Stability, $l_{2}$ -Gain, and Robust $H_{\infty }$ Control for Switched Systems via ${N}$ -Step-Ahead Lyapunov Function Approach

Unlike the traditional analysis and synthesis approach of a switched system that requires a monotonic decrease of Lyapunov function (LF), this paper investigates an N-step-ahead LF approach. Such an approach allows a non-monotonic behavior both at the switching instants and during the running time of each subsystem but guarantees an average decrease at every N sampling steps. The asymptotic stability criterion is improved as well as the capability of disturbance attenuation. By introducing a series of auxiliary variables, the future knowledge of states and exogenous noises can be properly used to derive sufficient conditions for the existence of a robust H_∞ controller in the form of a set of numerical testable conditions. Note that N has direct impact on the number of inequality constraints. The essential difficulty is to construct an exponential damping law of the decreasing points of LF, i.e., to find the joint point between the switching interval and the predictive horizon. Moreover, the relationship between N-step time difference of LF and switching rate, i.e., the average dwell time constraint, is thoroughly discussed. An ecology system is employed to demonstrate practical potentials of the presented design framework.