Nonlinear system stability boundary approximation by polytopes in state space†

Abstract

An algorithm that estimates the attracting basin boundary for a stable equilibrium point of a nonlinear system has been developed. This boundary is an important characterization of the global properties of a nonlinear dynamic system. The algorithm begins with a convex polytope that is entirely within the region of attraction and that surrounds the stable equilibrium point. The basin boundary of this stable equilibrium point is then approximated by a (possibly non-convex) polytope whose vertices are generated by integrating vertices of the initial polytope backwards in time. An accuracy test is applied on facets of the latter polytope. If any facet fails the test, then the approximation is refined by adding more vertices and facets to both the initial convex polytope and the final polytope. Three examples are presented to demonstrate the method. Examples include a third-order PID control system, a fourth-order balancing acrobot system, and a fourth-order lateral/directional aircraft system. For the PID position control system and for the acrobot, known limit cycles on the basin boundaries are plotted along with the approximate polytope boundary to show the reduced conservatism of the approximation. The algorithm can work with any order system, but computation time and memory requirements can grow rapidly with the state-space dimension.