Solutions to generalized Sylvester matrix equation by Schur decomposition

Transcription

1 International Journal of Systems Science Vol 8, No, May 007, 9 7 Solutions to generalized Sylvester matrix equation by Schur decomposition BIN ZHOU* and GUANG-REN DUAN Center for Control Systems and Guidance Technology, Harbin Institute of Technology, PO Box 41, Harbin 10001, P R China (Received 14 December 00; in final form 8 November 00) This note deals with the problem of solving the generalized Sylvester matrix equation AV EVF BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation The primary feature of this approach is that the matrix F is firstly transformed into triangular form by Schur decomposition and then unimodular transformation or singular value decomposition are employed The results can be easily extended to second order case and high order case and can provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory Keywords: Generalized Sylvester matrix equations; General parametric solutions; Unimodular transformation; Schur decomposition; Singular value decomposition 1 Introduction This note considers the generalized Sylvester matrix equation AV EVF BW, where A, E R nn, B R nr and F R pp are given matrices, while V R np and W R rp are matrices to be determined In the special case of E I, the equation reduces to AV VF BW: The equation () is closely related with many problems in conventional linear control systems theory, such as pole/eigenstructure assignment design (Gavin and Bhattacharya 198, Kwon and Youn 1987), Luenberger-type observer design (Luenberger 194, Tsui 1988), robust fault detection *Corresponding author zhoubinhit@1com ð1þ ðþ (Park and Rizzoni 1994, Duan and Patton 001), and so on, and has been investigated by several researchers (Tsui 1987, 199, Duan 199, 199, 199, Zhou and Duan 00) When dealing with eigenstructure assignment, observer design and model reference control for descriptor linear systems, the more generalized Sylvester matrix equation (1), with E being usually singular, is encountered In solving the generalized Sylvester matrix (1), finding the complete parametric solutions, that is, parametric solutions consisting of the maximum number of free parameters, is of extreme importance, since many problems, such as robustness in control system design, require full use of the design freedom For (1) with F being in Jordan form, complete parametric solutions have been proposed in Duan (199, 199) Under the R-controllability of the matrix triple ðe, A, BÞ, Duan (199) has given a complete and explicit solution which uses the right coprime factorization of the input-state transfer function ðse AÞ 1 B, while Duan (199) proposed a complete parametric solution which is not in a direct, explicit form but in a recursive form These existing solutions are directly applicable in problems like eigenstructure assignment since in such International Journal of Systems Science ISSN print/issn online ß 007 Taylor & Francis DOI: /

