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A Testbed for Networked Control Systems”

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1. Time 5 Figure 15 Simulation Result of the Switched System Then implemented and executed this switched system in C project which was consistent with the simulation References 1 The Baylor University Time Scales Group online Available http timescales org 2 S Hilger Ein MasskettenkalkAul mit Anwendung auf Zentrumsmannigfaltigkeiten 1988 3 Quanser Inc SRVO2 user manual 4 Quanser Inc Rotary Experiment 00 QuaRC Integration 5 Quanser Inc Rotary experiment 01 Modeling 6 Quanser Inc Rotary experiment 02 Position control 7 Quanser Inc Rotary experiment 03 Speed control 8 Quanser Inc Rotary experiment 07 Gantry control 3 21 Revised 07 27 2011 9 Quanser Inc Rotary Experiment 08 Self Erecting Inverted Pendulum Control 10 SysML org SysML online Available http sysml org 11 OpenModelica org OpenModelica online Available http www openmodelica org 12 VIATRA2 Developer Team VIATRA2 online Available http www eclipse org gmt VIATRA2 13 QNX Software Systems QNX neutrino RTOS online Available http www qnx
2. Actuator Dynamics Dynamicst gt JUL Trigger 2 3 4 xda TESTI T tau N m ZONES Vi Hi va Yin Cx n Dun x pe th dot rad s x n 1 Ax n Bu n Torque Voltage Actuator Electrical rotary pendulum system Scopes Sensor TrueTime Send TrueTime Receive Trigger Sensor Actuator 10 7 o me 5 o Select 5 Y 4 theta dot 6 de n at ule K State Space Observer Control gain K 1 Schedule TrueTime Send TrueTime Receive Network Schedule 9 Simulate Network Controller for Quanser Inverted Pendulum feontrellen s 8 TrueTime Network 11 Figure 12 NCS Simulink Model Horizontal Arm Position 8 T T T T T T T T 6r m 4r zi 2 o o 2 c 0 J 2 8 2 a Ab m reference position F 0 H 8 Il I i I i 0 2 4 6 8 10 12 14 16 18 20 time s Pendulum Position 2 T T T T T T T T T 9 o o 2 amp 9 o a 2 l l i i l I i l i 0 2 4 6 8 10 12 14 16 18 20 time s Figure 13 NCS Simulation Result 5 18 Last revised 07 27 2011 The pendulum system sends get the angle information and to the controller where the data rate is controlled by the sensor trigger The pendulum receives the control signal and writes it to the actuator The controller system receives the angle information calculates the control signal and sends it back to t
3. HH 7 1 7 2 MP M 7 1 7 3 1 e Ee c 7 2 8 CONCUSSION A A de ib a d eben extus 8 1 8 1 Observations OS NN 8 1 8 1 1 Comparisoriof Results 35 a AA AA oe Ree ee 8 1 8 1 2 Accuracy of Simulations iie dano re RAS 8 8 8 1 3 Timinglnaccutacy sce doeet ihe abide tese cessit arabes de paseo regne oaa nig OERe 8 9 8 1 4 Stability against Sampling Rate sessi nn nana nnns 8 4 8 2 Conclusions and Future Work sseessseeeeeeeeeneeeneen nennen eene nennen nennen 8 13 8 2 1 Abilities of The Test Bed oooconnncccnnnccccnonoccnnnncnnnnnancccnnnnonnnnno aa en nnne 8 13 pU MEM El A 8 13 8 2 3 F t re WOrk osito nare arn etl aa 8 13 Appendix A Parameter Definition Appendix B List of Hardware and Software Appendix C User Manual Appendix D How to Generate Arbitrary Time Scales Last revised 07 27 2011 1 Introduction This chapter introduces the background purpose and results of the project 1 1 Background The primary purpose of the project was to develop a test bed that can facilitate research on time scales theory and networked control theory Time Scales theory is a way to unify the seemingly disparate fields of discrete dynamical systems i e difference equations and continuous dynamical systems i e differential equations Time scale systems might best
4. time s Pendulum Position position degree o T li 70 75 80 85 90 95 time s Figure 18 Running Result Compile and run the model we can see that the controller can balance the pendulum Figure 18 records the arm and pendulum angles during an experiment The horizontal arm is tracking the input reference signal which is a square wave with amplitude of 10 and frequency of 0 1 Hz 5 5 Summary This chapter introduces how the pendulum controller is designed The general process is as follows First analyze the mathematical representation of the physical system Second design the control system block diagram based on control theory Third create the Simulink model based on the control block diagram and tune the parameters to make sure that the controller is working properly Forth 5 22 Last revised 07 27 2011 implement the executable controller by utilizing the Quanser library In the end compile and run it on hardware References 1 Quanser Inc Rotary Experiment 08 Self Erecting Inverted Pendulum Control 2 G Franklin and D Powell Digital Control of Dynamic Systems Addison Wesley publishing company 1980 3 R Dorf and R Bishop Modern Control Systems Prentice Hall 2008 4 Department of Automatic Control Lund University TrueTime toolbox online 2 0 beta 6 Available http www control Ith se user truetime 5 23 Last revised 07 27 2011
5. Constant4 Constant3 gt Product Figure 11 Non linear Pendulum Model 3 3 3 4 Design the Controller During the fall 2010 semester took the control theory class ECL 4337 which focused on the design of continuous control systems using the textbook Modern Control Systems 41 Since this project requires knowledge of digital control systems referenced the book Digital Control of Dynamic Systems 42 which describes how to design a digital feedback control system including a controller and an observer 3 13 alpha Revised 07 27 2011 3 3 3 5 Implement Simulink Controller Simulations implemented the following simulation models in Simulink e Simulation with the motor dynamics using a filter to estimate the velocity Figure 12 e Simulation with the motor dynamics using a state space observer to estimate the state variable x Figure 13 e NCS simulation Figure 14 For the rotary pendulum system the state variable x is defined as S D S Gm where is the pendulum position is the pendulum velocity is the horizontal arm position a is the arm velocity The state space representation of the inverted pendulum is x Ax Bu y Cx Du where u is the voltage applied to the DC motor The Simulink model for the pendulum system is shown in Figure 12 and Figure 13 The details of controller design will be introduced in Chapter 5 3 14 Revised 07 27 2011 oooo u oo
6. reference signal lo E RefGain 6 d A tau N m Vi V th dot rad s p Vi V Vm V Ly Select theta_dot Torque Voltage Actuator Dynamics Actuator Electrical xhat Dynamics Control gain K 4 y n Cx n Du n x n 1 Ax n Bu n rotary pendulum system Figure 12 Simulink Model with a Filter Observer Phy Ply d Scopes For our system however the encoder only has access to 0 and a and is missing O and so an observer is needed to estimate the velocities and d The observer can be implemented in two ways The observer subsystem 5 in Figure 12 utilizes a differentiator and a filter to estimate the velocities while the observer subsystem 5 in Figure 13 uses a state space observer to estimate the velocities The filter observer is specific to this application but need to investigate more general cases Therefore decided to use the state space observer 0000 oo reference signal m y_d gt a pe tau N m y E Vi V ahvi v Vm V y n Cx n Du n RefGain EA th dot rad s i i nM x n 1 Ax n Bu n Scopes B rotary pendulum system Select theta dot Torque Voltage Actuator Dynamics Actuator Electrical 6 Xhat Eila Control gain K vl Dynamics T State Space Observ
7. 6 C Project Though Quanser QUARC library and Simulink provides a fast way to construct a controller the drawback is that the sampling rate cannot be varied in the Simulink model because it is a fixed environment parameter This restricts the further research on the time scales control theory which requires the sampling rate to be altered during runtime The time scales can be generated by hardware i e utilizing an external signal generator However it is still difficult to directly control the signal generator in Simulink Therefore created a software solution by implementing the controller in C 6 1 C Version of the Balance Controller The C version of the pendulum controller is implemented by mimicking the Simulink model The C code implements all the functions that are in the executable Simulink model The executable Simulink model is redrawn in Figure 1 in which subsystems that pertain to the same functional group are circled in red circle The reference signal group generates a proper reference signal and applies the correct gain to eliminate the steady state error The controller group determines the status of the pendulum uses the observer to estimate the current state variable and applies the LAR feedback control law The Q4 board group directly accesses the Q4 board to read write data The C program can be modeled using SysML The Block Definition Diagram BDD describes the general structure of the system The BDD
8. A Testbed for Networked Control Systems Chengsen Song Master s Project Computer Science Department Baylor University August 2011 Project Report Chengsen Song July 27 2011 Table of Content Ti ulntrod ctiohi o eter ra DE Ree t i e ehe ed er oh date drid 1 1 1 1 Backgro nd uice tte dte ee e E E II UE 1 1 1 2 What Has Been Done zac e a e d e ERR dens rr Edere ic caos 1 1 1 3 Structure of the Final Report roi re E RR es 1 2 257 Related Work iste d er e is im ute ete t RO e PR RE debe IUS 2 1 2 1 WAC FOCI GION e 2 1 2 2 Networked Control System eae eto Pte ved 2 1 2 27 OVerVvIe Ww x ioter be e A p e te ehe ord t abest tee cette oem eed 2 1 2 2 27 Challenges x sehe dd aun ced un e BI 2 2 2 2 3 Improve the System Performance ooooocoooooonncncnnnnononnnnnnonnnononnnnnnnnnnnnnnr ono nnnnnnnnnnnnnnnnnnnnnos 2 3 2 3 System Simulation TOONS i rct A eH Ye eSATA e ee a e TREE Ys 2 4 2 3 1 Available Network Simulators oooooccnnnccnnnnccncnnonnnnnnonnnonanononnnn cnn cnn nennen nennen nnne 2 4 2 3 2 Available Control System SimulatorS oooooocccnconnnnnnnnoninoninnnnnanannnnnarnnnnononnonnnnnnnnnnninoos 2 5 2 3 3 Available Co Simulation Tools seesessseseseeeeeeeeenee nennen nnne nnne 2 6 2 4 Time Scales Theory irte t Ese i A 2 6 3 Project a edi 3 1 3 1 Introduction RE 3 1 3 1 1 Purpose and APPO Ci a i Urs 3 1 3 1 2 MM ede t a a 3 2 3 1 3 ACCOMPLISHMENTS id iaa 3 2 3 2 SUMIMEr 2
9. D O a d4 d b 0100 0 0 G5 Ay da b where 5 3 Last revised 07 27 2011 mr Aas ee 2 gl v uf orm rae Bam Sy m Ju mul ES amt p mur JR TN m l rB E A p M r J TN mj gu tmo Jum Ju mur isi m pl p Pas r J at ARAS gl p EM ISR E Sar ML Aldo Tm J E m2 J Quum Fam I mr JR arm p p arm p RR mj r AS 2 2 J crm pl d ul p tmu JR Given the numerical solution to each variable in Appendix A we can calculate the numerical solutions to the matrices The results are 0 0 1 0 0 0 0 1 0 53 1012 0 6586 0 6575 O 98 3814 0 6575 1 2182 0 0 B 274 4012 273 9627 1000 C 0100 J D 0 5 4 Last revised 07 27 2011 The advantage of the state space representation of the system is that it is easier to describe a MIMO system And for a state space system the controller design can be separated into two steps In the first step we assume that we can access all elements in the state variable x i e we have a full state feedback Based on this assumption we can design the full state feedback controller The second step deals with the fact that we are missing part of the elements in the state variable i e 6 and a To address this problem we design an observer to estimate the current state variable x based on the output from the system i e 0 and over time 5 1 3 Continuous to Discrete Domain Conversion Equation 6 gives the continuous model of the pendulum syste
10. e Investigate model transformation using VIATRA 12 e Read papers on network control system and identify research problems 3 2 3 What Did I Do 3 2 3 1 Quanser Labs On the software side followed the Quanser manual 3 and did Lab 0 3 and 7 8 This helped me understand how to install Quanser software how to create a Quanser Simulink model how to configure the target machine and how to compile and run a model On the hardware side worked with Dr Eisenbarth to setup the hardware Initially we had only one Windows XP PC used for both developing and running the controller However we found that sometimes the pendulum fell down after a short time Probably it is because that Windows XP is not a real time operating system Therefore Dr Eisenbarth added a second PC running the QNX Neutrino 13 real time operating system This is the current system configuration in which the QNX computer is only 3 3 Revised 07 27 2011 for running the controller and the Windows XP computer is only for developing the control algorithms The design process is that the controller is first simulated on Windows XP and then compiled downloaded and run on the QNX computer Since this configuration is setup the pendulum does not fall down again 3 2 3 2 SysML modeling We planned to use SysML to describe the system so read the book Systems Engineering with SysML UML 14 particularly Chapter 4 which describes the basic concepts and diagrams in
11. Appendix A Parameter Definition The parameters used in Matlab programs are listed in the following table Symbol Description Matlab Variable Value SI units a Pendulum angle a Pendulum speed a Pendulum acceleration 0 Arm angle Arm speed 6 Arm acceleration My Pendulum Mass with T fitting mp 0 127Kg L Distance from Pivot to Centre Ip 0 1556m Of Gravity r Full Length of the pendulum r 0 2159m Faria Moment of inertia acting seen Jarm 0 0020 from arm pivot Jo Moment of inertia acting seen Jp 0 0043 from pendulum pivot Barm Equivalent Viscous Damping Barm 0 0024 Nms rad Coefficient B Equivalent Viscous Damping Bp 0 0024 Nms rad Coefficient g Standard gravity g 9 81 m s Ng Gearbox efficiency eta_g 0 9 Nm Motor efficiency eta_m 0 69 K Motor torque constant Kt 7 68E 3 Nm Km Back emf constant Km 7 68E 3 V rad s Kg Total gearbox ratio Kg 70 Rm Motor armature resistance Rm 2 60 Im Motor current Tm The torque applied at the load gear Last revise 07 26 2011 Appendix B List of Hardware and Software The hardware and software used in this project are listed in the following tables Hardware Abbreviation Name Vender Website Q4 Data acquisition board Quanser Inc http quanser com Q4 terminal board QA terminal board Quanser Inc http quanser com UPM 1503 Power module
12. Henriksson B Lincoln J Eker and K Arzen How does control timing affect performance Analysis and simulation of timing using Jitterbug and TrueTime IEEE Control Systems Magazine vol 23 pp 16 30 2003 9 A Chamaken and L Litz Joint design of control and communication in wireless networked control systems A case study in American Control Conference 2010 pp 1835 1840 2 7 Last revise 07 27 2011 10 M S Hasan H Yu A Carrington and T C Yang Co simulation of wireless networked control systems over mobile ad hoc network using SIMULINK and OPNET ET Communications vol 3 pp 1297 1310 2009 11 E Weingartner H vom Lehn and K Wehrle A performance comparison of recent network simulators in IEEE International Conference on Communications 2009 pp 1 5 12 DARPA The network simulator ns 2 online Available http www isi edu nsnam ns 13 OMNeT Community OMNeT online Available http www omnetpp org 14 OPNET Technologies Inc OPNET modeler online Available http www opnet com solutions network rd modeler html 15 MathWorks Matlab Simulink online Available http www mathworks com products matlab 16 Math Works Real time workshop online Available http www mathworks com products simulink coder index html 17 OpenModelica org OpenModelica online Available http www openmodelica org 18 Department of Automatic Control Lund
13. Ts 10 ms This is because some packets are lost during the transmission so that the controller cannot calculate a correct control signal In this case the pendulum may become unstable We can also set the network type For example assume the network type is CAN with a data rate of 1 Mbps and no packet loss Figure 7 shows the simulation result when Ts 4 ms Though CAN has a much slower transmission rate 1Mpbs than Ethernet 100Mbps the pendulum is still balanced This is because that CAN bus is very suitable for real time communications systems Horizontal Arm Position 10 T T T T T T reference position 0 position degree a o a L 10 fi fi fi fi fi fi ji fi ji 0 2 4 6 8 10 12 14 16 18 20 time s Pendulum Position 3 T T T T T position degree eo us L k 0 2 4 6 8 10 12 14 16 18 20 time s Figure 7 NCS CAN 1Mbps Ts 4 ms loss rate 0 8 5 Last revised 07 27 2011 8 1 1 3 Switched System Simulation One of our objectives has been to investigate a stability of the time scales control system where the maximum sampling period Umax determines the Hilger circle with radius 1 max on the complex plane inside which the control system stays stable Even though the system not guaranteed to be stable outside the Hilger circle there is a conjecture that as long as the system does not stay outside the Hilger circle too
14. scales The implementation of the PendulumSystem update function is shown in Figure 4 If the fixed step length flag is to false the function calculates the Euclidian norm of the current estimated state variable then determine the next sampling period based on that norm if using time scales control determine next sampling period if lisFixedRate calculate norm of the vector double dp matrix_Xhat_ norm int index 0 if dp gt 2 nextTs constants TS5MS else nextTs constants TS20MS index 1 controller loadParameters index refpos loadParameters index Figure 4 int PendulumSystem update bool isFixedRate 6 4 Last revised 07 27 2011 int Ts mysys update true set timer expire time Ts itime it_value tv_sec 0 itime it_value tv_nsec Ts reload time Ts itime it_interval tv_sec 0 itime it interval tv nsec Ts timer settime timer id 0 amp itime NULL Figure 5 Calling PendulumSystem update 6 5 Last revised 07 27 2011 7 Creating your own controller When you need to create your own controller you can modify the existing Simulink and QNX C project 7 1 Step 1 The first step is to create a simulation model in Simulink This requires you to modify file sim statespace obeserver mdl as shown in Figure 1 theta ref d 1
