Jiashang Jiang

Yongxin Yuan

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 903–906Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comOptimal Control/Numerical AnalysisOn solutions of the matrix equations K X − E X F = BY andM X F 2 + D X F + K X = BYSur les solutions des équations matricielles K X − E X F = BY etM X F 2 + D X F + K X = BYYongxin Yuan, Jiashang JiangSchool of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 16 May 2012Accepted 9 October 2012Available online 22 October 2012Presented by Olivier PironneauThis note studies the solutions of generalized Sylvester equations K X − E X F = BY andM X F 2 + D X F + K X = BY , and obtains explicit solutions of the equations by using somematrix transformations and the minimal polynomial of the matrix F .© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette note on étudie les solutions des équations généralisées de Sylvester K X −E X F = BY et M F X2 + D X F + K X = BY , on donne des expressions explicites des solutionsde ces équations en utilisant des transformations matricielles et le polynôme minimal dela matrice F .© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionMany control problems, such as pole assignment [2,14,16], and eigenstructure assignment [8,12], can be represented bythe following second-order linear systemsMẍ(t) + Dẋ(t) + K x(t) = Bu(t), (1)where x(t) ∈ Rn is the state vector, u(t) ∈ Rq is the control vector and M , D , K and B are matrices of appropriate dimensions.In certain applications, the matrices M , D and K are called the mass, damping and stiffness matrices, respectively. It can beshown that the linear system (1) is closely related with a second-order Sylvester matrix equation and can be written asM X F 2 + D X F + K X = BY , (2)where M, D, K ∈ Cn×n, B ∈ Cn×q and F ∈ Cp×p are known matrices, X ∈ Cn×p and Y ∈ Cq×p are the matrices to be de-termined. When M = 0 and D = −E , the second-order Sylvester matrix equation (2) reduces to the generalized Sylvestermatrix equationK X − E X F = BY . (3)E-mail address: yuanyx_703@163.com (Y. Yuan).1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.crma.2012.10.011904 Y. Yuan, J. Jiang / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 903–906When M = 0, D = −In , B = In , Y = W , the second-order Sylvester matrix equation (2) becomes the normal Sylvester matrixequationK X − X F = W . (4)In addition, by substituting K = −F  in (4), the normal Sylvester matrix equation reduces to the well-known Lyapunovmatrix equationF  X + X F = −W . (5)All the equations mentioned above play an important role in various applied problems. Therefore, despite that numerousalgorithms were developed to solve these equations ([3–7,9–11,13,17] etc.), the development of some new algorithms is stillof importance. In this note, a simple method for solving Eq. (2) and Eq. (3) is presented by some matrix transformationsand the minimal polynomial of the matrix F , and the explicit solutions of the equations are provided.2. Matrix equation K X − E X F = BYIn this section, we discuss the solution of the matrix equation (3). To begin with, we give the following lemma [1]:Lemma 1. If L ∈ Cm×q, J ∈ Cm×p, then L Z = J has a solution Z ∈ Cq×p if and only if LL+ J = J . In this case, the general solution ofthe equation can be described as Z = L+ J + (Iq − L+L)U , where L+ represents the Moore–Penrose generalized inverse of the matrix L,and U ∈ Cq×p is an arbitrary matrix.It follows from Lemma 1 that the equation of (3) with unknown matrix Y has a solution if and only if(In − B B+)K X − (In − B B+)E X F = 0, (6)when the condition (6) is satisfied, the general solution