# Matrix Polynomials in the Theory of Linear Control Systems

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1 Matrix Polynomials in the Theory of Linear Control Systems Ion Zaballa Departamento de Matemática Aplicada y EIO

2 Matrix Polynomials in Linear Control

3 Matrix Polynomials in Linear Control Coefficient Matrices of highorder systems ( ) d T dt y(t) = V ( d x(t) = U ( dt d dt ) x(t) + W ) u(t) ( d dt ) u(t)

4 Matrix Polynomials in Linear Control Coefficient Matrices of highorder systems Matrix Polynomials Representing systems ( ) d T dt y(t) = V ( d x(t) = U ( dt d dt ) x(t) + W ) u(t) ( d dt ) u(t) ẋ(t) = Ax(t) + Bu(t)

5 Matrix Polynomials in Linear Control Coefficient Matrices of highorder systems Matrix Polynomials Representing systems Behaviours (Poldermann & Willems) ( ) d T dt y(t) = V ( d x(t) = U ( dt d dt ) x(t) + W ) u(t) ( d dt ) u(t) ẋ(t) = Ax(t) + Bu(t) ( ) d ( ) d P dt y(t) = Q dt u(t)

6 Polynomial System Matrices T ( ) d dt y(t) = V x(t) = U ( d dt ( ) d dt ) x(t) + W u(t) ( d dt ) u(t)

7 state vector Polynomial System Matrices ( ) ( ) d d T x(t) = U u(t) dt ( ) dt d y(t) = V x(t) + W dt ( d dt ) u(t)

8 state vector Polynomial System Matrices input or control vector ( ) ( ) d d T x(t) = U u(t) dt ( ) dt ( ) d d y(t) = V x(t) + W u(t) dt dt

9 state vector Polynomial System Matrices input or control vector ( ) ( ) d d T x(t) = U u(t) dt ( ) dt ( ) d d y(t) = V x(t) + W u(t) dt dt output or measurement vector

10 state vector Polynomial System Matrices input or control vector ( ) ( ) d d T x(t) = U u(t) dt ( ) dt ( ) d d y(t) = V x(t) + W u(t) dt dt output or measurement vector If R(λ) = R p λ p + R p 1 λ p R 1 λ + R 0 is a matrix polynomial: ( ) d d p x(t) d p 1 x(t) dx(t) R x(t) = R p dt dt p + R p 1 dt p R 1 + R 0 x(t) dt

11 state vector Polynomial System Matrices input or control vector ( ) ( ) d d T x(t) = U u(t) dt ( ) dt ( ) d d y(t) = V x(t) + W u(t) dt dt output or measurement vector If R(λ) = R p λ p + R p 1 λ p R 1 λ + R 0 is a matrix polynomial: ( ) d d p x(t) d p 1 x(t) dx(t) R x(t) = R p dt dt p + R p 1 dt p R 1 + R 0 x(t) dt [ ] [ ] T (s) U(s) x(s) = V (s) W (s) ū(s) [ ] 0, det T (s) 0 ȳ(s)

12 state vector Polynomial System Matrices input or control vector ( ) ( ) d d T x(t) = U u(t) dt ( ) dt ( ) d d y(t) = V x(t) + W u(t) dt dt output or measurement vector If R(λ) = R p λ p + R p 1 λ p R 1 λ + R 0 is a matrix polynomial: ( ) d d p x(t) d p 1 x(t) dx(t) R x(t) = R p dt dt p + R p 1 dt p R 1 + R 0 x(t) dt [ ] [ ] T (s) U(s) x(s) = V (s) W (s) ū(s) [ ] 0, det T (s) 0 ȳ(s) Polynomial System Matrix

13 ȳ(s) = V (s) x(s) + W (s)ū(s) = Transfer Function Matrix

14 Transfer Function Matrix T (s) x(s) = U(s)ū(s) ȳ(s) = V (s) x(s) + W (s)ū(s) =

15 Transfer Function Matrix T (s) x(s) = U(s)ū(s) ȳ(s) = V (s) x(s) + W (s)ū(s) = (V (s)t (s) 1 U(s) + W (s)) ū(s)

16 Transfer Function Matrix T (s) x(s) = U(s)ū(s) Transfer Function Matrix ȳ(s) = V (s) x(s) + W (s)ū(s) = (V (s)t (s) 1 U(s) + W (s)) ū(s)

17 Transfer Function Matrix T (s) x(s) = U(s)ū(s) Transfer Function Matrix ȳ(s) = V (s) x(s) + W (s)ū(s) = (V (s)t (s) 1 U(s) + W (s)) ū(s) When do two polynomial system matrices yield the same Transfer Function Matrix?

