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.
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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)
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)
Allgöwer, F., Findeisen, R., Nagy, Z.K.: Nonlinear model predictive control: from theory to application. J. Chin. Inst. Chem. Engrs. 35, 299–315 (2004)
Allgöwer, F., Chen, H.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998)
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)
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)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Basel (1997)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equation. Oxford Science Publications (1998)
Chaturantabut, S., Sorensen, D.C.: Discrete Empirical Interpolation for NonLinear Model Reduction. SIAM J. Sci. Comput. 32, 2737–2764 (2010)
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer, Berlin (2000)
Evans, L.C.: Partial differential equations. American Math. Society, Providence, Rhode Island (2008)
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)
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)
Ghiglieri, J., Ulbrich, S.: Optimal Flow Control Based on POD and MPC and an Application to the Cancellation of Tollmien-Schlichting Waves. Submitted (2012)
Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Springer, London (2011)
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)
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
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)
Ito, K., Kunisch, K.: Receding horizon control for infinite dimensional systems. ESAIM, Control, Optim. Calc. Var. 8, 741–760 (2002)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)
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)
Pannocchia, G., Rawlings, J.B., Wright, S.J.: Conditions under which suboptimal nonlinear MPC is inherently robust. In: 18th IFAC World Congress, Milan (2011)
Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, LLC (2009)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York (1980)
Sachs, E.W., Schu, M.: A-priori error estimates for reduced order models in finance. ESAIM: Math. Model. Numer. Anal. 47, 449–469 (2013)
Singler, J.R.: New POD expressions, error bounds, and asymptotic results for reduced order models of parabolic PDEs. Submitted (2013)
Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I–II. Q. Appl. Math. XVL, 561–590 (1987)
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and applications. Graduate Studies in Mathematics, Vol. 112, American Mathematical Society (2010)
Volkwein, S.: Lagrange-SQP techniques for the control constrained optimal boundary control for the Burgers equation. Comput. Optim. Appl. 26(253), 284 (2003)
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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
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DOI: https://doi.org/10.1007/s10444-014-9381-0
Keywords
- Dynamic programming
- Nonlinear model predictive control
- Asymptotic stability
- Suboptimal control
- Proper orthogonal decomposition