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Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. (English) Zbl 1143.65035
By extending the well-known Jacobi and Gauss-Seidel iterations for \({\mathbf{Ax}}= {\mathbf b}\), the iterative solutions of matrix equations \({\mathbf{AXB}}= {\mathbf F}\) and generalized Sylvester matrix equations \({\mathbf {AXB}}+{\mathbf {XB}}= {\mathbf F}\) (including the Sylvester equation \({\mathbf {AX}}+ {\mathbf{XB}}={\mathbf F}\) as a special case) are studied, and a gradient based and a least-squares based iterative algorithms for the solution is proved. It is proved that the iterative solution always converges to the exact solution for any initial values. The basic idea is to regard the unknown matrix \({\mathbf X}\) to be solved as the parameters of a system to be identified, and to obtain the iterative solutions by applying the hierarchical identification principle. Finally, the algorithms are tested and their effectiveness is shown using a numerical example.

65F30 Other matrix algorithms (MSC2010)
Full Text: DOI
[1] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press Baltimore, MD · Zbl 0865.65009
[2] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization, 44, 6, 2269-2284, (2006) · Zbl 1115.65035
[3] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems & control letters, 54, 2, 95-107, (2005) · Zbl 1129.65306
[4] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE transactions on automatic control, 50, 8, 1216-1221, (2005) · Zbl 1365.65083
[5] F. Ding, M. Li, J.Y. Dai, Hierarchical identification principle and a family of iterative methods, in: Proceedings of the 25th Chinese Control Conference, August 7-11, 2006, Harbin, Heilongjiang, PR China, pp. 418-422.
[6] Ljung, L., System identification: theory for the user, (1999), Prentice-Hall Englewood Cliffs, NJ
[7] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 2, 315-325, (2005) · Zbl 1073.93012
[8] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE transactions on automatic control, 50, 3, 397-402, (2005) · Zbl 1365.93551
[9] Ding, F.; Liu, P.X.; Shi, Y., Convergence analysis of estimation algorithms of dual-rate stochastic systems, Applied mathematics and computation, 176, 1, 245-261, (2006) · Zbl 1095.65056
[10] Ding, F.; Xiao, Y.S., A finite-data-window least squares algorithm with a forgetting factor for dynamical modeling, Applied mathematics and computation, 186, 1, 184-192, (2007) · Zbl 1113.93108
[11] Ding, F.; Chen, H.B.; Li, M., Multi-innovation least squares identification methods based on the auxiliary model for MISO systems, Applied mathematics and computation, 187, 2, 658-668, (2007) · Zbl 1114.93101
[12] Ding, F.; Chen, T., Identification of Hammerstein nonlinear ARMAX systems, Automatica, 41, 9, 1479-1489, (2005) · Zbl 1086.93063
[13] Ding, F.; Shi, Y.; Chen, T., Performance analysis of estimation algorithms of non-stationary ARMA processes, IEEE transactions on signal processing, 54, 3, 1041-1053, (2006) · Zbl 1373.94569
[14] Ding, F.; Chen, T., A gradient based adaptive control algorithm for dual-rate systems, Asian journal of control, 8, 4, 314-323, (2006)
[15] Ding, F.; Chen, X.W.; Wang, J.H., Performance analysis of auxiliary model based stochastic gradient parameter estimation for MIMO systems under weak conditions, Dynamics of continuous discrete and impulsive systems-series A-mathematical analysis, 13, Suppl, 993-997, (2006), (Part 2)
[16] Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1, 1-14, (2007) · Zbl 1140.93488
[17] Ding, F.; Chen, T.; Iwai, Z., Adaptive digital control of Hammerstein nonlinear systems with limited output sampling, SIAM journal on control and optimization, 45, 6, 2257-2276, (2007) · Zbl 1126.93034
[18] Ding, F.; Shi, Y.; Chen, T., Auxiliary model based least-squares identification methods for Hammerstein output-error systems, Systems & control letters, 5, 56, 373-380, (2007) · Zbl 1130.93055
[19] Mukaidani, H.; Xu, H.; Mizukami, K., New iterative algorithm for algebraic Riccati equation related to H infinity control problem of singularly perturbed systems, IEEE transactions on automatic control, 46, 10, 1659-1666, (2001) · Zbl 1006.93044
[20] Chu, K.E., The solution of the matrix equations AXB−CXD=E and (YA−DZ, YC−BZ)=(E,F), Linear algebra and its applications, 93, 93-105, (1987) · Zbl 0631.15006
[21] Xu, G.; Wei, M.; Zheng, D., On solutions of matrix equation AXB+CYD=F, Linear algebra and applications, 279, 1-3, 93-109, (1998) · Zbl 0933.15024
[22] Golub, G.H.; Nash, S.; Van Loan, C.F., A hessenberg – schur method for the matrix problem AX+XB=C, IEEE transactions on automatic control, 24, 6, 909-913, (1979) · Zbl 0421.65022
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