On Robustly Positively Invariant Sets and Coordinate Transformations for Discrete-time Nonlinear Systems: a Tutorial

This paper addresses the problem of characterizing analytically invariant sets for nonlinear discrete-time systems. Different notions of invariance are defined and the effect of state and input transformations on these invariant sets is studied. The main part of the paper considers finding robustly positively invariant sets for feedback linearizable (with respect to the disturbance input) systems. It is shown that taking the system into controller canonical form, simplifies the computations considerably. The ultimate goal is to find the minimal robustly invariant set, which is described through the notion of reachability. Finally, it is shown that convex invariant sets of discretized systems using the Euler forward discretization scheme are also invariant for the respective continuous-time system. The purpose of this article is to present some known as well as some new results, illustrated by simple examples, in a tutorial, self-contained form, invoking only basic set theoretic methods and coordinate transformations.