Common and multiple Lyapunov functions in stability analysis of nonlinear switched systems

Abstract

For switched systems with generally homogeneous functions as nonlinear right-hand sides, a method to construct the common Lyapunov function (CLF) from the Lyapunov functions for partial systems is proposed. Many physical laws are described in terms of such generally homogeneous functions. In stability theory these functions as the widest class of the first approximation equations are used. Theorems on asymptotic stability (including a case with perturbations) are formulated. For the arbitrarily nonlinear switched systems with pulses some necessary and sufficient conditions of invariance in terms of CLF as well as sufficient conditions in terms of multiple homomorphism instead of Lyapunov function are given. The homomorphic transformations enable one to reject the usual quasi-monotonicity condition of the right-hand sides of the reduced model, and to weaken the rest of conditions by permitting, for example, alternation of sign of homomorphic mapping. The conditions of existence of common quadratic LF (CQLF) for linear mechanical systems as well as theorems on existence of common vector Lyapunov functions (CVLF) for large-scale systems are proposed.