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User's guide for iREG

1. gt ad T t i 1 0 67 0 0 1 0 6 0 2 56 13 A rigorous analysis shows nevertheless that for any positive definite matrix F 0 F 0 gt 0 lim e t4 1 0 t oo The recursive least squares algorithm is an algorithm with a decreasing adaptation gain This is clearly seen if the estimation of a single parameter is considered In this case F t and t are scalars and 2 53 becomes F t 1 ramp SFO D Oen The same conclusion is obtained observing that F t 1 is the output of an integrator which has as input t oT t Since p t 7 t gt 0 one conclude that if t gt 0 in the average then F t 7 will tends towards infinity i e F t will tends towards zero The recursive least squares algorithm in fact gives less and less weight to the new prediction errors and thus to the new measurements Consequently this type of variation of the adaptation gain is not suitable for the estimation of time varying parameters and other variation profiles for the adaptation gain must therefore be considered Under certain conditions the adaptation gain is a measure of the evolution of the covariance of the parameter estimation error The least squares algorithm presented up to now for 6 t and t of dimension 2 may be generalized for any dimensions resulting from the description of discrete time systems of the form F t 4 1 1 dp 1 d B t t 2 57 u t Salt 2 57
2. 2 U t 2 Y t U t 2 89 Vi ytwt 1 2 90 UT t w t u t 1 2 91 Unfortunately as a consequence of the presence of noise this procedure cannot directly be applied in practice A more practical approach results from the observation that the rank test problem can be approached by the searching of 0 which minimize the following criterion for an estimated value of the order Vis N min X Y 9 R 292 where N is the number of data But this criterion is nothing else than an equivalent formulation of the least squares If the conditions for unbiased estimation using least squares are fulfilled 2 92 is an efficient way for assessing the order of the model since Vr Vr 1 0 when gt n In the mean time the objective of the identification is to estimate lower order models parsimony principle and therefore it is reasonable to add in the criterion 2 92 a term which penalizes the complexity of the model Therefore the criterion for order estimation will take the form CVrs N Vrs N S N 2 93 where typically S N 2 X N 2 94 X N in 2 94 is a function that decreases with N For example in the so called BICrs N criterion X N Si other choices are possible and the order is selected as the one which minimizes CVrs given by 2 93 Unfortunately the results are unsatisfactory in practice because in the majority of
3. Architecture tab It serves for the easier designing of digital RST controllers but only offers a limited although sufficient in most cases number of parameters that can be tuned You can e modify the fixed part of the S q7 polynomial by adding removing the integrator e modify the fixed part of the R q 1 polynomial by choosing whether to open or not the loop at 0 5 fs e modify the characteristic polynomial P q by changing the 1 regulation acceleration which is in fact the frequency of the dominant closed loop pole 2 robustification filter this is the value of the real multiple closed loop pole 3 or chose to include or not all plant stable poles by selecting the appropriate radio button from the Closed loop poles group he damping of these poles can be attenuated by a minimum value that can be specified here e modify the tracking model E fc by changing the 1 tracking acceleration which is the natural frequency for the tracking model 2 overshoot of the tracking model The button Controller design completed serves for easier utilization while using the automated mode of the program and switches to the Architecture tab where the user can save the controller or send it to the reduction part of the program 49 5 4 Figure 5 5 PI PID controller design tab active only while using the Continuous PI PID method It helps compute the parameters of a continuous PI or PID controller that is
4. A1 Ao as in Section 2 3 and the tolerance for computing the VGap distance of two sensitivity functions Method selection drop down list four method are available based on the CLIM algorithm with excitations added to the plant or controller input and also with or without filtered observation Section 6 Parameters of the PRBS signal frequency divider and number of bits in the shift register the program calculates the period of the signal nr 4094 in the given screen shot Start button will begin the identification of reduced order controller using the parameters provided by the user for the nominal system tolerance initial adaptation gain forgetting factor lambday and PRBS signal generator and the choosen method List of obtained controllers the first in the list is the nominal controller the others are the ones computed by the program the entries in the list that are marked with blue color will be shown in the Plots window 60 Controller Reduction Plots tab the nominal system s graphics are always shows so that the user can compare this with different reduced order controller Compute T by two possibilities are provided for calculating the polynomial T of the RST controller for the case of digital PID controller the T will be left unchanged T q R q All CL poles P B 1 T z is a polynomial with adjusted static gain containing all closed loop poles Gain P 1 B 1 T z 1 is a constant wit
5. AoA E 1 1 An O lt AD lt 1 2 71 The typical values being M 0 0 95 to 0 99 Ay 0 5 to 0 99 A1 t can be interpreted as the output of a first order filter 1 Ag 1 Aog7 with a unitary steady state gain and an initial condition A1 0 Relation 2 71 leads to a forgetting factor that asymptotically tends towards 1 The criterion minimized will be l J gt PG j y FH YP 2 72 1 i 1 j 15 As Au tends towards 1 for large i only the initial data are forgotten the adaptation gain tends towards a decreasing gain This type of profile is highly recommended for the model identification of stationary systems since it avoids a too rapid decrease of the adaptation gain thus generally resulting in an acceleration of the convergence by maintaining a high gain at the beginning when the estimates are at a great distance from the optimum Other types of evolution for A1 t can be considered For example oi t F t e t 1 ot t F t 9 t This forgetting factor depends upon the input output signals via t It automatically takes the value 1 if the norm of t t becomes null In the cases where the t sequence is such that the term o t F t o t is significative with respect to one the forgetting factor takes a lower value assuring good adaptation capabilities this is related to the concept of persistently exciting signal Another possible choice is At t 21 Ela
6. where A q l1 09 4p q 2 2 58 Ba bg bias q 5 2 59 2 57 can be written in the form NA nB y t 1 A aiylt 1 i b u t d i 1 07 t 2 60 i 1 i l in which e E m s anabi bns 2 61 am Lat w t na 1 u t d e u t d ng 4 1 2 62 The a priori adjustable predictor is given in the general case by NA nB PE 1 Y amp tut l i 0 b t ut d ic 1 i 1 i 1 6 t t 2 63 in which 87 t m i t mE na t bi t CN Ons t 2 64 and for the estimation of t the algorithm given in 2 51 through 2 54 is used with the appropriate dimension for 0 t p t and F t 2 3 Choice of the Adaptation Gain The recursive formula for the inverse of the adaptation gain F t 1 1 given by 2 52 is generalized by introducing two weighting sequences A1 t and A2 t as indicated below F t 1 M 0F 0 Ax t t t 0 A1 1 0 A t lt 2 F 0 gt 0 2 65 14 Note that A1 t and A2 t in 2 65 have the opposite effect A1 t lt 1 tends to increase the adaptation gain the gain inverse decreases A2 t gt 0 tends to decrease the adaptation gain the gain inverse increases For the purpose of this manual only the influence of A will be discussed as it is the chosen parameter for implementing different adaptation schemes Using the matrix inversion lemma given by 2 44 one obtains from 2 65 1 Leo POoWs FW PED xq 0
7. a gt 0 a 1 67 t F t t The forgetting factor tends towards 1 when the prediction error tends towards zero Conversely when a change occurs in the system parameters the prediction error increases leading to a forgetting factor less than 1 in order to assure a good adaptation capability Choice of the initial gain F 0 The initial gain F 0 is usually chosen as a diagonal matrix of the form given by 2 55 and respectively GI 0 F 0 2 73 0 i In the absence of initial information upon the parameters to be estimated typical value of initial estimates 0 a high initial gain GI is chosen A typical value is GI 1000 but higher values can be chosen If an initial parameter estimation is available resulting for example from a previous identification a low initial gain is chosen In general in this case GI lt 1 Since in standard RLS the adaptation gain decreases as the true model parameters are approached a significant measurement is its trace the adaptation gain may be interpreted as a measurement of the accuracy of the parameter estimation or prediction This explains the choices of F 0 proposed above Note that under certain hypotheses F t is effectively a measurement of the quality of the estimation This property can give indications upon the evolution of a parameter estimation procedure If the trace of F t did not decrease significantly the parameter estimation is in general poor This may
8. however this is never the case De RN 4 0 i gt 1 therefore one considers as a practical validation criterion extensively tested on applications RN 0 1 RN i lt T i21 2 79 where N is the number of samples This confidence interval corresponds to a 3 level of significance of the hypothesis test for Gaussian distribution Sharper confidence intervals can be defined Table 2 4 gives the values of the validation criterion for various N and various levels of significance The following remarks are important e An acceptable identified model has in general 1 8 RNG S N 1 gt 1 2 17 VN If several identified models have the same complexity number of parameters one chooses the model given by the methods that lead to the smallest FEN 1 e too good validation criterion indicates that model simplifications may be possible For simplicity s sake one can consider as a basic practical numerical value for the validation criterion value RN i lt 0 15 gt 1 If the level of the residual prediction errors is very low compared to the output level let us say more than 60 dB the whiteness tests lose their signification 3 Conversely for Gaussian data uncorrelation implies independence In this case RN i 0 i gt 1 implies independence between e t e t 1 i e the sequence of residuals e t is a Gaussian white noise 22 2 5 2 Validation of the Models Identified with Type
9. i b p 2 PT Titten hi Wed 1 2 3 4 5 6 7 8 Eti Mag dB c p 3 i MAL i reppin T 1 Mag dB 5 6 4 0 Frequency Hz Figure 2 8 Spectral density of a PRBS sequence fs 20 Hz a N28 p 1 b N28 p 2 c N 8 p 3 a a practical approach based on trial and error b estimation of d 4 ng directly from data Even in the case of using an estimation of the orders directly from data it is useful to see to what extend the two approaches are coherent We will discuss the two approaches next 2 5 5 A Practical Approach for Model Order Selection The plant disturbance model to be identified is of the form A q ult a B q ult w t where according to the structures chosen one has w t e t w t A q w t v t and u t are independent w t C q e t wt Siet The degrees of the polynomials A q 1 B q7 C q7 and Dia Il are respectively n4 np nco and np In order to start the parameter estimation methods n4 np and d must be specified For the methods which estimate also the noise model one needs in addition to specify nc and np A priori choice of n4 Two cases can be distinguished 1 Industrial plant temperature control flow rate concentration and so on For this type of plant in general Ta lt 3 and the value n4 2 which is very typical is a good starting value to choose 2 Electromechanical systems n4 results from the structural analy
10. n p 3 n 1 fu p un rr p An IT 8u p 3 n p 3 n 109994 I Yu ye 9 7 2 9 I Vvu 125 25 9 1 Yu 1 6 2 86 9 10SS0139 y 1o3oIp rd KC v 1p D I 9 Ta 2 72 q Tq 2 79 Tq 9 79 q tq 2 79 p To 2 9 o To 2 z Pp gt To 2 z 109994 MO 0 59 00 10 2 q 2 9 2 10 q 4 n 0 1o3ourereq qe1sn py ac nar a ngs f a nar I DOIN aston Wed AOTOA V suoryeA108qQ rg X nVIN IA PPO Am poxoy 1q o ndepy sup uo p xog YPM o q rreA TOJU9TINIISUT ta 10114 mdmo SUIUILIOS uorjeograuopr A1S10n291 Z 9 qer 21 Table 2 4 Confidence intervals for whiteness tests Level of Validation N 128 N 256 N 512 significance criterion 396 UM 0 192 0 136 0 096 596 UN 0 173 0 122 0 087 7 UN 0 16 0 113 0 08 Whiteness test Let e t be the centered sequence of the residual prediction errors centered measured value mean value One computes N R 0 SACH RN 0 s 2 77 N A R Belt 2 RN i us 2 78 i 1 0 soe with imax gt nA degree of polynomial A q which are estimations of the normalized autocorrelations If the residual prediction error sequence is perfectly white theoretical situation and the number of samples is very large N oo then RN 0 1 RN i 0 gt 1 In real situations
