Multivariable Control Systems ECSE 6460

1. Multivariable Control SystemsECSE 6460 Fall 2009 Lecture 27: 8 December 2009

2. Uncertain MIMO Plant Gp(s)

3. Frequency Response

4. Feedback Control Loop K(s) Gp(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 Unstructured Uncertainty Wo(s) + + K(s) G(s) WP(s) + + -

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

9. Robust Performance Assessment DP D N

10. Robust Performance Assessment D 0 0 DP N

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 D N D -1 In DK iteration, the K step consists of solving H infinity synthesis for weighted N.

13. New plant model with weights D D -1 P K The new plant model can be formed by including the weights.

14. Unstructured Uncertainty

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

16. First iteration See unstruct_ex.m

17. Second iteration See unstruct_ex.m

18. Structured Uncertainty

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

20. First iteration See struct_ex.m

21. Second iteration See struct_ex.m

22. Third iteration See struct_ex.m