Abstract
This paper investigates the problem of exponential stability in mean square sense for stochastic Markov jump systems with mixed time-varying delays and partly unknown transition rates. By employing a class of appropriate stochastic Lyapunov functionals, the analysis process of stability for stochastic Markov jump systems can be effectively carried out. Based on the linear matrix inequalities technique, the mean square exponential stability criteria are presented for stochastic Markov jump systems with partly unknown transition rates. Furthermore, by expanding this case to uncertain Markov jump systems, we derive the sufficient conditions for guaranteeing the stability of uncertain Markov jump systems. A numerical example is presented to illustrate the effectiveness of the proposed results.
Similar content being viewed by others
References
A. Chandrasekar, R. Rakkiyappan, J. Cao, Impulsive synchronization of Markovian jumping randomly coupled neural networks with partly unknown transition probabilities via multiple integral approach. Neural Netw. 70, 27–38 (2015)
J. Chen, K. Gu, V.L. Kharitonov, Stability of time-delay systems (Birkhauser, 2003)
Y. Chen, S. Fei, Y. Liu, Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback. Automatica 52, 242–247 (2015)
M.A. Davó, A. Baños, F. Gouaisbaut et al., Stability analysis of linear impulsive delay dynamical systems via looped-functionals. Automatica 81, 107–114 (2017)
L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, 1994)
W. Fei, L. Hu, X. Mao et al., Delay dependent stability of highly nonlinear hybrid stochastic systems. Automatica 82, 165–170 (2017)
Z. Fei, H. Gao, P. Shi, New results on stabilization of Markovian jump systems with time delay. Automatica 45(10), 2300–2306 (2009)
Q.L. Han, A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays. Automatica 40(10), 1791–1796 (2004)
Y. He, Y. Zhang, M. Wu et al., Improved exponential stability for stochastic Markovian jump systems with nonlinearity and time-varying delay. Int. J. Robust Nonlinear Control 20(1), 16–26 (2010)
P. Hinow, M. Mincheva, Linear stability of delayed reaction–diffusion systems. Comput. Math. Appl. 73(2), 226–232 (2017)
L. Hu, A. Yang, Fuzzy model-based control of nonlinear stochastic systems with time-delay. Nonlinear Anal. Theory Methods Appl. 71(12), e2855–e2865 (2009)
H. Huang, G. Feng, X. Chen, Stability and stabilization of Markovian jump systems with time delay via new Lyapunov functionals. IEEE Trans. Circuits Syst. I: Regul. Pap. 59(10), 2413–2421 (2012)
H. Huang, D.W.C. Ho, Y. Qu, Robust stability of stochastic delayed additive neural networks with Markovian switching. Neural Netw. 20(7), 799–809 (2007)
T.H. Lee, J.H. Park, S. Xu, Relaxed conditions for stability of time-varying delay systems. Automatica 75, 11–15 (2017)
F. Li, C. Du, C. Yang, W. Gui, Passivity-based asynchronous sliding mode control for delayed singular Markovian jump systems. IEEE Trans. Autom. Control 63(8), 2715–2721 (2018)
F. Li, P. Shi, C.C. Lim, L. Wu, Fault detection filtering for nonhomogeneous Markovian jump systems via a fuzzy approach. IEEE Trans. Fuzzy Syst. 26(1), 131–141 (2018)
L. Li, M. Shen, G. Zhang et al., \(H_{\infty }\) control of Markov jump systems with time-varying delay and incomplete transition probabilities. Appl. Math. Comput. 301, 95–106 (2017)
X. Li, H. Gao, K. Gu, Delay-independent stability analysis of linear time-delay systems based on frequency discretization. Automatica 70, 288–294 (2016)
B. Mu, H. Li, J. Ding et al., Consensus in second-order multiple flying vehicles with random delays governed by a Markov chain. J. Frankl. Inst. 352(9), 3628–3644 (2015)
G. Balas, R. Chiang, A. Packard, M. Safonov, Robust control toolbox for use with Matlab (The Mathworks, Natick, Massachusetts, 2005)
C.C. Shen, S.M. Zhong, New delay-dependent robust stability criterion for uncertain neutral systems with time-varying delay and nonlinear uncertainties. Chaos Solitons Fractals 40(5), 2277–2285 (2009)
H. Shen, F. Li, S. Xu, V. Sreeram, Slow state variables feedback stabilization for semi-Markov jump systems with singular perturbations. IEEE Trans. Autom. Control 63(8), 2709–2714 (2018)
