Abstract
The identification of reduced-order models from high-dimensional data is a challenging task, and even more so if the identified system should not only be suitable for a certain data set, but generally approximate the input-output behavior of the data source. In this work, we consider the input-output dynamic mode decomposition method for system identification. We compare excitation approaches for the data-driven identification process and describe an optimization-based stabilization strategy for the identified systems.
References
Alla, A., Kutz, J.N.: Nonlinear model order reduction via dynamic mode decomposition. SIAM J. Sci. Comput. 39(5), B778–B796 (2017). https://doi.org/10.1137/16M1059308
Amsallem, D., Farhat, C.: Stabilization of projection-based reduced-order models. Numer. Methods Eng. 91(4), 358–377 (2012). https://doi.org/10.1002/nme.4274
Annoni, J., Gebraad, P., Seiler, P.: Wind farm flow modeling using input-output dynamic mode decomposition. In: American Control Conference (ACC), pp. 506–512 (2016). https://doi.org/10.1109/ACC.2016.7524964
Annoni, J., Seiler, P.: A method to construct reduced-order parameter-varying models. Int. J. Robust Nonlinear Control 27(4), 582–597 (2017). https://doi.org/10.1002/rnc.3586
Antoulas, A.C.: Approximation of Large-Scale dynamical systems, Adv. Des. Control, vol. 6. Society of Industrial and Applied Mathematics Publications, Philadelphia (2005). https://doi.org/10.1137/1.9780898718713
Aström, K.J., Eykhoff, P.: System identification – a survey. Automatica 7(2), 123–162 (1971). https://doi.org/10.1016/0005-1098(71)90059-8
Brunton, B.W., Johnson, L.A., Ojemann, J.G., Kutz, J.N.: Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition. J. Neurosci. Methods 258, 1–15 (2016). https://doi.org/10.1016/j.jneumeth.2015.10.010
Burke, J.V., Overton, M.L.: Variational analysis of non-Lipschitz spectral functions. Math. Program. 90(2, Ser. A), 317–352 (2001). https://doi.org/10.1007/s102080010008
Chen, K.K., Tu, J.H., Rowley, R.W.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. Nonlinear Sci. 22(6), 887–915 (2012). https://doi.org/10.1007/s00332-012-9130-9
Curtis, F.E., Mitchell, T., Overton, M.L.: A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles. Optim. Methods Softw. 32(1), 148–181 (2017). https://doi.org/10.1080/10556788.2016.1208749
Fernando, K.V., Nicholson, H.: On the structure of balanced and other principal representations of SISO systems. IEEE Trans. Autom. Control 28(2), 228–231 (1983). https://doi.org/10.1109/TAC.1983.1103195
Holmes, P., Lumley, J.L., Berkooz, G., Rowley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9780511919701
Ionescu, T.C., Fujimoto, K., Scherpen, J.M.A.: Singular value analysis of nonlinear symmetric systems. IEEE Trans. Autom. Control 56(9), 2073–2086 (2011). https://doi.org/10.1109/TAC.2011.2126630
Katayama, K.: Subspace Methods for System Identification. Communications and Control Engineering. Springer, London (2005). https://doi.org/10.1007/1-84628-158-X
Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–381 (1931). http://www.pnas.org/content/17/5/315.full.pdf
Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. Society of Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9781611974508 (2016)
Lall, S., Marsden, J.E., Glavaški, S.: Empirical model reduction of controlled nonlinear systems. In: Proceedings of the IFAC World Congress, vol. F, pp. 473–478 (1999). https://doi.org/10.1016/S1474-6670(17)56442-3
Lewis, A.S., Overton, M.L.: Nonsmooth optimization via quasi-Newton methods. Math. Program. 141(1–2, Ser. A), 135–163 (2013). https://doi.org/10.1007/s10107-012-0514-2
Mezic, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309–325 (2005). https://doi.org/10.1007/s11071-005-2824-x
Mitchell, T.: GRANSO: GRadient-based Algorithm for Non-Smooth Optimization. http://timmitchell.com/software/GRANSO. See also [10]
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999). https://doi.org/10.1007/b98874
Oku, H., Fujii, T.: Direct subspace model identification of LTI systems operating in closed-loop. In: 43Rd IEEE Conference on Decision and Control, pp. 2219–2224 (2004). https://doi.org/10.1109/CDC.2004.1430378
Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15(1), 142–161 (2016). https://doi.org/10.1137/15M1013857
Proctor, J.L., Brunton, S.L., Kutz, J.N.: Generalizing Koopman Theory to Allow for Inputs and Control. arXiv:1602.07647, Cornell University. 1602.07647. Math.OC (2016)
Rowley, C.W., Dawson, S.T.M.: Model reduction for flow analysis and control. Annu. Rev. Fluid Mech. 49, 387–417 (2017). https://doi.org/10.1146/annurev-fluid-010816-060042
Rowley, C.W., Mezic, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–1127 (2009). https://doi.org/10.1017/S0022112009992059
Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/S0022112010001217
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). https://doi.org/10.3934/jcd.2014.1.391
Van Den Hof, P.M.J., Schrama, R.J.P.: An indirect method for transfer function estimation from closed loop data. Automatica 29(6), 1523–1527 (1993). https://doi.org/10.1016/0005-1098(93)90015-L
Van Overschee, P., De Moor, B.: N4SID: Numerical algorithms for state space subspace system identification. In: IFAC Proceedings Volumes, vol. 26, pp. 55–58 (1993). https://doi.org/10.1016/S1474-6670(17)48221-8
Viberg, M.: Subspace-based methods for the identification of linear time-invariant systems. Automatica 31(12), 1835–1851 (1995). https://doi.org/10.1016/0005-1098(95)00107-5
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Open access funding provided by Max Planck Society. The authors are grateful for the helpful feedback and comments provided by the two anonymous referees.
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Communicated by: Karsten Urban
Supported by the German Federal Ministry for Economic Affairs and Energy (BMWi), in the joint project: “MathEnergy – Mathematical Key Technologies for Evolving Energy Grids”, sub-project: Model Order Reduction (Grant number: 0324019B).
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Benner, P., Himpe, C. & Mitchell, T. On reduced input-output dynamic mode decomposition. Adv Comput Math 44, 1751–1768 (2018). https://doi.org/10.1007/s10444-018-9592-x
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DOI: https://doi.org/10.1007/s10444-018-9592-x
Keywords
- Dynamic mode decomposition
- Model reduction
- System identification
- Cross Gramian
- Optimization
Mathematics Subject Classification (2010)
- 93B30
- 90C99