Generalizing Koopman Theory to Allow for Inputs and Control

By Joshua L. Proctor, Steven L. Brunton, J. Nathan Kutz. SIAM Journal on Applied Dynamical Systems, 2016

425 citations recorded by the source corpus.

We develop a new generalization of Koopman operator theory that incorporates the e ects of inputs and control. Koopman spectral analysis is a theoretical tool for the analysis of nonlinear dynamical systems. Moreover, Koopman is intimately connected to dynamic mode decomposition (DMD), a method that discovers coherent, spatio-temporal modes from data, connects local-linear analysis to nonlinear operator theory, and importantly creates an equation-free architecture for the study of complex systems. For actuated systems, standard Koopman analysis and DMD are incapable of producing input-output models; moreover, the dynamics and the modes will be corrupted by external forcing. Our new theoretical developments extend Koopman operator theory to allow for systems with nonlinear input-output characteristics. We show how this generalization is rigorously connected to a recent development called dynamic mode decomposition with control. We demonstrate this new theory on nonlinear dynamical systems, including a standard susceptible-infectious-recovered model with relevance to the analysis of infectious disease data with mass vaccination (actuation).

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9d6fe89150401d69a90ae10dc9c4135ab406903c

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Citations in the corpus

Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems (2019, background): “The definition of the Koopman operator in the controlled setting depends on the type of input.(29) In this manuscript, we consider only closed-loop control and so the input is generated using a state-dependent control law, u = h(x).”
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).”
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].”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Recently, Koopman analysis has been used to develop nonlinear esti-5 mators [79], [80] and controllers [81], although establishing rigorous connections to control theory is an ongoing effort [82], [83], [84].”
Resurrecting Recurrent Neural Networks for Long Sequences (2023, background): “Several options exist for extending Koopman operator theory to systems with inputs (Kaiser et al., 2021; Korda and Mezić, 2020; Proctor et al., 2018; Surana, 2016).”
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition (2017, background): “[19, 20] discussed Koopman analysis of systems with control signals.”