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Iterative solutions to the linear matrix equation AXB + CXTD = E

Authors: Yongxin Yuan, Jiashang Jiang

Abstract:

In this paper the gradient based iterative algorithm is presented to solve the linear matrix equation AXB +CXTD = E, where X is unknown matrix, A,B,C,D,E are the given constant matrices. It is proved that if the equation has a solution, then the unique minimum norm solution can be obtained by choosing a special kind of initial matrices. Two numerical examples show that the introduced iterative algorithm is quite efficient.

Keywords: matrix equation, iterative algorithm, parameter estimation, minimum norm solution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079458

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References:


[1] G. R. Duan. Solutions to matrix equation AV + BW = V F and their application to eigenstructure assignment in linear systems. IEEE Transactions on Automatic Control, 38 (1993) 276-280.
[2] S. P. Bhattacharyya, E. De Souza. Pole assignment via Sylvester-s equation. Systems and Control Letters, 1 (1972) 261-263.
[3] G. R. Duan. On the solution to Sylvester matrix equation AV +BW = EV F. IEEE Transactions on Automatic Control, 41 (1996) 612-614.
[4] K. Zhou, J. Doyle, K. Glover. Robust and optimal control. Prentice-Hall, 1996.
[5] B. Zhou, G. R. Duan. An explicit solution to the matrix equation AX− XF = BY . Linear Algebra and its Applications, 402 (2005) 345-366.
[6] B. Zhou, G. R. Duan. A new solution to the generalized Sylvester matrix equation AV − EV F = BW. Systems & Control Letters, 55 (2006) 193-198.
[7] F. Ding, T. Chen. Iterative least squares solutions of coupled Sylvester matrix equations. Systems & Control Letters, 54 (2005) 95-107.
[8] H. Mukaidani, H. Xu, K. Mizukami. New iterative algorithm for algebraic Riccati equation related to H∞ control problem of singularly perturbed systems. IEEE Transactions on Automatic Control, 46 (2001) 1659-1666.
[9] I. Borno, Z. Gajic. Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems. Computers & Mathematics with Applications, 30 (1995) 1-4.
[10] C. T. Kelley. Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia, 1995.
[11] T. Mori, A. Derese. A brief summary of the bounds on the solution of the algebraic matrix equations in control theory. International Journal of Control, 39 (1984) 247-256.
[12] M. Mrabti, M. Benseddik. Unified type non-stationary Lyapunov matrix equations-simultaneous eigenvalue bounds. Systems & Control Letters, 18 (1995) 73-81.
[13] L. Xie, Y. Liu, H. Yang. Gradient based and least squares based iterative algorithms for matrix equations AXB + CXTD = F. Applied Mathematics and Computation, 217 (2010) 2191-2199.
[14] L. Xie, J. Ding, F. Ding. Gradient based iterative solutions for general linear matrix equations. Computers and Mathematics with Applications, 58 (2009) 1441-1448.
[15] F. Ding, T. Chen. On iterative solutions of general coupled matrix equations. SIAM Journal on Control and Optimization, 44 (2006) 2269- 2284.
[16] F. Ding, P. X. Liu, J. Ding. Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Applied Mathematics and Computation, 197 (2008) 41-50.
[17] J. Ding, Y. Liu, F. Ting. Iterative solutions to matrix equations of the form AiXBi = Fi. Computers and Mathematics with Applications, 59 (2010) 3500-3507.
[18] P. Lancaster, M. Tismenetsky. The Theory of Matrices. 2rd Edition. London: Academic Press, 1985.
[19] A. Ben-Israel, T. N. E. Greville. Generalized Inverses. Theory and Applications (second ed). New York: Springer, 2003.