Closed-form Solutions to the Matrix Equation AX EXF = BY with F in Companion Form

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1 International Journal of Automation and Computing 62), May 2009, DOI: /s Closed-form Solutions to the Matrix Equation AX EX BY with in Companion orm Bin Zhou Guang-Ren Duan Center for Control heory and Guidance echnology, Harbin Institute of echnology, Harbin , PRC Abstract: A closed-form solution to the linear matrix equation AX EX BY with X and Y unknown and matrix being in a companion form is proposed, and two equivalent forms of this solution are also presented he results 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 descriptor system theory he results proposed here are parallel to and more general than our early work about the linear matrix equation AX X BY Keywords: Linear matrix equations, closed-form solutions, right factorizations, descriptor linear systems, companion matrix 1 Introduction his paper considers the linear matrix equation AX EX BY 1) A, E R n n, B R n r, and R p p are given matrices while X R n p and Y R r p are matrices to be determined and matrix is in the following companion form 0 0 β β 1 2) 0 1 β In the special case of E I, 1) reduces to AX X BY 3) he linear matrix equation 3) is closely related with many problems in conventional linear control systems theory, such as pole/eigenstructure assignment design 1 5, Luenberger-type observer design 6 9, robust fault detection 10 12, regional pole assignment 13, robust partial pole-placement 14, constraint control 15 and so on, and has been investigated by several researchers 5, When dealing with eigenstructure assignment, observer design and model reference control for descriptor linear systems, the more general linear matrix equation 1), with E being usually singular, is encountered In solving the linear matrix equation 1), finding the complete parametric solutions, that is, parametric solutions consisting of the maximum number of free parameters, is of extreme importance his is because many problems, such as robustness in control system design, require full use of the design freedom or 1) with being in Jordan form, complete parametric solutions were proposed in 19, 20 Under the R-controllability of the matrix triple E, A, B), Duan 19 Manuscript received March 26, 2008; revised September 26, 2008 his work was supported by the Major Program of National Natural Science oundation of China No ) and Program for Changjiang Scholars and Innovative Research eam in University *Corresponding author address: binzhou@hiteducn; binzhou@msncom gave a complete and explicit solution which uses the right coprime factorization of the input-state transfer function se A) 1 B, while Duan 21 proposed a complete parametric solution which is not in a direct, explicit form but in a recursive form he advantages of setting in Jordan form in 3) were described again in 22, 23 to enable uniquely that the corresponding solutions are decoupled his is useful in the observer design because it enables the full realization of the critical robustness properties of state feedback control for most systems, and a systematic design of minimal order functional observers 22, 23 It is also useful in eigenvalue/vector assignment because the corresponding solutions are actually the eigenvector matrix 22, 23 In our recent work 20, when matrix is in the companion form as 2), we presented a closed-form solution to 3) he solution is expressed in terms of the controllability matrix of the matrix pair A, B), a symmetric operator matrix, and a parametric matrix in the Hankel matrix form h 1 h 2 h p h 2 h 3 h p+1 Hn, p) 4) h n h n+1 h n+ In this paper, we extend this result to the general case 1) and also provide this type of linear matrix equation closedform solutions involving the right coprime factorization of se A) 1 B and the parametric matrix in the form of 4) urthermore, the closed-form solutions given in 20 have the disadvantage that they may not be complete when A and have common eigenvalues In this paper, such restriction is removed if a certain condition is satisfied 2 Main results In this paper, we use A, adj A), det A), σ A), and A) to denote the transpose, adjoint, determinant, spectrum set, and the of matrix A, respectively; b ij is the i-th row and j-th column of matrix B; I p is the p p identity matrix, and 0 will be used as an r s null matrix when the dimensions are evident from the context We use

