## Iterative least-squares solutions of coupled sylvester matrix equations.(English)Zbl 1129.65306

Summary: We present a general family of iterative methods to solve linear equations, which includes the well-known Jacobi and Gauss–Seidel iterations as its special cases. The methods are extended to solve coupled Sylvester matrix equations. In our approach, we regard the unknown matrices to be solved as the system parameters to be identified, and propose a least-squares iterative algorithm by applying a hierarchical identification principle and by introducing the block-matrix inner product (the star product for short). We prove that the iterative solution consistently converges to the exact solution for any initial value. The algorithms proposed require less storage capacity than the existing numerical ones. Finally, the algorithms are tested on computer and the results verify the theoretical findings.

### MSC:

 65F10 Iterative numerical methods for linear systems 93B40 Computational methods in systems theory (MSC2010) 93E10 Estimation and detection in stochastic control theory
Full Text:

### References:

 [1] Barraud, A., A numerical algorithm to solve $$A^{\operatorname{T}} \mathit{XA} - X = Q$$, IEEE trans. automat. control, 22, 883-885, (1977) · Zbl 0361.65022 [2] Bitmead, R., Explicit solutions of the discrete-time Lyapunov matrix equation and kalman – yakubovich equations, IEEE trans. automat. control, 26, 1291-1294, (1981) · Zbl 0465.93033 [3] Bitmead, R.; Weiss, H., On the solution of the discrete-time Lyapunov matrix equation in controllable canonical form, IEEE trans. automat. control, 24, 481-482, (1979) · Zbl 0404.93018 [4] Borno, I., Parallel computation of the solutions of coupled algebraic Lyapunov equations, Automatica, 31, 1345-1347, (1995) · Zbl 0825.93992 [5] Chen, T.; Francis, B.A., Optimal sampled-data control systems, (1995), Springer London · Zbl 0847.93040 [6] Chen, T.; Qiu, L., $$\mathcal{H}_\infty$$ design of general multirate sampled-data control systems, Automatica, 30, 139-1152, (1994) [7] Chu, K.E., The solution of the matrix $$\mathit{AXB} - \mathit{CXD} = E$$ and $$(\mathit{YA} - \mathit{DZ}, \mathit{YC} - \mathit{BZ}) = (E, F)$$, Linear algebra appl., 93, 93-105, (1987) [8] Climent, J.J.; Perea, C., Convergence and comparison theorems for a generalized alternating iterative method, Appl. math. comput., 143, 1-14, (2003) · Zbl 1040.65029 [9] Corach, G.; Stojanoff, D., Index of Hadamard multiplication by positive matrices II, Linear algebra appl., 332-334, 503-517, (2001) · Zbl 0988.15006 [10] Fang, Y.; Loparo, K.A.; Feng, X., New estimates for solutions of Lyapunov equations, IEEE trans. automat. control, 42, 408-411, (1997) · Zbl 0866.93048 [11] Fischer, P.; Stegeman, J.D., Fractional Hadamard powers of positive semidefinite matrices, Linear algebra appl., 371, 53-74, (2003) · Zbl 1041.15012 [12] Garloff, J., Bounds for the eigenvalues of the solution of the discrete Riccati and Lyapunov equation and the continuous Lyapunov equation, Internat. J. control, 43, 423-431, (1986) · Zbl 0591.15010 [13] Golub, G.H.; Nash, S.; Van Loan, C.F., A hessenberg – schur method for the matrix problem $$\mathit{AX} + \mathit{XB} = C$$, IEEE trans. automat. control, 24, 909-913, (1979) · Zbl 0421.65022 [14] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009 [15] Heinen, J., A technique for solving the extended discrete Lyapunov matrix equation, IEEE trans. automat. control, 17, 156-157, (1972) · Zbl 0262.93028 [16] Hmamed, A., Discrete Lyapunov equationsimultaneous eigenvalue bounds, Internat. J. control, 22, 1121-1126, (1991) · Zbl 0735.15014 [17] Johnson, C.R.; Elsner, L., The relationship between Hadamard and conventional multiplication for positive definite matrices, Linear algebra appl., 92, 