Data-driven discovery of Koopman eigenfunctions for control (1707.01146v4)

Abstract: Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including 1) a nonlinear system with a known linear embedding, 2) a variety of Hamiltonian systems, and 3) a high-dimensional double-gyre model for ocean mixing.

• The paper introduces the KRONIC framework that transforms nonlinear dynamics into linear control models by leveraging validated Koopman eigenfunctions.
• It employs data-driven techniques, including regression and series expansions, to accurately compute and validate the essential Koopman eigenfunctions.
• Numerical results show KRONIC reduces predictive errors by up to 3 to 5 times over traditional methods, demonstrating scalability and robustness.

Essay on "Data-driven Discovery of Koopman Eigenfunctions for Control"

The paper "Data-driven discovery of Koopman eigenfunctions for control" describes a novel approach to transforming nonlinear dynamical systems into a form that is more tractable for control using linear systems theory. This method leverages the Koopman operator framework, which provides a linear representation of nonlinear dynamics via Koopman eigenfunctions. The focus of the paper is on improving predictive capabilities and control by embedding nonlinear dynamics in reduced-order models using validated Koopman eigenfunctions. The paper introduces the proposed Koopman Reduced Order Nonlinear Identification and Control (KRONIC) framework, which is demonstrated across a series of dynamic systems.

At the core of the paper is the observation that, while efforts have been made to use finite-dimensional approximations of the Koopman operator for controlling nonlinear dynamics, existing approaches often face closure issues. The authors address this by computing Koopman-invariant subspaces formed by eigenfunctions, which promise improved predictive power.

Intrinsically tied to the Koopman framework, this work requires data-driven techniques to approximate eigenfunctions, leveraging various regression methods and power series expansions. The authors illustrate that an essential part of this procedure is validating the eigenfunctions post-discovery to ensure fidelity. Noteworthy is the treatment of lightly damped eigenfunctions, which hold particular significance for control as they relate to persistent dynamics.

In terms of control, the paper shows how control can be represented directly within the eigenfunction coordinates, facilitating the potential utilization of optimal and robust control tactics commonly applied to linear systems. This transformation gives rise to KRONIC, a control architecture fabricated on these intrinsic coordinates.

The practical implications of the method are demonstrated with examples across several complex systems. These include a reduction problem for a nonlinear system with existing linear embedding solutions, an array of Hamiltonian systems characterizing energy conservation, and a model of ocean mixing represented through the high-dimensional double-gyre system.

Numerical results offer compelling evidence: KRONIC shows superior predictive performance thanks to its structure that inherently models the dynamics within a Koopman-invariant subspace, sometimes outperforming traditional approaches by a margin of 3 to 5 times in predictive error reduction in noisy environments. Additionally, KRONIC capitalizes on scalability and flexibility, especially when embedded within a larger machine learning and data-driven ecosystem.

The paper speculates on the potential this approach has to offer in the broad domain of artificial intelligence and control engineering. By effectively bridging nonlinear dynamical systems to linear control architectures through eigenfunction representation, the work opens promising avenues in areas like turbulence management, robotic control systems, and biointerfaces, to name a few.

Further developments could refine the eigenfunction discovery processes, possibly incorporating more advanced machine learning techniques and allowing for broader application scenarios, underscoring a neat integration with artificial intelligence's unfolding capabilities. However, the authors acknowledge current challenges in ensuring model stability and control applicability in varied unknown terrains, pointing toward future research in uncertainty quantification and robust nonlinear system control.