×

Observer-based finite-time \(H_\infty\) control of discrete-time Markovian jump systems. (English) Zbl 1270.93046

Summary: This paper investigates the observer-based finite-time \(H_\infty\) control problem for one family of discrete-time Markovian jump systems with time-varying norm-bounded disturbance. Firstly, the concepts of stochastic finite-time boundedness and stochastic \(H_\infty\) finite-time stabilization via observer-based state feedback are given. Then, under the assumption that the state vector is not available for feedback, an observer-based state feedback controller is designed to ensure stochastic finite-time stabilization or stochastic \(H_\infty\) finite-time stabilization via observer-based state feedback of the resulting closed-loop error discrete-time Markovian jump system. Sufficient criteria on the stochastic finite-time stabilization and stochastic \(H_\infty\) finite-time stabilization via observer-based state feedback are given in form of linear matrix inequalities with a fixed parameter, respectively. Finally, simulation examples are presented to illustrate the validity of the developed techniques.

MSC:

93B36 \(H^\infty\)-control
93E15 Stochastic stability in control theory
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Krassovskii, N. N.; Lidskii, E. A., Analytical design of controllers in systems with random attributes, Automat. Rem. Control, 22, 1021-1025 (1961), (1141-1146, 1289-1294) · Zbl 0104.36704
[2] Ji, Y.; Chizeck, H. J.; Feng, X.; Loparo, K. A., Stability and control of discrete-time jump linear-systems, Control Theory Adv. Technol., 7, 2, 247-270 (1991)
[3] Costa, O. L.V.; Fragoso, M. D., Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl., 179, 1, 154-178 (1993) · Zbl 0790.93108
[4] Mao, X., Stability of stochastic differential equations with Markovian switching, Stoch. Process. Appl., 79, 1, 45-67 (1999) · Zbl 0962.60043
[5] Shi, P.; Xia, Y.; Liu, G.; Rees, D., On designing of sliding mode control for stochastic jump systems, IEEE Trans. Automat. Control, 51, 1, 97-103 (2006) · Zbl 1366.93682
[6] Wu, L.; Shi, P.; Gao, H., State estimation and sliding mode control of Markovian jump singular systems, IEEE Trans. Automat. Control, 55, 5, 1213-1219 (2010) · Zbl 1368.93696
[7] Chizeck, H. J.; Willsky, A. S.; Castanon, D., Discrete-time Markovian-jump linear quadratic optimal-control, Int. J. Control, 43, 1, 213-231 (1986) · Zbl 0591.93067
[8] Costa, O. L.V.; de Paulo, W. L., Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems, Automatica, 43, 4, 587-597 (2007) · Zbl 1115.49021
[9] Xu, S.; Chen, T.; Lam, J., Robust \(H_\infty\) filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Trans. Automat. Control, 48, 5, 900-907 (2003) · Zbl 1364.93816
[10] Gao, H.; Lam, J.; Xie, L.; Wang, C., New approach to mixed \(H_2\) and \(H_\infty\) filtering for polytopic discrete-time systems, IEEE Trans. Signal Process., 53, 8, 3183-3191 (2005) · Zbl 1370.93274
[11] Liu, J.; Gu, Z.; Hu, S., \(H_\infty\) filtering for Markovian jump systems with time-varying delays, Int. J. Innov. Comput. Inf. Control, 7, 3, 1299-1310 (2011)
[12] Ding, Q.; Zhong, M., On designing \(H_\infty\) fault detection filter for Markovian jump linear systems with polytopic uncertainties, Int. J. Innov. Comput. Inf. Control, 6, 3(A), 995-1004 (2010)
[13] 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, 1, 95-100 (1998) · Zbl 0907.93062
