Stability of randomly switched diffusions

This paper provides a sufficient criterion for ε-moment stability (boundedness) and ergodicity for a class of systems comprising a finite set of diffusions among which switching is governed by a continuous time Markov chain. Stability/instability properties for each separate subsystem are assumed to be quantified by a Lyapunov function candidate and an associated growth rate equation. For the set of Lyapunov functions a compatibility criterion is assumed to be fulfilled bounding the ratio between pairs of Lyapunov functions. The established criterion is shown to be equivalent to an exact criterion for the almost sure convergence of an associated process bounding moments of the process under study. Examples are provided to illustrate the use of the established criterion.

[1]  El-Kébir Boukas,et al.  Stability of stochastic systems with jumps , 1996 .

[2]  John Lygeros,et al.  Stochastic Hybrid Systems: Research Issues and Areas , 2006 .

[3]  J. Liou.  Recurrence and transience of Gaussian diffusion processes , 1990 .

[4]  R. Tweedie.  Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space , 1975 .

[5]  John Leth,et al.  Comment on ‘Fault tolerance analysis for stochastic systems using switching diffusion processes’ by Yang, Jiang and Cocquempot , 2011, Int. J. Control.

[6]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[7]  Patrizio Colaneri,et al.  Almost Sure Stabilization of Uncertain Continuous-Time Markov Jump Linear Systems , 2010, IEEE Transactions on Automatic Control.

[8]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[9]  Youmin Zhang,et al.  Stochastic stability analysis for fault tolerant control systems with multiple failure processes , 2002, Int. J. Syst. Sci..

[10]  Bin Jiang,et al.  Fault tolerance analysis for stochastic systems using switching diffusion processes , 2009, Int. J. Control.

[11]  Kenji Kashima,et al.  Stability and stabilization of switching stochastic differential equations subject to probabilistic state jumps , 2010, 49th IEEE Conference on Decision and Control (CDC).

[12]  Patrizio Colaneri,et al.  On almost sure stability of continuous-time Markov jump linear systems , 2006, Autom..

[13]  K. Loparo,et al.  Stochastic stability properties of jump linear systems , 1992 .

[14]  Mark H. A. Davis.  Piecewise‐Deterministic Markov Processes: A General Class of Non‐Diffusion Stochastic Models , 1984 .

[15]  Qi He,et al.  Invariant density, Lapunov exponent, and almost sure stability of Markovian-regime-switching linear systems , 2011, J. Syst. Sci. Complex..

[16]  Frank Kozin,et al.  A survey of stability of stochastic systems , 1969, Autom..

[17]  X. Mao,et al.  Asymptotic stability in distribution of stochastic differential equations with Markovian switching , 2003 .

[18]  A. Morse,et al.  Stability of switched systems with average dwell-time , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[19]  Oswaldo Luiz do Valle Costa.  Stationary distributions for piecewise-deterministic Markov processes , 1990 .

[20]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[21]  P. Mandl.  Analytical treatment of one-dimensional Markov processes , 1968 .