to the equation of (3) with unknown matrix Y is given byY = B+(K X − E X F ) + (Iq − B+B)T ,where T ∈ Cq×p is an arbitrary matrix.LetP1 =(In − B B+)K , Q 1 =(In − B B+)E,then, the equation of (6) is equivalent toP1 X = Q 1 X F . (7)Applying the approach in [15], assume that the columns of the matrix [G1, H1] form the basis of the null space of[Q 1 ,−P1 ] (the matrices G1, H1 may be found using procedure null.m package MATLAB), then we haveG1 Q 1 = H1 P1. (8)Using the equality (8), we getG1 P1 X = G1 Q 1 X F = H1 P1 X F = H1 Q 1 X F 2. (9)LetP2 = G1 P1, Q 2 = H1 Q 1, (10)then the equation of (9) becomesP2 X = Q 2 X F 2. (11)Similarly, let the columns of the matrix [G2, H2] form the basis of the null space of [Q 2 ,−P2 ], that is,G2 Q 2 = H2 P2. (12)Using the equality (12), we haveP3 X = Q 3 X F 3,where P3 = G2 P2, Q 3 = H2 Q 2.Y. Yuan, J. Jiang / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 903–906 905A similar procedure can be used to construct the relation with higher degrees of the matrix F ,Pk X = Q k X F k, k = 1,2, . . . , (13)where Pk = Gk−1 Pk−1, Q k = Hk−1 Q k−1, and the columns of the matrix [Gk−1, Hk−1] form the basis of the null space of[Q k−1,−Pk−1], that is,Gk−1 Q k−1 = Hk−1 Pk−1, k = 2,3, . . . .It is easily seen thatPk = Gk−1Gk−2 · · · G2G1 P1, Q k = Hk−1 Hk−2 · · · H2 H1 Q 1. (14)Assume that the minimal polynomial of the matrix F ismF (λ) = λl + f1λl−1 + · · · + fl−1λ + fl. (15)Then, by (13), we have(Pl + f1 Hl−1 Pl−1 + f2 Hl−1 Hl−2 Pl−2 + · · · + fl−1 Hl−1 Hl−2 · · · H2 H1 P1 + fl Q l)X= Q l X(F l + f1 F l−1 + · · · + fl−1 F + fl I p) = 0.In summary of the above discussion and using Lemma 1, we have proved the following result:Theorem 1. Let P1 = (In − B B+)K , Q 1 = (In − B B+)E. Assume that the matrix [Gk−1, Hk−1] is of full row rank and satisfiesGk−1 Q k−1 = Hk−1 Pk−1 , k = 2,3, . . . , where Pk = Gk−1 Pk−1 , Q k = Hk−1 Q k−1 , k = 2,3, . . . . Let the minimal polynomial of thematrix F be given by (15). Set D = Pl + f1 Hl−1 Pl−1 + f2 Hl−1 Hl−2 Pl−2 +· · ·+ fl−1 Hl−1 Hl−2 · · · H2 H1 P1 + fl Q l, then the solutionof Eq. (3) can be expressed asX = (In − D+D)V , (16)Y = B+[K (In − D+D)V − E(In − D+D)V F ] + (Iq − B+B)T , (17)where V ∈ Cn×p , T ∈ Cq×p are arbitrary matrices.3. Matrix equation M X F 2 + D X F + K X = BYIn this section, we study the solution of the matrix equation (2). Using Lemma 1, the equation of (2) with unknownmatrix Y has a solution if and only if(In − B B+)(M X F 2 + D X F + K X) = 0, (18)when the condition (18) is satisfied, the general solution to the equation of (2) with unknown matrix Y is given byY = B+(M X F 2 + D X F + K X) + (Iq − B+B)T ,where T ∈ Cq×p is an arbitrary matrix.LetP̃1 =[−(In − B B+)K 00 (In − B B+)M], Q̃ 1 =[(In − B B+)D (In − B B+)M(In − B B+)M 0], (19)then, the equation of (18) is equivalent toP̃1[XX F]= Q̃ 1[XX F]F . (20)By a similar approach in Section 2, we haveP̃k[XX F]= Q̃ k[XX F]F k, (21)where the matrix [G̃k−1, H̃k−1] is of full row rank and is determined alternately by the following relations:G̃k−1 Q̃ k−1 = H̃k−1 P̃k−1, k = 2,3, . . . , (22)P̃k = G̃k−1 P̃k−1, Q̃ k = H̃k−1 Q̃ k−1, k = 2,3, . . . . (23)906 Y. Yuan, J. Jiang / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 903–906Assume that the minimal polynomial of the matrix F is given by (15). Then, by (21), we haveD̃[XX F]= 0, (24)where D̃ = P̃ l + f1 H̃l−1 P̃ l−1 + f2 H̃l−1 H̃l−2 P̃ l−2 + · · · + fl−1 H̃l−1 H̃l−2 · · · H̃2 H̃1 P̃1 + fl Q̃ l.LetD̃ = [P1,−Q 1].Then the equation of (24) is equivalent toP1 X = Q 1 X F , (7)and the solution is given by (16).By now, we have proved the following result:Theorem 2. Let P̃1, Q̃ 1 be given by (19). Assume