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From Hybrid Automata to DAE-Based Modeling

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Principles of Systems Design

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13660))

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. 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. 2.

    dAE stands for difference Algebraic Equation (related to discrete time), while DAE stands for Differential Algebraic Equation (related to continuous time).

  3. 3.

    Actually, this is an example of erroneous standardization of systems of equations. In [5, 6] it is proved that standardizing a system of equations by setting \(\varepsilon =0\) is correct if and only if doing so yields a structurally regular system of equations.

  4. 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. 5.

    To date, the following compilation tasks are implemented: multimode \(\varSigma \)-method, multimode initialization, and multimode Dulmage-Mendelsohn decomposition.

References

  1. Alur, R., Courcoubetis, C., Henzinger, T.A., Ho, P.-H.: Hybrid automata: an algorithmic approach to the specification and verification of hybrid systems. In: Grossman, R.L., Nerode, A., Ravn, A.P., Rischel, H. (eds.) HS 1991-1992. LNCS, vol. 736, pp. 209–229. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57318-6_30

    Chapter  Google Scholar 

  2. Benveniste, A., Bourke, T., Caillaud, B., Colaço, J., Pasteur, C., Pouzet, M.: Building a hybrid systems modeler on synchronous languages principles. Proc. IEEE 106(9), 1568–1592 (2018)

    Article  Google Scholar 

  3. Benveniste, A., Bourke, T., Caillaud, B., Pouzet, M.: Nonstandard semantics of hybrid systems modelers. J. Comput. Syst. Sci. 78(3), 877–910 (2012)

    Article  MATH  Google Scholar 

  4. Benveniste, A., Caillaud, B., Elmqvist, H., Ghorbal, K., Otter, M., Pouzet, M.: Multi-mode DAE models - challenges, theory and implementation. In: Steffen, B., Woeginger, G. (eds.) Computing and Software Science. LNCS, vol. 10000, pp. 283–310. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-91908-9_16

    Chapter  MATH  Google Scholar 

  5. Benveniste, A., Caillaud, B., Malandain, M.: The mathematical foundations of physical systems modeling languages. Ann. Rev. Control 50, 72–118 (2020)

    Article  MathSciNet  Google Scholar 

  6. Benveniste, A., Caillaud, B., Malandain, M.: Structural analysis of multimode DAE systems: summary of results. CoRR, abs/2101.05702 (2021)

    Google Scholar 

  7. Benveniste, A., et al.: The synchronous languages 12 years later. Proc. IEEE 91(1), 64–83 (2003)

    Article  Google Scholar 

  8. Broenink, J., Wijbrans, K.: Describing discontinuities in bond graphs. In: Proceedings of the 1st International Conference on Bond Graph Modeling. SCS Simulation Series, vol. 25, no. 2. (1993)

    Google Scholar 

  9. Caillaud, B., Malandain, M., Thibault, J.: Implicit structural analysis of multimode DAE systems. In: Ames, A.D., Seshia, S.A., Deshmukh, J. (eds.) HSCC 2020: 23rd ACM International Conference on Hybrid Systems: Computation and Control, Sydney, New South Wales, Australia, 21–24 April 2020, pp. 20:1–20:11. ACM (2020)

    Google Scholar 

  10. Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dassault Systèmes AB. Dymola official webpage. https://www.3ds.com/products-services/catia/products/dymola/. Accessed 01 June 2022

  12. Elmqvist, H.: A structured model language for large continuous systems, Ph.D. Lund University (1978)

    Google Scholar 

  13. Elmqvist, H., Mattsson, S.-E., Otter, M.: Modelica extensions for multi-mode DAE systems. In: Tummescheit, H., Arzèn, K.-E. (eds.) Proceedings of the 10th International Modelica Conference, Lund, Sweden. Modelica Association, September 2014

    Google Scholar 

  14. Elmqvist, H., Otter, M.: Modiamath webpage. https://modiasim.github.io/ModiaMath.jl/stable/index.html. Accessed 01 June 2022

  15. Fritzson, P., et al.: The OpenModelica integrated environment for modeling, simulation, and model-based development. Model. Identif. Control 41(4), 241–295 (2020)

    Article  Google Scholar 

  16. Heemels, W., Camlibel, M., Schumacher, J.: On the dynamic analysis of piecewise-linear networks. IEEE Trans. Circuits Syst. I-Regul. Pap. 49, 315–327 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Henzinger, T.A., Ho, P.-H.: HyTech: the cornell hybrid technology tool. In: Antsaklis, P., Kohn, W., Nerode, A., Sastry, S. (eds.) HS 1994. LNCS, vol. 999, pp. 265–293. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60472-3_14

    Chapter  Google Scholar 

  18. Höger, C.: Dynamic structural analysis for DAEs. In: Proceedings of the 2014 Summer Simulation Multiconference, SummerSim 2014, Monterey, CA, USA, 6–10 July 2014, p. 12 (2014)

    Google Scholar 

  19. Höger, C.: Elaborate control: variable-structure modeling from an operational perspective. In: Proceedings of the 8th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools, EOOLT 2017, Weßling, Germany, 1 December 2017, pp. 51–60 (2017)

    Google Scholar 

  20. Liberzon, D., Trenn, S.: Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability. Automatica 48(5), 954–963 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lindstrøm, T.: An invitation to nonstandard analysis. In: Cutland, N. (eds.) Nonstandard Analysis and its Applications, pp. 1–105. Cambridge Univ. Press, Cambridge (1988)

    Google Scholar 

  22. Mattsson, S.E., Söderlind, G.: Index reduction in differential-algebraic equations using dummy derivatives. Siam J. Sci. Comput. 14(3), 677–692 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pantelides, C.: The consistent initialization of differential-algebraic systems. SIAM J. Sci. Stat. Comput. 9(2), 213–231 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (2008)

    Google Scholar 

  25. Pryce, J.D.: A simple structural analysis method for DAEs. BIT 41(2), 364–394 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Robinson, A.: Nonstandard Analysis. Princeton Landmarks in Mathematics (1996). ISBN 0-691-04490-2

    Google Scholar 

  27. Thoma, J.: Introduction to bond graphs and their applications. In: Pergamon International Library of Science, Technology, Engineering and Social Studies. Pergamon Press (1975)

    Google Scholar 

  28. Trenn, S.: Distributional differential algebraic equations. Ph.D. thesis, Technischen Universität Ilmenau (2009)

    Google Scholar 

  29. Trenn, S.: Regularity of distributional differential algebraic equations. MCSS 21(3), 229–264 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Utkin, V.: Sliding mode control in mechanical systems. In: 20th International Conference on Industrial Electronics, Control and Instrumentation, IECON 1994, vol. 3, pp. 1429–1431, September 1994

    Google Scholar 

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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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Correspondence to Benoît Caillaud .

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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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