Abstract
In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, proper orthogonal decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model is obtained via a convex combination of these autonomous systems. Using a recent convergence result for extended dynamic mode decomposition (EDMD), convergence of the reduced objective function can be shown. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1-dimensional Burgers equation and the 2-dimensional Navier–Stokes equations as examples. Finally, an extension for online adaptivity is presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Albrecht, F., Haasdonk, B., Kaulmann, S., Ohlberger, M.: The localized reduced basis multiscale method. Proceedings of ALGORITHMY 2012, 393–403 (2012)
Arbabi, H., Korda, M., Mezic, I.: A data-driven Koopman model predictive control framework for nonlinear flows (2018). arXiv:1804.05291
Bellmann, R.E., Stuart, E.D.: Applied Dynamic Programming. Princeton University Press, Princeton (2015)
Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)
Bergmann, M., Cordier, L.: Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models. J. Comput. Phys. 227(16), 7813–7840 (2008)
Brunton, S.L., Brunton, B.W., Proctor, J.L., Kutz, J.N.: Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control. PLoS ONE 11(2), 1–19 (2016)
Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos 22 (2012)
Çimen, T.: State-dependent Riccati Equation (SDRE) control: a survey. In: IFAC Proceedings Volumes, vol. 41, pp. 3761–3775. IFAC (2008)
Egerstedt, M., Wardi, Y., Axelsson, H.: Transition-time optimization for switched-mode dynamical systems. IEEE Trans. Autom. Control 51(1), 110–115 (2006)
Egerstedt, M., Wardi, Y., Delmotte, F.: Optimal control of switching times in switched dynamical systems. In: 42nd IEEE International Conference on Decision and Control (CDC), pp. 2138–2143 (2003)
Elliott, D.: Bilinear Control Systems. Springer Science + Business Media, Berlin (2009)
Fahl, M.: Trust-region methods for flow control based on reduced order modelling. Ph.D. Thesis, University of Trier (2000)
Flaßkamp, K., Murphey, T., Ober-Blöbaum, S.: Discretized switching time optimization problems. In: 12th European Control Conference, pp. 3179–3184 (2013)
Grüne, L., Pannek, J.: Nonlinear Model Predictive Control, 2nd edn. Springer International Publishing, Berlin (2017)
Hanke, S., Peitz, S., Wallscheid, O., Klus, S., Böcker, J., Dellnitz, M.: Koopman operator based finite-set model predictive control for electrical drives (2018). arXiv:1804.00854
Hemati, M.S., Williams, M.O., Rowley, C.W.: Dynamic mode decomposition for large and streaming datasets. Phys. Fluids 26(111701), 1–6 (2014)
Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds.) Reduction of Large-Scale Systems, vol. 45, pp. 261–306. Springer, Berlin (2005)
Jasak, H., Jemcov, A., Tukovic, Z.: OpenFOAM : A C++ Library for Complex Physics Simulations. In: International Workshop on Coupled Methods in Numerical Dynamics, pp. 1–20 (2007)
Kaiser, E., Kutz, J.N., Brunton, S.L.: Data-driven discovery of Koopman eigenfunctions for control (2017). arXiv:1707.0114
Klus, S., Gelß, P., Peitz, S., Schütte, C.: Tensor-based dynamic mode decomposition. Nonlinearity 31(7), 3359–3380 (2018)
Klus, S., Koltai, P., Schütte, C.: On the numerical approximation of the Perron–Frobenius and Koopman operator. J. Comput. Dyn. 3(1), 51–79 (2016)
Klus, S., Nüske, F., Koltai, P., Wu, H., Kevrekidis, I., Schütte, C., Noé, F.: Data-driven model reduction and transfer operator approximation. J. Nonlinear Sci. 28(3), 985–1010 (2018)
Koopman, B.O.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)
Korda, M., Mezić, I.: Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica 93, 149–160 (2018)
Korda, M., Mezić, I.: On convergence of extended dynamic mode decomposition to the Koopman operator. J. Nonlinear Sci. 28(2), 687–710 (2018)
Kunisch, K., Volkwein, S.: Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102(2), 345–371 (1999)
Lasota, A., Mackey, M.C.: Chaos, fractals, and noise: stochastic aspects of dynamics. Applied Mathematical Sciences, vol. 97, 2nd edn. Springer, Berlin (1994)
Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G.: Model order reduction in fluid dynamics: challenges and perspectives. In: Quarteroni, A., Rozza, G. (eds.) Reduced Order Methods for Modeling and Computational Reduction, pp. 235–273. Springer, Cham (2014)
Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325 (2005)
Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357–378 (2013)
Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Phys. D Nonlinear Phenom. 197, 101–133 (2004)
Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer Science & Business Media, Berlin (2006)
Pardalos, P.M., Yatsenko, V.: Optimization and Control of Bilinear Systems. Springer, Berlin (2008)
Peitz, S.: Controlling nonlinear PDEs using low-dimensional bilinear approximations obtained from data (2018). arXiv:1801.06419
Peitz, S., Dellnitz, M.: A survey of recent trends in multiobjective optimal control surrogate models, feedback control and objective reduction. Math. Comput. Appl. 23(2) (2018)
Peitz, S., Klus, S.: Koopman operator-based model reduction for switched-system control of PDEs. Automatica 106, 184–191 (2019)
Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15(1), 142–161 (2015)
Proctor, J.L., Brunton, S.L., Kutz, J.N.: Generalizing Koopman Theory to allow for inputs and control. SIAM J. Appl. Dyn. Syst. 17(1), 909–930 (2018)
Qian, E., Grepl, M., Veroy, K., Willcox, K.: A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization. ACDL Technical Report TR16-3 (2016)
Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos 15(3), 997–1013 (2005)
Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)
Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)
Sirovich, L.: Turbulence and the dynamics of coherent structures part I: coherent structures. Q. Appl. Math. 45(3), 561–571 (1987)
Stellato, B., Ober-Blöbaum, S., Goulart, P.J.: Optimal control of switching times in switched linear systems. In: IEEE 55th Conference on Decision and Control, pp. 7228–7233 (2016)
Stellato, B., Ober-Blöbaum, S., Goulart, P.J.: Second-order switching time optimization for switched dynamical systems. IEEE Trans. Autom. Control. 62(10), 5407–5414 (2017)
Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44(1), 83–115 (2009)
Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014)
Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307–1346 (2015)
Zhu, F., Antsaklis, P.J.: Optimal control of hybrid switched systems: a brief survey. Discret. Event Dyn. Syst. 25(3), 345–364 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Peitz, S., Klus, S. (2020). Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-35713-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35712-2
Online ISBN: 978-3-030-35713-9
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)