Stochastic stability and robust control for sampled-data systems with Markovian jump parameters. (English) Zbl 1211.93131

Summary: The problems of stochastic stability and robust control for a class of uncertain sampled-data systems are studied. The systems consist of random jumping parameters described by finite-state semi-Markov process. Sufficient conditions for stochastic stability or exponential mean square stability of the systems are presented. The conditions for the existence of a sampled-data feedback control and a multirate sampled-data optimal control for the continuous-time uncertain Markovian jump systems are also obtained. The design procedure for robust multirate sampled-data control is formulated as linear matrix inequalities (LMIs), which can be solved efficiently by available software toolboxes. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed techniques.


93E14 Data smoothing in stochastic control theory
60K15 Markov renewal processes, semi-Markov processes
93D09 Robust stability
Full Text: DOI


[1] Aliyu, M. D.S.; Boukas, E. K., Robust \(H_\infty\) control for Markovian jump nonlinear systems, IMA J. Math. Control Inform., 17, 295-308 (2000) · Zbl 0989.93028
[2] Araki, M.; Ito, Y.; Hagiwara, T., Frequency response of sampled-data systems, Automatica, 32, 483-497 (1996) · Zbl 0861.93021
[3] Boukas, E. K., Stabilization of stochastic nonlinear hybrid systems, Int. J. Innovative Computing, Information and Control, 1, 131-141 (2005) · Zbl 1085.93026
[4] Boukas, E. K., Control of Systems with Controlled Jump Markov Disturbances, Control-Theory Adv. Tech., 9, 577-595 (1993)
[5] Bamieh, B.; Pearson, J. B., A general framework for linear periodic systems with application to \(H_\infty\) sampled-data control, IEEE Trans. Automat. Control, 37, 418-435 (1992) · Zbl 0757.93020
[6] Benjelloun, K.; Boukas, E. K.; Costa, O. L., \(H_\infty \)-control for linear time-delay systems with Markovian jumping parameters, J. Optim. Theory Appl., 105, 73-95 (2000) · Zbl 0971.93027
[7] Boukas, E. K.; Shi, P., Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters, Internat. J. Robust Nonlinear Control, 8, 1155-1167 (1998) · Zbl 0918.93060
[8] Chen, T.; Francis, B., Optimal Sampled-data Control Systems (1995), Springer · Zbl 0847.93040
[9] Chen, T.; Qiu, L., \(H_\infty\) design of general multirate sampled-data control systems, Automatica, 30, 1139-1152 (1994) · Zbl 0806.93038
[10] Costa, O. L.V.; Marques, R. P., Mixed \(H_2 / H_\infty \)-control of discrete-time Markovian jump linear systems, IEEE Trans. Automat. Control, 43, 95-100 (1998) · Zbl 0907.93062
[11] Dragan, V., The linear quadratic optimization problem for a class of singularly perturbed stochastic systems, Int. J. Innovative Computing, Information and Control, 1, 53-63 (2005)
[12] Dullerud, D. E.; Glover, K., Robust performance of periods systems, IEEE Trans. Automat. Control, 41, 1146-1159 (1996) · Zbl 0856.93034
[13] Feng, X.; Loparo, K. A.; Ji, Y.; Chizeck, H. J., Stochastic stability properties of jump linear systems, IEEE Trans. Automat. Control, 37, 38-53 (1992) · Zbl 0747.93079
[14] Fragoso, M. D.; Do Val, J. B.R.; Pinto, D. L., Jump linear \(H_\infty\) control: The discrete-time case, Control-Theory Adv. Tech., 10, 1459-1474 (1995)
[15] Gao, H.; Lam, J.; Xie, L.; Wang, C., New approach to mixed \(H_2 / H_\infty\) filtering for polytopic discrete-time systems, IEEE Trans. Signal Process., 53, 3183-3192 (2005) · Zbl 1370.93274
