Robust Invariant Sets Computation for Discrete-Time Perturbed Nonlinear Systems

In this article, we study the maximal robust invariant set estimation problem for discrete-time perturbed nonlinear systems within the optimal control framework. The maximal robust invariant set of interest is a set of all states such that every possible trajectory starting from it never violates a specified state constraint, regardless of actual disturbances. The maximal robust invariant set is shown to be the zero level set of the unique bounded solution to a Bellman type equation, which is a functional equation being widely used in discrete-time optimal control. Consequently, the maximal robust invariant set estimation problem is reduced to a problem of solving a Bellman type equation. This is the main contribution of this article. The uniqueness of bounded solutions enables us to solve the derived Bellman type equation using numerical methods such as the value iteration and policy iteration, which provide an approximation of the maximal robust invariant set. Finally, two examples demonstrate the performance of our Bellman equation based method.