Structural Methods for handling mode changes in multimode DAE systems

TL;DR

This paper introduces a novel mathematical and structural approach for handling mode changes in multimode DAE systems within physics-agnostic modeling languages like Modelica, enabling automated hot restarts even with impulses.

Contribution

It proposes a new mathematical framework and compile-time analysis for mode change handling in multimode DAE systems, improving automation and robustness.

Findings

Developed a mathematical meaning for hot restarts in mode changes.

Created an algorithm for structural and impulse analysis during mode transitions.

Enabled compile-time diagnostics for insufficiently specified mode changes.

Abstract

Hybrid systems are an important concept in Cyber-Physical Systems modeling, for which multiphysics modeling from first principles and the reuse of models from libraries are key. To achieve this, DAEs must be used to specify the dynamics in each discrete state (or mode in our context). This led to the development of DAE-based equational languages supporting multiple modes, of which Modelica is a popular standard. Mode switching can be time- or state-based. Impulsive behaviors can occur at mode changes. While mode changes are well understood in particular physics (e.g., contact mechanics), this is not the case in physics-agnostic paradigms such as Modelica. This situation causes difficulties for the compilation of programs, often requiring users to manually smooth out mode changes. In this paper, we propose a novel approach for the hot restart at mode changes in such paradigms. We propose a mathematical meaning for hot restarts (such a mathematical meaning does not exist in general), as well as a combined structural and impulse analysis for mode changes, generating the hot restart even in the presence of impulses. Our algorithm detects at compile time if the mode change is insufficiently specified, in which case it returns diagnostics information to the user.