Solutions to the generalized Sylvester matrix equations by a singular value decomposition

Journal of Control Theory Applications 2007 5 (4) 397 403 DOI 101007/s11768-006-6113-0 Solutions to the generalized Sylvester matrix equations by a singular value decomposition Bin ZHOU Guangren DUAN (Center for Control Theory Guidance Technology Harbin Institute of Technology Harbin Heilongjiang 150001 China) Abstract: In this paper solutions to the generalized Sylvester matrix equations AX XF BY MXN X T Y with A M R n n B T R n r F N R p p the matrices N F being in companion form are established by a singular value decomposition of a matrix with dimensions n (n + pr) The algorithm proposed in this paper for the euqation AX XF BY does not require the controllability of matrix pair (A B) the restriction that A F don t have common eigenvalues Since singular value decomposition is adopted the algorithm is numerically stable may provide great convenience to the computation of the solution to these equations can perform important functions in many design problems in control systems theory Keywords: Generalize Sylvester matrix equations; General solutions; Companion matrix; Singular value decomposition 1 Symbols notations In this paper we use B T rank(b) to denote the transpose the rank of matrix B respectively b ij is the i-th row j-th column of matrix B I p is the p p identity matrix 0 will be used as a r s matrix when the dimensions are evident from the context The symbol is to denote the Kronecker product We use A F r p which means that A is an r p matrix in the field F We use cola i q ip to denote a matrix in the form of cola i q ip A p A p+1 A q 1 A q use rowa i q ip to denote rowa i q ip colat i q ip T Further let A R n n B R n r we define the so-called Krylov matrix with matrix pair (A B) as follows: 2 Introduction Q c (A B k) cola i B k 1 The general solution to the generalized Sylvester matrix equation AX XF BY (1) A R n n B R n r F R p p are known is closely related with many problems in linear control systems theory such as eigenvalue assignment 1 2 observer design 3 eigenstructure assignment design 4 5 constrained control 6 etc has been studied by many authors (see 3 5 7 the references therein) When the matrix F is in Jordan form an attractive analytical restriction-free solution with explicit freedom is presented in 3 To obtain this solution one needs to carry out an orthonormal transformation compute a matrix inverse solve a series of linear equation groups Reference 5 proposes two solutions to the matrix equation also for the case that the matrix F is in Jordan form The first one is in an iterative form while the second is in an explicit parametric form To obtain the explicit solution proposed in 5 one needs to carry out a right coprime factorization of (si A) 1 B (when the eigenvalues of the Jordan matrix F are undetermined) or a series of singular value decomposition (when the eigenvalues of F are known) Generalization of this explicit solution to a more general case is considered in 4 7 When the matrix F is in companion form a very neat elegant general complete parametric solution to the generalized Sylvester matrix equation (1) is proposed in 8 The solution is expressed in terms of the controllability matrix of the matrix pair (A B) a symmetric operator matrix a parametric matrix in the Hankel matrix form Such a result can provide great convenience for many analysis design problems associated with the matrix equation (1) However this proposed solution has a disadvantage that when A F have common eigenvalues the given solution may be not complete This is not convenient to use in some problems Received 16 June 2006; revised 22 November 2006 This work was supported by the Chinese Outsting Youth Foundation (No69925308) Program for Changjiang Scholars Innovative Research Team in University