11. t 1 0 t 1 E e t g t i S 1 1 2 3 The validation method implements this principle It is made up of several steps 1 Creation of an I O file for the identified model using the same input sequence as for the system 2 Creation of files for the sequences y t g t e t system output model output residual output prediction error These files must contain at least 100 data in order that the test be significant 3 Uncorrelation test between the residual output prediction error sequence and the delayed prediction model output sequences Uncorrelation test Let y t and Q t be the centered output sequences of the plant and of the identified model respectively Let e t be the centered sequence of the residual output prediction errors centered measured values mean values One computes 1 N RG DO oi i 1 0 1 2 na 2 81 RN RU gt Fri 20 GEL eo i 0 1 2 na 2 82 which are estimations of the normalized cross correlations note that RN 0 Z 1 and nA is the degree of polynomial A q 1 If e t and g t i i gt 1 are perfectly uncorrelated theoretical situation then RN i 20 i 1 2 na 23 In practice one considers as a practical validation criterion 2 17 RN i lt i21 VN where N is the number of samples Table 2 4 can be used to find the numerical value of the validation criterion for various N and the level of signification of t
12. 1 Lambda0 0 95 1 1 000000 Figure 3 5 Parametric identification window different view 1 Identification Architecture PI PID Analysis Plots Tracking Reduc About iREG Step 4 Select method Closed Loop Output Error CLOE Step 5 Model complexity Delay 2 OrderofA 4 OrderofB 2 Ala kl q B g kk Ae wk Sa Mk Sie y I Ria pla Structure 5 dosed loop Step 7 Closed Loop Identification Controler R 0 000000 S 1 000000 Excitation superimposed to the reference E 1 000000 Step 6 Adaptation algorithm Initial adaptation gain Figure 3 6 Parametric identification window 38 different view 2 this is the denominator of the last identified valid plant model it can also be manually inputed by the user Recursive Maximum Likelihood RML the user has to specify some adaptation parameters Initial horizon and Contraction factor by default these have the values 50 respectively 1 00 Box Jenkins BJ and Generalized Closed Loop Output Error GCLOE are the only algorithms that use the order of a fourth polynomial D np Closed Loop Output Error with Filtered Observations FCLOE an initial plant model is necessary by default this is the last valid identified model but can also be manually inputed read from a file WinPIM or Matlab version 6 format or selected from the list of identified models 39 Part II Robust digit
13. 2 dei le da M Figure 3 4 Parametric identification window starting view 3 2 1 Parametric estimation window A basic view of this window is shown in Figure 3 4 More options can be visible as provided by the chosen algorithm and structure Figures 3 5 and 3 6 To ease the use this window has been divided into step which should be executed in the given order As can be observed the first to steps are included in the main tab of identification IO data set filtering and model order estimation Once in the parametric estimation window the user should select a structure for the model to be identified Step 3 e structures 1 4b are used for opened loop identification the input signal to the process is only the frequency rich signal used to excite the process in the scope of finding a correct model on a large frequency spectrum e g PRBS signal e structures 5 6b are used for closed loop identification the input signal to the process is a combination of PRBS and command given by the controller for closed loop identification the user has to specify a controller this can be done by choosing from a file WinREG or Matlab version 6 format by selecting the model that is currently loaded in the controller design part of iREG or by manually writing the coefficients of the R S and T polynomials in the appropriate text boxes also the user should specify the position where the excitation signal enters the closed loop s
14. 38 207090 als Next a certain number of choices for An E and their interpretations will be given A 1 Decreasing vanishing gain RLS In this case t Ay 1 A t 21 2 67 and EI 1 is given by 2 52 which leads to a decreasing adaptation gain The minimized criterion is that of 2 31 This type of profile is suited to the estimation of the parameters of stationary systems A 2 Constant forgetting factor In this case Ai t Au 0cA1 1 A2 t 2 1 2 68 The typical values for A are A1 0 95 to 0 99 The criterion to be minimized will be J t Y MT ty 67 He 1 2 69 i l The effect of A1 t lt 1 is to introduce increasingly weaker weighting on the old data i lt t This is why Ai is known as the forgetting factor The maximum weight is given to the most recent error This type of profile is suited to the estimation of the parameters of slowly time varying systems The use of a constant forgetting factor without the monitoring of the maximum value of F t causes problems in adaptive regulation if the o t 97 t sequence becomes null in the average steady state case because the adaptation gain will tend towards infinity In this case F t i Quyr t and F t i 1 F t For Ay lt 1 lim A1 oo and F t i will become asymptotically unbounded 1 00 A 8 Variable forgetting factor In this case A2 t A9 1 2 70 and the forgetting factor A t is given by M t
15. F t 1 F t T a 2 47 e t 1 y t 1 T telt 2 48 An equivalent form of this algorithm is obtained by introducing the expression of F t 1 given by 2 47 in 2 46 where t O F t 1 6 t e t 1 Foo gs 2 49 However from 2 7 2 8 and 2 49 one obtains e t 1 y t 1 O7 t 1 o t w t 1 t o t t 1 To t Pe Orig ei REED 2 50 1 7 0 F t o t 1 7 t F t o t which expresses the relation between the a posteriori prediction error and the a priori prediction error Using this relation in 2 49 an equivalent form of the parameter adaptation algorithm for the recursive least squares is obtained t 1 O t F t t e t 1 2 51 F t 1 71 F t t t 2 52 T F t 1 F t 1 DURAS im 2 53 y t 1 0 t t t TITO PES For the recursive least squares algorithm to be exactly equivalent to the non recursive least squares algorithm it must be started from a first estimation obtained at instant to dim t since normally F t given by 2 35 becomes nonsingular for t to In practice the algorithm is started up at t 0 by choosing F 0 i GDI 0 lt lt 1 2 55 a typical value being 0 001 GI 1000 It can be observed in the expression of F t 1 given by 2 37 that the influence of this initial error decreases with the time In this case one minimizes the following criterion 2 J t
16. a sequential minimization e sequential minimization of a criterion over a finite horizon starting at t 0 e relationship with the Kalman predictor However from the point of view of real time identification and adaptive control the parameter adaptation algorithms are supposed to operate on a very large number of measurements t oo Therefore it is necessary to examine the properties of parameter adaptation algorithms as t oo Specifically one should study the conditions which guarantee lim e t 1 2 0 i oo This corresponds to the study of the stability of parameter adaptation algorithms Conversely other parameter adaptation algorithms will be derived from the stability condition given above 2 5 Practical aspects of recursive open loop identification In this section the algorithms available in iREG for open loop plant model recusrive identification will be presented From a practical point of view plant model identification is a key starting point for designing a high perfomance linear controller Identification of dynamic systems is an experimental approach for determining a dynamic model of a system It includes four steps 1 Input output data acquisition under an experimental protocol 2 Selection or estimation of the model complexity structure 3 Estimation of the model parameters 4 Validation of the identified model structure of the model and values of the parameters As it will be seen the algorith
17. be changing the selection from the model s list e opening the parametric identification window this window Section 3 2 1 gives access to all parameters involved in the identification procedures the user can check and modify these values here to estimate a model while in advanced mode you have to open this window e the list of identified models shows the last 30 models that have been identified in the current session of the program the user can load a previous model by double clicking its name a window appears asking if the current plots should be kept for the purpose of comparing two or more models one should chose to keep the current graphics and use appropriate subplots for plotting the desired graphics of the loaded model another aspect of the model list is the name by witch a model is saved here this information consists of the structure used for identification the algorithm values of A forgetting factor and Ag and the order of the polynomials of that model e saving the model or using it to compute a robust digital controller the model can be saved in WinPIM or Matlab version 6 formats the model can also be send to the robust digital controller design part of iREG 36 Analysis Plots Tracking Reduc About REG Step 4 Select method Recursive Least Squares RLS Step 5 Model complexity Step 6 Adaptation algorithm Delay 2 Decreasing gain OrderofA 4 Initial adaptation gain 1000 OrderofB
18. cr pr pee Genulicauon Start Identification Architecture Controles Desin Band stop Filters Tracking About REG x unit x scale y scale Hz 0 linear 9 dB C rad s logarithmic linear Edit templates ratio f fs Freq1 Hz Mag 1 dB Load templates Plots control controller design Select subplot Freq2 Hz Mag2 dB Save templates Sup Syp Syp and Syb delay margin Ts A Grid ON Choose graphic to be ploted Modify Reset Figure 5 6 Analysis Plots tab for controlling the graphics associated with controller design This page also offers the possibility to modify the unit and scale of the x and y axis of the sensibility functions that are chosen for display Note that the modifications apply to all sensibility function graphs The Plots control controller design group is used for selecting different graphics to be displayed on each one of the four subplots available in the Analysis Plots tab of the Plots window Furthermore it is useful when one wants to show the grid or to modify the limits of the axes Note that these settings can be made separately for each of the four graphs displayed by using the drop down list One can also save as JPEG the selected graph by pressing the Save to JPEG button The Edit templates group can be used to modify the templates There are two ways of modifying the templates The first can be used independently for either Sup or Syp It
19. double click on one of them to modify its values To delete an unnecessary zero one can click the Delete button after previously double clicking on it in the list 47 Start Identification Architecture Analysis Plots Band stop Filters Tracking About REG Add Poles conjugate Natural frequency Hz 0 249500 oo d 0 800000 m val 0 300000 mlt 2 Multiple pole Total nr of poles 4 Model poles fn 0 095149 d 0 042142 n fn 0 250968 d 0 022103 Modulus margin 0 477 Delay margin 0 879 Ts 0 878781 M overshoot nat freq tr rise time s damp fn M tr Hr fixed part ofR Hs fixed part of S Open loop Add cplx zero V Integrator Add cplx pole val 1 000000 mlt 1 0 5fs 2200 i i Figure 5 3 Controller Design tab for advanced design of R and S polynomials 48 5 3 Automated digital RST controller design Acceleration for regulation loop 100 000000 Dominant pole design Include all plant stable poles Minimum damping V Integrator E Open loop at 0 5 fs Acceleration of tracking 100 000000 i Figure 5 4 Automated control design tab active only while in automated controller design mode It facilitates the designing of the polynomials R and S This tab Figure 5 4 will only be active while using the Digital RST automatic method selectable in the