15. 2 3 4 5 7 Ply 0000 oo D2R PE P tau N m Ply E vi v Vi V vm v gt Y M 0x m Duln arm reference Degreesto RefGain gt a jp th_dot rad s x n 1 Ax n Bu n Scopes position degrees Radians nu t dul St Select Torque Voltage Actuator Electrical oray pendulum sysiem Actuator Dynamics Dynamics theta_dot 8 vie Xhat ud Simulation model for the Quanser rotary inverted pendulum LQR Control Gain State Space Observer Use the State Space Observer to estimate the state Figure 1 sim_statespace_obeserver mdl You should keep block 1 4 5 and 6 Block 1 generates and outputs the reference angle Block 4 and 5 are for motor dynamics which converts the torque to voltage Block 6 is the physical model of the pendulum system which accepts voltage and outputs the arm and pendulum angles Then create your own controller by adding the controller blocks into the diagram 7 2 Step 2 After you get successful simulation results you can modify file exe statespace obeserver mdl 7 1 Last revised 07 27 2011 theta_ref y_d 1 2 nooo d inr gt D2R r Scopes arm reference Degrees to RefGain 3 position degrees Radians Mode Control rotary pendulum system B Executable model for the Quanser rotary inverted pendulum Use the State Space Observer to estimate the state vie ue LQR Control Gain State Space Observer
16. 7 9 10 11 24 AM l q sip mdl 1 8 10 9 33 AM setup_srv02_exp08_sip m 12 3 10 3 30 PM e SRY02_SIP_ABCD_eqns m 11 2 09 8 58 AM Figure 3 File structure for Quanser lab 8 5 Open SETUP_SRVO2_EXPO8_SIP m which is a Matlab file used to calculate the pendulum s physical parameters and construct a PD controller for the pendulum Run this file and the following results should appear in the Matlab command window C 5 Last revised 07 27 2011 SRV02 model parameters K 1 76 rad s V tau 0 0285 s SRV02 SIP balance control gains k 1 0 31623 N m rad k 2 1 9146 N m rad k 3 0 14956 N m rad s k 4 0 26268 N m rad s Here K represents steady state gain tau represents time constants and k represents the feedback gain of the PD controller 6 Open q_sesip_q4 mdl where sip stands for single inverted pendulum which contains the Simulink model as shown in Figure 4 SRV02 Inverted Pendulum Energy Based Swing Up and Balance Control Q u a RC o accelerate design Setpoint on j off po Swing Up Swing Up ON OFF Up to Enable Acceleration Factor for Swing Up 1 2027 Constant mu aid Swing Up Control SRVD2 E SIP Quarc High Gain Observer Figure 4 q_sesip_q4 mdl 7 Build the Simulink model by selecting QuaRC gt build within the Simulink window During the build process the graphical representation of the Simulink model is translated i
17. Control Simulation The sampling rate is an important parameter Generally a control system has a maximum sampling period Umax or minimal sampling rate Intuitively a controller cannot work too slowly to control the inverted pendulum To determine Umax by simulation we did several experiments where Ts was set to 5 ms 40 ms 20 ms and 10 ms The system s stability is shown in Figure 1 Since the pendulum is still stable when Ts 10 ms Figure 2 but is unstable when Ts 20 ms Figure 3 Therefore Umax must be in the range of 10 ms to 20 ms 8 1 1 2 Networked Control Simulation For an NCS we can compare the sampling rate the packet loss rate and the performance of different network 8 1 Last revised 07 27 2011 The first experiment is on the sampling rate of NCS Assume that the network type is Ethernet with a data rate of 100 Mbps and no packet loss When Ts 5 ms the simulation results is shown in Figure 4 in which the oscillation indicates that the LQR controller designed in Chapter 5 cannot balance the pendulum When Ts 4 ms the simulation results is shown in Figure 5 which shows that the pendulum is balanced However compared to the pendulum angle Figure 2 which is a local control the pendulum angle is not as smooth And comparing the maximum pendulum angle of the two figures we can see that the NCS is more than 2 degrees which is larger than the pendulum angle with the local control Therefore this result show
18. E it Data 1 3 Select theta_dot 6 Kila Control gain K Xhat Yh uM State Space Observer i Sensor Trigger Trigger TrueTime Send 9 3 4 Spring Semester 2011 3 4 1 Overview Figure 14 Simulink model of NCS Trigger Data 1 3 TrueTime Receive Controller 8 x n 1 Ax n Bu n A rotary pendulum system Scopest Data 1 2 xd UL Hp trigger TrueTime Send Sensor 7 1 Schedule TrueTime Network 11 Network Schedule During the spring 2011 semester implemented the controller that had simulated during the previous semester The first step was to implement it in Simulink After got the controller working 3 16 Revised 07 27 2011 found that Simulink had the limitation that the sampling rate could not be varied This restriction had to be overcome because we wanted to use the time scales theory which would require the sampling rate to be altered during runtime Dr Eisenbarth suggested that we attach a signal generator i e Agilent 33220A to the external interrupt pin on the Q4 board We could then use the Agilent software to generate arbitrary time scales This solution allowed us to predefine several time scales which could then be chosen at runtime However the time scales could not be changed after it is downloaded to the signal generator So implemented a softwar
19. This is the limitation of the encoders When powered on the encoders read O If the pendulum is at arbitrary position we cannot tell the exact angle However there is one deterministic position i e when the pendulum is static it must be in the straight down position which is 180 degrees Therefore the MC subsystem utilized this information When the model begins to execute the MC subsystem assumes that the pendulum is static and wait for the user to swing it up The controller can only work when the pendulum is close to the up right position The threshold is defined as 2 5 degrees If the pendulum angle is larger than the threshold the MC sub system outputs O to disable the controller Otherwise the MC subsystem outputs 1 to enable the controller The MC subsystem is designed as in Figure 15 CATCH ALPHA UP LIM Catching Pendulum Position Limit rad LE Select alpha lalpha_upl Switch enable WIICI Trigger Value Holds states once enabled Figure 15 Mode Control Subsystem The pendulum system sub system is the one that directly controls the O4 board and the power module Figure 17 shows the details of the subsystem The HIL initialize block will initialize the Q4 board when the model begins to execute The HIL Read Encoder block reads encoder values The Torque Voltage Actuator Dynamics and Actuator Electrical Dynamics subsystems convert the input torque to the desired voltage which will be output by the HIL
20. Xhat Figure 2 exe statespace obeserver mdl Block 1 subsystem 3 and subsystem 4 should be kept Block 1 is for reference position generation Subsystem 3 determines whether the pendulum can be controlled by the controller Subsystem 4 contains the Q4 board accessing blocks which read encoder values and write controls signal to the pendulum system 7 3 Step 3 To design new controller for the QNX C project users need to replace the existing Controller class O4 class and Reference class should be kept 7 2 Last revised 07 27 2011 8 Conclusion This chapter gives some experiment observations plus conclusion and possible future work 8 1 Observations Chapter 5 has illustrated both the simulation and execution experiments results where the pendulum is balanced For the controller many parameters could be tuned Therefore this section gives more experiment result and analyzes the observations 8 1 1 Comparison of Results The test bed provides simulation models for the local control system the networked control system and the switched system Each system has some parameters that can be tuned Section 8 1 1 1 shows the observations of tuning the sampling rate for local control system Section 8 1 1 2 shows the observations of tuning the sampling rate packet loss rate and network type for the NCS Section 8 1 1 3 shows the observations of tuning the switching threshold for the switched system 8 1 1 1 Local
21. a fixed rate false if updated at varying step length true if updated at fixed step length return the sampling period for the next step 7 int update bool isFixedRate C 23 Last revised 07 27 2011 5 2 Use the External Signal Generator The pendulum system can be controlled by the external signal generator To use this function follow these steps 1 2 3 4 Use cable to connect the output of the signal generator to the input of the signal buffer and connect the output of the signal buffer to the control port on the Q4 terminal board Power on the signal generator and set the output mode to square wave with frequency of 200Hz and amplitude 5 V Vpp equals to 5 V enable the signal buffer to output valid voltage signal to the Q4 terminal board For the detailed steps to set up the signal generator please refer to 33220A user manual Change the clock source Before you compile the model double click the rotary pendulum system subsystem In the new window double click the HIL Read Encoder Timebase HIL 1 block which pops up the Source Block Parameters dialogue Figure 24 Change the clock channel Click the button circled in red which pops up the channel selection dialogue The original channel is set to General Purpose counter Figure 25 Select the External Interrupt EXT INT channel on the left column and click gt gt button to switch it to the right column and switch the General Purpose c
22. a source on the left and leaves the tank via another pipe on the right at a rate controlled by a valve The liquid level in the tank must be maintained at a fixed level as closely as possible no matter what the input flow is levelSensor controller Figure 2 A Tank System model Tank ReadSignal tSensor Connector sensor reading tank level m ActSignal tActuator Connector actuator controlling input flow LiquidFlow qIn Connector flow m3 s through input valve LiquidFlow qOut Connector flow m3 s through output valuve parameter Real area unit m2 0 5 parameter Real flowGain unit m2 s 0 05 parameter Real minV 0 maxV 10 Limits for output valve flow Real h start 0 0 unit m Tank level equation assert minV gt 0 minV minimum Valve level must be gt 0 3 7 Revised 07 27 2011 der h qIn lflow qOut lflow area Mass balance equation qOut lflow LimitValue minV maxV flowGain tActuator act tSensor val h end Tank Figure 3 A Modelica Model The Modelica source code for the tank is shown in Figure 3 Note that each physical object is represented as an object or variable and equations represent continuous behaviors The out flow valve is controlled by a PI controller to keep the height of the liquid at 0 25 meters where the parameters for the PI controller are P 2 and 0 1 Th
23. and eventually fall 8 6 Last revised 07 27 2011 positions 0 15 0 1 0 05 o positions rad b 8 a 0 1 arm position 0 15 pendulum position 02 l l l 1 l l 0 2 4 6 8 10 12 14 Time s Sa period u 0 02 9 op CAOS e PERRO ERED ECOG cgo 0 M 0 2 4 6 8 10 12 14 Time s Figure 8 Switched System threshold 0 04 positions 0 15 T T 0 1 0 05 0 05 positions rad 0 1 arm position pendulum position 02 1 L 1 L l L 0 Sampling period u 0 0 9 02 SS s 0 015 0 01 pms Figure 9 Switched system threshold 0 1 8 7 Last revised 07 27 2011 8 1 2 Accuracy of Simulations By comparing Fig 11 and Fig 19 in Chapter 4 we can see that the result is different The simulation shows a smooth plot and no jitter which is unlike the real experiments result The following section discusses the potential reasons 8 1 2 1 Mathematical Model Limitation First the mathematical model in equation 1 and 2 in Chapter 5 may not be accurate The performance of the controller depends on the accuracy of the non linear model because the starting point of the controller design is to model the mathematical system model and get the non linear equations of motion The more accurate the physical model is the more accurate the pendulum is Besides the controller is designed based on the state space representation of
24. arm and pendulum angle under the varying sampling rate case Figure 23 shows the position and Ts when the horizontal arm is tracking a square wave reference input with amplitude 0 1 rad Here the above figure shows the arm position and pendulum position We can see that the arm can track the reference position and the pendulum angle is within 0 04 rad 2 3 degrees indicating that the pendulum is balanced The figure below shows the varying sampling period Ts switches between 5ms and 20ms When the reference input remains still the system can be sampled at 20ms while when the reference input changes system must be sampled at 5ms C 22 Last revised 07 27 2011 positions positions rad arm position 015P pendulum position E 0 2 l 0 2 4 6 8 10 12 14 Time s Sampling period u Figure 23 Switched system In the C project to test the switched system open main cpp and find the following line int Ts mysys update true Set the parameter to false and rebuild the project Here update is a member function of class PendulumSystem The parameter controls whether the controller is working under a fixed sampling rate The prototype of the update function is Qbrief Read encoders calculate control signal and write voltage out param bool is FixedRate indicates whether the controller is updated at
25. be understood as a bridge between discrete time and continuous time systems A networked control system is a control system that uses a shared network to transmit information It is widely used in today s life To read this project report it is assumed that the reader has a basic knowledge of control theory time scales theory and networked control systems Please refer to Chapter 3 of 1 Chapter 1 of 2 and 3 for detailed information 1 2 What Has Been Done In this project we developed a test bed for modeling and simulating the time scales control and networked control system for a rotary inverted pendulum 4 The test bed which allows users to develop simulate and run their own control algorithms could be used to implement and verify various theories The test bed provides both simulation and executable models for a rotary inverted pendulum The simulation models for both the local control system and the network control system were built using Matlab Simulink 5 The simulation environment enables a user to verify the control algorithms The Quanser QUARC library 6 was used to create an executable model which could directly access the 1 1 Last revised 07 27 2011 hardware components This executable model can be compiled and run on QNX 7 To implement a time scales control a user could choose to use an external signal generator or to implement a controller in C program would use the software timer on QNX The te
26. code the Quanser software creates a thread with fixed scheduling period on QNX Therefore it is difficult to generate and use time scales in Simulink 3 18 Revised 07 27 2011 3 4 3 3 Use external signal generator to generator time scales To solve Problem Two we attached a signal generator directly to the Q4 extension board to control the sampling rate via the external interrupt pin on the Q4 extension board To utilize this signal the sensor reading block must be configured to use an external interrupt as the clock After these hardware changes were made we did experiments to determine whether the Simulink model s sampling rate was in fact controlled by the external interrupt signal and the answer was yes 3 4 3 4 Implement the balance controller on QNX using C Though the time scales can be generated by external signal generator we could not directly control the signal generator from the Simulink model Therefore solved Problem Two by implementing a C version of the controller mimicking the Simulink model where the time scales was generated by software timer developed the project using QNX Momentics IDE To achieve a varying sampling rate use a software timer to control the sampling rate The QNX provides software timers that can be set When the timer expires the OS sends a signal to the controller program Upon receiving the signal the program directly accesses the O4 board to read sensor data and calculates the control
27. information i e reference input plant output control input among control system components sensor controller actuator etc using a shared network A graphical representation of this NCS model is shown in Figure 1 1 E Controller 1 v i SS Controller N Plant 1 pig Sensor T Plant M Figure 1 A Conceptual Model of NCS 2 1 Last revise 07 27 2011 NCSs have been popular and widely applied for many years because of their numerous advantages such as reduced wiring simpler installation ease of system diagnosis and maintenance and increased system edibility 2 2 2 Challenges There are challenges associated with the design of an NCS First there are various kinds of networks not all of which may be appropriate for a given situation Choosing an appropriate network may not be trivial Second packages transmitted through the network may suffer delay and loss which influences the performance of the control system Third the network may be vulnerable to external attacks Such a security issue is very important in critical applications 2 2 2 1 Network Technology Selection There are many types of networks that the designer can use Ethernet CAN 802 11 a b g bluetooth ZigBee Each type of network has its own advantages and disadvantages so the designer should select the most suitable one for the specific application 2 2 2 2 Network Delay and Packet Loss Compensation The network can introduce unrelia
28. is the continuous system to be discretized 7s is the sampling period and method is the discretization method which is chosen to be the zero order hold zoh in the project A B C D are constant matrices for pendulum system Creates a system pend ss A B C D then discrete the continuous system pend_d c2d pend Ts zoh Figure 2 Sample Code to Call c2d 5 2 Balance Controller Design Once the physical system has been represented we must determine the amount of feedback gain to apply to the system This section describes how to design the controller to balance the pendulum 5 6 Last revised 07 27 2011 5 2 1 Full State Feedback Controller Design To determine the feedback gain we first assume a full state feedback i e all the elements in the state variable x are known Based on this assumption the block diagram is shown in Figure 3 x n 41 Fx n Gu n Figure 3 Block Diagram of the Full State Feedback Control System u n Here we use the linear quadratic regulator LOR algorithm to design the state feedback controller For a continuous system x Ax Bu LOR theory states that the full state feedback control system satisfies the following criteria 3 1 The closed loop system is asymptotically stable 2 The cost function defined as J H x Qx u Ru dt is minimized with feedback control law u Kx where Q is a nonnegative definite matrix that penalizes the stat
