Dynamic Mode Decomposition with Control
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 of state and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).
This illustration outlines the three major components of applying DMDc. The top panel describes the collection of data from either a numerical, laboratory, or historical data and the curation of the data in to matrices for the methods.
Note, the figure in the historical plot is the data representing pre-vaccination Measles cases in the UK normalized similar to that found in. The middle panel outlines the procedure for DMD and DMDc for comparison. The bottom panel illustrates two practical applications of DMDc.
Conclusion: Methods such as DMDc will play an increasing role in the analysis of large-scale datasets from complex systems. DMD has already been applied to a significant number of applications in the fluid dynamics community and is expanding to a variety of other applications like background subtraction in video processing. We believe DMDc is poised to similarly excel as a tool for a diverse set of engineering applied science applications where control of the complex system is important.