20. is composed of 1 Filtering the data the select IO data set will be filtered by removing the mean centering the values on 0 and scaling making the inputs match the outputs from the maximum value point of view 2 The new set of data is saved with the same name as the initial one but with an autoID added to the end 3 A complexity estimation is run to find the orders of the model polynomials n4 ng and delay d 4 Three open loop identification algorithms are used Recursive Least Squares RLS Extended Recur sive Least Squares RELS and Output Error with Extended Prediction Model XOLOE for all of them an Decreasing gain adaptation is used with an initial gain of 1000 5 A best model is identified by comparing the largest values of the prediction error autocorrelations 6 A new identification rutine is run using the best model from the previous step with an Decreasint gain with variable forgetting factor adaptation algorithm A 0 97 and An 0 97 see Section 2 3 7 Based on the same criterion as before a new best model is selected from the two 8 If no valid model is found a new identification is done with an increased order of the C polynomial nc mazx na ng d only the Extended Recursive Least Squares RELS and Output Error with Extended Prediction Model XOLOE methods are used 9 In the end a check for common factors between polynomials A and B is done followed by a reiden tification if necessary further
21. or because the effect or the reduction of the signal noise ratio at high frequencies can be compensated by the use of appropriate identification techniques However it is recommended to choose p lt 4 Figure 2 8 shows the spectral density of PRBS sequences generated with N 8 for p 1 2 3 As one can see the energy of the spectrum is reduced in the high frequencies and augmented in the lower frequencies Furthermore for p 3 a whole occurs at f 3 Up to now we have been concerned only with the choice of the length and clock frequency of the PRBS however the magnitude of the PRBS must also be considered Although the magnitude of the PRBS may be very low it should lead to output variations larger than the residual noise level If the signal noise ratio is too low the length of the test must be augmented in order to obtain a satisfactory parameter estimation Note that in a large number of applications the significant increase in the PRBS level may be undesirable in view of the nonlinear character of the plants to be identified we are concerned with the identification of a linear model around an operating point 2 5 4 Model Order Selection In the absence of a clear a priori information upon the plant model structure two approaches can be used to select the appropriate orders for the values of d n4 np characterizing the input output plant model 25 TO Ir WI 0 1 2 3 4 5 6 7 8 9 10 Fm T IN hi Mag dB
22. parameter estimate at instant t obtained using t measurements First a parameter 0 must be estimated at instant t so that it minimizes the sum of the squares of the differences between the output of the plant and the output of the prediction model on a horizon of t measurements The value of t which minimizes the criterion 2 31 is obtained by seeking the value that cancels 5 J t 60 t 9J t _ i OT p i 1 E 22 06 6 t o i 1 9 i 1 0 2 33 From 2 33 taking into account that 07 t i let 1 eo 1 6 0 one obtains i E oi 1 19 i J A t gt y t oi 1 i i and left multiplying by2 gt gli 10 i J i 1 t is assumed that the matrix 37 dt 1 7 1 is invertible As it will be shown later this corresponds to an excitation condition 11 one obtains FOP Oi 2 34 in which F t Y iD i 1 2 35 This estimation algorithm is not recursive In order to obtain a recursive algorithm the estimation of t 1 is considered t 1 Ou 1 F t 1 gt y i e i 1 2 36 t 1 F t 1 7 2 pli 1 T i 1 F t 6 t 97 t 2 37 We can now express t 1 as a function of 6 t t 1 A t A t 1 2 38 From 2 37 one has 6 t 1 F t 1 ER y i i 1 y t 1 9 t 2 39 Taking into account 2 34 2 39 can be rewritten as t 1 F t 1 F t 6 t y t Dee 2 40 From 2 37 a
23. situations the conditions for unbiased parameter estimation using least squares are not fulfilled A solution would be to replace the matrix R fi by an instrumental variable matrix Z fi whose elements will not be correlated with the measurement noise Such an instrumental matrix Z f can be obtained by replacing in the matrix R the columns Y t 1 Y t 2 Y t 3 by delayed version of U t L i i e where L gt n Z R U t L 1 U t 1 U t L 2 U t 2 2 95 28 and therefore the following criterion is used for the order estimation 1 2nlog N CVp f N min lY t Zeg 2799 2 96 e N N and min CVp i 2 97 A typical curve of the evolution of the criterion 2 96 as a function of is shown in Figure 8 5 2 minimum Cen ae S a N lt penalty term I D D I I I I I t 2 7 Vp lt error term Figure 2 10 Evaluation of the criterion for order estimation Once an estimated order is selected one can apply a similar procedure to estimate a f d fug d from which A p and d are obtained 2 6 Practical aspects of recursive closed loop identification In practice it is sometimes very difficult or even impossible to carry an open loop identification experiment ex integrator behaviour open loop unstable In some situation a controller may already exist therefore there would be no reason to open the loop if one n
24. z 3 s z 1 With these notations we get e Ate 1 8o 2 1 HS G7 ys 27 Syp z E ET 4 11 PpPg z 1 s z 1 Ale 1 OH 1 Dr Ditz g z 1 where the filters F z Bg and F z 1 ER d consist of several second order notch filters alee band stop filters with limited attenuation simultaneously tuned The tuning means in fact searching for appropriate frequency characteristics of Fj z and F 27 Specifically in our case we are interested in frequency band stop with limited attenuation characteristics and thus the tuning concerns the frequency of band stop its bandwidth and the maximum attenuation in the band stop frequency 42 4 2 Digital controller design procedure Suppose that we dispose of the digital model G of the plant to be controlled The controller design consists of the following steps 1 Design specifications Determine desired closed loop and tracking performances The closed loop perfor mances such as robust stability disturbance rejection etc has to be expressed by some templates imposed on sensitivity functions The tracking properties include rise time maximal overshoot or settling time 2 Closed loop part R S design The sensitivity functions are shaped to satisfy design specifications to enter the frequency responses to imposed templates As we can see from the previous section one disposes with the following design parameter
25. 1 1 zt z7 and K is the nominal controller T R z K z 1 z S z 1 where Riz ro riz be drag F Steil l sz ck sng 78 6 2 l z lS z 1 The following sensitivity funtions will be usefull in this presentation e output sensitivity function 1 A z S z71 1 Sw TRG gt Pe 57 e input sensitivity function K A z 1 R z 1 ex EN EN Sec Jeer P z 1 e output sensitivity function with respect to an input disturbance dR 1 1 Sae G 2 z S z 1 KG P z 1 e complementary sensitivity function KG z 4p z l R z 1 SC ei Ek oque P z 1 where P z A 1 S z 2 B RO 1 The algorithms that will be used to find the reduced order controller are called Closed Loop Input Matching CLIM with excitation added to the controller input Figure 6 2 and Closed Loop Input Matching CLIM with excitation added to the plant input Figure 6 3 These are very similar to the Closed Loop Output Error CLOE algorithms used for recursive plant model identification in closed loop riUL X K u ep gt reduced order controller ep Ea E reduced order i controller PAA Figure 6 3 Closed Loop Input Matching with excitation added to the plant input AM amp Du gt K OX The signal x t is defi
26. 1 Validation of the Models Identified with Type I Methods This section is concerned with the validation of models identified using identification methods based on the whitening of the prediction error If the residual prediction error is a white noise sequence in addition to obtaining unbiased parameter estimates this also means that the identified model gives the best prediction for the plant output in the sense that it minimizes the variance of the prediction error On the other hand since the residual error is white and a white noise is not correlated with any other variable then all the correlations between the input and the output of the plant are represented by the identified model and what remains unmodelled does not depend on the input The principle of the validation method is as follows e If the plant disturbance structure chosen is correct i e representative of reality e If an appropriate identification method for the structure chosen has been used e If the degrees of the polynomials A q 1 B q C g 1 and the value of d delay have been correctly chosen the plant model is in the model set Then the prediction error e t asymptotically tends toward a white noise which implies Jim E e t e t i 0 i 1 2 3 1 2 3 The validation method implements this principle It is made up of several steps 1 Creation of an I O file for the identified model using the same input sequence as for the system 2
27. 221129 0 103510 S 1 000000 1 062358 0 196010 0 085613 0 000540 0 047500 Hr 1 000000 1 000000 Hs 1 000000 1 000000 Ts 0 050000 _Return without modifying data Figure 7 2 Model and controller parameters that are currently used to describe the nominal system It is recomended to chose Tolerance 0 001 Initial adaptation gain 1000 Forgetting factor Ai 0 95 1 Xo 0 5 1 PRBS frequency divider div 1 or 2 PRBS length of shift register 11 61 Bibliography 1 L D Landau and A Karimi Robust digital control using pole placement with sensitivity function shaping method nt J Robust and Nonlin Cont 1998 2 I D Landau and A Karimi A unified approach to model estimation and controller reduction duality and coherence European Journal of Control 2002 3 I D Landau A Karimi and A Constantinescu Direct controller order reduction by identification in closed loop Automatica 37 1689 1702 2001 4 I D Landau R Lozano M M Saad and A Karimi Adaptive control Springer London 2nd edition 2011 5 I D Landau and G Zito Digital Control Systems Design Identification and Implementation Springer London 2005 6 L Ljung and T Soderstr m Theory and practice of recursive identification The M I T Press Cambridge Massachusetts London England 1983 7 Hynek Prochzka and Ioan Dor Landau Pole placement with sensitivity function shaping using 2nd or
28. Acc 5 Compute Ty 6 Compute the parameters of the PI continuous controller Hpr s K 1 xc e If RapGain gt 0 25 K een eas k Ta 2 3 RapGain 0 3 43 4 3 2 The KLV method for the computation of PID controllers 1 2 Define Go the processes amplification at the zero frequency Define the frequency at which the total phase shit reaches 135 as w135 and the amplification for which this phase shift is obtained as G35 Define the acceleration factor Fact Acc as a value between 25 and 200 Ts 21 0 W135 There are three ways to specify the filtered component of the derivative a N imposed Tz 1 x A 44 FactAcc Ty 0 b amp imposed Ty 44 Ty c 44 20 Ts Ty Compute the parameters of the PID continuous controller Hprp s K 1 x gt Ka s 1 e K 4 FactAcc Zi V 2 Gr135 e Ka Ts 44 Chapter 5 How to use the application This part of the program is composed of 7 tabs Architecture Controller Design Automated control design PI PID Analysis Plots Band stop Filters and Tracking They will pe presented in the next sections 5 1 Architecture tab This is the main tab of Controller Design part of the software Figure 5 1 me dentification Start Identification C Digital RST automatic Continuous PI PID Load plant model Save controller and model To MAT v6 File To WinREG File To NX RST F
29. Creation of a prediction error file for the identified model minimum 100 data 3 Whiteness uncorrelatedness test on the prediction errors sequence also known as residual prediction errors 18 D 2 10 29A UOI AIOSQ p I 2 o2 Toig uonejdepy T 2 6 T 2 T 17 TIOT19ISOd Y IOIIG I 1 2 6 1 9 poud e uonorpoiq 2 6 1 t 2 59 tU Houojsod e yndyng 2 O 4 0 1 Hof LOL e iojotporq 1 2u p D gt 1 2u p 9 iw dui NR x m T 8u p 3 n p 3 n 1 8u p 3 n p 3 n 109994 r Vu 26 6 G r Vu p A 0 59 Yu 2 e 10859139Y 10j0tpoiq Ou Vu xeu Hu PE 2 74 QU 70 H 2 uqa 9 29 q T a 79 ng tq 2 79 o To 2 72 Dp etr 9 72 Vup gt gt gt Tp 3 2 109994 uQ q 0 221 0 9 MD 0 221 9 19 2 799 72 10 1oyourereq o qegsnfpy s 25 tay r n I DO N SSION WRI HO IOX PPOW uonorpe1q STAY sorenbS 3seoT STA serenbg seo pepuejxq tu 10117 ndo popuoyxH dAISINDOY SUIT JLIOZ S uorjeogrquepr SAISINIIH T Z ALL 19 dis 1 2 T Ari H Yo 10 Yo 79 9722 Yo 10999 VOLYRA19SO I 1 2 10115 uorjejdepy I 1 2 8 19 2 LIOLI93s0d e IOI T of 1 9 4 1 9 uoud e uonorpoiq NOT 79 0 1 D