18 Strict System Equivalence Unimodular: det = c 0 [ M(s) 0 X(s) I p ] n m [ ] [ ] [ ] n T 1 (s) U 1 (s) N(s) Y (s) T2 (s) U = 2 (s) p V 1 (s) W 1 (s) 0 I m V 2 (s) W 2 (s)

19 Strict System Equivalence Unimodular: det = c 0 [ ] [ M(s) 0 T 1 (s) X(s) I p V 1 (s) ] [ ] [ U 1 (s) N(s) Y (s) T = 2 (s) W 1 (s) 0 I m V 2 (s) ] U 2 (s) W 2 (s)

20 Strict System Equivalence Unimodular: det = c 0 [ ] [ M(s) 0 X(s) I p T 1 (s) V 1 (s) U 1 (s) W 1 (s) ] [N(s) ] Y (s) 0 I m = [ T 2 (s) V 2 (s) U 2 (s) W 2 (s) ]

21 Strict System Equivalence Unimodular: det = c 0 [ M(s) ] [ 0 T 1 (s) X(s) V 1 (s) I p U 1 (s) W 1 (s) ] [ ] [ N(s) Y (s) = 0 I m T 2 (s) V 2 (s) U 2 (s) W 2 (s) ]

22 Strict System Equivalence [ M(s) 0 X(s) I p ] n m [ ] [ ] [ ] n T 1 (s) U 1 (s) N(s) Y (s) T2 (s) U = 2 (s) p V 1 (s) W 1 (s) 0 I m V 2 (s) W 2 (s) V 1 (s)t 1 (s) 1 U 1 (s) + W 1 (s) = V 2 (s)t 2 (s) 1 U 2 (s) + W 2 (s)

23 Strict System Equivalence [ M(s) 0 X(s) I p ] n m [ ] [ ] [ ] n T 1 (s) U 1 (s) N(s) Y (s) T2 (s) U = 2 (s) p V 1 (s) W 1 (s) 0 I m V 2 (s) W 2 (s)? V 1 (s)t 1 (s) 1 U 1 (s) + W 1 (s) = V 2 (s)t 2 (s) 1 U 2 (s) + W 2 (s)

24 Coprime Matrix Polynomials A(s) = Ã(s) C(s) B(s) = B(s) C(s) Right Common Factor A(s), B(s) right coprime C(s) unimodular A(s), B(s) right coprime [ ] A(s) B(s) e [ ] In 0 A(s), B(s) left coprime [ A(s) B(s) ] e [ In 0 ]

25 Coprimeness and Strict System Equivalence If G(s) F(s) p m, (T (s), U(s), V (s), W (s)) Realization of G(s) if G(s) = W (s) + V (s)t (s) 1 U(s). order= deg(det T (s)) (T (s), U(s), V (s), W (s)) realization of least order (T (s), U(s)) left coprime and (T (s), V (s)) right coprime [ ] [ ] T1 (s) U If P 1 (s) = 1 (s) T2 (s) U, P V 1 (s) W 1 (s) 2 (s) = 2 (s) polynomial system matrices of least V 2 (s) W 2 (s) order P 1 (s) s.s.e. P 2 (s) V 1 (s)t 1 (s) 1 U 1 (s) + W 1 (s) = V 2 (s)t 2 (s) 1 U 2 (s) + W 2 (s)

26 Systems in State-Space Form (Σ) { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) P (s) = [ sin A ] B C 0

27 (Σ) Systems in State-Space Form { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) P (s) = [ sin A ] B C 0 Controllability in [t 0, t 1 ]: x 0, x 1 R n, u defined in [t 0, t 1 ] such that the solution of the I.V.P. ẋ(t) = Ax(t) + Bu(t), x(t 0 ) = x 0, satisfies x(t 1 ) = x 1. x x 1 x 0 t 0 t 1 t

