Abstract
Hybrid automata combine finite state models with continuous variables that are governed by differential equations. Hybrid automata are used to model systems in a wide range of domains such as automotive control, robotics, electronic circuits, systems biology, and health care. Numerical simulation approximates the evolution of the variables with a sequence of points in discretized time. This highly scalable technique is widely used in engineering and design, but it is difficult to simulate all representative behaviors of a system. To ensure that no critical behaviors are missed, reachability analysis aims at accurately and quickly computing a cover of the states of the system that are reachable from a given set of initial states. Reachability can be used to formally show safety and bounded liveness properties. This chapter outlines the major concepts and discusses advantages and shortcomings of the different techniques.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Althoff. Reachability analysis of nonlinear systems using conservative polynomialization and non-convex sets. In Hybrid systems: computation and control (HSCC'19), pages 173–182. ACM, 2013.
M. Althoff and B. H. Krogh. Avoiding geometric intersection operations in reachability analysis of hybrid systems. In Hybrid Systems: Computation and Control (HSCC'lli), pages 4 & -54. ACM, 2012.
M. Althoff, , H. Krogh, and O. Stursberg. Analyzing reachability of linear dynamic systems with parametric uncertainties. In A. Rauh and E. Auer, editors, Modeling, Design, and Simulation of Systems with Uncertainties. Springer, 2011.
R. AIur. Formal verification of hybrid systems. In S. Chakraborty, A. Jerraya, S. , Baruah, and S. Fischmeister, editors, EMSOFT, pages 273–278. ACM, 2011.
R. Alur, C. Courcoubetis, N. Halbwachs, T. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138:3–34, 1995.
R. Alur, C. Courcoubetis, T. A. Henzinger, and P.-H. Ho. Hybrid automata: An algorithmic approach to the specification and verification of hybrid systems. In Hybrid Systems, LNCS 736, pages 209–229. Springer, 1993.
E. Asarin, T. Dang, and A. Girard. Hybridization methods for the analysis of nonlinear systems. Acta Inf., 43(7):451–476, 2007.
E. Asarin, T. Dang, O. Maler, and O. Bournez. Approximate reachability analysis of piecewise-linear dynamical systems. In Hybrid Systems: Computation and Control (HSCC’00), volume 1790 of LNCS, pages 20–31. Springer, 2000.
E. Asarin, T. Dang, O. Maler, and R. Testylier. Using redundant constraints for refinement. In Automated Technology for Verification and Analysis, pages 37–51. Springer, 2010.
R. Bagnara, P. M. Hill, and E. Zaffanella. The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems. Science of Computer Programming, 72(1–2):3–21, 2008.
O. Bouissou, A. Chapoutot, S. Mimram, and B. Strazzulla. Set-based simulation for design and verification of simulink models. In Embedded Real Time Software and Systems (ERTS’tf), 2014.
O. Bouissou, S. Mimram, and A. Chapoutot. Hyson: Set-based simulation of hybrid systems. In RSP, pages 79–85. IEEE, October 2012.
D. Bruck, H. Elmqvist, S. E. Mattsson, and H. Olsson. Dymola for multiengineering modeling and simulation. In Proceedings of Modelica, 2002.
R. P. Canale and S. C. Chapra. Numerical methods for engineers. Mc Graw Hill, New York, 1998.
C. G. Cassandras and J. Lygeros. Stochastic hybrid systems. CRC Press, 2006.
C. Chase, J. Serrano, and P. J. Ramadge. Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. Automatic Control, IEEE Transactions on, 38(1):70–83, 1993.
X. Chen, E. Abraham, and S. Sankaranarayanan. Taylor model flowpipe construction for non-linear hybrid systems. In RTSS, pages 183–192. IEEE Computer Society, 2012.
A. Chutinan and B. H. Krogh. Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In F. W. Vaandrager and J. H. van Schuppen, editors, HSCC, volume 1569 of LNCS, pages 76–90. Springer, 1999.
T. Dang and R. Testylier. Reachability analysis for polynomial dynamical systems using the bernstein expansion. Reliable Computing, 17(2):128–152, 2012.
A. Donze. Breach, a toolbox for verification and parameter synthesis of hybrid systems. In Computer Aided Verification, pages 167–170. Springer, 2010.
J. Eker, J. W. Janneck, E. A. Lee, J. Liu, X. Liu, J. Ludvig, S. Neuendorffer, S. Sachs, and Y. Xiong. Taming heterogeneity-the ptolemy approach. Proceedings of the IEEE, 91(1):127–144, 2003.
G. E. Fainekos, A. Girard, and G. J. Pappas. Temporal logic verification using simulation. In Formal Modeling and Analysis of Timed Systems, pages 171–186. Springer, 2006.
G. Frehse. PHAVer: algorithmic verification of hybrid systems past HyTech. STTT, 10(3):263–279, 2008.
G. Frehse, C. L. Guernic, A. Donze, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang, and O. Maler. SpaceEx: Scalable verification of hybrid systems. In G. Gopalakrishnan and S. Qadeer, editors, CAV, LNCS. Springer, 2011.
