Shooting Methods for Locating Grazing Phenomena in Hybrid Systems

Hybrid systems are typified by strong coupling between continuous dynamics and discrete events. For such piecewise smooth systems, event triggering generally has a significant influence over subsequent system behaviour. Therefore it is important to identify situations where a small change in parameter values alters the event triggering pattern. The bounding case, which separates regions of (generally) quite different dynamic behaviour, is referred to as grazing. At a grazing point, the system trajectory makes tangential contact with an event triggering hypersurface. The paper formulates conditions governing grazing points. Both transient and periodic behaviour are considered. The resulting boundary value problems are solved using shooting methods that are applicable for general nonlinear hybrid (piecewise smooth) dynamical systems. The grazing point formulation underlies the development of a continuation process for exploring parametric dependence. It also provides the basis for an optimization technique that finds the smallest parameter change necessary to induce grazing. Examples are drawn from power electronics, power systems and robotics, all of which involve intrinsic interactions between continuous dynamics and discrete events.