J-SPECTRAL FACTORIZATION

Transcription

1 J-SPECTRAL FACTORIZATION of Regular Para-Hermitian Transfer Matrices Qing-Chang Zhong School of Electronics University of Glamorgan United Kingdom

2 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 2/42

3 Notations G(s) = A B C D = D + C(sI A) 1 B G (s) = G T ( s) = G( s ) = J p,q = I p 0 0 I q A B : the signature matrix C D Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 3/42

4 Some definitions A transfer matrix Λ(s) is called a para-hermitian matrix if Λ (s) = Λ(s). A transfer matrix W(s) is a J-spectral factor of Λ(s) if W(s) is bistable and Λ(s) = W (s)jw(s). Such a factorization of Λ(s) is referred to as a J-spectral factorization. A matrix W(s) is a J-spectral cofactor of a matrix Λ(s) if W(s) is bistable and Λ(s) = W(s)JW (s). Such a factorization of Λ(s) is referred to as a J-spectral cofactorization. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 4/42

5 Background J-spectral factorization plays an important role in H control of finite-dimensional systems and H control of infinite-dimensional systems as well. The necessary and sufficient condition has been well understood. The J-spectral factorizations involved in the literature are done for matrices in the form G JG, mostly with a stable G. For the case with an unstable G, the following three steps can be used to find the J-spectral factor of G JG: (i) to find the modal factorization of Λ = G JG; (ii) to construct a stable G such that Λ = G JG ; (iii) to derive the J-spectral factor of G JG, i.e., of G JG. The problem is that, in some cases, a para-hermitian transfer matrix Λ is given in the form of a state-space realization and cannot be explicitly written in the form G JG, e.g., in the context of H control of time-delay systems. In order to use the above-mentioned results, one would have to find a G such that Λ = G JG. It would be advantageous if this step could be avoided. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 5/42

6 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 6/42

7 Regular para-hermitian transfer matrices Theorem 1 A given square, minimal, rational matrix Λ(s), having no poles or zeros on the jω-axis including, is a para-hermitian matrix if and only if a minimal realization can be represented as A R B (1) Λ = E A C C B D where D = D, E = E and R = R. Such a para-hermitian transfer matrix is called regular. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 7/42

8 General state-space form Theorem 1 says that a regular para-hermitian transfer matrix Λ realized in the general state-space form Hp B (2) Λ = Λ D C Λ can always be transformed into the form of (1) after a certain similarity transformation. the canonical form of regular para-hermitian transfer matrices. Notations: H p : the A-matrix of Λ. H z : the A-matrix of Λ 1 (H z = H p B Λ D 1 C Λ ). Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 8/42

9 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 9/42

10 Projections For a given nonsingular matrix partitioned as M N, denote the projection onto the subspace ImM along the subspace ImN by P. Then, PM = M, PN = 0. P M N = M 0. Hence, the projection matrix P is P = M 0 M N 1. Similarly, the projection Q onto the subspace Im N along the subspace ImM is 1 1 Q = 0 N M N = N 0 N M. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 10/42

11 Properties of projections (i) P + Q = I; (ii) PQ = 0; (iii) P 2 = P and Q 2 = Q; (iv) ImP = ImM; (v) M 0 M N 1 M 0 = M 0 ; (vi) 0 N M N 1 0 N = 0 N ; (vii) M 0 M N 1 0 N = 0. If M T N = 0, i.e., the projection is orthogonal, then the projections reduce to P = M(M T M) 1 M T and Q = N(N T N) 1 N T. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 11/42

12 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 12/42

13 J-spectral factorization for the full set Via similarity transformations with two matrices Via similarity transformations with one matrix Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 13/42

14 Triangular forms of H p and H z Assume that a para-hermitian matrix Λ as given in (2) is minimal and has no poles or zeros on the jω-axis including. There always exist nonsingular matrices p and z (e.g. via Schur decomposition) such that? 0 (3) 1 p H p p =? A + and (4) 1 z H z z = A? 0?, where A + is antistable and A is stable (A + and A have the same dimension). Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 14/42

15 With two matrices z and p Lemma 1 Λ admits a J p,q -spectral factorization for some unique J p,q (where p is the number of the positive eigenvalues of D and q is the number of the negative eigenvalues of D) iff I 0 (5) = z p 0 is nonsingular. If this condition is satisfied, then a J spectral factor is formulated as (6) W = I 0 J p,q D W C Λ 1 H p I 0 I 0 I I 0 1 B Λ D W where D W is a nonsingular solution of D W J p,qd W = D., Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 15/42