398 B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 such as pole assignment eigenstructure assignment In this note we also consider the generalized Sylvester matrix (1) with F being a companion matrix Different from the method proposed in 8 in this note when the matrix F is in the companion form the generalized Sylvester matrix equation (1) is firstly converted into a matrix equation in vector form as à x 0 (2) à Rn (n+pr) Such a linear equation can be efficiently solved by a singular value decomposition of matrix à With some relations between the original matrix variables X Y the new vector variable x solutions to the original generalized Sylvester matrix equation can be immediately obtained It is worth pointing out that the generalized Sylvester matrix equation (1) can always be converted into the vector form as (2) with à I p A F T I n I p B x rowrowx i p i1 rowy i p i1 However in this case à Rpn (pn+pr) the dimension pn (pn+pr) is obviously higher than n (n+pr) which may cause some numerical problems The generalized Sylvester matrix equation AX EXF BY (3) is closely related with many synthesis problems in descriptor linear systems theory In solving the generalized Sylvester matrix (3) it is extremely important to find the complete parametric solutions that is parametric solutions consisting of the maximum number of free parameters since many problems such as robustness in control system design require full use of the design freedom For (3) with F being in Jordan form 4 has proposed a complete parametric solution However this solution is not in a direct explicit form but in a recursive form Also when the matrix F is assumed in Jordan form the matrix triple (E A B) is assumed to be R-controllable 7 has given a complete explicit solution which uses the right coprime factorization of the input-state transfer function (se A) 1 B When F is an arbitrary matrix 9 gives a complete explicit solution also involving the right coprime factorization of the input-state transfer function (se A) 1 B such results are also extended to a more general case in 10 In this paper we consider this problem in another way Now we give the following lemma Lemma 1 The generalized Sylvester matrix equation (3) A E R n n B R n r F R p p are known the matrix pair (E A) is regular is equivalent to the generalized Sylvester matrix equation MXN X T Y (4) with M (γe A) 1 E N γi F (5) T (γe A) 1 B γ is an arbitrary scalar such that (γe A) is nonsingular Proof Since the matrix pair (E A) is regular there exists a scalar γ such that (γe A) is nonsingular Premultiplying the equation (3) by (γe A) 1 produces (γe A) 1 AX (γe A) 1 EXF (γe A) 1 BY (6) Let M (γe A) 1 E note that γm (γe A) 1 A γ (γe A) 1 E (γe A) 1 A (γe A) 1 (γe A) I we have (γe A) 1 A γm I So equation (6) is equivalent to (γm I) X MXF T Y or MX (γi F ) X T Y Let N γi F then the above equation is reduced to (4) The above Lemma 1 shows that the solutions to the generalized Sylvester matrix equation (3) can be immediately obtained while the solutions to the generalized Sylvester matrix equation (4) are gotten Similar to the generalized Sylvester matrix equation (1) the generalized Sylvester matrix equation (4) can also be solved by the same process described above As shown in 8 the assumption that the matrix F is in companion form does not lose generality However the companion matrix may have several forms such as 0 0 β 0 1 0 β 1 C 1 (β) 0 1 β p 1 β p 1 1 0 C 3 (β) β 1 0 1 β 0 0 0 C 2 (β) C1 T (β) C 4 (β) C3 T (β) The following lemma shows that they are similar to some special transformation matrix

B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 399 Lemma 2 The four types of companion matrices are similar to each other ie C 3 (β) EC 1 (β) E C 2 (β) S1 1 (β) C 1 (β) S 1 (β) C 4 (β) S 1 (β) E 1 C 1 (β) S 1 (β) E E col e i 1 ip with e i the i-th column of the identity matrix I p β 1 I r β 2 I r β p 1 I r I r β 2 I r β 3 I r I r 0 S r (β) (7) β p 1 I r I r 0 I r 0 0 0 This above lemma shows that we need only to consider the case F C 1 (β) (8) 3 The main results 31 The generalized Sylvester matrix equation AX XF BY In this section we consider the generalized Sylvester matrix equation (1) Theorem 1 The generalized Sylvester matrix equation (1) is equivalent to { rowxi p i2 rowai p 1 i1 x 1 Π(A B)rowy i p 1 i1 β(a)x 1 cola i B p 1 S r(β)rowy i p i1 (9) S r (β) is defined as (7) B 0 0 0 AB B 0 Π (A B) (10) A p 3 B B 0 A p 2 B A p 3 B AB B Proof Note that the generalized Sylvester matrix equation (1) is equivalent to the following series of equations: Ax 1 x 2 By 1 Ax 2 x 3 By 2 Ax p 1 x p By p 1 Ax p + p β i 1 x i By p i1 (11) which are also equivalent to x 2 Ax 1 By 1 x 3 A 2 x 1 1 A i By 2 i x p A p 1 x 1 p 2 A i By p 1 i Ax p + p β i 1 x i By p i1 (12) Substituting the first p 1 equations in (12) into the last equation in (12) yields A(A p 1 x 1 p 2 A i By p 1 i ) + p β i 1 x i i1 A p x 1 p 2 A i+1 By p 1 i + β 0 x 1 + p β i 1 A i 1 x 1 i 2 A j By i 1 j i2 j0 A p x 1 + p 1 β i A i x 1 p 1 A i By p i p 2 i1 β p 1 A i By p i 1 1 β 2 A i By 2 i 0 β 1 A i By 1 i By p which is also equivalent to β(a)x 1 p 1 A i By p i + p 2 β p 1 A i By p i 1 + + 1 β 2 A i By 2 i + 0 β 1 A i By 1 i cola i B p 1 S r(β)coly i p This is the second equation in (9) The first equation in (9) is equivalent to the first p 1 equations in (11) Note that the above processes are invertible The proof is then completed Lemma 3 The degree of freedom in the solution to the generalized Sylvester matrix equation (1) is ϖ pr + n ω (13) ω rankβ(a) Q c (A B p) Proof It follows from the proof of Theorem 1 that the generalized Sylvester matrix equation (1) is equivalent to the second equation in (9) first p 1 equations in (11) ie (1) is equivalent to Ax 1 x 2 By 1 Ax 2 x 3 By 2 Ax p 1 x p By p 1 β(a)x 1 cola i B p 1 S r(β)rowy i p i1

