From Electrical Switched Networks to Hybrid Automata

  • Alessandro Cimatti
  • Sergio Mover
  • Mirko SessaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9995)


In this paper, we propose a novel symbolic approach to automatically synthesize a Hybrid Automaton (HA) from a switched electrical network. The input network consists of a set of physical components interconnected according to some reconfigurable network topology. The underlying model defines a local dynamics for each component in terms of a Differential-Algebraic Equation (DAE), and a set of network topologies by means of discrete switches. Each switch configuration induces a different topology, where the behavior of the system is a Hybrid Differential-Algebraic Equations.

Two relevant problems for these networks are validation and reformulation. The first consists of determining if the network admits an Ordinary Differential Equations (ODE) that describes its dynamics; the second consists of obtaining such ODE from the initial DAE. This step is a key enabler to use existing formal verification tools that can cope with ODEs but not with DAEs.

Since the number of network topologies is exponential in the number of switches, first, we propose a technique based on Satisfiability Modulo Theories (SMT) that can solve the validation problem symbolically, avoiding the explicit enumeration of the topologies. Then, we show an SMT-based algorithm that reformulates the network into a symbolic HA. The algorithm avoids to explicitly enumerate the topologies clustering them by equivalent continuous dynamics.

We implemented the approach with several optimizations and we compared it with the explicit enumeration of configurations. The results demonstrate the scalability of our technique.


