Set‐invariance‐based interpretations for the L1 performance of nonlinear systems with non‐unique solutions

Abstract

AbstractThis article is concerned with tackling the performance analysis problem of continuous and piecewise continuous nonlinear systems with non‐unique solutions by using the involved arguments of set‐invariance principles. More precisely, this article derives a sufficient condition for the performance of continuous nonlinear systems in terms of the invariant set. With respect to the case such that solving a nonlinear differential equation is difficult and thus an employment of the invariant set‐based sufficient condition is a non‐trivial task, we also derive another sufficient condition through the extended invariance domain approach. Because this extended approach characterizes set‐invariance properties in terms of the corresponding vector field and an extended version of contingent cones, the performance analysis problem could be solved without considering both the explicit solutions for the differential equation and the relevant solution uniqueness. These arguments associated with the performance of continuous systems are further extended to the involved case of piecewise continuous nonlinear systems, and we establish parallel results relevant to the set‐invariance principles obtained for the continuous nonlinear systems. Finally, numerical examples are provided to demonstrate the effectiveness as well as the applicability of the overall results derived in this article.