Matrix Polynomials in the Theory of Linear Control Systems

Size: px
Start display at page:

Download "Matrix Polynomials in the Theory of Linear Control Systems"

Transcription

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

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1

1 2 3 1 1 2 x = + x 2 + x 4 1 0 1 (d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which

More information

Linear Control Systems

Linear Control Systems Chapter 3 Linear Control Systems Topics : 1. Controllability 2. Observability 3. Linear Feedback 4. Realization Theory Copyright c Claudiu C. Remsing, 26. All rights reserved. 7 C.C. Remsing 71 Intuitively,

More information

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

More information

G(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s).

G(s) = Y (s)/u(s) In this representation, the output is always the Transfer function times the input. Y (s) = G(s)U(s). Transfer Functions The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input, i.e., Y (s)/u(s). Denoting this ratio by G(s), i.e.,

More information

EE 580 Linear Control Systems VI. State Transition Matrix

EE 580 Linear Control Systems VI. State Transition Matrix EE 580 Linear Control Systems VI. State Transition Matrix Department of Electrical Engineering Pennsylvania State University Fall 2010 6.1 Introduction Typical signal spaces are (infinite-dimensional vector

More information

22 Matrix exponent. Equal eigenvalues

22 Matrix exponent. Equal eigenvalues 22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

R mp nq. (13.1) a m1 B a mn B

R mp nq. (13.1) a m1 B a mn B This electronic version is for personal use and may not be duplicated or distributed Chapter 13 Kronecker Products 131 Definition and Examples Definition 131 Let A R m n, B R p q Then the Kronecker product

More information

CONTROLLER DESIGN USING POLYNOMIAL MATRIX DESCRIPTION

CONTROLLER DESIGN USING POLYNOMIAL MATRIX DESCRIPTION CONTROLLER DESIGN USING POLYNOMIAL MATRIX DESCRIPTION Didier Henrion, Centre National de la Recherche Scientifique, France Michael Šebek, Center for Applied Cybernetics, Faculty of Electrical Eng., Czech

More information

University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7. Review University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

More information

Matrix Differentiation

Matrix Differentiation 1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

More information

MATRICES WITH DISPLACEMENT STRUCTURE A SURVEY

MATRICES WITH DISPLACEMENT STRUCTURE A SURVEY MATRICES WITH DISPLACEMENT STRUCTURE A SURVEY PLAMEN KOEV Abstract In the following survey we look at structured matrices with what is referred to as low displacement rank Matrices like Cauchy Vandermonde

More information

Matlab and Simulink. Matlab and Simulink for Control

Matlab and Simulink. Matlab and Simulink for Control Matlab and Simulink for Control Automatica I (Laboratorio) 1/78 Matlab and Simulink CACSD 2/78 Matlab and Simulink for Control Matlab introduction Simulink introduction Control Issues Recall Matlab design

More information

Presentation 3: Eigenvalues and Eigenvectors of a Matrix

Presentation 3: Eigenvalues and Eigenvectors of a Matrix Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues

More information

Structural Issues in Control Design for Linear Multivariable Systems

Structural Issues in Control Design for Linear Multivariable Systems Structural Issues in Control Design for Linear Multivariable Systems by Steven Ronald Weller B.E. (Hons. I), M.E. A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of

More information

3.1 State Space Models

3.1 State Space Models 31 State Space Models In this section we study state space models of continuous-time linear systems The corresponding results for discrete-time systems, obtained via duality with the continuous-time models,

More information

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns

4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows

More information

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V.

is in plane V. However, it may be more convenient to introduce a plane coordinate system in V. .4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

Normal forms of regular matrix polynomials via local rank factorization

Normal forms of regular matrix polynomials via local rank factorization Dipartimento di Scienze Statistiche Sezione di Statistica Economica ed Econometria Massimo Franchi Paolo Paruolo Normal forms of regular matrix polynomials via local rank factorization DSS Empirical Economics

More information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS Bull Austral Math Soc 77 (2008), 31 36 doi: 101017/S0004972708000038 COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V EROVENKO and B SURY (Received 12 April 2007) Abstract We compute

More information

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14

The determinant of a skew-symmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14 4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with