2 70 B Zhou and G-R Duan problems the matrix F is originally required to be in Jordan form However, in some other problems the matrix F is an arbitrary square matrix Although we can apply some similarity transformation to transform F into a Jordan form, such a process is not desirable because it not only gives additional computational load but also may result in numerically unreliable solutions (Wilkinson 199) It is well-known that, for any square matrix F, there exists an unitary matrix U C pp such that F USU H, where S C pp is an upper triangular matrix in the form of 0 s p ðþ s 1 s 1 s 1p 1 s 1p 0 s s s p S, ð4þ 0 4 s p 1 s p 1p 7 and such a process is numerically reliable (Wilkinson 19) So, in this note, without loss of generality, we consider the generalized Sylvester matrix equation (1) by originally assuming that the matrix F is already in such triangular form (4) Main results With the special structure of matrix F, the generalized Sylvester matrix equation (1) can be rewritten as the following series of linear equations: A s i E B þ Xi 1 s ji Ev j, i 1; ; ; p, ðþ there exist two unimodular matrices Ps ðþ and Qs ðþ such that Ps ðþ A se Let Qs ðþbe partitioned as Qs ðþ B N # 1ðÞ s N ðþ s D 1 ðþ s D ðþ s Qs ðþ 0 I : ðþ, N 1 ðþr s nr ½Š, s then we have the following theorem: Theorem 1: Let ðe, A, BÞ be R-controllable and F S be in the form of (4) Then the complete parametric # N # 1ðs i Þ þ Xi 1 D 1 ðs i Þ s ji ð7þ # N ðs i ÞPs ð i ÞE v j, v 0 0, D ðs i ÞPs ð i ÞE ð8þ where C r, i 1,,, p, are some arbitrary vectors Proof: We firstly show that the vectors,, i 1,,, p, satisfying () can be expressed in the form of (8) Note that () is equivalent to A s i E B Xi 1 s ji Ev j, i 1; ; ; p: Premultiplying (9) by Ps ð i Þ and using (), produces 0 I Q 1 ðs i Þ Ps ð i Þ Xi 1 s ji Ev j : ð9þ ð10þ with,, i 1,,, p, in the form of V v 1 v v p, W w1 w w p : If we denote Q 1 ðs i Þ l i, ð11þ 1 Solutions based on unimodular transformation Under the R-controllability of matrix triple ðe, A, BÞ, ie (Duan 199), rank A se B n, 8s C, then the equation (10) is equivalent to l i Ps ð i Þ Xi 1 s ji Ev j : ð1þ

3 Solutions to generalized Sylvester matrix equation by Schur decomposition 71 Substituting (1) into (11), yields # # Qs ð i Þ l i Qs ð i Þ Ps ð i Þ Xi s ji Ev j N # 1ðs i Þ N ðs i Þ D 1 ðs i Þ D ðs i Þ Ps ð i Þ Xi s ji Ev j N # 1ðs i Þ þ N # ðs i ÞPs ð i ÞE X i 1 s ji v j : D 1 ðs i Þ D ðs i ÞPs ð i ÞE Secondly, we show that the vectors,, i 1,,, p, given by (8) satisfy the generalized Sylvester matrix equation () This can be validated by direct substitution œ It follows from the above theorem that this type of solution is given in iterative way In the following, we will give a direct formula Firstly, we introduce the so-called path function pathði, j, Rs ðþþ with respect to polynomial Rs ðþ: For example and pathð1, 4, Rs ðþþ s 14 I þ s 1 s 4 Rs ð Þþs 1 s 4 Rs ð Þ þ s 1 s s 4 Rs ð ÞRs ð Þ, where pathði, j, Rs ðþþ denotes a path from i to j with respect to RðÞN s ðþps s ðþe, and C r, i 1,,, p, are some arbitrary vectors The proof of this theorem can be done by direct calculation Remark 1: The generalized Sylvester matrix equation () is a special form of (1), thus complete parametric solutions to () can also be easily obtained Theorem 1 can be easily extended to the secondorder and high-order case Consider the following second-order generalized Sylvester matrix equation MVF þ DVF þ KV BW, ð1þ where M, D, K R nn, B R nr, F C pp are given and the matrix F is in the form of (4) and F is in the form of s 1 p 1 p 1p 1 p 1p 0 s p p p F : 0 4 s p 1 p 7 p 1p 0 s p When dealing with certain control problems, such as pole assignment (Rincon 199, Chu and Datta 199), eigenstructure assignment (Inman and Kress 1999, Duan and Liu 00) and observer design, of the second-order linear system, such generalized Sylvester matrix equation (1) is often encountered With the special structure of matrix F, we have pathð,, Rs ðþþ s I þ s s Rs ð Þþs 4 s 4 Rs ð 4 Þ þ s s Rs ð Þþs s 4 s 4 Rs ð 4 ÞRs ð Þ þ s s s