G. Frehse, R. Kateja, and C. Le Guernic. Flowpipe approximation and clustering in space-time. In Hybrid systems: computation and control (HSCC’13), pages 203212. ACM, 2013.
A. Girard. Reachability of uncertain linear systems using zonotopes. In M. Morari and L. Thiele, editors, HSCC, volume 3414 of LNCS, pages 291–305. Springer, 2005.
A. Girard, C. L. Guernic, and O. Maler. Efficient computation of reachable sets of linear time-invariant systems with inputs. In J. P. Hespanha and A. Tiwari, editors, HSCC, volume 3927 of LNCS, pages 257–271. Springer, 2006.
A. Girard and G. Zheng. Verification of safety and liveness properties of metric transition systems. ACM Transactions on Embedded Computing Systems (TECS), 11(S2):54, 2012.
M. R. Greenstreet. Verifying safety properties of differential equations. In Computer Aided Verification, pages 277–287. Springer, 1996.
C. L. Guernic and A. Girard. Reachability analysis of hybrid systems using support functions. In A. Bouajjani and O. Maler, editors, CAV, volume 5643 of LNCS, pages 540–554. Springer, 2009.
N. Halbwachs, Y.-E. Proy, and P. Raymond. Verification of linear hybrid systems by means of convex approximations. In International Static Analysis Symposium, SAS’94, Namur (Belgium), September 1994.
T. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: A model checker for hybrid systems. Software Tools for Technology Transfer, pages 110–122, 1997.
T. A. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: A model checker for hybrid systems. In O. Grumberg, editor, CAV, volume 1254 of LNCS, pages 460–463. Springer, 1997.
T. A. Henzinger, P.-H. Ho, and H. Wong-Toi. Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control, 43:540–554, 1998.
T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata? Journal of Computer and System Sciences, 57:94–124, 1998.
P.-H. Ho. Automatic Analysis of Hybrid Systems. PhD thesis, Cornell University, Aug. 1995. Technical Report CSD-TR95–1536.
A. A. Julius, G. E. Fainekos, M. Anand, I. Lee, and G. J. Pappas. Robust test generation and coverage for hybrid systems. In Hybrid Systems: Computation and Control, pages 329–342. Springer, 2007.
W. Kuhn. Rigorously computed orbits of dynamical systems without the wrapping effect. Computing, 61(1):47–67, 1998.
A. B. Kurzhanski and P. Varaiya. Dynamics and Control of Trajectory Tubes. Springer, 2014.
A. A. Kurzhanskiy and P. Varaiya. Ellipsoidal toolbox (et). In Decision and Control, 2006 45th IEEE Conference on, pages 1498–1503. IEEE, 2006.
C. Le Guernic. Reachability analysis of hybrid systems with linear continuous dynamics. PhD thesis, Universite Grenoble 1 – Joseph Fourier, 2009.
A. V. Lotov, V. A. Bushenkov, and G. K. Kamenev. Interactive Decision Maps, volume 89 of Applied Optimization. Kluwer, 2004.
O. Maler. Algorithmic verification of continuous and hybrid systems. In Int. Workshop on Verification of Infinite-Stale System (Infinity), 2013.
MapleSoft. Maplesim 7: Advanced system-level modeling. http://www.maplesoft.com/products/maplesim, 2015.
MathWorks. Mathworks simulink: Simulation et model-based design, Mar. 2014. www.mathworks.fr/products/simulink.
S. E. Mattsson, H. Elmqvist, and M. Otter. Physical system modeling with mod- elica. Control Engineering Practice, 6(4):501–510, 1998.
P. Prabhakar and M. Viswanathan. A dynamic algorithm for approximate flow computations. In E. Frazzoli and R. Grosu, editors, HSCC, pages 133–142. ACM, 2011.
W. H. Press. Numerical recipes 3rd edition: The art of scientific computing. Cambridge University Press, 2007.
S. Sankaranarayanan, T. Dang, and F. Ivancic. Symbolic model checking of hybrid systems using template polyhedra. In Tools and Algorithms for the Construction and Analysis of Systems, pages 188–202. Springer, 2008.
P. Tabuada. Verification and Control of Hybrid Systems: A Symbolic Approach. Springer, 2009.
F. Zhang, M. Yeddanapudi, and P. Mosterman. Zero-crossing location and detection algorithms for hybrid system simulation. In IFAC World Congress, pages 7967–7972, 2008.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Fachmedien Wiesbaden
About this chapter
Cite this chapter
Frehse, G. (2015). An Introduction to Hybrid Automata, Numerical Simulation and Reachability Analysis. In: Drechsler, R., Kühne, U. (eds) Formal Modeling and Verification of Cyber-Physical Systems. Springer Vieweg, Wiesbaden. https://doi.org/10.1007/978-3-658-09994-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-658-09994-7_3
Published:
Publisher Name: Springer Vieweg, Wiesbaden
Print ISBN: 978-3-658-09993-0
Online ISBN: 978-3-658-09994-7
eBook Packages: Computer ScienceComputer Science (R0)