Stochastic Stability and Stabilization of Semi-Markov Jump Linear Systems with Uncertain Transition Rates
Keywords:Semi-Markov Jump Linear System, Stochastic Stability, Stabilization, Time-Varying Transition Rates, Uncertainty, Lyapunov Function.
This paper investigates the problems of robust stochastic stability and stabilization for a general class of continuous-time semi-Markovian jump linear systems (S-MJLSs). The main contribution of the research is to eliminate the limitations of the traditional S-MJLSs with precisely available information by introducing a system with uncertain, time-varying transition rates (TRs) of the jump process, in addition to the imperfect information on the system dynamic matrices. The new system is called the general uncertain semi-Markov jump linear system (GUS-MJLS); it does not contain certain values of the transition rates, but includes nominal time-dependent values in addition to bounded deviations. It is suitable to describe a broader class of dynamical systems with estimated information and modeling errors and also covers the concepts of Markov jump linear system (MJLSs) with time-constant and certain TRs. For this system, the stability is firstly analyzed through the multiple stochastic Lyapunov function approach. Then, based on the stability results, a robust state-feedback controller is formulated. To deal with the time-dependent TRs, a sojourn-time fractionizing technique is used and numerically testable conditions are developed. Finally, discussions on reducing the conservativeness of the robust theorems are provided. The theoretical results are successfully tested on an industrial continuous stirred tank reactor (CSTR) subject to stochastically varying environmental conditions. Comparative simulations are also provided to show the superiority of the presented framework and design method to the existing ones.
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