2 B Zhou and G R Duan / Closed-form Solutions to the Matrix Equation AX EX BY with 205 row{a i} q ip to denote a matrix in the form of row {A i} q ip A p A p+1 A q 1 A q, p q and use col {A i} q ip, p q to denote col {A i} q ip A p A p+1 respectively We use I p, q to denote the set {p, p + 1,, q} A number λ is an eigenvalue of a matrix pair E, A) if it satisfies det λe A) 0 he Kronecker product of two matrices A and B is denoted by A B Right factorization of A se) 1 B involves two polynomial matrices N s) R n r s and D s) R r r s satisfying A se) 1 B N s) D 1 s) which can also be rewritten as A q A se) N s) BD s) 0 5) In the sequel, we need the expanding of N s) and D s), ie, N s) ω N is i, N i R n r 6) D s) ω D is i, D i R r r in which ω max {deg N s)), deg D s))} 21 Complete parametric solutions he main result of this paper is given as the following theorem whose proof is given in Appendix heorem 1 Let N s) and D s) be two polynomial matrices satisfying 5) hen, the closed-form parametric solutions to the linear matrix equation 1) are given by X row {N i} ω H ω + 1, p) 7) Y row {D i} ω H ω + 1, p) h i R r, i I 1, p are group of arbitrary vectors, and the vectors h j R r, j I p + 1, ω + p are determined recursively by h p+j β ih i+j, j I 1, ω 8) Next, we focus on establishing some equivalent forms for 7) We first denote X row {x i} p, Y row {yi}p 9) With the notation H ω + 1, p), the following recursive solution can be obtained Corollary 1 he solutions given by heorem 1 are equivalent to x k ω N ih i+k, k I 1, p 10) y k ω D ih i+k h i R r, i I 1, p are group of arbitrary vectors, and the vectors h j R r, j I p + 1, ω + p are determined recursively by 8) It follows from heorem 1 that the free parameters in the solutions are really h i R r, i I 1, p while h j R r, j I p + 1, ω + p are determined recursively by 8) In the following, we will give an equivalent expression which only involves the free parameters h i R r, i I 1, p he proof is also given in Appendix heorem 2 Denote r hen, solutions given by heorem 1 are equivalent to 1) If ω + 1 p, then x i row {N i i 1 r h y i row {D i r i 1 h, i I 1, p 11) h col {h j} p j1 Rrp is an arbitrary vector and N i 0, D i 0, i I p, ω + 1 2) If ω + 1 > p, then x i row { N i i 1 r h y i row { D i r i 1 h, i I 1, p 12) h col {h j} p j1 Rrp is an arbitrary vector, and row { N i row {Ni} + ω p N p+i i r row { D i row {Di} + ω p D p+i r i with 13) row {β i 14) We now face a question: under what condition, the solutions given by heorems 1 and 2 are complete? he following proposition, whose proof is given in Appendix, answers this question Proposition 1 If E, A, B) is R-controllable, ie, ) se A B n then the parametric solutions given by 7) are complete if and only if N s) r, s σ ) 15) D s) Remark 1 We note that the matrix equation 1) can be used for pole assignment problems in both continuous-time and discrete-time linear systems see, eg, 24) herefore, the explicit solutions proposed in this paper can also be used for both continuous-time and discrete-time linear systems 22 Solutions in special forms When the right factorization N s) and D s) is specially chosen as N s) adj se A) n 1 R is i D s) det se A) n 16) α is i

