A Decompositional Proof Scheme for Automated Convergence Proofs of Stochastic Hybrid Systems

  • Jens Oehlerking
  • Oliver Theel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5799)

Abstract

In this paper, we describe a decompositional approach to convergence proofs for stochastic hybrid systems given as probabilistic hybrid automata. We focus on a concept called “stability in probability”, which implies convergence of almost all trajectories of the stochastic hybrid system to a designated equilibrium point. By adapting classical Lyapunov function results to the stochastic hybrid case, we show how automatic stability proofs for such systems can be obtained with the help of numerical tools. To ease the load on the numerical solvers and to permit incremental construction of stable systems, we then propose an automatable Lyapunov-based decompositional framework for stochastic stability proofs. This framework allows conducting sub-proofs separately for different parts of the automaton, such that they still yield a proof for the entire system. Finally, we give an outline on how these decomposition results can be applied to conduct quantitative probabilistic convergence analysis, i.e., determining convergence probabilities below 1.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Branicky, M.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control 43(4), 475–482 (1998)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Pettersson, S.: Analysis and Design of Hybrid Systems. PhD thesis, Chalmers University of Technology, Gothenburg (1999)Google Scholar
  3. 3.
    Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming, Series B (96), 293–320 (2003)Google Scholar
  4. 4.
    Heemels, M., Weiland, S., Juloski, A.: Input-to-state stability of discontinuous dynamical systems with an observer-based control application. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 259–272. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Oehlerking, J., Theel, O.: Decompositional construction of Lyapunov functions for hybrid systems. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 276–290. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Shiryaev, A.N.: Probability, 2nd edn. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Kushner, H.J.: Stochastic stability. Lecture Notes in Mathematics, vol. (249), pp. 97–124 (1972)Google Scholar
  8. 8.
    Loparo, K.A., Feng, X.: Stability of stochastic systems. In: The Control Handbook, pp. 1105–1126. CRC Press, Boca Raton (1996)Google Scholar
  9. 9.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Society for Industrial and Applied Mathematics, SIAM (1994)Google Scholar
  10. 10.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  11. 11.
    Borchers, B.: CSDP, a C library for semidefinite programming. Optimization Methods and Software 10(1), 613–623 (1999), https://projects.coin-or.org/Csdp/ CrossRefMathSciNetGoogle Scholar
  12. 12.
    Romanko, O., Pólik, I., Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones (1999), http://sedumi.ie.lehigh.edu

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jens Oehlerking
    • 1
  • Oliver Theel
    • 1
  1. 1.Department of Computer ScienceUniversity of OldenburgOldenburgGermany

Personalised recommendations