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

This article provides a synthesis method of the ℒ1$$ {\mathcal{L}}_1 $$ 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 ℒ1$$ {\mathcal{L}}_1 $$ performance of piecewise continuous nonlinear systems, that is, the ℒ∞$$ {\mathcal{L}}_{\infty } $$ 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 ℒ1$$ {\mathcal{L}}_1 $$ 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 ℒ1$$ {\mathcal{L}}_1 $$ 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 ℒ1$$ {\mathcal{L}}_1 $$ controller.