Semantics of multi-mode DAE systems

Albert Benveniste 1 Timothy Bourke 2, 3 Benoît Caillaud 1 Marc Pouzet 2, 3
1 HYCOMES - Modélisation hybride & conception par contrats pour les systèmes embarqués multi-physiques
IRISA-D4 - LANGAGE ET GÉNIE LOGICIEL, Inria Rennes – Bretagne Atlantique
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 continuous and discrete dynamics and sensitivity to the discretization scheme. Modular modeling, where subsystems models can be simply assembled with no rework, calls for using Differential Algebraic Equations (DAE). In turn, DAE are strictly more difficult than ODE. They require sophisticated pre-processing using various notions of index before they can be submitted to a solver. In this report we study some fundamental issues raised by the modeling and simulation of hybrid systems involving DAEs. The objective of this work is to serve for the evolution and the design of future releases of the Modelica language for such systems. We focus on the following questions: * What is the proper notion of index for a hybrid DAE system? * What are the primitive statements needed for a DAE hybrid systems modeler? The differentiation index for DAE explicitly relies on everything being differentiable. Therefore, generalizations to hybrid systems must be done with caution. We propose relying on non-standard analysis for this. Non-standard analysis formalizes differential equations as discrete step transition systems with infinitesimal time basis. We can thus bring hybrid DAE systems to their nonstandard form, where the notion of difference index can be firmly used. From this study, general hints for future releases of Modelica can be drawn.
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Albert Benveniste, Timothy Bourke, Benoît Caillaud, Marc Pouzet. Semantics of multi-mode DAE systems. 2013. ⟨hal-00938891⟩

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