Abstract
Reachability analysis techniques aim to compute which states a dynamical system can enter. The analysis of systems described by nonlinear differential equations is known to be particularly challenging. Hybridization methods tackle this problem by abstracting nonlinear dynamics with piecewise linear dynamics around the reachable states, with additional inputs to ensure overapproximation. This reduces the analysis of a system with nonlinear dynamics to the one with piecewise affine dynamics, which have powerful analysis methods. In this paper, we present improvements to the hybridization approach based on a dynamics scaling model transformation. The transformation aims to reduce the sizes of the linearization domains, and therefore reduces overapproximation error. We showcase the efficiency of our approach on a number of nonlinear benchmark instances, and compare our approach with Flow*.
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References
Althoff, M.: Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In: Proceedings of the 16th International Conference on Hybrid Systems: Computation and Control, pp. 173–182. ACM (2013)
Althoff, M., et al.: Arch-comp18 category report: continuous and hybrid systems with linear continuous dynamics. In: Proceedings of the 5th International Workshop on Applied Verification for Continuous and Hybrid Systems, pp. 23–52 (2018)
Althoff, M., Le Guernic, C., Krogh, B.H.: Reachable set computation for uncertain time-varying linear systems. In: Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control, pp. 93–102. ACM (2011)
Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: Proceedings of the 47th IEEE Conference on Decision and Control (2008)
Asarin, E., Dang, T., Girard, A.: Reachability analysis of nonlinear systems using conservative approximation. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 20–35. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36580-X_5
Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43(7), 451–476 (2007)
Azuma, S., Imura, J., Sugie, T.: Lebesgue piecewise affine approximation of nonlinear systems. Nonlinear Anal. Hybrid Syst. 4(1), 92–102 (2010)
Bak, S., Bogomolov, S., Althoff, M.: Time-triggered conversion of guards for reachability analysis of hybrid automata. In: Abate, A., Geeraerts, G. (eds.) FORMATS 2017. LNCS, vol. 10419, pp. 133–150. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-65765-3_8
Bak, S., Bogomolov, S., Schilling, C.: High-level hybrid systems analysis with Hypy. In: ARCH@ CPSWeek, pp. 80–90 (2016)
Bak, S., Duggirala, P.S.: Hylaa: a tool for computing simulation-equivalent reachability for linear systems. In: Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control, pp. 173–178. ACM (2017)
Bak, S., Tran, H.D., Johnson, T.T.: Numerical verification of affine systems with up to a billion dimensions (2018). arXiv preprint arXiv:1804.01583
Bogomolov, S., Forets, M., Frehse, G., Podelski, A., Schilling, C., Viry, F.: Reach set approximation through decomposition with low-dimensional sets and high-dimensional matrices. In: 21th International Conference on Hybrid Systems: Computation and Control, HSCC 2018, pp. 41–50. ACM (2018)
Bogomolov, S., Forets, M., Frehse, G., Potomkin, K., Schilling, C.: JuliaReach: a toolbox for set-based reachability. In: 22nd ACM International Conference on Hybrid Systems: Computation and Control, HSCC 2019, pp. 39–44. ACM (2019)
Borwein, J., Lewis, A.S.: Convex Analysis and Nonlinear Optimization Theory and Examples. Springer, New York (2010). https://doi.org/10.1007/978-0-387-31256-9
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). https://doi.org/10.1007/978-3-642-39799-8_18
Chen, X., Sankaranarayanan, S.: Decomposed reachability analysis for nonlinear systems. In: 2016 IEEE Real-Time Systems Symposium (RTSS), pp. 13–24. IEEE (2016)
Dang, T., Le Guernic, C., Maler, O.: Computing reachable states for nonlinear biological models. In: Degano, P., Gorrieri, R. (eds.) CMSB 2009. LNCS, vol. 5688, pp. 126–141. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03845-7_9
Dang, T., Maler, O., Testylier, R.: Accurate hybridization of nonlinear systems. In: Proceedings of the 13th ACM International Conference on Hybrid Systems: Computation and Control, pp. 11–20. ACM (2010)
Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 167–170. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14295-6_17
Duggirala, P.S., Mitra, S., Viswanathan, M., Potok, M.: C2E2: a verification tool for stateflow models. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 68–82. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_5
Franzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. J. Satisfiability Boolean Model. Comput. 1, 209–236 (2007)
Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_30
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). https://doi.org/10.1007/978-3-540-31954-2_19
Gurung, A., Deka, A.K., Bartocci, E., Bogomolov, S., Grosu, R., Ray, R.: Parallel reachability analysis for hybrid systems. In: 14th ACM-IEEE International Conference on Formal Methods and Models for System Design, MEMOCODE 2016, pp. 12–22. ACM-IEEE (2016)
Han, Z., Krogh, B.H.: Reachability analysis of nonlinear systems using trajectory piecewise linearized models. In: 2006 American Control Conference, p. 6. IEEE (2006)
Henzinger, T.A., Ho, P.H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Trans. Autom. Control 43(4), 540–554 (1998)
Johnson, T.T., Green, J., Mitra, S., Dudley, R., Erwin, R.S.: Satellite rendezvous and conjunction avoidance: case studies in verification of nonlinear hybrid systems. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 252–266. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32759-9_22
Klipp, E., Herwig, R., Kowald, A., Wierling, C., Lehrach, H.: Systems Biology in Practice: Concepts, Implementation and Application. Wiley, Hoboken (2008)
Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \({\delta }\)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46681-0_15
Le Guernic, C.: Reachability analysis of hybrid systems with linear continuous dynamics. Ph.D. thesis, Université Joseph-Fourier-Grenoble I (2009)
Le 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). https://doi.org/10.1007/978-3-642-02658-4_40
Le Guernic, C., Girard, A.: Reachability analysis of linear systems using support functions. Nonlinear Anal. Hybrid Syst. 4(2), 250–262 (2010)
Li, C., Chen, L., Aihara, K.: Synchronization of coupled nonidentical genetic oscillators. Phys. Biol. 3(1), 37 (2006)
Li, D., Bak, S., Bogomolov, S.: Reachability analysis of nonlinear systems using hybridization and dynamics scaling: Proofs. Technical report CS-TR-1534, Newcastle University (2020)
Matthias, A., Ahmed, E.G., Bastian, S., Goran, F.: Report on reachability analysis of nonlinear systems and compositional verification. https://cps-vo.org/node/24199
Prigogine, I., Balescu, R.: Phénomènes cycliques dans la thermodynamique des processus irréversibles. Bull. Cl. Sci. Acad. R. Belg 42, 256–265 (1956)
Rand, R., Holmes, P.: Bifurcation of periodic motions in two weakly coupled van der pol oscillators. Int. J. Non-Linear Mech. 15(4–5), 387–399 (1980)
Smith, A.P., Muñoz, C.A., Narkawicz, A.J., Markevicius, M.: Kodiak: an implementation framework for branch and bound algorithms (2015)
van der Walt, S., Colbert, S.C., Varoquaux, G.: The NumPY array: a structure for efficient numerical computation. Comput. Sci. Eng. 13(2), 22–30 (2011)
Acknowledgments
This research was supported in part by the Air Force Office of Scientific Research under award numbers FA2386-17-1-4065 and FA9550-19-1-0288. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force.
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Li, D., Bak, S., Bogomolov, S. (2020). Reachability Analysis of Nonlinear Systems Using Hybridization and Dynamics Scaling. In: Bertrand, N., Jansen, N. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2020. Lecture Notes in Computer Science(), vol 12288. Springer, Cham. https://doi.org/10.1007/978-3-030-57628-8_16
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