Rs ð ÞRs ð Þþs 4 s 4 s Rs ð ÞRs ð 4 Þ þ s s 4 s 4 s Rs ð ÞRs ð 4 ÞRs ð Þ: Then we have the following result Theorem : Let ðe, A, BÞ be R-controllable and F S be in the form of (4) Then the complete parametric N 1ðs i Þ D 1 ðs i Þ þ N ðs i ÞPs i D ðs i ÞPs ð i ÞE ð ÞE X i 1 pathði, j, Rs ðþþn 1 s j fj, Ms i þ Ds i þ K vi B Xi 1 s ji D þ p ji M vj : Also, when ðm, D, K, BÞ is R-controllable, ie, rank Ms þ Ds þ K B n, 8s C, there exist two unimodular matrices Ps ðþand Qs ðþsuch that Ps ðþ Ms þ Ds þ K B Qs ðþ 0 I : Let Qs ðþbe partitioned as (7), we have Theorem : Let ðm, D, K, BÞ be R-controllable and F S be in the form of (4) Then the complete parametric

4 7 B Zhou and G-R Duan # # # N 1 ðs i Þ N ðs i ÞPs ð i Þ X i 1 s ji D þ p ji M vj, D 1 ðs i Þ D ðs i ÞPs ð i Þ where v 0 0, and C r, i 1,,, p, are some arbitrary vectors However, Theorem cannot be easily extended to this case Solutions to the high-order case in the form of X m i0 A i VF i BW, ð14þ corresponding to Theorem can also be easily obtained and thus are omitted here Solutions based on singular value decomposition It follows from the above subsection that the polynomial matrices Ps ðþ and Qs ðþ satisfying () is pivotal for constructing the parametric solutions to this type of generalized Sylvester matrix equations However, such pair of polynomial matrices Ps ðþ and Qs ðþ are generally difficult to obtain since they involve symbolic operations Indeed, the unimodular transformation can be replaced by singular value decomposition which is accepted numerically reliable When the R-controllability of the matrix triple ðe, A, BÞ is guaranteed, for arbitrary eigenvalue of matrix F, ie, s i, i 1,,, p, we have rank½ A s i E BŠ n: Then there exist two series of unitary matrices P i and Q i, i 1,,, p, such that P i A s i E B Q i 0 i, i 1,,, p, ð1þ Theorem 4: Let ðe, A, BÞ be R-controllable and F S be in the form of (4) Then the complete parametric # # N i1 þ Xi 1 D i1 s ji N i 1 # i P i E v j, v 0 0, D i 1 i P i E where C r, i 1,,, p, are some arbitrary vectors Remark : Solutions to the second-order generalized Sylvester matrix equation (1) corresponding to Theorem 4 can also be easily obtained Remark : It is well-known that the numbers of computational operations of SVD for a matrix A R nm, m > n and Schur decomposition for a matrix B R nn is about 4m n þ n and n respectively: Thus the total number of computational operations is about Denoting flop p 4ðn þ rþ n þ n þ p : Rs ð i Þ R i N i 1 i P i E, and using the path function defined in the above subsection, we have the following result which is parallel to Theorem Theorem : Let ðe, A, BÞ be R-controllable and F S be in the form of (4) Then the complete parametric where i is an invertible diagonal matrix Partition Q i as # N i1 N i Q i, N i1 C nr, i 1,,, p: D i1 D i Then (1) is equivalent to N i1 N i 1 # i P i A se B D i1 D i 1 i i 1,,, p: 0 I, ð1þ Comparing (1) with (), we have the following result parallel to Theorem 1 N i1 D i1 þ N # i 1 i P i E X i 1 D i 1 i P i E pathði, j, Rs ðþþn j1 f j, where C r, i 1,,, p, are some arbitrary vectors Real solutions by using singular value decomposition In some applications one needs real matrices V and W So in this subsection we will implement real arithmetic It is well-known that one can use orthogonal reduction on the matrix F to upper quasitriangular

