Switching and impulsive control algorithms for nonlinear hybrid dynamical systems

Abstract

Control algorithms are developed for physical processes modeled as hybrid dynamical systems (HDSs). In this framework, a HDS is a nonlinear switched system of ordinary differential equations (ODEs) coupled with impulsive equations. Switching and impulsive control is applied with two performance goals in mind: First, a high-frequency switching control method is provided to drive a HDS state to the origin while only requiring the HDS state intermittently. Attractivity of the origin is proved under a shell bisection algorithm; a high-frequency switching control rule is designed for this purpose. Second, a state-dependent switching control strategy is derived for when the transient behavior of the HDS is of interest. Finite-time stabilization is guaranteed under a so-called minimum rule algorithm; for each HDS mode, the state space is divided into different control regions and a switching control rule is constructed to switch between controllers whenever a boundary is reached. The theoretical tools used in this article include the Campbell–Baker–Hausdorff formula, multiple Lyapunov functions, and average dwell-time conditions.