Lyapunov function computation for autonomous systems with complex dynamic behavior

Abstract

A computational approach is presented in this paper to construct local Lyapunov functions for autonomous dynamical systems with multiple isolated locally asymptotically stable (such as point-like, periodic, or strange) attractors. We consider systems of nonlinear ODEs, where the right-hand-side of the dynamic equations is given in the form of rational functions (i.e., fraction of polynomials). The Lyapunov function is searched in a parameterized quadratic form of rational terms of the state variables. The quadratic decomposition of the rational state-dependent inequalities is performed using the linear fractional transformation (LFT) and further algebraic/numeric simplification steps. Unlike the sum of squares (SOS) approach, the sufficient linear matrix inequality (LMI) conditions for the Lyapunov function are formulated only locally on a compact polytopic subset of the state space, which allows indefinite matrix solutions for the quadratic decomposition. The local solution is enforced using affine annihilators with matrix Lagrange multipliers. Alongside the typical Lyapunov conditions, further boundary LMI constraints are prescribed using Finsler’s lemma to ensure the required geometric properties of the Lyapunov function. The results are illustrated on four planar benchmark models having either multiple locally stable equlibria or a limit cycle, and on the Lorenz system.