29. of the controller 5 3 1 Simulink Simulation Model The Simulink representation of the complete model is shown in Figure 8 in which each rectangle is a Simulink block or subsystem For example the block rotary pendulum fully characterizes the pendulum system The State Space Observer subsystem characterizes the Observer The detail of the State Space Observer subsystem is shown in Figure 9 theta ref 7 d 1 2 4 4 5 e Ply 0000 oo D2R P r P tau N m ely n vi v phi v vm v e Y0 ex m Du n arm reference Degrees to RefGain a I JP th dot rad s x n 1 Ax n Bu n Scopes position degrees Radians o rotal ndulum system Select Torque Voltage Actuator Electrical ly penauium syste theta dot Actuator Dynamics Dynamics 9 8 ve Xhat uje Simulation model for the Quanser rotary inverted pendulum LQR Control Gain Use the State Space Observer to estimate the state State Space Observer Figure 8 Simulink Model with Motor Dynamics 5 14 Last revised 07 27 2011 C2 0 P y n Cx n Du n u Lp P x inet Ax n Bu n gt k gt Vector Discrete State Space1 Gain Concatenate Xhat Figure 9 State Space Observer 5 3 2 Parameters Parameters that can be adjusted include the sampling period the Q and R matrices and the poles angles Configure the system with the following parameters e sampling peri
30. of the applications of time scale control theory References 1 R A Gupta and M Chow Networked Control System Overview and Research Trends IEEE Transactions on Industrial Electronics vol 57 pp 2527 2535 2010 2 L Samaranayake M Leksell and S Alahakoon Relating sampling period and control delay in distributed control systems in The International Conference on Computer as a Tool 2005 pp 274 277 3 C Lai and P Hsu Design the Remote Control System With the Time Delay Estimator and the Adaptive Smith Predictor EEE Transactions on Industrial Informatics vol 6 pp 73 80 2010 4 X Dong and Q Zhang Stability of singular NCS with time varying delay and state feedback in International Conference on Measuring Technology and Mechatronics Automation 2010 pp 473 476 5 E C Martins and F G Jota Design of Networked Control Systems With Explicit Compensation for Time Delay Variations EEE Transactions on Systems Man and Cybernetics Part C Applications and Reviews vol 40 pp 308 318 2010 6 Y Uchimura and H Shimano Network based control with compensation of time varying delay and modeling error in 35th Annual Conference of IEEE on Industrial Electronics 2009 pp 3013 3018 7 Y Xia G P Liu M Fu and D Rees Predictive control of networked systems with random delay and data dropout Control Theory amp Applications IET vol 3 pp 1476 1486 2009 8 A Cervin D
31. of the pendulum system is shown in Figure 2 The PendulumSystem class is the top level class that is composed of three main classes Reference Signal Controller and Q4 The PendulumSystem class stores the state variable and other variables that are exchanged between different classes and are frequently updated The reason is that the difference equation x n 1 Fx n Gu n indicates that the next pendulum state variable x n 1 is 6 1 Last revised 07 27 2011 calculated based on the current input u n and the current state variable x n which should be saved Therefore designed the PendulumSystem class to be the top class that keeps track of everything theta ref 0000 oo P D2R Scopes arm reference Degrees to position degrees Radians Reference Executable model for the Quanser rotary inverted pendulum Use the State Space Observer to estimate the state la ra State Space Observer Controller Figure 1 Executable Simulink Mode The internal structure is best described using the Internal Block Diagram IBD The advantage of SysML is that we can show the interfaces and data flow between each subsystem As shown in Figure 3 the Reference Signal class outputs angle data i e the horizontal arm position to the Controller block The Q4 block outputs encoder data in the form of a Matrix to the Controller block Based on
32. result of local control system in Figure 7 250 ain qOut gt Clock Displa Step tActuator tSensor sane Pet Subsystem Trigger Scopet 1 2 Data JUL gt Sonor TrueTime Send Trigger Sensor 1 Schedule TrueTime Receive Actuator Network Schedule TrueTime Network 0 25 Plz Constant Discrete PID Controller TrueTime Send Controller Ny f Computational Delay TrueTime Receive Controller Figure 8 Simulink Model of the Networked Control System tank liquid height 0 7 T T T T 0 47 7 height m 0 3F 4 0 17 4 1 L L 0 50 100 150 200 250 time s Figure 9 Simulation Result of the Networked Control System 3 3 3 3 Model Rotary pendulum and Design Controller At the beginning of the fall semester read the Quanser manual 9 to understand how to model the pendulum dynamics The dynamic equations can be constructed by using the Lagrangian 40 3 11 Revised 07 27 2011 created a non linear Simulink model Figure 11 based on the dynamic equations to represent the pendulum system The pendulum system is a non linear system But in order to simplify the problem we linearize the equations by assuming that the pendulum angle is close to O when balanced This results in a state space representation which
33. router and they both obtain their respective IP addresses via DHCP However once the IP address of the Target is defined this address must be manually in set in the Simulink model see section 3 4 The connections among the hardware components are shown in Figure 1 e The analog output channel O on the O4 terminal board port 1 transmits DC motor control signal and is routed to the UPM from D A port port 2 e The UPM to load port port 3 emits enough current to drive the DC motor and is first connected to a pushdown break button which in turn is routed to the DC motor port on SRVO2 port 4 The break button in a blue box with a red button is used as an emergency stop C 1 Last revised 07 27 2011 Normally the switch is closed In case of emergency e g pendulum rotating wildly you can push and hold the red button to temporarily cut off electricity to stop the motor then turn off the UPM e The arm encoder on SRVO2 port 6 is connected to the encoder input channel O port 5 on the Q4 terminal board e The pendulum encoder port 8 is connected to the encoder input channel 1 port 7 on the Q4 terminal board Figure 1 Wiring Diagram 1 3 Use Source Code Repository We have used a Subversion repository to manage the files associated with this project in conjunction with the QNX Momentics IDE The repository contains several Matlab Simulink simulation files an executable Simulink model and a QNX C project We can
34. sensors to get the arm and pendulum velocity so that we can get complete state variable and make the system a full state feedback system e Develop the time scales observer If we keep the current hardware configuration which is not a full state feedback system a time scales observer is needed if we want to use the time scales control 8 13 Last revised 07 27 2011 e Use wireless transmitters to directly transmit sensor data and control signal between the plant and the controller which implements an NCS e Apply the time scales theory on NCS One NCS co design method is adaptive sampling period selection which means that the controller chooses proper sampling period based on current network condition 3 4 It has been proved that the adaptive sampling rate is feasible 4 References 1 Quanser Inc SRVO2 user manual 2 B Liu Effect of finite word length on the accuracy of digital filters a review IEEE Transactions on Circuit Theory vol 18 pp 670 677 1971 3 F Xia L Ma C Peng Y Sun and J Dong Cross Layer Adaptive Feedback Scheduling of Wireless Control Systems Sensors vol 8 pp 4265 4281 2008 4 A Gravagne J M Davis J J Dacunha and R J Marks Bandwidth reduction for controller area networks using adaptive sampling in Proceedings of the 2004 IEEE International Conference on Robotics and Automation New Orleans LA 2004 pp 5250 5255 8 14 Last revised 07 25 2011
35. signal We can set the timer expiration time after it expires so that the sampling rate can vary This gives us the flexibility to generate time scales while the controller is running At the beginning had some difficulty compiling the project But after set the correct library header path and linked to the correct Quanser library the compilation was successful 3 4 3 5 Investigate the switch system stability One of our objectives is to investigate the stability of the system when it switches from a stable state to an unstable state and vice versa From time scale theory we know that there exists a 44 and corresponding Hilger circle associated with the control system 4 is the largest sampling period that a digital controller can use and still guarantee stability Assume that the system s graininess is y the 3 19 Revised 07 27 2011 Hilger circle has a diameter equal to the reciprocal of graininess and is tangent to the imaginary axis 45 If the system poles stay within the Hilger circle the system will remain stable As the sampling period 4 varies the Hilger circle changes The system poles may go outside the Hilger circle if e is too large However there is a conjecture that as long as the system does not stay outside the Hilger circle too long the system is still stable To demonstrate this simulated what happened when the system switched forth and back inside and outside of the Hilger circle First us
36. these information the Controller calculates the voltage to be applied to the DC motor and outputs it to the Q4 block 6 2 Last revised 07 27 2011 bdd Package PendulumStructure q4 refpos oller ontro Figure 2 BDD of pendulum system ibd block PendulumSystem E ange ref controller RefAngle angle m Voltage PendAngk Matrix ntroller q4 ItemFlow ItemFlow 4 volt Voltage enc Matrix r Matrix T Voltage Figure 3 IBD of PendulumSystem block 6 3 Last revised 07 27 2011 6 2 Varying Sampling Rate Varying sampling rate is achieved by using a QNX software timer The QNX provides software timers whose expiration time could be set during runtime At each timer expiration the QNX OS sends a signal to the C grogram which is waiting for that signal Upon receiving the signal the main function will call the update function in the PendulumSystem class which reads encoder data calculates the control signal voltage and outputs the voltage to the Q4 class The PendulumSystem update function accepts an input parameter of Boolean type which indicates whether the controller runs at a fixed rate or a variable rate The return value indicates what the next sampling period is Figure 5 so the software time can be set In this way the timer expiration time it set at each step so that the sampling rate is varying This gives us the flexibility to generate time
37. use Eclipse IDE or the QNX Momentics IDE which is also Eclipse based to access the repository C 2 Last revised 07 27 2011 To access these files for the first time assuming that the repository has already been created on the server follow these steps 1 Create a new Eclipse workspace or use an existing workspace 2 Define a new repository locations within the IDE as svntssh USERNAME wind ecs baylor edu home grad group qnx repository and provide your user name and password to access the repository 3 Once connected you should see the repository structure similar to Figure 2 e The controller folder contains the Simulink project and the C project e The documents folder contains reports created in the project e The projectReport folder contains all the project reports e The weeklyReports folder contains Chengsen Song s weekly reports since July 2010 C 3 Last revised 07 27 2011 4 Navigate and highlight the controller gt trunk gt controller folder to check out the controller Right click the mouse then select check out Other folder can be similarly checked out 4 amp controller 236 a trunk236 4 amp controller 236 c 217 2 simulink 236 B project 63 branches 62 amp tags 62 4 2 documents 260 a trunk260 3 docs2010 109 3 docs2011 260 2 manual 195 B project 133 branches 64 Y tags64 a amp FinalReport 262 3 appendices 259 amp chapters 262 B project
38. with Matlab and Simulink Matlab is a textual programming language whereas Simulink is a graphical programming language that allows the user to define a system as a collection of interconnected building blocks where the blocks are manipulated graphically In this project we developed our own Simulink version of a controller which then served as the architecture for a controller that we implemented in C executed without using Matlab or Simulink 4 2 3 QNX QNX is a realtime operating system RTOS from QNX Software Systems We are currently running the QNX Neutrino RTOS A RTOS is necessary here because the timing constraints are critical The C development system for QNX is their QNX Momentics Tool Suite an Eclipse based integrated development environment In addition to its editing feature the tool suite can give developers an at a glance view of realtime interactions and memory profiles 4 5 Last revised 07 27 2011 5 Pendulum Controller Design This chapter introduces how to design a controller to balance the rotary inverted pendulum 5 1 Physical model The rotary pendulum is an unstable system that requires a controller to keep the pendulum at its upright position The controller incorporates a physical model for the pendulum i e mathematical equations that characterizing its structure and behavior This section describes those equations 5 1 1 Non Linear Equations of Motion The simplified diagram of the rota
39. 010 wre c 3 3 Me UI ies 3 3 3 2 2 Objectives O 3 3 3 2 3 What Did LDO ii nh A aca adios 3 3 3 3 Fall Semester 2010 ctt eto eb ett ie ederet 3 6 9 3 T OVerVIQWS N E E A A AE E EAE AA i tene st weds e etude decet uuu ete eue 3 6 A et tene et metet eee 3 6 3 3 3 What Did DO siete ciel iii aaa 3 6 3 4 Spring Semester 2011 s e A ese ee esaet eas ee ee a ete aee a 3 16 SAT OVerVie Wax oer edet dete AE A 3 16 34 2 A oce o hene dai ad REDE 3 17 3 4 3 What Did DO iiia eite t ern tp eene e eine en ear ERI EE atn 3 17 4 The System Configuration eterne pet e ree Fade a leia datada 4 1 4 1 Hardware CER 4 1 ATI GombpUters eee reser etie A nk den oo raa Cea dan ra a ehe a Une ade 4 1 4 1 2 SRVO2 E servo plant ctn bem a nitro ido as 4 1 4 1 3 UPM 1503 Power Module sseessssesseeseeeee enne nennen eene nennen nennen ntes risen 4 3 4 1 4 Quanser Q4 board cenae rode dee eed radio HERREN cates 4 4 4 2 kac oesheeeres 4 4 42 1 Quanser library aiit o or t e ada t EN EE GN as 4 4 4 2 2 Matlab and Simulink ite ihr teo ee RU ore be RR eea eaii aieiaa 4 5 42 3 QNX p an fedt drea e e egets hid tesa cir teen e e er e pedis 4 5 5 Pendulum Controller Design useessesseseeeeee ener nnn nnn nannten nennen ener 5 1 5 1 Physical model uu Re A eite ont peste teint E at ER aoe 5 1 5 1 1 Non Linear Equations of Motion ccooncnccu
40. 08 2 M Bohner and A Peterson Dynamic Equations on Time Scales Birkhauser 2001 3 R A Gupta and M Chow Networked Control System Overview and Research Trends IEEE Transactions on Industrial Electronics vol 57 pp 2527 2535 2010 4 Quanser Inc Rotary Experiment 08 Self Erecting Inverted Pendulum Control 5 MathWorks Matlab Simulink online Available http www mathworks com products matlab 6 Quanser Inc QUARC 2 1 overview online Available http www quanser com english html quarc fs overview htm 7 QNX Software Systems QNX neutrino RTOS online Available http www qnx com products neutrino rtos index html 8 Agilent Technologies Agilent 33220A User s Guide online Available http cp literature agilent com litweb pdf 33220 90002 pdf 1 3 Last revise 07 27 2011 2 Related Work 2 1 Introduction In this chapter the related research on networked control system NCS and time scales theory is introduced 2 2 Networked Control System This section introduces the research topics in NCS 2 2 1 Overview A control system aims to regulate the behavior of other systems Traditional control systems are feedback control systems in which the feedback signals are transmitted through cable or wires When a traditional feedback control system uses a network to connect the controller to the system it is called a Networked Control System NCS More specifically an NCS exchanges
41. 2 time s Figure 14 Quanser Lab 8 Running Result Fs 300Hz 8 12 Last revised 07 27 2011 8 2 Conclusions and Future Work 8 2 1 Abilities of The Test Bed The test bed developed in this project including the simulator and controller could help users understand control systems The experiments that we have done using the test bed have shown its ability to test time scales control and network control theories More importantly the models in the test bed provide useful blocks with interfaces that can be reused so that uses could embed their own controllers which is helpful for future researches In conclusion the test bed developed in this project can help facilitate the future research of developing and verifying the time scales control theory as well as network control theory 8 2 2 Limitations Some restrictions have limited the experiments that we can do using this test bed First we lack the time scales control theory for a non fullstate feedback system For the pendulum system where the arm and pendulum velocities are not directly sensed by sensors it is not a full state feedback system The traditional pole placement algorithm does not work quite well in this case Second current hardware configuration cannot send senor data directly to controller so we have not implemented a real NCS yet 8 2 3 Future Work Future research could involve the following areas e Embed velocity sensors in the pendulum system Add velocity
