Constructing Piecewise Polynomial Lyapunov Functions for Local Stability of Nonlinear Systems Using Handelman's Theorem

TL;DR

This paper introduces a convex, linear programming-based method to construct piecewise polynomial Lyapunov functions for nonlinear systems, improving computational efficiency and applicability to arbitrary polytopes.

Contribution

It presents the first use of Handelman's theorem combined with polytope decomposition to create piecewise polynomial Lyapunov functions for stability analysis.

Findings

Efficient linear programming approach for Lyapunov function construction.

Applicable to arbitrary convex polytopes containing the equilibrium.

Demonstrated effectiveness on the reverse-time Van Der Pol oscillator.

Abstract

In this paper, we propose a new convex approach to stability analysis of nonlinear systems with polynomial vector fields. First, we consider an arbitrary convex polytope that contains the equilibrium in its interior. Then, we decompose the polytope into several convex sub-polytopes with a common vertex at the equilibrium. Then, by using Handelman's theorem, we derive a new set of affine feasibility conditions -solvable by linear programming- on each sub-polytope. Any solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. This is the first result which utilizes Handelman's theorem and decomposition to construct piecewise polynomial Lyapunov functions on arbitrary polytopes. In a computational complexity analysis, we show that for large number of states and large degrees of the Lyapunov function, the complexity of the proposed feasibility problem is less than the complexity of certain semi-definite programs associated with alternative methods based on Sum-of-Squares or Polya's theorem. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of the reverse-time Van Der Pol oscillator.