The almost surely (a.s.) exponential stability is studied for semi-Markovian switched stochastic systems with randomly impulsive jumps. We start from the case that switches and impulses occur synchronously, in which the impulsive switching signal is a semi-Markovian process. For the case that switches and impulses occur asynchronously, the impulsive arrival time sequence and the types of jump maps are driven by a renewal process and a Markov chain, respectively. By applying the multiple Lyapunov function approach, sufficient conditions of exponential stability a.s. are obtained based upon the ergodic property of semi-Markovian process. The validity of the proposed theoretical results is demonstrated by a numerical example.
This work was supported by National Natural Science Foundation of China (Grant No. 11571322).
[1] Cong S. A Result on Almost Sure Stability of Linear Continuous-Time Markovian Switching Systems. IEEE Trans Automat Contr, 2018, 63: 2226-2233 CrossRef Google Scholar
[2] Chatterjee D, Liberzon D. Stabilizing Randomly Switched Systems. SIAM J Control Optim, 2011, 49: 2008-2031 CrossRef Google Scholar
[3] Wang B, Zhu Q. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. Syst Control Lett, 2017, 105: 55-61 CrossRef Google Scholar
[4] Wang B, Zhu Q. Stability analysis of semi-Markov switched stochastic systems. Automatica, 2018, 94: 72-80 CrossRef Google Scholar
[5] Wang B, Zhu Q. A Note on Sufficient Conditions of Almost Sure Exponential Stability for Semi-Markovian Jump Stochastic Systems. IEEE Access, 2019, 7: 49466-49473 CrossRef Google Scholar
[6] Wu X, Tang Y, Cao J. Stability Analysis for Continuous-Time Switched Systems With Stochastic Switching Signals. IEEE Trans Automat Contr, 2018, 63: 3083-3090 CrossRef Google Scholar
[7] Hespanha J P, Teel A R. Stochastic Impulsive Systems Driven by Renewal Processes: Extended Version. Technical Report, University of California, Santa Barbara, 2005. http://www.ece.ucsb.edu/hespanha/techreps.html. Google Scholar
[8] Hu Z, Yang Z, Mu X. Stochastic input-to-state stability of random impulsive nonlinear systems. J Franklin Institute, 2019, 356: 3030-3044 CrossRef Google Scholar
[9] Shen Y, Wu Z G, Shi P. Dissipativity based fault detection for 2D Markov jump systems with asynchronous modes. Automatica, 2019, 106: 8-17 CrossRef Google Scholar
[10] Wu Z G, Dong S, Shi P. Reliable Filter Design of Takagi-Sugeno Fuzzy Switched Systems With Imprecise Modes.. IEEE Trans Cybern, 2019, : 1-11 CrossRef PubMed Google Scholar
[11] Cassandras C G, Lygeros J. Stochastic hybrid systems: research issues and areas. In: Stochastic Hybrid Systems. Boca Raton: CRC Press, 2007. 1--14. Google Scholar
[12] Liu K, Fridman E, Johansson K H. Networked Control With Stochastic Scheduling. IEEE Trans Automat Contr, 2015, 60: 3071-3076 CrossRef Google Scholar
[13] Teel A R, Subbaraman A, Sferlazza A. Stability analysis for stochastic hybrid systems: A survey. Automatica, 2014, 50: 2435-2456 CrossRef Google Scholar
[14] Guan Z H, Hill D J, Yao J. A HYBRID IMPULSIVE AND SWITCHING CONTROL STRATEGY FOR SYNCHRONIZATION OF NONLINEAR SYSTEMS AND APPLICATION TO CHUA'S CHAOTIC CIRCUIT. Int J Bifurcation Chaos, 2006, 16: 229-238 CrossRef Google Scholar
[15] Matsuoka Y, Saito T. Rich Superstable Phenomena in a Piecewise Constant Nonautonomous Circuit with Impulsive Switching. IEICE Trans Fundamentals Electron Commun Comput Sci, 2006, E89-A: 2767-2774 CrossRef Google Scholar
[16] Ren W, Xiong J. Lyapunov Conditions for Stability of Stochastic Impulsive Switched Systems. IEEE Trans Circuits Syst I, 2018, 65: 1994-2004 CrossRef Google Scholar
[17] Vaidyanathan S, Volos C K, Pham V T. Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation. J Eng Sci Technol Rev, 2015, 8: 174--184. Google Scholar