28 (Σ) Systems in State-Space Form { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) P (s) = [ sin A ] B C 0 Controllability in [t 0, t 1 ]: x 0, x 1 R n, u defined in [t 0, t 1 ] such that the solution of the I.V.P. ẋ(t) = Ax(t) + Bu(t), x(t 0 ) = x 0, satisfies x(t 1 ) = x 1. x x 1 x 0 t 0 t 1 (Σ) controllable (A, B) controllable t

29 Controllability, Observability and Coprimeness (Σ) { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) (A, B) controllable is equivalent to: P (s) = [ sin A ] B C 0 rank [ B AB A n 1 B ] = n, or ( [sin si n A and B are left coprime A B ] e [ In 0 ])

30 Controllability, Observability and Coprimeness (Σ) { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) (A, B) controllable is equivalent to: P (s) = [ sin A ] B C 0 rank [ B AB A n 1 B ] = n, or ( [sin si n A and B are left coprime A B ] e [ In 0 ]) Observability in [t 0, t 1 ]: The value of y in [t 0, t 1 ] determines the state at t 0, x(t 0 ), and so the vector function x(t) in [t 0, t 1 ]. (Σ) observable (A, C) observable (A T, C T ) controllable.

31 Controllability, Observability and Coprimeness (Σ) { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) (A, B) controllable is equivalent to: P (s) = [ sin A ] B C 0 rank [ B AB A n 1 B ] = n, or ( [sin si n A and B are left coprime A B ] e [ In 0 ]) Observability in [t 0, t 1 ]: The value of y in [t 0, t 1 ] determines the state at t 0, x(t 0 ), and so the vector function x(t) in [t 0, t 1 ]. (Σ) observable (A, C) observable (A T, C T ) controllable. [ ] sin A B P (s) = is of least order if and only if (A, B) controllable C 0 and (A, C) observable.

32 Transfer Function Matrix From now on: { ẋ(t) = Ax(t) + Bu(t) (Σ) y(t) = I n x(t) P (s) = Transfer Function Matrix: G(s) = (si n A) 1 B [ sin A ] B I n 0 s 0

33 Transfer Function Matrix From now on: { ẋ(t) = Ax(t) + Bu(t) (Σ) y(t) = I n x(t) P (s) = Transfer Function Matrix: G(s) = (si n A) 1 B lcm{denominators of G(s)} [ sin A ] B I n 0 s 0 G(s) = Ñ(s)( d(s) I n) 1

34 Transfer Function Matrix From now on: { ẋ(t) = Ax(t) + Bu(t) (Σ) y(t) = I n x(t) P (s) = Transfer Function Matrix: G(s) = (si n A) 1 B lcm{denominators of G(s)} [ sin A ] B I n 0 s Remove right common factors from Ñ(s) and d(s)i n G(s) = Ñ(s)( d(s) I n) 1 = N(s)D(s) 1 0

35 Transfer Function Matrix From now on: { ẋ(t) = Ax(t) + Bu(t) (Σ) y(t) = I n x(t) P (s) = Transfer Function Matrix: G(s) = (si n A) 1 B lcm{denominators of G(s)} If (A, B) controllable [ sin A ] B I n 0 s Remove right common factors from Ñ(s) and d(s)i n G(s) = Ñ(s)( d(s) I n) 1 = N(s)D(s) 1 [ ] sin A B sse I n 0 I n m D(s) I m 0 N(s) 0 0 (n m)

36 Polynomial Matrix Representations [ ] [ ] [ ] U(s) 0 sin A B V (s) Y (s) = X(s) I n I n 0 0 I m I n m D(s) I m 0 N(s) 0 U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m ( )

37 Polynomial Matrix Representations [ ] [ ] [ ] U(s) 0 sin A B V (s) Y (s) = X(s) I n I n 0 0 I m I n m D(s) I m 0 N(s) 0 U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m X(s) = [ V 1 (s) Y (s) ] U(s), N(s) = Y (s)d(s) + V 2 (s), where V (s) = [ V 1 (s) V 2 (s) ], V 2 n m ( )

38 Polynomial Matrix Representations [ ] [ ] [ ] U(s) 0 sin A B V (s) Y (s) = X(s) I n I n 0 0 I m I n m D(s) I m 0 N(s) 0 U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m X(s) = [ V 1 (s) Y (s) ] U(s), N(s) = Y (s)d(s) + V 2 (s), where V (s) = [ V 1 (s) V 2 (s) ], V 2 n m ( ) A Polynomial Matrix Representation of a controllable system (A, B) is any non-singular Matrix Polynomial D(s) such that ( ) is satisfied for some unimodular matrices U(s), V (s) and some matrix Y (s).