  1. 1.
    Agrawal, A., Simon, G., Karsai, G.: Semantic translation of simulink/stateflow models to hybrid automata using graph transformations. Electron. Notes Theoret. Comput. Sci. 109, 43–56 (2004). Proceedings of the Workshop on Graph Transformation and Visual Modelling Techniques (GT-VMT2004). CrossRefzbMATHGoogle Scholar
  2. 2.
    Akers, A., Gassman, M., Smith, R.: Hydraulic Power System Analysis. Fluid Power and Control. CRC Press, Boca Raton (2006). CrossRefzbMATHGoogle Scholar
  3. 3.
    Bae, K., Kong, S., Gao, S.: SMT encoding of hybrid systems in dReal. In: Frehse, G., Althoff, M. (eds.) 1st and 2nd International Workshop on Applied verification for Continuous and Hybrid Systems, ARCH14 2015. EPiC Series in Computing, vol. 34, pp. 188–195. EasyChair, Manchester (2015)Google Scholar
  4. 4.
    Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Satisfiability, pp. 825–885 (2009).
  5. 5.
    Benner, P.: Large-scale Networks in Engineering and Life Sciences. Springer, New York (2014)CrossRefGoogle Scholar
  6. 6.
    Cimatti, A., Mover, S., Sessa, M.: From electrical switched networks to hybrid automata (extended version). In: Fitzgerald, J., et al. (eds.) FM 2016. LNCS, vol. 9995, pp. 164–181. Springer, Heidelberg (2016). CrossRefGoogle Scholar
  7. 7.
    Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: HyComp: an SMT-based model checker for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 52–67. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_4 Google Scholar
  8. 8.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: ETAPS 2013, pp. 93–107. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-36742-7_7 CrossRefzbMATHGoogle Scholar
  9. 9.
    Cimatti, A., Mover, S., Tonetta, S.: A quantifier-free SMT encoding of non-linear hybrid automata. In: FMCAD, pp. 187–195 (2012).
  10. 10.
    Dang, T., Donzé, A., Maler, O.: Verification of analog and mixed-signal circuits using hybrid system techniques. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 21–36. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-30494-4_3 CrossRefGoogle Scholar
  11. 11.
    Frehse, G., Krogh, B.H., Rutenbar, R.A., Maler, O.: Time domain verification of oscillator circuit properties. Electron. Notes Theoret. Comput. Sci. 153(3), 9–22 (2006). doi: 10.1016/j.entcs.2006.02.019 CrossRefGoogle Scholar
  12. 12.
    Gario, M., Micheli, A.: pysmt: a solver-agnostic library for fast prototyping of smt-based algorithms. In: SMT Workshop (2015)Google Scholar
  13. 13.
    Henzinger, T.A.: The theory of hybrid automata. In: Proceedings of 11th Annual IEEE Symposium on Logic in Computer Science, New Brunswick, New Jersey, USA, 27–30 July 1996, pp. 278–292 (1996).
  14. 14.
    Janschek, K.: Mechatronic Systems Design: Methods, Models, Concepts. Springer Science & Business Media, Berlin (2011)zbMATHGoogle Scholar
  15. 15.
    Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \(\delta \)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_15 Google Scholar
  16. 16.
    Lee, H.L., Althoff, M., Hoelldampf, S., Olbrich, M., Barke, E.: Automated generation of hybrid system models for reachability analysis of nonlinear analog circuits. In: The 20th Asia and South Pacific Design Automation Conference, ASP-DAC 2015, Chiba, Japan, 19–22 January 2015, pp. 725–730 (2015).
  17. 17.
    Manamcheri, K., Mitra, S., Bak, S., Caccamo, M.: A step towards verification and synthesis from simulink/stateflow models. In: Proceedings of the 14th ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2011, Chicago, IL, USA, 12–14 April 2011, pp. 317–318 (2011).
  18. 18.
    Massarini, A., Reggiani, U., Kazimierczuk, M.K.: Analysis of networks with ideal switches by state equations. IEEE Trans. Circ. Syst. I: Fundam. Theory Appl. 44(8), 692–697 (1997)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mathworks, T.: Simscape power systems.
  20. 20.
    Minopoli, S., Frehse, G.: SL2SX translator: from simulink to spaceex models. In: Proceedings of the 19th International Conference on Hybrid Systems: Computation and Control, HSCC 2016, Vienna, Austria, 12–14 April 2016, pp. 93–98 (2016).
  21. 21.
    Mover, S., Cimatti, A., Tiwari, A., Tonetta, S.: Time-aware relational abstractions for hybrid systems. In: EMSOFT, pp. 14:1–14:10 (2013).
  22. 22.
    Nguyen, L.V., Johnson, T.T.: Benchmark: DC-to-DC switched-mode power converters (buck converters, boost converters, and buck-boost converters). In: Frehse, G., Althoff, M. (eds.) ARCH14 2015, 1st and 2nd International Workshop on Applied Verification for Continuous and Hybrid Systems. EPiC Series in Computing, vol. 34, pp. 19–24. EasyChair (2015)Google Scholar
  23. 23.
    Nuzzo, P., Xu, M., Ozay, N., Finn, J.B., Sangiovanni-Vincentelli, A., Murray, R., Donze, A., Seshia, S.: A contract-based methodology for aircraft electric power system design. IEEE Access.
  24. 24.
    Riaza, R.: Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  25. 25.
    SAE International: AIR 6110 - Contiguous Aircraft/System Development Process Example (2011)Google Scholar
  26. 26.
    Skaar, D.L.: Using the superposition method to formulate the state variable matrix for linear networks. IEEE Trans. Educ. 44(4), 311–314 (2001)CrossRefGoogle Scholar
  27. 27.
    Tiwari, A.: HybridSAL relational abstracter. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 725–731. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31424-7_56 CrossRefGoogle Scholar
  28. 28.
    Tiwari, A.: Time-aware abstractions in HybridSal. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 504–510. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-21690-4_34 CrossRefGoogle Scholar
  29. 29.
    Zaki, M.H., Tahar, S., Bois, G.: Formal verification of analog and mixed signal designs: survey and comparison. In: 2006 IEEE North-East Workshop on Circuits and Systems, pp. 281–284, June 2006Google Scholar
  30. 30.
    Zhang, Y., Sankaranarayanan, S., Somenzi, F.: Piecewise linear modeling of nonlinear devices for formal verification of analog circuits. In: FMCAD, pp. 196–203 (2012).

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Alessandro Cimatti
    • 1
  • Sergio Mover
    • 2
  • Mirko Sessa
    • 1
    • 3
    Email author
  1. 1.Fondazione Bruno KesslerTrentoItaly
  2. 2.University of Colorado BoulderBoulderUSA
  3. 3.University of TrentoTrentoItaly

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