SIAM J. Control. Optim. | 2021
Synthesizing Robust Domains of Attraction for State-Constrained Perturbed Polynomial Systems
Abstract
In this paper we propose a convex programming based method to compute robust domains of attraction for state-constrained perturbed polynomial continuous-time systems. The robust domain of attraction is a set of states such that every trajectory starting from it will approach the equilibrium while never violating the specified state constraint, irrespective of the actual perturbation. With Kirszbraun s extension theorem for Lipschitz maps, we first characterize the interior of the maximal robust domain of attraction for state-constrained polynomial systems as the strict one sub-level set of the unique viscosity solution to a generalized Zubov s equation. Instead of solving this Zubov s equation based on traditional grid-based numerical methods, we synthesize robust domains of attraction via solving semi-definite programs, which are constructed from the generalized Zubov s equation. A robust domain of attraction could be obtained by solving a single semi-definite program, rendering our method simple to implement. We further show that the existence of solutions to the constructed semi-definite program is guaranteed and there exists a sequence of solutions such that their strict one sub-level sets inner-approximate the interior of the maximal robust domain of attraction in measure under appropriate assumptions. Finally, we evaluate our semi-definite programming based method on three case studies.