Abstract
In this paper, we define relational abstractions of hybrid systems. A relational abstraction is obtained by replacing the continuous dynamics in each mode by a binary transition relation that relates a state of the system to any state that can potentially be reached at some future time instant using the continuous dynamics. We construct relational abstractions by reusing template-based invariant generation techniques for continuous systems described by Ordinary Differential Equations (ODE). As a result, we abstract a given hybrid system as a purely discrete, infinite-state system. We apply k-induction to this abstraction to prove safety properties, and use bounded model-checking to find potential falsifications. We present the basic underpinnings of our approach and demonstrate its use on many benchmark systems to derive simple and usable abstractions.
Sankaranarayanan’s work has been supported by NSF Career grant CNS-0953941. Tiwari’s work supported in part by DARPA under Contract No. FA8650-10-C-7078, NSF grants CSR-0917398 and SHF:CSR-1017483.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Abate, A., Tiwari, A., Sastry, S.: Box invariance in biologically-inspired dynamical systems. Automatica 45(7), 1601–1610 (2009)
Alur, R., Dang, T., Ivančić, F.: Counter-example guided predicate abstraction of hybrid systems. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 208–223. Springer, Heidelberg (2003)
Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43, 451–476 (2007)
Berdine, J., Chawdhary, A., Cook, B., Distefano, D., O’Hearn, P.W.: Variance analyses from invariance analyses. In: POPL, pp. 211–224. ACM, New York (2007)
Berz, M., Makino, K.: Performance of Taylor Model Methods for Validated Integration of ODEs. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds.) PARA 2004. LNCS, vol. 3732, pp. 65–73. Springer, Heidelberg (2006)
Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Springer, Heidelberg (2008)
Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12(3), 299–328 (1991)
Colón, M.A., Sankaranarayanan, S., Sipma, H.B.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420–432. Springer, Heidelberg (2003)
Colón, M.A., Sipma, H.B.: Synthesis of linear ranking functions. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 67–81. Springer, Heidelberg (2001)
Cousot, P., Cousot, R.: Abstract Interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: ACM Principles of Programming Languages, pp. 238–252 (1977)
Dang, T., Maler, O., Testylier, R.: Accurate hybridization of nonlinear systems. In: HSCC 2010, pp. 11–20. ACM, New York (2010)
Dang, T., Salinas, D.: Image Computation for Polynomial Dynamical Systems Using the Bernstein Expansion. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 219–232. Springer, Heidelberg (2009)
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)
Frehse, G.: PHAVer: Algorithmic Verification of Hybrid Systems Past HyTech. STTT 10(3) (June 2008)
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)
Guernic, C.L., Girard, A.: Reachability analysis of linear systems using support functions. Nonlinear Analysis: Hybrid Systems 4(2), 250–262 (2010)
Gulwani, S., Jain, S., Koskinen, E.: Control-flow refinement and progress invariants for bound analysis. In: PLDI (2009)
Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 190–203. Springer, Heidelberg (2008)
Gupta, A., Majumdar, R., Rybalchenko, A.: From tests to proofs. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 262–276. Springer, Heidelberg (2009)
Halbwachs, N., Proy, Y.-E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. In: FMSD, vol. 11(2), pp. 157–185 (1997)
Henzinger, T.A.: The theory of hybrid automata. In: LICS 1996, pp. 278–292. IEEE, Los Alamitos (1996)
Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control 43, 540–554 (1998)
Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. In: Lynch, N.A., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 202–214. Springer, Heidelberg (2000)
Meiss, J.D.: Differential Dynamical Systems. SIAM publishers, Philadelphia (2007)
Mysore, V., Piazza, C., Mishra, B.: Algorithmic algebraic model checking II: Decidability of semi-algebraic model checking and its applications to systems biology. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 217–233. Springer, Heidelberg (2005)
Nielson, F., Nielson, H.R., Hankin, C.: Principles of Program Analysis. Springer, Heidelberg (1999)
Oishi, M., Mitchell, I., Bayen, A.M., Tomlin, C.J.: Invariance-preserving abstractions of hybrid systems: Application to user interface design. IEEE Trans. on Control Systems Technology 16(2) (March 2008)
Platzer, A., Clarke, E.: Computing differential invariants of hybrid systems as fixedpoints. Formal Methods in Systems Design 35(1), 98–120 (2009)
Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS, pp. 32–41. IEEE, Los Alamitos (2004)
Podelski, A., Wagner, S.: Model checking of hybrid systems: From reachability towards stability. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 507–521. Springer, Heidelberg (2006)
Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 477–492. Springer, Heidelberg (2004)
Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation based abstraction refinement. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 573–589. Springer, Heidelberg (2005)
Rushby, J., Lincoln, P., Owre, S., Shankar, N., Tiwari, A.: Symbolic analysis laboratory (SAL). Cf, http://www.csl.sri.com/projects/sal/
Sankaranarayanan, S., Dang, T., Ivančić, F.: Symbolic model checking of hybrid systems using template polyhedra. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 188–202. Springer, Heidelberg (2008)
Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Fixed point iteration for computing the time elapse operator. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 537–551. Springer, Heidelberg (2006)
Sheeran, M., Singh, S., Stålmarck, G.: Checking safety properties using induction and a SAT-solver. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 108–125. Springer, Heidelberg (2000)
Sturm, T., Tiwari, A.: Verification and synthesis using real quantifer elimination (2011) (submitted)
Tiwari, A.: Approximate reachability for linear systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 514–525. Springer, Heidelberg (2003)
Tiwari, A.: HybridSAL: A tool for abstracting HybridSAL specifications to SAL specifications (2007)
Tiwari, A.: Abstractions for hybrid systems. Formal Methods in Systems Design 32, 57–83 (2008)
Weispfenning, V. In: Applied Algebra and Error-Correcting Codes (AAECC)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sankaranarayanan, S., Tiwari, A. (2011). Relational Abstractions for Continuous and Hybrid Systems. In: Gopalakrishnan, G., Qadeer, S. (eds) Computer Aided Verification. CAV 2011. Lecture Notes in Computer Science, vol 6806. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22110-1_56
Download citation
DOI: https://doi.org/10.1007/978-3-642-22110-1_56
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22109-5
Online ISBN: 978-3-642-22110-1
eBook Packages: Computer ScienceComputer Science (R0)