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Hu, Lisheng; Shi, Peng; Huang, Biao

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

J. Math. Anal. Appl. 313, No. 2, 504-517 (2006).
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.

Cited in 23 Documents

MSC:

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

Keywords:

Markovian jump system; stochastic stability; robust control; sampled-data system; linear matrix inequality

Software:

MR and LTV Synthesis Tools
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\textit{L. Hu} et al., J. Math. Anal. Appl. 313, No. 2, 504--517 (2006; Zbl 1211.93131)
Full Text: DOI

References:

[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
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