Advertisement

Theory of Optimal Control Using Bisimulations

  • Mireille Broucke
  • Maria Domenica Di Benedetto
  • Stefano Di Gennaro
  • Alberto Sangiovanni-Vincentelli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)

Abstract

We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem: to synthesize optimal enabling conditions for switching between locations in which the control is constant. An algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non-Zenoness of trajectories. We show that the discrete value function converges to the viscosity solution of the Hamilton-Jacobi-Bellman equation as a discretization parameter tends to zero.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Bardi and I. Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston, 1997.MATHGoogle Scholar
  2. 2.
    V.G. Boltyanskii. Sufficient conditions for optimality and the justification of the dynamic programming method. SIAM Journal of Control, 4, pp. 326–361, 1966.CrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Branicky, V. Borkar, S. Mitter. A unified framework for hybrid control: model and optimal control theory. IEEE Trans. AC, vol. 43, no. 1, pp. 31–45, January, 1998.MATHMathSciNetGoogle Scholar
  4. 4.
    M. Broucke. A geometric approach to bisimulation and verification of hybrid systems. In Hybrid Systems: Computation and Control, F. Vaandrager and J. van Schuppen, eds., LNCS 1569, p. 61–75, Springer-Verlag, 1999.CrossRefGoogle Scholar
  5. 5.
    I. Capuzzo Dolcetta and L.C. Evans. Optimal switching for ordinary differential equations. SIAM J. Control and Optimization, vol. 22, no. 1, pp. 143–161, January 1984.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M Crandall, P. Lions. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., vol. 277, no. 1, pp. 1–42, 1983.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W.H. Fleming, R.W. Rishel. Deterministic and stochastic optimal control. Springer-Verlag, New York, 1975.MATHGoogle Scholar
  8. 8.
    O. Maler, A Pnueli, J. Sifakis. On the synthesis of discrete controllers for timed systems. In Proc. STACS’ 95, E.W. Mayr and C. Puech, eds. LNCS 900, Springer-Verlag, p. 229–242, 1995.Google Scholar
  9. 9.
    J. Raisch. Controllability and observability of simple hybrid control systems-FDLTI plants with symbolic measurements and quantized control inputs. International Conference on Control’ 94, IEE, vol. 1, pp. 595–600, 1994.CrossRefGoogle Scholar
  10. 10.
    J. Stiver, P. Antsaklis, M. Lemmon. A logical DES approach to the design of hybrid control systems. Mathemtical and computer modelling. vol. 23, no. 11-1, pp. 55–76, June, 1996.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    H.S. Witsenhausen. A class of hybrid-state continuous-time dynamic systems. IEEE Trans. AC, vol. 11, no. 2, pp. 161–167, April, 1966.Google Scholar
  12. 12.
    H. Wong-Toi. The synthesis of controllers for linear hybrid automata. In Proc. 36th IEEE Conference on Decision and Control, pp. 4607–4612, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Mireille Broucke
    • 1
  • Maria Domenica Di Benedetto
    • 1
    • 2
  • Stefano Di Gennaro
    • 2
  • Alberto Sangiovanni-Vincentelli
    • 1
  1. 1.Dept. of Electrical Engineering and Computer SciencesUniversity of California at BerkeleyUSA
  2. 2.Dip. di Ingegneria ElettricaUniversità di L’AquilaL’AquilaItaly

Personalised recommendations