Constructive Approximations of the Zeno Dynamics in Affine Switched Systems: the Projection Based Approach

This paper deals with a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno dynamics. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real-world processes do not present the corresponding behavior, mathematical models of some interconnected engineering systems may be Zeno due to the corresponding formal abstraction. We propose a useful approximative approach to Zeno dynamics, namely, a projection based characterization of this phenomena. A resulting trajectory associated with the Zeno dynamics can finally be described as a result of a specific dynamic projection procedure applied to the original model. We use here the projected dynamic systems methodology. The obtained formal description provides an effective theoretic basis for a constructive treatment of the Zeno behaviour. We also discuss an application of the proposed technique to the conventional sliding mode control processes.