Towards a Geometric Theory of Hybrid Systems

Abstract

Abstract. We propose a framework for a geometric theory of hybrid systems. Given a deterministic, non-blocking hybrid system, we introduce the notion of its hybrifold with the associated hybrid flow on it. This enables us to study hybrid systems from a global geometric perspective as (generally non-smooth) dynamical systems. This point of view is adopted in studying the Zeno phenomenon. We show that it is due to nonsmoothness of the hybrid flow. We introduce the notion of topological equivalence of hybrid systems and locally classify isolated Zeno states in dimension two.

Structural stability of hybrid systems1

“…Furthermore, each continuous time orbit always exits a domain through a guard which lies on the boundary of the domain, and enters it through the image of a reset map which is a homeomorphism and takes values on the boundary of the "next" domain. For a more detailed and precise formulation of the notion of regularity, see [17] and [18]. whenever both sides are defined.…”

“…The existence of is ensured by regularity of [17,18]. Collapse each equivalence class to a point to obtain the quotient space ).…”

“…If the hybrid system satisfies this property, then it can be shown that . For details, please see [17].…”

“…Recently we suggested a geometric approach for the analysis of a class of hybrid systems [17,18]. We introduced the notions of the hybrifold and hybrid flow in order to study the hybrid system as a single piecewise smooth dynamical system.…”

“…A standard equilibrium on the boundary of a domain, however, is not locally structurally stable. The main contribution of the paper is to show that Zeno states, which necessarily lie on the boundary of a domain [18], are locally structurally stable. This means that it may be difficult to remove Zenoness by perturbing the system.…”