400 B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 which can be rewritten as rowxi p i1 Ξ 1 Ξ 2 0 rowy i p i1 β (A) 0 0 0 0 A I 0 0 B 0 Ξ 1 0 A I B 0 0 0 A I 0 0 Q c (A B p) S r (β) 0 0 0 0 Ξ 2 0 0 0 0 B 0 Through some simplifications we have β (A) Qc (A B p) S r (β) 0 rankξ rank 0 0 I (p 1)n (p 1)n + ω So according to linear equation theory the degree of freedom in the solution is ϖ np + pr ω (p 1)n pr + n ω The following corollary is immediately obtained according to the above lemma Corollary 1 The degree of freedom in the solution to the generalized Sylvester matrix (1) is rp if one of the following statements holds: 1) The matrix A F don t have common eigenvalues 2) The matrix pair (A B) is controllable p 1 n r Note that the degree of freedom in the second equation in (9) is ϖ pr + n rankβ(a) Q c (A B p) (14) With Lemma 3 (14) we find that the degree of freedom in the solution to (1) is equal to the degree of freedom in the solution to the second equation in (9) It follows from this fact that we need only to solve the second equation in (9) Note that this equation is in the form of (2) By linear equation theory the following theorem is deduced Theorem 2 All the solutions to the generalized Sylvester matrix equation (1) are given by the second equation in (9) x 1 y i i 1 2 p are given by x 1 V 12 f (15) row y i p i1 Sr 1 (β) V 22 f R ϖ is an arbitrary vector V11 V 12 V V 12 R n ϖ V 22 R pr ϖ (16) V 21 V 22 satisfies the following singular value decomposition: U T β(a) cola i B p 1 Σω ω 0 V 0 0 with Σ ω ω an invertible matrix Proof We firstly show that the expression (15) satisfies the second equation in (9) Let In 0 0 V 0 Sr 1 (β) f substitute it into the second equation in (9) then we get U β (A) col A i B p 1 S r (β) U β (A) col A i B p 1 0 V f Σω ω 0 0 0 0 0 f This shows that is the solution to the second equation in (9) Furthermore using the notation (16) we have In 0 0 V 0 Sr 1 (β) f In 0 V11 V 12 0 0 Sr 1 (β) V 21 V 22 f V 11 V 12 0 Sr 1 (β) V 21 Sr 1 (β) V 22 f V 12 f Sr 1 (β) V 22 f The last expression is equivalent to (15) This shows that (15) satisfies the second equation in (9) Secondly we show the solutions given by (15) are complete To prove this note that f R ϖ is an arbitrary vector with the number of its elements equal to the degree of freedom in the solutions so we need only to show rank V 12 S 1 r (β) V 22 ϖ This is obvious since V 12 V 22 are two parts of the unitary matrix V Sr 1 (β) is nonsingular The formula given in the above theorem involves the in-

B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 401 verse of the matrix S r (β) However it can be calculated by explicit recursive algorithm as proposed in the following lemma The proof is omitted here Lemma 4 Let S r (β) be defined as (7) then Sr 1 (β) col ( F T) i b p 1 I r b 0 0 1 T By applying Theorem 1 we can get the following result regarding solution to the normal Sylvester matrix equation AX XF BG (17) Corollary 2 Let A F do not have common eigenvalues then the unique solution to the normal Sylvester matrix equation (17) is given by { x1 β (A) 1 col A i B p 1 S r (β) row g i p i1 row x i p i2 row A i p 1 i1 x 1 Π (A B) row g i p 1 i1 32 The generalized Sylvester matrix equation MXN X T Y The results proposed in this subsection are quite similar to the above subsection We firstly give the following parallel theorem to Theorem 1: Theorem 3 The generalized Sylvester matrix equation (4) with N F C 1 (β) is equivalent to { rowxi 1 ip 1 rowmp 1 i1 x p+π(m T )rowy i 1 ip 1 β (M)x p colm i T p 1 S r (β)rowy i 1 ip (18) Π(M T ) is defined as (10) β (s) p β i s p i β p 1 I r 0 0 0 0 β p 2 I r β p 3 I r β 0 I r S r (β) 0 β p 3 I r 0 (19) 0 β 0 I r 0 0 Proof The generalized Sylvester matrix equation (4) is equivalent to Mx 2 x 1 T y 1 Mx 3 x 2 T y 2 Mx p x p 1 T y p 1 M p β i 1 x i x p T y p which can also be rewritten as x p 1 Mx p T y p 1 x p 2 M 2 x p 1 M i T y p 2+i x 1 M p 1 x p p 2 M i T y 1+i M p β i 1 x i x p T y p i1 By substituting x i i 1 2 p 1 into the last equation we obtain x p M p 1 β i 1 M p i x p p i 1 M j T y i+j i1 β p 1 Mx p T y p j0 By some simplifications the above equation is equivalent to the second equation in (18) The equivalence between (4) (18) can be similarly proved as that in Theorem 1 Similar to Lemma 3 we have the following corollary: Corollary 3 The degree of freedom in the solution to the generalized Sylvester matrix equation (4) is ϖ pr + n ω ω rankβ (M) col M i T p 1 S r (β) The parallel solutions to the generalized Sylvester matrix equation (4) are given in the following theorem with its proof omitted Theorem 4 The complete parametric solutions to the generalized Sylvester matrix equation (4) is characterized by the second equation in (18) x p y i i 1 2 p are given by x p row y i p i1 V12 V 22 f (20) f R ϖ is an arbitrary vector V11 V 12 V V 12 R n ϖ V 22 R pr ϖ V 21 V 22 satisfies the following singular value decomposition: U T β (M) colm i T p 1 S r (β)v Σω ω 0 0 0 with Σ ω ω being an invertible matrix Remark 1 Similar to Corollary 2 we can also obtain the unique solution to the following normal Sylvester matrix equation: MXN X T G