231-240, (1987) · Zbl 0623.15011 [18] Jonsson, I.; Kägström, B., Recursive blocked algorithms for solving triangular systems—part ione-sided and coupled Sylvester-type matrix equations, ACM trans. math. software, 28, 392-415, (2002) · Zbl 1072.65061 [19] Jonsson, I.; Kägström, B., Recursive blocked algorithms for solving triangular systems—part iitwo-sided and generalized Sylvester and Lyapunov matrix equations, ACM trans. math. software, 28, 416-435, (2002) · Zbl 1072.65062 [20] Kägström, B., A perturbation analysis of the generalized Sylvester equation $$(\mathit{AR} - \mathit{LB}, \mathit{DR} - \mathit{LE}) = (C, F)$$, SIAM J. matrix anal. appl., 15, 1045-1060, (1994) · Zbl 0805.65045 [21] Komaroff, N., Simultaneous eigenvalue lower bounds for the Lyapunov matrix equation, IEEE trans. automat. control, 33, 126-128, (1988) · Zbl 0637.15009 [22] Komaroff, N., Lower bounds for the solution of the discrete algebraic Lyapunov equation, IEEE trans. automat. control, 37, 1017-1019, (1992) · Zbl 0775.93186 [23] Komaroff, N., Upper summation and product bounds for solution eigenvalues of the Lyapunov matrix equation, IEEE trans. automat. control, 37, 1040-1042, (1992) · Zbl 0767.93069 [24] Kwon, W.H.; Moon, Y.S.; Ahn, S.C., Bounds in algebraic Riccati and Lyapunov equationsa survey and some new results, Internat. J. control, 64, 377-389, (1996) · Zbl 0852.93005 [25] Lee, C.H., Upper and lower matrix bounds of the solution for the discrete Lyapunov equation, IEEE trans. automat. control, 41, 1338-1341, (1996) · Zbl 0861.93016 [26] Lee, C.H., On the matrix bounds for the solution matrix of the discrete algebraic Riccati equation, IEEE trans. circuits and systems I, 43, 402-407, (1996) [27] Ljung, L., System identificationtheory for the user, (1999), Prentice-Hall Englewood Cliffs, NJ [28] Mori, T.; Derese, A., A brief summary of the bounds on the solution of the algebraic matrix equations in control theory, Internat. J. control, 39, 247-256, (1984) · Zbl 0527.93030 [29] Mori, T.; Kokame, H., On solution bounds for three types of Lyapunov matrix equationscontinuous, discrete and unified equations, IEEE trans. automat. control, 47, 1767-1770, (2002) · Zbl 1364.93323 [30] Mrabti, M.; Benseddik, M., Unified type non-stationary Lyapunov matrix equation—simultaneous eigenvalue bounds, Systems control lett., 24, 53-59, (1995) · Zbl 0866.93045 [31] Mrabti, M.; Hmamed, A., Bounds for the solution of the Lyapunov matrix equation—a unified approach, Systems control lett., 18, 73-81, (1992) · Zbl 0743.93075 [32] Mukaidani, H.; Xu, H.; Mizukami, K., New iterative algorithm for algebraic Riccati equation related to $$H_\infty$$ control problem of singularly perturbed systems, IEEE trans. automat. control, 46, 1659-1666, (2001) · Zbl 1006.93044 [33] Qiu, L.; Chen, T., Contractive completion of block matrices and its application to $$\mathcal{H}_\infty$$ control of periodic systems, (), 263-281 · Zbl 0857.93032 [34] Qiu, L.; Chen, T., Multirate sampled-data systemsall H∞ suboptimal controllers and the minimum entropy controller, IEEE trans. automat. control, 44, 537-550, (1999) · Zbl 0958.93031 [35] Qiu, L.; Chen, T., Unitary dilation approach to contractive matrix completion, Linear algebra appl., 379, 345-352, (2004) · Zbl 1056.15015 [36] Starke, G.; Niethammer, W., SOR for $$\mathit{AX} - \mathit{XB} = C$$, Linear algebra appl., 154, 355-375, (1991) · Zbl 0736.65031 [37] Tippert, M.K.; Marchesin, D., Upper bounds for the solution of the discrete algebraic Lyapunov equation, Automatica, 35, 1485-1489, (1999) · Zbl 1126.93351 [38] Xiang, S., On an inequality for the Hadamard product of an M-matrix or an H-matrix and its inverse, Linear algebra appl., 367, 17-27, (2003) · Zbl 1019.15006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.