On robust stabilization of Markovian jump systems with uncertain switching probabilities. (English) Zbl 1093.93026

Summary: This brief paper is concerned with the robust stabilization problem for a class of Markovian jump linear systems with uncertain switching probabilities. The uncertain Markovian jump system under consideration involves parameter uncertainties both in the system matrices and in the mode transition rate matrix. First, a new criterion for testing the robust stability of such systems is established in terms of linear matrix inequalities. Then, a sufficient condition is proposed for the design of robust state-feedback controllers. A globally convergent algorithm involving convex optimization is also presented to help construct such controllers effectively. Finally, a numerical simulation is used to illustrate the developed theory.


93D21 Adaptive or robust stabilization
93E15 Stochastic stability in control theory
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[2] Boukas, E. K.; Shi, P.; Benjelloun, K., On stabilization of uncertain linear systems with jump parameters, International Journal of Control, 72, 9, 842-850 (1999) · Zbl 0958.93083
[3] Costa, O. L.V.; Val, J. B.R.; Geromel, J. C., Continuous-time state-feedback \(H_2\)-control of Markovian jump linear system via convex analysis, Automatica, 35, 259-268 (1999) · Zbl 0939.93041
[4] de Farias, D. P.; Geromel, J. C.; do Val, J. B.R.; Costa, O. L.V., Output feedback control of Markov jump linear systems in continuous-time, IEEE Transactions on Automatic Control, 45, 5, 944-949 (2000) · Zbl 0972.93074
[5] do Val, J. B.R.; Geromel, J. C.; Goncalves, A. P.C., The \(H_2\)-control for jump linear systemscluster observations of the Markov state, Automatica, 38, 343-349 (2002) · Zbl 0991.93125
[6] El Ghaoui, L.; Oustry, F.; Rami, M. A., A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42, 8, 1171-1176 (1997) · Zbl 0887.93017
[7] El Ghaoui, L.; Rami, M. A., Robust state-feedback stabilization of jump linear systems via LMIs, International Journal of Robust and Nonlinear Control, 6, 9-10, 1015-1022 (1996) · Zbl 0863.93067
[8] Feng, X.; Loparo, K. A.; Ji, Y.; Chizeck, H. J., Stochastic stability properties of jump linear systems, IEEE Transactions on Automatic Control, 37, 1, 38-53 (1992) · Zbl 0747.93079
[9] Ji, Y.; Chizeck, H. J., Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control, 35, 7, 777-788 (1990) · Zbl 0714.93060
[10] Leibfritz, F., An LMI-based algorithm for designing suboptimal static \(H_2 / H_\infty\) output feedback controllers, SIAM Journal on Control and Optimization, 39, 6, 1711-1735 (2001) · Zbl 0997.93032
[11] Lee, Y. S.; Moon, Y. S.; Kwon, W. H.; Park, P. G., Delay-dependent robust \(H_\infty\) control for uncertain systems with a state-delay, Automatica, 40, 65-72 (2004) · Zbl 1046.93015
[12] Mao, X., Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Transactions on Automatic Control, 47, 10, 1604-1612 (2002) · Zbl 1364.93685
[13] Mariton, M., Jump linear systems in automatic control (1990), Marcel Dekker: Marcel Dekker New York
[14] Shi, P.; Boukas, E. K.; Agarwal, R. K., Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Transactions on Automatic Control, 44, 8, 1592-1597 (1999) · Zbl 0986.93066
[15] 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
[16] Zhang, L.; Huang, B.; Lam, J., \(H_\infty\) model reduction of Markovian jump linear systems, Systems & Control Letters, 50, 103-118 (2003) · Zbl 1157.93519
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