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Robust stability and \(H_\infty \) control for uncertain discrete Markovian jump singular systems with mode-dependent time-delay. (English) Zbl 1163.93026
Summary: The robust stochastic stability, stabilization and \(H_\infty \) control for mode-dependent time-delay discrete Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent transformation and by introducing new state vectors, the singular system is transformed into a standard linear system, and delay-dependent Linear Matrix Inequalities (LMIs) conditions for the mode-dependent time-delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with \(\gamma\)-disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust \(H_\infty\) control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper.

MSC:
93D09 Robust stability
93B36 \(H^\infty\)-control
60J75 Jump processes (MSC2010)
93E15 Stochastic stability in control theory
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[1] Boukas EK, Liu ZK. Robust stability and H control of discrete-time jump linear systems with time-delays: an LMI approach. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, December 1999; 1527-1532.
[2] Boukas, Robust H control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control 46 (12) pp 1918– (2001)
[3] Boukas, Delay-dependent robust stability and H control of jump linear systems with time-delay, International Journal of Control 74 pp 329– (2001) · Zbl 1015.93069
[4] Cao, Stochastic stabilizability and H control for discrete-time jump linear systems with time delay, Journal of the Franklin Institute 336 pp 1263– (1999) · Zbl 0967.93095
[5] Cao, Robust H control of uncertain Markovian jump systems with time-delay, IEEE Transactions on Automatic Control 45 (1) pp 77– (2000)
[6] Chen, Delay-dependent stability and H control of uncertain discrete-time Markovian jump systems with mode-dependent time delays, Systems and Control Letters 52 pp 361– (2004) · Zbl 1157.93438
[7] Shi, Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE Transactions on Automatic Control 44 (11) pp 2139– (1999) · Zbl 1078.93575
[8] Fridman, A descriptor system approach to H control of linear time-delay systems, IEEE Transactions on Automatic Control 47 (2) pp 253– (2002) · Zbl 1364.93209
[9] Fridman, An improved stabilization method for linear time-delay systems, IEEE Transactions on Automatic Control 47 (11) pp 1931– (2002)
[10] Wu, New delay-dependent stability criteria and stabilizing method for neutral systems, IEEE Transactions on Automatic Control 49 (12) pp 2266– (2004) · Zbl 1365.93358
[11] Dai, Singular Control Systems (1989) · Zbl 0669.93034
[12] Campbell, Singular Systems of Differential Equations (1994)
[13] Luenberger, Singular dynamic Leontief systems, Econometrics 45 (32) pp 991– (1977) · Zbl 0368.90029
[14] Silva, Looking for nonnegative solutions of a Leontief dynamic model, Linear Algebra and its Applications 364 pp 281– (2003) · Zbl 1044.15014
[15] Hill, Stability theory for differential-algebraic systems with application to power system, IEEE Transactions on Circuits and Systems 37 (11) pp 1416– (1990)
[16] Sastry, Jump behavior of circuits and systems, IEEE Transactions on Circuits and Systems CAS-28 pp 1109– (1981) · Zbl 0476.93036
[17] Boukas, Control of Singular Systems with Random Abrupt Changes (2008) · Zbl 1251.93001
[18] Ma S, Cheng Z. An LMI approach to robust stabilization for uncertain discrete-time singular systems. Proceedings of the 41st IEEE CDC, Las Vegas, Nevada, U.S.A., December 2002; 1090-1095.
[19] Xu, Robust stability and stabilization of discrete singular systems: an equivalent characterization, IEEE Transactions on Automatic Control 49 (4) pp 568– (2004) · Zbl 1365.93375
[20] Shi, On H control design for singular continuous-time delay systems with parametric uncertainties, Nonlinear Dynamics and Systems Theory 4 (1) pp 59– (2004) · Zbl 1073.93020
[21] Xu, Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Transactions on Automatic Control 47 pp 1122– (2002) · Zbl 1364.93723
[22] Yue, Reliable H control of uncertain descriptor systems with multiple delays, IEE Proceeding Control Theory and Applications 150 (6) pp 557– (2003)
[23] Ma S, Cheng Z. Delay-dependent robust stabilization for uncertain discrete-time singular systems with time-delay. Proceedings of the Sixth World Congress on Intelligent Control and Automation, Dalian, China, June 2006; 2081-2085.
[24] Ma, Delay-dependent robust H control for uncertain discrete-time singular systems with time-delay, Journal of Computational and Applied Mathematics 217 (1) pp 194– (2008) · Zbl 1142.93011
[25] Xu, Robust H control for discrete singular systems with state delay and parameter uncertainty, Dynamics of Continuous, Discrete, Impulsive Systems 11 (3) pp 497– (2002)
[26] Boukas, Static output feedback control for stochastic hybrid systems: LMI approach, Automatica 42 (1) pp 183– (2006) · Zbl 1121.93365
[27] Boukas, On stability and stabilizability of singular stochastic systems with delay, Journal of Optimization Theory and Applications 127 (2) pp 249– (2005) · Zbl 1101.93077
[28] Xu, Robust Control and Filtering of Singular Systems (2006)
[29] Lam, Robust H control of descriptor discrete time Markovian jump systems, International Journal of Control 80 (3) pp 374– (2007)
[30] Xie, Robust H control for linear systems with norm-bounded time-varying uncertainty, IEEE Transactions on Automatic Control 37 pp 1188– (1992) · Zbl 0764.93027
[31] Petersen, A stabilization algorithm for a class of uncertain linear systems, Systems and Control Letters 8 pp 351– (1987) · Zbl 0618.93056
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