimates of the available range of actuation with the possibility of not fully utilising plant capacity in some operating scenaria e The controller in the inner loop may send a signal informing the controller in the outer loop when it is in saturation and whether it is at its maximum or minimum value The controller in the outer loop may then stop the integration if this would move the controller output in the wrong direction e Use the velocity form of the controller provided the reference signal for the inner loop is calculated as present plant output change in reference from outer loop If the reference signal is calculated as reference at last time step change in reference from outer loop windup may still occur e For PI controllers use the implementation shown in Fig 1 1 where the plant input used in the outer loop is the plant measurement for the inner loop Note that the two latter anti windup schemes above both require a clear timescale separation between the loops th
333. y water As it was however mentioned above in case of regular maintenance and storing such problems will not occur Acknowledgements The work has been partially supported by the Slovak Grant Agency grants No VG 1 0656 09 and VG 1 0369 10 It was also supported by a grant No NIL I 007 d from Iceland Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism This project is also co financed from the state budget of the Slovak Republic 6 Laboratory Model of Coupled Tanks 241 References Zilka V 2007 Nonlinear controllers for a fluid tank system Master s thesis Fac ulty of Electrical Engineering and IT Slovak University of Technology Bratislava Slovakia in Slovak 242 Vladim r ilka and Mikul Huba Comments Remarks 6 Laboratory Model of Coupled Tanks 243 Comments Remarks 244 Vladim r ilka and Mikul Huba Comments Remarks 6 Laboratory Model of Coupled Tanks 245 Comments Remarks Chapter 7 Constrained PID Control Tasks for Coupled Tanks Control Vladim r ilka and Mikul Huba Abstract This chapter treats basic issues of linear and non linear controller s design applied to control of hydraulic plant of availabe three tank system Controllers based on approximation of the given non linear system by the first order linear system by step responses are proposed and compared with controllers based on the analytical linearization of the no

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