42. 2 which will display the pendulum and arm positions in scopes to indicate whether the pendulum is balanced Note that some parameters can be tuned You can try to tune these parameters until the results show that the pendulum can be balanced After the simulation results become acceptable you can run the executable model to actually control the system This model is compiled and the binary code can be downloaded and run on the QNX target machine The relevant files are C 7 Last revised 07 27 2011 la exe filter observer mdl ba exe_statespace_observer mdl 5 postscript_local m 5 postscript_ncs m 5 postscript_timescales_local m Readme txt PA setup m 5 setup_calc_conversion_constants m 5 setup_config_sp m LA setup config srv02 m setup_SRVO2_SIP_ABCD_eqns m la sim NCS mdl ba sim_statespace_observer mdl amp sim switched system m sim timescales m sim timescales generator m ba sim_timescales_local mdl a sim timescales NCS mdl Figure 5 Files contained in the quanser pendulum control folder The setup m files are Matlab files used for calculating the pendulum s physical parameters determining control gains and other parameters that are used by the controller The setup m file is the main setup file which calls the other setup m files The mdl files are Simulink models which can be used to simulate the system See section 3 3 for how these files should be used After a simulation the pos
43. 216 3 projectReport 214 gt reverselJulll 225 4 2 weeklyReports 82 4 trunk82 4 amp weeklyReports 82 gt Archive 82 E weekly reports fall2010 59 3 weekly reports spring2011 59 gt weekly reports summer2010 70 B project 59 2 branches 58 Y tags61 B Plugins log 225 Ci ROOT 262 i REVISION Figure 2 SVN Repository 2 RuntheQuanser Example Lab 8 Quanser provides an example Lab 8 to demonstrate how to balance then pendulum This section introduces how to run this Quanser example Before running the Quanser Lab 8 log on the Host and the Target Since the Host runs Windows 7 computer use your user name assign by the administrator e g ece01 song For the QNX computer log in as the super user with the password quarc target Note that the keyboard is shared between the host and target computers To use the keyboard on either computer hit the Scroll Lock key twice to toggle between the computers C 4 Last revised 07 27 2011 After logging on both Host and Target follow these steps 1 On the Host running Windows 7 double click the VMware Player desktop icon to display the VMware Player window Then Choose open a virtual machine You can find the virtual Windows XP Professional located in CNVMNXP PRO Then select poweron this virtual machine 2 On the Target run Launch gt QUARC gt Console An empty console should then appear 3 On the Host switch to the Windows XP virtual machine 4 Start M
44. 4 e matrix test 162 svn ssh songc wir 1 trunk162 gt Binaries 3 controller 162 ER Archives gt c 162 A Includes e 192 168 0 99 77 gy doc 146 E cpp New Ex include 77 amp simulink lib 77 project Check Out Pa 45 branches 62 4d Find Check Out As n constants h 148 Y tags 62 ales Controller cpp 148 amp documents 158 ty Controller h 148 3 projectReport 161 of Cut main cpp 148 e e 82 Paste Matrix cpp 149 Cm ROOT 162 X Delete R Matrix h 149 i REVISIONS Copy URL fj Observer cpp 151 Refactor gt n Observer h 150 ih Parameters K h 98 Show History hy Parameters Observer h 123 Show Properties y Parameters RefGain h 117 j PendulumSystem cpp 148 d ij PendulumSystem h 148 Add Revision Link Ir Q4 cpp 148 Export Q4 h 148 Import fj Reference cpp 148 Compare With A hy Reference h 148 Ly common mk 77 Refresh Makefile 77 Figure 18 check out C project Figure 19 Project structure C 19 Last revised 07 27 2011 4 After checking out the code switch to the C perspective You should see the file structure shown in Figure 19 Select the matrix_test project and select Project gt Build Project from the IDE menu to build the project 5 Switch to the Target On the desktop use the right mouse button and select terminal which will display a terminal window In the window type qconn and press enter The qconn daemon is the target agent that communicates w
45. K Save the file and run the simulation You can get a result similar to Figure 13 You can find that there are jitters in the arm and pendulum positions However the pendulum angle remains within 3 degrees showing that the pendulum is still balanced Similarly you can select different network types r ny Source Block Parameters TrueTime Network mm Real Time Network mask link z Parameters Network type CSMA CD Ethernet Network number 1 Number of nodes 3 Data rate bits s 100000000 m Minimum frame size bits 80 Loss probability 0 1 Initial seed Ma Y Show Schedule output port 4 A A Figure 12 TrueTime Network Parameters C 14 Last revised 07 27 2011 Horizontal Arm Position 8 T T T T T T T T T position degree reference position i 0 2 4 6 8 10 12 14 16 18 20 time s Pendulum Position 3 T T T T T T T T position degree o N A o 10 12 14 16 18 20 time s Figure 13 NCS Simulation Result loss rate 0 02 3 4 Run the Local Control Executable Model Our local control executable model implements the same controller as the local control simulation model This model is compiled by Matlab and the QUARC library on the Host and the binary code is downloaded to the QNX machine Whenever you create a new executable model you must set the target file
46. ON controller simulink quanser_pendulum_control 4 Choose File gt Open to open the setup m file For NCS you will need a reasonably high sampling rate or a reasonably low sampling period By trial and error we found that 0 004 s is the largest sampling period that will allow the system to maintain stability Therefore set the sampling period to a value less than or equal to 0 004 6 sampling period Ts 0 004 5 Click the button in the red circle or hit F5 to run the setup m file This will calculate the constants and matrices that will be used to represent the physical characteristics of the pendulum system The Matlab command window will show the control gain matrix 6 Open sim NCS mdl shown in Figure 10 1 2 3 4 xad ua tau N m oo vi li mm Y Oxin Du n x p th dot rad s x n 1 Ax n Bu n reference signal RefGain 2 Torque Voltage Actuator Electrical rotary pendulum system Scopes Actuator Dynamics Dynamicst K Data 152 1 2 Data JUL Hp Trigger cele TrueTime Send TrueTime Receive ngger Sensor Actuator 10 7 u a a 5 o Select Mich Y 4 a theta_dot ule Data Kr Data 1 3 i K State Space Observer 1 3 ontrol gain TEES Trigger 1 Schedule TrueTime Send TrueTime Receive eta 9 Co
47. Quanser Inc http quanser com SRVO2 E Servo plant Quanser Inc http quanser com DI 604 Router D Link http d link com default aspx 33220A Signal Generator Agilent http www home agilent com agilen Technologies t home jspx lc eng amp cc US Software Name Vender Version Website Matlab The Mathworks Inc R2010a http www mathworks com QUARC Quanser Inc 2 1 http quanser com Momentics IDE QNX software Systems 4 7 http www qnx com QNX Neutrino RTOS QNX software Systems NA http www qnx com VMware Player WMware Inc 3 1 3 http www vmware com Agilent lO Libraries Agilent Technologies 16 1 http www agilent com Suite IntuiLink Waveform Agilent Technologies 1 6 http www agilent com Editor B 1 Last revised 07 27 2011 Appendix C Test Bed User Manual 1 Set Up the System 1 1 Install Software Here we assume that the Host runs Windows 7 and the Target runs QNX Software installed on the Host includes Matlab VMware player the QUARC library and QNX Momentics IDE On the target the QUARC library is installed Because the QUARC library does not support Windows 7 we have to install a VMware Player on the Host so that we could run a virtual Windows XP computer on which the QUARC library is also installed When the QUARC library supports Windows 7 then VMware will not be necessary 1 2 Connect Wires The Host and Target computers communicate through an Ethernet connection They are connected to a
48. SysML The Systems Modeling Language SysML is a general purpose modeling language for system engineering applications It supports the specification analysis design verification and validation of a broad range of systems and systems of systems Such systems may include hardware software information processes personnel and facilities 10 Here SysML can be used to describe the structure of a control system Artisan Studio 15 is a tool that supports SysML modeling and became familiar with it by completing one of its tutorials 16 constructed a SysML model to represent the water distiller example from Chapter 15 of 14 Also constructed a SysML model of the pendulum system including Block Definition Diagram BDD Internal Block Diagram IBD activities diagrams and state charts 3 2 3 3 Modelica and Simulink modeling We investigated the possibility to use Modelica in conjunction with SysML or Simulink to model the system Modelica 17 is a non proprietary object oriented equation based language used to model complex physical systems OpenModelica 11 is an open source Modelica simulation environment followed the OpenModelica Users Guide 18 to learn how to simulate a system and replicated the tank example in Chapter 15 of 19 3 4 Revised 07 27 2011 Simulink 20 is an environment for multi domain simulation and model based design for dynamic and embedded systems It provides an interactive graphical env
49. University TrueTime toolbox online 2 0 beta 6 Available http www control Ith se user truetime 19 UC Berkeley EECS department Ptolemy II online Available http ptolemy berkeley edu ptolemyll 20 Wikipedia org Time scale calculus online Available http en wikipedia org wiki Time scales calculus 21 B Jackson A General Linear Systems Theory on Time Scales Transforms Stability and Control 2007 22 B Allen Experimental Investigation of a Time Scales Linear Feedback Control Theorem 2007 2 8 Revised 07 27 2011 3 Project Summary 3 1 Introduction In this project we developed a test bed to model and simulate embedded control systems This test bed is designed to e help users understand digital control systems e provide a platform for users to define their own controller e provide a platform for users to simulate their own controller e provide a platform for users to apply their own controller on hardware e provide a platform for users to design and verify time scales controllers This document explains the purpose of the project what and how things have been done It is organized in chronological order summarizing the following semesters summer 2010 fall 2010 and spring 2011 Each section identifies the objectives and how those objectives were met 3 1 1 Purpose and Approach The primary purpose of the project is to develop a test bed that can facilitate researches on time scal
50. Varying Sampling Rate Simulation One of our objectives is to investigate the stability of the system when it switches from a stable state to an unstable state and vice versa We know that the Hilger circle associated with Maa determines the area that the system stays stable However there is a conjecture that as long as the system does not stay outside the Hilger circle too long the system is still stable C 21 Last revised 07 27 2011 To demonstrate this did a simulation to determine what happened when the system switched forth and back inside and outside of the Hilger circle First use the Simulink model to find the sampling rates 1 that will balance and 2 that will not balance the pendulum Let Ts represents the sampling period By doing several experiments find that when Ts lt 10ms the pendulum is stable and when Ts 20ms the pendulum is unstable Second investigate the case where the control system switches between the stable and unstable case For the simulation create a Matlab program to model the switching mechanism where the switch criterion is based on the Euclidian norm of difference between the reference vector and the state variable Ss 7 5ms ifllref x I gt 0 04 20ms ifllref xII lt 0 04 To run this simulation file perform steps 1 3 in section 3 2 Then open file sim switched system m and run it When simulation finishes a picture similar to Figure 23 will show the results of the
51. Write Analog block to the power amplifier 5 20 Last revised 07 27 2011 0 Quarc No Swing Up_ P enable E 7 Switch HIL Initialize HIL 1 940 HIL iau N m Maq O Write tau N m a T Analog u a HIL Write Analog P tp th dot rad s Mn MP HILO ees A Torque Voltage Actuator Electrical eta_do Actuator Dynamics Dynamics Dm vm W HIL roel Encoder Timebase eZ Encoder Calibration Y HIL Read Encoder radicount o alpha alpha up Hp Timebase HIL 1 gun Calculate Upright Angle HIL Watchdog HIL Watchdog Stop with Error HIL 1 Stopped by watchdog Figure 16 Pendulum System Subsystem The Calculate Upright Angle subsystem Figure 17 is used to convert the pendulum angles from its encoder values When the pendulum is initially at its straight down position the encoder reads 0 which is mapped to 180 degrees When the pendulum is swung up the encoder reads either 2048 or 2048 which is mapped to O degree This subsystem apply the modulo operation to the input angle and subtract the offset angle to get the absolution angle R gt ha op Si Mod Angle Math alpha up rad Function Offset DIS rad Figure 17 Calculate Upright Angle Subsystem 5 21 Last revised 07 27 2011 Horizontal Arm Position position degree reference position 6
52. assume that the reference input is a step signal which means that r n 1 for n20 For the step signal the z transform is R z z z 1 we can apply the final value theorem to 1 36 and get lim y k lim z DY z lim z DH I F GK NGz z 1 38 NH I F GK G and we determine the numerical value of the reference gain N from 38 by calculating matrix U 5 12 Last revised 07 27 2011 U H I F GK G 39 and N a 40 Ud where U 1 represents the first row of matrix U Therefore the resulting control system block diagram with a reference signal is designed in Figure Plant x n 1 Fx n Gu n m Observer amp n 1 F amp n Gu n L y n H amp n Figure 7 Block Diagram of the Control System with Reference Signal The observer can be implemented based on its state space representation which is II gt gt yo a e amp AX 41 lt lt Il Cr ya o e where the state variable X is the estimation of x Note that in Figure 7 the observer has two inputs u and y so define the input for the observer as u u i and the other matrices are y 5 13 Last revised 07 27 2011 A A LC B B L C oO o o Fr O O c or c c oo oc m I ooo c ooo c ooo c 5 3 Simulation Section 5 2 gives the theoretical solution to the controller Based on the control block diagram in Figure 7 we can implement the control system in Simulink and simulate the performance
53. atlab on the Windows XP virtual machine Change the current working directory to C Mprojects MATLAB SRVO2 Lab Files Exp08 Inverted Pendulum Lab Files To do this click the button in the circle shown in Figure 6 Then in the popup dialogue find the respective folder and you should see a list similar to Figure 3 Current Directory da E Exp08 Inverted Pendulum Lab Files Y amp C Name Date Modified FurutaEisenbarth quarc qnx x86 9 13 10 4 16 PM FurutaEisenbarth quarc windows 9 13 10 4 16 PM a sesip pot q4 quarc windows 9 13 10 4 16 PM D q sesip q4 quarc qnx x86 6 13 11 3 10 PM D a sesip q4 quarc windows 11 30 10 3 38 PM a sesip a4 raccel rtw 9 13 10 4 16 PM a sesip quarc qnx x86 12 9 10 2 29 PM D q sesip quarc windows 12 9 10 2 26 PM read encoder quarc qnx x86 2 23 11 4 38 PM srj 2 16 11 5 04 PM write motor read encoder quarc qnx x86 2 24 11 10 23 AM calc conversion constants m 11 2 09 8 58 AM e calculate_qr m 1 7 10 5 28 PM E config sp m 11 2 09 8 58 AM e config srv02 m 11 2 09 8 58 AM d_model_param m 11 2 09 8 58 AM E FurutaEisenbarth rt qnx_x86 7 15 10 1 39 PM i q_sesip mdl 6 18 10 12 02 PM E q_sesip mdl autosave 12 9 10 2 29 PM O a_sesip rt qnx_x86 12 9 10 2 29 PM w q_sesip_pot_q4 mdl 6 18 10 3 24 PM E a sesip pot q4 rt windows 6 18 10 3 40 PM l q_sesip_q4 mal 6 13 11 3 10 PM O a sesip a4 rt qnx x86 6 13 11 3 10 PM O a sesip q4 rt windows
54. ble nondeterministic levels of service in terms of delay jitter and packet loss These are problems that are not normally associated with traditional feedback control systems A common approach is to let the controller solve the problems Based on the type of delay there are three solutions 2 1 If the delay is constant the control target can be modeled in a state space equation using the present input and the past inputs Here standard Linear Time Invariant LTI control theory can be applied 2 If delay is variable and the delay statistics for the network are available and there is a finite number of delay states then delay can be treated as a jump linear system driven by an underlying Markov Chain 3 If no delay statistics data are available or if the delay 2 2 Last revise 07 27 2011 statistics of the states of the Markov chain does not remain constant then either robust controller techniques or predictor based methods can be applied Usually the controller applies compensation algorithms to compensate for the delay Different mathematical heuristic or statistical based approaches are used for delay compensation in NCSs such as the optimal stochastic method the queuing buffering method the robust control method 2 7 This project does not provide our own delay analysis and delay compensation However the test bed has the ability to introduce arbitrary delay and packet loss rate so that the researchers can investigate th
55. ciency K is total gearbox ratio K is motor torque 8 constant K is back emf constant and V is motor voltage Rearrange 3 we get m R Tn K K 0 4 7 Mn K K i m Equation 4 shows that the output voltage depends on the torque z and the arm velocity 6 5 2 Last revised 07 27 2011 5 1 2 State Space Representation Equations 1 and 2 show that the pendulum system is a non linear system However non linear systems are difficult to analyze so we simplify the problem by linearizing the equations and get a collection of linear equations that approximates of the non linear equations 1 Consequently we can define a state variable and utilize the state space representation which is a standard form to describe multi in multi out dynamic system The state variable is defined as 5 S 2 amp Gm where is the horizontal arm angle is the horizontal arm velocity a is the pendulum angle and is the pendulum velocity Note that the encoders in the pendulum system only get the arm and pendulum angles Consequently only part of the state variable x is known and we will deal with it later The state space representation of the inverted pendulum is x Ax Bu y Cx Du Bl where u is the voltage applied to the DC motor In 6 A is a 4x4 square matrix B is a 4x1 matrix C is a 2x4 matrix and D is a 2x1 matrix The analytical solutions to these matrices are 0 0 1 0 0 0 0 0 1 0 1000 0 A B C
