Conic Abstractions for Hybrid Systems
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Abstract
Despite researchers’ efforts in the last couple of decades, reachability analysis is still a challenging problem even for linear hybrid systems. Among the existing approaches, the most practical ones are mainly based on bounded-time reachable set over-approximations. For the purpose of unbounded-time analysis, one important strategy is to abstract the original system and find an invariant for the abstraction. In this paper, we propose an approach to constructing a new kind of abstraction called conic abstraction for affine hybrid systems, and to computing reachable sets based on this abstraction. The essential feature of a conic abstraction is that it partitions the state space of a system into a set of convex polyhedral cones which is derived from a uniform conic partition of the derivative space. Such a set of polyhedral cones is able to cut all trajectories of the system into almost straight segments so that every segment of a reach pipe in a polyhedral cone tends to be straight as well, and hence can be over-approximated tightly by polyhedra using similar techniques as HyTech or PHAVer. In particular, for diagonalizable affine systems, our approach can guarantee to find an invariant for unbounded reachable sets, which is beyond the capability of bounded-time reachability analysis tools. We implemented the approach in a tool and experiments on benchmarks show that our approach is more powerful than SpaceEx and PHAVer in dealing with diagonalizable systems.
Keywords
Affine system Hybrid system Reachability analysis Conic abstraction Discrete abstractionNotes
Acknowledgments
This work was partly supported by the Austrian Science Fund (FWF) under grants S11402-N23 (RiSE/SHiNE) and Z211-N23 (Wittgenstein Award) and by the ARC project DP140104219 (Robust AI Planning for Hybrid Systems).
References
- 1.Henzinger, T.: The theory of hybrid automata. In: Proceedings of IEEE Symposium on Logic in Computer Science, pp. 278–292 (1996)Google Scholar
- 2.Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T., Ho, P., Nicollin, X., Olivero, A., Sifakis, J., Yovine, S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(1), 3–34 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_18 CrossRefGoogle Scholar
- 4.Dang, T., Maler, O.: Reachability analysis via face lifting. In: Henzinger, T.A., Sastry, S. (eds.) HSCC 1998. LNCS, vol. 1386, pp. 96–109. Springer, Heidelberg (1998). doi: 10.1007/3-540-64358-3_34 CrossRefGoogle Scholar
- 5.Kloetzer, M., Belta, C.: Reachability analysis of multi-affine systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 348–362. Springer, Heidelberg (2006). doi: 10.1007/11730637_27 CrossRefGoogle Scholar
- 6.Prabhakar, P., Viswanathan, M.: A dynamic algorithm for approximate flow computations. In: HSCC, pp. 133–142 (2011)Google Scholar
- 7.Lal, R., Prabhakar, P.: Bounded error flowpipe computation of parameterized linear systems. In: 2015 International Conference on Embedded Software (EMSOFT 2015), Amsterdam, Netherlands, 4–9 October 2015, pp. 237–246 (2015)Google Scholar
- 8.Kong, H., Bogomolov, S., Schilling, C., Jiang, Y., Henzinger, T.A.: Safety verification of nonlinear hybrid systems based on invariant clusters. In: HSCC, ser. (HSCC 2017), pp. 163–172. ACM, New York (2017)Google Scholar
- 9.Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Vaandrager, F.W., Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 76–90. Springer, Heidelberg (1999). doi: 10.1007/3-540-48983-5_10 CrossRefGoogle Scholar
- 10.Asarin, E., Bournez, O., Dang, T., Maler, O.: Approximate reachability analysis of piecewise-linear dynamical systems. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 20–31. Springer, Heidelberg (2000). doi: 10.1007/3-540-46430-1_6 CrossRefGoogle Scholar
- 11.Kurzhanski, A., Varaiya, P.: Ellipsoidal techniques for reachability analysis: internal approximation. Syst. Contr. Lett. 41(3), 201–211 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Botchkarev, O., Tripakis, S.: Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 73–88. Springer, Heidelberg (2000). doi: 10.1007/3-540-46430-1_10 CrossRefGoogle Scholar
- 13.Stursberg, O., Krogh, B.H.: Efficient representation and computation of reachable sets for hybrid systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 482–497. Springer, Heidelberg (2003). doi: 10.1007/3-540-36580-X_35 CrossRefGoogle Scholar
- 14.Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31954-2_19 CrossRefGoogle Scholar
