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UC ECE Working Paper TR 138/8/92/ECE, Draft 1

ADB 1 9/28/92

Modeling Discrete-Event Systems Using
Partial Difference Equations Albert D. Baker
Electrical and Computer Engineering Department University of Cincinnati
Cincinnati, OH 45221
Partial difference equation models of discrete-event systems are motivated by reviewing continuous-time and discrete-time system models. The solution to these partial difference equations is presented. This solution may be simplified in some special cases. These models allow for discrete-event systems to be combined and decomposed. A coordination construct between parallel discrete-event systems is presented whereby controlled and closed-loop discreteevent systems can be represented. This coordination construct allows for a new form of instability which is discussed. Other remaining issues in the use of partial difference equations to model discrete-event systems are raised. From a systems theory point of view, there are three basic types of systems: continuoustime systems, discrete-time systems, and discrete-event systems. The generally accepted models for continuous-time and discrete-time systems are similar, providing closed-form solutions to system state for all time. Models for discrete-event systems are not generally agreed upon. Heretofore, no discrete-event-systems model has provided closed-form solutions similar to those available for continuous- and discrete-time systems. This paper shows that partial difference equation models can provide closed-form solutions to discrete-event system models. The use of partial difference equation models is first motivated by reviewing the use of differential equations for continuous-time systems and difference equations for discrete-time systems. 1. MODEL FORMULATION

Continuous-time systems are parameterized by the passage of continuous time. For these systems, time is an element of the set of nonnegative real numbers, t ? R+ = [0 ? ?). The current state of a continuous-time system is determined by the current position in continuous time and the initial state of the system. Continuous time systems are usually described by differential equations or Laplace transforms. The standard state-space representation of a linear, time-invariant, continuous-time system is usually given by linear, first-order differential equations of the form:

? x(t) = Ax(t) + Bu(t)

, [1]