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Optimization-Based Verification and Stability Characterization of Piecewise Affine and Hybrid Systems

  • Alberto Bemporad
  • Fabio Danilo Torrisi
  • Manfred Morari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)

Abstract

In this paper, we formulate the problem of characterizing the stability of a piecewise affine (PWA) system as a verification problem. The basic idea is to take the whole IR n as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semi-global stability by restricting the set of initial conditions to an (arbitrarily large) bounded set X(0), and label as “asymptotically stable in T steps” the trajectories that enter an invariant set around the origin within a finite time T, or as “unstable in T steps” the trajectories which enter a set X inst of (very large) states. Subsets of X(0) leading to none of the two previous cases are labeled as “non-classifiable in T steps”. The domain of asymptotical stability in T steps is a subset of the domain of attraction of an equilibrium point, and has the practical meaning of collecting the initial conditions from which the settling time to a specified set around the origin is smaller than T. In addition, it can be computed algorithmically in finite time. Such an algorithm requires the computation of reach sets, in a similar fashion as what has been proposed for verification of hybrid systems. In this paper we present a substantial extension of the verification algorithm presented in [6] for stability characterization of PWA systems, based on linear and mixed-integer linear programming. As a result, given a set of initial conditions we are able to determine its partition into subsets of trajectories which are asymptotically stable, or unstable, or non-classifiable in T steps.

Keywords

Hybrid System Outer Approximation Hybrid Automaton Switching Sequence Multiple Lyapunov Function 
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.

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References

  1. [1]
    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 A.P. Ravn R.L. Grossman, A. Nerode and H. Rischel, editors, Hybrid Systems, volume 736 of Lecture Notes in Computer Science, pages 209–229. Springer Verlag, 1993.Google Scholar
  2. [2]
    A. Asarin, O. Maler, and A. Pnueli. On the analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science, 138:35–65, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Balluchi, M. Di Benedetto, C. Pinello, C. Rossi, and A. Sangiovanni-Vincentelli. Hybrid control for automotive engine management: the cut-off case. In T.A. Henzinger and S. Sastry, editors, Hybrid Systems: Computation and Control, volume 1386 of Lecture Notes in Computer Science, pages 13–32. Springer Verlag, 1998.Google Scholar
  4. [4]
    A. Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. IEEE Trans. Automatic Control, to appear. http://control.ethz.ch/.
  5. [5]
    A. Bemporad and M. Morari. Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3):407–427, March 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    A. Bemporad and M. Morari. Verification of hybrid systems via mathematical programming. In F.W. Vaandrager and J.H. van Schuppen, editors, Hybrid Systems: Computation and Control, volume 1569 of Lecture Notes in Computer Science, pages 31–45. Springer Verlag, 1999.CrossRefGoogle Scholar
  7. [7]
    A. Bemporad, M. Morari, V. Dua, and E. N. Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Technical Report AUT99-16, Automatic Control Lab, ETH Zürich, Switzerland, 1999.Google Scholar
  8. [8]
    A. Bemporad and F.D. Torrisi. Inner and outer approximation of polytopes using hyper-rectangles. Technical Report AUT00-02, Automatic Control Lab, ETH Zurich, 2000.Google Scholar
  9. [9]
    V.D. Blondel and J.N. Tsitsiklis. Complexity of stability and controllability of elementary hybrid systems. Automatica, 35:479–489, March 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    M. S. Branicky. Studies in hybrid systems: modeling, analysis, and control. PhD thesis, LIDS-TH 2304, Massachusetts Institute of Technology, Cambridge, MA, 1995.Google Scholar
  11. [11]
    M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automatic Control, 43(4):475–482, April 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    K. Fukuda. cdd/cdd+ Reference Manual. Institute for operations Research ETH-Zentrum, ETH-Zentrum, CH-8092 Zurich, Switzerland, 0.61 (cdd) 0.75 (cdd+) edition, December 1997.Google Scholar
  13. [13]
    E.G. Gilbert and I. Kolmanovsky. Maximal output admissible sets for discrete-time systems with disturbance inputs. In Proc. American Contr. Conf., pages 2000–2005, 1995.Google Scholar
  14. [14]
    E.G. Gilbert and K. Tin Tan. Linear systems with state and control constraints: the theory and applications of maximal output admissible sets. IEEE Trans. Automatic Control, 36(9):1008–1020, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    T.A. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: a model checker for hybrid systems. Software Tools for Technology Transfer, 1:110–122, 1997.zbMATHCrossRefGoogle Scholar
  16. [16]
    M. Johannson and A. Rantzer. Computation of piece-wise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automatic Control, 43(4):555–559, 1998.CrossRefGoogle Scholar
  17. [17]
    M. Kantner. Robust stability of piecewise linear discrete time systems. In Proc. American Contr. Conf., pages 1241–1245, Evanston, IL, USA, 1997.Google Scholar
  18. [18]
    Y. Kesten, A. Pnueli, J. Sifakis, and S. Yovine. Integration graphs: a class of decidable hybrid systems. In R.L. Grossman, A. Nerode, A.P. Ravn, and H. Rischel, editors, Hybrid Systems, volume 736 of Lecture Notes in Computer Science, pages 179–208. Springer Verlag, 1993.Google Scholar
  19. [19]
    S. Kowalewski, O. Stursberg, M. Fritz, H. Graf, I. Hoffmann, J. Preußig, M. Remelhe, S. Simon, and H. Treseler. A case study in tool-aided analysis of discretely controlled continuos systems: the two tanks problem. In Hybrid Systems V, volume 1567 of Lecture Notes in Computer Science, pages 163–185. Springer-Verlag, 1999.CrossRefGoogle Scholar
  20. [20]
    D. Liberzon and A.S. Morse. Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19(5):59–70, October 1999.CrossRefGoogle Scholar
  21. [21]
    J. Lygeros, D.N. Godbole, and S. Sastry. A game theoretic approach to hybrid system design. In R. Alur and T. Henzinger, editors, Hybrid Systems III, volume 1066 of Lecture Notes in Computer Science, pages 1–12. Springer Verlag, 1996.CrossRefGoogle Scholar
  22. [22]
    J. Lygeros, C. Tomlin, and S. Sastry. Controllers for reachability specifications for hybrid systems. Automatica, 35(3):349–370, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    R. Raman and I. E. Grossmann. Relation between milp modeling and logical inference for chemical process synthesis. Computers Chem. Engng., 15(2):73–84, 1991.CrossRefGoogle Scholar
  24. [24]
    E. Sontag. From linear to nonlinear: Some complexity comparisons. In Proc. 34th IEEE Conf. on Decision and Control, pages 2916–2920, 1995.Google Scholar
  25. [25]
    E. D. Sontag. Nonlinear regulation: The piecewise linear approach. IEEE Trans. Automatic Control, 26(2):346–358, April 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    E.D. Sontag. Interconnected automata and linear systems: A theoretical framework in discrete-time. In R. Alur, T.A. Henzinger, and E.D. Sontag, editors, Hybrid Systems III-Verification and Control, number 1066 in Lecture Notes in Computer Science, pages 436–448. Springer-Verlag, 1996.Google Scholar
  27. [27]
    M.L. Tyler and M. Morari. Propositional logic in control and monitoring problems. Automatica, 35(4):565–582, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    V. I. Utkin. Variable structure systems with sliding modes. IEEE Trans. Automatic Control, 22(2):212–222, April 1977.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Alberto Bemporad
    • 1
  • Fabio Danilo Torrisi
    • 1
  • Manfred Morari
    • 1
  1. 1.Automatic Control LaboratorySwiss Federal Institute of TechnologyZürichSwitzerland

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