Computing invariant sets of discrete-time nonlinear systems via state immersion

In this paper, we propose a method for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we will focus on the maximal invariant set. Some special types of nonlinear systems can be considered as the projection of a higher dimensional linear system with a state immersion transformation. For such systems, the equivalence between invariant sets of the nonlinear system and its linear equivalent can be also established, which allows to characterize the maximal invariant set of the nonlinear system using a lifted linear model. For general nonlinear systems, we will use linear approximations because equivalent linear models cannot be achieved exactly. To handle mismatch errors, we tighten the constraint set of the lifted linear model, which will lead to an invariant inner approximation of the maximal invariant set