3 206 International Journal of Automation and Computing 62), May 2009 we can get a special form of the solutions given in heorem 1 Corollary 2 If E, A) and have no common eigenvalues, then the complete parametric solutions to the linear matrix equation 1) are given by X row {R i} n 1 H n, p) Y r α, β) H n, p) 17) h i R r, i I 1, p are a group of arbitrary vectors, and the vectors h j R r, j I p + 1, n + p 1 are determined recursively by 8), and r α, β) row {α i} n 1 αn 0 row {β i Proof We need only to show the expression of Y According to heorem 1, we have Y row {α i} n H n + 1, p) 18) It follows from the recursive relation 8) that H n + 1, p) I nr 0 row {β i Substituting 19) into 18), produces Y row {α i} n row {α i} n 1 αn r α, β) H n, p) I nr 0 row {β i 0 row {β i H n, p) H n, p) 19) H n, p) We now consider the special case 3) When the right factorization N s) and D s) satisfying A si) N s) BD s) 0 are specially chosen as N s) adj si A) n 1 R is i D s) det si A) n α is i, α n 1 20) we can get a special form of the solutions given in heorem 1, which is just the main result given in 20 Corollary 3 If A and have no common eigenvalues and p n, then the complete parametric solutions to the linear matrix equation 3) are given by { X Q c A, B S r α) H n, p) 21) Y r α, β) H n, p) H n, p) is defined by 4) with its elements satisfying the condition 8) in heorem 1, and { } n 1 Q c A, B row A i B r α, β) row {α i} n 1 0 row {β i S rα) α 1 α 2 α n 1 α 2 α 3 α n 1 Proof It follows from 25 that row{r i} n 1 Q c A, B S r α), Q c A, B is the controllability matrix of the matrix pair A, B) hen by using Corollary 2, we get 21) 3 Conclusions A closed-form solution to the linear matrix equation AX EX BY, with being a companion matrix, is established, which possesses the following features: 1) It is in a closed form and can be immediately obtained as soon as a pair of right coprime polynomial matrices satisfying 5) is derived; 2) It gives all the degrees of freedom to the equation, which are represented by parameter vectors h i, i I 1, p; 3) It does not require the eigenvalues of matrix to be prescribed, and thus allows matrix to be set undetermined and used as a parameter matrix Due to the above advantages, the provided solution may play important roles in descriptor linear system theory Acknowledgment he authors would like to thank Prof Chia Chi sui for his helpful suggestions on the manuscript, which have helped to improve the quality of the paper Appendix Proof of heorem 1 We only need to show that solutions given by 7) satisfy the linear matrix equation 1) o this end, it suffices to show Ax i Ex i+1 By i Ax p + p Ex iβ i 1 By p, i I 1, p 1 A1) x i N jh i+j, y i D jh i+j A2) Substituting 6) into 5) and equating both sides of the coefficient matrix of s i, we get the following series of equations: AN 0 + BD 0 0 AN i + EN i 1 + BD i 0, i I 1, ω EN ω 0 A3) irst, substituting A2) into A1) and using the above

4 B Zhou and G R Duan / Closed-form Solutions to the Matrix Equation AX EX BY with 207 series of equations A3) yields Ax i Ex i+1 AN jh i+j EN jh i+j+1 AN 0h i + AN 0h i + AN j EN j 1 h i+j EN ωh i+ω+1 j1 BD jh i+j j1 BD jh i+j By i, i I 1, p 1 Second, with the recursive relation 8), we obtain Ax p + A p Ex iβ i 1 N jh p+j + p Ev iβ i 1 A N jh p+j + E N j p h i+jβ i 1 AN j EN j 1 h p+j + AN 0h p j1 BD jh p+j + BDh p By p j1 Proof of heorem 2 Denote H i col {h j} i+ ji ; then H i+1 r H i and it follows that H i i 1 r H1, i I 0, p A4) When ω + 1 p, N i 0, D i 0, i I p, ω + 1 and the i-th column of matrix H is denoted by H i hen, we have x i row {N j or equivalently, f i x i y i Hi, yi row {Dj} Hi A5) col {f i} p col {Π N, D) Hi}p, Π N, D) Substituting A4) into A6), we obtain row {N j row {D j } i 1 p col {f i} p {Π col N, D) r H 1 A6) A7) A8) he above equation is equivalent to 11) when H 1 is denoted by h When ω + 1 > p, by direct calculation and noting 14), we have N ph p+1 N p col {h i} p N p+1h p+2 N p+1 col {h i} p+1 i2 Np+1 r col {h i} p N ωh ω+1 N ω ω p r col {hi} p hus, according to 7), we get x 1 row {N i} ω col {hi}ω+1 N ih i+1 + N ih i+1 ip j p N ih i+1 + N j r col {hi} p jp ω row {N i + j