Delay-dependent robust \(H _{\infty }\) admissibility and stabilization for uncertain singular system with Markovian jumping parameters. (English) Zbl 1169.93420
Summary: This paper investigates the problem of delay-dependent robust \(H _{\infty }\) admissibility and stabilization for uncertain singular time delay systems with Markovian jumping parameters. The considered systems are not necessarily assumed to be regular and impulse-free. In terms of the linear matrix inequality approach, a delay-dependent stochastic admissibility criterion is given to ensure that the nominal system is regular, impulse-free and stochastically stable. Based on this criterion, the problem is solved. A numerical example is provided to demonstrate the efficiency of the proposed methods in this paper.
MSC:
| 93E15 | Stochastic stability in control theory |
| 60J75 | Jump processes (MSC2010) |
| 93C41 | Control/observation systems with incomplete information |
| 93B36 | \(H^\infty\)-control |
| 15A39 | Linear inequalities of matrices |
Keywords:
Markovian jumping parameters; singular control system; uncertainty; robust \(H _{\infty }\) admissibility; linear matrix inequalityReferences:
| [1] | E.K. Boukas, On robust stability of singular systems with random abrupt changes. Nonlinear Anal. Theory Methods Appl. 63, 301–310 (2005) · Zbl 1089.34046 · doi:10.1016/j.na.2005.03.110 |
| [2] | E.K. Boukas, Stabilization of stochastic singular nonlinear hybrid systems. Nonlinear Anal. 64, 217–228 (2006) · Zbl 1090.93048 · doi:10.1016/j.na.2005.05.066 |
| [3] | E.K. Boukas, S. Xu, J. Lam, On stability and stabilizability of singular stochastic systems with delays. J. Optim. Theory Appl. 127, 249–262 (2005) · Zbl 1101.93077 · doi:10.1007/s10957-005-6538-5 |
| [4] | Y.Y. Cao, J. Lam, Robust H control of uncertain Markovian jump systems with time-delay. IEEE Trans. Autom. Control 45, 77–85 (2000) · Zbl 0983.93075 · doi:10.1109/9.827358 |
| [5] | W.H. Chen, J.X. Xu, Z.H. Guan, Guaranteed cost control for uncertain Markovian jump systems with mode-dependent time-delays. IEEE Trans. Autom. Control 48, 2270–2277 (2003) · Zbl 1364.93369 · doi:10.1109/TAC.2003.820165 |
| [6] | W.H. Chen, Z.H. Guan, X. Lu, Delay-dependent output feedback stabilisation of Markovian jump system with time-delay. IEE Proc. Control Theory Appl. 151, 561–566 (2004) · doi:10.1049/ip-cta:20040844 |
| [7] | W.H. Chen, Z.H. Guan, P. Yu, Delay-dependent stability and H control of uncertain discrete-time Markovian jump systems with mode-dependent time delays. Syst. Control Lett. 52, 361–376 (2004) · Zbl 1157.93438 · doi:10.1016/j.sysconle.2004.02.012 |
| [8] | W.H. Chen, Z.H. Guan, X. Lu, Passive control synthesis for uncertain Markovian jump systems with multiple mode-dependent time-delays. Asian J. Control 7, 135–143 (2005) · doi:10.1111/j.1934-6093.2005.tb00382.x |
| [9] | L. Dai, Singular Control Systems (Springer, Berlin, 1989) · Zbl 0669.93034 |
| [10] | Y.M. Fu, G.R. Duan, Robust guaranteed cost observer for uncertain descriptor time-delay systems with Markovian jumping parameters. Acta Autom. Sinica 31, 479–483 (2005) |
| [11] | Y.M. Fu, G.R. Duan, Q. Lei, Robust guaranteed cost observer design for uncertain descriptor systems with state delays and Markovian jumping parameters. Int. J. Math. Control Inf. 23, 403–412 (2006) · Zbl 1113.93044 |
| [12] | Z.W. Gao, S.X. Ding, Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems. Automatica 43, 912–920 (2007) · Zbl 1117.93019 · doi:10.1016/j.automatica.2006.11.018 |
| [13] | Z.W. Gao, S.X. Ding, Fault estimation and fault-tolerant control for descriptor systems via proportional, multiple-integral and derivative observer design. IET Control Theory Appl. 1, 1208–1218 (2007) · doi:10.1049/iet-cta:20060389 |