39 Some Consequences of the Definition. I If D(s) is a PMR of (A, B), D(s)W (s) is also a PMR of (A, B) for any unimodular W (s). U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m U(s) [ si n A B ] [ ] [ ]. V (s) W (s) Y (s) In m 0 0 = 0 I m 0 D(s)W (s) I m W (s) = Diag(I n m, W (s)).

40 Some Consequences of the Definition. I If D(s) is a PMR of (A, B), D(s)W (s) is also a PMR of (A, B) for any unimodular W (s). U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m U(s) [ si n A B ] [ ] [ ]. V (s) W (s) Y (s) In m 0 0 = 0 I m 0 D(s)W (s) I m W (s) = Diag(I n m, W (s)). If D 1 (s), D 2 (s) are PMRs of (A 1, B 1 ) and (A 2, B 2 ) : D 2 (s) = D 1 (s)w (s) (A 2, B 2 ) = (P 1 A 1 P, P 1 B 1 ). W (s) unimodular, P invertible.

41 Some Consequences of the Definition. II If D(s) = D l s l + D l 1 s l D 1 s + D 0 (si n A) 1 B = N(s)D(s) 1 (si n A) 1 BD(s) = N(s) A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 ( ) (A, B) controllable and ( ) D(s) PMR of (A, B)

42 Some Consequences of the Definition. II If D(s) = D l s l + D l 1 s l D 1 s + D 0 (si n A) 1 B = N(s)D(s) 1 (si n A) 1 BD(s) = N(s) A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 ( ) (A, B) controllable and ( ) D(s) PMR of (A, B) si n A is a linearization of D(s) [ ] [ ] [ V (s) Y (s) U(s) si n A B = 0 I m I n m 0 0 D(s) 0 I m ]

43 Some Consequences of the Definition. II If D(s) = D l s l + D l 1 s l D 1 s + D 0 (si n A) 1 B = N(s)D(s) 1 (si n A) 1 BD(s) = N(s) A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 ( ) (A, B) controllable and ( ) D(s) PMR of (A, B) si n A is a linearization of D(s) [ ] [ ] [ V (s) Y (s) U(s) si n A B = 0 I m I n m 0 0 D(s) 0 I m ] Given D(s), non-singular, is there, for any linearization si n A of D(s), a control matrix B such that (A, B) is controllable and D(s) is a PMR of (A, B)?

44 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n

45 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B b 2 b 3 b 4

46 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB b 2 b 3 b 4 Ab 2 Ab 4

47 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4

48 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B A 3 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4 A 3 b 4

49 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B A 3 B A 4 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4 A 3 b 4 A 4 b 4

50 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B A 3 B A 4 B A 5 B A 6 B A 7 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4 A 3 b 4 A 4 b 4

51 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B A 3 B A 4 B A 5 B A 6 B A 7 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4 l 1 = 0, l 2 = 2, l 3 = 1, l 4 = 5, l 5 = 0 k 1 = 5, k 2 = 2, k 3 = 1, k 4 = 0, k 5 = 0 A 3 b 4 A 4 b 4 Controllability Indices

52 Controllability indices Assume (A, B) controllable: rank [ B AB A n 1 B ] = n n = 8, m = 5 B AB A 2 B A 3 B A 4 B A 5 B A 6 B A 7 B b 2 b 3 b 4 Ab 2 Ab 4 A 2 b 4 l 1 = 0, l 2 = 2, l 3 = 1, l 4 = 5, l 5 = 0 k 1 = 5, k 2 = 2, k 3 = 1, k 4 = 0, k 5 = 0 A 3 b 4 A 4 b 4 Controllability Indices Controllability indices of (A, B)= minimal indices of s [ I n 0 ] [ A B ] Matlab

53 PMRs and Controllability indices unordered controllability indeces s l s l 2 0 D(s) = D hc + D..... lc (s). 0 0 s l m det D hc 0 degree of i-th column l i

