A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

  • Davide M. Raimondo
  • Stefano Riverso
  • Sean Summers
  • Colin N. Jones
  • John Lygeros
  • Manfred Morari
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 417)

Abstract

In this chapter an algorithm for nonlinear explicit model predictive control is presented. A low complexity receding horizon control law is obtained by approximating the optimal control law using multiscale basis function approximation. Simultaneously, feasibility and stability of the approximate control law is ensured through the computation of a capture basin (region of attraction) for the closed-loop system. In a previous work, interval methods were used to construct the capture basin (feasible region), yet this approach suffered due to slow computation times and high grid complexity.

In this chapter, we suggest an alternative to interval analysis based on zonotopes. The suggested method significantly reduces the complexity of the combined function approximation and verification procedure through the use of DC (difference of convex) programming, and recursive splitting. The result is a multiscale function approximation method with improved computational efficiency for fast nonlinear explicit model predictive control with guaranteed stability and constraint satisfaction.

Keywords

Expense Cyan Bravo Gridding 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adjiman, C., Floudas, C.: Rigorous convex underestimators for general twice-differentiable problems. Journal of Global Optimization 9(1), 23–40 (1996)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alamo, T., Bravo, J., Camacho, E.F.: Guaranteed state estimation by zonotopes. Automatica 41(6), 1035–1043 (2005)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Alamo, T., Bravo, J., Redondo, M., Camacho, E.: A set-membership state estimation algorithm based on DC programming. Automatica 44(1), 216–224 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Aubin, J.: Viability Theory. Birkhauser Boston Inc., Cambridge (1991)MATHGoogle Scholar
  5. 5.
    Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Berz, M., Makino, K.: Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by shrink wrapping. International Journal of Differential Equations and Applications 10(4), 385–403 (2005)MathSciNetGoogle Scholar
  7. 7.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994)MATHCrossRefGoogle Scholar
  8. 8.
    Canale, M., Fagiano, L., Milanese, M.: Set membership approximation theory for fast implementation of model predictive control laws. Automatica 45(1) (2009)Google Scholar
  9. 9.
    Canale, M., Fagiano, L., Milanese, M., Novara, C.: Set membership approximations of predictive control laws: The tradeoff between accuracy and complexity. In: European Control Conference (ECC 2009), Budapest, Hungary (2009)Google Scholar
  10. 10.
    Chen, F.L., Allgöwer, F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34(10), 1205–1218 (1998)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Combastel, C.: A state bounding observer for uncertain non-linear continuous-time systems based on zonotopes. In: Proc. 44th IEEE Conf. Decision and Control (CDC 2005), and 2005 European Control Conf (CDC-ECC 2005), Seville, Spain, pp. 7228–7234 (2005)Google Scholar
  12. 12.
    Delanoue, N., Jaulin, L., Hardouin, L., Lhommeau, M.: Guaranteed characterization of capture basins of nonlinear state-space systems. In: Informatics in Control, Automation and Robotics: Selected Papers from the International Conference on Informatics in Control, Automation and Robotics 2007 (ICINCO 2007), pp. 265–272. Springer, Heidelberg (2008)Google Scholar
  13. 13.
    Donzé, A., Clermont, G., Legay, A., Langmead, C.: Parameter synthesis in nonlinear dynamical systems: Application to systems biology. In: Proc. 13th Annual Int. Conf. Research in Computational Molecular Biology, pp. 155–169 (2009)Google Scholar
  14. 14.
  15. 15.
    Feednetback, www.feednetback.eu
  16. 16.
    Fukuda, K.: From the zonotope construction to the Minkowski addition of convex polytopes. Journal of Symbolic Computation 38(4), 1261–1272 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Horst, R., Thoai, N.: DC programming: Overview. Journal of Optimization Theory and Applications 103(1), 1–43 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Houska, B., Ferreau, H., Diehl, M.: ACADO toolkit—An open-source framework for automatic control and dynamic optimization. Optimal Control Applications and Methods 32(3), 298–312 (2011)MATHCrossRefGoogle Scholar
  19. 19.
    Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40(2), 293–300 (2004)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Johansen, T.A., Petersen, I., Slupphaug, O.: On explicit suboptimal LQR with state and input constraints. In: Proc. 39th IEEE Conf. Decision and Control (CDC 2000), Sydney, Australia, pp. 662–667 (2000)Google Scholar
  21. 21.
    Kearfott, R.: Rigorous global search: Continuous problems, vol. 13. Kluwer Academic Publishers, Dordrecht (1996)MATHGoogle Scholar
  22. 22.
    Kuhn, W.: Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61(1), 47–67 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kvasnica, M., Grieder, P., Baotić, M.: Multi-Parametric Toolbox, MPT (2004), http://control.ee.ethz.ch/mpt/
  24. 24.
    Makino, K.: Rigorous analysis of nonlinear motion in particle accelerators. Ph.D. thesis, Michigan State University (1998)Google Scholar
  25. 25.
    Moore, R.E.: Interval Analysis, vol. 60. Prentice Hall, Englewood Cliffs (1966)MATHGoogle Scholar
  26. 26.
    Pin, G., Filippo, M., Pellegrino, A., Parisini, T.: Approximate off-line receding horizon control of constrained nonlinear discrete-time systems. In: Proc. European Control Conference, Budapest, Hungary, pp. 2420–2431 (2009)Google Scholar
  27. 27.
    Rump, S.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999), http://www.ti3.tu-harburg.de/rump/ Google Scholar
  28. 28.
    Saint-Pierre, P.: Approximation of the viability kernel. Applied Mathematics and Optimization 29(2), 109–187 (1994)MathSciNetGoogle Scholar
  29. 29.
    Seron, M.M., Goodwin, G.C., Dona, J.A.D.: Geometry of model predictive control including bounded and stochastic disturbances under state and input constraints. Tech. rep., University of Newcastle, Newcastle, Australia (2000)Google Scholar
  30. 30.
    Summers, S., Jones, C.N., Lygeros, J., Morari, M.: A multiscale approximation scheme for explicit model predictive control with stability, feasibility, and performance guarantees. In: Proc. IEEE Conf. Decision and Control (CDC 2009), Shanghai, China, pp. 6328–6332 (2009)Google Scholar
  31. 31.
    Summers, S., Raimondo, D.M., Jones, C.N., Lygeros, J., Morari, M.: Fast explicit nonlinear model predictive control via multiresolution function approximation with guaranteed stability. In: Proc. 8th IFAC Symp. Nonlinear Control Systems (NOLCOS 2010), Bologna, Italy (2010), http://control.ee.ethz.ch/index.cgi?page=publications;action=details;id=3557
  32. 32.
    Tuy, H.: DC optimization: theory, methods and algorithms. In: Handbook of global optimization, pp. 149–216 (1995)Google Scholar
  33. 33.
    Wan, J., Vehi, J., Luo, N.: A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality. Journal of Global Optimization 44(3), 395–407 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer London 2012

Authors and Affiliations

  • Davide M. Raimondo
    • 1
  • Stefano Riverso
    • 2
  • Sean Summers
    • 1
  • Colin N. Jones
    • 1
  • John Lygeros
    • 1
  • Manfred Morari
    • 1
  1. 1.Automatic Control LaboratoryETHZürichSwitzerland
  2. 2.Dipartimento di Informatica e SistemisticaLaboratorio di Identificazione e Controllo dei Sistemi Dinamici, Universit‘a degli Studi di PaviaPaviaItaly

Personalised recommendations