Abstract
We propose a new approximation technique for Hybrid Automata. Given any Hybrid Automaton H, we call Approx(H,k) the Polynomial Hybrid Automaton obtained by approximating each formula φ in H with the formulae φ k obtained by replacing the functions in φ with their Taylor polynomial of degree k. We prove that Approx(H,k) is an over–approximation of H. We study the conditions ensuring that, given any ε > 0, some k 0 exists such that, for all k>k 0, the “distance” between any vector satisfying φ k and at least one vector satisfying φ is less than ε. We study also conditions ensuring that, given any ε > 0, some k 0 exists such that, for all k > k 0, the “distance” between any configuration reached by Approx(H,k) in n steps and at least one configuration reached by H in n steps is less than ε.
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References
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. Theor. Comput. Sci. 138(1), 3–34 (1995)
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
Alur, R., Henzinger, T.A., Ho, P.H.: Automatic symbolic verification of embedded systems. IEEE Trans. Software Eng. 22(6), 181–201 (1996)
Alur, R., Henzinger, T.A., Lafferriere, G., Pappas, G.J.: Discrete abstractions of hybrid systems. Proc. IEEE 88(7), 971–984 (2000)
Asarin, E., Bournez, O., Dang, T., Maler, O., Pnueli, A.: Effective synthesis of switching controllers for linear systems. Proc. IEEE 88(7), 1011–1025 (2000)
Bayen, A.M., Crück, E., Tomlin, C.J.: Guaranteed overapproximations of unsafe sets for continuous and hybrid systems: Solving the Hamilton-Jacobi equation using viability techniques. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 90–104. Springer, Heidelberg (2002)
Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximation. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 76–90. Springer, Heidelberg (1999)
Fränzle, M.: Analysis of hybrid systems: An ounce of realism can save an infinity of states. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 126–140. Springer, Heidelberg (1999)
Fränzle, M.: What will be eventually true of polynomial hybrid automata. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 340–359. Springer, Heidelberg (2001)
Henzinger, T.A., Ho, P.H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Trans. Automat. Contr. 43(4), 540–554 (1998)
Henzinger, T.A., Kopke, P.W., Puri, A., Varaiya, P.: What’s decidable about hybrid automata? J. Comput. Syst. Sci. 57(1), 94–124 (1998)
Henzinger, T.A., Majumdar, R.: Symbolic model checking for rectangular hybrid systems. In: Schwartzbach, M.I., Graf, S. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 142–156. Springer, Heidelberg (2000)
Kurzhanski, A.B., Varaiya, P.: Reachability under uncertainty. In: Proc IEEE Conference on Decision and Control (2002)
Lafferriere, G., Pappas, G.J., Sastry, S.: O-minimal hybrid systems. Math. Contr. Sign. Syst. 13(1), 1–21 (2000)
Lafferriere, G., Pappas, G.J., Yovine, S.: A new class of decidable hybrid systems. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 137–151. Springer, Heidelberg (1999)
Lafferriere, G., Pappas, G.J., Yovine, S.: Reachability computation for linear hybrid systems. In: Proc. IFAC World Congress, pp. 7–12 (1999)
Mitchell, I., Bayen, A.M., Tomlin, C.J.: Validating a Hamilton-Jacobi approximation to hybrid system reachable sets. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 418–432. Springer, Heidelberg (2001)
Pnueli, A., Sifakis, J. (Eds.): Special issue on hybrid systems. Theor. Comput. Sci. 138(1) (1995)
Puri, A., Borkar, V., Varaiya, P.: ε-approximation of differential inclusions. In: Alur, R., Sontag, E.D., Henzinger, T.A. (eds.) HS 1995. LNCS, vol. 1066, pp. 362–376. Springer, Heidelberg (1996)
Tomlin, C., Mitchell, I., Bayen, A., Oishi, M.: Computational techniques for the verification and control of hybrid systems. Proc. IEEE 91(7), 986–1001 (2003)
Yazarel, H., Pappas, G.J.: Geometric programming relaxations for non linear system reachability. In: Proc. American Control Conference (2004)
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Lanotte, R., Tini, S. (2005). Taylor Approximation for Hybrid Systems. In: Morari, M., Thiele, L. (eds) Hybrid Systems: Computation and Control. HSCC 2005. Lecture Notes in Computer Science, vol 3414. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31954-2_26
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DOI: https://doi.org/10.1007/978-3-540-31954-2_26
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