The 2 by 2 Spectral Nevanlinna Pick Controller Design Problem

1. The 2 by 2 Spectral Nevanlinna Pick Controller Design Problem Fang-Bo Yeh and Huang-Nan Huang Department of Mathematic Tunghai University

2. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

3. Introduction • -norm thestructured singular value is a powerful tool in robust control.. • Spectral norm is the lower bound of -norm, and  norm is its upper bound. H control is too conservative. • No define theory for -synthesis. • SNP interpolation theory is developed with aims to solve this problem. • Formulate controller synthesis into SNP interpolation problem. • Design -controller using SNP theory: 2 by 2 case.

4. P K A + - S Robust Control Problem Design K such that is internally stable and track r under the influence: 1. perturbations in system model 2. disturbance in actuator 3. sensor noise

5. Da P0(s) + Type of Uncertainties • Real parametric uncertainty: e.g. a given plant • Unstructured uncertainty: unmodeled dynamics 1. Additive type -

6. Dm Dm P0(s) P0(s) + + Type of Uncertainties • Unstructured uncertainty: unmodeled dynamics 2. Multiplicative type –

7. P K A + + - S + W1 W2 P W3 K A + + - S + Robust Control Problem, Again r: reference input d: disturbance n: noise Design Philosophy: “Shaping” i.e. filtering W1, W2, W3

8. D G D M K Structured Uncertainty Robust stability: (w=0,z=0) M+D is stable Robust performance: Design K such that (i) M+D is stable (ii)

9. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

10. D M -Analysis and Synthesis Consider a matrix MCnn (the plant) and Cn  n the structured uncertainty set. Uncertainty • Definition of  = the smallest  that causes M “instability”

11. Bounds on  - When(S=1,F=0,r1=n) , (S=0,F=1,m1=n), the equality hold. - * Lower bound always holds, but the set of r(UM) is not convex, * Upper bound holds when 2S+F≤3.

12. M D D M Linear Fractional Transformation(LFT) Let M be a complex matrix of the form Define the lowerLFT Fl as Define the upperLFT Fu as

13. D M Robust Stability using -Synthesis - Norm • Let S denote the set of real-rational, proper, stable transfer matrices. Let Robust Stability The loop shown is well-posed and internally stable for all S with ||||<1 if and only if

14. D DF DRP M M Robust Performance D M Robust Performance For all S with ||||<1, the loop shown is well-posed, internally stable, and || Fu(M, ) ||<1 if and only if

15. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

16. G K M Problem Description Find K such that where G is chosen, respectively, as • nominal performance (D=0): • robust stability only: • robust performance:

17. Q Parameterization where By using lower bound on m: we arrive at new problem: Find Q such that SpectralModel Matching Problem

18. Solve Interpolation Problem Let pi, i=1,2,…,n be the RHP poles of T2(G12) , T3(G21); zj, j=1,2,…,mbe the RHP zeros of T2(G12) , T3(G21). The problem becomes find analytic functionF on RHP satisfying the interpolation conditions: Spectral NP Interpolation Problem

19. Remark for Q Once F is solved, the Q is computed as following: • T2, T3 are square and invertible, • T2 is left invertible, T3 is right invertible, hence there exists such that and then

20. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

21. Spectral NP Interpolation Problem Given distinct points l1, l2,…, lninside open unit disk D and W1, W2,…, WnCmm find an analytic mm matrix function F such that Define then

22. Existence of the function F (Bercovici, Foias & Tannenbaum,1989) Such a function F exists if and only if there exists invertible mm matrices Mi, i=1,…,n such that Difficulty: there are mmn unknowns in Mi, i=1,…,n. Pick Matrix for HNP problem: Choose Mi=I.

23. Existence of F (m=2) (Agler & Young, 2001) Such a function F exists if and only if there exist b1,…,bn,c1,…,cn such that where Note: there are only 2n unknowns instead of 2  2  n.

24. SNP Interpolation Problem: n=m=2 case four complex unknowns a, b, c, and d. If exists R such that Define the symmetrized bidisc with two unknowns s and p.

25. Problem Transformation

26. Modified SNP Interpolation Problem Agler & Young, 2000

27. Solution of Modified SNP Problem Given 1, 2D,(s1,p1),(s1,p2)G2 find analytic function  , ()G2, 2Dsuch that Alger-Yeh-Young Theorem 2003 : Suppose 0 is defined by and i, sisatisfy then the solution () =(s(), p()) with where

28. Symmetrized Bidisc

29. Geometry of

30. Geometry of G2

31. Geometry of G2

32. Spectral Interpolation Find Analytic function such that

33. is a Complex Geodesic of through Main Idea Smallest

34. Totally geodesic disc Isometry

35. Caratheodory Distance : Caratheodory distance

36. Conclusion :

37. Kobayashi Distance : Kobayashi distance is a corresponding extremal

38. known： • Schwarz Lemma： • Lempert’ Lemma： If is convex , then

39. Complex Geodesic of complex geodesic of : or where

40. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

41. D G K Algorithm of -Synthesis via SNP Theory • First transform the robust performance problem tothemodel matching form

42. Algorithm of -Synthesis via SNPT (cont’d) • Modify the problem to the situation such that we can use the solution of SNP problem. • Solve the SNP problem for the function F . • Find the controller K. • Iterate for the desired K.

43. Outline • Introduction • -Analysis and Synthesis • Problem Description • Spectral NP Interpolation Theory: 2 by 2 case • Algorithm of -Synthesis via SNP Theory • Numerical Examples • Conclusions

44. Numerical Examples • Real parameter uncertainty • Dynamical uncertainty: SISO case • Dynamical uncertainty: 2 Input 2 output case

45. d G Real Parameter Uncertainty Plant: Structured uncertainty notation: Closed loop system: since Solve and

46. Dp Wu P Wp + + K D G K Dynamic Uncertainty Block Diagram: Standard Notation: Closed loop system:

47. Dynamic Uncertainty (cont’d) Q parameterization: Model Matching: Interpolation conditions: let zi, pj be the zeros and poles of P(s), then Restriction: M(s) must be 22matrices and the total number of i+j, must be 2.

48. Dynamic Uncertainty: SISO case Plant: Closed loop system: Interpolation Condition: Choose bj,cj=16/33, the existence of  is guaranteed. Remark: for all SISO system