402 B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 according to Theorem 3 4 Examples To illustrate the proposed method we give the following two examples Example 1 We firstly consider a generalized Sylvester matrix equation in the form of (1) with 05 21 32 A 01 12 3 201 25 05 02 0 B 0 1 0 12 F 1 21 05 02 By applying singular value decomposition to the matrix β (A) B AB we get 0082 0218 0468 0589 V 12 0060 0064 0273 0567 0357 0769 0094 0179 0893 0230 0013 0033 0200 0530 0046 0064 V 22 0069 0057 0716 0459 0136 0141 0427 0287 For simplicity we select f 01 02 03 04 T obtain 0589 1563 X 0567 1447 0459 0998 Y 0287 0667 0179 0037 Example 2 We further consider a generalized Sylvester matrix equation in the form of (4) with 101 203 26 M 51 0001 006 932 501 001 01 03 T 0 1 0 13 N 1 22 01 12 By applying singular value decomposition to the matrix β (A) colm i T p 1 S r (β) we get 0004 0082 0009 0253 V 12 0007 0008 0039 0426 0009 0131 0002 0636 1000 0001 0 0006 0001 0986 0 0064 V 22 0 0001 1000 00147 0006 0065 0007 0588 We specially choose f 01 02 03 04 T obtain 0197 0115 X 0343 0157 0520 0280 0305 0103 Y 0225 0172 References 1 S P Bhattacharyya E de Souza Pole assignment via Sylvester equationj Systems & Control Letters 1982 1(4): 261 263 2 L H Keel J A Fleming S P Bhattacharyya Minimum norm pole assignment via Sylvester s equationj Linear Algebra Its Role in Systems Theory AMS Contemporary Mathematics 1985 47(3): 265 272 3 C C Tsui A complete analytical solution to the equation T A F T LC its applicationsj IEEE Transactions on Automatic Control 1987 32(8): 742 744 4 G Duan Solution to matrix equation AV + BW EV F eigenstructure assignment for descriptor systemsj Automatica 1992 28(3): 639 643 5 G Duan Solutions to matrix equation AV + BW V F their application to eigenstructure assignment in linear systemsj IEEE Transactions on Automatic Control 1993 38(2): 276 280 6 A Saberi A A Stoorvogel P Sannuti Control of Linear Systems with Regulation Input ConstraintsM// Series of Communications Control Engineering New York: Springer- Verlag 1999 7 G Duan On the solution to Sylvester matrix equation AV + BW EV F J IEEE Transactions on Automatic Control 1996 41(4): 612 614 8 B Zhou G Duan An explicit solution to the matrix equation AX XF BY J Linear Algebra Its Applications 2005 402(1): 345 366 9 B Zhou G Duan A new solution to the generalized Sylvester matrix equation AV EV F BW J Systems & Control Letters 2006 55(3): 193 198 10 G Duan B Zhou Solution to the second-order Sylvester matrix equation MV F 2 + DV F + KV BW J IEEE Transactions on Automatic Control 2006 51(5): 805 809

B ZHOU et al / Journal of Control Theory Applications 2007 5 (4) 397 403 403 Bin ZHOU was born in HuBei Province China in 1981 He received the Bachelor s degree from the Department of Control Science Engineering at Harbin Institute of Technology Harbin China in 2004 He is now a graduate student in the Center for Control Systems Guidance Technology in Harbin Institute of Technology His current research interests include linear systems theory constrained control systems E-mail: zhoubinhit@163com Guang-Ren DUAN was born in Heilongjiang Province 1962 He received his BSc degree in Applied Mathematics both his M S PhD degrees in Control Systems Theory He is currently the Director of the Center for Control Systems Guidance Technology at Harbin Institute of Technology His main research interests include robust control eigenstructure assignment descriptor systems E-mail: grduan@hiteducn