Structural Analysis of Multi-Mode DAE Systems

Albert Benveniste 1 Benoît Caillaud 1 Marc Pouzet 2, 3 Hilding Elmqvist 4 Martin Otter 5
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
Inria de Paris, DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique
Abstract : Multi-mode DAE systems constitute the mathematical model supporting physical modeling languages such as Modelica. Multi-mode DAE systems are systems of equations of the form "if guard_i do fi(the xj and derivatives of them) = 0" where guard_i is a predicate involving system variables and guarding the DAE fi(...) = 0. Single-mode (i.,e., usual) DAE systems face the issue of differentiation index, originating from the possible existence of so-called “latent constraints”. DAE systems having index 2 are not comfortable to solvers and those having index larger than 3 are problematic or even out of reach. Unfortunately, complex Modelica models may lead to such cases, even more so when composed with the control software. The Structural Analysis of DAE systems, of which the popular Pantelides algorithm is representative, is a symbolic pre-processing preparing the simulation for DAE systems of large index. Unlike for single-mode DAE systems, however, no theory exists that can support the structural analysis of multi-mode DAE systems. As a practical consequence, Modelica compilers handle some models and refuse other ones, including some models that are sound and useful to consider, from physical and practical points of view. Worse, there is no clear definition of the class of Modelica models that can/cannot be handled. As we said, the reason for this is the lack of rigourous mathematical support for the structural analysis of multi-mode DAE systems. It is generally considered that the structural analysis is mode-dependent. This, however, tells nothing about how mode changes must be handled. Even more so when mode changes occur in cascades. In this paper we develop a comprehensive mathematical approach to the structural analysis of multimode DAE systems. We first observe that a single-mode DAE system and its explicit first order Euler approximation possess identical structural analyses, regardless of the step size. We thus propose to take a step size that is infinitesimal in the sense of non-standard analysis, so that the Euler scheme is no longer an approximation but rather a re-interpretation of the original system, in the non-standard domain. Building on top of this, we are able to propose a full mathematical study of the structural analysis of multi-mode DAE systems and we show how simulation code can be deduced from this analysis.
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Submitted on : Monday, July 11, 2016 - 1:38:33 PM
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Albert Benveniste, Benoît Caillaud, Marc Pouzet, Hilding Elmqvist, Martin Otter. Structural Analysis of Multi-Mode DAE Systems. [Research Report] RR-8933, Inria. 2016, pp.32. ⟨hal-01343967v1⟩

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