Timed Relational Abstractions for Sampled Data Control Systems

  • Aditya Zutshi
  • Sriram Sankaranarayanan
  • Ashish Tiwari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7358)


In this paper, we define timed relational abstractions for verifying sampled data control systems. Sampled data control systems consist of a plant, modeled as a hybrid system and a synchronous controller, modeled as a discrete transition system. The controller computes control inputs and/or sends control events to the plant based on the periodically sampled state of the plant. The correctness of the system depends on the controller design as well as an appropriate choice of the controller sampling period.

Our approach constructs a timed relational abstraction of the hybrid plant by replacing the continuous plant dynamics by relations. These relations map a state of the plant to states reachable within the sampling time period. We present techniques for building timed relational abstractions, while taking care of discrete transitions that can be taken by the plant between samples. The resulting abstractions are better suited for the verification of sampled data control systems. The abstractions focus on the states that can be observed by the controller at the sample times, while abstracting away behaviors between sample times conservatively. The resulting abstractions are discrete, infinite-state transition systems. Thus conventional verification tools can be used to verify safety properties of these abstractions. We use k-induction to prove safety properties and bounded model checking (BMC) to find potential falsifications. We present our idea, its implementation and results on many benchmark examples.


Hybrid System Network Control System Discrete Transition Autonomous Transition Hybrid Automaton 
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.


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    Arney, D., Jetley, R., Jones, P., Lee, I., Sokolsky, O.: Formal methods based development of a PCA infusion pump reference model: Generic infusion pump (GIP) project. In: Proc. High Confidence Medical Devices, Software Systems and Medical Device Plug and Play Interoperability (2007)Google Scholar
  3. 3.
    Asarin, E., Dang, T., Girard, A.: Hybridization methods for the analysis of nonlinear systems. Acta Informatica 43, 451–476 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berdine, J., Chawdhary, A., Cook, B., Distefano, D., O’Hearn, P.W.: Variance analyses from invariance analyses. In: POPL, pp. 211–224. ACM (2007)Google Scholar
  5. 5.
    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic Model Checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Bochev, P., Markov, S.: A self-validating numerical method for the matrix exponential. Computing 43(1), 59–72Google Scholar
  7. 7.
    Bozzano, M., Cimatti, A., Tapparo, F.: Symbolic Fault Tree Analysis for Reactive Systems. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 162–176. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Collins, G.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  10. 10.
    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)Google Scholar
  11. 11.
    Dang, T., Maler, O., Testylier, R.: Accurate hybridization of nonlinear systems. In: HSCC 2010, pp. 11–20. ACM (2010)Google Scholar
  12. 12.
    de Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Dolzmann, A., Sturm, T.: REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dutertre, B., de Moura, L.: The YICES SMT solver. Cf, (last viewed January 2009)
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Frehse, G., Le 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)CrossRefGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Goldsztejn, A.: On the exponentiation of interval matrices. Preprint (Working Paper) # hal-00411330, version 1. Cf (2009),
  19. 19.
    Guernic, C.L., Girard, A.: Reachability analysis of linear systems using support functions. Nonlinear Analysis: Hybrid Systems 4(2), 250–262 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gulwani, S., Jain, S., Koskinen, E.: Control-flow refinement and progress invariants for bound analysis. In: PLDI (2009)Google Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    Halbwachs, N., Proy, Y.-E., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Formal Methods in System Design 11(2), 157–185 (1997)CrossRefGoogle Scholar
  23. 23.
    Henzinger, T.A., Ho, P.-H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control 43, 540–554 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Jeannet, B., Halbwachs, N., Raymond, P.: Dynamic Partitioning in Analyses of Numerical Properties. In: Cortesi, A., Filé, G. (eds.) SAS 1999. LNCS, vol. 1694, pp. 39–50. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  25. 25.
    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)CrossRefGoogle Scholar
  26. 26.
    Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, New York (1995)CrossRefGoogle Scholar
  27. 27.
    Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45(1), 161–208 (2003)CrossRefGoogle Scholar
  28. 28.
    Moore, R., Kearfott, R.B., Cloud, M.: Introduction to Interval Analysis. SIAM (2009)Google Scholar
  29. 29.
    Oppenheimer, E.P., Michel, A.N.: Application of interval analysis techniques to linear systems. II. the interval matrix exponential function. IEEE Trans. on Circuits and Systems 35(10), 1230–1242 (1988)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS, pp. 32–41. IEEE Computer Society (2004)Google Scholar
  31. 31.
    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)CrossRefGoogle Scholar
  32. 32.
    Rushby, J., Lincoln, P., Owre, S., Shankar, N., Tiwari, A.: Symbolic analysis laboratory (sal). Cf,
  33. 33.
    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)CrossRefGoogle Scholar
  34. 34.
    Sankaranarayanan, S., Homaei, H., Lewis, C.: Model-Based Dependability Analysis of Programmable Drug Infusion Pumps. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 317–334. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  35. 35.
    Sankaranarayanan, S., Tiwari, A.: Relational Abstractions for Continuous and Hybrid Systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 686–702. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  36. 36.
    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)CrossRefGoogle Scholar
  37. 37.
    Silva, B.I., Richeson, K., Krogh, B.H., Chutinan, A.: Modeling and verification of hybrid dynamical system using checkmate. In: ADPM 2000 (2000),
  38. 38.
    Sturm, T., Tiwari, A.: Verification and synthesis using real quantifier elimination. In: ISSAC, pp. 329–336. ACM (2011)Google Scholar
  39. 39.
    Tiwari, A.: Abstractions for hybrid systems. Formal Methods in Systems Design 32, 57–83 (2008)zbMATHCrossRefGoogle Scholar
  40. 40.
    Tiwari, A., SRI International: HybridSAL: A tool for abstracting HybridSAL specifications to SAL specifications. Cf (2007)
  41. 41.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. In: Applied Algebra and Error-Correcting Codes (AAECC), vol. 8, pp. 85–101 (1997)Google Scholar
  42. 42.
    Zhang, W., Branicky, M.S., Phillips, S.M.: Stability of networked control systems. IEEE Control Systems Magazine 21, 84–99 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aditya Zutshi
    • 1
  • Sriram Sankaranarayanan
    • 1
  • Ashish Tiwari
    • 2
  1. 1.University of ColoradoBoulderUSA
  2. 2.SRI InternationalMenlo ParkUSA

Personalised recommendations