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Optimization Based Stabilization of Nonlinear Control Systems

  • Lars Grüne
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4818)

Abstract

We present a general framework for analysis and design of optimization based numerical feedback stabilization schemes utilizing ideas from relaxed dynamic programming. The application of the framework is illustrated for a set valued and graph theoretic offline optimization algorithm and for receding horizon online optimization.

Keywords

Model Predictive Control Short Path Problem Nonlinear Control System Feedback Stabilization Recede Horizon Control 
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.
    Bardi, M., Capuzzo Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1,2. Athena Scientific, Belmont, MA (1995)zbMATHGoogle Scholar
  3. 3.
    Camilli, F., Grüne, L., Wirth, F.: Control Lyapunov functions and Zubov’s method. In: SIAM J. Control Optim. (to appear, 2008)Google Scholar
  4. 4.
    Camilli, F., Grüne, L., Wirth, F.: A regularization of Zubov’s equation for robust domains of attraction. In: Treleaven, P.C., Nijman, A.J., de Bakker, J.W. (eds.) PARLE 1987. LNCS, vol. 258, pp. 277–290. Springer, Heidelberg (1987)Google Scholar
  5. 5.
    Grimm, G., Messina, M.J., Tuna, S.E., Teel, A.R.: Model predictive control: for want of a local control Lyapunov function, all is not lost. IEEE Trans. Automat. Control 50(5), 546–558 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Grüne, L.: Homogeneous state feedback stabilization of homogeneous systems. SIAM J. Control Optim. 38, 1288–1314 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grüne, L.: An adaptive grid scheme for the discrete Hamilton–Jacobi–Bellman equation. Numer. Math. 75(3), 319–337 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grüne, L., Junge, O.: A set oriented approach to optimal feedback stabilization. Syst. Control Lett. 54(2), 169–180 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grüne, L., Junge, O.: Approximately optimal nonlinear stabilization with preservation of the Lyapunov function property. In: Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, Louisiana (2007)Google Scholar
  10. 10.
    Grüne, L., Junge, O.: Global optimal control of perturbed systems. J. Optim. Theory Appl. 136 (to appear, 2008)Google Scholar
  11. 11.
    Grüne, L., Nešić, D.: Optimization based stabilization of sampled–data nonlinear systems via their approximate discrete–time models. SIAM J. Control Optim. 42, 98–122 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grüne, L., Nešić, D., Pannek, J.: Model predictive control for nonlinear sampled–data systems. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358, Springer, Heidelberg (1989)Google Scholar
  13. 13.
    Grüne, L., Rantzer, A.: On the infinite horizon performance of receding horizon controllers. In: Preprint, Universitat Bayreuth, IEEE Trans. Automat. Control (2006) (to appear, 2008) www.math.uni-bayreuth.de/~lgruene/publ/infhorrhc.html
  14. 14.
    Jadbabaie, A., Hauser, J.: On the stability of receding horizon control with a general terminal cost. IEEE Trans. Automat. Control 50(5), 674–678 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40(2), 293–300 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Junge, O., Osinga, H.M.: A set oriented approach to global optimal control. ESAIM Control Optim. Calc. Var. 10(2), 259–270 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kreisselmeier, G., Birkhölzer, T.: Numerical nonlinear regulator design. IEEE Trans. Autom. Control 39(1), 33–46 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lincoln, B., Rantzer, A.: Relaxing dynamic programming. IEEE Trans. Autom. Control 51, 1249–1260 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    von Lossow, M.: A min-max version of Dijkstra’s algorithm with application to perturbed optimal control problems. In: Proceedings of the GAMM Annual meeting, Zürich, Switzerland (to appear, 2007)Google Scholar
  20. 20.
    Rantzer, A.: Relaxed dynamic programming in switching systems. IEE Proceedings — Control Theory and Applications 153, 567–574 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tuna, E.S.: Optimal regulation of homogeneous systems. Automatica 41(11), 1879–1890 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Lars Grüne
    • 1
  1. 1.Mathematical InstituteUniversity of BayreuthBayreuthGermany

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