More information

1 Cubic Hermite Spline Interpolation

1 Cubic Hermite Spline Interpolation cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite

More information

x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3

x + y + z = 1 2x + 3y + 4z = 0 5x + 6y + 7z = 3 Math 24 FINAL EXAM (2/9/9 - SOLUTIONS ( Find the general solution to the system of equations 2 4 5 6 7 ( r 2 2r r 2 r 5r r x + y + z 2x + y + 4z 5x + 6y + 7z 2 2 2 2 So x z + y 2z 2 and z is free. ( r

More information

EC9A0: Pre-sessional Advanced Mathematics Course

EC9A0: Pre-sessional Advanced Mathematics Course University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

More information

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1

ASEN 3112 - Structures. MDOF Dynamic Systems. ASEN 3112 Lecture 1 Slide 1 19 MDOF Dynamic Systems ASEN 3112 Lecture 1 Slide 1 A Two-DOF Mass-Spring-Dashpot Dynamic System Consider the lumped-parameter, mass-spring-dashpot dynamic system shown in the Figure. It has two point

More information

Lecture 5 Rational functions and partial fraction expansion

Lecture 5 Rational functions and partial fraction expansion S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions pole-zero plots partial fraction expansion repeated poles nonproper rational functions

More information

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations.

Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3. Stability and pole locations. Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Dr Glenn Vinnicombe HANDOUT 3 Stability and pole locations asymptotically stable marginally stable unstable Imag(s) repeated poles +

More information

1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams

1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A = MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

More information

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

More information

Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8

Bindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8 Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e

More information

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

More information

Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability. p. 1/?

Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability. p. 1/? Nonlinear Systems and Control Lecture # 15 Positive Real Transfer Functions & Connection with Lyapunov Stability p. 1/? p. 2/? Definition: A p p proper rational transfer function matrix G(s) is positive

More information

Linearly Independent Sets and Linearly Dependent Sets

Linearly Independent Sets and Linearly Dependent Sets These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation

More information

Solution of Linear Systems

Solution of Linear Systems Chapter 3 Solution of Linear Systems In this chapter we study algorithms for possibly the most commonly occurring problem in scientific computing, the solution of linear systems of equations. We start

More information

Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology.

Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology. International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology. Patricia

More information

Lecture 5: Singular Value Decomposition SVD (1)

Lecture 5: Singular Value Decomposition SVD (1) EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse?

MathQuest: Linear Algebra. 1. Which of the following matrices does not have an inverse? MathQuest: Linear Algebra Matrix Inverses 1. Which of the following matrices does not have an inverse? 1 2 (a) 3 4 2 2 (b) 4 4 1 (c) 3 4 (d) 2 (e) More than one of the above do not have inverses. (f) All

More information

LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS

LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS This is a reprint from Revista de la Academia Colombiana de Ciencias Vol 8 (106 (004, 39-47 LEVERRIER-FADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS by J Hernández, F Marcellán & C Rodríguez LEVERRIER-FADEEV

More information

Nonlinear Algebraic Equations Example

Nonlinear Algebraic Equations Example Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities

More information

C 1 x(t) = e ta C = e C n. 2! A2 + t3

C 1 x(t) = e ta C = e C n. 2! A2 + t3 Matrix Exponential Fundamental Matrix Solution Objective: Solve dt A x with an n n constant coefficient matrix A x (t) Here the unknown is the vector function x(t) x n (t) General Solution Formula in Matrix

More information

A note on companion matrices

A note on companion matrices Linear Algebra and its Applications 372 (2003) 325 33 www.elsevier.com/locate/laa A note on companion matrices Miroslav Fiedler Academy of Sciences of the Czech Republic Institute of Computer Science Pod

More information

Michigan State University Department of Electrical and Computer Engineering. ECE 280: ELECTRICAL ENGINEERING ANALYSIS (Spring 2008) Overview

Michigan State University Department of Electrical and Computer Engineering. ECE 280: ELECTRICAL ENGINEERING ANALYSIS (Spring 2008) Overview Michigan State University Department of Electrical and Computer Engineering ECE 280: ELECTRICAL ENGINEERING ANALYSIS (Spring 2008) Overview Staff Lectures Instructor (First Half of Course: 01/07/08 02/22/08)

More information

Real Roots of Quadratic Interval Polynomials 1

Real Roots of Quadratic Interval Polynomials 1 Int. Journal of Math. Analysis, Vol. 1, 2007, no. 21, 1041-1050 Real Roots of Quadratic Interval Polynomials 1 Ibraheem Alolyan Mathematics Department College of Science, King Saud University P.O. Box:

More information

Optimal Control. Lecture Notes. February 13, 2009. Preliminary Edition

Optimal Control. Lecture Notes. February 13, 2009. Preliminary Edition Lecture Notes February 13, 29 Preliminary Edition Department of Control Engineering, Institute of Electronic Systems Aalborg University, Fredrik Bajers Vej 7, DK-922 Aalborg Ø, Denmark Page II of IV Preamble

More information

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE?

WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? WHICH LINEAR-FRACTIONAL TRANSFORMATIONS INDUCE ROTATIONS OF THE SPHERE? JOEL H. SHAPIRO Abstract. These notes supplement the discussion of linear fractional mappings presented in a beginning graduate course

More information

Congestion Control of Active Queue Management Routers Based on LQ-Servo Control

Congestion Control of Active Queue Management Routers Based on LQ-Servo Control Congestion Control of Active Queue Management outers Based on LQ-Servo Control Kang Min Lee, Ji Hoon Yang, and Byung Suhl Suh Abstract This paper proposes the LQ-Servo controller for AQM (Active Queue

More information

Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor

Chapter 12 Modal Decomposition of State-Space Models 12.1 Introduction The solutions obtained in previous chapters, whether in time domain or transfor Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology 1 1 c Chapter

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

Mean value theorem, Taylors Theorem, Maxima and Minima.

Mean value theorem, Taylors Theorem, Maxima and Minima. MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

More information

Time-Domain Solution of LTI State Equations. 2 State-Variable Response of Linear Systems

Time-Domain Solution of LTI State Equations. 2 State-Variable Response of Linear Systems 2.14 Analysis and Design of Feedback Control Systems Time-Domain Solution of LTI State Equations Derek Rowell October 22 1 Introduction This note examines the response of linear, time-invariant models

More information

Low Dimensional Representations of the Loop Braid Group LB 3

Low Dimensional Representations of the Loop Braid Group LB 3 Low Dimensional Representations of the Loop Braid Group LB 3 Liang Chang Texas A&M University UT Dallas, June 1, 2015 Supported by AMS MRC Program Joint work with Paul Bruillard, Cesar Galindo, Seung-Moon

More information

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

More information

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS

COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS COMMUTATIVITY DEGREES OF WREATH PRODUCTS OF FINITE ABELIAN GROUPS IGOR V. EROVENKO AND B. SURY ABSTRACT. We compute commutativity degrees of wreath products A B of finite abelian groups A and B. When B

More information

LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

More information

Chapter 14: Analyzing Relationships Between Variables

Chapter 14: Analyzing Relationships Between Variables Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 14: Analyzing Relationships Between

More information

ECE 516: System Control Engineering

ECE 516: System Control Engineering ECE 516: System Control Engineering This course focuses on the analysis and design of systems control. This course will introduce time-domain systems dynamic control fundamentals and their design issues

More information

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

More information

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Numerical Methods for Differential Equations Course objectives and preliminaries Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis

More information

Chapter 6. Orthogonality

Chapter 6. Orthogonality 6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

More information

Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable. Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

More information

Lecture 2 Matrix Operations

Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

More information

The Geometry of Polynomial Division and Elimination

The Geometry of Polynomial Division and Elimination The Geometry of Polynomial Division and Elimination Kim Batselier, Philippe Dreesen Bart De Moor Katholieke Universiteit Leuven Department of Electrical Engineering ESAT/SCD/SISTA/SMC May 2012 1 / 26 Outline

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited

The Matrix Elements of a 3 3 Orthogonal Matrix Revisited Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, Three-Dimensional Proper and Improper Rotation Matrices, I provided a derivation

More information

Explicit inverses of some tridiagonal matrices

Explicit inverses of some tridiagonal matrices Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,

More information

1 Solving LPs: The Simplex Algorithm of George Dantzig

1 Solving LPs: The Simplex Algorithm of George Dantzig Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

More information

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

More information

Generalized Inverse of Matrices and its Applications

Generalized Inverse of Matrices and its Applications Generalized Inverse of Matrices and its Applications C. RADHAKRISHNA RAO, Sc.D., F.N.A., F.R.S. Director, Research and Training School Indian Statistical Institute SUJIT KUMAR MITRA, Ph.D. Professor of