16 Proof: The existence condition Formulae (3) and (4) mean that 0 0 H p p = I p I A +, and I I H z z = 0 z A 0. 0 I Hence, p and I z span the antistable 0 eigenspace M of H p and the stable eigenspace M of H z, respectively. It is well known that there exists a J-spectral factorization iff M M = {0}. This is equivalent to that the given in (5) is nonsingular. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 16/42

17 Derivation of the factor When the condition holds, there exists a projection P onto M along M. The projection matrix P is I 0 P = With this projection formula, a J-spectral factor can be found as PH p P PB Λ W = J p,q DW C. ΛP D W Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 17/42

18 Simplification of the factor Apply a similarity transformation with, then W = 1 PH p I J p,q D W C Λ I PB Λ D W. After removing the unobservable states by deleting the second row and the second column, W becomes W = I 0 1 PH p I 0 J p,q D W C Λ I 0 I 0 1 PB Λ D W. Since I 0 1 P = I 0 1, W can be further simplified as given in (6). Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 18/42

19 With one common matrix In general, z p. However, these two can be the same. Theorem 2 Λ admits a J-spectral factorization if and only if there exists a nonsingular matrix such that A p 1 0 A H p =? A p, 1 z H z =? + 0 A z + where A z and Ap are stable, and A z + and Ap + are antistable. When this condition is satisfied, a J-spectral factor W is given in (6). Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 19/42

20 Proof Sufficiency. It is obvious according to Lemma 1. In this case, z = p =. Necessity. If there exists a J-spectral factorization then the given in (5) is nonsingular. This does satisfy the two formulae in Theorem 2. Since this is in the same form as that in Lemma 1, the J-spectral factor is the same as that in (6). Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 20/42

21 The bistability of W The A-matrix of W is and that of W 1 is They are all stable. I 0 1 H p I 0 I 0 1 H z I 0 = A p = A z. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 21/42

22 Interpretation This theorem says that a J-spectral factorization exists if and only if there exists a common similarity transformation to transform H p (H z, resp.) into a 2 2 lower (upper, resp.) triangular block matrix with the (1, 1)-block including all the stable modes of H p (H z, resp.). Once the similarity transformation is done, a J- spectral factor can be formulated according to (6). If there is no such a similarity transformation, then there is no J-spectral factorization. Since the similarity transformation corresponds to the change of state variables, this theorem might provide a way to further understand the structure of H control. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 22/42

23 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 23/42

24 J-spectral factorization for a smaller subset Λ = A R B E A C C B D In this case, H p = A R E A, H z = A R E A B C D 1 C B.= A z E z R z A z. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 24/42

25 Theorem 3 For a Λ characterized in Theorem 1, assume that: (i) (E, A) is detectable and E is sign definite; (ii) (A z, R z ) is stabilizable and R z is sign definite. Then the two ARE I L o H p L o = 0, L c I H z I = 0 I L c always have unique symmetric solutions L o and L c. Λ(s) has a J p,q -spectral factorization iff det(i L o L c ) 0. If so, A + L W = o E B + L o C J p,q D W (B L c + C)(I L o L c ) 1 D W where D W is nonsingular and D W J p,qd W = D., Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 25/42

26 Proof In this case, z = I 0 L c I, p = I L o 0 I and = I L c L o I. According to Lemma 1, there exists a J-spectral factorization iff is nonsingular, i.e., det(i L o L c ) 0. Substitute into (6) and apply a similarity transformation with (I L o L c ) 1, then W = = I L o H p I L c (I L o L c ) 1 J p,q D W (C + B L c )(I L o L c ) 1 I L o H p I 0 J p,q D W (C + B L c )(I L o L c ) 1 B + L o C D W B + L o C D W, where the first ARE in the theorem was used. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 26/42

27 The A-matrix of W 1 A + L o E + (B + L o C )D 1 W J p,q 1 D W (B L c + C)(I L o L c ) 1 = I L o H z I (I L o L c ) 1 L c (I L o L c ) 1 I L o H z I L c = (I L o L c ) 1 I L o H z I L c + L o L c I H z I L c = I 0 H z I L c, where the means similar to and the second ARE in Theorem 3 was used. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 27/42

28 The dual case Theorem 4 For a Λ characterized in Theorem 1, assume that: (i) (A, R) is stabilizable and R is sign definite; (ii) (E z, A z ) is detectable and E z is sign definite. Then the two ARE L c I H p I = 0, I L o H z L o L c I = 0 always have unique symmetric solutions L c and L o. In this case, Λ(s) has a J p,q -spectral factorization iff det(i L o L c ) 0. If so, W(s) = A + RL c (I L o L c ) 1 (B + L o C )D W J p,q, B L c + C D W where D W is nonsingular and D W J p,q D W = D. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 28/42

