Advertisement

Automation and Remote Control

, Volume 67, Issue 3, pp 350–360 | Cite as

Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary

  • V. A. Baturin
  • E. V. Goncharova
  • F. L. Pereira
  • J. B. Sousa
Determinate Systems

Abstract

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.

PACS number

02.30.Yy 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Sousa, J. and Pereira, F., Some Questions about Hybrid Systems, Proc. Eur. Control Conf., 2001, pp. 3879–3886.Google Scholar
  2. 2.
    Clarke, F.H., A Proximal Characterization of the Reachable Set, Syst. Control Lett., 1996, vol. 27(3), pp. 195–197.CrossRefzbMATHGoogle Scholar
  3. 3.
    Graettinger, T. and Krogh, B., Hyperplane Method for Reachable State Estimation for Linear Time-Invariant Systems, J. Optim. Theory Appl., 1991, vol. 69, pp. 555–588.CrossRefMathSciNetGoogle Scholar
  4. 4.
    Kurzhanski, A.B., Ellipsoidal Calculus for Estimation and Control, Boston: Birkhauser, 1997.Google Scholar
  5. 5.
    Brogliato, B., Nonsmooth Impact Mechanics: Models, Dynamics and Control, Berlin: Springer-Verlag, 1996.Google Scholar
  6. 6.
    Clark, C., Clarke, F., and Munro, G., The Optimal Exploitation of Renewable Stocks, Econometrica, 1979, vol. 47, pp. 25–47.Google Scholar
  7. 7.
    Marec, J.P., Optimal Space Trajectories, New York: Elsevier, 1979.Google Scholar
  8. 8.
    Arutyunov, A., Dykhta, V., and Pereira, F., Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without A Priori Normality Assumptions, FEUP Int. Report, 2002.Google Scholar
  9. 9.
    Pereira, F. and Silva, G., Necessary Conditions of Optimality for Vector-valued Impulsive Control Problems, Syst. Control Lett., 2000, vol. 40, pp. 205–215.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Dykhta, V.A. and Samsonyuk, O.N., Optimal’noe impul’snoe upravlenie s prilozheniyami (Optimal Pulse Control with Applications), Moscow: Fizmatlit, 2000.Google Scholar
  11. 11.
    Miller, B.M., The Generalized Solutions of Nonlinear Optimization Problems with Impulse Control, SIAM J. Control Optim., 1996, vol. 34, pp. 1420–1440.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Osnovy teorii optimal’nogo upravleniya (Elements of Optimal Control Theory), Krotov, V.F., Ed., Moscow: Nauka, 1973.Google Scholar
  13. 13.
    Bressan, A. and Rampazzo, F., Impulsive Control Systems with Commutative Vector Fields, J. Optim. Theory Appl., 1991, vol. 71(1), pp. 67–83.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Zavalishchin, S.T. and Sesekin, A.N., Impul’snye protsessy: modeli i prilozheniya (Pulse Processes: Models and Applications), Moscow: Nauka, 1991.Google Scholar
  15. 15.
    Dykhta, V.A. and Derenko, N.V., Numerical Methods of Solving Impulse Control Problems based on the Generalized Stationary Condition, Tr. Vseross. Nauchn. Shk.: Komp’yuternaya logika, algebra i intellektnoe upravlenie. Problemy analiza ustoichivosti razvitiya i strategicheskoi stabil’nosti (Proc. All Russian Scientific School: Computer Logic, Algebra, and Intelligent Control. Problems of Analysis of Stability Development and Strategic Stability), Irkutsk, 1994, vol. 2, pp. 59–70.Google Scholar
  16. 16.
    Baturin, V.A. and Goncharova, E.V., An Algorithm for Impulsive Control Problems, Proc. 10th IEEE Mediterranean Conf. Control Automation, Lisbon, 2002.Google Scholar
  17. 17.
    Baturin, V.A. and Verkhozina, I.O., Relaxation Improvments Methods in Unbounded Linear Control Problems, Tr. 12 Baikal’skoi mezhdynar. konf. “Metody optimizatsii i ikh prilozheniya” (Proc. 12th Baikal Int. Conf. Optimization Methods and Their Application), Irkutsk, 2001, vol. 2, pp. 83–87.Google Scholar
  18. 18.
    Baturin, V.A. and Urbanovich, D.E., Priblizhennye metody optimal’nogo upravleniya, osnovannye na printsipe rasshireniya (Approximate Optimal Control Methods based on the Extension Principle), Novosibirsk: Nauka, 1997.Google Scholar
  19. 19.
    Gurman, V.I., Baturin, V.A., Danilina, E.V., et al., Novye metody uluchsheniya upravlyaemykh protsessov (New Refinement Methods for Control Processes), Novosibirsk: Nauka, 1987.Google Scholar
  20. 20.
    Pereira, F. and Sousa, J., On the Approximation of the Reachable Set Boundary, Proc. Controlo 2000, Guimaraes, 2000, pp. 642–646.Google Scholar
  21. 21.
    Baturin, V.A. and Goncharova, E.V., Refinement Methods based on Approximate Reachable Set Representation, Avtom. Telemekh., 1999, no. 11, pp. 19–29.Google Scholar
  22. 22.
    Silva, G.N. and Vinter, R.B., Necessary Conditions for Optimal Impulsive Control Problems, SIAM J. Control Optim., 1997, vol. 35, no. 6, pp. 1829–1846.CrossRefMathSciNetGoogle Scholar
  23. 23.
    Silva, G.N. and Vinter, R.B., Measure Differential Inclusions, J. Math. Anal. Appl., 1996, vol. 202, pp. 727–746.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Optimal Control Problems), Moscow: Nauka, 1985.Google Scholar
  25. 25.
    Gaishun, I.V., Vpolne razreshimye mnogomernye differentsial’nye uravneniya (Completely Solvable Multidimensional Differential Equations), Minsk: Nauka i Tekhnika, 1983.Google Scholar
  26. 26.
    Massel’, L.V., Boldyrev, E.A., Gornov, A.Yu., et al., Integratsiya informatsionnykh tekhnologii v systemnykh issledovaniyakh energetiki (Integration of Information Technologies into Energetics Systems Research), Novosibirsk: Nauka, 2003.Google Scholar
  27. 27.
    Gurman, V.I., Printsip rasshireniya v zadachakh optimal’nogo upravleniya (The Extension Principle in Optimal Control Problems), Moscow: Fizmatlit, 1997.Google Scholar

Copyright information

© Pleiades Publishing, Inc. 2006

Authors and Affiliations

  • V. A. Baturin
    • 1
  • E. V. Goncharova
    • 1
  • F. L. Pereira
    • 2
  • J. B. Sousa
    • 2
  1. 1.Institute of System Dynamics and Control Theory, Siberian BranchRussian Academy of SciencesIrkutskRussia
  2. 2.Institute for Systems and Robots, Engineering FacultyUniversity of PortoPortugal

Personalised recommendations