56. co design approach which provides a solution that involves improvement of both the control system and the communication system 8 11 The key idea of co design approach is to optimize the control performance by adjusting the control and the network algorithms based on network statistics For example A Chamaken presents a co design approach for the inverted pendulum system 9 The authors develop and implement two controllers with control laws considering network statistics Three different MAC protocols are used in conjunction with two different controllers to stabilize the inverted pendulum The result shows that the optimal control performance can be achieved at the control layer and the communication layer 2 3 System Simulation Tools System simulation tools can be used to simulate the performance of the NCS before the implementation Simulation is often easier than implementation Especially when designing the controller many parameters in control algorithms need fine tunes In this case simulation can save time and speed up the process Since the NCS consists of the control system and network system first we consider using separate simulation tools for each system However this approach is not efficient enough Hence we decide to use the TrueTime toolbox as the co simulation tool 2 3 1 Available Network Simulators Ns 12 is a discrete event simulator and is widely used in network research Ns provides a substantial support for the s
57. com products neutrino rtos index html 14 T Weilkiens Systems Engineering with SysML UML USA MORGAN KAUFMANN 2006 15 Atego Artisan studio online Available http www atego com 16 Artisan Software Artisan Studio Studio Tutorial 2009 17 The Modelica Association Modelica online Available https www modelica org 18 P Fritzson 2009 OpenModelica users guide online Available http www ida liu se labs pelab modelica OpenModelica releases 1 6 0 doc OpenModelicaUsersGuid 19 P Fritzson Principles of Object Oriented Modeling and Simulation with Modelica 2 1 Wiley IEEE Press 2004 20 MathWorks Matlab Simulink online Available http www mathworks com products matlab 21 OptXware Research amp Development LLC The Viatra l Model Transformation Framework Users Guide 22 T Johnson Integrating Models and Simulations of Continuous Dynamic System Behavior into SysML 2008 23 G P Liu D Rees and S C Chai Design and practical implementation of networked predictive control systems Presented at Networking Sensing and Control 2005 Proceedings 2005 IEEE 24 L Wu and X Hao A novel optimal controller design and evaluation for networked control systems with time variant delays in 2010 International Conference on Measuring Technology and Mechatronics Automation 2010 pp 261 264 25 A Onat T Naskali E Parlakay and O Mutluer 2010 Control over imperfec
58. d on the Target s PCI bus and is connected to the external terminal board via a flat cable The terminal board provides ports for digital encoder input analog input analog 4 3 Last revised 07 27 2011 output and digital input For this project we have only used the digital encoder input and the analog input ports Digital Input Encoder Input Figure 6 Quanser Q4 Terminal Board 4 2 Software 4 2 1 Quanser Library Quanser QUARC library allows control prototyping and hardware in the loop testing It is integrated with Simulink and the Real Time Workshop RTW The QUARC library extends the code generation capabilities of RTW by adding a new set of Targets such as Windows and QNX x86 The Target OS setting determines the code generated by the RTW This allows the user to compile the C source code 4 4 Last revised 07 27 2011 generated from the model to link it with the appropriate libraries for the particular Target platform and to download the code to the Target via an Ethernet connection between the Host and the Target The QUARC library supports data acquisition cards from other manufacturers such as National Instruments The library also provides a driver for the Q4 board so that the controller can use high level functions from the QUARC library to complete low level lO tasks 4 2 2 Matlab and Simulink We have used Matlab as our basic development platform because the Quanser hardware and software works seamlessly
59. dow double click both scopes to see the results 6 Because the output is displayed in two scopes it is difficult to compare and analyze We can display the results in a single figure by running the postscript local m file which creates the diagram in Figure 9 C 10 Last revised 07 27 2011 Horizontal Arm Position 8 T T T T T T reference position o position degree o time s Pendulum Position 1 5 T T T T T T T iE Es A o o o S e 0 2 o 2 0 5 1 15 2 4 6 8 10 12 14 16 18 20 time s Figure 9 Local Control Simulation Result 3 3 Run the NCS Model We created a network control system NCS simulation model by using the TrueTime toolbox to simulate the network Before you run the simulation you must first set up the TrueTime environment 1 Start Matlab and switch the current working directory to the following directory located in your workspace YOUR WORKSPACE LOCATION controller simulink truetime 2 0 beta6 examples 2 Open the setup m file Edit the first line to identify the TrueTime toolbox kernel location For example if your workspace location is C Song wks research then edit the first line as C 11 Last revised 07 27 2011 setenv TTKERNEL C Song wks research controller simulink truet ime 2 0 beta6 kernel 3 Switch the current working directory to folder YOUR_WORKSPACE_LOCATI
60. e F H P Click the button in the red circle or hit F5 to run the setup m file This will calculate the constants and matrices representing the pendulum system The Matlab command window will then show the control gain matrices File Edit Tex Go Cell Tools Debug Desktop Window Help OEE s2OC S2 MAesnG eaeanras Figure 7 Run setup m Open the sim_statespace_obeserver mdl fiel which represents the entire simulation model including the feedback controller observer and reference signal C 9 Last revised 07 27 2011 1 theta_ref 0000 oo D2R m arm reference Degrees to RefGain position degrees Radians nu Select theta dot P tau N m Po at pth dot rad s 4 Vi V 1 39 Vi V Vm V Torque Voltage Actuator Electrical Actuator Dynamics 8 vie Dynamics gt y_d 6 y n Cx n Du n Mi x n 1 Ax n Bu n Scopes rotary pendulum system Khe LQR Control Gain Xhat T State Space Observer Figure 8 Simulink Simulation Model Simulation model for the Quanser rotary inverted pendulum Use the State Space Observer to estimate the state 5 Run the model When finished you can see the simulation results via the Scopes To do this double click the Scopes subsystem The details of this subsystem are shown in a new window which contains two scopes In the new win
61. e and execute C and C code from Simulink diagrams The Quanser QUARC library is based on RTW which can generate and compile C code for various targets Modelica is a non proprietary object oriented equation based language used to model complex physical systems OPENMODELICA 17 is an open source Modelica based modeling and simulation environment intended for industrial and academic usage Both tools can be used to model a dynamic system The difference is that Modelica is equation based so a user must provide explicit ordinary differential equations to represent a system while Simulink uses blocks to represent mathematic operations 2 5 Last revise 07 27 2011 2 3 3 Available Co Simulation Tools TrueTime 18 is a Matlab Simulink based simulator for real time control systems TrueTime facilitates the co simulation environment of controller task execution in real time kernels network transmissions and continuous plant dynamics Ptolemy II 19 is an open source software framework supporting experimentation with actor oriented design The Ptolemy Project has developed directors supporting process networks PN discrete events DE dataflow SDF synchronous reactive SR rendezvous based models 3 D visualization and continuous time models In this project we chose TrueTime toolbox working with Simulink as a co simulation tool The main reason of the tool selection is that Ptolemy II with limited available resources is not tha
62. e control system diagram is shown in Figure 4 TankPI cOut E tSensor tActuator iconos Figure 4 A Tank Simulink Model with a Continuous PI Controller To model the system first created the tank subsystem Here replaced the ordinary differential equations in Modelica with Simulink blocks The corresponding equation and blocks are marked in Figure 5 with the associated equations shown in the figure 3 8 Revised 07 27 2011 r Ml tank Subsystem O El mem File Edit View Simulation Format Tools Help Diac les 2 12 A p Ps Noma Z2 gs 2 b der h qIn Iflow qOut Iflow area Gain1 Integrator Saturation qOut lflow LimitV alue minV maxV lowGain tActuator act Ready Figure 5 Simulink Subsystem A continuous PI controller was designed to control the liquid level in the tank Figure 6 shows the Simulink control system diagram The Subsystem in Figure 6 is connected to the PI controller to form a closed loop system pan qOut o TS Step tActuator tSensor Subsystem Scopel PI s a 0 25 PID Controller1 Add Constant Figure 6 Simulink Model of the Local Control System 3 9 Revised 07 27 2011 tank liquid height 0 7 T T height m 0 50 100 150 200 250 300 time s Figure 7 Simulation Result of the Local Control System To simulate the tank system we assume that there is an input liquid flow which is the input signal i e
63. e controller receives a reference input r n and 27 becomes Xx n 1 F GK LH amp n Ly n Mr n u n Kx n Nr n controller 28 where M a matrix and N a vector are to be determined To determine M and N the criterion is that r n does not have any influence on the system which means that the state error e n x n X n is independent of r n Since x n 1 Fx n Gu n Fx n G K amp n Nr n 29 Fx n GKx n NGr n and amp n 1 F GK LH amp n Ly n Mr n F GK LH amp n LHx n Mr n nd 29 30 we get x n4 1 amp 1 F LH x n amp 1 NG M r n 31 Substituting e n R n X n we get e n 1 F LH e NG M r n 32 5 11 Last revised 07 27 2011 In 32 if r n has no effect on e n it must be true that NG M 0 therefore M NG 33 Since G is known the only thing we need to determine is N the reference gain If we assume that x n X n for a sufficient large n which indicates that the estimated state converges to the real pendulum state after sufficient long time then we get x n 1 Fx n Gu n Fx n G Nr n Kx n 34 Fx n G Nr n Kx n F GK x n NGr n Take the z transform on both sides zX z F GK X z NGR z 35 Solve X z and get X z zI F GK NGR z 36 Consequently the z transform of the output y n is Y z HX x 37 H 1 F GK NGR z If we
64. e parameters D 3 Last revised 07 25 2011 Send Abitrary Waveform Waveform Parameters Reset Parameters Frequency kHz 1 0000000 Amplitude V p p 0 1000 Offset Vdc roo Defaults for Waveform Current from Instrument Reset Waveform Location on Instrument Volatile Memory C Memory Name VOLATILE Send to Instrument Waveform and Parameters C Waveform Parameters Only Figure 5 Send Arbitrary Waveform 9 Edit parameters and click Send button A dialogue will prompt the estimated transferring time Click OK 10 The edited waveform will be sent to the signal generator through Ethernet After the transmission is done the signal generator will output the desired waveform References 1 Agilent Technologies Agilent 33220A User s Guide online Available http cp literature agilent com litweb pdf 33220 90002 pdf 2 Agilent Technologies IO libraries suite 16 1 online Available http www home agilent com agilent product jspx cc US amp lc eng amp nid 34466 977662 amp amp cc US amp lc eng 3 Agilent Technologies IntuiLink waveform editor online Available http www home agilent com agilent editorial jspx ckey 1000000918 epsg sud amp id 1000000918 epsg sud amp ni d 536902257 536881980 02 amp lc eng amp cc IN
65. e solution which was a C representation of the controller running on QNX QNX provides software timers that can be used in the C project When a timer expires the program directly accesses the Q4 board and calculates the control signal So we can set the timer expiration time at each step which gives us the flexibility to change the sampling rate During this time to understand time scales theory on control systems read Benjamin Allen s thesis Experimental investigation of a time scales linear feedback control theorem 43 and Billy Jackson s dissertation A General Linear Systems Theory on Time Scales Transforms Stability and Control 44 3 4 2 Objectives The objectives of this semester were to e Implement my controller in Simulink e Implement my controller in C e Study the time scales theory e Investigate the stability of switched systems 3 4 3 WhatDid I Do 3 4 3 1 Implement the balance controller in Simulink implemented an executable model in Simulink which can be compiled downloaded and run on QNX However when first implemented the controller the controller did not balance the pendulum It 3 17 Revised 07 27 2011 seemed that the motor did not output enough torque After investigating the Quanser Lab 8 and each sub system in the Quanser example found that had failed to include the motor dynamic sub system in my model After adding the motor dynamics sub system which converts the torque to a volta
66. e variable x in the cost function and R is a positive definite matrix that penalizes the control input u in the cost function Note that it follows that the LQR based control design requires the availability of all the elements in the state variable x The feedback control gain K given by K R B P 14 where P is found by solving the continuous time algebraic Riccait equation A P PA PBR B P Q 0 15 5 7 Last revised 07 27 2011 Similarly for a discrete linear system described by x n 1 Ax n Bu n the cost function J J Y x Qx Qe u Ru k 16 and the feedback gain K can be calculated as 2 K R B PB B PA 17 where P is the unique positive definite solution to the discrete time algebraic Riccati equation P Q A P PB R B PB B P A 18 We can use the Matlab function d qr to calculate K The syntax is K S e dlqr A B Q R where A and B is the matrix of discrete time state space system model Q and R are the parameters defined in the cost function Sample code is shown in Figure 4 Note that F and G are the same variables used in 11 Design digital LOR controller x 0 2 weighting factor for the pendulum position y 0 4 weighting factor for the pendulum angle Q x 0 0 0 O y 0 0 0000 000 0 R 5 K controlgain dlqr F G Q R Figure 4 Sample Code to Call dlqr 5 2 2 Observer Design As indicated above we do not have direct access to the complete state variable Consequent
67. ed the Simulink model to find the sampling rates 1 that would balance and 2 that would not balance the pendulum Let Ts represents the sampling period By doing several experiments found that when Ts lt 10 ms the pendulum is stable and when Ts 20 ms the pendulum is unstable Second investigated the case where the control system switched between the stable and unstable case For the simulation created a Matlab program to model the switching mechanism where the switch criterion was based on the norm of difference between the reference vector and the state variable Ss 5ms ifllref x li gt 0 04 20ms ifllref xIl lt 0 04 Figure 15 shows the position and Ts when the horizontal arm is tracking a square wave reference input with amplitude 0 1 rad Here the above figure shows the arm position and pendulum position We can see that the arm can track the reference position and the pendulum angle is within 0 04 rad indicating that the pendulum is balanced The figure below shows the switching sampling period where Ts switches between 5ms and 20ms When the reference input remains unchanged the system is sampled at 20ms However when the reference input changes system must be sampled at 5ms 3 20 Revised 07 27 2011 015 arm position pendulum position Sampling period u
68. eir own various algorithms 2 2 2 3 Network Security Enhancement Network medium particularly wireless medium is susceptible to easy interception Research in NCS was initiated from the concern for security and convenience in hazardous environments such as nuclear reactor power plants and military applications In all these applications security is of the utmost concern The research in this project does not include NCS security issues from the control system s perspective The future research should consider these issues 2 2 3 Improve the System Performance The delay and packet loss problems can be addressed either within the control system or within the network system or in both systems The first solution is to address the problem within the control system Given the network technology the designer can analyze the statistics of the network Based on this information the designer will design the suitable controller that can minimize the influence of the network The second solution is to improve the network technology to minimize the delay and loss rate so that the network is compatible for any type of controller Such methods include 1 NCS Scheduling approaches and 2 Dynamic bandwidth allocation DBA approaches 1 2 3 Last revise 07 27 2011 The above solutions only consider part of the system either the control system or the network system which is not complete and efficient Consequently some researchers focus on
69. er Figure 13 Simulink Model with a State Space Observer 3 15 Revised 07 27 2011 The above models Figure 12 and Figure 13 represent local control systems To simulate a networked control system and investigate its performance created an NCS model Figure 14 using a TrueTime Network block which can be configured to be one of several networks e g Ethernet CAN and various parameters such as bit rate loss rate In Figure 14 subsystems 1 6 are the same as the local control system Figure 12 and Figure 13 But the system now is divided into two isolated systems the controller system and the pendulum system In the pendulum system blocks 1 4 7 10 the sensors block 7 get the position information and sends it to the controller where the data rate is controlled by the sensor trigger The pendulum receiver block 10 receives the control signal and writes it to the actuator In the controller system subsystems 5 8 9 the controller receiver block 8 receives the position information calculates the control signal and sends it back to the actuator through controller sender block 9 2 4 oooo oo reference signal RefGain gt tau N m th dot rad s Vi V I vi V Vm V Torque Voltage Actuator Dynamics Actuator Electrical Dynamics1 y n Cx n Du n TrueTime Receive Actuator 10