p N j r col {h i} p row { N i h Similarly, we can obtain jp x j row { N i row { N i col {hi}p+j 1 ij r j 1 h, j I 2, p row { N i is given by 13) he vectors yi, i I 1, p can also be obtained via the same procedure hus, we have col {f i} p {Π col N, D ) } i 1 p r h A9) with Π N, D ) row { N j row { D j A10) Expression A9) shows 12) Proof of Proposition 1 It follows from 5 that when E, A, B) is R-controllable, the degree of freedom in the solution is rp Since the degree of freedom of the solution is already rp, we need only to show that { } i 1 p ) col Π N, D) r rp A11) when ω + 1 p, and { col Π N, D ) } i 1 p ) r rp A12) when ω + 1 > p, Π N, D) and Π N, D ) are respectively defined as A7) and A10) Case 1 ω + 1 p In this case, it is obvious that A11) is equivalent to the observability of the matrix pair r, Π N, D) ) Next, we will show that the matrix pair r, Π N, D) ) is observable if and only if 15) holds

5 208 International Journal of Automation and Computing 62), May 2009 Construct a linear system in the form of { ẋ Ãx + Bu y Cx A13) Ã r, C Π N, D), B 0 0 It is obvious that Ã, B) is controllable since it is in a controllable form 26 Note that the transfer function of the linear time-invariant system A13) is 1 s) β s) rp Π N, D) col {s i Π N, D) col {s i Ñ s) D 1 s) A14) Ñ s) Π N, D) col { } s i col {N s), D s)} D s) diag {β s), β s),, β s) } Since deg det D ) ) s) dim Ã, it follows from linear system theory that the linear system A13) is a minimal realization of the transfer function A14) ) if and only if the polynomial matrix pair Ñ s), D s) is right coprime 26, ie, Ñ s) rp, s C A15) D s) With the special structure of D s), A15) holds if and only if N s) Ñ s) rp, s σ ) D s) A16) urthermore, the linear time-invariant system A13) is a minimal realization of the transfer function A14) if and only if Ã, B) is controllable and Ã, C) is observable Since Ã, B) has been in controllable form, the linear system A13) is a minimal realization of the transfer function A14) if and only if Ã, C) is observable ogether with A16), the completeness of the proposed solutions in heorem 1 holds in this case Case 2 ω + 1 > p In this case, similarly to the case ω + 1 p, we can also yield the condition A16), ie, the solutions given by 12) are complete if and only if Note that N s) D s) rp, s σ ) N s) row { N i col { s i ω p N is i + N p+i r i col {s i A17) A18) Since s σ ), using the Cayley-Hamilton theorem yields col {s i s p hus, in view of the above equation, we obtain i r col {s i i 1 r col { } s i βcol { } s i i 1 { } p r col s i i 1 s r col {s i s p+i Substituting the above series of equations into A18), we obtain ω p N s) N is i + N p+is p+i N is i N s), Similarly, we can also show that D s) D s), s σ ) s σ ) A19) A20) Equations A19) and A20) imply that A17) is equivalent to 15) So the completeness of the proposed solutions in heorem 1 also holds in this case References 1 R K Gavin, S P Bhattacharyya Robust and Wellconditioned Eigenstructure Assignment via Sylvester s Equation Optimal Control Applications and Methods, vol 4, no 3, pp , B H Kwon, M J Youn Eigenvalue-generalized Eigenvector Assignment by Output eedback IEEE ransactions on Automatic Control, vol 32, no 5, pp , J Lam, W Y Yan Pole Assignment with Optimal Spectral Conditioning Systems & Control Letters, vol 29, no 5, pp , J Lam, W Y Yan, Hu Pole Assignment with Eigenvalue and Stability Robustness International Journal of Control, vol 72, no 13, pp , G R Duan Solutions to Matrix Equation AV + BW V and heir Application to Eigenstructure Assignment in Linear Systems IEEE ransactions on Automatic Control, vol 38, no 2, pp , D G Luenberger Observing the State of a Linear System IEEE ransactions on Military Electronics, vol 8, no 2, pp 74 80, D G Luenberger An Introduction to Observers IEEE ransactions on Automatic Control, vol 16, no 6, pp , B Zhou, G R Duan, Y L Wu Parametric Approach for the Normal Luenberger unction Observer Design in Second-order Descriptor Linear Systems International Journal of Automation and Computing, vol 5, no 2, pp , 2008

