Discrete Event Simulation of Hybrid Systems

Instead of discretising the time and using a BDF-based method for the numerical computation of a continuous-time model one may think of quantising the state variables. That is, instead of using a multistep method to compute an approximation of the value x(f +i) of a state variable x at time tjc+i, the question then is at what time the state x will deviate from its current value x (tk) by more than a given quantum A Q. In other words, the task is to find the smallest time step h so that 

2 Bond Graph Representations of Hybrid System Models 

This approach replaces the traditional discrete time based computation of continuous time models by a discrete event simulation that advances the time from the time point of an event to the time of the next event which is attractive for the computation of hybrid models because discrete events, i.e. discontinuous mode changes, and the continuous time behaviour during system modes can be uniformly processed in the framework of the well-known Discrete EVent System (DEVS) specification introduced by Ziegler [47, 48]. Moreover, the DEVS formalism is supported by software libraries such as adevs [49, 50] or simulation environments such as PowerDEVS [51, 52], 

An atomic DEVS model is defined as a tuple of sets and functions (cf. Appendix A.3). If the output events of an atomic DEVS model are converted into input events of another atomic DEVS model, i.e. if atomic DEVS models are coupled, then the result defines a new DEVS model. That is, complex systems can be modelled in the DEVS framework in a hierarchical manner. The DEVS formalism is widely used in computer science. Its application to the numerical solution of continuous-time models, however, is much less common. 

The quantised state system (QSS) method introduced by Kofman [53] allows for a discrete event simulation of hybrid systems. The method starts from the observation that a piecewise constant trajectory can be represented by sequences of events. The reader is referred to the literature, e.g. references [1, 53, 54] for details. In the following, only the basic idea is outlined in a simplified manner. 

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An attractive feature of a discrete event simulation of hybrid systems is that the simulation time advances from discrete event to discrete event. Eor the QSS method, discontinuities in the inputs and the quantised variables dictate the time advance. No iteration is necessary to locate the time point of a discontinuity allowing for an efficient processing of models with discontinuities. Discrete event simulation using the quantised-based integrations needs much less simulation steps than a numerical integration method of comparable accuracy based on time-discretisation. Accordingly, computational costs are saved. Nevertheless, there are still some problems to be tackled with the QSS approach as detailed in [1, Chap. 12.11]. 

Kofman, E. (2004). Discrete event simulation of hybrid systems. SIAM Journal on Scientific Computing, 25(5), 1771-1797, from http //dx.doi.org/10.1137/S1064827502418379. 

Venkateswaran, J., Son, Y., and Jones, A. Hierarchical production planning using a hybrid system dynamic-discrete event simulation architecture. In Proceedings of the 2004 Winter Simulation Conference, volume 2, pages 1094-1102, 2004. 

The outlined quantised state integration can be applied to systems of coupled ODEs with piecewise constant input functions. Moreover, it can be extended so that hybrid DAE systems can be solved by discrete event simulation [54],... 

Others define 3 and 4 as discrete event simulation as follows a discrete event simulation is one in which the state of a model changes at only a discrete, but possibly random, set of simulated time points. Mixed or hybrid systems with both discrete and continuous change do exist. Actually simulation packages tiy to include both. 

The last important evolution of PrODHyS is the integration of a dynamic hybrid simulation kernel (Ferret et al., 2004 Olivier et al., 2006, 2007). Indeed, the nature of the studied phenomena involves a rigorous description of the continuous and discrete dynamic. The use of Differential and Algebraic Equations (DAE) systems seems obvious for the description of continuous aspects. Moreover the high sequential aspect of the considered systems justifies the use of Petri nets model. This is why the Object Differential Petri Nets (ODPN) formalism is used to describe the simulation model associated with each component. It combines in the same structure a set of DAE systems and high level Petri nets (defining the legal sequences of commutation between states) and has the ability to detect state and time events. More details about the formalism ODPN can be found in previous papers (Ferret et al., 2004). 

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