70. es a PD controller while our controller is a LOR controller 8 10 Last revised 07 27 2011 Horizontal Arm Position 8 T T T T reference position 9 BF E F amp 4F Ej 5 a 2r 7 Es D 4 2 1 1 1 1 1 60 62 64 66 68 70 72 time s Pendulum Position 1 2 T T T T T 1 LA 0 8 7 3 0 6 E g 04 4 S 02 4 5 2 p 4 0 2 E 0 4 4 0 6 60 62 64 66 68 70 72 time s Figure 11 Running Results Fs 200 Hz Horizontal Arm Position 15 T T T T FF reference position TE 10r 2 gt Et amp OF 5L 10 L 1 L 1 70 75 80 85 90 time s Pendulum Position 2 T T T T T aa 15r 4 tf 4 05 a z E or J 0 5 d AP 2 1 1 L 1 1 1 5 75 80 85 90 95 100 time s Figure 12 Running Results Fs 240 Hz 8 11 Last revised 07 27 2011 Horizontal Arm Position 6 T T T T T T T reference position 5 position degree N 50 51 52 53 54 55 56 57 58 59 60 time s Pendulum Position position degree Figure 13 Quanser Lab 8 Running Result Fs 200Hz Horizontal Arm Position T T T T T ir PATA position H 18 10 a position degree o 5 1 fo 102 104 106 108 110 112 time s Pendulum Position 3 T T T T T EP 2r ad F pp 7 c S T 7 a Th 4 A A e O A 400 102 104 106 108 110 11
71. es theory and networked control theory The Baylor University Time Scales Group 1 has done considerable researches on the time scales theory The theory of time scales was introduced by Stefan Hilger s in his Ph D dissertation 2 and subsequent papers as a way to unify the seemingly disparate fields of discrete dynamical systems i e difference equations and continuous dynamical systems i e differential equations Time scale systems might best be understood as the continuum bridge between discrete time and continuous time systems The test bed allows users to develop and simulate control algorithms and to generate arbitrary time scales to regulate sampling rates so it can be used to implement and verify the time scales theories 3 1 Revised 07 27 2011 The secondary purpose of this project is to understand digital control systems Also this project the examples and the project report should be helpful for people who are learning control theory We have chosen the rotary inverted pendulum as an example system because it is an unstable system and it is a commonly used example in the literature In particular we use the Quanser SRVO2 3 pendulum system which consists of a horizontal arm a vertical pendulum a gear chain and a DC servo motor as shown in Figure 1 The detailed system configuration will be introduced in Chapter 4 Figure 1 SRVO2 Rotary Inverted Pendulum System 3 1 2 Timeline e This project started dur
72. esult shows that the controller which uses the state space observer can balance the pendulum over a large range of sampling rate To generate various sampling frequencies we use the external signal generator The controller was designed at the sampling rate of 200 Hz Follow Appendix C section 5 2 and configure the system to use the external signal generator Run the controller When the pendulum is balanced adjust the output frequency of the signal generator The observation is that if the frequency increased or decreased too much the pendulum will unstable By tuning the frequency we find that the system is stable within 140 Hz to 240Hz Figure 11 and Figure 12 show the arm and pendulum angles when sampling rate is 200 Hz and 240 Hz respectively From Figure 12 we can see a larger swing of the arm which indicates instability The Qanser Lab 8 Appendix C Section 2 provides a different observer It utilizes a differentiator and a low pass filter to estimate the velocity which means that the observer calculates the velocity base on equation v n s n s n 1 T then process v n using a low pass filter By repeating the above experiment it is shown that this observer has a larger range of stability which is 140 Hz to 300 Hz Figure 13 and Figure 14 show the arm and pendulum angles when sampling rate is 200 Hz and 300 Hz respectively The difference between these two models is due to the different structures The Quanser Lab 8 provid
73. for the model But this step must be done only once 1 In the Simulink model on the Host select QUARC gt options 2 In the configuration parameters dialogue choose Real Time Workshop on the left column of the window where you can set various simulation parameters for the model C 15 Last revised 07 27 2011 3 In target selection click browse and choose quarc_qnx_x86 tlc in the popup dialogue as shown in Figure 14 which identifies the kind of target to build Configuration Parameters discrete_statespace Configuration Active q Target selection Solver i file quarc_qnx_x86 tlc Data Import Export A quarc am l Jl Browse Optimization Lanquage c vi Description QUARC QNX x86 Target Build process TLC options Makefile configuration Savi v Generate makefile Hardware Implementation Model Referencing imulation Target Template makefile quarc default tmf Symbols SS Make command make_rtw Select objective Unspecified ES Check model before generating code Off v Check model ems J mb ao Figure 14 RTW Configuration To set up the communicate between the Host and the Target you need to specify the Target IP address Switch to the Target On the desktop use the right mouse button and select terminal which will display a terminal window In the window type ifconfig and press enter The network information will be shown Reco
74. ge signal that is applied to the motor the controller can balance the pendulum 3 4 3 2 Read Ben s thesis and Jackson s dissertation on time scales theory read Benjamin Allen s thesis Experimental investigation of a time scales linear feedback control theorem 43 and Billy Jackson s dissertation A General Linear Systems Theory on Time Scales Transforms Stability and Control 44 Based on Ben s Matlab code implemented a Matlab program to simulate the time scales controller However there were two problems when we considered the time scales control 1 How to design the observer Currently time scales control theory can only solve full state feedback control problem In the pendulum system we only have position information but no velocity information So our system has only partial state information Also the conventional observer technique generally does not work when using time scales In my simulation calculated the observer matrices at each step using the conventional pole placement algorithm The simulation result showed that the pendulum would converge only after a long time At this point we lack the theory so this problem remains unsolved 2 How to vary the sampling rate while a model is running To use time scales control we need to vary the sampling rate In Simulink the sampling rate is a system parameter that should be set before simulation Once it is set we cannot change it Besides when compiled into executable
75. gear drive connected to a DC servo motor The horizontal arm which rotates in the horizontal plane is mounted on the outside gear of the gear drive The gear drive is driven by the DC motor and has two configuration options low gear configuration left in Figure 3 and high gear configuration right in Figure 3 This project used the high gear configuration The SRVO2 E is equipped with two optical incremental digital encoders that measure the horizontal arm position and the pendulum position respectively The encoders provide a high resolution i e 4096 counts per revolution in quadrature mode The encoder attached to the arm shaft measures the horizontal angle of the arm defined as O The other encoder attached to the pendulum hinge measures the vertical angle of the pendulum defined as shown in Figure 2 Both a and are measured in radians Figure 2 Rotary Inverted Pendulum 4 2 Last revised 07 27 2011 Figure 3 Gear Configuration 4 1 3 UPM 1503 Power Module The UPM 1503 Universal Power Module contains a 12 volt power supply analog sensor input and power amplified analog output It is used to drive the DC motor Universal Power Module Figure 4 UPM 1503 4 1 4 Quanser Q4 board The Quanser O4 board is really two boards a Data Acquisition DAQ board shown in Figure 5 and an external terminal board shown in Figure 6 These boards connect the controller to the pendulum system The DAQ board is installe
76. he actuator through controller sender block 9 Simulate the network control system using the same parameters listed in 5 3 2 and the simulation result is shown in Figure 13 Compare Figure 13 and Figure 11 we can see that the maximum pendulum angle in Figure 13 is 1 5 degrees which is larger than that in Figure 11 This may be because of the delay effect in the network 5 4 Simulink Executable Model Using Quanser QUARC library we can compile a Simulink model into a QNX executable image that can be used to control the actual system This executable model is implemented in Figure 14 The structure of the executable model was developed based on the Simulink model shown in Figure 8 rotary pendulum system theta ref py 1 2 y TR D2R Scopes arm reference Degrees to RefGain 3 position degrees Radians 5b E Control Executable model for the Quanser rotary inverted pendulum Use the State Space Observer to estimate the state We Xhat A u LQR Control Gain State Space Observer Figure 14 Executable Simulink Model Compare Figure 14 with Figure 8 we see that a Mode Control MC subsystem is present in Figure 14 The MC receives the output of the pendulum and output an enable signal which controls when the balance controller begins to execute The MC subsystem is necessary because that the encoders only 5 19 Last revised 07 27 2011 give the relative angle not the absolution angle
77. imulation of TCP routing and multicast protocols over wired and wireless local and satellite networks It is an open source tool and is available from http www isi edu nsnam ns 2 4 Last revise 07 27 2011 OMNeT 13 is a discrete event simulation environment Its primary application is the simulation of communication networks OMNeT offers an Eclipse based IDE and a graphical runtime environment OMNeT is free for academic and non profit use It is available from http www omnetpp org OPNET Modeler 14 is a commercial tool developed by OPNET Inc Modeler incorporates a broad suite of protocols and technologies and includes a development environment to enable the modeling of all network types and technologies e g VolP TCP OSPFv3 MPLS IPv6 A performance comparison of recent network simulators including ns 2 ns 3 OMNet can be found in 11 2 3 2 Available Control System Simulators Matlab 15 is a common development environment for a variety of engineering applications Simulink 15 is an environment for multi domain simulation and Model Based Design MBD for dynamic and embedded systems Simulink provides an interactive graphical environment and a customizable set of block libraries that can be used to design simulate implement and test a variety of time varying systems including communications controls signal processing video processing and image processing Real Time Workshop RTW 16 can generat
78. ing the summer 2010 semester when we set up the hardware and indentified the research problem e In the fall 2010 semester we analyzed the dynamic model designed and simulated the pendulum controller in Simulink Also we re configured the system with the controller running QNX e Inthe spring 2011 semester we implemented the controller in Simulink and then in C 3 1 3 Accomplishments We have completed the following tasks e Work through Quanser lab O 4 lab 1 5 lab 2 6 lab 3 7 lab 7 8 lab 8 9 e Identify relevant research papers e Understand how to model the pendulum system using Matlab Simulink 3 2 Revised 07 27 2011 e Understand how to design the controller e Construct the controller in Simulink and then simulate the system e Construct the controller in C 3 2 Summer 2010 3 2 1 Overview We started the project in the summer of 2010 Dr Eisenbarth set up the rotary pendulum hardware for the Quanser SRVO2 with one PC running Windows XP for both developing and running the controller Quanser provides eight labs that demonstrate how to utilize the hardware and software worked on doing these labs as well as indentifying the research problems 3 2 2 Objectives Our objectives during that time were to e Understand how to use the Quanser software and hardware e Carry out the rotary pendulum experiments provided by Quanser e Investigate SysML 10 modeling e Investigate Modelica 11 modeling and simulation
79. ironment and a customizable set of block libraries that enable users to design simulate implement and test various applications including communications controls signal processing video processing and image processing Both tools can be used to model a dynamic system The difference is that Modelica is equation based so that a user must provide explicit ordinary differential equations to represent a system On the contrary Simulink uses blocks to represents mathematic operations 3 2 3 4 VIATRA2 Model transformation is important in model driven engineering In this project model transformation is used to transfer the Simulink model into an executable binary code So how to do model transformation is one of the research problems to be investigated VIATRA2 Visual Automated model TRAnsformations 12 framework provides a general purpose support for the entire life cycle of engineering model transformations In particular it provides a means to uniformly represent models and metamodels and a high performance transformation engine To understand the framework read the VIATRA 2 Model Transformation Framework User s Guide 21 Chapter 1 3 And did VIATRA2 hello world and Transforming UML activity diagram into a Petri net tutorial But at current stage model transformation has not been done yet The VIATRA2 tool has not been used 3 2 3 5 Other relevant research Thomas Johnson s thesis Integrating Models and Simulations of Continuou
80. is a linear system an approximation of the actual non linear system RH Constant Subsystem rad2deg1 Figure 10 Simulink Rotary Pendulum Model The performance of the linear approximation can be analyzed by comparing it with the respective non linear Simulink model created the first Simulink model Figure 10 that utilizes the non linear model Initially the simulation initial condition sets the pendulum angle to 0 01 rad which is a small non zero value representing an unbalanced condition The simulation sets the input to O volts simulating the free falling nature of the pendulum Similarly created the second Simulink model with the non linear subsystem replaced by a state space subsystem After running the simulations compared the outputs arm position and pendulum position of the two models The result should indicate that the output of the linear model matches the non linear model when the pendulum angle is small and the output of the linear model deviates from the non linear model when the pendulum angle becomes large However my non linear block simulation result showed an unchanged pendulum angle which was incorrect Since the non linear model was not used in the project did not investigate this issue further 3 12 Revised 07 27 2011 Integrator1 e theta dot sin alpha Trigonometric Function alpha_doty 2 rigonometric Function1 Integrator2
81. ith the host 6 In the Host select Run gt Run If you run it for the first time a dialogue will ask you to select the type of application Select C C QNX Application Figure 20 and click OK because the target OS is QNX In the following dialogue Figure 21 select the first binary file and click OK 7 At this point the controller should be running Figure 22 Turn on the power amplifier then manually swing the pendulum to the upright position Once the pendulum is close to the vertical position the controller begins to work to balance the pendulum 8 To stop the controller click the red button in Figure 22 E y Run As S Select a way to run matrix_test C C QNX Application C3 C C QNX Application Dialog Local C C Application Description Description not available Figure 20 Choose a way to run the project C 20 Last revised 07 27 2011 y QNX QConn Application mI Choose a binary to run Binaries i rie I s matrix test g Qualifier ble matrix test x86 o matrix test o C Saas Figure 21 Choose a binary to run Ee Problems Tasks B Console X o Properties a X amp Ex BE EE LS a matrix test C C QNX QConn 1P tmp matrix_testsong13075631005981 on 192 168 0 99 pid 471079 6 8 11 2 58 PM board opened running Figure 22 Project is running 5 Varying Sampling Rate 5 1
82. l e Implement the Simulink model that simulates the balance controller e Investigate networked control systems e Implement an NCS controller for the rotary pendulum e Determine how to use the TrueTime toolbox to simulate wireless networks 3 3 3 What Did I Do 3 3 3 1 Investigate NCS simulation tools searched the Internet for network and control systems simulation tools and found several relevant types co simulation tools e g TrueTime 35 Ptolemy II 38 PiccSIM 39 network simulators e g 3 6 Revised 07 27 2011 OPNET 37 NS 2 34 NS 3 34 OMNeT 36 and control system simulation tools e g Simulink 20 and Modelica 11 evaluated these tools based on their capabilities the related programming languages the operating system that is assumed the compatibilities of working with other tools and required resources After evaluating each tool found that the TrueTime toolbox would work well for NCS co simulation because the TrueTime toolbox provides Simulink blocks that are targeted for network simulation making it easy to set up an NCS environment in Simulink 3 3 3 2 Practice Simulink modeling replicated the tank example from page 385 of Principles of Object Oriented Modeling and Simulation with Modelica 2 1 19 The original model is represented in Modelica and created an equivalent model in Simulink Figure 2 shows the structure of the tank system Liquid enters the tank through a pipe from
