×

Solutions to a family of matrix equations by using the Kronecker matrix polynomials. (English) Zbl 1181.15020

Using the standard technique of Kronecker products, the authors give closed form solutions to the real generalized Sylvester matrix equation \(\sum_i A_iXF^i + \sum_k B_kYF^k = \sum_j E_jRF^j\), where \(A_i\), \(B_k\), \(E_j\) and \(F\), \(R\) are known matrices and the matrix pair \((X,Y)\) is the solution. A generalization of this equation is also considered.

MSC:

15A24 Matrix equations and identities
93C05 Linear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Duan, G. R., Solutions to matrix equation \(AV + BW = VF\) and their application to eigenstructure assignment in linear systems, IEEE Transactions on Automatic Control, 38, 2, 276-280 (1993) · Zbl 0775.93098
[2] Duan, G. R., The solution to the matrix equation \(AV + BW = EVJ + R\), Applied Mathematics Letters, 17, 1197-1204 (2004) · Zbl 1065.15015
[3] Duan, G. R., Parametric eigenstructure assignment in high-order linear systems, International Journal of Control, Automation and Systems, 3, 3, 419-429 (2005)
[4] Duan, G. R., Two parametric approaches for eigenstructure assignment in second-order linear systems, Journal of Control Theory and Applications, 1, 59-64 (2003) · Zbl 1260.93073
[5] Duan, G. R.; Liu, G. P., Complete parametric approach for eigenstructure assignment in a class of second-order linear systems, Automatica, 38, 4, 725-729 (2002) · Zbl 1009.93036
[6] Duan, G. R.; Liu, G. P.; Thompson, S., Eigenstructure assignment design for proportional-integral observers: continuous-time case, IEE Proceedings: Control Theory & Applications, 148, 3, 263-267 (2001)
[7] G.R. Duan, B. Zhou, Solution to the equation \(\mathit{MVF}^2 + \mathit{DVF} + \mathit{KV} = \mathit{BW} \); G.R. Duan, B. Zhou, Solution to the equation \(\mathit{MVF}^2 + \mathit{DVF} + \mathit{KV} = \mathit{BW} \)
[8] Duan, G. R.; Zhou, B., Solution to the second-order Sylvester matrix equation \(MVF^2 + DVF + KV = BW\), IEEE Transactions on Automatic Control, 51, 5, 805-809 (2006) · Zbl 1366.15011
[9] Zhou, B.; Duan, G. R., An explicit solution to the matrix equation \(AX - XF = BY\), Linear Algebra and Its Applications, 402, 345-366 (2005) · Zbl 1076.15016
[10] Zhou, B.; Duan, G. R., A new solution to the generalized Sylvester matrix equation \(AV - EVF = BW\), Systems & Control Letters, 55, 3, 193-198 (2006) · Zbl 1129.15300
[11] Zhou, B.; Duan, G. R., On the generalized Sylvester mapping and matrix equations, Systems & Control Letters, 57, 3, 200-208 (2008) · Zbl 1129.93018
[12] B. Zhou, G.R. Duan, Parametric approach for the normal Luenberger function observer design in second-order linear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 1423-1428.; B. Zhou, G.R. Duan, Parametric approach for the normal Luenberger function observer design in second-order linear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 1423-1428.
[13] Lei, Y.; Liao, A.-P., A minimal residual algorithm for the inconsistent matrix equation \(AXB = C\) over symmetric matrices, Applied Mathematics and Computation, 188, 1, 499-513 (2007) · Zbl 1131.65038
[14] Kaabi, A.; Kerayechian, A.; Toutounian, F., A new version of successive approximations method for solving Sylvester matrix equations, Applied Mathematics and Computation, 186, 1, 638-645 (2007) · Zbl 1121.65044
[15] Gavin, K. R.; Bhattacharyya, S. P., Robust and well-conditioned eigenstructure assignment via Sylvester’s equation, Optimal Control Application and Methods, 4, 205-212 (1983) · Zbl 0512.93035
[16] Kwon, B. H.; Youn, M. J., Eigenvalue-generalized eigenvector assignment by output feedback, IEEE Transactions on Automatic Control, 32, 5, 417-421 (1987) · Zbl 0611.93030
[17] Luenberger, D. G., An introduction to observers, IEEE Transactions on Automatic Control, 16, 596-602 (1971)
[18] Park, J.; Rizzoni, G., An eigenstructure assignment algorithm for the design of fault detection filters, IEEE Transactions on Automatic Control, 39, 1521-1524 (1994) · Zbl 0825.93822
[19] Tsui, C. C., New approach to robust observer design, International Journal of Control, 47, 745-751 (1988) · Zbl 0636.93030
[20] Tsui, C. C., A complete analytical solution to the equation \(TA - FT = LC\) and its applications, IEEE Transactions on Automatic Control, 32, 742-744 (1987) · Zbl 0617.93009
[21] Tsui, C. C., On the solution to matrix equation \(TA - FT = LC\) and its applications, SIAM Journal of Matrix Analysis and Applications, 14, 1, 34-44 (1993) · Zbl 0768.15009
[22] Wang, Q.-W.; Chang, H.-X.; Ning, Q., The common solution to six quaternion matrix equations with applications, Applied Mathematics and Computation, 198, 1, 209-226 (2008) · Zbl 1141.15016
[23] F. Rincon, Feedback stabilization of second-order models, Ph.D. dissertation, Northern Illinois University, De Kalb, Illinois, USA, 1992.; F. Rincon, Feedback stabilization of second-order models, Ph.D. dissertation, Northern Illinois University, De Kalb, Illinois, USA, 1992.
[24] Kim, Y.; Kim, H. S., Eigenstructure assignment algorithm for mechanical second-order systems, Journal of Guidance, Control and Dynamics, 22, 5, 729-731 (1999)
[25] Chu, E. K.; Datta, B. N., Numerically robust pole assignment for second-order systems, International Journal of Control, 64, 4, 1113-1127 (1996) · Zbl 0850.93318
[26] Inman, D. J.; Kress, A., Eigenstructure assignment algorithm for second-order systems, Journal of Guidance, Control and Dynamics, 22, 5, 729-731 (1999)
[27] Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation, 197, 1, 41-50 (2008) · Zbl 1143.65035
[28] Ramadan, M. A.; El-Sayed, E. A., On the matrix equation \(XH = HX\) and the associated controllability problem, Applied Mathematics and Computation, 186, 1, 844-859 (2007) · Zbl 1126.93012
[29] Wu, A. G.; Duan, G. R., Kronecker maps and Sylvester-polynomial matrix equations, IEEE Transactions on Automatic Control, 52, 5, 905-910 (2007) · Zbl 1366.93190
[30] Huang, L., The explicit solutions and solvability of linear matrix equations, Linear Algebra and its Applications, 311, 195-199 (2000) · Zbl 0958.15008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.