On the index of multi-mode DAE Systems (also called Hybrid DAE Systems)

Albert Benveniste 1 Timothy Bourke 2 Benoît Caillaud 1 Marc Pouzet 3
1 HYCOMES - Modélisation hybride & conception par contrats pour les systèmes embarqués multi-physiques
Inria Rennes – Bretagne Atlantique , IRISA-D4 - LANGAGE ET GÉNIE LOGICIEL
2 Parkas - Parallélisme de Kahn Synchrone
DI-ENS - Département d'informatique de l'École normale supérieure, Inria Paris-Rocquencourt, CNRS - Centre National de la Recherche Scientifique : UMR 8548
Abstract : Hybrid systems modelers exhibit a number of difficulties related to the mix of continuousand discrete dynamics and sensitivity to the discretization scheme. Modular modeling, where subsystemsmodels can be simply assembled with no rework, calls for using Differential Algebraic Equations (DAE).In turn, DAE are strictly more difficult than ODE. In most modeling and simulation tools, beforesimulation can occur, sophisticated pre-processing is applied to DAE systems based on the notion ofdifferentiation index. Graph based algorithms such as the one originally proposed by Pantelides areefficient at finding the index, structurally (i.e., outside some exceptional values for the system parameters).The differentiation index for DAE explicitly relies on everything being differentiable. Therefore,extensions to hybrid systems must be done with caution—to our knowledge, no such extension exists.We propose to rely on non-standard analysis for this. Non-standard analysis formalizes differentialequations as discrete step transition systems with infinitesimal time basis. We can thus bring hybridDAE systems to their non-standard form, where the notion of difference index can be firmly used—thedifference index of a difference Algebraic Equation (dAE) is an easy transposition of the differentiationindex, in which forward shift replaces differentiation. We prove that the differentiation index of a DAEis structurally equal to the difference index of its non-standard interpretation, which is a dAE. We canthus propose the difference index of the non-standard semantics of a hybrid DAE system, as a consistentextension of both the differentiation index of DAE and the difference index of dAE.It turns out that the index theory for (discrete time) dAE systems is interesting in itself and raises newissues. We discuss graph based algorithms à la Pantelides for computing the dAE index and discussexamples.
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Albert Benveniste, Timothy Bourke, Benoît Caillaud, Marc Pouzet. On the index of multi-mode DAE Systems (also called Hybrid DAE Systems). [Research Report] RR-8630, Inria. 2014, pp.30. ⟨hal-01084069⟩

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