30. II Methods This section is concerned with the validation of models obtained using the identification methods based on the decorrelation between the observations and the prediction errors The uncorrelation between the observations and the prediction error leads to an unbiased parameter estimation However since the observations include the predicted output which depends on the input the uncorrelation between the residual prediction error and the observations implies also that the residual prediction error is uncorrelated with the input The interpretation of this fact is that the residual prediction error does not contain any information depending upon the input i e all the correlations between the input and the output of the plant are represented by the identified model The principle of the validation method is as follows e If the disturbance is independent of the input gt E w t u t 0 e If the model disturbance structure chosen is correct i e representative of the reality e If an appropriate identification method has been used for the chosen structure If the degrees of the polynomials A q B q7 and the value of d delay have been correctly chosen the plant model is in the model set then the predicted outputs g t 1 9 t 2 generated by a model of the form output error type predictor Aug Toi a B q ult 2 80 and the prediction error t are asymptotically uncorrelated which implies N Set
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32. User s guide for iREG Identification and Controller Design and Reduction Ioan Dor Landau and Tudor Bogdan Airimitoaie December 14 2013 Contents 1 About iREG 3 LI Sotware requirements osos FOR kao RE RE e c he kus m ede Ge ew W ac E do w s W 3 LX Cpu Ej ci o o s ioii e de OF ED PROD BAA DR Oe hdd a ee e dS 3 L3 Description of the software s i osa oem A a w a a RR P RR a W 3 I Identification 5 2 Basic theory 6 2l Gradient AMECA y ox cR RS RR Rubus Wee Re Ge Gack deu Ree Ge ae eG 7 2 1 1 Improved Gradient Algorithm o aa s racea wna k kak wenn a wu e n e iaria 9 2 2 Recursive Least Squares Algorithm 10 2 3 Choice of the Adaptation Gain 14 2 4 Some Remarks on the Parameter Adaptation Algorithms eee 16 2 5 Practical aspects of recursive open loop identification eee eee e e 17 2 5 1 Validation of the Models Identified with Type I Methods 18 2 5 2 Validation of the Models Identified with Type II Methods 23 2 5 3 Selection of the Pseudo Random Binary Sequence aaaea 24 2 04 Model Order Selection vu 2c kn u w gui kem ems a AE w la ee e w Re 25 2 5 5 A Practical Approach for Model Order Selection eee eee 26 2 5 6 Direct Order Estimation from Data 28 2 6 Practical aspects of recursive closed loop identification 29 3 How to use the application 32 3 1 Automatic use of the Identification tab aoaaa a 32 3 2 Advanc
33. adaptation gain plays an important role in the performances of the parameter adaptation algorithm and it may be constant or time varying The problem addressed in this chapter is the synthesis and analysis of parameter adaptation algorithms in a deterministic environment 2 1 Gradient Algorithm The aim of the gradient parameter adaptation algorithm is to minimize a quadratic criterion in terms of the prediction error Consider the discrete time model of a plant described by y t 1 ary t bru t 07 t 2 1 where the unknown parameters a and b form the components of the parameter vector 0 gr a4 51 2 2 and 9 t y t u t 2 3 is the measurement vector The adjustable prediction model will be described in this case by jf t 1 HE UO a tult b t u t OT HOM 2 4 where t 1 is termed the a priori predicted output depending on the value of the estimated parameter vector at instant t OT t t bit 2 5 As it will be shown later it is very useful to consider also the a posteriori predicted output computed on the basis of the new estimated parameter vector at t 1 OU 1 which will be available somewhere between t 1 and t 2 The a posteriori predicted output will be given by t 1 KEE DPE 0 41 t 1 y t by t 1 u t 6 t 1 t 2 6 One defines an a priori prediction error f t 1 y t 1 00 t 1 2 7 sense of adaptation
34. al controller design Chapter 4 Basic theory This chapter presents the key aspects of designing robust digital controllers 4 1 Notation A standard digital pole placement configuration using a polynomial controller denoted R S is shown in Fig ure 4 1 The plant model G z is of the form Bla biz 1 b z 2 G 1 d d nB 4 1 z E A z 1 E eo a EE Qs 42 nA y t controller model t r t E u t y t Bn Am T E 1 S B A b t The R S part of the controller has the transfer function R 27 Ro z Hm z 7 S 271 So z Hs z where Hp z and Hs z 1 denote the fixed parts of the controller either imposed by the design or introduced in order to shape the sensitivity functions Ro z 1 and So z 1 are solutions of the Bezout equation poles of the closed loop 4 2 Ar Ds z Hs z z 4B z Ry z Hg z Pp z Pp z 1 4 3 SS P z 1 where P z represents the desired closed loop poles Pp 27 defines the dominant poles specified and Pp z 1 defines the auxiliary poles which in part can be specified by design specifications and the remaining part is introduces in order to shape the sensitivity functions The tracking part T z of the controller is used to compensate the closed loop dynamic in such way that 1 the entire system transfer function from r t to y t has the dynamic of the reference model SS The polyno
35. also discretized to the digital RST form Continuous PI PID design using the KLV method Start Identification Architecture PI PID Analysis Plots Tracking About iREG Controller type Derivative filter Td N Loose Continuous PID controller N Imposed Td N Acceleration factor 25 200 100 000000 I Without i N 5 000000 Proportional k 0 011129 Integral Ti 0 904563 Derivative Td 0 000000 Derivative filter Td N 0 000000 so that it can be analysed This tab Figure 5 5 will only be active while using the Continuous PI PID method selectable in the Architecture tab It can be used to compute the parameters of a continuous PI or PID controller The program automatically computes the discretized R q and S q 1 which it uses to compute the graphics Also the program only saves the discretized R and S and the values of P I D and filter of D can be read in this tab In this tab the user can switch between PI and PID change the acceleration factor for the KLV method see 4 3 for more details read the values of the parameters for the continuous regulator P I D and the derivative of D change the mode for computing the derivative s filter 50 5 5 Analysis Plots Fig 5 6 shows how this tab looks like Using this you can select the graphics that will be displayed on the Analysis plots tab of the Plot Window and modify their characteristics X yaar
36. at step 2 Finally if it is necessary the sampling period can be modified by clicking on the Sampling period button see Figure 3 1 35 e selecting a different IO data set from the list that displays all the available data set it should be pointed out that by filtering the data a new data set is created that will be saved in this list then the user can return to a previous data set by double clicking its name e modifying the parameters of the identification plots the user can select which graphs will appear and in what subplot of the Identification Plots tab from the Plots window each part of iREG iden tification robust digital controller design and controller order reduction has its one tab s in the Plots window and change the limits and grids of each graphic e IO data set filtering once there is at least one IO data set in the memory of the program the user has the following options for modifying the selected data Select range creates a new IO data set that contains only a part of the original one Remove DC component creates a new IO data set that is centered on zero Add offset creates a new IO data set that based on the original one by adding a scalar value to all it s components the values added to the input and output can be different Multiply by gain creates a new IO data set by multiplying all values of the original set with a scalar value the value used for the inputs can be diffe
37. consists of choosing a starting frequency and a ending frequency together with a starting magnitude and a ending one The template will then be modified by drawing a straight line between these two points The second way to modify the templates is choosing a new delay margin This will modify both Syp and Sup The delay margin should be given as a multiple of the sampling period Ts Once modified the templates can be saved for latter use 51 5 6 Band stop Filters Start Identification Architecture Controller Design Analysis Plots Attenuation Amplification 1 000000 Enable Sup filters Enable filter indicators Frequency Hz 0 250000 Filters for Syp Filters for Sup Add fiter for Sup Figure 5 7 Band stop Filters tab used for designing second order notch filters This window is used for the design of filters It is possible to add filters for Syp and Sup Each filter has three parameters that can be adjusted e Attenuation Amplification of the filter e Frequency at which the filter operates e Damping of denominator adjusts the width of the filter The list box on the right shows all filters that have been created They are grouped by the sensitivity function that they influence directly either Syp or Sup The filters on either one of the sensitivity functions can be enabled or disabled by clicking one of the appropriate check boxes The Enable filter indicators will put
38. d 2 8 it results that e t 1 y t 1 H t 1 y t 1 07 t D t 2 22 and respectively sel e t 1 50 t 1 as t 2 23 Introducing 2 23 in 2 21 the parameter adaptation algorithm of 2 11 becomes t 1 OU F t e t 1 2 24 This algorithm depends on e t 1 which is a function of 0 t 1 For implementing this algorithm e t 1 must be expressed as a function of e t 1 i e e t 1 f 0 t t e t 1 2 22 can be rewritten as t 1 y t 1 TOGE OE 1 AATA 2 25 The first two terms of the right hand side correspond to e t 1 and from 2 24 one obtains t 1 A t F t e t 1 2 26 which enables to rewrite 2 25 as ett 1 e t 1 gi t F t e t 1 2 27 from which the desired relation between e t 1 and t 1 is obtained e t 1 17 prey 2 28 and the algorithm of 2 24 becomes F t e t 1 t 1 t 1 T DF2 D 2 29 which is a stable algorithm irrespective of the value of the gain matrix F positive definite The division by 1 dl t F t introduces a normalization with respect to F and t which reduces the sensitivity of the algorithm with respect to F and t In this case the equation for the evolution of the parametric error t is given by T 64 1 1 TRO r AHIT 2 30 and the eigenvalues of A t w
39. d test controller and Load test model these group can be used to test a combination of model controller This offers the possibility to plot different graphs and to check the robustness in dicators modulus and delay margins There are another two buttons available e System parameters switches the current Architecture tab to one that displays all current parameters from the controller design part of the program Figure 5 2 e Send do Controller Reduction the controller that is currently being designed will be sent to controller reduction tool together with the plant model that has been used here Section 7 This allows for the design of a controller with a smaller number of parameters and a digital PID controller 5 2 Advanced digital RST controller design This tab can be viewed in Figure 5 3 It is very important to know that this tab is deactivated while using the Digital RST automatic or Continuous PI PID methods and will not be visible There are three parts of this tab 1 Add Poles is used to modify the closed loop poles and is situated in the upper part of the screen In the right the poles of the closed loop system and of the plant are displayed The program offers the possibility to add remove complex real closed loop poles e to add a real multiple pole you have to click on the New pole button Once you have added the new pole you can modify it to make it a complex pole by clicking on the radio buttons fro
40. der digital notch filters Automatica 39 6 1103 1107 2003 8 A Voda and I D Landau The autocalibration of pi controllers based on two freqeuncy measurements International Journal of Adaptive Control and Signal Processing 9 395 421 1995 9 A Voda and I D Landau A method for the auto calibration of pid controllers Automatica 31 41 53 1995 62