29 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 29/42

30 Appl.: Λ = G JG with a stable G Here, G(s) = A B does not have poles or zeros on the jω-axis including and J = J C D is a signature matrix. The realization of Λ = G JG is Λ = A 0 B C JC A C JD D JC B D JD. In this case, H p = A 0 C JC A H z = A 0 C JC A B C JD (D JD) 1 D JC B. Since A is stable, there is no similarity transformation needed to bring H p into the triangular form. Hence, p can be chosen as p = I 0. 0 I Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 30/42

31 Since p = I 0 0 I, a in the following form = X 1 0 X 2 I is possible to bring H z into the upper triangular form. According to Theorem 2, Λ admits a J- spectral factorization iff is non-singular. This is equivalent to that X 1 is non-singular. When X 1 is non-singular, then a further similarity transformation with X can be done. 0 I This gives another, with X = X 2 X 1 1, as = I 0. Substitute this into Theorem X I 2, then I 0 X I H z I 0 X I = Az? 0 A z +. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 31/42

32 This means X I H z I X = 0, A z = I 0 H z I X is stable. Hence, Λ admits a J-spectral factorization iff the above ARE has a stabilizing solution X. When this condition holds, the J-spectral factor can be easily found, according to Theorem 2, as W = = I 0 J p,q D W I 0 H p I 0 I X I X I 0 D JC B I 0 I X I 0 A B, J p,q D W (D JC + B X) D W I 0 I 0 X I D W B C JD where D W is nonsingular and D W J p,qd W = D JD. This result has been well documented in the literature. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 32/42

33 Outline Notations and definitions Regular para-hermitian matrices Properties of projections J-spectral factorization for the full set J-spectral factorization for a smaller set Applications Numerical examples Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 33/42

34 Numerical example I Λ(s) = A minimal realization of Λ is Λ = 0 s 1 s+1 s+1 s Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 34/42

35 H p = , H z = = Apparently, there does not exist a common similarity transformation to make the (1, 1)-elements of H p and H z all stable. Hence, this Λ does not admit a J-spectral factorization. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 35/42

36 Numerical example II Λ(s) = s2 4 s s 2 1 s 2 4 A minimal realization of Λ is Λ = Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 36/42

37 H p = and H z = By doing similarity transformations, it is easy to transform H p and H z into a lower (upper, resp.) triangular matrix with the first two diagonal elements being negative. In order to bring H p into a lower triangular matrix, two steps are used. The first step is to bring it into an upper triangular matrix and the second step is to group the stable modes, as shown below. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 37/42

38 with p1 = with p2 = This is a lower triangular matrix, to which H p is similarly transformed with p = p1 p2. In general, a third step is needed to make the non-zero elements, if any, in the upper-right area to 0. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 38/42

39 Similarly, H z is transformed into an upper triangular matrix 1 z H z z = , with z = can be obtained by combining the first two columns of z and the last two columns of p as: = z = , p = This is nonsingular and, hence, there exists a J-spectral factorization. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 39/42

40 The D-matrix D = This gives J p,q = of Λ has one positive eigenvalue and a negative eigenvalue and D W = such that D W J p,qd W = D. According to (6), a J-spectral factor of Λ is found as W = , which can be described in transfer matrix as W(s) = s+1 0 s+2 s+2 0 s+1. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 40/42

41 One step further It is worth noting that a corresponding J-spectral factor for Λ(s) = Λ(s) = s 2 4 s s2 1 s 2 4 with J p,q = is W(s) = W(s) = s+2 0 s+1 0 s+1 s+2 because J p,q = J p,q. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 41/42

42 Summary A class of regular invertible para-hermitian transfer matrices is characterized and then the J-spectral factorization problem is revisited using the elementary tool, similarity transformations. A transfer matrix Λ in this class admits a J-spectral factorization if and only if there exists a common nonsingular matrix to similarly transform the A-matrices of Λ and Λ 1, resp., into 2 2 lower (upper, resp.) triangular block matrices with the (1,1)-block including all the stable modes of Λ (Λ 1, resp.). One possible way to obtain this matrix involves two steps: (i) separately bringing the A-matrices into triangular forms and (ii) combining the corresponding columns of the matrices used in the two similarity transformations. Another possible way is to simultaneously triangularize the two A-matrices. However, this is not a trivial problem. The resulting J-spectral factor is formulated in terms of the original realization of Λ. When a transfer matrix meets additional conditions, there exists a J-spectral factorization if and only if a coupling condition related to the stabilizing solutions of two ARE holds. Two numerical examples are given to illustrate the theory, which is the key to solve the delay-type Nehari problem. Q.-C. ZHONG: J-SPECTRAL FACTORIZATION OF REGULAR PARA-HERMITIAN TRANSFER MATRICES p. 42/42

43 Qing-Chang Zhong Robust Control of Time-Delay Systems Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo