Accurate GDEVS modelling : discrete event integrator.
J. C. Carmona , A. Naamane, and N. Giambiasi
LSIS CNRS UMR 6168, Marseille, France
http ://wwww.lsis.org
ABSTRACT
In this paper we shall present our recent work on eventoriented
modelling. In particular we shall propose a continuous
model of an integrator using piecewise linear inputoutput
trajectories leading to a reduced order parameterized
model. We shall show that this fact contributes to easily
describe the state transitions of the system du to external or
internal events. In particular we shall focus on the accuracy
and the simplicity of the method and the natural and convenient
treatment of the delicate problem of external discontinuities.
KEY WORDS
Discrete systems. Discrete event model : DEVS, GDEVS.
Piecewise linear trajectory.
1 Introduction
Simulation models require to start with a formal specification
of the problem, independent of the simulation implementation
details. From a general point of view of dynamic
systems, one can specify a model in the time domain in
one of the three following paradigms : a set of differential
equations, a discrete time formalism and discrete event formalism,
even though in a careful analysis, one can distinguish
a fourth paradigm : the difference equation paradigm.
See [1] for a more detailed presentation.
It is well-known that continuous simulation, by defi-
nition, is capable of representing "soft changes" in a system
behavior. Furthermore discrete event techniques clearly
promise faster execution speed. In the history of discrete
event simulation, Zeigler [2] and [3] has pioneered a formalism
for the specification of discrete event models with explicit
time allowing the representation the dynamic systems
behavior with constant piecewise input-output trajectories.
This event oriented approach the so-called DEVS formalism
(Discrete Event Specification), naturally provides effi-
cient solutions in particular in the case of systems with discontinuities
in their input-output behavior. In this situation,
while continuous and discrete time simulation techniques
may generate errors, DEVS abstractions seem to be a natural
choice. Nevertheless, in case of higher-order discontinuities
the implicit choice of piecewise constant trajectories
leads to rather important inaccuracies. Therefore GDEVS
(Generalized DEVS) using higher-order piecewise inputoutput
trajectories seems to be a relevant alternative solution
to this problem. Moreover the choice of piecewise polynomial
input-output trajectories leads to rather simple parameterized
model of these signals, namely the polynomial
parameters or all linear combinations of them more appropriate.
See [4] for more details. Moreover the GDEVS techniques
offer the ability to develop a uniform approach to
model hybrid systems (abstraction closer to real systems),
i.e. composed of both continuous and discrete components
[5].
In this context it was interesting to model basic continuous
component of dynamic systems such as an integrator,
in a way that facilitates the transposition to a GDEVS
model. Consequently our approach is clearly an event
oriented approach in the sense of the choice of the specific
times used are only du to behavior changes of the studied
system and is not based on a prior choice. The underlying
objective is to strictly satisfy a given accuracy.
In this paper we shall propose an event-oriented continuous
model of an integrator using piecewise linear inputoutput
trajectories. We shall show that the state transitions
of the system du to external or internal events is therefore
easily formalized in this context. In this sense we shall facilitate
the description the GDEVS model related to this
system and its insertion in an hybrid system model.
Moreover it was interesting to efficiently solve the delicate
problem of input discontinuities. Therefore the assessment
of a simple formalism able to easily treat such
discontinuities widely motivated our work. In the same vein
we deliberately adopted a simulation oriented approach.
This paper is organized as follow : after this introduction
section, we shall present the basic background necessary
to the GDEVS framework. Then we shall present the
main results of our work : the event oriented model of an
integrator system in continuous time. Then examples shall
illustrate the accuracy of the method, its flexibility on input
event occurrence and its capability of treating input discontinuities.
Finally some conclusions and directions of research
will be presented in a last section.
2 Review of the GDEVS formalism
We assume that input-output trajectories are piecewise
linear segments :
x(t) = si(t - ti) + Xi 8t 2 [ti, ti+1]
entirely parameterized by the triplet (ti, si,Xi).
.....