83. long the system will remain stable It should be possible to investigate this controller using the test bed where the system would switch from inside to outside of the Hilger circle and vice versa The matlab file sim_switched_system m simulates the above idea see in section 5 1 in Appendix C The switching threshold is based on the norm of the difference between the pendulum angle and the reference angle i e l ref x ll where ll x ll denotes the Euclidean norm of the vector x If the difference is less than the threshold the controller will work at a longer sampling period 20 ms otherwise it will work at a shorter sampling period 5 ms For example we set the switching threshold 5ms ifllref x l gt 0 04 to 0 04 i e T and the arm follows the reference signal of a square with 20ms ifllref xllx 0 04 a frequency of 0 1 Hz and an amplitude of 0 1 rad The results are shown in Figure 8 When the reference signal changes the controller works at a shorter sampling period and when the reference signal is stable the pendulum tends to work at a longer sampling period Figure 9 shows the experimental results with a threshold 0 1 where the pendulum is still stable However the arm angle is less smooth compared to the arm angle in Figure 9 Therefore this threshold controls how long the controller stays inside outside the Hilger circle If the threshold is too large the pendulum may stay outside the Hilger circle too long
84. ly we must estimate the missing values which can be done using an observer 3 The general form of the state variable feedback compensator is amp AX 4 Bu L y C 19 5 8 Last revised 07 27 2011 where X denotes the estimate of the state variable x and Lis a matrix representing a collection of coefficients to be determined If we define the estimation error e as e x X 20 and 6 19 we get X amp Ax Bu Af Bu L y C amp A x AX L Cx Cx 21 A LO yx amp Consequently A LC e 22 Control theory indicates that if all the poles i e roots of system 22 locate inside the unit circle the system is stable Therefore given a vector P of desired self conjugate closed loop pole locations the matrix L can be determined by calling Matlab function place The syntax is K place A B p where A and B are the matrices of discrete time state space system model and p is the vector for pole locations In the discrete domain the observer design is similar The equation for the observer is amp n41 F amp n Gu n L y n H amp n 23 and the estimation error can be defined as e n x n amp n 24 Then 11 23 we get e n 1 2 amp n41 x n 41 Fx n Gu n FX n Gu n L y n CX n F LH x n amp n F LH e n 25 Last revised 07 27 2011 We can also call place to determine the matrix L as shown in Figure 5 where F and H represe
85. m However when the controller is implemented on computers it becomes a discrete i e digital controller The system matrices need to be re calculated when the controller is converted from continuous domain to discrete domain 2 This section discusses how to do the conversion Consider the state space model 6 At t f given initial condition X x 1 the solution to the ordinary differential equation x Ax Bu is t x t 2 e xq eM Bu r dr 7 to In digital domain the system would normally be sampled at a constant rate F where F is the inverse of the sampling period T Let f nT and t n 1 T then 1 7 becomes S n DT x n DT e x nT eM Bulr dr 8 nT If we use a zero order hold ZOH sampler then u 7 u t u nT To facilitate the solution for a ZOH with no delay let y n DT 7 9 then we have 5 5 Last revised 07 27 2011 x 1 1 T ex nT ed Bu nT 10 In 10 if we define F e G e dnB Jo 1 11 H C J D then we get x n 1 Fx n Gu n 12 y n Hx n Ju n Compare 11 with 6 we can see that 11 is the standard difference equations for a discrete system Using an infinite exponential series expansion matrix G can be represented as oo ATUS 13 A DP x Matlab can be used to convert a continuous system to a discrete system using the function c2d as shown in Figure 2 The syntax of c2d is sysd c2d sys Ts method where sys
86. nto C code The Target Language Compiler then compiles the C code and generates an executable binary specific to QNX which is afterwards downloaded to the target 8 Turn on the UPM power module The switch is on the back towards the top 9 Run the program on the Target In the Simulink model window on the Host select QuaRC gt start The controller will cause the pendulum to swing up and then balance the pendulum C 6 Last revised 07 27 2011 10 Stop the program In the Simulink model window on the Host select QuaRC gt stop 11 When finished power off the power amplifier the Windows XP virtual machine and the QNX machine 3 Run Our Simulink Model We create our own Simulink model of the pendulum system which can be used to simulate the system or actually control the inverted pendulum when it is near the vertical position This model can be used as the basis for designing and fine tuning other controllers 3 1 Introduction The models are located in the following address which should be stored in your personal workspace YOUR WORKSPACE LOCATIONNcontrollerNsimulinkNquanser pendulum control The Simulink simulation model uses the LQR algorithm to control the pendulum It is a different controller from the PD controller in Quanser Lab 8 The executable Simulink model uses the same LOR controller which has several new blocks to directly access the hardware and handle the data Run the simulation model following section 3
87. ntroller NIISQuIS Simulate Network Controller for Quanser Inverted Pendulum 8 TrueTime Network 11 Figure 10 NCS Simulation C 12 Last revised 07 27 2011 7 8 Run the model When finished you can see the simulation results by double clicking the Scopes subsystem The details of this subsystem are shown in a new window containing two scopes In the new window double click each scope to see the results As before these plots are displayed in two scopes so that it is difficult to compare and analyze Then open the postscript_ncs m file and run it to display both plots in a single window Figure 11 If you compare Figure 9 and Figure 11 you can see that the maximum pendulum position is smaller than 1 5 degrees in the local control case and larger than 1 5 degrees in the NCS case due to the delay effect of the NCS Horizontal Arm Position 8 T T T T T T T T position degree reference position 6 0 time s Pendulum Position 2 T T T T T T T T position degree e o a P un time s Figure 11 NCS Simulation Result C 13 Last revised 07 27 2011 Many network parameters in the NCS model can be changed For example you can set the packet loss rate to see its effect Double click the TrueTime Network block block 11 in Figure 10 Then in the TrueTime Network Parameters dialogue set Loss probability to 0 02 Click O
88. nts the matrices in 1 11 and F represents the transpose of F oe pole locations 0 3 0 3 0 9 0 91 pole placement place F H P Figure 5 Sample Code to Call place FU oe p Therefore the block diagram of the control system based on 23 is shown in Figure 6 Sensor x n 1 Fx n Gu n Control Law amp n 1 FX n Gu n L y n H amp n Figure 6 Block Diagram of the Control System without Reference Signal 5 2 3 Controller with Reference Signal Input A limitation of the above controller is that it does not accept a reference input In the general case a reference input is a desired output that the system will converge to Recall that for the rotary 9 inverted pendulum the output y is Which has 2 elements In order to keep the pendulum a balanced the pendulum angle should be 0 all the time so that the only variable that we can control is the horizontal arm angle In this case we only set the reference input to a scalar This section discusses how to design the controller in the presence of a reference input 5 10 Last revised 07 27 2011 Let us first repeat the plant 12 and controller equations 23 where the plant refers to the system being controlled i e the pendulum x n 1 Fx n ae 26 y n Hx n R n 1 T GK LHB amp G EO contre 27 u n K amp n Therefore if we want to control the horizontal arm angle th
89. od T 0 001s e LQR algorithm matrices Q 0 2000 00 400 0000 0000 R 5 e Poles are at 0 3 0 3 0 9 0 9 Once we have the Simulink model the controller can be can simulated The first experiment tests whether the controller can balance the pendulum Set the reference signal to 0 which means that we keep the horizontal arm at a fixed angle of O degree Assume that the initial state of the pendulum is 0 1 0 1 0 5 0 5 X which means that initially the vertical pendulum and the horizontal arm are at 0 1 rad 5 7 degrees and their velocities are 0 5 rad s Apparently the pendulum deviates from the up right position and is rotating so it is not balanced The controller works at the sampling rate of 1KHz Figure 10 shows the arm and pendulum angles changing over time From the figure we can see that at t 0 the values of a 5 15 Last revised 07 27 2011 and are 5 7 Within 3 seconds both a and Oconverges to 0 which means that the pendulum is balanced Horizontal Arm Position 6 T T T T I sl reference position 0 position degree 2 l l l j l 0 0 5 1 1 5 2 2 5 3 time s Pendulum Position 6 T T T T T position degree 0 0 5 1 1 5 2 2 5 3 time s Figure 10 Simulation Result when X0 0 1 0 1 0 5 0 5 The second experiment tests whether the horizontal arm angle can follow the reference signal Set the refe
90. ork Control Ethernet 100Mbps Ts 5 ms loss rate 0 8 3 Last revised 07 27 2011 Horizontal Arm Position 10 position degree o De reference position 10 1 fi 1 fi fi 9 0 4 6 8 10 12 14 16 18 20 time s Pendulum Position 3 T T T T T T T I 2r 9 1 J E 5 3 gl ar zl 3 L it L L ji L L L 0 2 4 6 8 10 12 14 16 18 20 time s Figure 5 Network Control Ethernet 100Mbps Ts 4 ms loss rate 0 Horizontal Arm Position 10 T T T T T T T T 5r zi T o o o er 0 E 2 m o a 5r a reference position 0 10 L L L L 1 L I I 0 4 6 8 10 12 14 16 18 20 time s Pendulum Position T T T T I a T o 5 4 o 2 3 9 gt o a fi fi fi fi 10 12 14 16 18 20 time s Figure 6 NCS Ethernet 100Mbps Ts 2 ms loss rate 2 Last revised 07 27 2011 TrueTime toolbox provides the ability to adjust the network parameters The following experiments are on other network parameters For example we can set the packet loss rate for a TrueTime network Still assume that the network type is Ethernet with a data rate of 100Mbps The sensor data is transmitted to the controller every 2 ms When the packet loss rate is 296 the simulation result is shown in Figure 6 We can see that the pendulum angle has some random pattern compared to Figure 2 Local Control
91. ounter to the left The new result should be the same as in Figure 26 C 24 Last revised 07 27 2011 Source Block Parameters HIL Read En Cc max ceil 1 qc_get_step_size 1 Wve Figure 24 Source Block Parameters for the HIL Read Encoder Select channels System timer limited to 1 msec Watchdog counter WATCHDOG External Interrupt EXT INT Figure 25 Select channels C 25 Last revised 07 27 2011 Select channels Description Select the desired channels from the list of available channels and click the Add gt gt button to add them to the list of channels selected Use the Remove button to remove channels from the list Reorder the selected channels using the Move Up and Move Down buttons Channel selection Channels available Channels selected System timer limited to 1 msec External Interrupt EXT_INT Watchdog counter WATCHDOG General purpose counter COLINTER Select All Clear All Move Up Move Down Reorder Image Conflict Channel Information None Mame General purpose counter COUNTER Number 1 Figure 26 Select external interrupt as the new clock channel 5 Repeat the steps as described in section 3 4 6 When the controller is running the Simulink model is updated at the frequency of the external signal generator You can slightly adjust the output frequency of the signal generator to see the perfo
92. qin Assume that the input flow maintains at 0 02 m s from O to 150 second and then increases to 0 06 m s after 150 second The height of liquid in the tank is shown in Figure 7 At t 0 the height is 0 and the liquid begins to flow into the tank After the height is over 0 25 m the PI controller adjusts the valve to maintain the height at 0 25 m At t 150 the input liquid flow increases so here is an overshoot in the height But the PI controller maintains the liquid level to 0 25 m After successfully creating the Simulink model and running the simulation embedded a TrueTime network block in the model and completed an NCS as shown in Figure 8 The upper collection of blocks represents the tank sensor and actuator system and the lower collection of blocks represents the controller system These two collections communicate through TrueTime send and receive blocks which are configured to use the Ethernet protocol The tank system uses a Sensor Trigger to sample the liquid height and sends the liquid height information through the TrueTime Sensor block On the controller side the TrueTime Receiver Controller block receives the data calculates the control signal and sends it through the TrueTime Send Controller block The TrueTime Receive Actuator block receives the 3 10 Revised 07 27 2011 control signal and adjusts the actuator accordingly The simulation result is shown in Figure 9 which is identical to the simulation
93. rd the IP address For example the IP address in current configuration is 192 168 0 98 This address is fixed if the Ethernet cable remains in the same port on the router Then switch to Host In the Simulink model select QUARC preferences In the pop up dialogue input the information in each tab of the dialogue shown in Figure 15 C 16 Last revised 07 27 2011 QUARC Preferences Preferences Set the default preferences for QUARC QUARC Preferences Preferences Set the default preferences for QUARC Preferences Set the default preferences for QUARC Target Target Type Logging Build Instructions Target Type Loosing Build Instructions There is a default model URI for each target type Select a target type first and then fill in the default model URI The default model URI may contain the following format specifiers Sm the model name Targettype qnx x85 Default model URI tcpip 192 168 0 99 17001 There is a default target URI for each communication protocol Select a protocol first and then fill in the default target URI The default target URI may contain the following format specifiers which are used to derive the target URI from the model URI 36m the model name h the hostname from the model URI p the port from the model URI not usually used 360 the options from the model URI not usually used Default URI Protocol tcpip Default targe
94. rence input to a square wave with an amplitude of 0 1 rad 5 73 degrees and a frequency of 0 1 Hz Figure 11 shows the results of the pendulum angle and arm angle We can see that the arm angle tracks the reference input and keeps the pendulum angle within 1 5 degrees which indicates the pendulum is balanced 5 16 Last revised 07 27 2011 Horizontal Arm Position 8 T T T T T T T T T position degree qu reference position 0 2 4 6 8 10 12 14 16 18 20 time s Pendulum Position 1 5 T T T T T T T T T position degree time s Figure 11 Pendulum System Simulation Result 5 3 3 Networked Control System Model To simulate a networked control system a model was created Figure 12 using the TrueTime toolbox 4 The TrueTime Network block number 11 in Figure 12 can be configured to be one of several networks e g Ethernet CAN and its bit rate or loss rate In Figure 12 subsystems 1 6 are the same as those of the control system in Figure 8 i e the local control system But it is divided into two isolated systems the controller system and the pendulum system The pendulum system blocks 1 4 communicates with the controller blocks 5 6 8 using the facilities of the TrueTime send receive blocks blocks 7 10 5 17 Last revised 07 27 2011 oo reference signal RefGain 1 2 Data
95. rmance comparison of recent network simulators in EEE International Conference on Communications 2009 pp 1 5 34 DARPA The network simulator ns 2 online Available http www isi edu nsnam ns 35 Department of Automatic Control Lund University TrueTime toolbox online 2 0 beta 6 Available http www control Ith se user truetime 36 OMNeT Community OMNeT online Available http www omnetpp org 37 OPNET Technologies Inc OPNET modeler online Available http www opnet com solutions network_rd modeler html 38 UC Berkeley EECS department Ptolemy II online Available http ptolemy berkeley edu ptolemyll 39 Wireless Sensor Systems group Aalto University PiccSIM online Available http wsn tkk fi en software piccsim 40 M Calkin Lagrangian and Hamiltonian Mechanics World Scientific 1996 41 R Dorf and R Bishop Modern Control Systems Prentice Hall 2008 3 23 Revised 07 27 2011 42 G Franklin and D Powell Digital Control of Dynamic Systems Addison Wesley publishing company 1980 43 B Allen Experimental Investigation of a Time Scales Linear Feedback Control Theorem 2007 44 B Jackson A General Linear Systems Theory on Time Scales Transforms Stability and Control 2007 45 Baylor Time Scales Group The Hilger complex plane online Available http marksmannet com TimeScales Time Scales Tutorial index files 5 html 3 24 La
96. rmance of the controller C 26 Last revised 07 25 2011 Appendix D How to generate arbitrary time scales This appendix introduces how to generate arbitrary time scales using the Agilent 33220A signal generator 1 Before start you need to install the Agilent IO Libraries Suite 16 1 2 and IntuiLink Waveform Editor 3 on the Host Agilent 33220A is required to be connected to the router where the Host and Target are connected In current configuration Agilent 33220A will obtain IP address by DHCP 1 Poweron Agilent 33220A 2 Power on the Windows 7 Host machine Run Start gt All Programs gt Agilent lO Libraries Suite Agilent Connection Expert A window similar to Figure 1 is shown System tasks Refresh all cA eceo1 Add an instrument gt pam en gu add an interface 5 2 LAN TCPIPO E 3 Mz 192 168 0 101 More Information BB pxio S6 usso 9 How do I add an instrument 9 How do I get drivers Where can I find programming samples 16 0 14518 0 r al Agilent Connection Expert Erima File Edit View YO Configuration Tools Help RefreshAl WP Und al EE Interactive 10 MZ Add Instrument JA Add Interface Update Drivers X De Task Guide Instrument I O on this PC System Name ECE01 32 bit 64 bit 1 providers installed Figure 1 Agilent Connection Expert 3 Click Add Instrument and pop up the dialogue shown in Figure 2 Click OK The software will search for in