that the matrix [G̃k−1, H̃k−1] is of full row rank and satisfies G̃k−1 Q̃ k−1 = H̃k−1 P̃k−1 ,k = 2,3, . . . , where P̃k = G̃k−1 P̃k−1 , Q̃ k = H̃k−1 Q̃ k−1 , k = 2,3, . . . . Let the minimal polynomial of the matrix F be given by (15). SetD̃ = P̃ l + f1 H̃l−1 P̃ l−1 + f2 H̃l−1 H̃l−2 P̃ l−2 + · · · + fl−1 H̃l−1 H̃l−2 · · · H̃2 H̃1 P̃1 + fl Q̃ l and then partition D̃ as D̃ = [P1,−Q 1]. Thenthe solution of Eq. (2) can be expressed asX = (In − D+D)V , (25)Y = B+[M(In − D+D)V F 2 + D(In − D+D)V F + K (In − D+D)V ] + (Iq − B+B)T , (26)where D = Pl + f1 Hl−1 Pl−1 + f2 Hl−1 Hl−2 Pl−2 + · · · + fl−1 Hl−1 Hl−2 · · · H2 H1 P1 + fl Q l , and V ∈ Cn×p , T ∈ Cq×p are arbitrarymatrices.References[1] A. Ben-Israel, T.N.E. Greville, Generalized Inverse: Theory and Applications, Wiley, New York, 1974.[2] E.K. Chu, B.N. Datta, Numerically robust pole assignment for second-order systems, Internat. J. Control 64 (1996) 1113–1127.[3] M. Dehghan, M. Hajarian, Efficient iterative method for solving the second-order Sylvester matrix equation E V F 2 − AV F − C V = BW , IET ControlTheory Appl. 3 (2009) 1401–1409.[4] F. Ding, T. Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett. 54 (2005) 95–107.[5] G.R. Duan, Solution to matrix equation AV + BW = E V F and eigenstructure assignment for descriptor systems, Automatica 28 (1992) 639–643.[6] G.R. Duan, Solution to matrix equation AV + BW = V F and their application to eigenstructure assignment in linear systems, IEEE Trans. Aurora.Control 38 (1993) 276–280.[7] G.R. Duan, On the solution to Sylvester matrix equation AV − BW = E V F , IEEE Trans. Aurora. Control 41 (1996) 612–614.[8] G.R. Duan, Two parametric approaches for eigenstructure assignment in second-order linear systems, J. Control Theory Appl. 1 (2003) 59–64.[9] G.R. Duan, G.P. Liu, S. Thompson, Eigenstructure assignment design for proportional-integral observers: continuous-time case, IEE Proc. Control TheoryAppl. 148 (2001) 263–267.[10] G.R. Duan, B. Zhou, Solution to the second-order Sylvester matrix equation M V F 2 + D V F + K V = BW , IEEE Trans. Automat. Control 51 (2006) 805–809.[11] K.R. Gavin, S.P. Bhattacharyya, Robust and well-conditioned eigenstructure assignment via Sylvester’s equation, Optimal Control Appl. Methods 4 (1983)205–212.[12] D.J. Inman, A. Kress, Eigenstructure assignment using inverse eigenvalue methods, J. Guid. Contr. Dynam. 18 (1995) 625–627.[13] A. Jameson, Solution of the equation A X − X B = C by inversion of an M × M or N × N matrix, SIAM J. Appl. Math. 16 (1968) 1020–1023.[14] Y. Kim, H.S. Kim, Eigenstructure assignment algorithm for mechanical second-order systems, J. Guid. Contr. Dynam. 22 (1999) 729–731.[15] W.-W. Lin, S.-F. Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 38(2006) 26–39.[16] F. Rincon, Feedback stabilization of second-order models, PhD dissertation, Northern Illinois University, De Kalb, Illinois, USA, 1992.[17] B. Zhou, G.R. Duan, A new solution to the generalized Sylvester matrix equation AV − E V F = BW , Systems Control Lett. 55 (2006) 193–198.

On solutions of the matrix equations $ KX-EXF=BY$ and $ MX{F}^{2}+DXF+KX=BY$

Yuan, Yongxin

Jiang, Jiashang

Numérisation de Documents Anciens Mathématiques

http://www.numdam.org/item/10.1016/j.crma.2012.10.011.pdf

On solutions of the matrix equations $KX-EXF=BY$ and $MX{F}^{2}+DXF+KX=BY$

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On solutions of the matrix equations KX−EXF=BY KX-EXF=BYKX−EXF=BY and MXF2+DXF+KX=BY MX{F}^{2}+DXF+KX=BYMXF2+DXF+KX=BY

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On solutions of the matrix equations $KX-EXF=BY$ and $MX{F}^{2}+DXF+KX=BY$