Computation of Lyapunov functions for smooth nonlinear systems using convex optimization

Abstract

It is shown that for smooth nonlinear systems conditions for the existence of a Lyapunov function that guarantees uniform exponential stability can be formulated as linear inequalities defined pointwise in the state space when assuming a general linearly parameterized class of smooth non-quadratic Lyapunov-function candidates. Hence, computation of the Lyapunov function involves the solution of a convex large-scale optimization problem using linear or quadratic programming. The optimization criterion can for example be selected to find a Lyapunov function which predicts fast decay rate or large region of attraction. Analysis of the tradeoff between accuracy and computational complexity as well as possible conservativeness of the procedure is given particular attention. The procedure is illustrated using numerical examples.