Abstract
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.
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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)
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)
Alur, R.: Timed Automata. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 8–22. Springer, Heidelberg (1999)
Asarin, E., Bouajjani, A.: Perturbed Turing Machines and Hybrid Systems. In: LICS, pp. 269–278 (2001)
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)
Brogan, W.L.: Modern Control Theory, 3rd edn. Prentice-Hall, Inc., Upper Saddle River (1991)
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)
Cimatti, A., Roveri, M., Traverso, P.: Strong planning in non-deterministic domains via Model Checking. In: AIPS, pp. 36–43 (1998)
Frehse, G.: Phaver: algorithmic verification of Hybrid Systems past Hytech. Int. J. Softw. Tools Technol. Transf. 10(3), 263–279 (2008)
Fu, M., Xie, L.: The sector bound approach to quantized feedback control. IEEE Trans. on Automatic Control 50(11), 1698–1711 (2005)
Henzinger, T., Ho, P.H., Wong-Toi, H.: Hytech: A model checker for Hybrid Systems. STTT 1(1), 110–122 (1997)
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)
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)
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)
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)
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)
Pola, G., Girard, A., Tabuada, P.: Approximately bisimilar symbolic models for nonlinear control systems. Automatica 44(10), 2508–2516 (2008)
Tronci, E.: Automatic synthesis of controllers from formal specifications. In: ICFEM, pp. 134–143. IEEE (1998)
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)
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Mari, F., Melatti, I., Salvo, I., Tronci, E. (2012). Undecidability of Quantized State Feedback Control for Discrete Time Linear Hybrid Systems. In: Roychoudhury, A., D’Souza, M. (eds) Theoretical Aspects of Computing – ICTAC 2012. ICTAC 2012. Lecture Notes in Computer Science, vol 7521. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32943-2_19
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DOI: https://doi.org/10.1007/978-3-642-32943-2_19
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