On the computation of local invariant sets for nonlinear systems

The importance of invariant sets in control is due to the fact that they define a region of the space where stability, and potentially asymptotic convergence, are assured. For this reason many control design strategies are related to the computation of an invariant set. This is particularly the case for receding horizon based strategies as model predictive control. This paper presents a method for computing a convex invariant set for nonlinear systems. Using properties of D.C. functions, which are functions that can be expressed as difference of two convex functions, and the fact that any continuous nonlinear function can be expressed as D.C. functions, or, at least, well approximated by them, we propose an algorithm for computing a polyhedral invariant set in which no global optimization problem has to be solved. The proposed strategy is guaranteed to provide a non-empty local invariant set provided the nonlinear system is locally stable.