A Unifying Approach for Studying the Behaviour and Control of Perturbations in Variable Structure Systems

The behaviour of perturbations around trajectories of variable structure systems is studied and a sensitivity formula is derived. This formula is used to adapt classical shooting algorithms for computing periodic solutions to piecewise smooth systems. The influence of switching boundaries on the stability of periodic solutions is studied within a control theory approach. 1 Introduction In this paper we discuss some properties of variable structure systems. We use this general term to denote systems which switch between several configurations, each of them modelled by an (non) autonomous ODE. The switching of the structure can occur when a trajectory of the system crosses a (possibly time-varying) boundary (figure 1-a) or when the value of a discrete state changes. This usually occurs when the continuous state satisfies a spatial condition. In the latter case the systems are usually denoted by hybrid systems due to the mixture of discrete and continuous states. The difference betwee..