# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
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.