On the construction of Lyapunov functions using the sum of squares decomposition

A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.

1 Introduction

Stability of dynamical systems plays a very important role in control system analysis and design, Unlike the case of linear systems, proving stability of equilibria of nonlinear systems is more complicated. A sufficient condition is the existence of a Lyapunov function [1]: a positive definite function defined in some region of the state space containing the equilibrium point whose derivative along the system trajectories is negative semi-definite. This is Lyapunov's direct method, which even though addresses exactly and in a simple way the important issue of stability, it does not provide any coherent methodology for constructing such a function. Lyapunov's indirect method that investigates the local stability of the equilibria, is inconclusive when the linearized system has imaginary axis eigenvalues. Other methodologies to determine the stability properties of the equilibria of nonlinear systems (such as exhaustive simulations, Linear Parameter Varying (LPV) techniques, Integral Quadratic Constraint (IQC) formulations [2] etc) are sometimes quite conservative.