41. e number of cells of the shift register Figure 2 6 summation modulo 2 Figure 2 6 Generation of a PRBS of length 2 1 31 sampling periods presents the generation of a PRBS of length 31 2 1 obtained by means of a 5 cells shift register Note that at least one of the N cells of the shift register should have an initial logic value different from zero one generally takes all the initial values of the N cells equal to the logic value 1 Table 2 5 gives the structure enabling maximum length PRBS to be generated for different numbers of cells Note also a very important characteristic element of the PRBS the maximum duration of a PRBS impulse is equal to N number of cells This property is to be considered when choosing a PRBS for system identification In order to correctly identify the steady state gain of the plant dynamic model the duration of at least one of the pulses e g the maximum duration pulse must be greater than the rise time g of the plant The maximum duration of a pulse being N T the following condition results N T gt tg 2 83 which is illustrated in Figure 2 8 From condition 2 83 one determines N and therefore the length of the sequence which is 2 1 Furthermore in order to cover the entire frequency spectrum generated by a particular PRBS the length of a test must be at least equal to the length of the sequence In a large number of cases the duration of the test L is chos
42. e output sequences For text files one can select the appropriate column number for the input and the output sequences Each drop down box gives the possibility to select a number from 1 to the total number of columns If the total number of columns obtained in iREG is different from the expected one after visual inspection of the text file make sure that the right column separators are used as indicated at step 2 5 Finally if it is necessary the sampling period can be modified by clicking on the Sampling period button see Figure 3 1 selecting a different IO data set from the list of all available ones modifying the parameters of the identification plots the user can select which graphs will appear and in what subplot of the Identification Plots tab from the Plots window each part of iREG iden tification robust digital controller design and controller order reduction has its one tab s in the Plots window and change the limits and grids of each graphic analyzing the identified models the user can plot different graphs in the subplots of the identification tab in the Plots window among these there are a few that can be used to analyze the model these are strictly related to the selected model from the model list therefore the user can analyze different models be changing the selection from the model s list Start parameter estimation this will begin the identification procedure available in the automated mode this
43. e parameter adaptation algorithm PAA which drives the parameters of the adjustable prediction model from the data acquired on the system at each sampling instant This algorithm has a recursive structure i e the new value of the estimated parameters is equal to the previous value plus a correcting term which will depend on the most recent measurements In general a parameter vector is defined Its components are the different parameters that should be estimated The parameter adaptation algorithms generally have the following structure New estimated Previous estimated Adaptation parameters parameters gain vector vector matrix Measurement Prediction error x function x function vector scalar This structure corresponds to the so called integral type adaptation algorithms the algorithm has memory and therefore maintains the estimated value of the parameters when the correcting terms become null The algo rithm can be viewed as a discrete time integrator fed at each instant by the correcting term The measurement function vector is generally called the observation vector The prediction error function is generally called the adaptation error As will be shown there are more general structures where the integrator is replaced by other types of dynamics i e the new value of the estimated parameters will be equal to a function of the previous parameter estimates eventually over a certain horizon plus the correcting term The
44. ed use of the Identification tab o sa w a ua css e r Ea Rm RA 35 3 2 1 Parametric estimation window 2 2 2 2 4 s 3T II Robust digital controller design 40 4 Basic theory 41 s R Noti y Kil suwas a BR A m s RB TUTTI 41 42 Digital controller design procedure sedora e a e wa wa kuwan eres 43 4 3 Continuous PI PID controller design oi sgag asa q a ga w Ua w w Q s q p 8 e e q W UQ 43 4 3 1 The KLV method for the omputation of PI controllers 43 4 3 2 The KLV method for the computation of PID controllers 44 5 How to use the application 45 eet dabo Q uu u 6 hada s w BARRA PE OS OE EUR eps RU a uku tup e de ed GS 45 5 2 Advanced digital RST controller design 46 5 9 Automated control Agen nu nc xm Robo RR ove oco w e EU US POR GU UR A E RR w w x 49 54 Contingous PI EID controller uos uo Y Eoo RR eR cem mE HR eee ow gs 50 tubo Ausus PIG o Da Ge die Robe RR a E RU E E a 51 5 6 Band stop Filters 5 7 Tracking III Controller reduction 6 Basic theory 7 How to use the application 56 57 60 Chapter 1 About REG iREG Identification and Robust Digital Controller Design is a software product designed in Visual C It is a user friendly application with graphical user interface GUI for identifying SISO Single Input Single Output processes and designing robust digital controllers for SISO plants There are various routines for identification in opened and closed loo
45. eeds to find a better model for improving the existing controller A basic scheme for closed loop identification of a plant with a RST controller is presented in Figure 2 11 Figure 2 11 Closed loop identification scheme with PRBS added to the plant input The objective is to estimate the parameters of the plant model defined by 2 57 2 59 The output of the plant oparating in closed loop is given by y t 1 A y t B u t d Av t 1 6T y t Av t 1 2 98 where u t is the plant input y t is the plant output w t is the output disturbance noise and 29 where r t is the external excitation added to the control input 2 99 Two more options are available for the position where the external excitation enters the closed loop system In this case in 2 99 r t will be replaced by r t e excitation added to the reference e excitation added to the measurement For a fixed value of the estimated parameters the predictor of the closed loop is described by g t 1 A G t Droit d 67 t where T i Beni bus g t 9 t t n4 D t d t ng 1 d at S ll rale The closed loop prection output error is defined as eor t 1 y t 1 t 1 2 100 2 101 2 102 2 103 2 104 The parameter adaptation algorithm remains the same as in the open loop recursive identification 2 51 2 54 Table 2 6 summarizes
46. en equal to the length of the sequence If the duration of the test is specified it must therefore be ensured that 2N 1 T lt L L test duration 2 84 24 Table 2 5 Generation of maximum length PRBS Number of Cells Sequence Length Bits Added N L 2N 1 B and B 2 3 Land 2 3 7 1 and 3 4 15 3 and 4 5 31 3 and 5 6 63 5 and 6 7 127 4 and 7 8 255 2 3 4 and 8 9 511 5 and 9 10 1023 7 and 10 Figure 2 7 Choice of a maximum duration of a pulse in a PRBS Note that the condition 2 83 can result in fairly large values of N corresponding to sequence lengths of prohibitive duration either because T is very large or because the system to be identified may well evolve during the duration of the test This is why in a large number of practical situations a submultiple of the sampling frequency is chosen as the clock frequency for the PRBS If ferns P ipo iae 2 85 then condition 2 83 becomes p N T gt tg 2 86 This approach is more interesting than the increase of the sequence length by increasing N in order to satisfy 2 83 Note that dividing the clock frequency of the PRBS will reduce the frequency range corresponding to a constant spectral density in the high frequencies while augmenting the spectral density in the low frequencies In general this will not affect the quality of identification either because in many cases when this solution is considered the plant to be identified has a low band pass
47. eria This is linked on the one hand to the complexity of the controller which will depend on n4 and ng but equally to the robustness of the identified model with respect to the operating conditions A first approach to estimate the values of N4 max and np max is to use the RLS and to study the evolution of the variance of the residual prediction errors i e the evolution of N 1 E 0 gt t R 0 E et Nel as a function of the value of n4 np A typical curve is given in Figure 5 8 1 R o d the good value presence of noise ME absence of noise 1 2 3 4 5s 6 7 Spar DA DB Figure 2 9 Evolution of the variance of residual errors as a function of the number of model parameters In theory if the example considered is simulated and noise free the curve should present a neat elbow followed by a horizontal segment which indicates that the increase in parameter number does not improve the performance In practice this elbow is not neat because measurement noise is present The practical test used for determining na ng is the following consider first n4 npg and the corresponding variance of the residual errors R 0 Consider now n na 1 ng and the corresponding variance of the residual errors R 0 If R 0 gt 0 8R 0 it is unwise to increase the degree of na same test with n ng 1 With the choice that results for n4 and np the model identified by the RLS does not necessarily verify the va
48. fter post multiplying both sides by t one gets FETH F t 1 t t T t 0 t 2 41 and 2 40 becomes 6 t 1 F t 1 F t 1 14 O y t 1 8 Dace 2 42 Taking into account the expression of e t 1 given by 2 13 the result is t 1 F t 1 t e t 1 2 43 The adaptation algorithm of 2 43 has a recursive form similar to the gradient algorithm given in 2 15 except that the gain matrix F t 1 is now time varying since it depends on the measurements it automatically corrects the gradient direction and the step length A recursive formula for F t 1 remains to be given from the recursive formula F t 1 given in 2 37 This is obtained by using the matriz inversion lemma Matrix Inversion Lemma Let F be a n x n dimensional nonsingular matriz R a m x m dimensional nonsingular matriz and H a n x m dimensional matriz of maximum rank then the following identity holds F HR HT F FH R X H FH HTF 2 44 12 Proof By direct multiplication one finds that F FH R HTFH HT FF HR HT I For the case of 2 37 one chooses H t R 1 and one obtains from 2 37 and 2 44 Freide pn Wee 1 9t t F t o t and putting together the different equations a first formulation of the recursive least squares RLS parameter adaptation algorithm PAA is given below F t 1 F t 0 t 1 0 t F t 1 t e t 1 2 46 T
49. g plots All the tracking plots are displayed in the third tab of the Plots Window called Tracking Plots An example of this window can be seen in Figure 5 10 You can see in this figure how the perturbation is rejected with the current controller 54 Magnitude Mognitude System output y t step response Command u t step response Magnitude 2 000 o Desired trajectory y t step response Magnitude Figure 5 10 Tracking Plots tab of the Plots window 55 Part III Controller reduction Chapter 6 Basic theory v t p t r K u G Ki y Figure 6 1 Nominal closed loop system This chapter addresses the problem of directly estimating the parameters of a reduced order digital controller using a closed loop type identification algorithm The algorithm minimizes the closed loop plant input error between the nominal closed loop system and the closed loop system using the reduced order controller 3 and 2 It is assumed that a plant model if necessary validated in closed loop with the nominal controller is available In the given nominal system from Figure 6 1 the G is the identified plant model Un z z 1 Re A z 1 where Biz bizt bnp z 1B z 1 A z 1 z na D 6
50. gra dient ho a Figure 2 2 Principle of the gradient method and an a posteriori prediction error e t 1 y t 1 t 1 2 8 The objective is to find a recursive parameter adaptation algorithm with memory The structure of such an algorithm is i R t 1 t AG t 1 OE FOE t eO t 1 2 9 The correction term f 6 t 6 t t 1 must depend solely on the information available at the instant t 1 when y t 1 is acquired last measurement y t 1 0 t and a finite amount of information at times t t 1 t 2 t n The correction term must enable the following criterion to be minimized at each step min J t 1 e t UP 2 10 HO A solution can be provided by the gradient technique If the iso criterion curves J constant are represented in the plane of the parameters a1 bi concentric closed curves are obtained around the minimum value of the criterion which is reduced to the point a1 bi corresponding to the parameters of the plant model As the value of J const increases the iso criterion curves move further and further away from the minimum This is illustrated in Figure 2 2 In order to minimize the value of the criterion one moves in the opposite direction of the gradient to the corresponding iso criterion curve This will lead to a curve corresponding to J const of a lesser value as is shown in Figure 2 2 The corresponding parametric adaptation algorithm will ha