6 B Zhou and G R Duan / Closed-form Solutions to the Matrix Equation AX EX BY with C C sui New Approach to Robust Observer Design International Journal of Control, vol 47, no 3, pp , J Park, G Rizzoni An Eigenstructure Assignment Algorithm for the Design of ault Detection ilters IEEE ransactions on Automatic Control, vol 39, no 7, pp , J Chen, R J Patton, H Y Zhang Design of Unknown Input Observers and Robust ault Detection ilters International Journal of Control, vol 63, no 1, pp , G R Duan, R J Patton Robust ault Detection Using Luenberger-type Unknown Input Observers A Parametric Approach International Journal of Systems, vol 32, no 4, pp , J Lam, H K am Regional Pole Assignment with Eigenstructure Robustness International Journal of Systems Science, vol 28, no 5, pp , J Lam, H K am Robust Partial Pole-placement via Gradient low Optimal Control Applications and Methods, vol 18, no 5, pp , A Saberi, A A Stoorvogel, P Sannuti Control of Linear Systems with Regulation and Input Constraints, in series of Communications and Control Engineering, Springer-Verlag, New York, USA, A Jameson Solution of the Equation AX + XB C by Inversion of an M M or N N Matrix SIAM Journal on Applied Mathematics, vol 16, no 5, pp , C C sui A Complete Analytical Solution to the Equation A LC and Its Applications IEEE ransactions on Automatic Control, vol 32, no 8, pp , C C sui On the Solution to Matrix Equation A LC and Its Applications SIAM Journal of Matrix Analysis and Applications, vol 14, no 1, pp 34 44, G R Duan On the Solution to Sylvester Matrix Equation AV + BW EV IEEE ransactions on Automatic Control, vol 41, no 4, pp , B Zhou, G R Duan An Explicit Solution to the Matrix Equation AX X BY Linear Algebra and its Applications, vol 402, pp , G R Duan Solution to Matrix Equation AV + BW EV and Eigenstructure Assignment for Descriptor Systems Automatica, vol 28, no 3, pp , C C sui An Overview of Applications and Solutions of a undamental Matrix Equation Pair Journal of lin Institute, vol 341, no 6, pp , C C sui Modern Control Systems Design heories New Development, 2nd Edition, Science Press, Beijing, PRC, 2007 in Chinese) 24 D Z Zheng Linear System heory, singhua University Press, Beijing, PRC, 2002 in Chinese) 25 G R Duan, B Zhou An Explicit Solution to Right actorization with Application in Eigenstructure Assignment Journal of Control heory and Applications, vol 3, no 3, pp , Kailath Linear Systems, Prentice-Hall, Inc, Englewood Cliffs, NJ, USA, 1980 Bin Zhou received his Bachelor degree from the Department of Control Science and Engineering at Harbin Institute of echnology, Harbin, PRC, in 2004 He was a research associate in the Department of Mechanical Engineering, University of Hong Kong, PRC, from December 2007 to March 2008 He is currently a Ph D candidate at the same institute In 2009, he joined Harbin Institute of echnology as a lecturer His research interests include constrained control systems and nonlinear systems theory Guang-Ren Duan received his B Sc degree in applied mathematics, M Sc and Ph D degrees in control systems theory rom 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of echnology, he became a professor of control systems theory in 1991 Dr Duan visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998, and worked at the Queen s University of Belfast, UK, from October 1998 to October 2002 Since August 2000, he has been elected specially employed professor at Harbin Institute of echnology sponsored by the Cheung Kong Scholars Program of the Chinese government He is currently the Director of the Centre for Control Systems and Guidance echnology at Harbin Institute of echnology His main research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control, and magnetic bearing control

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