Discrete Time Systems with Event-Based Dynamics: Recent Developments in Analysis and Synthesis Methods

Abstract

1.1 Definitions and basic properties Discrete event systems (DES) constitute a specific subclass of discrete time systems whose dynamic behavior is governed by instantaneous changes of the system state that are triggered by the occurrence of asynchronous events. In particular, the characteristic feature of discrete event systems is that they are discrete in both their state space and in time. The modeling formalism of discrete event systems is suitable to represent man-made systems such as manufacturing systems, telecommunication systems, transportation systems and logistic systems (Caillaud et al. (2002); Delgado-Eckert (2009c); Dicesare & Zhou (1993); Kumar & Varaiya (1995)). Due to the steady increase in the complexity of such systems, analysis and control synthesis problems for discrete event systems received great attention in the last two decades leading to a broad variety of formal frameworks and solution methods (Baccelli et al. (1992); Cassandras & Lafortune (2006); Germundsson (1995); Iordache & Antsaklis (2006); Ramadge & Wonham (1989)). The literature suggests different modeling techniques for DES such as automata (Hopcroft & Ullman (1979)), petri-nets (Murata (1989)) or algebraic state space models (Delgado-Eckert (2009b); Germundsson (1995); Plantin et al. (1995); Reger & Schmidt (2004)). Herein, we focus on the latter modeling paradigm. In a fairly general setting, within this paradigm, the state space model can be obtained from an unstructured automaton representation of a DES by encoding the trajectories in the state space in an n-dimensional state vector x(k) ∈ Xn at each time instant k, whose entries can assume a finite number of different values out of a non-empty and finite set X. Then, the system dynamics follow