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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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.

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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) +

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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) +

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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)

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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”

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ Nov 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. -

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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.

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

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

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

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

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

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

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

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

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

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

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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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 

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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.

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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.

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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.

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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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,

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

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

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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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

F.B. Yeh & H.N. Huang, Dept. of Mathematics, Tunghai Univ 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, Web: 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.

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