The L1 controller synthesis for piecewise continuous nonlinear systems via set invariance principles

This article provides a synthesis method of the L-1 controller for piecewise continuous nonlinear systems in terms of set invariance principles. Following the arguments of Filippov's differential equations, solutions of piecewise continuous differential equations are first characterized by set-valued functions. Based on such characteristics of solutions equipped with set-valued functions, the L-1 performance of piecewise continuous nonlinear systems, that is, the Script capital L-& INFIN; gain from the associated disturbance to regulated output, is formulated. To clarify the existence of a controller such that the resulting closed-loop system satisfies the L-1 performance in the presence of piecewise continuity, the so-called generalized controlled invariance domain is proposed. Taking this advanced argument allows us to establish a synthesis procedure for the L-1 controller by showing a lower-semicontinuity of the set-valued functions involved in the generalized controlled invariance domain. Finally, some numerical examples are provided to evaluate the validity of the existence condition for the L-1 controller.