| [14] | Y. He, Q.-G. Wang, L. Xie, C. Lin, Further inprovement of free-weighting matrices technique for systems with time-varying delay. IEEE Trans. Autom. Control 52, 293–299 (2007) · Zbl 1366.34097 · doi:10.1109/TAC.2006.887907 |
| [15] | J. Lam, Z. Shu, S. Xu, E.-K. Boukas, Robust H control of descriptor discrete-time Markovian jump systems. Int. J. Control 80, 374–385 (2007) · Zbl 1120.93057 · doi:10.1080/00207170600999322 |
| [16] | C. Lin, Q.-G. Wang, T.H. Lee, Robust normalization and stabilization of uncertain descripter systems with norm-bounded perturbations. IEEE Trans. Autom. Control 50, 515–520 (2005) · Zbl 1365.93451 · doi:10.1109/TAC.2005.844908 |
| [17] | X. Mao, Exponential stability of stochastic delay internal systems with Markovian switching. IEEE Trans. Autom. Control 47, 1604–1612 (2002) · Zbl 1364.93685 · doi:10.1109/TAC.2002.803529 |
| [18] | X. Mao, J. Lam, S. Xu, H. Gao, Razumikhin method and exponential stability of hybrid stochastic delay interval systems. J. Math. Anal. Appl. 314, 45–66 (2006) · Zbl 1127.60072 |
| [19] | Z. Wang, D.W.C. Ho, Filtering on nonlinear time-delay stochastic systems. Automatica 39, 101–109 (2003) · Zbl 1010.93099 · doi:10.1016/S0005-1098(02)00178-4 |
| [20] | Z. Wang, H. Qiao, K.J. Burnham, On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters. IEEE Trans. Autom. Control 47, 640–646 (2002) · Zbl 1364.93672 · doi:10.1109/9.995042 |
| [21] | Z. Wang, Y. Liu, L. Yu, X. Liu, Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356, 346–352 (2006) · Zbl 1160.37439 · doi:10.1016/j.physleta.2006.03.078 |
| [22] | Z. Wang, J. Lam, X. Liu, Filtering for a class of nonlinear discrete-time stochastic systems with state delays. J. Comput. Appl. Math. 210, 153–163 (2007) · Zbl 1152.93053 · doi:10.1016/j.cam.2006.02.009 |
| [23] | Z. Wang, S. Lauria, J. Fang, X. Liu, Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos Solitons Fractals 32, 62–72 (2007) · Zbl 1152.34058 · doi:10.1016/j.chaos.2005.10.061 |
| [24] | Z. Wang, F. Yang, D.W.C. Ho, X. Liu, Robust variance-constrained H control for stochastic systems with multiplicative noises. J. Math. Anal. Appl. 328, 487–502 (2007) · Zbl 1117.93068 · doi:10.1016/j.jmaa.2006.05.067 |
| [25] | G. Wei, Z. Wang, H. Shu, K. Fraser, X. Liu, Robust filtering for gene expression time series data with variance constraints. Int. J. Comput. Math. 84, 619–633 (2007) · Zbl 1116.62100 · doi:10.1080/00207160601134433 |
| [26] | G. Wei, Z. Wang, H. Shu, Nonlinear H control of stochastic time-delay systems with Markovian switching. Chaos Solitons Fractals 35, 442–451 (2008) · Zbl 1138.93061 · doi:10.1016/j.chaos.2006.05.015 |
| [27] | Z. Wu, W. Zhou, Delay-dependent robust H control for uncertain singular time-delay systems. IET Control Theory Appl. 1, 1234–1241 (2007) · doi:10.1049/iet-cta:20060446 |
| [28] | M. Wu, Y. He, J.-H. She, G.-P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 40, 1435–1439 (2004) · Zbl 1059.93108 · doi:10.1016/j.automatica.2004.03.004 |
| [29] | L. Xie, C.E. Souza, Robust H control for linear systems with norm-bounded time-varying uncertainties. IEEE Trans. Autom. Control 14, 1188–1191 (1992) · Zbl 0764.93027 · doi:10.1109/9.151101 |
| [30] | J. Xiong, J. Lam, H. Gao, D.W.C. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities. Automatica 41, 897–903 (2005) · Zbl 1093.93026 · doi:10.1016/j.automatica.2004.12.001 |
| [31] | S. Xu, J. Lam, Robust Control and Filtering of Singular Systems (Springer, Berlin, 2006) · Zbl 1114.93005 |
| [32] | S. Xu, C. Yang, H state feedback control for discrete singular systems. IEEE Trans. Autom. Control 45, 1405–1409 (2000) · Zbl 0990.93018 · doi:10.1109/9.880625 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