[14] Wang, Z.; Huang, L.; Yang, X., \(H_\infty\) performance for a class of uncertain stochastic nonlinear Markovian jump systems with time-varying delay via adaptive control method, Appl. Math. Model., 35, 4, 1983-1993 (2011) · Zbl 1217.93159
[15] Seiler, P.; Sengupta, R., A bounded real lemma for jump systems, IEEE Trans. Automat. Control, 48, 9, 1651-1654 (2003) · Zbl 1364.93223
[16] Shi, P.; Boukas, E. K.; Agarwal, R. K., Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE Trans. Automat. Control, 44, 11, 2139-2144 (1999) · Zbl 1078.93575
[17] Souza, C. E., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Trans. Automat. Control, 51, 5, 836-841 (2006) · Zbl 1366.93479
[18] Karan, M.; Shi, P.; Kaya, C. Y., Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems, Automatica, 42, 12, 2159-2168 (2006) · Zbl 1104.93056
[19] Boukas, E. K.; Shi, P., Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters, Int. J. Robust Nonlinear Control, 8, 13, 1155-1167 (1998) · Zbl 0918.93060
[20] Nakura, G., Stochastic optimal tracking with preview by state feedback for linear discrete-time markovian jump systems, Int. J. Innov. Comput. Inf. Control, 6, 1, 15-27 (2010)
[21] Yin, Y.; Shi, P.; Liu, F., Gain scheduled PI tracking control on stochastic nonlinear systems with partially known transition probabilities, J. Franklin Inst., 348, 4, 685-702 (2011) · Zbl 1227.93127
[22] Kushner, H. J., Stochastic Stability and Control (1967), Academic Press: Academic Press New York · Zbl 0244.93065
[23] Costa, O. L.V.; Fragoso, M. D.; Marques, R. P., Discrete-Time Markov Jump Linear Systems (2005), Springer: Springer London · Zbl 1081.93001
[24] Amato, F.; Ariola, M.; Dorato, P., Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 9, 1459-1463 (2001) · Zbl 0983.93060
[25] EI-Gohary, A.; AI-Ruzaiza, A. S., Optimal control of non-homogenous prey-predator models during infinite and finite-time intervals, Appl. Math. Comput., 146, 2-3, 495-508 (2003) · Zbl 1026.92044
[26] EI-Gohary, A., Optimal control of an angular motion of a rigid body during infinite and finite-time intervals, Appl. Math. Comput., 141, 2-3, 541-551 (2003) · Zbl 1041.70022
[28] Weiss, L.; Infante, E. F., Finite time stability under perturbing forces and on product spaces, IEEE Trans. Automat. Control, 12, 1, 54-59 (1967) · Zbl 0168.33903
[29] D’Angelo, H., Linear Time-Varying Systems: Analysis and Synthesis (1970), Allyn and Bacon: Allyn and Bacon Boston · Zbl 0202.08502
[30] Kushner, H. J., Finite-time stochastic stability and the analysis of tracking systems, IEEE Trans. Automat. Control, 11, 2, 219-227 (1966)
[31] Zhang, W.; An, X., Finite-time control of linear stochastic systems, Int. J. Innov. Comput. Inf. Control, 4, 3, 689-696 (2008)
[32] Amato, F.; Ambrosino, R.; Ariola, M.; Cosentino, C., Finite-time stability of linear time-varying systems with jumps, Automatica, 45, 5, 1354-1358 (2009) · Zbl 1162.93375
[33] Ambrosino, R.; Calabrese, F.; Cosentino, C.; De, T. G., Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Automat. Control, 54, 4, 861-865 (2009) · Zbl 1367.93425
[34] Amato, F.; Ariola, M.; Cosentino, C., Finite-time control of discrete-time linear systems: analysis and design conditions, Automatica, 46, 5, 919-924 (2010) · Zbl 1191.93099
[35] Garcia, G.; Tarbouriech, S.; Bernussou, J., Finite-time stabilization of linear time-varying continuous systems, IEEE Trans. Automat. Control, 54, 2, 364-369 (2009) · Zbl 1367.93060