More information

A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved

A Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of Matrix-Valued Functions of a Real Variable

More information

BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam

BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad

More information

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

More information

Typical Linear Equation Set and Corresponding Matrices

Typical Linear Equation Set and Corresponding Matrices EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of

More information

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI

Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:

More information

Technische Universität Ilmenau Institut für Mathematik

Technische Universität Ilmenau Institut für Mathematik Technische Universität Ilmenau Institut für Mathematik Preprint No. M 13/08 Zero dynamics and stabilization for linear DAEs Thomas Berger 2013 Impressum: Hrsg.: Leiter des Instituts für Mathematik Weimarer

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

The Method of Least Squares

The Method of Least Squares The Method of Least Squares Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract The Method of Least Squares is a procedure to determine the best fit line to data; the

More information

HEAVISIDE CABLE, THOMSON CABLE AND THE BEST CONSTANT OF A SOBOLEV-TYPE INEQUALITY. Received June 1, 2007; revised September 30, 2007

HEAVISIDE CABLE, THOMSON CABLE AND THE BEST CONSTANT OF A SOBOLEV-TYPE INEQUALITY. Received June 1, 2007; revised September 30, 2007 Scientiae Mathematicae Japonicae Online, e-007, 739 755 739 HEAVISIDE CABLE, THOMSON CABLE AND THE BEST CONSTANT OF A SOBOLEV-TYPE INEQUALITY Yoshinori Kametaka, Kazuo Takemura, Hiroyuki Yamagishi, Atsushi

More information

Dynamic Eigenvalues for Scalar Linear Time-Varying Systems

Dynamic Eigenvalues for Scalar Linear Time-Varying Systems Dynamic Eigenvalues for Scalar Linear Time-Varying Systems P. van der Kloet and F.L. Neerhoff Department of Electrical Engineering Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands

More information

ORDINARY DIFFERENTIAL EQUATIONS

ORDINARY DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.

More information

A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

More information

Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

More information

COMPOUND MATRICES AND THREE CELEBRATED THEOREMS

COMPOUND MATRICES AND THREE CELEBRATED THEOREMS COMPOUND MATRICES AND THREE CELEBRATED THEOREMS K. K. NAMBIAR AND SHANTI SREEVALSAN ABSTRACT. Apart from the regular and adjugate compounds of a matrix an inverse compound of a matrix is defined. Theorems

More information

OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS

OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NON-liFE INSURANCE BUSINESS By Martina Vandebroek

More information

7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix

7. LU factorization. factor-solve method. LU factorization. solving Ax = b with A nonsingular. the inverse of a nonsingular matrix 7. LU factorization EE103 (Fall 2011-12) factor-solve method LU factorization solving Ax = b with A nonsingular the inverse of a nonsingular matrix LU factorization algorithm effect of rounding error sparse

More information

Root Computations of Real-coefficient Polynomials using Spreadsheets*

Root Computations of Real-coefficient Polynomials using Spreadsheets* Int. J. Engng Ed. Vol. 18, No. 1, pp. 89±97, 2002 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 2002 TEMPUS Publications. Root Computations of Real-coefficient Polynomials using Spreadsheets* KARIM

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

More information

The integrating factor method (Sect. 2.1).

The integrating factor method (Sect. 2.1). The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable

More information

LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA. September 23, 2010 LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

More information

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10 Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

More information

Nonlinear Least Squares Data Fitting

Nonlinear Least Squares Data Fitting Appendix D Nonlinear Least Squares Data Fitting D1 Introduction A nonlinear least squares problem is an unconstrained minimization problem of the form minimize f(x) = f i (x) 2, x where the objective function

More information

5.3 Determinants and Cramer s Rule

5.3 Determinants and Cramer s Rule 290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given

More information

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression

The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every

More information

On the Geometry and Parametrization of Almost Invariant Subspaces and Observer Theory

On the Geometry and Parametrization of Almost Invariant Subspaces and Observer Theory On the Geometry and Parametrization of Almost Invariant Subspaces and Observer Theory Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades der Bayerischen Julius-Maximilians-Universität

More information

Solutions to Linear Algebra Practice Problems

Solutions to Linear Algebra Practice Problems Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the

More information

MAT188H1S Lec0101 Burbulla

MAT188H1S Lec0101 Burbulla Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

More information