Skip to main content
Log in

Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this article a stabilizing feedback control is computed for a semilinear parabolic partial differential equation utilizing a nonlinear model predictive (NMPC) method. In each level of the NMPC algorithm the finite time horizon open loop problem is solved by a reduced-order strategy based on proper orthogonal decomposition (POD). A stability analysis is derived for the combined POD-NMPC algorithm so that the lengths of the finite time horizons are chosen in order to ensure the asymptotic stability of the computed feedback controls. The proposed method is successfully tested by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alla, A., Falcone, M.: An adaptive POD approximation method for the control of advection-diffusion equations. In: Kunisch, K., Bredies, K., Clason, C., von Winckel, G. (eds.) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol. 164, pp. 1–17. Birkhäuser, Basel (2013)

    Chapter  Google Scholar 

  2. Alla, A., Falcone, M.: A time-adaptive pod method for optimal control problems. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, pp 245–250 (2013)

  3. Allgöwer, F., Findeisen, R., Nagy, Z.K.: Nonlinear model predictive control: from theory to application. J. Chin. Inst. Chem. Engrs. 35, 299–315 (2004)

    Google Scholar 

  4. Allgöwer, F., Chen, H.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998)

    Article  MATH  Google Scholar 

  5. Altmüller, N., Grüne, L.: A comparative stability analysis of Neumann and Dirichlet boundary MPC for the heat equation. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, pp. 1161–1166 (2013)

  6. Altmüller, N., Grüne, L., Worthmann, K.: Receding horizon optimal control for the wave equation. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 3427–3432, Atlanta (2010)

  7. Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  8. Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equation. Oxford Science Publications (1998)

  9. Chaturantabut, S., Sorensen, D.C.: Discrete Empirical Interpolation for NonLinear Model Reduction. SIAM J. Sci. Comput. 32, 2737–2764 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer, Berlin (2000)

    Book  Google Scholar 

  11. Evans, L.C.: Partial differential equations. American Math. Society, Providence, Rhode Island (2008)

  12. Findeisen, R., Allgöwer, F.: An introduction to nonlinear model predictive control. In: Scherer, C.W., Schumacher, J.M. (eds.) Summerschool on The Impact of Optimization in Control. Dutch Institute of Systems and Control, DISC (2001)

  13. Findeisen, R., Allgöwer, F.: The quasi-infinte horizon approach to nonlinear model predictive control. In: Zinober, A., Owens, D. (eds.) Nonlinear and Adaptive Control, Lecture Notes in Control and Information Sciences, pp. 89–105. Springer-Verlag, Berlin (2002)

    Google Scholar 

  14. Ghiglieri, J., Ulbrich, S.: Optimal Flow Control Based on POD and MPC and an Application to the Cancellation of Tollmien-Schlichting Waves. Submitted (2012)

  15. Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Springer, London (2011)

    Book  MATH  Google Scholar 

  16. Grüne, L., Panneck, J., Seehafer, M., Worthmann, K.: Analysis of unconstrained nonlinear MPC schemes with time varying control horizon. SIAM J. Control. Optim. 48, 4938–4962 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gubisch, M., Volkwein, S.: Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control. Submitted (2013). http://kops.ub.uni-konstanz.de/handle/urn:nbn:de:bsz:352-250378

  18. Holmes, P., Lumley, J.L., Berkooz, G., Romley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, 2nd edn. Cambridge University Press (2012)

  19. Ito, K., Kunisch, K.: Receding horizon control for infinite dimensional systems. ESAIM, Control, Optim. Calc. Var. 8, 741–760 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kunisch, K., Volkwein, S., Xie, L.: HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3, 701–722 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pannocchia, G., Rawlings, J.B., Wright, S.J.: Conditions under which suboptimal nonlinear MPC is inherently robust. In: 18th IFAC World Congress, Milan (2011)

  24. Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, LLC (2009)

  25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)

    MATH  Google Scholar 

  26. Sachs, E.W., Schu, M.: A-priori error estimates for reduced order models in finance. ESAIM: Math. Model. Numer. Anal. 47, 449–469 (2013)

    Article  MathSciNet  Google Scholar 

  27. Singler, J.R.: New POD expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. Submitted (2013)

  28. Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I–II. Q. Appl. Math. XVL, 561–590 (1987)

    MathSciNet  Google Scholar 

  29. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and applications. Graduate Studies in Mathematics, Vol. 112, American Mathematical Society (2010)

  30. Volkwein, S.: Lagrange-SQP techniques for the control constrained optimal boundary control for the Burgers equation. Comput. Optim. Appl. 26(253), 284 (2003)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Stefan Volkwein.

Additional information

Communicated by: Editors of Special Issue on MoRePas

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alla, A., Volkwein, S. Asymptotic stability of POD based model predictive control for a semilinear parabolic PDE. Adv Comput Math 41, 1073–1102 (2015). https://doi.org/10.1007/s10444-014-9381-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9381-0

Keywords

Mathematics Subject Classifications (2010)

Navigation