×

zbMATH — the first resource for mathematics

Iterative solutions of coupled discrete Markovian jump Lyapunov equations. (English) Zbl 1139.60334
Summary: Two iterative methods are given to solve coupled discrete Markovian jump Lyapunov equations arising from the stability analysis of discrete-time Markovian jump systems. When such equations have unique positive definite solutions, a sufficient condition for the convergence of iterates for one iterative method is derived. When such equations only have unique solutions, but not necessarily positive definite, the above iterative method may fail and an alterative iterative method is provided by using the solutions of discrete-time Lyapunov equations as its intermediate steps. A sufficient condition is also presented for the convergence of the iterates.

MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
39A10 Additive difference equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boukas, E.K.; Liu, Z.K., Robust \(H_\infty\) control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE transactions on automatic control, 46, 12, 1918-1924, (2001) · Zbl 1005.93050
[2] 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
[3] de Souza, C.E.; Fragoso, M.D., \(H_\infty\) filtering for discrete-time linear systems with Markovian jumping parameters, International journal of robust and nonlinear control, 13, 14, 1299-1316, (2003) · Zbl 1043.93016
[4] Gao, H.; Lam, J.; Xu, S.; Wang, C., Stabilization and \(H_\infty\) control of two-dimensional Markovian jump systems, IMA journal of mathematical control and information, 21, 4, 377-392, (2004) · Zbl 1069.93007
[5] Ji, Y.; Chizeck, H.J., Controllability, observability and discrete-time Markovian jump linear quadratic control, International journal of control, 48, 2, 481-498, (1988) · Zbl 0669.93007
[6] Wang, Z.; Lam, J.; Liu, X., Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE transactions on circuits and systems-II: analog and digital signal processing, 51, 5, 262-268, (2004)
[7] Wang, Z.; Lam, J.; Liu, X., Robust filtering for discrete-time Markovian jump delay systems, IEEE signal processing letters, 11, 8, 659-662, (2004)
[8] Xu, S.; Chen, T.; Lam, J., Robust \(H_\infty\) filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE transactions on automatic control, 48, 5, 900-907, (2003) · Zbl 1364.93816
[9] Zhang, L.; Huang, B.; Lam, J., \(H_\infty\) model reduction of Markovian jump linear systems, Systems & control letters, 50, 2, 103-118, (2003) · Zbl 1157.93519
[10] Kubrusly, C.S.; Costa, O.L.V., Mean square stability conditions for discrete stochastic bilinear systems, IEEE transactions on automatic control, 30, 11, 1082-1087, (1985) · Zbl 0591.93061
[11] 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
[12] Ding, F.; Chen, T., Iterative least-squares solutions of coupled Sylvester matrix equations, Systems & control letters, 54, 2, 95-107, (2005) · Zbl 1129.65306
[13] Berman, A.; Plemmons, R.J., ()
[14] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[15] Chen, M.; Li, X., An estimation of the spectral radius of a product of block matrices, Linear algebra and its applications, 379, 267-275, (2004) · Zbl 1043.15012
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.