A state event detection algorithm for numerically simulating hybrid systems with model singularities

This article describes an algorithm for detecting the occurrence of events, which signify discontinuities in the first derivative of the state variables, while simulating a set of nonsmooth differential equations. Such combined-discrete continuous systems arise in many contexts and are often referred to as hybrid systems, switched systems, or nonsmooth systems. In all cases, the state events are triggered at simulated times which generate states corresponding to the zeros of some algebraic “event” function. It has been noted that all existing simulators are prone to failure when these events occur in the neighborhood of model singularities---regions of the state space where the right-hand side of the differential equation is undefined. Such model singularities are often the impetus for using nonsmooth models in the first place. This failure occurs because existing algorithms blindly attempt to interpolate across singular regions, checking for possible events after the fact. The event detection algorithm described here overcomes this limitation using an approach inspired by feedback control theory. A carefully constructed extrapolation polynomial is used to select the integration step size by checking for potential future events, avoiding the need to evaluate the differential equation in potentially singular regions. It is shown that this alternate approach gives added functionality with little impact on the simulation efficiency.