A Lyapunov-Metzler condition for almost sure stability of Markov jump linear systems
System, Structure and Control, Volume # 3 | Part# 1
Colaneri, Patrizio; de Souza, Valeska M.
Digital Object Identifier (DOI)
Markov jump systems,switched systems,Lyapunov functions,stochastic stability
In this paper we study the almost sure stability of continuous-time jump linear systems with a finite-state Markov form process. A new sufficient condition is derived based on an extended Lyapunov-Metzler inequality. This condition is proven to be less conservative with respect to the existing sufficient conditions. The inequalities generalize those associated with mean square stability and establish a clear link between the two notions. The problem of almost sure stabilizability and detectability is finally discussed.
 Y. Fang and K.A. Loparo (2002). On the relationship between the sample path and moment Lyapunov exponents for jump linear systems. IEEE Trans. on Automat. Control, vol. AC- 47, n. 9, pp. 1556-1560.  P. Bolzern, P. Colaneri, and G. De Nicolao (2005). Almost sure stability of continuous-time Markov jump linear systems: A randomized approach. In Proc. 16th IFAC World Congress, Praha, Czech Republic, July 3-8.  P. Bolzern, P. Colaneri, and G. De Nicolao (2006). On almost sure stability of continuous-time Markov jump linear systems. Automatica, 42, pp. 983-988.  M. Mariton (1990). Jump Linear Systems in Automatic Control. Marcel Dekker, New York.  Y. Fang and K.A. Loparo (2002). Stabilization of continuous-time jump linear systems. IEEE Trans. on Automat. Control, vol. AC-47, n. 10, pp. 1590-1603.  L. Arnold, W. Kleimann, E. Oeljeklaus (1986). Lyapunov Exponents of linear stochastic systems. in Lyapunov Exponents, L. Arnold and V. Wihstutz eds., Lecture Notes in Mathematics vol. 1186, Springer-Verlag, Berlin, pp. 85- 125.  H.J. Chizeck and A.S. Willsky and D. Castanon. (1986). Discrete-time Markovian-jump quadratic optimal control, Int. J. Control, Vol. 43, pp. 213-231.  J.C. Geromel, P. Colaneri (2006). Stability and stabilization of continuous-time switched systems. SIAM J. contr. And Opt., Vol. 45, N. 5, pp. 1915-1930.  R. Krtolica and U. Ozguner and H. Chan and H. Goktas and J. Winckelman and M. Liubakka (1991). Stability of linear feedback systems with random communication delays, Proc. of 1991 American Control Conference, Boston, USA, pp. 2648-2653.  Y. Fang and K.A. Loparo and X. Feng (1991). Modeling issues for the control systems with communication delays, Ford Motor Co., Research Rep.  M. Gomez-Puig and J.G. Montalvo (1997). A new indicator to assess the credibility of the EMS, European Economic Review, Vol. 41, pp. 1511-1535.  M. Tanelli, B. Picasso, P. Bolzern and P. Colaneri (2006). On Robust Almost Sure stabilization of continuous-Time Markov jump linear systems. Proc. of the Conference on decision and Control, San Diego, USA, pp.  X. Feng and K.A. Loparo and Y. Ji and H.J. Chizeck (1992). Stochastic stability properties of jump linear systems, IEEE Trans. on Automatic Control, Vol. AC-37, 1, pp. 38-53.  Y. Fang and K.A. Loparo and X. Feng (1994). Almost sure and d-moment stability of jump linear systems, Int. J. Control, Vol. 59, 5, pp. 1281-1307.  X. Feng and K.A. Loparo (1991). A non-random spectrum for Lyapunov exponents of linear stochastic systems, Stochastic Analysis. Applications, Vol. 9, N. 1, pp. 25-40.