Real-Time Systems

, Volume 53, Issue 3, pp 327–353 | Cite as

Timed-automata abstraction of switched dynamical systems using control invariants

  • Patricia Bouyer
  • Nicolas Markey
  • Nicolas Perrin
  • Philipp Schlehuber-Caissier


The development of formal methods for control design is an important challenge with potential applications in a wide range of safety-critical cyber-physical systems. Focusing on switched dynamical systems, we propose a new abstraction, based on time-varying regions of invariance (control funnels), that models behaviors of systems as timed automata. The main advantage of this method is that it allows for the automated verification and reactive controller synthesis without discretizing the evolution of the state of the system. Efficient and analytic constructions are possible in the case of linear dynamics whereas bounding funnels with conjectured properties based on numerical simulations can be used for general nonlinear dynamics. We demonstrate the potential of our approach with three examples.


Switched dynamical systems Timed automata Cyber-physical systems 



This work has been partly supported by ERC Starting Grant EQualIS (FP7-308087) and by European FET Project Cassting (FP7-601148).


  1. Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235MathSciNetCrossRefzbMATHGoogle Scholar
  2. Asarin E, Maler O, Pnueli A (1995) Reachability analysis of dynamical systems having piecewise-constant derivatives. Theor Comput Sci 138(1):35–65MathSciNetCrossRefzbMATHGoogle Scholar
  3. Asarin E, Maler O, Pnueli A, Sifakis J (1998) Controller synthesis for timed automata. In: IEEE SSSC’98. Elsevier, New York, pp 469–474Google Scholar
  4. Aubin JP (1988) Viability tubes. In: Modelling and adaptive control. Springer, New York, pp. 27–47Google Scholar
  5. Behrmann G, David A, Larsen KG, Håkansson J, Pettersson P, Yi W, Hendriks M (2006) Uppaal 4.0. In: IEEE quantitative evaluation of systems (QEST’06), pp. 125–126Google Scholar
  6. Behrmann G, Cougnard A, David A, Fleury E, Larsen KG, Lime D (2007) UPPAAL-Tiga: time for playing games! In: International conference on computer aided verification (CAV’07). Lecture notes in computer science, vol. 4590. Springer, New York, pp. 121–125Google Scholar
  7. Bouyer P, Dufourd C, Fleury E, Petit A (2004) Updatable timed automata. Theor Comput Sci 321(2–3):291–345MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bouyer P, Fahrenberg U, Larsen KG, Markey N (2011) Quantitative analysis of real-time systems using priced timed automata. Commun ACM 54(9):78–87Google Scholar
  9. Bouyer P, Markey N, Perrin N, Schlehuber-Caissier P (2015) Timed-automata abstraction of switched dynamical systems using control funnels. In: FORMATS’15. Lecture notes in computer science, vol. 9268. Springer, New York, pp. 60–75Google Scholar
  10. David, A, Grunnet JD, Jessen JJ, Larsen KG, Rasmussen JI (2012) Application of model-checking technology to controller synthesis. In: IEEE formal methods and models for co-design (FMCO’10). Lecture notes in computer science, vol. 6957. Springer, New York, pp. 336–351Google Scholar
  11. DeCastro J, Kress-Gazit H (2014) Synthesis of nonlinear continuous controllers for verifiably-correct high-level, reactive behaviors. IJRR 34(3):378–394Google Scholar
  12. Duggirala PS, Mitra S, Viswanathan M (2013) Verification of annotated models from executions. In: IEEE international conference on embedded software (EMSOFT’13), pp 1–10Google Scholar
  13. Frazzoli E, Dahleh MA, Feron E (2005) Maneuver-based motion planning for nonlinear systems with symmetries. IEEE Trans Robot 21(6):1077–1091CrossRefGoogle Scholar
  14. Fu J, Topcu U (2015) Computational methods for stochastic control with metric interval temporal logic specifications. Tech. Rep. arXiv:1503.07193
  15. Julius AA, Pappas GJ (2009) Trajectory based verification using local finite-time invariance. In: International conference on hybrid systems: computation and control (HSCC’09). Lecture notes in computer science, vol. 5469. Springer, New York, pp 223–236Google Scholar
  16. Koiran P, Cosnard M, Garzon M (1994) Computability with low-dimensional dynamical systems. Theor Comput Sci 132(1):113–128MathSciNetCrossRefzbMATHGoogle Scholar
  17. Le Ny JL, Pappas GJ (2012) Sequential composition of robust controller specifications. In: IEEE international conference on robotics and automation (ICRA’12), pp. 5190–5195Google Scholar
  18. Liu J, Prabhakar P (2014) Switching control of dynamical systems from metric temporal logic specifications. In: IEEE international conference on robotics and automation (ICRA’14), pp. 5333–5338Google Scholar
  19. Majumdar A, Ahmadi AA, Tedrake R (2013) Control design along trajectories with sums of squares programming. In: IEEE international conference on robotics and automation (ICRA’13), pp. 4054–4061Google Scholar
  20. Majumdar A, Tedrake R (2013) Robust online motion planning with regions of finite time invariance. In: International workshop on the algorithmic foundations of robotics (WAFR’12). STAR, vol 86. Springer, New York, pp 543–558Google Scholar
  21. Maler O, Batt G (2008) Approximating continuous systems by timed automata. In: Proceedings of formal methods in systems biology (FMSB’08). LNBI, vol 5054. Springer, New York, pp. 77–89Google Scholar
  22. Maler O, Manna Z, Pnueli A (1992) From timed to hybrid systems. In: Real-time: theory in practice. Lecture notes in computer science, vol 600. Springer, New York, pp 447–484Google Scholar
  23. Mason MT (1985) The mechanics of manipulation. In: IEEE international conference on robotics and automation (ICRA’85), vol 2, pp 544–548Google Scholar
  24. Prajna S, Papachristodoulou A, Wu F (2004) Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach. In: 5th Asian IEEE control conference, vol 1, pp 157–165Google Scholar
  25. Quottrup MM, Bak T, Zamanabadi RI (2004) Multi-robot planning : a timed automata approach. In: IEEE international conference on robotics and automation (ICRA’04), vol 5, pp 4417–4422Google Scholar
  26. Sloth C, Wisniewski R (2010) Timed game abstraction of control systems. Tech. Rep. arXiv:1012.5113
  27. Sloth C, Wisniewski R (2013) Complete abstractions of dynamical systems by timed automata. Nonlinear Anal 7(1):80–100MathSciNetzbMATHGoogle Scholar
  28. Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  29. Tedrake R, Manchester IR, Tobenkin M, Roberts JW (2010) LQR-trees: feedback motion planning via sums-of-squares verification. IJRR 29(8):1038–1052Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Patricia Bouyer
    • 1
  • Nicolas Markey
    • 1
  • Nicolas Perrin
    • 2
    • 3
  • Philipp Schlehuber-Caissier
    • 2
  1. 1.LSV – CNRS & ENS CachanUniversité Paris-SaclayCachanFrance
  2. 2.Sorbonne Universités, UPMC Univ Paris 06, UMR 7222, ISIRParisFrance
  3. 3.CNRS, UMR 7222, ISIRParisFrance

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