1. Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 27: 8 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. From previous lecture 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 performance a la unstructured uncertainty a la structured uncertainty

6. Today’s class How to design a controller that minimizes the structured singular value. This is a sub-optimal solution, but works well in practice. The technique is called DK iteration (see dksyn function in MATLAB).

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

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

9. Robust Performance Assessment N  PP

10. N  PP  

11. N as a ‘function’ of K P K Note that, from previous classes, we know how to design K so as to minimize the H-infinity norm of N.

12. Robust Performance Assessment N D D In DK iteration, the K step consists of solving H infinity synthesis for weighted N.

13. New plant model with weights P K The new plant model can be formed by including the weights. D D

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

16. First iteration See unstruct_ex.m

17. Second iteration See unstruct_ex.m

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

20. First iteration See struct_ex.m

21. Second iteration See struct_ex.m

22. Third iteration See struct_ex.m