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Gradient based iterative algorithm for solving coupled matrix equations. (English) Zbl 1159.93323
Summary: This paper is concerned with iterative methods for solving a class of coupled matrix equations including the well-known coupled Markovian jump Lyapunov matrix equations as special cases. The proposed method is developed from an optimization point of view and contains the well-known Jacobi iteration, Gauss-Seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle, as special cases. We have provided analytically a necessary and sufficient condition for the convergence of the proposed iterative algorithm. Simultaneously, the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form. The proposed approach requires less computation and is numerically reliable as only matrix manipulation is required. Some other existing results require either matrix inversion or special matrix products. Numerical examples show the effectiveness of the proposed algorithm.
MSC:
93B40Computational methods in systems theory
93C05Linear control systems
15A18Eigenvalues, singular values, and eigenvectors
60J75Jump processes
Software:
KELLEY
References:
[1]Zhou, B.; Duan, G. R.: An explicit solution to the matrix equation AX-XF=BY, Linear algebra and its applications 402, No. 1, 345-366 (2005) · Zbl 1076.15016 · doi:10.1016/j.laa.2005.01.018
[2]Zhou, B.; Duan, G. R.: A new solution to the generalized Sylvester matrix equation AV-EVF=BW, Systems control letters 55, No. 3, 193-198 (2006) · Zbl 1129.15300 · doi:10.1016/j.sysconle.2005.07.002
[3]Zhou, B.; Duan, G. R.: On the generalized Sylvester mapping and matrix equations, Systems control letters 57, No. 3, 200-208 (2008) · Zbl 1129.93018 · doi:10.1016/j.sysconle.2007.08.010
[4]Duan, G. R.: On the solution to Sylvester matrix equation AV+BW=EVF, Institute of electrical and electronics engineers. Transactions on automatic control 41, No. 4, 612-614 (1996) · Zbl 0855.93017 · doi:10.1109/9.489286
[5]Qiu, L.; Chen, T.: Unitary dilation approach to contractive matrix completion, Linear algebra and its applications 379, 345-352 (2004) · Zbl 1056.15015 · doi:10.1016/S0024-3795(03)00577-9
[6]Chen, T.; Qiu, L.: H design of general multirate sampled-date control systems, Automatica 30, 1139-1152 (1994) · Zbl 0806.93038 · doi:10.1016/0005-1098(94)90210-0
[7]Ding, F.; Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations, Systems control letters 54, No. 2, 95-107 (2005) · Zbl 1129.65306 · doi:10.1016/j.sysconle.2004.06.008
[8]Mukaidani, H.; Xu, H.; Mizukami, K.: New iterative algorithm for algebraic Riccati equation related to H control problem of singularly perturbed systems, Institute of electrical and electronics engineers. Transactions on automatic control 46, 1659-1666 (2001) · Zbl 1006.93044 · doi:10.1109/9.956068
[9]Borno, I.; Gajic, Z.: Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems, Computers mathematics with applications 30, No. 7, 1-4 (1995) · Zbl 0837.93075 · doi:10.1016/0898-1221(95)00119-J
[10]Ding, F.; Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations, Institute of electrical and electronics engineers. Transactions on automatic control 50, No. 8, 1216-1221 (2005)
[11]Kelley, C. T.: Iterative methods for linear and nonlinear equations, (1995)
[12]Mori, T.; Derese, A.: A brief summary of the bounds on the solution of the algebraic matrix equations in control theory, International journal of control 39, 247-256 (1984) · Zbl 0527.93030 · doi:10.1080/00207178408933163
[13]Mrabti, M.; Benseddik, M.: Unified type non-stationary Lyapunov matrix equations–simultaneous eigenvalue bounds, Systems control letters 18, 73-81 (1995)
[14]Wang, Q.; Lam, J.; Wei, Y.; Chen, T.: Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and mathematics with applications 55, No. 4, 843-850 (2008) · Zbl 1139.60334 · doi:10.1016/j.camwa.2007.04.031
[15]I. Jonsson, B. Kägström, Recursive blocked algorithms for solving triangular systems–Part I: one-sided and coupled Sylvester-type matrix equations, Association for Computing Machinery. Transactions on Mathematical Software 28, 392–415 · Zbl 1072.65061 · doi:10.1145/592843.592845
[16]I. Jonsson, B. Kägström, Recursive blocked algorithms for solving triangular systems–Part II: Two-sided and generalized Sylvester and Lyapunov matrix equations, Association for Computing Machinery. Transactions on Mathematical Software 28 416–435 · Zbl 1072.65062 · doi:10.1145/592843.592846
[17]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, No. 1, 41-50 (2008) · Zbl 1143.65035 · doi:10.1016/j.amc.2007.07.040
[18]Ding, F.; Chen, T.: On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization 44, No. 6, 2269-2284 (2006) · Zbl 1115.65035 · doi:10.1137/S0363012904441350
[19]Costa, O. L. V.; Fragoso, M. D.: Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of mathematical analysis and applications 179, No. 1, 154-178 (1993) · Zbl 0790.93108 · doi:10.1006/jmaa.1993.1341
[20]Zhou, B.; Lam, J.; Duan, G.: Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations, Computers and mathematics with applications 56, 3070-3078 (2008) · Zbl 1165.15304 · doi:10.1016/j.camwa.2008.07.037
[21]Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A.: State–space solutions to standard H2 and H control problems, Institute of electrical and electronics engineers. Transactions on automatic control 34, No. 8, 831-847 (1989) · Zbl 0698.93031 · doi:10.1109/9.29425
[22]Zhou, K.; Doyle, J.; Glover, K.: Robust and optimal control, (1996) · Zbl 0999.49500
[23]Zhou, B.; Li, Z. -Y.; Duan, G.; Wang, Y.: Weighted least squares solutions to general coupled Sylvester matrix equations, Journal of computational and applied mathematics 224, No. 2, 759-776 (2009) · Zbl 1161.65034 · doi:10.1016/j.cam.2008.06.014