[18] Li X, Li P, Wang Q. Input/output-to-state stability of impulsive switched systems. Syst Control Lett, 2018, 116: 1-7 CrossRef Google Scholar
[19] Liu J, Liu X, Xie W C. Class- estimates and input-to-state stability analysis of impulsive switched systems. Syst Control Lett, 2012, 61: 738-746 CrossRef Google Scholar
[20] Kobayashi H, Mark B L, Turin W. Probability, Random Processes and Statistical Analysis. New York: Cambridge University Express, 2012. Google Scholar
[21] Ross S M. Stochastic Processes. Hoboken: John Wiley & Sons Inc., 1996. Google Scholar
[22] Hou Z, Dong H, Shi P. Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching. ANZIAM J, 2007, 49: 231-241 CrossRef Google Scholar
[23] Zong G, Ren H. Guaranteed cost finite?time control for semi?Markov jump systems with event?triggered scheme and quantization input. Int J Robust NOnlinear Control, 2019, 29: 5251-5273 CrossRef Google Scholar
[24] Qi W, Zong G, Karimi H R. $\mathscr~{L}_\infty$ Control for Positive Delay Systems With Semi-Markov Process and Application to a Communication Network Model. IEEE Trans Ind Electron, 2019, 66: 2081-2091 CrossRef Google Scholar
[25] Qi W, Zong G, Karim H R. Observer-Based Adaptive SMC for Nonlinear Uncertain Singular Semi-Markov Jump Systems With Applications to DC Motor. IEEE Trans Circuits Syst I, 2018, 65: 2951-2960 CrossRef Google Scholar
[26] Huang J, Shi Y. Stochastic stability and robust stabilization of semi-Markov jump?linear systems. Int J Robust NOnlinear Control, 2013, 23: 2028-2043 CrossRef Google Scholar
[27] Ning Z, Zhang L, Lam J. Stability and stabilization of a class of stochastic switching systems with lower bound of sojourn time. Automatica, 2018, 92: 18-28 CrossRef Google Scholar
[28] Sun W, Guan J, Lu J. Synchronization of the Networked System With Continuous and Impulsive Hybrid Communications.. IEEE Trans Neural Netw Learning Syst, 2019, : 1-12 CrossRef PubMed Google Scholar
[29] Xu Y, He Z. Stability of impulsive stochastic differential equations with Markovian switching. Appl Math Lett, 2014, 35: 35-40 CrossRef Google Scholar
[30] Prandini M, Jianghai Hu M. Application of Reachability Analysis for Stochastic Hybrid Systems to Aircraft Conflict Prediction. IEEE Trans Automat Contr, 2009, 54: 913-917 CrossRef Google Scholar
[31] Luo S, Deng F, Zhao X. Stochastic stabilization using aperiodically sampled measurements. Sci China Inf Sci, 2019, 62: 192201 CrossRef Google Scholar
[32] Qi W, Zong G, Karimi H R. Sliding Mode Control for Nonlinear Stochastic Singular Semi-Markov Jump Systems. IEEE Trans Automat Contr, 2020, 65: 361-368 CrossRef Google Scholar
[33] Zhang S, Xiong J, Liu X. Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching. Sci China Inf Sci, 2018, 61: 070211 CrossRef Google Scholar
[34] Wu S J, Zhou B. Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-lipschitz conditions. Acta Math Sin-Engl Ser, 2011, 27: 519-536 CrossRef Google Scholar
[35] Mao X R, Yuan C. Stochastic Differential Equations With Markovian Switching. London: Imperial College Press, 2006. Google Scholar
[36] Deng F, Luo Q, Mao X. Stochastic stabilization of hybrid differential equations. Automatica, 2012, 48: 2321-2328 CrossRef Google Scholar
[37] Pardoux E. Markov Processes and Applications. Hoboken: John Wiley & Sons Inc., 2008. Google Scholar
Figure 1
(Color online) The state trajectories of the three subsystems of system
Figure 2
(Color online) The switching signals (a) and state trajectories (b) of semi-Markovian switched system
Figure 3
(Color online) The switching signals and impulsive jumps (a), and state trajectories (b) of semi-Markovian switched system
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