54 PMRs and Controllability indices unordered controllability indeces s l s l 2 0 D(s) = D hc + D..... lc (s). 0 0 s l m det D hc 0 degree of i-th column l i Matrix polynomials with this property are called column proper or column reduced

55 PMRs and Controllability indices unordered controllability indeces s l s l 2 0 D(s) = D hc + D..... lc (s). 0 0 s l m det D hc 0 degree of i-th column l i Matrix polynomials with this property are called column proper or column reduced Let A(s) F[s] m m be a non-singular matrix polynomial. Then there is U(s) F[s] m m, unimodular, such that D(s) = A(s)U(s) is a column proper matrix. In general, D(s) is not unique but all have the same column degrees up to reordering.

56 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A

57 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A D(s) = A(s)U(s) D(s) = D hc Diag(s l1, s l2,..., s lm )+ D lc (s) column proper

58 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A D(s) = A(s)U(s) D(s) = D hc Diag(s l1, s l2,..., s lm )+ D lc (s) column proper D(s) = Diag(s l1, s l2,..., s lm ) + D 1 hc D lc(s) column proper

59 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A D(s) = A(s)U(s) D(s) = D hc Diag(s l1, s l2,..., s lm )+ D lc (s) column proper (A c, B c ) MATLAB example D(s) = Diag(s l1, s l2,..., s lm ) + D 1 hc D lc(s) column proper

60 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A D(s) = A(s)U(s) D(s) = D hc Diag(s l1, s l2,..., s lm )+ D lc (s) column proper (A c, B c ) MATLAB example D(s) = Diag(s l1, s l2,..., s lm ) + D 1 hc D lc(s) column proper (A c, B c D 1 hc )

61 Column Proper Matrix Polynomials Given A(s), choose a linearization si n A D(s) = A(s)U(s) D(s) = D hc Diag(s l1, s l2,..., s lm )+ D lc (s) column proper (A c, B c ) (A c, B c D 1 hc ) MATLAB example A and A c linearizations of A(s): A = P 1 A cp D(s) = Diag(s l1, s l2,..., s lm ) + D 1 hc D lc(s) column proper B = P 1 B c D 1 hc A(s) is a Polynomial Matrix Representation of (A, B) and l 1,... l m are its (unordered) controllability indices.

62 Wiener-Hopf Factorization and Indices A 1 (s), A 2 (s) F[s] m n Wiener-Hopf equivalent (at on the left): A 2 (s) = B(s) A 1 (s) U(s) Biproper: invertible lim B(s) s Unimodular: lim s a U(s) invertible, a C

63 Wiener-Hopf Factorization and Indices A 1 (s), A 2 (s) F[s] m n Wiener-Hopf equivalent (at on the left): A 2 (s) = B(s) A 1 (s) U(s) Biproper: invertible lim B(s) s Unimodular: lim s a U(s) invertible, a C A(s)U(s) = D hc Diag(s l 1,..., s lm ) + D lc (s) = [D hc + D lc (s) Diag(s l 1,..., s lm l )] Diag(s 1,..., s lm ) }{{} B(s) F pr(s) m m lim s B(s) = D hc invertible

64 Wiener-Hopf Factorization and Indices A 1 (s), A 2 (s) F[s] m n Wiener-Hopf equivalent (at on the left): A 2 (s) = B(s) A 1 (s) U(s) Biproper: invertible lim B(s) s Unimodular: lim s a U(s) invertible, a C A(s)U(s) = D hc Diag(s l 1,..., s lm ) + D lc (s) = [D hc + D lc (s) Diag(s l 1,..., s lm l )] Diag(s 1,..., s lm ) }{{} B(s) F pr(s) m m lim s B(s) = D hc invertible A(s) W H Diag(s k 1, s k 2,..., s km ) k 1 k 2 k m = Wiener-Hopf factorization indices of A(s)

65 Brunovsky-Kronecker canonical form Diag(s k 1, s k 2,..., s km ) is a PMR of (A c, B c ) A c = Diag F k i k i, B c = Diag. F k i 1, 1 i m s s [ I n 0 ] [ 0 s ] se. A c B c Diag 0 0 s F k i (k i +1) : 1 i m s 1 Kronecker canonical form

