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On reduced input-output dynamic mode decomposition

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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.

Keywords

Dynamic mode decomposition Model reduction System identification Cross Gramian Optimization 

Mathematics Subject Classification (2010)

93B30 90C99 

Notes

Acknowledgements

Open access funding provided by Max Planck Society. The authors are grateful for the helpful feedback and comments provided by the two anonymous referees.

References

  1. 1.
    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 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    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 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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 CrossRefGoogle Scholar
  6. 6.
    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 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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 CrossRefGoogle Scholar
  8. 8.
    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 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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 CrossRefMATHGoogle Scholar
  12. 12.
    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 CrossRefMATHGoogle Scholar
  13. 13.
    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 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Katayama, K.: Subspace Methods for System Identification. Communications and Control Engineering. Springer, London (2005).  https://doi.org/10.1007/1-84628-158-X CrossRefMATHGoogle Scholar
  15. 15.
    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 CrossRefMATHGoogle Scholar
  16. 16.
    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)
  17. 17.
    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
  18. 18.
    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 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    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 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mitchell, T.: GRANSO: GRadient-based Algorithm for Non-Smooth Optimization. http://timmitchell.com/software/GRANSO. See also [10]
  21. 21.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999).  https://doi.org/10.1007/b98874 CrossRefMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    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 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)
  25. 25.
    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 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010).  https://doi.org/10.1017/S0022112010001217 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    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 MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    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
  31. 31.
    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 MathSciNetCrossRefMATHGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

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