A Hybrid Bounding Method for Computing an Over-Approximation for the Reachable Set of Uncertain Nonlinear Systems

Abstract

In this paper, we show how to compute an over-approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relies on state-of-the-art validated numerical integration methods. The second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty. Since the derived bracketing systems are piecewise Ck-differentiable functions, validated numerical integration methods cannot be used directly. Hence, our contribution resides in the use of hybrid automata to model the bounding systems. We give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set. The computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques. Sufficient conditions are given which ensure the epsiv-practical stability of the enclosures given by our hybrid bounding method. Two examples are also given which show that the performance of our method is very promising.