Robust invariant sets generation for state-constrained perturbed polynomial systems

In this paper we study the problem of computing robust invariant sets for state-constrained perturbed polynomial systems within the Hamilton-Jacobi reachability framework. A robust invariant set is a set of states such that every possible trajectory starting from it never violates the given state constraint, irrespective of the actual perturbation. The main contribution of this work is to describe the maximal robust invariant set as the zero level set of the unique Lipschitz-continuous viscosity solution to a Hamilton-Jacobi-Bellman (HJB) equation. The continuity and uniqueness property of the viscosity solution facilitates the use of existing numerical methods to solve the HJB equation for an appropriate number of state variables in order to obtain an approximation of the maximal robust invariant set. We furthermore propose a method based on semi-definite programming to synthesize robust invariant sets. Some illustrative examples demonstrate the performance of our methods.