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Nonfragile \(H_\infty\) control for stochastic systems with Markovian jumping parameters and random packet losses. (English) Zbl 1406.93109

Summary: This paper is concerned with the nonfragile \(H_\infty\) control problem for stochastic systems with Markovian jumping parameters and random packet losses. The communication between the physical plant and controller is assumed to be imperfect, where random packet losses phenomenon occurs in a random way. Such a phenomenon is represented by a stochastic variable satisfying the Bernoulli distribution. The purpose is to design a nonfragile controller such that the resulting closed-loop system is stochastically mean square stable with a guaranteed \(H_\infty\) performance level \(\gamma\). By using the Lyapunov function approach, some sufficient conditions for the solvability of the previous problem are proposed in terms of linear matrix inequalities (LMIs), and a corresponding explicit parametrization of the desired controller is given. Finally, an example illustrating the effectiveness of the proposed approach is presented.

MSC:

93B36 \(H^\infty\)-control
93E03 Stochastic systems in control theory (general)
60J75 Jump processes (MSC2010)
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[1] Arnold, L., Stochastic Differential Equations: Theory and Applications (1974), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0278.60039
[2] Wu, L.; Zheng, W. X.; Gao, H., Dissipativity-based sliding mode control of switched stochastic systems, IEEE Transactions on Automatic Control, 58, 3, 785-791 (2013) · Zbl 1369.93585 · doi:10.1109/TAC.2012.2211456
[3] Gard, T. C., Introduction to Stochastic Differential Equations. Introduction to Stochastic Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 114 (1988), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 0628.60064
[4] Gao, H.; Lam, J.; Wang, C., Robust energy-to-peak filter design for stochastic time-delay systems, Systems & Control Letters, 55, 2, 101-111 (2006) · Zbl 1129.93538 · doi:10.1016/j.sysconle.2005.05.005
[5] Wu, Z.-G.; Shi, P.; Su, H.; Chu, J., Dissipativity analysis for discrete-time stochastic neural networks with time-varying delays, IEEE Transactions on Neural Networks and Learning Systems, 24, 3, 345-355 (2013) · doi:10.1109/TNNLS.2012.2232938
[6] Shen, H.; Xu, S.; Song, X.; Shi, G., Passivity-based control for Markovian jump systems via retarded output feedback, Circuits, Systems, and Signal Processing, 31, 1, 189-202 (2012) · Zbl 1242.93040 · doi:10.1007/s00034-011-9328-3
[7] Shen, H.; Wang, Z.; Huang, X.; Wang, J., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute (2013) · Zbl 1290.93110 · doi:10.1016/j.jfranklin.2013.02.031
[8] Zhao, H.; Chen, Q.; Xu, S., \(H_\infty\) guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays, Journal of the Franklin Institute, 346, 10, 945-957 (2009) · Zbl 1185.93036 · doi:10.1016/j.jfranklin.2009.05.007
[9] Krasovskiĭ, N. N.; Lidskiĭ, È. A., Analytical design of controllers in systems with random attributes—Part I, Automation and Remote Control, 22, 1021-1025 (1961) · Zbl 0104.36704
[10] Boukas, E.-K.; Liu, Z.-K., Deterministic and Stochastic Time-Delay Systems (2002), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0998.93041
[11] Costa, O. L. V.; Fragoso, M. D.; Todorov, M. G., Continuous-Time Markov Jump Linear Systems. Continuous-Time Markov Jump Linear Systems, Probability and Its Applications (2013), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1277.60003 · doi:10.1007/978-3-642-34100-7
[12] He, S.; Liu, F., Unbiased \(H_\infty\) filtering for neutral Markov jump systems, Applied Mathematics and Computation, 206, 1, 175-185 (2008) · Zbl 1152.93052 · doi:10.1016/j.amc.2008.08.046
[13] He, S.; Liu, F., Robust finite-time stabilization of uncertain fuzzy jump systems, International Journal of Innovative Computing, Information and Control, 6, 9, 3853-3862 (2010)
[14] Shen, H.; Xu, S.; Zhou, J.; Lu, J., Fuzzy \(H_\infty\) filtering for nonlinear Markovian jump neutral systems, International Journal of Systems Science, 42, 5, 767-780 (2011) · Zbl 1233.93091 · doi:10.1080/00207721003790351
