Synthesizing switching logic using constraint solving

VMCAI 2009

Abstract

For a system that can operate in multiple different modes, we define the switching logic synthesis problem as follows: given a description of the dynamics in each mode of the system, find the conditions for switching between the modes so that the resulting system satisfies some desired properties. In this paper, we present an approach for solving the switching logic synthesis problem in the case when (1) the dynamics in each mode of the system are given using differential equations and, hence, the synthesized system is a hybrid system, and (2) the desired property is a safety property. Our approach for solving the switching logic synthesis problem, called the constraint-based approach, consists of two steps. In the first constraint generation step, the synthesis problem is reduced to satisfiability of a quantified formula over the theory of reals. In the second constraint solving step, the quantified formula is solved. This paper focuses on constraint generation. The constraint generation step is based on the concept of a controlled inductive invariant. The search for controlled inductive invariant is cast as a constraint solving problem. The controlled inductive invariant is then used to arrive at the maximally liberal switching logic. We prove that the synthesized switching logic always gives us a well-formed and safe hybrid system. When the system, the safety property, and the controlled inductive invariant are all expressed only using polynomials, the generated constraint is an $${\exists\forall}$$ formula in the theory of reals, whose satisfiability is decidable.

Keywords

Formal methods Controller synthesis Hybrid systems

References

1. 1.
Alur R., Courcoubetis C., Halbwachs N., Henzinger T.A., Ho P.-H., Nicollin X., Olivero A., Sifakis J., Yovine S.: The algorithmic analysis of hybrid systems. Theor. Comput. Sci. 138(3), 3–34 (1995)
2. 2.
Asarin E., Bournez O., Dang T., Maler O., Pnueli A.: Effective synthesis of switching controllers for linear systems. Proc. IEEE 88(7), 1011–1025 (2000)
3. 3.
Blanchini F.: Set invariance in control. Automatica 35, 1747–1767 (1999)
4. 4.
Burns K., Gidea M.: Differential Geometry and Topology: With a view to dynamical systems. Chapman & Hall, London (2005)
5. 5.
Chaudhuri, S., Solar-Lezama, A.: Smooth interpretation. In: ACM Conference on Programming Language Design and Implementation PLDI (2010)Google Scholar
6. 6.
Colón, M.: Schema-guided synthesis of imperative programs by constraint solving. In: LOPSTR, pp. 166–181 (2004)Google Scholar
7. 7.
Cury J., Krogh B., Niinomi T.: Supervisory controllers for hybrid systems based on approximating automata. IEEE Trans. Aut. Control 43, 564–568 (1998)
8. 8.
Gulwani, S., Srivastava, S., Venkatesan, R.: Program analysis as constraint solving. In: Proceedings of ACM Conference on Programming Language Design and Implementation PLDI, pp. 281–292 (2008)Google Scholar
9. 9.
Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: CAV, volume 5123 of LNCS, pp. 190–203. Springer (2008)Google Scholar
10. 10.
Hong, H.: Quantifier elimination procedure by cylindrical algebraic decomposition (1995). http://www.gwdg.de/~cais/systeme/saclib, http://www.eecis.udel.edu/~saclib/
11. 11.
Jha, S., Gulwani, S., Seshia, S., Tiwari, A.: Synthesizing switching logic for safety and dwell-time requirements. In: ACM/IEEE International Conference on Cyber-Physical Systems, ICCPS (2010)Google Scholar
12. 12.
Koo, T., Sastry, S.: Mode switching synthesis for reachability specification. In: Proceedings of HSCC 2001, LNCS 2034, pp. 333–346 (2001)Google Scholar
13. 13.
Liberzon D., Morse A.S.: Benchmark problems in stability and design of switched systems. IEEE Control Syst. Mag. 19, 59–70 (1999)
14. 14.
Lustig, Y., Vardi, M.: Synthesis from component libraries. In: Proc. FoSSaCS, pp. 395–409 (2009)Google Scholar
15. 15.
Manna Z., Waldinger R.: A deductive approach to program synthesis. ACM TOPLAS 2(1), 90–121 (1980)
16. 16.
Manon, P., Valentin-Roubinet, C.: Controller synthesis for hybrid systems with linear vector fields. In: Proceedings of IEEE Symposium on Intell. Control, pp. 17–22 (1999)Google Scholar
17. 17.
Moor, T., Raisch, J.: Discrete control of switched linear systems. In: Proceedings of European Control Conference on ECC’99 (1999)Google Scholar
18. 18.
Platzer A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Log. Comput. 20(1), 309–352 (2010) Advance Access published on (November 18 2008)
19. 19.
Prajna, S., Jadbabaie, A.: Safety verification of hybrid systems using barrier certificates. In: Proceedings of HSCC, volume 2993 of LNCS, pp. 477–492 (2004)Google Scholar
20. 20.
Prajna, S., Jadbabaie, A., Pappas, G.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Automat. Contr. 52(8) (2007)Google Scholar
21. 21.
Sankaranarayanan, S., Sipma, H., Manna, Z.: Constructing invariants for hybrid systems. In: Proceedings of HSCC, volume 2993 of LNCS, pp. 539–554 (2004)Google Scholar
22. 22.
Shapiro E.Y.: Algorithmic Program DeBugging. MIT Press, Cambridge (1983)Google Scholar
23. 23.
Solar-Lezama, A., Tancau, L., Bodík, R., Seshia, S., Saraswat, V.: Combinatorial sketching for finite programs. In: ASPLOS (2006)Google Scholar
24. 24.
Taly, A., Gulwani, S., Tiwari, A.: Synthesizing switching logic using constraint solving. In: Proceedings of 10th International Conference on Verification, Model Checking and Abstract Interpretation, VMCAI, volume 5403 of LNCS, pp. 305–319. Springer (2009)Google Scholar
25. 25.
Taly, A., Tiwari, A.: Deductive verification of continuous dynamical systems. In: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2009), volume 4 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 383–394. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2009)Google Scholar
26. 26.
Tarski A.: A Decision Method for Elementary Algebra and Geometry. 2nd edn. University of California Press, California (1948)
27. 27.
Tomlin C., Lygeros L., Sastry S.: A game-theoretic approach to controller design for hybrid systems. Proc. IEEE 88(7), 949–970 (2000)