Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays. (English) Zbl 1351.93163
Summary: This paper deals with the problem of stability analysis of nonlinear singular systems with Markovian jumping parameters and mode-dependent interval time varying delays. New delay-dependent stability conditions are derived in terms of linear matrix inequalities (LMIs) by constructing a mode-dependent Lyapunov-Krasovskii functional and using some integral inequalities. Numerical examples are presented to illustrate the usefulness and less conservativeness of the proposed theoretical results.
MSC:
| 93E15 | Stochastic stability in control theory |
| 93D09 | Robust stability |
| 60J27 | Continuous-time Markov processes on discrete state spaces |
| 34K20 | Stability theory of functional-differential equations |
| 34K50 | Stochastic functional-differential equations |
Keywords:
non-linear singular systems; linear matrix inequality; Lyapunov-Krasovskiĭ functional; Markovian jump; mode-dependent delayReferences:
| [1] | Dai, L., Singular control systems (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0669.93034 |
| [2] | Silva, M. S.; Lima, T. P.D., Looking for non-negative solutions of a Leontief dynamic model, Linear Algebra Appl, 364, 286-316 (2003) · Zbl 1044.15014 |
| [3] | Campbell SLC. Singular systems of differential equations. In: Preprints of IFAC conference on control of independent systems, Belfort, France; 1997. p. 501-6.; Campbell SLC. Singular systems of differential equations. In: Preprints of IFAC conference on control of independent systems, Belfort, France; 1997. p. 501-6. |
| [4] | Muller, P. C., Linear mechanical descriptor systems: identification, analysis and design (1980), Pitman: Pitman London |
| [5] | Hale, J. K.; Verduyn Lunel, S. M., Introduction to functional differential equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0787.34002 |
| [6] | Gu, K.; Kharitonov, L.; Chen, J., Stability of time delay systems (2003), Birkhauser: Birkhauser Boston · Zbl 1039.34067 |
| [7] | Kolmanovskii, V. B.; Nosov, V. R., Stability of functional differential equations (1986), Academic press: Academic press New York · Zbl 0593.34070 |
| [8] | Xu, S. Y.; Lam, J.; Yang, C. W., Robust \(H_∞\) control for uncertain singular systems with state delay, Int J Robust Nonlinear Control, 13, 1213-1223 (2003) · Zbl 1039.93019 |
| [9] | Fridman, E., Stability of linear descriptor systems with delay: a Lyapunov-based approach, J Math Anal Appl, 273, 24-44 (2002) · Zbl 1032.34069 |
| [10] | Xu, S. Y.; Van Dooren, P.; Stefan, R.; Lam, J., Robust stability and stabilization for singular systems with state delay and parameter uncertainty, IEEE Trans Automat Control, 47, 1122-1228 (2002) · Zbl 1364.93723 |
| [11] | Wu, Z.; Zhou, W., Delay-dependent robust \(H_∞\) control for uncertain singular time-delay systems, IET Control Theory Appl, 1, 1234-1241 (2007) |
| [12] | Yang, F.; Zhang, Q. L., Delay-dependent \(H_∞\) control for linear descriptor systems with delay in state, J Control Theory Appl, 3, 76-84 (2005) |
| [13] | Zhu, S.; Zhang, C.; Chen, Z.; Feng, J., Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems, IEEE Trans Automat Control, 52, 880-885 (2007) · Zbl 1366.93478 |
| [14] | Xu, S.; Lam, J.; Zou, Y., An improved characterization of bounded realness for singular delay systems and its applications, Int J Robust Nonlinear Control, 18, 263-277 (2008) · Zbl 1284.93117 |
| [15] | Lu, R.; Su, H.; Chu, J.; Xue, A., A simple approach to robust D-stability analysis for uncertain singular delay systems, Asian J Control, 11, 411-419 (2009) |
| [16] | Kim, J. H., Delay-dependent robust and non-fragile guaranteed cost control for uncertain singular systems with time-varying state and input delays, Int J Control Automat Syst, 7, 357-364 (2009) |
| [17] | Boukas, E. K., Delay-dependent robust stabilizability of singular linear systems with delays, Stoch Anal Appl, 27, 637-655 (2009) · Zbl 1167.93396 |
| [18] | Haidar, A.; Boukas, E. K., Exponential stability of singular systems with multiple time-varying delays, Automatica, 45, 539-545 (2009) · Zbl 1158.93347 |
| [19] | Xu, S.; Lam, J.; Mao, X., Delay-dependent \(H_∞\) control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Trans Circuits Syst I, 54, 2070-2078 (2007) · Zbl 1374.93134 |
| [20] | Zhang, B.; Xu, S.; Zou, Y., Output feedback stabilization for delayed large-scale stochastic systems with Markovian jumping parameters, Asian J Control, 11, 457-460 (2009) |
