1. Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 26: 4 December 2009

2. Uncertain MIMO Plant G p (s)

3. Frequency Response

4. Feedback Control Loop G p (s)K(s) + - practically zero steady state error for step reference rise time less than 1 second overshoot less than 5% robustly for any value of c

5. Menu for the day Representing the uncertainty a la unstructured uncertainty a la structured uncertainty Assessment of robust stability a la unstructured uncertainty a la structured uncertainty Assessment of robust stability a la unstructured uncertainty a la structured uncertainty

7. Unstructured Uncertainty G (s) Wo (s)  (s) G (s) Wi (s)  (s) + + + + Output uncertainty Input uncertainty

8. Output Uncertainty G (s) Wo (s)  (s) + + Output uncertainty We need to select a weight such that:

9. Input Uncertainty We need to select a weight such that: G (s) Wi (s)  (s) + + Input uncertainty

10. Weight Wo(s) Refer to uncert1.m

11. Structured Uncertainty

12. G (s) Ws (s)  + + relax to

13. Relaxation from real to complex uncertainty

15. Assessment of Robustness In this topic, we first design a controller for the nominal system, then we assess the robustness properties of the system. Two control designs are explored, decentralized inverse-based, and H-infinity synthesis. G p (s)K(s) + -

16. Decentralized Inverse Based Based on nominal plant model Refer to decent.m

17. Nominal Performance

18. H  synthesis for nominal plant Use S/T synthesis with: Refer to hinf_ex.m

19. Nominal Performance

20. Robust Stability Assessment K(s) + - G (s) Wo (s)  (s) + + Unstructured Uncertainty

21. Robust Stability Assessment K(s) - G (s) Wo (s) + +

22. Robust Stability Assessment Refer to decent.m

23. Robust Stability Assessment Refer to hinf_ex.m

24. Robust Stability Assessment K(s) + - G (s) Ws (s)  + + Structured Uncertainty

25. Robust Stability Assessment K(s) + - G (s) Ws (s) + + Structured Uncertainty

26. Structured Singular Value Peak at 0.130 For decentralized controller

27. Perturbed system responses

28. Structured Singular Value Peak at 0.150

29. Perturbed system responses

31. Nominal Performance Refer to hinf_ex.m

32. Robust Performance Assessment K(s) + - G (s) Wo (s) + + Unstructured Uncertainty + W P (s)

33. Robust Performance Assessment K(s) + - G (s) Wo (s) + + Unstructured Uncertainty + W P (s)

34. Robust Performance Assessment N  PP

36. K(s) + - G (s) Ws (s) + + Structured Uncertainty + W P (s)

37. Robust Performance Assessment K(s) + - G (s) Ws (s) + + Structured Uncertainty + W P (s)

38. Robust Performance Assessment N  PP