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Model reduction for discrete-time Markov jump Lur’e systems with time-varying delays in a unified framework. (English) Zbl 1395.93138
Summary: This paper addresses the model reduction problem for a class of Markov jump Lur’e systems with piecewise homogeneous transition probabilities and time-varying delays in discrete-time domain. The purpose of this paper is to solve the \(H_\infty\), \(l_2\)-\(l_\infty\), passivity and dissipativity model reduction problems in discrete-time domain in a unified framework by using an extended dissipativity performance index. By constructing a mode-dependent Lyapunov-Krasovskii functional with the sector condition assumption for the nonlinearities, the sufficient conditions on the existence of desired reduced-order models are derived in terms of linear matrix inequalities, which ensures that the resulting error system is stochastically stable and has a prescribed performance index. A numerical example is presented to show the effectiveness of the proposed theories.

MSC:
93B11 System structure simplification
93C55 Discrete-time control/observation systems
93B36 \(H^\infty\)-control
93C10 Nonlinear systems in control theory
60J75 Jump processes (MSC2010)
93E15 Stochastic stability in control theory
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[1] Costa, O.; Fragoso, M.; Marques, R., Discrete-time Markov jump linear systems, (2006), Springer London · Zbl 1081.93001
[2] Blair, W.; Sworder, D., Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria, Int. J. Control, 21, 5, 833-841, (1975) · Zbl 0303.93084
[3] 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 Trans. Autom. Control, 45, 5, 944-949, (2000) · Zbl 0972.93074
[4] Sworder, D. D.; Rogers, R. O., An LQG solution to a control problem with solar thermal receiver, IEEE Trans. Autom. Control, 28, 971-978, (1983)
[5] Lee, J. W.; Dullerud, G. E., Optimal disturbance attenuation for discrete-time switched and Markovian jump linear systems, SIAM J. Control Optim., 45, 4, 1329-1358, (2006) · Zbl 1123.93045
[6] Costa, O.; Fragoso, M., Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl., 179, 1, 154-178, (1993) · Zbl 0790.93108
[7] de Souza, C. E., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Trans. Autom. Control, 51, 5, 836-841, (2006) · Zbl 1366.93479
[8] Pan, S.; Sun, J.; Zhao, S., Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers, Automatica, 44, 11, 2954-2958, (2008) · Zbl 1152.93509
[9] Xu, S.; Chen, T.; Lam, J., Robust \(H_\infty\) filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Trans. Autom. Control, 48, 5, 900-907, (2003) · Zbl 1364.93816
[10] Goncalves, A.; Fioravanti, A.; Geromel, J., \(H_\infty\) filtering of discrete time Markov jump linear systems through linear matrix inequalities, IEEE Trans. Autom. Control, 54, 6, 1347-1351, (2009) · Zbl 1367.93643
[11] Fragoso, M.; Baczynski, J., Optimal control for continuous-time linear quadratic problems with infinite Markov jump parameters, SIAM J. Control Optim., 40, 1, 270-297, (2001) · Zbl 1058.93058
[12] Boukas, E.; Liu, Z., Robust \(H_\infty\) control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Trans. Autom. Control, 46, 12, 1918-1924, (2001) · Zbl 1005.93050
[13] Cao, Y.; Lam, J., Robust \(H_\infty\) control of uncertain Markovian jump systems with time-delay, IEEE Trans. Autom. Control, 45, 1, 77-83, (2000) · Zbl 0983.93075
[14] Aberkane, S., Stochastic stabilization of a class of nonhomogeneous Markovian jump linear systems, Syst. Control Lett., 60, 3, 156-160, (2011) · Zbl 1210.93078
[15] Zhang, L., \(H_\infty\) estimation for discrete-time piecewise homogeneous Markov jump linear systems, Automatica, 45, 1, 2570-2576, (2009) · Zbl 1180.93100
[16] Song, G.; Zhang, Y.; Xu, S., Stability and l_2-gain analysis for a class of discrete-time nonlinear Markovian jump systems with actuator saturation and incomplete knowledge of transition probabilities, IET Control Theory Appl., 6, 17, 2716-2723, (2012)
[17] C. Gonzaga, O. Costa, Stochastic stability for discrete-time Markov jump Lur׳e systems, in: Proceedings of the 52th IEEE Conference on Decision and Control, Palazzo dei Congressi, Florence, Italy, 2013, pp. 5993-5998.
