$$\mathscr {L}_1$$-Gain and Control Synthesis

Abstract

AbstractThis chapter treats the problems of stochastic stability, \(\mathscr {L}_1\)-gain and control synthesis for positive semi-Markov jump systems (S-MJSs). The system under consideration involves semi-Markov stochastic process related to Weibull distribution. The main motivation for this chapter is that the positive condition sometimes needs to be considered in S-MJSs and the controller design methods in the existing works have some conservation. To deal with these problems, some sufficient conditions for stochastic stability of positive S-MJSs are established by implying the linear co-positive Lyapunov function. Then, some sufficient conditions for \(\mathscr {L}_1\)-gain constraint are also presented, upon which, a state feedback controller is designed by decomposing the controller gain matrix such that the resulting closed-loop system is positive and stochastically stable with \(\mathscr {L}_1\)-gain performance in the form of standard linear programming. The advantages of the new framework lie in the following facts: (i) the weak infinitesimal operator is derived for S-MJSs under the constraint of positive condition and (ii) the less conservative stabilizing controller is designed to achieve the desired control performance. Finally, numerical examples are given to demonstrate the validity of the main results.