Piecewise quadratic common Lyapunov functions and the absolute stability of single variable Lur'e systems

Abstract

One of the main factors motivating the study of switched systems is the rapidly developing area of switching control. A switched control system is a type of hybrid dynamic system consisting of continuous and/or discrete time processes interfaced with some decision making process, which estimates the current active environment and selects the appropriate controller. The type of dynamic system that we are interested in is the single input, single output Lur'e system. Sufficient conditions for the absolute stability of single variable Lur'e systems are presented. For general Lur'e systems, sufficient conditions for the existence of a piecewise quadratic common Lyapunov function are shown. For 2D Lur'e systems, a piecewise quadratic common Lyapunov function exists if the 45° criterion holds.