[18] Geromel, J. C.; Deaecto, G. S., Stability analysis of lur׳e-type switched systems, IEEE Trans. Autom. Control, 59, 11, 3046-3050, (2014) · Zbl 1360.93529
[19] Gonzaga, C. A.C.; Costa, O. L.V., Stochastic stabilization and induced l_2-gain for discrete-time Markov jump lur׳e systems with control saturation, Automatica, 50, 9, 2397-2404, (2014) · Zbl 1297.93175
[20] Zhang, Y.; Ou, Y.; Zhou, Y., Observer-based l_2-\(l_\infty\) control for discrete-time nonhomogeneous Markov jump lur׳e systems with sensor saturations, Neurocomputing, 162, 141-149, (2015)
[21] Khalil, H. K., Nonlinear systems, (2002), Prentice Hall New Jersey
[22] Wang, Z.; Liu, Y.; Liu, X., \(H_\infty\) filtering for uncertain stochastic time-delay systems with sector-bounded nonlinearities, Automatica, 44, 5, 1268-1277, (2008) · Zbl 1283.93284
[23] Wu, L.; Zheng, W., Weighted \(H_\infty\) model reduction for linear switched systems with time-varying delay, Automatica, 45, 1, 186-193, (2009) · Zbl 1154.93326
[24] Du, B.; Lam, J.; Zou, Y., Stability and stabilization for Markovian jump time-delay systems with partially unknown transition rates, IEEE Trans. Circuits Syst. I—Regul. Pap., 60, 2, 341-351, (2013)
[25] Guo, X. G.; Yang, G. H., Delay-dependent reliable \(H_\infty\) filtering for sector-bounded nonlinear continuous-time systems with time-varying state delays and sensor failures, Int. J. Syst. Sci., 43, 1, 117-131, (2012) · Zbl 1259.93117
[26] Zhu, Y.; Zhang, Q.; Wei, Z.; Zhang, L., Robust stability analysis of Markov jump standard genetic regulatory networks with mixed time-delays and uncertainties, Neurocomputing, 110, 13, 44-50, (2013)
[27] Dong, H.; Wang, Z.; Gao, H., Distributed \(H_\infty\) filtering for a class of Markovian jump nonlinear time-delay systems over lossy sensor networks, IEEE Trans. Ind. Electron., 60, 10, 4665-4672, (2013)
[28] You, J.; Yin, S.; Yu, Z., Robust estimation for discrete time-delay Markov jump systems with sensor non-linearity and missing measurements, IET Control Theory Appl., 8, 5, 330-337, (2014)
[29] Moore, B., Principal component analysis in linear systemscontrollability, observability, and model reduction, IEEE Trans. Autom. Control, 26, 1, 17-31, (1981)
[30] Glover, K., All optimal Hankel-norm approximations of linear multivariable systems and their \(L_\infty \operatorname{-error}\) bounds, Int. J. Control, 39, 6, 1115-1193, (1984) · Zbl 0543.93036
[31] Zhang, L.; Lam, J., On H_2 model reduction of bilinear systems, Automatica, 38, 2, 205-216, (2002) · Zbl 0991.93020
[32] Lam, J.; Gao, H.; Xu, S.; Wang, C., \(H_\infty\) and \(L_2 / L_\infty\) model reduction for system input with sector nonlinearities, J. Optim. Theory Appl., 125, 1, 137-155, (2005) · Zbl 1062.93020
[33] Wang, Q.; Lam, J.; Xu, S.; Gao, H., Delay-dependent and delay-independent energy-to-peak model approximation for systems with time-varying delay, Int. J. Syst. Sci., 6, 8, 445-460, (2005) · Zbl 1121.93020
[34] Y. Li, Y. Qu, H. Gao, C. Wang, Robust L_1 model reduction for uncertain stochastic systems with state delay, in: 2005 American Control Conference, Portland, OR, USA, 2005, pp. 2602-2607.
[35] Yang, W.; Zhang, L.; Shi, P.; Zhu, Y., Model reduction for a class of nonstationary Markov jump linear systems, J. Frankl. Inst., 349, 2445-2460, (2012) · Zbl 1287.93017
[36] Q. Wang, J. Lam, \(H_\infty\) model approximation for discrete-time Markovian jump systems with mode-dependent time delays, in: Proceedings of the 44th IEEE Conference on Decision and Control, and 2005 European Control Conference, CDC-ECC׳05, Seville, Spain, 2005, pp. 6122-6127.
[37] Zhang, L.; Boukas, E. K.; Shi, P., \(H_\infty\) model reduction for discrete-time Markov jump linear systems with partially known transition probabilities, Int. J. Control, 82, 2, 343-351, (2009) · Zbl 1168.93319
[38] Zhang, B.; Zheng, W.; Xu, S., Filtering of Markovian jump delay systems based on a new performance index, IEEE Trans. Circuits Syst. I—Regul. Pap., 60, 5, 1250-1263, (2013)
[39] Dynkin, E., Theory of Markov processes, (2006), Dover Mineola, NY, (reprint of 1961 edition) · Zbl 1116.60001
[40] Shen, H.; Park, J. H.; Zhang, L., Robust extended dissipative control for sampled-data Markov jump systems, Int. J. Control, 87, 8, 1549-1564, (2014) · Zbl 1317.93171
[41] Zhang, L.; Shi, P.; Wang, C., Robust \(H_\infty\) filtering for switched linear discrete-time systems with polytopic uncertainties, Int. J. Adapt. Control Signal Process., 20, 6, 291-304, (2006) · Zbl 1127.93324
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