Control of a Ball and Beam System Wei Wang School of Mechanical Engineering The University of Adelaide South Australia 5005 AUSTRALIA Submitted for the degree of Advanced Master on the 5rd June, 2007 Abstract The ball and beam system can usually be found in most university control labs since it is relatively easy to build, model and control theoretically. The system includes a ball, a beam, a motor and several sensors. The basic idea is to use the torque generated from motor to the control the position of the ball on the beam. The ball rolls on the beam freely. By employing linear sensing techniques, the information from the sensor can be taken and compared with desired positions values. The difference can be fed back into the controller, and then in to the motor in order to gain the desired position. The mathematical model for this system is inherently nonlinear but may be linearized around the horizontal region. This simplified linearised model, however, still represents many typical real systems, such as horizontally stabilizing an airplane during landing and in turbulent airflow. By considering real plant problems such as the sensor noise and actuator saturation, the controllers of the system become more efficient and robust. In this thesis, a theoretical analysis of the ball and beam system is conducted by employing physical laws and linear approximation. A performance comparison among the different control techniques is presented in both simulation and experimental conditions. A system identification (ID) process is essential in control design - i - processes. Different ID techniques have been tested in this system, and an accuracy comparison is presented. - ii - Statement of Originality To the best of my knowledge and belief, all the material presented in this thesis, except where otherwise referenced, is my own original work, and has not been published previously for the award of any other degree and diploma in any university. If accepted for the award of the degree of advanced master, I consent that this thesis be made available for loan and photocopying Wei Wang - iii - Acknowledgements I would like to thank all the people who have helped towards the work in this thesis. I am especially grateful to Associate Professor Ben Cazzolato for his supervision and continued support. Without his help, this thesis would never be possible. I would also like to give thanks to staff in the electronic lab and mechanical workshop, Silvio De Ieso, Joel walker, and George Osborne for their help with building the rig, sensors and interfaces for the system. Many thanks are given to Dorothy Missingham for helping me write skillfully. I would like give most thanks to my parents, YaoPing Wang and JianPing Jiang who give me so much support, both in emotional and financial. Finally, I would like to express my grateful to Lili Wang for her selfless support all the time. - iv - Contents Contents Abstract..........................................................................................................................i Statement of Originality............................................................................................iii Acknowledgements.....................................................................................................iv Contents........................................................................................................................v 1 Introduction...........................................................................................................1 1.1 Introduction...............................................................................................1 1.2 Literature review.....................................Error! Bookmark not defined. 1.3 Aims..........................................................................................................7 1.4 Assumptions..............................................................................................8 1.5 Conclusion................................................................................................8 2 Potential Design Solutions..................................................................................10 2.1 Configuration..........................................................................................10 2.2 Transmission mechanism........................................................................12 2.2.1 Reducing Gearbox...........................................................................12 2.2.2 Transmission belts............................................................................13 2.2.3 No transmission...............................................................................14 2.3 Materials selection..................................................................................14 2.4 System Overview....................................................................................15 2.5 Sensors....................................................................................................18 2.5.1 Position sensor.................................................................................18 2.5.2 Angle sensor.....................................................................................20 2.6 Motor.......................................................................................................21 - v - Contents 2.7 Control hardware and software...............................................................22 2.8 Conclusion..............................................................................................23 3 Theoretical Analysis............................................................................................24 3.1 Mathematical model................................................................................25 3.1.1 Analysis of the force balance of the ball..........................................26 3.1.2 Analysis of the torque balance of the motor and beam....................27 3.1.3 DC motor equation...........................................................................28 3.1.4 Summary of dynamics.....................................................................29 3.2 State space model....................................................................................29 3.3 Model for simulation in ‘Matlab’...........................................................32 3.4 Simplified model with four states...........................................................35 3.5 Conclusion..............................................................................................38 4 Simulation & Experiment..................................................................................40 4.1 Hardware and software setup..................................................................40 4.2 Introduction to the control structures......................................................41 4.2.1 Full state feedback control...............................................................42 4.2.2 Full state estimator control...............................................................42 4.2.3 Reduced order estimator control......................................................43 4.2.4 Command tracking...........................................................................45 4.3 Introduction to gain calculation methods................................................51 4.3.1 Pole placement method....................................................................51 4.3.2 LQR method.....................................................................................51 4.3.3 LQG method....................................................................................52 4.4 Full state feedback control......................................................................53 4.4.1 Pole placement method....................................................................55 4.4.1.1 Simulation.................................................................................56 4.4.1.2 Experiment................................................................................58 4.4.2 LQR method.....................................................................................61 4.4.2.1 Simulation.................................................................................62 4.4.2.2 Experiment................................................................................64 - vi - Contents 4.5 Full state estimate control.......................................................................66 4.5.1 Pole placement method....................................................................67 4.5.1.1 Simulation.................................................................................68 4.5.1.2 Experiment................................................................................69 