Structural Methods for handling mode changes in multimode DAE systems

TL;DR

The paper tackles the fundamental challenge of mode changes in multimode DAE systems by developing a mathematically grounded hot restart mechanism. It fuses structural analysis (incidence graphs, Σ-method, Dulmage–Mendelsohn decomposition) with impulse handling via a carefully constructed mode-change array and a rescaling framework that assigns impulse orders to variables and equations. The authors prove correctness and provide algorithmic procedures to synthesize restart conditions that are deterministic, consistent, and invariant-preserving, with bounds (K_*,K^*) guiding the search for solutions. They demonstrate the approach on a cup-and-ball-like example and discuss scalability and future work toward modular, all-modes-at-once analysis for Modelica-style multimode DAEs.

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.

Figures (11)

• Figure 1: Cup-and-ball example: Mode change array $A_0$. In the last column we point the facts and the disabled conflicting equations; black equations are enabled. Triple $(e_1,e_2,k_1")$ is structurally nonsingular, with a one-to-one matching ${\color{blue}{\cal M}}=\{(e_1,{\color{blue}x"}),(e_2,{\color{blue}{\lambda}}),(k_1",{\color{blue}y"})\}$ between enabled equations and variables.
• Figure 4: Zero-crossing in discrete time. The detection instant is in red and the first instant of the new mode is in blue.
• Figure 5: Cup-and-ball example: Mode change array with $K{=}2$. $\mathcal{X}^-$ is the same as for $K{=}1$. Facts and conflicts are pointed in green and red, respectively. The subsystem in black is structurally nonsingular, with a perfect matching ${\color{blue}{\cal M}}$ highlighted in blue. In the right most column we indicate the origin of each equation: for example, $t_*{+}\varepsilon$ indicates that the corresponding equation originates from the $1$-shifted discretized dynamics.
• Figure 6: Cup-and-ball example with $K{=}2$: Rescaling calculus. The rescaling equation for $f_7$ corresponds to rescaling equation (\ref{['ksdjfhsgdkfjgh']}) for nonlinear functions.
• Figure 9: Cup-and-ball example with exogenous mode change: mode change array associated to $\gamma:\hbox{\sc f}\rightarrow\hbox{\sc t}$, with $K=2$. We show here ${\color{blue}{\cal M}}$. The dynamics belonging to previous mode is not shown. Conflicts are shown in red.

Theorems & Definitions (32)

• Definition 1
• Proposition 1: Implicit Function Theorem
• Definition 2
• Definition 3
• Lemma 1
• Definition 4: $\sim$ -- closed system
• Lemma 2
• Lemma 3
• Definition 5: facts
• Definition 6: mode change array