Dynamic Mode Decomposition with Control

By Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz. SIAM Journal on Applied Dynamical Systems, 2014

1,067 citations recorded by the source corpus.

We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. W...

Subfields: Civil and Structural Engineering, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1556de28dd5a7033ebca57b78e923ee019e40680

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Citations in the corpus

Formulas for Data-Driven Control: Stabilization, Optimality, and Robustness (2019, methodology): “’s fundamental lemma and the dynamic mode decomposition [31], a numerical procedure for recovering state and control matrices of a linear system from its trajectories.”
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, methodology): “For a related work on extending Koopman operator methods to controlled dynamical systems, see [2, 18, 19, 24].”
Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders (2020, background): “In order to reduce costs, there has recently been signicant interest in the reduced-order modeling (ROM) of such systems [1, 2, 3, 4, 5, 6, 7]. As such, this eld nds extensive application in control [8, 9, 10, 11], multi-delity optimization [12], uncertainty quantication [13, 14] and data-assimilati”
Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems (2017, methodology): “Various algorithms have extended these methods using dynamic and extended dynamic mode decomposition, both for autonomous and controlled systems [7], [8], [9].”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, background): “[19], [20] discussed Koopman analysis of systems with control signals.”
Chaos as an intermittently forced linear system (2016, methodology): “Augmenting DMD with nonlinear measurements may enrich the model44, but there is no guarantee that the resulting models will be closed under the Koopman operator39.”
Deep Dynamical Modeling and Control of Unsteady Fluid Flows (2018, methodology): “presented several methods for estimating A and B given matrices X̃ and Ỹ [28].”
Modal Analysis of Fluid Flows: An Overview (2017, background): “DMD has recently been extended to handle such systems [124].”
Behavioral systems theory in data-driven analysis, signal processing, and control (2021, background): “Independently, this approach is also known as dynamic mode decomposition (DMD) in the nonlinear dynamics and fluids communities (Proctor et al., 2016).”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, methodology): “MD if formulated in discrete-time, with linear library elements in , and without a sparsity-promoting L1 penalty term, i.e. =0. DMD was recently extended to include actuation inputs by Proctor et al [32], to disambiguate the effect of internal dynamics and control. In dynamic mode decomposition with”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, positive): “The theory also allows for critical enabling theoretical augmentations that can take advantage of compression and sparsity [22, 7, 16], multiresolution/multiscale phenomenon [27], denoising [9, 18], data fusion [52], extended and kernel DMD [51, 50], and control [36].”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, methodology): “The initial success of this algorithm in the context of fluid mechanics motivated an ongoing line of research on data-driven analysis of high-dimensional and complex systems using Koopman operator theory and, consequently, a large number of DMD-type algorithms have been proposed in recent years for ”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “Similar to how DMD was extended to include inputs and control, Koopman analysis has recently been extended to include inputs and control (Proctor et al., 2016b).”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “The DMD algorithm has also been extended to include sparsity and compressed sensing [23, 40, 46, 55, 113], actuation and control [92], multi-resolution analysis [69], de-noising [33, 50], and streaming variants [51].”
Multiresolution Dynamic Mode Decomposition (2015, background): “DMD Control: A second important direction revolves around recent innovations of the DMD with control [24], which is capable of disambiguating between the underlying dynamics and the eﬀects of actuation, resulting in accurate input-output models.”