51. h an appropriate value so that the static gain of the closed loop system is equal to one Perturbation parameters the user has the possibility to check the reduced order controller on a step reference magnitude 1 and with a step disturbance the time of appearence and amplitude of the per turbation can be chosen by the user the effect on the closed loop system with nominal reduced controller can be viewed in the Plots window Controller Reduction Plots tab Theoretical image of the two systems with nominal and reduced order controllers this image is in accor dance with the algorithm used for estimation Compute Digital PID this button can be used as an alternative to Start it will compute two digital PID controllers using the filtered versions of the CLIM algorithms excitation on controller respectively plant input the controllers are of second order with integrator the T q7 polynomial is equal to R q and 1 the reference modes is absent q S 1 so that the graphics will show the actual performances if implemented on a PID control structure Save selected controllers this will save the selected controller s to a single Matlab version 6 file in the future there will be also the possibility to save in NX RST format n SSS Controller reduction data A 1 000000 1 601555 1 865776 1 493516 0 886836 B 0 000000 0 000000 0 000000 0 304177 0 400380 R 0 110531 0 352889 0 156603 0 295384 0
52. he test All the comments made in the previous section apply also in this case In particular the basic practical numerical value for the validation criterion which is RN i lt 0 15 i2 1 is worth remembering his test is also used when one would like to compare models identified with Type I method with models identified with Type II method 2 5 3 Selection of the Pseudo Random Binary Sequence The correct parameter estimation requires the use of a rich signal persistently exciting signal Pseudo random binary sequences PRBS offer on the one hand a signal with a large frequency spectrum approaching the white noise and on the other hand they have a constant magnitude This allows to define precisely the level of the instantaneous stress on the process or actuator PRBS are sequences of rectangular pulses modulated in width that approximate a discrete time white noise and thus have a spectral content rich in frequencies They owe their name pseudo random to the fact that they are characterized by a sequence length within which the variations in pulse width vary randomly but that over a large time horizon they are periodic the period being defined by the length of the sequence In the practice of system identification one generally uses just one complete sequence The PRBS are generated by means of shift registers with feedback implemented in hardware or software The maximum length of a sequence is 2 1 in which N is th
53. ics and modify their appearance Band stop Filters used to implement filters for the precise shaping of the sensibility functions Tracking for designing the tracking part of a digital controller reference model and polynomial T Reduc can be used to reduce the order of a RST controller About REG information regarding the software name version authors and copyright Part I Identification Chapter 2 Basic theory The recursive parameter estimation principle for sampled models is illustrated in Figure 2 1 DISCRETIZED PLANT Adjustable Discrete time Model estimated Parameter model a Adaptation parameters Algorithm Figure 2 1 Parameter estimation principle A discrete time model with adjustable parameters is implemented on the computer The error between the system output at instant t y t and the output predicted by the model g t called plant model error or prediction error is used by the parameter adaptation algorithm which at each sampling instant will modify the model parameters in order to minimize this error in the sense of a certain criterion The input to the system is a low level and frequency rich signal generated by the computer for the case of plant model identification in open loop or a combination of this and the signal generated by the controller in the case of closed loop identification The key element for implementing the on line estimation of the plant model parameters is th
54. identification Plots control identification plots Options for controlling Select subplot Choose graphic to be ploted the identification Load I O Data Name of data IO data loading options Grid on xmin 0 000000 ymin 0 000000 E Limits on xmax 5 000000 y max 0 150000 List of transformations that can be applied to the currently selected IO data set tep 1 Edit selected IO data with a filter from the list Have you removed the DC component List of IO data sets in List of the models that the memory of the have been identified program Using this you can save Methods for estimating the selected model or the complexity of the use itto compute a model based on the controller with iRDCD dest ran orders nA nB and Figure 3 3 Tab for the advanced mode of using the identification algorithms delay d mode The main window Figure 3 3 offers control over e switching between advanced and automated modes the tab will alternate view from Figure 3 3 to Figure 3 1 and back e loading IO data sets to types of input output data sequences can be used in iREG basic text files dat trt or c extensions and Matlab s mat files All input output sequences loaded in REG are shown in the List of I O data sets see lower left box in Figure 3 1 The next steps present the procedure for loading I O data sequences in iREG 1 First
55. igure 3 3 and back e loading I O data sets to types of input output data sequences can be used in iREG basic tezt files dat trt or c extensions and Matlab s mat files All input output sequences loaded in REG are shown in the List of I O data sets see lower left box in Figure 3 1 The next steps present the procedure for loading I O data sequences in iREG 1 First the user should introduce a short name for the I O data sequence that is going to be loaded in iREG in the Name of data text box if this is left empty not recommended the name of the file on the disk will be used as name for the I O data sequence 32 2 If you are planing to load data from a text file it is important to open it first in a text editor e g Notepad to see which column corresponds to the input and which to the output Note that the accepted column separators are space tab semicolon and Z If you are loading data from a Matlab mat file make sure that it has been saved in the Matlab version 6 format the proper command to use in Matlab is save v6 file name variables 3 Click on the Load I O data button select the appropriate type of file that you want to load search for the file on the disk and than click Ok 4 A new dialog box opens which is used for selecting the appropriate input output data from the given file For mat files one can select by their names the variables for the sampling time Ts the input and th
56. ill always be inside or on the unit circle but this is not enough to conclude upon the stability of the algorithm 2 2 Recursive Least Squares Algorithm When using the Gradient Algorithm e t 1 is minimized at each step or to be more precise one moves in the quickest decreasing direction of the criterion with a step depending on F The minimization of e t 1 at 10 each step does not necessarily lead to the minimization of 5 e 4 1 i l on a time horizon as is illustrated in Figure 2 4 In fact in the vicinity of the optimum if the gain is not low enough oscillations may occur around the minimum On the other hand in order to obtain a satisfactory convergence speed at the beginning when the optimum is far away a high adaptation gain is preferable In fact the least squares algorithm offers such a variation profile for the adaptation gain The same equations as in the gradient algorithm are considered for the plant the prediction model and the prediction errors namely 2 1 through 2 8 The aim is to find a recursive algorithm of the form of 2 9 which minimizes the least squares criterion ros J t DM F t de DP 2 31 The term 6 t i 1 corresponds to 07 t i 1 i t yG 1 br t u i 1 2 32 bi a Figure 2 4 Evolution of an adaptation algorithm of the gradient type Therefore this is the prediction of the output at instant i i lt t based on the
57. lidation criterion Therefore while keeping the values of n4 and np other structures and methods must be tried out in order to obtain a valid model If after all the methods have been tried none is able to give a model that satisfies the validation criterion then n4 and ng must be increased For a more detailed discussion of various procedures for the estimation of n4 max and np max 27 Initial Choice of nc and np Noise Model As a rule nc np n4 is chosen 2 5 6 Direct Order Estimation from Data To introduce the problem of order estimation from data we will start with an example Assume that the plant model can be described by y t ary t 1 byu t 1 2 87 and that the data are noise free The order of this model is n n4 ng 1 Question Is any way to test from data if the order assumption is correct To do so construct the following matrix y t y t 1 u t 1 util yt 2 u e 2 Y t RW 2 88 y t 2 y t 3 u t 3 Clearly if the order of the model given in Eq 2 87 is correct the vector Y t will be a linear combination of the columns of R 1 Y t R 1 0 with 67 a1 b1 and the rank of the matrix will be 2 instead of 3 If the plant model is of order 2 or higher the matrix 2 88 will be full rank Of course this procedure can be extended for testing the order of a model by testing the rank of the matrix Y t R where Sai Y t 1 U t 1 Y t
58. lines on the graphs of Syp and Sup at the frequencies of the filters that are enabled as in Figure 5 8 To modify one filter first you have to double click on it in the list of filters Afterwards it can be modified or removed 52 Modulus margin 0 485 Syp Output Sensitivity Function AAA AA 0 tE 3 m i 2 10 7 E 9 20 J 3 E 514 5 1 er ee ee ee ee re 2 4 6 8 10 Frequency Hz Figure 5 8 Output sensitivity function with filters and indicators 53 5 7 Tracking identification LA Architecture Controler Design Analysis Plots Band stop Fiters Tracking about REG Axis properties E Grid ON Rising time s 14 000000 Limits ON Max overshoot 1 730854 x min Simulation time s x max Tevaluation Gain P 1 B 1 Tracking model V Activate deactivate reference model Nat freq f fs 0 064650 i Damping 0 800000 Figure 5 9 Tracking tab for computing the tracking part of a controller reference model and polynomial T This tab Figure 5 9 is used for designing the tracking part of the controller polynomial T and the reference model 2 For the 7 polynomial there are three possible calculation methods that can be chosen from a dropdown list called T evaluation see reference 5 for more information e All CL poles Pd Pf B 1 T z is a polynomial with adjusted static gain containing all cl
59. m the upper left part of the screen Complex conjugate poles Multiple pole e one can also add a complex conjugate pole from the internal model of the plant by double clicking on the desired one from the list of plant poles 46 The values of the poles can be modified by moving the cursors or by direct typing in the edit boxes Furthermore the value of two complex conjugate poles can be calculated from the overshoot and rise time of a second order system You just need to select a complex pole write in the values for the overshoot M in percents and the rise time tr in seconds and click the nat freqg damp fn M tr button To modify and or delete an existing closed loop pole you first have to double click on it in the list After that it will be selected and one can either modify its values or delete it The program offers the possibility to add as much fixed part to the S and R polynomials as needed One just has to modify them in the corresponding sections Hr fixed part of R and Hs fixed part of S are used for modifying the fixed part of S and H polynomials One can easily add remove integrator or open the loop at 0 5 by clicking on the appropriate check box The Add cpl zero and Add real zero buttons are used for adding as much fixed part to the controller as needed As for the closed loop poles there is a list of zeros for each polynomial In this list one can easily see what has been added to the controller and can
