Generalized discrete event abstraction of continuous systems: GDEVS formalism

Abstract

In this paper, we propose to model basic continuous component of dynamic systems in a way that facilitate the transposition to a G-DEVS model, which is a paradigm that offers 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. In that, our approach is clearly a discrete event approach where the choice of the time interval between two steps of calculation is based on the behavior changes of the process and no longer constant and/or a priori given, the underlying objective being to strictly satisfy to a given accuracy with a low computational cost. More precisely, we present a generalized discrete event model of an integrator using polynomial descriptions of input–output trajectories. We shall show its great capability of easily handling the delicate problem of input discontinuities, and a detailed comparison with classical discrete time simulation methods, will demonstrate its relevant properties. Several examples, including a complete hybrid system, will illustrate our results.

Introduction

The development of a simulation model requires one to start with a formal specification that is independent of the simulation implementation details. According to [1], and in particular [2], starting from a general specification of a dynamic system, one can specify a simulation model in one of the three following paradigms: a set of differential equations, a discrete time formalism or a discrete event formalism.

There is a fourth paradigm that emerges from a careful analysis of the role of time: difference equation formalism. In other words, following the observing of a real system, one may choose any one of these paradigms to specify a simulation model. In theory, one has to choose the paradigm that would appear to best solve the problem. It may be pointed out that the choice of the paradigm is often based on historical practices within the given domain. Realizing that discrete event formalisms and discrete event simulation techniques have attained certain maturity, it is logical to promote their utilization in every field, wherever appropriate, including design automation, computer sciences, and manufacturing.

While the analog computers of the 1950s and 1960s were capable of modelling continuous systems, the representation and execution on today’s digital computer requires the use of discrete event or discrete time techniques. Continuous simulation is based on discrete time techniques, and is capable of representing “soft changes,” i.e. little changes between two consecutive discrete time values, in a system behavior. In contrast, in discrete event formalisms, the events encapsulate “abrupt changes” in a system behavior, which constitutes the source of the inaccuracies in continuous simulation.

A key characteristic of a classical discrete event abstraction [3], [4] is that it utilizes piecewise constant input–output trajectories to develop a discrete event model of a continuous dynamic system. Given a dynamic system, the trajectories are extracted through threshold sensors and utilized in synthesizing a discrete event model.

We have proposed to generalize the abstraction beyond the constant input–output trajectories [5], [6]. This new approach is termed GDEVS (Generelized Discret Event Specification) [7], [8]. In GDEVS (Generalized Discrete EVent Specification) the input, output and state trajectories are organized through piecewise polynomial segments. The utilization of arbitrary polynomial functions for segments promises higher accuracies in modeling continuous processes as classical discrete event abstractions that, in turn, permit faster execution on host computers in contrast to continuous simulations. The GDEVS abstraction has also been utilized to model and simulate, for the first time, logic gates with fuzzy delays [9].

Moreover the G-DEVS paradigm offers 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 [10].

In this context it was interesting to model basic continuous component of dynamic systems in a way that facilitate the transposition to a G-DEVS model. Consequently our approach is clearly a discrete event approach where the choice of the time interval between two steps of calculation is based on the behaviour changes of the process and no longer constant and a priori given. The underlying objective is to strictly satisfy to a given accuracy.

In this paper we propose a generalized discrete event model of an integrator using piecewise linear input–output trajectories. We will demonstrate that the state transitions of the system due to external or internal events is therefore easily formalized in G-DEVS. We will show how the output errors of the G-DEVS model remain smaller than a given tolerance and the possibility to change this tolerance according to different strategic choices. In effect it can be interesting to set this accuracy index at a high level in particular when the dynamics of the system is important. On the other hand, in some particular case it would be preferable to relax this condition. The preceding property will definitely distinguish this approach to the fixed step classical methods. Moreover the remarkable behaviour on input discontinuities will be emphasized, and a detailed comparison with classical discrete time approaches such as the basic Euler method is widely discussed, leading to the perspective of an efficient use of this approach in real-time simulation. However, we shall show how using G-DEVS paradigm, we avoid the problem of convenient synchronization protocols in either mixed-mode simulation or distributed simulation. Several examples, including an hybrid-system illustrates our main results.