1. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 Fang-Bo Yeh and Huang-Nan Huang Department of Mathematic Tunghai University The 2 by 2 Spectral Nevanlinna Pick Controller Design Problem

2. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 2 Outline Introduction Introduction  - Analysis and Synthesis  - Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions

3. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 3 Introduction  -norm the is a powerful tool in robust control..  -norm the structured 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. Spectral norm is the lower bound of  -norm, and  norm is its upper bound. H  control is too conservative. No define theory for  -synthesis. No define theory for  -synthesis. SNP interpolation theory is developed with aims to solve this problem. SNP interpolation theory is developed with aims to solve this problem. Formulate controller synthesis into SNP interpolation problem. Formulate controller synthesis into SNP interpolation problem. Design  -controller using SNP theory: 2 by 2 case. Design  -controller using SNP theory: 2 by 2 case.

4. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 4 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 K A +  S P

5. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 5 Type of Uncertainties Real parametric uncertainty: e.g. a given plant Real parametric uncertainty: e.g. a given plant Unstructured uncertainty: unmodeled dynamics Unstructured uncertainty: unmodeled dynamics 1. Additive type - aa P0(s)P0(s) +

6. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 6 Type of Uncertainties Unstructured uncertainty: unmodeled dynamics Unstructured uncertainty: unmodeled dynamics 2. Multiplicative type – mm P0(s)P0(s) + mm P0(s)P0(s) +

7. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 7 Robust Control Problem, Again r: reference input d: disturbance n: noise K A +  S P + + Design Philosophy: “Shaping” i.e. filtering W 1, W 2, W 3 K A +  S P + + W3W3 W2W2 W1W1

8. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 8 Structured Uncertainty  M K G  Robust stability: (w =0,z =0 ) M+  is stable Robust performance: Design K such that (i) M+  is stable (ii)

9. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 9 Introduction Introduction  -Analysis and Synthesis  -Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

10. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 10  -Analysis and Synthesis Definition of  Consider a matrix M  C n  n (the plant) and  C n  n the structured uncertainty set. Uncertainty  M = the smallest  that causes M “instability”

11. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 11 - When (S=1,F=0,r 1 =n), (S=0,F=1,m 1 =n), the equality hold. Bounds on  * Lower bound always holds, but the set of  (UM) is not convex, * Upper bound holds when 2S+F≤3. -

12. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 12 Linear Fractional Transformation(LFT) Let M be a complex matrix of the form  M  M Define the lower LFT F l as Define the upper LFT F u as

13. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 13 - Norm Robust Stability using  -Synthesis -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  M

14. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 14 Robust Performance Robust Performance For all  S  with ||  ||  <1, the loop shown is well-posed, internally stable, and || F u ( M,  ) ||  <1 if and only if  M  M FF  RP M

15. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 15 Introduction Introduction  -Analysis and Synthesis  -Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

16. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 16 Problem Description Find K such that where G is chosen, respectively, as nominal performance (  =0): robust stability only: robust performance: K G M

17. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 17 where By using lower bound on  we arrive at new problem: Find Q such that Spectral Model Matching Problem Q Parameterization

18. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 18 Spectral NP Interpolation Problem Interpolation Problem Let p i, i =1,2,…, n be the RHP poles of T 2 ( G 12 ), T 3 ( G 21 ); z j, j =1,2,…, m be the RHP zeros of T 2 ( G 12 ), T 3 ( G 21 ). The problem becomes find analytic function F on RHP satisfying the interpolation conditions: Solve

19. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 19 Remark for Q Once F is solved, the Q is computed as following: T 2, T 3 are square and invertible, T 2 is left invertible, T 3 is right invertible, hence there exists such that and then

20. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 20 Introduction Introduction  -Analysis and Synthesis  -Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

21. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 21 Spectral NP Interpolation Problem Given distinct points     …  n  inside open unit disk D and W   W   …  W n  C m  m find an analytic m  m matrix function F such that Define then

22. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 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 M i, i =1,…, n such that Difficulty: there are m  m  n unknowns in M i, i =1,…, n. Pick Matrix for H  NP problem: Choose M i = I.

23. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 23 Existence of F (m=2) (Agler & Young, 2001) Such a function F exists if and only if there exist b 1,…, b n, c 1,…, c n such that Note: there are only 2  n unknowns instead of 2  2  n. where

24. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 24 SNP Interpolation Problem: n = m =2 case four complex unknowns a, b, c, and d. If exists R such that with two unknowns s and p. Define the symmetrized bidisc

25. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 25 Problem Transformation

26. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 26 Modified SNP Interpolation Problem Agler & Young, 2000

27. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 27 Solution of Modified SNP Problem Alger-Yeh-Young Theorem 2003 : Suppose  0 is defined by where then the solution  ( ) =( s ( ), p ( )) with Given 1, 2  D, ( s 1,p 1 ),( s 1,p 2 )  2 find analytic function ,  ( )  2,  2  D such that and i, s i satisfy

28. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 28 Symmetrized Bidisc

29. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 29 Geometry of

30. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 30 Geometry of  2

31. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 31 Geometry of  2

32. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 32 Spectral Interpolation Find Analytic function such that

33. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 33 Main Idea Smallest Smallest is a Complex Geodesic of through

34. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 34 Totally geodesic disc Isometry

35. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 35 Caratheodory Distance : Caratheodory distance : Caratheodory distance

36. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 36 Conclusion :

37. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 37 Kobayashi Distance : Kobayashi distance : Kobayashi distance is a corresponding extremal

38. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 38 known ： 1. 1. Schwarz Lemma ： 2. 2. Lempert’ Lemma ： If is convex, then

39. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 39 Complex Geodesic of complex geodesic of : complex geodesic of : or or where where

40. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 40 Introduction Introduction  -Analysis and Synthesis  -Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  Synthesis via SNP Theory Algorithm of  Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

41. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 41 Algorithm of  -Synthesis via SNP Theory First transform the robust performance problem to the model matching form First transform the robust performance problem to the model matching form K G 

42. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 42 Algorithm of  -Synthesis via SNPT (cont’d) Modify the problem to the situation such that we can use the solution of SNP problem. Modify the problem to the situation such that we can use the solution of SNP problem. Solve the SNP problem for the function F. Solve the SNP problem for the function F. Find the controller K. Find the controller K. Iterate for the desired K. Iterate for the desired K.

43. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 43 Introduction Introduction  -Analysis and Synthesis  -Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

44. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 44 Numerical Examples Real parameter uncertainty Real parameter uncertainty Dynamical uncertainty: SISO case Dynamical uncertainty: SISO case Dynamical uncertainty: 2 Input 2 output case Dynamical uncertainty: 2 Input 2 output case

45. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 45 Plant: Structured uncertainty notation:  G Closed loop system: Solve since and Real Parameter Uncertainty

46. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 46 Block Diagram: Closed loop system: K pp WuWu + P WpWp + Dynamic Uncertainty Standard Notation: K G 

47. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 47 Q parameterization: Dynamic Uncertainty (cont’d) Model Matching: Interpolation conditions: let z i, p j be the zeros and poles of P ( s ), then Restriction: M ( s ) must be 2  2 matrices and the total number of i+j, must be 2.

48. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 48 Plant: Closed loop system: Dynamic Uncertainty: SISO case Interpolation Condition: Remark: for all SISO system Choose b j,c j  16/33, the existence of  is guaranteed.

49. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 49

50. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 50

51. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 51 Algorithm: Step 1: transform RHP into unit disk via s =(1+ )/(1  ) and z eros of T 2 are Step 2:  1 =1,the interpolation conditions are 1. Solve Plant: Model matching problem: Dynamic Uncertainty: 2x2 case

52. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 52 2. Compute Dynamic Uncertainty: 2x2 case (cont’d) leads to 3. 4. The Möbius transform formation is 5.

53. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 53 Dynamic Uncertainty: 2x2 case (cont’d) and choose 6. Let we have 7. Replace with ( s -1)/( s +1). For example when g =0,

54. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 54

55. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 55 Step 3:  1 =1/2,repeat again.

56. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 56 Step 4:  1 =1/3,repeat again.

57. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 57 Introduction Introduction  - Analysis and Synthesis  - Analysis and Synthesis Problem Description Problem Description Spectral NP Interpolation Theory: 2 by 2 case Spectral NP Interpolation Theory: 2 by 2 case Algorithm of  -Synthesis via SNP Theory Algorithm of  -Synthesis via SNP Theory Numerical Examples Numerical Examples Conclusions Conclusions Outline

58. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 58 Conclusions J.Agler, F.B.Yeh, N.J.Young Realization of Functions Into the Symmetrised Bidisc, Operator Theory: Advances and Applications,Vol.143,p1-37, 2003. Web: http://www.math.thu.edu.tw/~fbyeh The transformation from the  - synthesis problem to a spectral model matching problem is given. The transformation from the  - synthesis problem to a spectral model matching problem is given. Propose a algorithm using the SNP theory to solve Mu- synthesis. Propose a algorithm using the SNP theory to solve Mu- synthesis. With the development of SNP theory, this method could be used more practically. With the development of SNP theory, this method could be used more practically.

59. F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ. 2004.Nov.8 59 Comments and Questions. Thanks for your attention.