Abstract
Tom Henzinger was among the co-founders of the paradigm of hybrid automata in 1992. Hybrid automata possess different locations, holding different ODE-based dynamics; exit conditions from a location trigger transitions, resulting in starting conditions for the next location. A large research activity was developed in the formal verification of hybrid automata; this paradigm still grounds popular commercial tools such as Stateflow for Simulink.
However, modeling from first principles of physics requires a different approach: balance equations and conservation laws play a central role, and elementary physical components come with no prespecified input/output profile. All of this leads to grounding physical modeling on DAEs (Differential Algebraic Equations, of the form \(f(x',x,v)=0\)) instead of ODEs. DAE-based modeling, implemented for example in the Modelica language, allows for modularity and reuse of models.
Unsurprisingly, DAE-based hybrid systems (also known as multimode DAE systems) emerge as the central paradigm in multiphysics modeling. Despite the growing popularity of modeling tools based on this paradigm, fundamental problems remain in the handling of multiple modes and mode changes—corresponding to multiple locations and transitions in hybrid automata. Deep symbolic analyses (grouped under the term “structural analysis” in the related community), grounded on solid foundations, are required to generate simulation code. This paper reviews the issues related to multimode DAE systems and proposes algorithms for their analysis. Computer science is instrumental in these works, with a lot to offer to the simulation scientific community.
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Notes
- 1.
Numerical singularity occurs when the Jacobian matrix of the system is singular; in this specific case, this can only occur when \(\partial f_1(\omega _1,\tau _1) / \partial \tau _1 +\partial f_2(\omega _2,\tau _2) / \partial \tau _2=0\). Unlike structural singularity, numerical singularity depends on numerical values of both coefficients and variables.
- 2.
dAE stands for difference Algebraic Equation (related to discrete time), while DAE stands for Differential Algebraic Equation (related to continuous time).
- 3.
- 4.
This equation-centric viewpoint was first used in the clock calculus and conditional dependency graph used in the compilation of the Signal synchronous language [7].
- 5.
To date, the following compilation tasks are implemented: multimode \(\varSigma \)-method, multimode initialization, and multimode Dulmage-Mendelsohn decomposition.
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Acknowledgement
This work was supported by the FUI ModeliScale DOS0066450/00 French national grant (2018–2021) and the Inria IPL ModeliScale large scale initiative (2017–2021, https://team.inria.fr/modeliscale/).
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Benveniste, A., Caillaud, B., Malandain, M. (2022). From Hybrid Automata to DAE-Based Modeling. In: Raskin, JF., Chatterjee, K., Doyen, L., Majumdar, R. (eds) Principles of Systems Design. Lecture Notes in Computer Science, vol 13660. Springer, Cham. https://doi.org/10.1007/978-3-031-22337-2_1
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