## 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:

 93B40 Computational methods in systems theory (MSC2010) 93C05 Linear systems in control theory 15A18 Eigenvalues, singular values, and eigenvectors 60J75 Jump processes (MSC2010)

KELLEY
Full Text:

### References:

  Zhou, B.; Duan, G. R., An explicit solution to the matrix equation $$A X - X F = B Y$$, Linear Algebra and its Applications, 402, 1, 345-366 (2005)  Zhou, B.; Duan, G. R., A new solution to the generalized Sylvester matrix equation $$A V - E V F = B W$$, Systems & Control Letters, 55, 3, 193-198 (2006)  Zhou, B.; Duan, G. R., On the generalized Sylvester mapping and matrix equations, Systems & Control Letters, 57, 3, 200-208 (2008) · Zbl 1129.93018  Duan, G. R., On the solution to Sylvester matrix equation $$A V + B W = E V F$$, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 41, 4, 612-614 (1996) · Zbl 0855.93017  Qiu, L.; Chen, T., Unitary dilation approach to contractive matrix completion, Linear Algebra and Its Applications, 379, 345-352 (2004) · Zbl 1056.15015  Chen, T.; Qiu, L., $$H_\infty$$ design of general multirate sampled-date control systems, Automatica, 30, 1139-1152 (1994) · Zbl 0806.93038  Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems & Control Letters, 54, 2, 95-107 (2005) · Zbl 1129.65306  Mukaidani, H.; Xu, H.; Mizukami, K., New iterative algorithm for algebraic Riccati equation related to $$H_\infty$$ control problem of singularly perturbed systems, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 46, 1659-1666 (2001) · Zbl 1006.93044  Borno, I.; Gajic, Z., Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems, Computers & Mathematics with Applications, 30, 7, 1-4 (1995) · Zbl 0837.93075  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, 8, 1216-1221 (2005) · Zbl 1365.65083  Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations (1995), SIAM: SIAM Philadelphia · Zbl 0832.65046  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  Mrabti, M.; Benseddik, M., Unified type non-stationary Lyapunov matrix equations—simultaneous eigenvalue bounds, Systems & Control Letters, 18, 73-81 (1995) · Zbl 0866.93045  Wang, Q.; Lam, J.; Wei, Y.; Chen, T., Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and Mathematics with Applications, 55, 4, 843-850 (2008) · Zbl 1139.60334  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  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  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, 1, 154-178 (1993) · Zbl 0790.93108  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  Doyle, J. C.; Glover, K.; Khargonekar, P. P.; Francis, B. A., State-space solutions to standard $$H_2$$ and $$H_\infty$$ control problems, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, 34, 8, 831-847 (1989) · Zbl 0698.93031  Zhou, K.; Doyle, J.; Glover, K., Robust and Optimal Control (1996), Prentice-Hall  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, 2, 759-776 (2009) · Zbl 1161.65034
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.