Constructing Piecewise Polynomial Lyapunov Functions Over Arbitrary Convex Polytopes Using Handelman Basis

We introduce a new algorithm to check the local stability and compute the region of attraction of isolated equilibria 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 equality and inequality feasibility conditions -solvable by linear programmingon each subpolytope. The solution to this feasibility problem yields a piecewise polynomial Lyapunov function on the entire polytope. To the best of the authors’ knowledge, this is the first effort that utilizes Handelman’s theorem to construct piecewise polynomial Lyapunov functions on sub-divided 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 the semi-definite programs associated with Sum-of-Squares and Polya’s algorithms. Using different types of convex polytopes, we assess the accuracy of the algorithm in estimating the region of attraction of the equilibrium point of reverse-time Van Der Pol oscillator.