97. ry inverted pendulum Use the State Space Observer to estimate the state Y uM LQR Control Gain State Space Observer Xhat Figure 16 Simulink Executable Model Experiments have shown that the pendulum is balanced using this balance controller as shown in Figure 17 Figure 17 Experiment Result 4 QNX C Project We also defined a C version of the controller We used our Simulink model to define the basic architecture and translated each Simulink subsystem into a C equivalent C 18 Last revised 07 27 2011 4 1 Execute the Controller The C code was developed on QNX Momentics IDE on the Host running Windows 7 To check out the source code and to run the program follow these steps 1 Launch QNX Momentics IDE 4 7 installed on the Host When you run it for the first time you must choose a workspace location After that you can use the current location 2 Access the SVN repository The IDE is Eclipse based you must define the SVN repository location the same as the steps 1 3 in Section 2 3 3 Switch to the SVN repositories perspective after you have successfully defined the location Browse the repository Select the two folders in C folder Right click the mouse and click check out shown in Figure 18 Project Explorer i EL O svnessh songcwind ecs baylor edu l y 192 168 0 99 77 svn ssh songc wir 2 controller 162
98. ry inverted pendulum is shown in Figure 1 The polar coordinate system is used to describe the inverted pendulum system where q is polar angle of the vertical pendulum and is the polar angle of the horizontal arm with the directions shown in Figure 1 These angles are measured by the encoders which increases when the arm pendulum rotates Counter Clockwise CCW The physical model can be analyzed by using the Lagrangrian The nonlinear equations of motions are 1 2 ngu E 1x2 2 2 2 2 2m l GOsin a cosa m l rsina m r m l m l cosa O T m l ra cosa T B 0 and 2rFAA e A RED m 1 0 sina cosa m l r0cosa J m l 4 m gl sina B 0 2 5 1 Last revised 07 27 2011 a gt 0CCw am X Jo gt ocow Figure 1 Rotary Inverted Pendulum where m is the pendulum mass D is the distance from pendulum pivot to centre of gravity r is Full Length of the pendulum J is moment of inertia acting seen from arm pivot 7 is the torque applied at the load gear For a complete listing of the symbols used in the equations 1 and 2 please refer to Appendix A In 1 z with unit Nm is the torque applied to the gear It is generated by the servo motor Given the torque it is necessary to convert the torque to a voltage that will be applied to the DC motor By analyzing the motor dynamics 1 we have ae NNK K A E K K 0 Tn s R 3 m where N is gearbox efficiency 7 is motor effi
99. s Dynamic System Behavior into SysML 22 describes how SysML and Modelica can be used in concert The objective of his research was to use graph patterns and transformation rules to integrate models of continuous 3 5 Revised 07 27 2011 dynamic system behaviors represented using Modelica with Sys ML models This would provide a more comprehensive modeling approach used IEEExplore to locate papers relevant to my project based on the keyword Networked Control System NCS selected about 20 papers read through each one and selected the interesting ones and added them to my RefWorks repository 23 28 In particular found that NCS co design which considers the characteristics of both the network and control system and solves the problem as an integrated system is very interesting 29 33 and focused on it during the fall 2010 semester 3 3 Fall Semester 2010 3 3 1 Overview During the fall 2010 semester set up the rotary pendulum simulation environment in Simulink and developed the controller The simulation results demonstrated that the controller could successfully balance the pendulum also investigated how to simulate an Network Control System NCS After investigating several different tools 11 20 34 39 I finally chose the TrueTime toolbox 35 3 3 2 Objectives The objectives of the fall semester were to e Understand how to model the dynamics of the rotary pendulum e Understand the Quanser Simulink mode
100. s that an NCS requires a higher sampling rate than a corresponding local control system Ts Stability 5ms Stable 10 ms Stable 20 ms Unstable 40 ms Unstable Figure 1 Stability against Sampling Period Horizontal Arm Position 8 T T T T reference position 6 0 4 position degree o time s Pendulum Position 2 T T T position degree time s Figure 2 Local Control Ts 10 ms 8 2 Last revised 07 27 2011 Horizontal Arm Position x 10 2 T T 0 pl A E 9 4r 2 S 0r 7 5 amp ab 10H reference position 1 0 12 l 1 J 1 0 2 4 8 10 12 14 16 18 20 time s x10 Pendulum Position 0 5 T T a 0 Y 5 05H 3 S T 1 3 a 15h 2b 25 l l l i l 0 2 4 8 10 12 14 16 18 20 time s Figure 3 Local Control Ts 20 ms Horizontal Arm Position 100 T T T T T I I reference position ame 50r T o EQU 2 oM IN I M tW a o a 50 F 100 fi i fi fi fi fi fi 0 2 4 8 10 12 14 16 18 20 time s Pendulum Position 100 T T T T T T a 50r T o o o e 0 2 Ez o a 50r 100 1 1 1 1 1 0 2 4 8 10 12 14 16 18 20 time s Figure 4 Netw
101. st bed provides the basic structure and interface for time scales and networked controllers For example researchers could incorporate time scales theory into these models and controllers so that control algorithms could be verified or tailored for efficiency 1 3 Structure of the Final Report The structure of the project report is as follows Chapter 2 introduces the background and the related work Chapter 3 is the project summary which records what we have done from summer 2010 through summer 2011 Chapter 4 describes the hardware and software configuration for the test bed which shows how to setup the system Chapter 5 presents the controller design for the rotary inverted pendulum and the implementation of our simulation environment Chapter 6 summarizes the implementation of the C controller Chapter 7 introduces how to implement a new controller based on the existing example Chapter 8 introduces the experiment observations and conclusions Appendix A lists the parameters including their symbols names and values that are used in defining the pendulum mathematical model Appendix B records the hardware and software in current configuration Appendix C constitutes the user manual for the test bed Appendix D shows how to generate arbitrary time scales using an Agilent 33220A 8 signal generator and software 1 2 Last revised 07 27 2011 References 1 R Dorf and R Bishop Modern Control Systems Prentice Hall 20
102. st revised 07 27 2011 4 The System Configuration The hardware and software that we have used in the project are described in this chapter For a detailed list of the hardware and software plus contact information please refer to Appendix B 4 1 Hardware The hardware includes the following components a host computer Host running Windows 7 on which a virtual machine was installed to run Windows XP a target computer Target running QNX Neutrino Realtime Operating System a Quanser Q4 data acquisition board and a terminal board a UPM 1503 Power Module and a SRVO2 E servo plant and pendulum Rotary Pendulum encoders l 5 voltage Rotary Power Module Pendulum Q4 Board Figure 1 Hardware Configuration 4 1 1 Computers The Host is used to develop the control algorithm and to run a simulation using the Matlab Simulink environment By utilizing the Quanser software a Simulink model can be compiled on the Host computer and the executable can be downloaded and run on the Target The Target runs the QNX realtime operating system and is used to execute the controller 4 1 2 SRVO2 E Servo Plant The Rotary Pendulum module is from Quanser Corporation It consists of a horizontal arm and a vertical pendulum attached to a Quanser SRVO2 E servo plant as shown in Figure 2 4 1 Last revised 07 27 2011 The Quanser SRVO2 E servo plant is the base of the pendulum system It has a metal frame that houses a
103. struments on local Ethernet D 1 Last revised 07 25 2011 ug Add Instrument 00000 Interface Name Description Add LAN instr TCPIPO LAN interface COMI ASRL1 Available RS 232 serial interface COMS ASRL3 Available RS 232 serial interface PXIO Available PXI interface USBO Unavailable USB interface s Interface status information Instruments may be added to this interface Figure 2 Add Instrument 4 Once the instrument is found a dialogue will show Figure 3 Click OK Instrument Automatically find and Web Page identify local instruments A m local instrument is one on Find Again Figure 3 Instrument found Last revised 07 25 2011 5 Run Start gt All Programs gt Agilent IntuiLink gt Waveform generator gt Waveform editor This software is used to design waveform that will be generated by the signal generator AJ Agilent IntuiLink Waveform Editor File Edit View SampleWaveForms Math Communications Tools Window Help DE H amp a CA GSNNUMNA SW RW QA E Select mode Select X 2518 Y 0 894 Length 0 TCPIPO 192 168 0 101 INSTO INSTR Figure 4 Run Waveform editor 6 Edit wave form using the graphic tool After finishing editing you can select to save the waveform 7 Onthe 33220A push the Arb button 8 In Waveform editor select Communication gt Send waveform A dialogue will show Figure 5 You can edit th
104. t On the contrary Simulink simulation uses double precision variables so that no angle information is lost The finite wordlength effect could be analyzed using the DSP theory 2 The simulation models can be improved to consider the finite wordlength effect 8 1 3 Timing Inaccuracy The C project uses software time to realize varying timer interrupt Though QNX provides higher timing accuracy than Windows XP it still has some jitter and latency Figure 1 shows the kernel event tracing of the C program captured by QNX Momentics IDE The vertical line records the time that the thread receives a message from the timer interrupt From the figure we can see that the interval between two messages is actually 6 998 ms not 5 ms which is preset Besides the interval is not fixed ranging from 6 997 ms to 7 002ms which indicates the jitter of the software timer Timeline gi Bi hel X A or A METER Mer cy MEE LL Mr umm 1 505sec 1 preg 1 551sec 6 998 ms 1 599sec y 3 Thread 9 o ef Figure 10 Kernal Event Tracing Hardware signal generator provides the most accuracy timing But currently we cannot directly control the signal generator in Simulink environment Therefore future research could investigate how to improve the hardware to make it controllable 8 9 Last revised 07 27 2011 8 1 4 Stability against Sampling Rate Another experiment we did is to check the controller s stability against different sampling rate The r
105. t URI tpi 92 168 0 99 17000 Model Target Instructions The default target type used by QUARC may be set here New models will be configured automatically to use this target type Default target type qnx x86 OK Jl Cancel Jl Help Defaults OK jl Cancel JI Help Defaults OK Hi Cancel fi Help Defaults Figure 15 Set Target IP address Perform steps 1 3 in section 3 2 and compile and run the model follow these steps 1 On the Target run Launch gt QUARC gt Console This will run the QUARC program that will communicate with the Host 2 On the Host Select menu QUARC compile The Simulink model will be compiled and automatically downloaded to the QNX machine When finished the QNX console will prompt model model name downloaded 3 Turn on the UPM power amplifier 4 On the Host choose QUARC gt start to run the program 5 Manually swing the pendulum up to the vertical position The controller will then balance the pendulum 6 On the Host select QuaRC stop to stop the program 7 Turn off the power amplifier the Host and the Target C 17 Last revised 07 27 2011 theta_ref EA 1 2 4 y mE p D2R A gt C 7 gt Scopes arm reference Degreesto RefGain 3 position degrees Radians gt E Control rotary pendulum system Executable model for the Quanser rota
106. t networks Model based predictive networked control systems IEEE Transactions on Industrial Electronics 99 pp 1 1 3 22 Revised 07 27 2011 26 N J Ploplys P A Kawka and A G Alleyne 2004 Closed loop control over wireless networks Control Systems Magazine IEEE 24 3 pp 58 71 27 Y Wang S X Ding H Ye and G Wang A New Fault Detection Scheme for Networked Control Systems Subject to Uncertain Time Varying Delay IEEE Transactions on Signal Processing vol 56 pp 5258 5268 2008 28 Y Zhang Q Zhong and L Wei Stability of networked control systems with communication constraints in Control and Decision Conference 2008 CCDC 2008 Chinese 2008 pp 335 339 29 A Cervin D Henriksson B Lincoln J Eker and K Arzen How does control timing affect performance Analysis and simulation of timing using Jitterbug and TrueTime EEE Control Systems Magazine vol 23 pp 16 30 2003 30 X Diao ME 452 course project Il rotary inverted pendulum 2006 31 A Chamaken and L Litz Joint design of control and communication in wireless networked control systems A case study in American Control Conference 2010 pp 1835 1840 32 M S Hasan H Yu A Carrington and T C Yang Co simulation of wireless networked control systems over mobile ad hoc network using SIMULINK and OPNET ET Communications vol 3 pp 1297 1310 2009 33 E Weingartner H vom Lehn and K Wehrle A perfo
107. t widely used whereas Matlab Simulink provides many useful tools for designing control systems and TrueTime toolbox works perfectly with Simulink to simulate the network system 2 4 Time Scales Theory In mathematics time scale calculus is a unification of the theory of difference equations with that of differential equations unifying integral and differential calculus with the calculus of finite differences Consequently it provides formalism for describing hybrid discrete continuous dynamical systems 20 It has applications in any field that requires simultaneous modeling of discrete and continuous data One goal of this project is to provide a test bed including a simulation environment and an experiment system for researchers to investigate the time scales theory on control systems The time scales theory is important because conventional control system is classified as either continuous system or discrete system Time scales theory helps build the connection However control theory on time scales has not been much developed Such a test bed will be useful for future researches Billy Jackson 21 examined linear systems theory in the arbitrary time scale setting by considering Laplace transforms stability controllability observability and realizability Benjamin Allen 22 described 2 6 Last revise 07 27 2011 the design and implementation of a simulator and real time controller useful for experimentation with and demonstration
108. the system It undergoes the linear approximation process of the non linear model With loss of information this process also reduces the accuracy 8 1 2 2 Motor Model Second the motor model also has limitations SRV 02 incorporates a Faulhaber 2338S006 Coreless DC Motor with dynamic model shown in equation 4 in Chapter 5 The motor parameters can be found in the data sheet 1 However different motors could have slightly different parameters Besides the motor has a minimal input voltage which means that the motor only rotates when the input is larger than this threshold In Simulation the horizontal arm moves whatever the input voltage is Due to this reason once the pendulum is near balanced position in real experiment controller only outputs a small voltage in which case the arm will not move After a few cycles the pendulum will not reach the desired position In this case a large voltage will be generated and it pushes the arm to deviate from the balanced position 8 8 Last revised 07 27 2011 8 1 2 3 Finite Wordlength Effect Another reason could be the finite wordlength effect The pendulum system uses encoders to get angle information With a resolution of 4096 each encoder works as a 14 bit A D converter so that there is a certain amount of uncertainty introduced during quantization From digital signal processing theory we know that this will cause the finite wordlength effect which means that some information is los
109. tscript m files can be used for drawing figures of the simulation results The exe mdl files can be compiled downloaded and executed on the QNX computer 3 2 Run the Local Control Model To run the simulation follow these steps 1 Start Matlab and navigate to the current working directory To do this click the button in the circle shown in Figure 6 Then in the popup dialogue navigate to YOUR WORKSPACE LOCATIONNcontrollerNsimulinkNquanser pendulum control C 8 Last revised 07 27 2011 2 3 4 4h MATLAB 7 10 0 R2010a File Edit Debug Parallel Desktop Window Help HeG 4 amp a v Q r E Q Current Folder CAUsersVsongc ECS 005 Documents MATLAB Y Figure 6 Change current working directory Choose File gt Open to open the setup m file This file first calculates the pendulum s physical system parameters then creates an LQR controller and the state space observer You can adjust several variables to design the controller including The sampling period which controls how fast the controller samples the system 5 sampling period Ts 0 005 The Qand R matrices for the LQR algorithm Design digital LOR controller x 0 2 sweighting factor for the cart position y 0 4 weighting factor for the pendulum angle Q x 00 0 Oy 0 0 0000 0000 R 5 K_controlgain dlqr F G Q R The pole locations for the pole placement algorithm P 70 3 0 3 0 9 0 9 L plac
110. uoonooooonnnnnnnnnnnnnnononncnnnnnnnnnnnnnnnnr nnns nans nsn nnn 5 1 5 1 2 State Space Representation cccccccssssssssseeeeccecceccesausesecseseeeseceesesaeaeueseseeeeseseeeeeeeaes 5 3 5 1 3 Continuous to Discrete Domain Conversi n seuran 5 5 5 2 Balance Controller Design cooccnncnncocuoonooooononnnnnnononnnnnnnnononnnnnnnannnnnno nono ono nnnnnnnnnnnnnnnnnnnenaness 5 6 5 2 1 Full State Feedback Controller Design oonooococococnonononnnnnnnonononononnnnnnnnnnnona ono n ono nnnnnnnnoos 5 7 5 2 2 Observer Desig Nire en erae ee ere eR he RUD TU 5 8 5 2 3 Controller with Reference Signal Input oocccccccncnnccnnnninnninnnananonanononononnnononanononoss 5 10 5 3 SIMULATION x cocer ec ette eh ed ede tue V e x dee de etd tete verts 5 14 5 3 1 Simulink Simulation Model essere nnne nennen 5 14 5 3 2 Paramete rSn anen ECT 5 15 5 3 3 Networked Control System Model ooooooooccccncccncnnccococonononanananonano non ononnnnnnnnnnnnnnnncninnos 5 17 5 4 Simulink Executable Model eese eee eene nennen enne 5 19 5 5 SUMMA o dee eoe rede ee cabaret rev E OU ie 5 22 ME Go ados AA E S mm 6 1 6 1 C Version of the Balance Controller esee 6 1 6 2 Varying Sampling Rate 5e mE eth rae stb ste n e edes 6 4 7 Creating your own controller esses nnnnn nnns nnns nsns nennen nennen nnns 7 1 7 1 je

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