Stochastic stability and robust stabilization of semi-Markov jump linear systems

The semi-Markov jump linear system (S-MJLS) is more general than the Markov jump linear system (MJLS) in modeling some practical systems. Unlike the constant transition rates in the MJLS, the transition rates of the S-MJLS are time varying. This paper focuses on the robust stochastic stability condition and the robust control design problem for the S-MJLS with norm-bounded uncertainties. The infinitesimal generator for the constructed Lyapunov function is first derived. Numerically solvable sufficient conditions for the stochastic stability of S-MJLSs are then established in terms of linear matrix inequalities. To reduce the conservativeness of the stability conditions, we propose to incorporate the upper and lower bounds of the transition rate and meanwhile apply a new partition scheme. The robust state feedback controller is accordingly developed. Simulation studies and comparisons demonstrate the effectiveness and advantages of the proposed methods. Copyright (c) 2012 John Wiley & Sons, Ltd.