Level-set approach for Reachability Analysis of Hybrid Systems under Lag Constraints

TL;DR

This paper introduces a level-set method for reachability analysis of hybrid systems with lag constraints, using quasi-variational inequalities and numerical schemes, demonstrated on electric vehicle range extension.

Contribution

It develops a novel approach combining level-set methods and quasi-variational inequalities for hybrid systems with lag constraints, with proven convergence and practical numerical examples.

Findings

Proved a comparison principle ensuring uniqueness of the solution.

Developed finite difference schemes with convergence guarantees.

Applied method to electric vehicle range extension problem.

Abstract

This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite differences schemes for solving the system of inequalities is proposed and the convergence of the numerical solution towards the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module -- the range extender -- as an extra source of energy in addition to its main source -- a high voltage battery. The reachability study of this system is used to establish the maximum range of a simple vehicle model.