60. mial T z is considered to have three basic forms e T contains all closed loop poles given by the polynomial P AS BR and its static gain is adjusted so that the static gain of the transfer function from y t to y t is 1 Hence T z 4 4 41 e T contains dominant closed loop poles given by the polynomial Pp and its static gain is adjusted so the static gain of the transfer function from y t to y t is 1 Hence Pp z7 Pr 1 Torte B 4 5 e T is a gain with the value Mis m 4 6 1 The reference model Lu is considered to be a second order transfer function with dynamics defined by natural frequency and damping Sensitivity function shaping is the chosen method for assuring the desired controller and closed loop performances The considered sensitivity functions are e The output sensitivity function Sy 27 47 e The input sensitivity function Bor J 4 8 e The complementary sensitivity function Di 1 Ro z 1 AR z 1 P z 1 Se 4 9 where Syp is shaped to obtain a sufficient closed loop robust stability the shaping of Sup allows to limit controller gain and hence actuator effort and Sun shaping help to limit noise sensitivity of the closed loop and it serves to fix a desired closed loop tracking performance More details can be found in 5 1 7 We can now introduce the following parameterization s e s z7 Hs z ws Peter Pier n z 1 SC Pe z 1 Pl
61. more the program check for integrator or derivator in the identified model 10 All information regarding these steps is present in the final message from the program e g Fig ure 3 2 the program gives the possibility to use the best model for designing a robust digital controller in the automatic mode as presented in Section 5 3 33 e saving the model or using it to compute a robust digital controller the model can be saved in WinPIM or Matlab version 6 formats the model can also be send to the robust digital controller design part of iREG E Identification and Controller Design and Reduction CA The best model has been identified with RELS variable forgetting factor The orders of the model are nA 4 nB 2 nC 4 delay 2 Parameters A 1 000000 1 601555 1 865776 1 493516 0 886836 B 0 000000 0 000000 0 000000 0 304177 0 400380 C 1 000000 0 737723 0 283304 0 038273 0 080312 Model is VALID Continue with automatic controller design Figure 3 2 Example of message at the end of the automatic identification procedure 34 3 2 Advanced use of the Identification tab In this section the functionality of the advanced identification mode is presented This tab is presented in Fig ure 3 3 There are 2 windows used for controlling the different parameters involved in the advanced identification Start Identification About iRDCD Switch between automated and advanced
62. ms which will be used for parameter estimation will depend on the assumptions made on the noise disturbing the measurements assumptions which have to be confirmed by the model validation 6 4 5 It is important to emphasize that no one single plant disturbance structure exists that can describe all the situations encountered in practice Furthermore there is no parameter estimation algorithms that may be used with all possible plant disturbance structures such that the estimated parameters are always unbiased Furthermore due to the lack of a priori information the input output data acquisition protocol may be initially inappropriate All these causes may lead to identified models which do not pass the validation test and therefore the identifi cation should be viewed as an iterative process as illustrated in Figure 2 5 17 I 0 Data Acquisition under an Experimental Protocol Y Model Complexity Estimation or Selection Choice of the Noise Model Parameter Estimation Y Model Validation Control Design Figure 2 5 The iterative process of identification Tables 2 1 2 3 summarizes a number of significant recursive parameter estimation techniques They all use PAA of the form 1 6 t F 0 9 w t 1 2 74 F t 1 M t F t MIND t 2 75 0 lt A 0 lt 1 0 lt A s lt 2 F 0 gt 0 F toaF 0 0 o oo v t 1 PUT 2 76 1 TO FOI 2 5
63. ned as x t r t y t in Figure 6 2 and z t G r t u t in Figure 6 3 Using this we get the a priori input t 1 f t z t RE z 1 8 0 OF H t and the a posteriori input t 1 67 t 1 t 58 where for the scheme of Figure 6 2 amp t 1 z e t B z t d A z 1 r t and for the scheme of Figure 6 3 we have amp t 1 A 271 a t B z t d B z7 r t d The unknowns controller parameters are T t 81 ns o t Fnr E and the measurements available values T t t t ns 1 t 1 t ng 1 The a priori closed loop input error will be given by t 4 1 u t 1 t 1 and the a posteriori by Zerf 1 u t 1 a t 1 For each of the algorithms excitation added to the controller plant input another filtered version FCLIM can be considered by taking _ Ae 6 SC belt 6 3 while in the case of the nonfiltered algorithm CLIM b t c t 6 4 The parameter adaptation algorithm PAA is described by these equations 62 t 1 t FO Q cr t 1 6 5 FUCE 1 A 0F 0 209 09 t 6 6 0 lt t 150 AQ t 2 F 0 gt 0 u eer t 1 U FOO d 59 Chapter 7 How to use the application Use this button to load a controller model from a file the file PRBS signal is does not need to necessa
64. ng to Figure 2 2 If the adaptation gain is large near the optimum one can move away from this minimum instead of getting closer The following analysis will allow to establish necessary conditions upon the adaptation gain in order to avoid instability Consider the parameter error defined as A t t 0 2 16 From Eqs 2 1 and 2 4 it results e t 4 1 y t 1 9 E 1 eT 6 07 eelt 9 0 6 t 2 17 Subtracting 0 in the two terms of 2 15 and using 2 17 one gets A t 1 O t FOLOTTE I F t o t 0 t A t 6 t 2 18 D 2 18 corresponds to a time varying dynamical system A necessary stability condition but not sufficient is that the eigen values of A t be inside the unit circle at each instant t This leads to the following condition for the choice of the adaptation gain as F al a lt raa 2 19 2 1 1 Improved Gradient Algorithm In order to assure the stability of the PAA for any value of the adaptation gain a or of the eigenvalues of the gain matrix P the same gradient approach is used but a different criterion is considered min J t 1 e t 1 2 20 0 t 1 1A symmetric square matrix F is termed positive definite if zT Fx gt 0 for all z Z 0 z R In addition i all the terms of the main diagonal are positive ii the determinants of all the principals minors are positive The equation SUUM naa 2 21 260 t 1 66 t 1 is then obtained From 2 6 an
65. occur in system identification when the level and type of input used are not appropriate 2 4 Some Remarks on the Parameter Adaptation Algorithms The parameter adaptation algorithms PAA presented up to now integral type have the form A 1 A2 1 F t e t e t 1 E 1 6 0 Flt De 9 1 O 1 re Pat where t is the vector of estimated parameters and F t 1 t e t 1 represents the correcting term at each sample F t is the adaptation gain constant or time varying t is the observation vector and e t 4 1 is the a priori prediction error or in general the adaptation error i e the difference between the measured output at the instant t 1 and the predicted output at t 1 based on the knowledge of OU 16 There is always a relationship between the a priori prediction adaptation error e t 1 and the a posteriori prediction adaptation error e t 1 defined on the basis of the knowledge of 0 t 1 This relation is u e t 4 1 REG Several types of updating formulas for the adaptation gain can be used in connection with the type of the parameter estimation problem to be solved systems with fixed or time varying parameters with or without availability of initial information for the parameter to be estimated The PAA examined up to now are based on e t 4 1 e minimization of a one step ahead criterion e off line minimization of a criterion over a finite horizon followed by
66. ormat Controller test Load test controller Load test model Figure 5 1 Architecture main tab of controller design This tab is divided in the following groups of buttons e Method there are three approaches available for designing a controller One can chose to design a digital RST controller and for this there are 2 available methods advanced and automated or chose to design a continuous PI PID controller e Load plant model one can use this button to load the polynomials A and B and the delay of a process that has been identified The program reads Matlab v6 and WinPIM files 45 Sampling period Ts Modify Ts Polynomial A 1 000000 1 601555 1 865776 1 493516 0 886836 PolynomialB 0 000000 0 000000 0 000000 0 304177 0 400380 Polynomial R 0 372340 0 138969 1 398277 1 959669 0 639385 Polynomial 5 1 000000 0 159249 0 950121 0 179292 0 288663 Polynomial T 0 155379 Polynomial Am 1 000000 1 160804 0 384219 Polynomial Bm 0 000000 0 129456 0 093959 Condition Number 17 903716 can Delay s Overshoot 95 Time constant s Second order model from rise time and overshoot First order model from gain delay and T Figure 5 2 System parameters window e Save controller and model offers the possibility to save the designed digital controller to one of these file formats Matlab v6 WinREG and NX RST e Loa
67. osed loop poles e Dominant CL poles Pd Pf 1 B 1 T z is a polynomial with adjusted static gain containing one complex pair of roots corresponding to closed loop dominant poles defined by Pu e Gain P 1 B 1 T z 1 is a constant with an appropriate value so that the static gain of the closed loop system is equal to one The reference model can be enabled or disabled By disabling it the reference s model transfer function becomes equal to 1 B z Am 27 1 If enabled it s transfer function is obtained by discretization of a continuous second order transfer function with normalized natural frequency and damping given by the user in the Tracking model group of this tab Note that in the automated controller design mode one can only enable disable the reference model but it s dynamics cannot be modified The polynomial T 27 can be changed as in advanced controller design mode Rising time gives the system rising time time to reach 9096 of the final stationary value in seconds Max overshoot represents the relative difference between the maximum value and the final stationary value in percents 96 Simulation time shows the total simulation time This value can be edited and changed Perturbation group can be used to add a step perturbation to the output of the system Axis properties group can be used in a way similar to the Plots control group of the Analysis plots tab but with effect on the trackin
68. p The controller design procedure is the combined pole placement with sensitivity functions shaping The theoretical background behind these procedures can be found in 2 and 5 The program was developed in GIPSA LAB Grenoble by T B Airimitoaie under the supervision of Professor LD Landau It has commenced as an interactive software for the design of robust digital controllers during a final year project T B Airimitoaie Interactive C program for the design of robust digital controllers and continued with the addition of open loop and closed loop identification routines during the PhD thesis of T B Airimitoaie 1 1 Software requirements The iREG program was developed under Visual C environment and needs one of Microsofts Operating Systems to work properly The software can read or write Matlab version 6 files mat without an existing installation of the Matlab software 1 2 Getting started e To run iREG one has to double click the icon entitled iREG exe A window like the one in Fig 1 1 will appear e Next you should select the proper button depending on what you want to do identify a plant model design a controller robust digital or continuous or compute a reduced order controller e The program offers the possibility to switch between English and French interface 1 8 Description of the software After double clicking the program icon 2 windows will be displayed the main window Figure 1 1 witch gives access to all
