Timed and Hybrid Models

Abstract

In this chapter we concentrate on timed models of DES. We also explore what happens when a system combines time-driven dynamics with event-driven dynamics giving rise to what are referred to as hybrid systems, which were introduced at the end of Chap. 1. The simplest instance of time-driven dynamics is the case of adjoining one or more clocks to the untimed DES model, resulting in a timed model. In the case of timed DES models, the sample paths are no longer specified as event sequences {e ₁ e ₂, …} or state sequences {x ₀,x ₁,….}, but they must include some form of timing information. For example, let t _k , k= 1, 2, …, denote the time instant when the kth event and state transition occurs (with t ₀ given); then a timed sample path of a DES may be described by the sequence {(x ₀,t ₀), (x ₁ t ₁), …}. Similarly, a timed sequence of events would look like {(e ₁ t ₁), (e ₂,t ₂), …}. Creating a framework for timed DES models will enable us to address questions such as “how many events of a particular type can occur in a given time interval?”, “is the time interval between occurrences of two different events always greater than a given lower bound?” or “how long does the system spend in a given state?” These issues are of critical importance in analyzing the behavior of many classes of DES since they provide us with particularly useful measures of system performance. The next step is to consider more complicated time-driven dynamics than clocks. Namely, at each discrete state of the system, we can associate a set of differential equations that describe the evolution of continuous variables of interest. This brings us to the realm of hybrid systems.