66 Brunovsky-Kronecker canonical form Diag(s k 1, s k 2,..., s km ) is a PMR of (A c, B c ) A c = Diag F k i k i, B c = Diag. F k i 1, 1 i m s s [ I n 0 ] [ 0 s ] se. A c B c Diag 0 0 s F k i (k i +1) : 1 i m s 1 Kronecker canonical form (A c, B c ) = system in Brunovsky form

67 Feedback equivalence (A, B) has controllability indices k 1 k 2 k m s [ I n 0 ] [ A B ] se s [ I n 0 ] [ A c ] B c P (s [ I n 0 ] [ A B ] )T = s [ I n 0 ] [ A c ] B c P [ A B ] [ P 1 ] 0 = [ A R Q c ] B c (A c, B c ) = (P AP 1 + P BF, P BQ)

68 Feedback equivalence (A, B) has controllability indices k 1 k 2 k m s [ I n 0 ] [ A B ] se s [ I n 0 ] [ A c ] B c P (s [ I n 0 ] [ A B ] )T = s [ I n 0 ] [ A c ] B c P [ A B ] [ P 1 ] 0 = [ A R Q c ] B c (A c, B c ) = (P AP 1 + P BF, P BQ) Feedback equivalence Any controllable system is feedback equivalent to a system in Brunovsky form

69 Summarizing D(s) is a Polynomial Matrix Representation of a controllable (A, B) U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m (si n A) 1 B = N(s)D(s) 1, N(s), D(s) right coprime A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 (D(s) = D l s l + D l 1 s l D 0 )

70 Summarizing D(s) is a Polynomial Matrix Representation of a controllable (A, B) U(s) [ si n A B ] [ ] [ ] V (s) Y (s) In m 0 0 = 0 I m 0 D(s) I m (si n A) 1 B = N(s)D(s) 1, N(s), D(s) right coprime A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 (D(s) = D l s l + D l 1 s l D 0 ) (A, B) D(s) (P 1 AP, P 1 B) D(s)U(s) Similarity Right Equivalence (P 1 AP + P 1 BF, P 1 BQ) B(s)D(s)U(s) Feedback Equivalence Wiener-Hopf Equivalence Controllability Indices Left Wiener-Hopf indices

71 Matrix Polynomials with non-singular Leading Coefficient non-singular D(s) = D l s l + D l 1 s l D 1 s + D 0 B(s) = D l + D l 1 s D 1 s l+1 + D 0 s l biproper ( lim B(s) = D l ) s m {}}{ B(s) 1 D(s) = s l I m ( l, l,..., l) = Wiener-Hopf indices of D(s)

72 Matrix Polynomials with non-singular Leading Coefficient non-singular D(s) = D l s l + D l 1 s l D 1 s + D 0 B(s) = D l + D l 1 s D 1 s l+1 + D 0 s l biproper ( lim B(s) = D l ) s m {}}{ B(s) 1 D(s) = s l I m ( l, l,..., l) = Wiener-Hopf indices of D(s) D(s) Polynomial Matrix Representation of (A, B) rank [ B AB A lm 1 B ] = lm and A l BD l + A l 1 BD l ABD 1 + BD 0 = 0 m {}}{ Since ( l, l,..., l) = Controllability indices of (A, B) rank [ B AB A lm 1 B ] = rank [ B AB A l 1 B ] (A, B)= Standard Pair of D(s)

73 References H. H. Rosenbrock, State-space and Multivariable Theory, Thomas Nelson and Sons, London, T. Kailath, Linear Systems, Prentice Hall, New Jersey, P. A. Fuhrmann, A Polynomial Approach to Linear Algebra, Springer, New York, 1996 J. W. Polderman, J. C. Willems, Introduction to Mathematical System Theory. A behavioral Approach Springer, New York, P. Fuhrmann, J. C. Willems, Factorization indices at infinity for rational matrix functions, Integral Equations Operator Theory, 2/3, (1979), pp

74 References I. Gohberg, M. A. Kaashoek, F. van Schagen, Partially Specified Matrices and Operators: Classification, Completion, Applications, Bikhäuser, Basel, K. Clancey, I. Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser, Basel, I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982 and SIAM, Philadelphia, I. Zaballa, Controllability and Hermite indices of matrix pairs, Int. J. Control, 68 (1), (1997), pp

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