69. rent from the value used for the outputs Differentiate the output creates a new IO data set that has the same inputs but the outputs are filtered by 1 271 Integrate the input creates a new IO data set that has the same outputs but the inputs are filtered 1 b deeg Differentiate the input creates a new IO data set that has the same outputs but the inputs are filtered by 1 z Apply digital filter creates a new IO data set by filtering with a user provided digital filter the user can choose to filter only the inputs only the outputs or both the name field from the IO data loading group can also be used for naming the filtered data e estimating the complexity of the process the two options available under this drop down list will return the values of N n4 ng and delay d these values should be known before starting any identification algorithm because they give the number of parameters that need to be estimated the program will first give the estimated value of N max na ng d the user can check this value and modify it if necessary based on this the other three will be calculated e analyzing the identified models the user can plot different graphs in the subplots of the identification tab in the Plots window among these there are a few that can be used to analyse the model these are strictly related to the selected model from the model list therefore the user can analyse different models
70. ry for contain both computing simulated data on the nominal system View the parameters Kate parameters of the model and the Tolerance Amplitude 0 250 nominal controller Time s 1 500 Parameters of the step 2 perturbation viewable Adaptation algorithm in the Plots window parameters Theoretical image of Splectihe method par d oc 6 14 the nominal and estimating reduced 1 a k 6 1404 3 reduced closed loop order controllers S 6 1687 0 8573 5 7963 0 8752 systems Los Y i 5 5089 1 9241 Button to start the 0 0654 0 1029 1 8652 reduction procedure 0 3856 0 1013 0 2730 6 1415 0 7890 List of obtained controllers click to select and view sensibility functions comparison Use this button instead of Start to compute PID type reduced order controllers Button to save the Method for computing reduced order the polynomial T Compute T by Gain P 1 8 1 controller s Figure 7 1 Controller reduction tab Figure 7 1 gives an explained view over the controller reduction tab of iREG The different parts of this tab will be described next Load model controller use this to load a model and or a controller if not imported directly from the controller design part of REG Section 5 1 Parameters model controller will open a window like the one in Figure 7 2 Adaptation algorithm parameters initial adaptation gain GI as described by 2 73 forgetting factor
71. s e Pp polynomial of desired dominant the slowest closed loop poles P polynomial of desired auxiliary closed loop poles e Hg fixed part of the controller numerator e Hs fixed part of the controller denominator e Fy second order digital notch filters on Syp e L second order digital notch filters on Sup which allows us to shape appropriately the sensitivity functions Syp Sup and Syb 3 Tracking part design H the tracking properties are not satisfied by closed loop controller part R S the tracking part has to be designed One has to choose an appropriate structure of T and to design the reference model mm corresponding to the desired tracking performances For reference model adjusting the natural frequency and damping of the reference model denominator are modified 4 3 Continuous PI PID controller design The following theory has been extracted from 9 and 8 4 3 1 The KLV method for the omputation of PI controllers 1 Define Go the processes amplification at the zero frequency 2 Define the frequency at which the total phase shit reaches 135 as w135 and the amplification for which this phase shift is obtained as G35 3 Define the acceleration factor Fact Acc as a value between 25 and 200 4 Compute the gain ratio RapGain in the following manner Gias Go a RapGain lt 0 1 gt a 1 15 b 0 1 RapGain lt 0 25 gt a 1 c 0 25 RapGain gt a 1 15 RapGain 0 75 a w13s5 Fact
72. sis of the system Example Flexible Robot Arm with two vibration modes In this case na 4 is chosen since a second order is required to model a vibratory mode Initial choice of d and ng If no knowledge of the time delay is available d 0 is chosen as an initial value If a minimum value is known an initial value d din is chosen If the time delay has been underestimated during identification the first coefficients of B q 1 will be very low ng must then be chosen so that it can both indicate the presence of the time delays and identify the transfer function numerator ng dmax dmin 2 is then chosen as the initial value At least two coefficients are required in B q7 because of the fractional delay which is often present in appli cations If the time delay is known ng gt 2 is chosen but 2 remains a good initial value 26 Determination of time delay d first approximation One identifies using the RLS Recursive Least Squares The estimated numerator will be of the form Bla bq bog byg If KE a b 0 is considered and time delay d is increased by 1 d dmin 1 since if 64 0 B q 1 q7 baq b3q If b lt 0 15 bdi 1 t 1 2 di time delay d is increased by d d din di After these modifications identification is restarted Determination of the NA max and NB max The aim is to obtain the simplest possible identified model that verifies the validation crit
73. t tab Figure 1 1 There are 2 possible modes for doing identification in iREG automatic Figure 3 1 and advanced Figure 3 3 3 1 Automatic use of the Identification tab The automated identification tab is presented in Figure 3 3 The main objective is that of simplifying as much as possible the identification procedure while maintaining good performance For more advanced features see Section 3 2 The main features of the Identification tab are Switch between automated and Plots control identification plots advanced Eee Select subplot Choose graphic to be ploted identification gen Options for controlling the identification m graphics IO data loading Tagen xmin 0 000000 ymin 0 018348 options E Limits on xmax 5 000000 ymax 0 150000 Double dick to select Pushing this button stimatic 0 ARX RLS 1 00 1 00 4 20 2 will start the Gottes Re automated S identification List of the models that El have been identified 3 ARMAX RELS 0 97 0 97 4 2 procedure using the selected IO data set List of IO data sets in the memory of the program This drop down list offers the options for saving the selected model or using it in the Use selected model to robust controller design part of iRDCD Figure 3 1 Tab for the automated mode of using the identification algorithms e switching between advanced and automated modes the tab will alternate view from Figure 3 1 to F
74. the characteristics of the algorithms used for recursive plant model identification in closed loop 30 Old Do 26 2 NOS Mo Old 2 109994 _UOVAI OS O T4 2 72 T 3 4 10114 uonejdepy IT 26 I 26 I 17 TIOT19ISOd e IOI 1 17 6 1 1 o word e uonorpe1iq 2 6 1 17 8 1 2 6 HIOLI0jsOd indmo DAD 19 1 2 6 moude z1ojrpeiq p ag Dip 2 7 1092 709 4 D 2 DEN 1 qu dor 2o nat 67 n I Hu 2 123 2 4193 I Hu 7 793 94 123 ru pa 2 n 1 u p aye p In 1 u p ae p a e 1 fu p 30 Ip Un 109994 t Yu 36 6 7 Ta Yu 396 96 I 79 NT Vu 98 96 0 4 10859439Y 10j0tpoiq ve tp 1P verde 9 Tol 249 9 tq 1 19 CM 2 12 ch Ta 2 19 DU 9 2 10 D 2 G q 2 9 29 Dia H 59 2 0 1ojourereq 9 qe3sn py sr Hai ab Hai a Hai I Po N 9stoN 13ue d 10199 HO TOX 10194 10114 3ndjn 10114 yndyno 10114 yndyno 1010 door peso pozi e1ouor door p so p pu 1xq door pasalo po4o3 tT 10114 ding door paso p surqjriog e uorneoyguopr doot posojo AAISINIIY 9 z q I 31 Chapter 3 How to use the application The identification tab Figure 3 1 appears after selecting the Identification button from the Star
75. the functionality of the program and the plots window witch shows the graphics Each window is organized in tabs The main window contains the following tabs e Start starting tab of the application the different functionalities of the program identification controller design or controller reduction can be activated from this tab furthermore it an be used to save load the workspace e Identification this tab gives access to all identification routines implemented in iREG and can also be used to modify the graphics associated with the identification part of this software Controller Identification Controller design Controller reduction Help gt Open Manual REG pdf EN Save workspace Load workspace Show Plots Window Figure 1 1 Start window of the application Architecture includes all necessary functionality represents the initial tab for designen a controller one can select the type of controller to be design digital RST or continuous PI PID load the plant model save the controller or test an already computed controller Controller Design offers all necessary tools to design and properly adjust all parameters of a robust digital controller Automated controller Design offers only the basic tools to design and adjust a robust digital controller PI PID offers the possibility of implementing a continuous PI or PID controller Analysis Plots can be used to display graph
76. the user should introduce a short name for the I O data sequence that is going to be loaded in iREG in the Name of data text box if this is left empty not recommended the name of the file on the disk will be used as name for the I O data sequence If you are planing to load data from a text file it is important to open it first in a text editor e g Notepad to see which column corresponds to the input and which to the output Note that the accepted column separators are space tab semicolon and Z If you are loading data from a Matlab mat file make sure that it has been saved in the Matlab version 6 format the proper command to use in Matlab is save v6 filename variables Click on the Load I O data button select the appropriate type of file that you want to load search for the file on the disk and than click Ok A new dialog box opens which is used for selecting the appropriate input output data from the given file For mat files one can select by their names the variables for the sampling time Ts the input and the output sequences For text files one can select the appropriate column number for the input and the output sequences Each drop down box gives the possibility to select a number from 1 to the total number of columns If the total number of columns obtained in iREG is different from the expected one after visual inspection of the text file make sure that the right column separators are used as indicated
77. ve the form 5I t 1 t 1 t F 570 2 11 where F al a gt 0 is the matrix adaptation gain I unitary diagonal matrix and AU 1 66 t is the gradient of the criterion given in 2 10 with respect to 0 t From 2 10 one obtains 18J t 1 _ e t 1 o 2 55 rm e t 1 2 12 But P S e t Lt 1 2 y t 1 y t gtt 4 1 T Helt 2 13 e c t 1 ng t 2 14 Introducing 2 14 in 2 12 the parameter adaptation algorithm of 2 11 becomes t 1 A t PAD t 1 2 15 where F is the matrix adaptation gain There are two possible choices 1 F al a gt 0 2 F gt 0 positive definite matrix The resulting algorithm has an integral structure Therefore it has memory for it 1 0 6 t 1 t The geometrical interpretation of the PAA of 2 15 is given in Figure 2 3 The correction is done in the direction of the observation vector which in this case is the measurement vector in the case F al a gt 0 or within 90 deg around this direction when F gt 0 a positive definite matrix may cause a rotation of a vector for less than 90 deg ov Fo te t 1 F al Fo te t 1 F gt 0 Q t Figure 2 3 Geometrical interpretation of the gradient adaptation algorithm The parameter adaptation algorithm given in 2 15 presents instability risks if a large adaptation gain respec tively a large a is used This can be understood by referri
78. ystem excitation added to the reference controller output or measurement see Section 2 6 for theoretical explanations Each structure has it s own set of algorithms from each the user should select in Step 4 The common set of parameters needed for each of the algorithms is given by the orders of the model s polynomials Step 5 n4 np delay d and for some other algorithms also nc and np and the adaptation algorithm s parameters Step 6 The adaptation algorithms are Decreasing gain Decreasing gain with constant forgetting factor and Decreasing gain with variable forgetting factor Based on what is chosen the user is given the option to modify the Initial adaptation gain An Forgetting factor and An Lambda0 see Section 2 3 4 1 A 2 and A 3 for more information Some of the identification algorithms need that the user specifies some further parameters presented next e Output Error with Filtered Observations FOLOE needs a filter denominator to be specified by default 37 Architecture PI PID Analysis Plots Tracking Reduc About iREG Step 4 Select method Recursive Maximum Likelihood RML Step 5 Model complexity Delay 2 Order of A Order of B Order of C Structure 3 Y Step 7 RML adaptation parameters Initialization horizon 50 Contraction factor Step 6 Adaptation algorithm Decreasing gain with var forgetting fact Initial adaptation gain Forgetting factor 0 95

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