H. Shen, F. Li, H. Yan, H.R. Karimi, H.K. Lam, Finite-time event-triggered \({H} _\infty \) control for TS fuzzy Markov jump systems. IEEE Trans. Fuzzy Syst. 26(5), 3122–3135 (2018)
H. Shen, L. Su, J.H. Park, Reliable mixed \(H_{\infty }\) passive control for T-S fuzzy delayed systems based on a semi-Markov jump model approach. Fuzzy Sets Syst. 314, 79–98 (2017)
D.Y. Wang, L.S. Li, Mean-square stability analysis of discrete-time stochastic Markov jump recurrent neural networks with mixed delays. Neurocomputing 189, 171–178 (2016)
J. Wang, K. Liang, X. Huang, Z. Wang, H. Shen, Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback. Appl. Math. Comput. 328, 247–262 (2018)
T. Wang, T. Li, G. Zhang et al., Further triple integral approach to mixed-delay-dependent stability of time-delay neutral systems. ISA Trans. 70, 116–124 (2017)
Y. Wang, X. Sun, J. Zhao, Stabilization of a class of switched stochastic systems with time delays under asynchronous switching. Circuits Syst. Signal Process. 32(1), 347–360 (2012)
Y.E. Wang, X.M. Sun, F. Mazenc, Stability of switched nonlinear systems with delay and disturbance. Automatica 69, 78–86 (2016)
Z. Wang, Y. Liu, X. Liu, Exponential stabilization of a class of stochastic system with Markovian jump parameters and mode-dependent mixed time-delays. IEEE Trans. Autom. Control 55(7), 1656–1662 (2010)
T. Wu, F. Li, C. Yang, W. Gui, Event-based fault detection filtering for complex networked jump systems. IEEE/ASME Trans. Mechatron. 23(2), 497–505 (2018)
L.H. Xie, Output feedback \(H_{\infty }\) control of systems with parameter uncertainty. Int. J. Control 63(4), 741–750 (1996)
D. Yao, R. Lu, Y. Xu, H. Ren, Observer-based sliding mode control of Markov jump systems with random sensor delays and partly unknown transition rates. Int. J. Syst. Sci. 48(14), 2985–2996 (2017)
S. Zhai, X.S. Yang, Exponential stability of time-delay feedback switched systems in the presence of asynchronous switching. J. Frankl. Inst. 350(1), 34–49 (2013)
W. Zhang, Y. Tang, X. Wu et al., Stochastic stability of switched genetic regulatory networks with time-varying delays. IEEE Trans. Nanobioscience 13(3), 336–342 (2014)
X.M. Zhang, Q.L. Han, A. Seuret et al., An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay. Automatica 84, 221–226 (2017)
Y. Zhang, Y. Ou, X. Wu et al., Resilient dissipative dynamic output feedback control for uncertain Markov jump Lure systems with time-varying delays. Nonlinear Anal. Hybrid Syst. 24, 13–27 (2017)
P. Zhao, Practical stability, controllability and optimal control of stochastic Markovian jump systems with time-delays. Automatica 44(12), 3120–3125 (2008)
X.Y. Zhao, F.Q. Deng, Moment stability of nonlinear discrete stochastic systems with time-delays based on H-representation technique. Automatica 50(2), 530–536 (2014)
J. Zhou, H. Dong, J. Feng, Event-triggered communication for synchronization of Markovian jump delayed complex networks with partially unknown transition rates. Appl. Math. Comput. 293, 617–629 (2017)
Q. Zhu, F. Xi, X. Li, Robust exponential stability of stochastically nonlinear jump systems with mixed time delays. J. Optim. Theory Appl. 154(1), 154–174 (2012)
S. Zhu, Q.L. Han, C. Zhang, \(L_{1}\)-stochastic stability and \(L_{1}\)-gain performance of positive Markov jump linear systems with time-delays: necessary and sufficient conditions. IEEE Trans. Autom. Control 62(7), 3634–3639 (2017)
Acknowledgements
The authors would like to express their sincere gratitude to the anonymous referees and Prof. Zhanjie Song for many valuable suggestions and comments that helped to improve the paper. This work was supported by the National Natural Science Foundation of China (No. 91746107).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cui, K., Zhu, J. & Li, C. Exponential Stabilization of Markov Jump Systems with Mode-Dependent Mixed Time-Varying Delays and Unknown Transition Rates. Circuits Syst Signal Process 38, 4526–4547 (2019). https://doi.org/10.1007/s00034-019-01085-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-019-01085-2