5 Solutions to generalized Sylvester matrix equation by Schur decomposition 7 Schur form, ie, S 0 S 01 S 0q 1 S 0q 0 S 1 S 1 S 1q S, 0 S q 1 S 4 q 1q 7 0 S q where S 0 R mm is in the form of (4), S i R, i 1,,, q, has a pair of conjunctive eigenvalues ðe i, e i Þ, S j, k R, j 1,,, q 1, k j þ, j þ,, q, and S 0,k R p, k 1,,, q Obviously, we have m þ q p: Corresponding to the structure of S, we decompose V and W as 8 V V 0 V 1 V q, Vi 1, >< i 1,,, q W W 0 W 1 W q, Wi 1, >: i 1,,, q: With this special structure of S, the equation (1) with F S becomes and AV 0 EV 0 S 0 BW 0, AV i EV i S i BW i þ E Xi 1 V j S ji, i 1,,, q: j0 ð17þ ð18þ Note that equation (17) is also in the form of (1) and its solution can be gotten by applying Theorems 1,, 4 and where the free parameters are real Thus V 0 and W 0 are real matrices When V 0 and W 0 are obtained, we consider the first equation in (18), ie, AV 1 EV 1 S 1 BW 1 þ EV 0 S 01 : ð19þ Since S 1 has a pair of conjunctive eigenvalues, we have # e 1 0 S 1 U 1 U e, U 1 u 1 u 1 C : ð0þ 1 Substituting (0) into (19) and simplifying produces e 1 0 A v 11 v 1 U1 E v 11 v 1 U1 0 e 1 B w 11 w 1 U1 þ D 1 : ð1þ where D 1 d 1 d1 EV0 S 01 U 1 : Denote ( 11 1 v11 v 1 U1! 11! 1 w11 w 1 U1, then (1) becomes A 11 e 1 E 11 B! 11 þ d 1 A 1 e 1 E 1 B! 1 þ d 1 : ðþ Obviously, ð 11,! 11 Þ satisfies the first equation in () if and only if ð 11,! 11 Þsatisfies the second equation in () So in the following we need only to consider the first equation and then set ð 1,! 1 Þ ð 11,! 11 Þ: ðþ The first equation in equation () can be equivalently rewritten as 11 A e 1 E B d 1 :! 11 ð4þ Note that equation (4) is in the form of (9) Similar to the proof of Theorem 1, we can obtain the following result Theorem : Let ðe, A, BÞ be R-controllable Then the complete parametric solutions to (4) are given by 11! 11 N 1ðe 1 Þ f 1 þ N ðe 1 ÞPe ð 1 Þ D 1 ðe 1 Þ D ðe 1 ÞPe ð 1 Þ d 1 : ðþ where f 1 R r is an arbitrary vector Using this result, we have: Corollary 7: Let ðe, A, BÞ be R-controllable Then the complete parametric solutions to (1) are given by ( v 11 v U 1 1 w 11 w 1!11! 1 U 1 1, where ð 11,! 11 Þ is given in () and ð 1,! 1 Þ is determined by () Furthermore, the vectors j, j, i 1,, j 1,, are all real

6 74 B Zhou and G-R Duan Now V 0 and V 1 are obtained and are all real, we consider the second equation in (18), ie, AV EV S BW þ D, where D EV 0 S 0 þ EV 1 S 1 : Similar to the above, we can get real solutions V and W : Repeat the above process for p times, and we can get real solutions V i, W i, i 1,,, p: Remark 4: Theorem is obtained corresponding to Theorem 1 However, for numerical stability consideration, results corresponding to Theorem 4 by using singular value decomposition can also be easily obtained Remark : The above method can also be applied on the second-order and high order generalized Sylvester matrix equations (1) and (14) Example Consider a generalized Sylvester matrix equation in the form of (1) with the parameters (Duan 199) E , A 1 7 4, 0 1 B : 0 1 It is easy to verify that ðe, A, BÞ is R-controllable The polynomial matrices Ps ðþ and Qs ðþ satisfying () are given by Qs ðþ 1 s , Ps ðþi : 4 s 1 s Without loss of generality, we assume F s 1 s 1, 0 s then the complete parametric solutions to the generalized Sylvester matrix equation (1) can be expressed as v 1 w 1 v w 1 0 s f 1, 4 s s f þ s 1 f 1 : 4 s s 1 s where f 1, f R are arbitrary vectors Acknowledgment This work is supported by the Chinese Outstanding Youth Foundation under Grant No 9908 and Program for Changjiang Scholars and Innovative Research Team in University References NF Almuthairi and S Bingulac, On coprime factorization and minimal