[16] Gao, H.; Wang, C., Delay-dependent robust \(H_\infty\) and \(L_2-L_\infty\) filtering for a class of uncertain nonlinear time-delayed systems, IEEE Trans. Automat. Control, 48, 1661-1666 (2003) · Zbl 1364.93210
[17] Hagiwara, T.; Araki, M., Design of a stable state feedback controller based on the multirate sampling of the plant output, IEEE Trans. Automat. Control, 33, 812-819 (1988) · Zbl 0648.93043
[19] Hu, L.-S.; Cao, Y.-Y.; Shao, H.-H., Constrained robust sampled-data control for nonlinear uncertain systems, Internat. J. Robust Nonlinear Systems, 12, 447-464 (2002) · Zbl 1026.93035
[20] Hu, L.-S.; Lam, J.; Cao, Y.-Y.; Shao, H.-H., LMI approach to robust \(H_2\) sampled-data control for linear uncertain systems, IEEE Trans. Syst. Man Cyber. Part B, 33, 149-155 (2003)
[21] Halevi, Y.; Ray, A., Integrated communication and control systems: Part I—analysis and part II—design consideration, ASME J. Dyn. Syst. Meas. Contr., 110, 367-381 (1988)
[22] Ji, Y.; Chizeck, H. J.; Feng, X.; Loparo, K. A., Stability and control of discrete-time jump linear systems, Control-Theory Adv. Tech., 7, 247-270 (1991)
[23] Kabamba, P. T.; Hara, S., Worst case analysis and design of sampled-data control systems, IEEE Trans. Automat. Control, 38, 1337-1357 (1993) · Zbl 0787.93068
[24] Khammash, M. H., Necessary and sufficient conditions for the robustness of time-varying systems with applications to sampled-data systems, IEEE Trans. Automat. Control, 38, 49-57 (1993) · Zbl 0777.93018
[25] Kushner, H. J., Stochastic Stability and Control (1967), Academic Press: Academic Press New York · Zbl 0178.20003
[26] Lall, S. G.; Dullerud, G. E., An LMI solution to the robust synthesis problem for multi-rate sampled-data systems, Automatica, 37 (2001) · Zbl 1031.93121
[27] Nguang, S. K.; Shi, P., Fuzzy \(H\)-infinity output feedback control of nonlinear systems under sampled measurements, Automatica, 39, 2169-2174 (2003) · Zbl 1041.93033
[28] Nguang, S. K.; Shi, P., \(H_\infty\) filtering of nonlinear sampled-data systems, Automatica, 36, 303-310 (2000) · Zbl 0943.93041
[29] Qiu, L.; Chen, T., \(H_2\)-optimal design of multirate sampled-data systems, IEEE Trans. Automat. Control, 39, 2506-2511 (1994) · Zbl 0825.93436
[30] Shi, P., Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Automat. Control, 43, 1022-1027 (1998) · Zbl 0951.93050
[31] Shi, P., Robust control of linear continuous time-delay systems with finite discrete jumps and norm-bounded uncertainties, Internat. J. Systems Sci., 29, 1381-1392 (1998) · Zbl 1065.93510
[32] Shi, P.; Boukas, E. K., \(H_\infty\) control for Markovian jumping linear systems with parametric uncertainty, J. Optim. Theory Appl., 95, 75-99 (1997) · Zbl 1026.93504
[33] Shi, P.; Boukas, E. K.; Agarwal, R. K., Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delays, IEEE Trans. Automat. Control, 44, 2139-2144 (1999) · Zbl 1078.93575
[34] Sivashankar, N.; Khargonekar, P. P., Robust stability and performance analysis of sampled-data systems, IEEE Trans. Automat. Control, 38, 58-69 (1993) · Zbl 0773.93069
[35] Voulgaris, P. G.; Bamieh, B., Optimal \(H_\infty\) and \(H_2\) control of hybrid multirate systems, Systems Control Lett., 20, 249-261 (1993) · Zbl 0781.93062
[36] Yamamoto, Y.; Khargonekar, P. P., Frequency response of sampled-data systems, IEEE Trans. Automat. Control, 41, 166-176 (1996) · Zbl 0842.93050
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.