Homogeneous Rational Lyapunov Functions for Performance Analysis of Switched Systems

Abstract

An old open problem in continuous-time switched linear systems consists of determining the H2 norm and the root mean square (RMS) gain. Indeed, existing methods allow one to find only (generally) conservative upper bounds of these performance indexes. This talk addresses this problem by introducing a novel class of Lyapunov functions, called homogeneous rational Lyapunov functions (HRLFs). It is shown that sufficient conditions for establishing upper bounds of the sought performance indexes in the case of arbitrary switching can be given in terms of linear matrix inequality (LMI) feasibility tests by searching for an HRLF of chosen degree. Moreover, it is shown that these conditions are also necessary by searching for an HRLF of degree sufficiently large. Hence, the talk continues by considering the case of switching with dwell time constraints, showing that analogous LMI conditions can be obtained for this case by searching for a family of HRLFs mutually constrained by the dwell time specification.