Undecidability of Quantized State Feedback Control for Discrete Time Linear Hybrid Systems

  • Federico Mari
  • Igor Melatti
  • Ivano Salvo
  • Enrico Tronci
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7521)


We show that the existence of a quantized controller for a given Discrete Time Linear Hybrid System (DTLHS) is undecidable. This is a relevant class of controllers since control software always implements a quantized controller. Furthermore, we investigate the relationship between dense time modelling and discrete time modelling by showing that any Rectangular Hybrid Automaton (and thus, any Timed Automaton) can be modelled as a DTLHS.


Control Problem Weak Solution Strong Solution Transition Relation Label Transition System 
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  1. 1.
    Agrawal, M., Thiagarajan, P.S.: The Discrete Time Behavior of Lazy Linear Hybrid Automata. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 55–69. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.A., Ho, P.H., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of Hybrid Systems. Theoretical Computer Science 138(1), 3–34 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Alur, R.: Timed Automata. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 8–22. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Asarin, E., Bouajjani, A.: Perturbed Turing Machines and Hybrid Systems. In: LICS, pp. 269–278 (2001)Google Scholar
  5. 5.
    Bemporad, A., Morari, M.: Verification of Hybrid Systems via Mathematical Programming. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 31–45. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Brogan, W.L.: Modern Control Theory, 3rd edn. Prentice-Hall, Inc., Upper Saddle River (1991)zbMATHGoogle Scholar
  7. 7.
    Cassez, F., Henzinger, T.A., Raskin, J.-F.: A Comparison of Control Problems for Timed and Hybrid Systems. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 134–148. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Cimatti, A., Roveri, M., Traverso, P.: Strong planning in non-deterministic domains via Model Checking. In: AIPS, pp. 36–43 (1998)Google Scholar
  9. 9.
    Frehse, G.: Phaver: algorithmic verification of Hybrid Systems past Hytech. Int. J. Softw. Tools Technol. Transf. 10(3), 263–279 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fu, M., Xie, L.: The sector bound approach to quantized feedback control. IEEE Trans. on Automatic Control 50(11), 1698–1711 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Henzinger, T., Ho, P.H., Wong-Toi, H.: Hytech: A model checker for Hybrid Systems. STTT 1(1), 110–122 (1997)zbMATHCrossRefGoogle Scholar
  12. 12.
    Henzinger, T.A., Kopke, P.W.: Discrete-time Control for Rectangular Hybrid Automata. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 582–593. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. 13.
    Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about Hybrid Automata? J. of Computer and System Sciences 57(1), 94–124 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Larsen, K.G., Pettersson, P., Yi, W.: Uppaal: Status & Developments. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 456–459. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Mari, F., Melatti, I., Salvo, I., Tronci, E.: Synthesis of Quantized Feedback Control Software for Discrete Time Linear Hybrid Systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 180–195. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Minsky, M.L.: Recursive unsolvability of Post’s problem of ”tag” and other topics in theory of Turing Machines. The Annals of Mathematics 74(3), 437–455 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Pola, G., Girard, A., Tabuada, P.: Approximately bisimilar symbolic models for nonlinear control systems. Automatica 44(10), 2508–2516 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Tronci, E.: Automatic synthesis of controllers from formal specifications. In: ICFEM, pp. 134–143. IEEE (1998)Google Scholar
  19. 19.
    Vidal, R., Schaffert, S., Shakernia, O., Lygeros, J., Sastry, S.: Decidable and semi-decidable controller synthesis for classes of Discrete Time Hybrid Systems. In: CDC, pp. 1243–1248. IEEE Computer Society (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Federico Mari
    • 1
  • Igor Melatti
    • 1
  • Ivano Salvo
    • 1
  • Enrico Tronci
    • 1
  1. 1.Computer Science DepartmentSapienza University of RomeItaly

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