4.5.2 LQR and LQG methods...................................................................71 4.5.2.1 Simulation.................................................................................72 4.5.2.2 Experiment................................................................................74 4.6 Reduced order system (LQR and LQG methods)...................................76 4.6.1 Simulation........................................................................................78 4.6.2 Experiment.......................................................................................80 4.7 Conclusion..............................................................................................82 5 System Identification..........................................................................................83 5.1 Introduction to system identification (ID)..............................................84 5.1.1 ‘Free structured parameterizations’ approach..................................85 5.1.2 ‘Grey box’ approach.........................................................................86 5.1.3 ‘Black box’ approach.......................................................................87 5.2 System identification of the open-loop plant (simulation).....................88 5.3 System identification of the closed-loop system....................................92 5.3.1 Simulation........................................................................................93 5.3.2 Experiment.......................................................................................97 5.4 ID from full state estimator plant..........................................................100 5.4.1 Simulation......................................................................................101 5.4.2 Experiment.....................................................................................105 5.5 ID from reduced order system..............................................................107 5.5.1 Simulation......................................................................................109 5.5.2 Experiment.....................................................................................113 5.6 Conclusion............................................................................................115 6 Considerations in the Experiment...................................................................117 6.1 Sensor calibration..................................................................................117 6.1.1 Position sensor calibration.............................................................118 - vii - Contents 6.1.2 Encoder calibration........................................................................120 6.2 System parameter measurement...........................................................121 6.2.1 Moment of inertia of the beam.......................................................121 6.2.2 System damping.............................................................................123 6.3 Motor compensation.............................................................................125 6.4 Conclusion............................................................................................130 7 Conclusions........................................................................................................132 7.1 Conclusions...........................................................................................132 7.2 Recommendations for further work......................................................134 7.2.1 Model the friction between the ball and resistive wires................134 7.2.2 Other advanced control techniques................................................134 References................................................................................................................................135 Appendix...................................................................................................................138 A Matlab M-files for simulation and experiment..............................................138 A.1 Full state feedback & estimator control by the pole placement method138 A.2 Full state feedback & estimator control by the LQR method...............141 A.3 Reduced order observers by the LQR method......................................144 B Matlab M-files for system identification.........................................................147 B.1 ID the open loop plant...........................................................................147 B.1.1 Main program.................................................................................147 B.1.2 M-file for ‘grey box’ approach.......................................................151 B.2 ID full state feedback............................................................................152 B.2.1 Main program.................................................................................152 B.2.2 M-file for ‘grey box’ approach.......................................................157 B.3 ID full state observers...........................................................................158 B.4 ID Reduced observers...........................................................................163 C DC motor (RH-11D-3001-E100-AO)...............................................................169 C.1 DC motor (RH-11D-3001)....................................................................169 C.2 Digital encoder (E100) for the DC motor.............................................171 C.3 Mechanical drawing of the DC motor with EncoderError! Bookmark not defined. - viii - Contents D Mechanical drawings........................................................................................173 D.1 Solid model of the entire ball and beam system...................................173 D.2 Base.......................................................................................................174 D.3 Support..................................................................................................175 D.4 Link.......................................................................................................176 D.5 Beam.....................................................................................................178 E Circuit Diagrams...............................................................................................181 F The ball and beam system Manual..................................................................183 F.1 Introduction...........................................................................................183 F.2 Hardware equipments...........................................................................183 F.2.1 DC motor.......................................................................................184 F.2.2 Sensors...........................................................................................184 F.2.3 Power Amplifier.............................................................................184 F.2.4 Data acquisition boards (Quanser Q4 board).................................185 F.3 Software tools.......................................................................................188 F.3.1 Simulink.........................................................................................188 F.3.2 Wincon...........................................................................................190 F.4 Operation of instrumentation................................................................192 F.4.1 Connections....................................................................................193 F.4.2 Caring for the system.....................................................................194 G System identification results............................................................................196 G.1 Nominal plant........................................................................................196 G.2 Simulation results..................................................................................197 G.2.1 ID from Open-loop plant...............................................................197 G.2.2 ID from closed-loop plant..............................................................198 G.2.3 ID from full state observer plant....................................................199 G.2.4 ID from reduced order observer plant............................................199 G.3 Experiment results................................................................................200 G.3.1 ID from closed loop plant..............................................................200 G.3.2 ID from full state observer plant....................................................201 - ix -