[36] Yang, Y.; Li, J. M.; Chen, G. P., Finite-time stability and stabilization of nonlinear stochastic hybrid systems, J. Math. Anal. Appl., 356, 1, 338-345 (2009) · Zbl 1163.93033
[37] Yang, D.; Cai, K., Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems, Expert Syst. Appl., 37, 10, 6963-6967 (2010)
[38] He, S.; Liu, F., Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities, Appl. Math. Model., 35, 6, 2631-2638 (2011) · Zbl 1219.93143
[39] Meng, Q.; Shen, Y., Finite-time \(H_\infty\) control for linear continuous system with norm-bounded disturbance, Commun. Nonlinear Sci. Numer. Simul., 14, 1043-1049 (2009) · Zbl 1221.93066
[40] Shen, Y.; Li, C., LMI-based finite-time boundedness analysis of neural networks with parametric uncertainties, Neurocomputing, 71, 4-6, 502-507 (2008)
[41] Amato, F.; Ariola, M., Finite-time control of discrete-time linear systems, IEEE Trans. Automat. Control, 50, 5, 724-729 (2005) · Zbl 1365.93182
[42] Zhang, Y.; Cheng, W.; Mu, X.; Liu, C., Stochastic \(H_\infty\) finite-time control of discrete-time systems with packet loss, Math. Probl. Eng., 2012, 897481, 1-15 (2012)
[43] Amato, F.; Ariola, M.; Dorato, P., Finite-time stabilzation via dynamic output feedback, Automatica, 42, 2, 337-342 (2006) · Zbl 1099.93042
[44] Zhang, Y.; Liu, C.; Mu, X., Robust finite-time \(H_\infty\) control of singular stochastic systems via static output feedback, Appl. Math. Comput., 218, 9, 5629-5640 (2012) · Zbl 1238.93121
[45] Zhang, Y.; Liu, C.; Mu, X., Robust finite-time stabilization of uncertain singular Markovian jump systems, Appl. Math. Model., 36, 10, 5109-5121 (2012) · Zbl 1252.93130
[46] Lin, X.; Du, H.; Li, S., Finite-time boundedness and \(L_2\)-gain analysis for switched delay systems with norm-bounded disturbance, Appl. Math. Comput., 217, 12, 5982-5993 (2011) · Zbl 1218.34082
[47] Syrmos, V. L.; Abdallah, C. T.; Dorato, P.; Grigoriadis, K., Static output feedback-A survey, Automatica, 33, 2, 125-137 (1997) · Zbl 0872.93036
[48] Boukas, E. K., Static output feedback control for stochastic hybrid systems: LMI approach, Automatica, 42, 1, 183-188 (2006) · Zbl 1121.93365
[49] Chen, J., Robust \(H_\infty\) output dynamic observer-based control of uncertain time-delay systems, Chaos Solitons Fract., 31, 2, 391-403 (2007) · Zbl 1142.93329
[50] Lee, K. H.; Lee, J. H.; Kwon, W. H., Sufficient LMI conditions for \(H_\infty\) output feedback stabilization of linear discrete-time systems, IEEE Trans. Automat. Control, 51, 4, 675-680 (2006) · Zbl 1366.93505
[51] He, S.; Liu, F., Observer-based finite-time control of time-delayed jump systems, Appl. Math. Comput., 217, 6, 2327-2338 (2010) · Zbl 1207.93113
[52] Liu, H.; Shen, Y.; Zhao, X., Delay-dependent observer-based \(H_\infty\) finite-time control for switched systems with time-varying delay, Nonlinear Anal. Hybrid Syst., 6, 3, 885-898 (2012) · Zbl 1244.93045
[53] Zhang, Y.; Cheng, W.; Mu, X.; Guo, X., Observer-based finite-time \(H_\infty\) control of singular Markovian jump systems, J. Appl. Math., 2012, 205727, 1-19 (2012)
[54] Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V., Linear Matrix Inequality in Systems and Control Theory. Linear Matrix Inequality in Systems and Control Theory, SIAM Studies in Applied Mathematics (1994), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA
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.