[15] Wu, Z.-G.; Shi, P.; Su, H.; Chu, J., Asynchronous \(l_2 - l_\infty\) filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities, Automatica (2013) · Zbl 1417.93317 · doi:10.1016/j.automatica.2013.09.041
[16] Wu, Z.-G.; Su, H.; Shi, P.; Chu, J., Analysis and Synthesis of Singular Systems with Time-Delays. Analysis and Synthesis of Singular Systems with Time-Delays, Lecture Notes in Control and Information Sciences, 443 (2013), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1362.93001 · doi:10.1007/978-3-642-37497-5
[17] Zhang, B.; Zheng, W. X.; Xu, S., Filtering of Markovian jump delay systems based on a new performance index, IEEE Transactions on Circuits and Systems I, 60, 5, 1250-1263 (2013) · Zbl 1468.94288 · doi:10.1109/TCSI.2013.2246213
[18] Shen, H.; Xu, S.; Song, X.; Luo, J., Delay-dependent robust stabilization for uncertain stochastic switching systems with distributed delays, Asian Journal of Control, 11, 5, 527-535 (2009) · doi:10.1002/asjc.133
[19] Shen, H.; Xu, S.; Lu, J.; Zhou, J., Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays, Journal of the Franklin Institute, 349, 5, 1665-1680 (2012) · Zbl 1254.93148 · doi:10.1016/j.jfranklin.2011.11.011
[20] Costa, O. L. V.; de Oliveira, A., Optimal mean-variance control for discrete-time linear systems with Markovian jumps and multiplicative noises, Automatica, 48, 2, 304-315 (2012) · Zbl 1260.93173 · doi:10.1016/j.automatica.2011.11.009
[21] He, S.; Liu, C.-L., Finite-time \(H_\infty\) fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback, IEEE Transactions on Fuzzy Systems, 20, 4, 605-614 (2012) · doi:10.1109/TFUZZ.2011.2177842
[22] Ma, H.; Jia, Y., \(H_2\) control of discrete-time periodic systems with Markovian jumps and multiplicative noise, International Journal of Control, 86, 10, 1837-1849 (2013) · Zbl 1312.93115 · doi:10.1080/00207179.2013.797108
[23] Niu, Y.; Ho, D. W. C.; Wang, X., Sliding mode control for Itô stochastic systems with Markovian switching, Automatica, 43, 10, 1784-1790 (2007) · Zbl 1119.93063 · doi:10.1016/j.automatica.2007.02.023
[24] Yang, G.-H.; Che, W.-W., Non-fragile \(H_\infty\) filter design for linear continuous-time systems, Automatica, 44, 11, 2849-2856 (2008) · Zbl 1152.93365 · doi:10.1016/j.automatica.2008.03.018
[25] Dong, H.; Wang, Z.; Ho, D. W. C.; Gao, H., Robust \(H_\infty\) fuzzy output-feedback control with multiple probabilistic delays and multiple missing measurements, IEEE Transactions on Fuzzy Systems, 18, 4, 712-725 (2010) · doi:10.1109/TFUZZ.2010.2047648
[26] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory. Linear Matrix Inequalities in System and Control Theory, Studies in Applied and Numerical Mathematics, 15 (1994), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 0816.93004 · doi:10.1137/1.9781611970777
[27] Yang, R.; Shi, P.; Liu, G.-P., Filtering for discrete-time networked nonlinear systems with mixed random delays and packet dropouts, IEEE Transactions on Automatic Control, 56, 11, 2655-2660 (2011) · Zbl 1368.93734 · doi:10.1109/TAC.2011.2166729
[28] Xu, S.; Lam, J., Robust Control and Filtering of Singular Systems. Robust Control and Filtering of Singular Systems, Lecture Notes in Control and Information Sciences, 332 (2006), Berlin, Germany: Springer, Berlin, Germany · Zbl 1114.93005
[29] Xu, S.; Chen, T., Robust \(H_\infty\) control for uncertain stochastic systems with state delay, IEEE Transactions on Automatic Control, 47, 12, 2089-2094 (2002) · Zbl 1364.93755 · doi:10.1109/TAC.2002.805670
[30] Shen, H.; Song, X.; Wang, Z., Robust fault-tolerant control of uncertain fractional-order systems against actuator faults, IET Control Theory & Applications, 7, 9, 1233-1241 (2013) · doi:10.1049/iet-cta.2012.0822
[31] Zhang, Z.; Xu, S.; Shen, H., Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance, International Journal of Robust and Nonlinear Control, 20, 15, 1723-1738 (2010) · Zbl 1204.93062 · doi:10.1002/rnc.1544
[32] Zhang, Z.; Shao, H.; Wang, Z.; Shen, H., Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems, Applied Mathematics and Computation, 218, 14, 7614-7621 (2012) · Zbl 1250.34045 · doi:10.1016/j.amc.2012.01.028
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