Structural Analysis of Multimode DAE Systems: summary of results

TL;DR

This paper reviews and extends the mathematical theory of structural analysis for Differential Algebraic Equation systems used in physical modeling languages, focusing on regularity, index, and handling of mode changes.

Contribution

It consolidates scattered knowledge on structural analysis, extends Pryce's Sigma-method to non-square systems, and discusses analysis of systems with existential quantifiers.

Findings

Extended Sigma-method to non-square systems

Collected and unified background on structural analysis

Analyzed systems with existential quantifiers

Abstract

Modern modeling languages for general physical systems, such as Modelica, Amesim, or Simscape, rely on Differential Algebraic Equations (DAEs), i.e., constraints of the form f(\dot{x},x,u)=0. This drastically facilitates modeling from first principles of the physics, as well as model reuse. In recent works [RR-9334], we presented the mathematical theory needed to establish the development of compilers and tools for DAE-based physical modeling languages on solid mathematical grounds.At the core of this analysis sits the so-called *structural analysis*, whose purpose, at compile time, is to either identify under- and over-specified subsystems (if any), or to rewrite the model in a form amenable of existing DAE solvers, including the handling of mode change events. The notion of "structure" collects, for each mode and mode change event, the variables and equations involved, as well as the *latent equations* (additional equations redundant with the system), needed to prepare the code submitted to the solver. The notion of DAE *index* (the minimal number of times any equation has to be possibly differentiated) is part of this structural analysis. This report complements [RR-9334] by collecting all the needed background on structural analysis. The body of knowledge on structural analysis is large and scattered, which also motivated us to collect it in a single report.We first explain the primary meaning of structural analysis of systems of equations, namely the study of their regularity or singularity in some generic sense. We then briefly review the body of graph theory used in this context. We develop some extensions, for which we are not aware of any reference, namely the structural analysis of systems of equations with existential quantifiers. For the structural analysis of DAE systems, we focus on John Pryce's Sigma-method, that we both summarize and extend to non-square systems. The uses of these tools and methods in [RR9334] are highlighted in this report.