realization of transfer-function matrices using pseudoobservability concept, Int J Sys Sci,, pp , 1994 TGJ Beelen and GW Veltkamp, Numerical computation of a coprime factorization of a transfer-function matrix, Sys Contr Lett, 9, pp 81 88, 1987 J Chen, R Patton and H Zhang, Design unknown input observers and robust fault detection filters, Int J Contr,, pp 8 10, 199 EK Chu and BN Datta, Numerically robust pole assignment for second-order systems, Int J Contr, 4, pp , 199 GR Duan, Solution to matrix equation AV þ BW EV F and eigenstructure assignment for descriptor systems, Automatica, 8, pp 9 4, 199 GR Duan, Solutions to matrix equation AV þ BW V F and their application to eigenstructure assignment in linear systems, IEEE Trans Autom Contr, 8, pp 7 80, 199 GR Duan, On the solution to Sylvester matrix equation AV þ BW EV F, IEEE Trans Autom Contr, AC-41, pp 1 14, 199 GR Duan and GP Liu, Complete parametric approach for eigenstructure assignment in a class of second-order linear systems, Automatica, 8, pp 7 79, 00 GR Duan and RJ Patton, Robust fault detection using Luenberger-type unknown input obsververs a parametric approach, Int J Sys Sci,, pp 40, 001 KR Gavin and SP Bhattacharyya, Robust and well-conditioned eigenstructure assignment via Sylvester s equation, Proc Amer Contr Conf, 198 BH Kwon and MJ Youn, Eigenvalue-generalized eigenvector assignment by output feedback, IEEE Trans Autom Contr, AC-, pp , 1987 DJ Inman and A Kress, Eigenstructure assignment algorithm for second-order systems, J Guid, Contr Dyn,, pp 79 71, 1999

7 Solutions to generalized Sylvester matrix equation by Schur decomposition 7 Y Kim and HS Kim, Eigenstructure assignment algorithm for mechanical second-order systems, J Guid, Contr Dyn,, pp 79 71, 1999 DG Luenberger, Observing the state of a linear system, IEEE Trans Mil, Electr, MI-8, pp 74 80, 194 DG Luenberger, An introduction to observers, IEEE Trans Autom Contr, 1, pp 9 0, 1971 J Park and G Rizzoni, An eigenstructure assignment algorithm for the design of fault detection filters, IEEE Trans Autom Contr, 9, pp 11 14, 1994 F Rincon, Feedback stabilization of second-order models, PhD dissertation, Northern Illinois University, De Kalb, Illinois, USA, 199 CC Tsui, A complete analytical solution to the equation TA FT LC and its applications, IEEE Trans Autom Contr, AC-, pp , 1987 CC Tsui, New approach to robust observer design, Int J Contr, 47, pp 74 71, 1988 CC Tsui, On the solution to matrix equation TA FT LC and its applications, SIAM J Matr Anal App, 14, pp 4 44, 199 JH Wilkinson, Algebraic Eigenvalue Problem, New York: Oxford University Press, 19 B Zhou and GR Duan, An explicit solution to the matrix equation AX XF BY, Lin Alg App, 40, pp 4, 00 Bin Zhou was born in HuBei Province, China, in 1981 He received the Bachelor s degree from the Department of Control Science and Engineering at Harbin Institute of Technology, Harbin, China, in 004 He is now a graduate student in the Center for Control Systems and Guidance Technology in Harbin Institute of Technology His current research interests include linear systems theory and constrained control systems Guang-Ren Duan was born in Heilongjiang Province, 19 He received his BSc degree in Applied Mathematics, and both his MSc and PhD degrees in Control Systems Theory He is currently the Director of the Center for Control Systems and Guidance Technology at Harbin Institute of Technology His main research interests include robust control, eigenstructure assignment and descriptor systems