Advertisement

Explicit NMPC Using mp-QP Approximations of mp-NLP

  • Alexandra Grancharova
  • Tor Arne Johansen
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 429)

Abstract

A numerical algorithm for approximate multi-parametric nonlinear programming (mp-NLP) is developed. The algorithm locally approximates the mp-NLP with a multi-parametric quadratic program (mp-QP). This leads to an approximate mp-NLP solution that is composed from the solution of a number of mp-QP solutions. The method allows approximate solutions to nonlinear optimization problems to be computed as explicit piecewise linear functions of the problem parameters. In control applications such as nonlinear constrained model predictive control this allows efficient online implementation in terms of an explicit piecewise linear state feedback without any real-time optimization.

Keywords

Model Predictive Control Sequential Quadratic Programming Binary Search Tree Nonlinear Model Predictive Control Polyhedral Region 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bemporad, A., Filippi, C.: Suboptimal explicit RHC via approximate multiparametric quadratic programming. J. Optimization Theory and Applications 117, 9–38 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bemporad, A., Filippi, C.: An algorithm for approximate multiparametric convex programming. Computational Optimization and Applications 35, 87–108 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-dynamic programming. Athena Scientific, Belmont (1998)Google Scholar
  5. 5.
    Domínguez, L.F., Narciso, D.A., Pistikopoulos, E.N.: Recent advances in multiparametric nonlinear programming. Computers and Chemical Engineering 34, 707–716 (2010)CrossRefGoogle Scholar
  6. 6.
    Domínguez, L.F., Pistikopoulos, E.N.: Quadratic approximation algorithm for multiparametric nonlinear programming problems. Technical report. Imperial College London (2009)Google Scholar
  7. 7.
    Fiacco, A.V.: Introduction to sensitivity and stability analysis in nonlinear programming. Academic Press, Orlando (1983)zbMATHGoogle Scholar
  8. 8.
    Gravdahl, J.T., Egeland, O.: Compressor surge control using a close-coupled valve and backstepping. In: Proceedings of the American Control Conference, Albuquerque, NM, vol. 2, pp. 982–986 (1997)Google Scholar
  9. 9.
    Grancharova, A., Johansen, T.A.: Approximate explicit model predictive control incorporating heuristics. In: Proceedings of IEEE International Symposium on Computer Aided Control System Design, Glasgow, Scotland, U.K., pp. 92–97 (2002)Google Scholar
  10. 10.
    Grancharova, A., Johansen, T.A., Tøndel, P.: Computational aspects of approximate explicit nonlinear model predictive control. In: Findeisen, R., Allgöwer, F., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. LNCIS, vol. 358, pp. 181–192. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Greitzer, E.M.: Surge and rotating stall in axial flow compressors, part i: Theoretical compression system model. J. Engineering for Power 98, 190–198 (1976)CrossRefGoogle Scholar
  12. 12.
    Guddat, J., Guerra Vazquez, F., Jongen, H.T.: Parametric optimization: Singularities, pathfollowing and jumps. Wiley (1990)Google Scholar
  13. 13.
    Johansen, T.A.: On multi-parametric nonlinear programming and explicit nonlinear model predictive control. In: Proceedings of the IEEE Conference on Decision and Control, Las Vegas, NV, vol. 3, pp. 2768–2773 (2002)Google Scholar
  14. 14.
    Johansen, T.A., Grancharova, A.: Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Trans. Automatic Control 48, 810–815 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, London (1980)Google Scholar
  17. 17.
    Levitin, E.S.: Perturbation theory in mathematical programming. Wiley (1994)Google Scholar
  18. 18.
    Mangasarian, O.L., Rosen, J.B.: Inequalities for stochastic nonlinear programming problems. Operations Research 12, 143–154 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Narciso, D.: Developments in nonlinear multiparametric programming and control. PhD thesis. London, U.K. (2009)Google Scholar
  20. 20.
    Nocedal, J., Wright, S.J.: Numerical optimization. Springer, New York (1999)zbMATHCrossRefGoogle Scholar
  21. 21.
    Parisini, T., Zoppoli, R.: A receding-horizon regulator for nonlinear systems and a neural approximation. Automatica 31, 1443–1451 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Parisini, T., Zoppoli, R.: Neural approximations for multistage optimal control of nonlinear stochastic systems. IEEE Trans. on Automatic Control 41, 889–895 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Pistikopoulos, E.N., Georgiadis, M.C., Dua, V.: Multi-parametric programming: Theory, algorithms, and applications. Wiley-VCH (2007)Google Scholar
  24. 24.
    Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Mathematical Programming 70, 159–172 (1995)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rockafellar, R.T.: Convex analysis. Princeton University Press, New Jersey (1970)zbMATHGoogle Scholar
  26. 26.
    Spjøtvold, J., Kerrigan, E.C., Jones, C.N., Tøndel, P., Johansen, T.A.: On the facet-to-facet property of solutions to convex parametric quadratic programs. Automatica 42, 2204–2209 (2006)CrossRefGoogle Scholar
  27. 27.
    Tøndel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39, 743–749 (2003)CrossRefGoogle Scholar
  28. 28.
    Tøndel, P., Johansen, T.A., Bemporad, A.: An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica 39, 489–497 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of System Engineering and RoboticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Department of Engineering CyberneticsNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations