Synthesizing Robust Domains of Attraction for State-Constrained Perturbed Polynomial Systems

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.