McFarlane-Glover loop shaping method for a ball-and-beam mechatronic system

1. McFarlane-Glover loop shaping method for a ball-and-beam mechatronic system Jayanth Ganapathiraju

2. Relevance of Ball and Beam • Modern Industrial processes are intrinsically unstable. • Chemical Processes: Control of exothermic chemical reactions(run away reactions). • Power Generation: Position control of the plasma(High voltage transmission). • Aerospace: Control of rocket during take off. • Control of unstable systems is thus critically important. Ball and Beam system replicates these systems.

3. Ball and Beam System Specifications: SRV-02 servo, high gear ratio(for backlash),Quanser board, MATLAB-SIMULINK

4. SYSTEM MODEL Simplified Model • Ball acceleration proportional to gravity. • Assumes no friction.

5. SYSTEM MODEL(contd.) • Real Ball and Beam has additional components: • Dynamic components of motor. • Coulomb friction in moving parts and dead zone. • Saturation in motor input amplifier. • Noise: Ball position sensor. • Feedback: linearizing effect, reduce impact of motor dynamics.

6. CONTROLLERS • Unstable open loop response of system. • Hence, phase advance to stabilize system. • Controller Options: • PD control • Phase Lead Compensation • State Observer with State feedback • LQR • LQG • Sliding Mode and Variable structure control • Fuzzy Control • Robust Control

7. Robust Loop shaping • McFarlane-Glover’s technique • Modern H∞ optimization approach. • Incorporate simple performance/robustness tradeoff. • Based on concepts from classical Bode plot methods. • Multivariable • Robust-stability guaranteed in face of plant perturbations and uncertainties.

8. Robust Loop Shaping(contd.) • Step 1 • Augment system with weighting functions W1 and W2. W1 G W2 Augmented System

9. Robust loop Shaping(contd.)

10. Robust Loop Shaping(contd.) • Step 2 • Check if design index γ < 5, else goto Step 1 • Synthesize H∞ optimal controller. W1 G W2 K∞

11. Robust Loop Shaping (contd.) • Final Controller K=W1.K∞.W2 G W1 K∞ W2 K

12. Robust Loop Shaping (contd.) • Some guidelines for Loop shaping • The Loop transfer function has low gain around the frequency of the modulus of any RHP zero. • The Loop transfer function should have a large gain around the frequency of modulus of RHP pole. • The Loop transfer function should not have a large slope near the crossover frequency.

13. LSDP Tool (contd.)

14. Controller Design • Design index value γ = 2.5457 which indicates a good design. • W1=5(s+1)/(0.3s+1) • W2=1 • H∞ controller is designed using MATLAB.

15. SIMULATION • Upon Simulation, the following response is obtained,

16. Response Plots Response for output Response for control input

17. SIMULINK MODEL FOR REAL PLANT

18. Performance on the actual plant Using Statefeedback control

19. Performance on the actual plant • Using Robust Control

20. Plots Input Sensitivity Control Sesitivity Complementary Sensitivity Input Disturbance Sensitivity

21. Plots (contd.) Servo Command Servo Angle

22. Plots (contd.) Error

23. Plots (contd.) Voltage input profile Position of Ball

24. CONCLUSION • A robust controller has been designed for the Ball and Beam system which is essentially an open loop unstable plant. • McFarlane-Glover method is procedurally easy and practical, this method implicitly minimizes the input, output and control sensitivity functions which results in a better perfromance than other phase advance control techniques. • An attempt to implement Robust control theory(which appears abstract) in practice and show that it works.