
UC ECE Working Paper TR 138/8/92/ECE, Draft 1
ADB 1 9/28/92
Modeling DiscreteEvent Systems
Using
Partial Difference Equations
Albert D. Baker
Electrical and Computer Engineering Department
University of Cincinnati
Cincinnati, OH 45221
Partial difference equation models of discreteevent systems are motivated by reviewing
continuoustime and discretetime system models. The solution to these partial difference
equations is presented. This solution may be simplified in some special cases. These models
allow for discreteevent systems to be combined and decomposed. A coordination construct
between parallel discreteevent systems is presented whereby controlled and closedloop 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
discreteevent systems are raised.
From a systems theory point of view, there are three basic types of systems: continuoustime
systems, discretetime systems, and discreteevent systems. The generally accepted models
for continuoustime and discretetime systems are similar, providing closedform solutions to
system state for all time. Models for discreteevent systems are not generally agreed upon.
Heretofore, no discreteeventsystems model has provided closedform solutions similar to those
available for continuous and discretetime systems. This paper shows that partial difference
equation models can provide closedform solutions to discreteevent system models. The use of
partial difference equation models is first motivated by reviewing the use of differential
equations for continuoustime systems and difference equations for discretetime systems.
1. MODEL FORMULATION
Continuoustime 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 continuoustime 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 statespace representation of a linear, timeinvariant, continuoustime system is usually given by linear, firstorder differential equations of the form:
? x(t) = Ax(t) + Bu(t)
, [1]