| [21] | Chen, B.; Li, H.; Shi, P.; Lin, C.; Zhou, Q., Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays, Inform Sci, 179, 2851-2860 (2009) · Zbl 1165.93341 |
| [22] | Zhang, Y.; Xu, S.; Zhang, B., Robust output feedback stabilization for uncertain discrete-time fuzzy Markovian jump systems with time-varying delays, IEEE Trans Fuzzy Syst, 17, 411-420 (2009) |
| [23] | Wang G, Zhang Q, Yang C. Dissipative control for singular Markovian jump systems with time delay. Optim Control Appl Methods. doi:10.1002/oca.1004; Wang G, Zhang Q, Yang C. Dissipative control for singular Markovian jump systems with time delay. Optim Control Appl Methods. doi:10.1002/oca.1004 · Zbl 1301.93151 |
| [24] | Ma, S.; Boukas, E. K., Robust \(H_∞\) filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay, IET Control Theory Appl, 3, 351-361 (2009) |
| [25] | Shen, H.; Xu, S.; Song, X.; Chu, Y., Delay-dependent \(H_∞\) filtering for stochastic systems with Markovian switching and mixed mode-dependent delays, Nonlinear Anal Hybrid Syst, 4, 122-133 (2010) · Zbl 1179.93164 |
| [26] | Ma, Q.; Xu, S.; Zou, Y.; Lu, J., Stability of stochastic Markovian jump neural networks with mode-dependent delays, Neurocomputing, 74, 2157-2163 (2011) |
| [27] | Guang, Z. H.; Jie, F.; Dong, M. T.; Cheng, T. S., Robust stability analysis for Markovian jumping stochastic neural networks with mode-dependent time-varying interval delay and multiplicative noise, Chinese Phys B, 18, 3325-3336 (2009) |
| [28] | Wang, H. J.; Xue, A. K.; Lu, R. Q., Absolute stability criteria for a class of nonlinear singular system with time delay, Nonlinear Anal, 70, 621-630 (2009) · Zbl 1168.34048 |
| [29] | Ding, Y.; Zhong, S.; Chen, W., A delay-range-dependent uniformly asymptotic stability criterion for a class of nonlinear singular systems, Nonlinear Anal Real World Appl, 12, 1152-1162 (2011) · Zbl 1218.93078 |
| [30] | Li, H.; Chen, B.; Zhou, Q.; Qian, W., Robust stability for uncertain delayed fuzzy Hopfield neural networks with Markovian jumping parameters, IEEE Trans Syst Man Cybernet B: Cybernet, 39, 94-102 (2009) |
| [31] | He, Y.; Zhang, Y.; Wu, M.; She, J. H., Improved exponential stability for stochastic Markovian jump systems with nonlinearity and time-varying delay, Int J Robust Nonlinear Control, 20, 16-26 (2010) · Zbl 1192.93125 |
| [32] | Sun, J.; Liu, G. P.; Chen, J., Delay-dependent stability and stabilization of neutral time-delay systems, Int J Robust Nonlinear Control, 19, 1364-1375 (2009) · Zbl 1169.93399 |
| [33] | Zhang, X. M.; Han, Q. L., A delay decomposition approach to delay-dependent stability for linear systems with time-varying delays, Int J Robust Nonlinear Control, 19, 1922-1930 (2009) · Zbl 1185.93106 |
| [34] | Li, H.; Gao, H.; Shi, P., New passivity analysis for neural networks with discrete and distributed delays, IEEE Trans Neural Networks, 31, 1842-1847 (2010) |
| [35] | Wu, Z. G.; Su, H. Y.; Chu, J., Improved results on delay-dependent \(H_∞\) control for singular time-delay systems, Acta Automat Sinica, 35, 1101-1106 (2009) · Zbl 1212.93088 |
| [36] | Cao, Y. Y.; Lam, J.; Hu, L. S., Delay-dependent stochastic stability and \(H_∞\) analysis for time-delay systems with Markovian jumping parameters, J Franklin Inst, 340, 423-434 (2003) · Zbl 1040.93068 |
| [37] | Wu, J.; Chen, T. W.; Wang, L., Delay-dependent robust stability and \(H_∞\) control for jump linear systems with delays, Syst Control Lett, 55, 939-948 (2006) · Zbl 1117.93072 |
| [38] | Boukas, E. K.; Liu, Z. K.; Shi, P., Delay-dependent stability and output feedback stabilisation of Markov jump system with time-delay, IEE Proc Control Theory Appl, 149, 379 (2002) |
| [39] | Cao, Y. Y.; Hu, L. S.; Xue, A. K., A new delay-dependent stability condition and \(H_∞\) control for jump time-delay system, (Proceedings of American control conference, vol. 5 (2004), IEEE: IEEE Boston, USA), 4183-4188 · Zbl 1220.35150 |
| [40] | Zhao, X. D.; Zeng, Q. S., Delay-dependent stability analysis for Markovian jump systems with interval time-varying-delays, Int J Automat Comput, 7, 224-229 (2010) |
| [41] | Wu, Z.; Su, H.; Chu, J., Delay-dependent stabilization of singular Markovian jump systems with state delay, J Control Theory Appl, 7, 231-236 (2009) · Zbl 1212.93318 |
| [42] | Yoneyama, J., Robust stability for descriptor systems with time-varying delay, Appl Math Sci, 4, 977-989 (2010) · Zbl 1209.93161 |
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