Dynamical Systems Background

Abstract

This chapter introduces some basic preliminaries about stability analysis of dynamical systems, in particular continuous-time, time-invariant, nonlinear systems. First, the concept of equilibrium point is introduced, hence providing the notion of local asymptotical stability and its characterization based on LFs. Then, the definition of DA of an equilibrium point is provided, which is the set of initial conditions from which the state converges to such an equilibrium point. It is recalled how inner estimates can be obtained by looking for sublevel sets of LFs included in the region where the temporal derivatives are negative, moreover basic properties of such estimates are reviewed. The chapter hence proceeds by considering uncertain nonlinear systems, in particular nonlinear systems with coefficients depending on a time-invariant uncertain vector constrained in a given set of interest. The concept of common equilibrium point is introduced, hence providing the notion of robust local asymptotical stability and its characterization based on common and parameter-dependent LFs. Then, the definition of RDA of a common equilibrium point is provided, which is the set of initial conditions from which the state converges to such an equilibrium point for all admissible uncertainties. It is recalled how inner estimates can be obtained by looking for either sublevel sets of common LFs or parameter-dependent sublevel sets of parameter-dependent LFs.