- 15.Girard, A., Guernic, C., Maler, O.: Efficient computation of reachable sets of linear time-invariant systems with inputs. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 257–271. Springer, Heidelberg (2006). doi: 10.1007/11730637_21 CrossRefGoogle Scholar
- 16.Guernic, C., Girard, A.: Reachability analysis of hybrid systems using support functions. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 540–554. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02658-4_40 CrossRefGoogle Scholar
- 17.Jiang, Y., Song, H., Wang, R., Gu, M., Sun, J., Sha, L.: Data-centered runtime verification of wireless medical cyber-physical system. IEEE Trans. Ind. Inform. PP(99), 1 (2016)Google Scholar
- 18.Jiang, Y., Zhang, H., Li, Z., Deng, Y., Song, X., Gu, M., Sun, J.: Design and optimization of multiclocked embedded systems using formal techniques. IEEE Trans. Ind. Electron. 62(2), 1270–1278 (2015)CrossRefGoogle Scholar
- 19.Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: HyTech: a model checker for hybrid systems. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 460–463. Springer, Heidelberg (1997). doi: 10.1007/3-540-63166-6_48 CrossRefGoogle Scholar
- 20.Frehse, G.: Phaver: algorithmic verification of hybrid systems past hytech. Int. J. Softw. Tools Technol. Transfer 10(3), 263–279 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Batt, G., Belta, C., Weiss, R.: Temporal logic analysis of gene networks under parameter uncertainty. Trans. Autom. Contr. 53(Special Issue), 215–229 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Alur, R., Dang, T., Ivančić, F.: Progress on reachability analysis of hybrid systems using predicate abstraction. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 4–19. Springer, Heidelberg (2003). doi: 10.1007/3-540-36580-X_4 CrossRefGoogle Scholar
- 23.Tiwari, A., Khanna, G.: Series of abstractions for hybrid automata. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 465–478. Springer, Heidelberg (2002). doi: 10.1007/3-540-45873-5_36 CrossRefGoogle Scholar
- 24.Tiwari, A.: Abstractions for hybrid systems. Formal Methods Syst. Des. 32(1), 57–83 (2008)CrossRefzbMATHGoogle Scholar
- 25.Roohi, N., Prabhakar, P., Viswanathan, M.: Hybridization based CEGAR for hybrid automata with affine dynamics. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 752–769. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49674-9_48 CrossRefGoogle Scholar
- 26.Sogokon, A., Ghorbal, K., Jackson, P.B., Platzer, A.: A method for invariant generation for polynomial continuous systems. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 268–288. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49122-5_13 CrossRefGoogle Scholar
- 27.Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43(7), 451–476 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 28.Henzinger, T., Wong-Toi, H.: Linear phase-portrait approximations for nonlinear hybrid systems. Hybrid Syst. III, 377–388 (1996)Google Scholar
- 29.Frehse, G., Guernic, C., Donzé, A., Cotton, S., Ray, R., Lebeltel, O., Ripado, R., Girard, A., Dang, T., Maler, O.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22110-1_30 CrossRefGoogle Scholar
- 30.Frehse, G.: PHAVer: algorithmic verification of hybrid systems past HyTech. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 258–273. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31954-2_17 CrossRefGoogle Scholar
- 31.Doyen, L., Henzinger, T.A., Raskin, J.-F.: Automatic rectangular refinement of affine hybrid systems. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 144–161. Springer, Heidelberg (2005). doi: 10.1007/11603009_13 CrossRefGoogle Scholar
- 32.Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic press, Amsterdam (2012)zbMATHGoogle Scholar
- 33.Kong, H., Bartocci, E., Bogomolov, S., Grosu, R., Henzinger, T.A., Jiang, Y., Schilling, C.: Discrete abstraction of multiaffine systems. In: Cinquemani, E., Donzé, A. (eds.) HSB 2016. LNCS, vol. 9957, pp. 128–144. Springer, Cham (2016). doi: 10.1007/978-3-319-47151-8_9 Google Scholar
- 34.Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Scalable analysis of linear systems using mathematical programming. In: Cousot, R. (ed.) VMCAI 2005. LNCS, vol. 3385, pp. 25–41. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-30579-8_2 CrossRefGoogle Scholar
- 35.GLPK (GNU linear programming kit). www.gnu.org/software/glpk
- 36.Frehse, G., Kateja, R., Le Guernic, C.: Flowpipe approximation and clustering in space-time. In: Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 203–212. ACM (2013)Google Scholar
- 37.Fehnker, A., Ivančić, F.: Benchmarks for hybrid systems verification. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 326–341. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24743-2_22 CrossRefGoogle Scholar