Skip to main content
SpringerLink
Log in

Timed-automata abstraction of switched dynamical systems using control invariants

  • Published:
Real-Time Systems Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. In this paper, we mostly consider state spaces that describe the position and velocity of systems controlled in acceleration. The continuity of trajectories in the state space ensures that the position is always a continuously differentiable function of time.

  2. One should always choose the reference trajectory such that there exists a margin between the reference values and the limit imposed by (8) since otherwise the convergence is very slow.

References

  • Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235

    Article  MathSciNet  MATH  Google Scholar 

  • Asarin E, Maler O, Pnueli A (1995) Reachability analysis of dynamical systems having piecewise-constant derivatives. Theor Comput Sci 138(1):35–65

    Article  MathSciNet  MATH  Google Scholar 

  • Asarin E, Maler O, Pnueli A, Sifakis J (1998) Controller synthesis for timed automata. In: IEEE SSSC’98. Elsevier, New York, pp 469–474

  • Aubin JP (1988) Viability tubes. In: Modelling and adaptive control. Springer, New York, pp. 27–47

  • 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. 125–126

  • 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. 121–125

  • Bouyer P, Dufourd C, Fleury E, Petit A (2004) Updatable timed automata. Theor Comput Sci 321(2–3):291–345

    Article  MathSciNet  MATH  Google Scholar 

  • Bouyer P, Fahrenberg U, Larsen KG, Markey N (2011) Quantitative analysis of real-time systems using priced timed automata. Commun ACM 54(9):78–87

  • 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. 60–75

  • 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. 336–351

  • DeCastro J, Kress-Gazit H (2014) Synthesis of nonlinear continuous controllers for verifiably-correct high-level, reactive behaviors. IJRR 34(3):378–394

    Google Scholar 

  • Duggirala PS, Mitra S, Viswanathan M (2013) Verification of annotated models from executions. In: IEEE international conference on embedded software (EMSOFT’13), pp 1–10

  • Frazzoli E, Dahleh MA, Feron E (2005) Maneuver-based motion planning for nonlinear systems with symmetries. IEEE Trans Robot 21(6):1077–1091

    Article  Google Scholar 

  • Fu J, Topcu U (2015) Computational methods for stochastic control with metric interval temporal logic specifications. Tech. Rep. arXiv:1503.07193

  • 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 223–236

  • Koiran P, Cosnard M, Garzon M (1994) Computability with low-dimensional dynamical systems. Theor Comput Sci 132(1):113–128

    Article  MathSciNet  MATH  Google Scholar 

  • Le Ny JL, Pappas GJ (2012) Sequential composition of robust controller specifications. In: IEEE international conference on robotics and automation (ICRA’12), pp. 5190–5195

  • 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. 5333–5338

  • 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. 4054–4061

  • 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 543–558

  • 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. 77–89

  • 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 447–484

  • Mason MT (1985) The mechanics of manipulation. In: IEEE international conference on robotics and automation (ICRA’85), vol 2, pp 544–548

  • 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 157–165

  • 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 4417–4422

  • Sloth C, Wisniewski R (2010) Timed game abstraction of control systems. Tech. Rep. arXiv:1012.5113

  • Sloth C, Wisniewski R (2013) Complete abstractions of dynamical systems by timed automata. Nonlinear Anal 7(1):80–100

    MathSciNet  MATH  Google Scholar 

  • Sontag ED (1998) Mathematical control theory: deterministic finite dimensional systems. Springer, New York

    Book  MATH  Google Scholar 

  • Tedrake R, Manchester IR, Tobenkin M, Roberts JW (2010) LQR-trees: feedback motion planning via sums-of-squares verification. IJRR 29(8):1038–1052

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. LSV – CNRS & ENS Cachan, Université Paris-Saclay, Cachan, France

    Patricia Bouyer & Nicolas Markey

  2. Sorbonne Universités, UPMC Univ Paris 06, UMR 7222, ISIR, 75005, Paris, France

    Nicolas Perrin & Philipp Schlehuber-Caissier

  3. CNRS, UMR 7222, ISIR, 75005, Paris, France

    Nicolas Perrin

Authors
  1. Patricia Bouyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolas Markey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nicolas Perrin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Philipp Schlehuber-Caissier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicolas Perrin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouyer, P., Markey, N., Perrin, N. et al. Timed-automata abstraction of switched dynamical systems using control invariants. Real-Time Syst 53, 327–353 (2017). https://doi.org/10.1007/s11241-016-9262-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11241-016-9262-3

Keywords

Search

Navigation