Data-driven discovery of Koopman eigenfunctions for control

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 presents the KRONIC framework that identifies Koopman eigenfunctions from data to transform nonlinear systems into a linear representation suitable for control.
• It employs techniques like Extended Dynamic Mode Decomposition, sparse regression, and power series expansions to approximate eigenfunctions and capture persistence in system dynamics.
• Applications to Hamiltonian systems and the double-gyre ocean model demonstrate enhanced control accuracy and robustness in managing complex nonlinear behavior.

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

Introduction to Koopman Operator and KRONIC

The paper addresses the challenge of reformulating nonlinear dynamics into a linear framework using the Koopman operator approach. The Koopman operator provides a linear embedding of nonlinear systems, enabling the application of linear control methods to handle complex nonlinear dynamics. This paper introduces the concept of Koopman Reduced Order Nonlinear Identification and Control (KRONIC), which is a methodology to identify and control nonlinear systems by leveraging data-driven discovery of Koopman eigenfunctions. These eigenfunctions provide intrinsic coordinates along which dynamics can be described linearly.

Identifying Koopman Eigenfunctions

Koopman eigenfunctions are crucial for constructing a linear representation of nonlinear dynamics. The identification of these eigenfunctions from data involves approximating the Koopman operator's action on a set of measurement functions. This paper discusses using techniques like Extended Dynamic Mode Decomposition (EDMD) for approximating Koopman eigenfunctions and addresses closure issues that arise when finite-dimensional approximations fail to fully capture the system's dynamics. The paper highlights the importance of validating these eigenfunctions to ensure they accurately describe the system.

Data-Driven Approximation Techniques

The paper presents methods for approximating Koopman eigenfunctions using data-driven regression techniques. These methods include using the infinitesimal generator of the Koopman operator combined with sparse regression and power series expansions to discover eigenfunctions that can be directly validated. This approach is useful for identifying "lightly damped" eigenfunctions, which correspond to nearly conserved quantities and persistent dynamics, making them valuable for control tasks.

Applications of KRONIC Framework

The KRONIC framework is applied to several complex systems, demonstrating its utility in both theoretical derivations and real-world applications:

1. Nonlinear Systems with Known Linear Embeddings: The framework effectively identifies and controls systems with known linear representations.
2. Hamiltonian Systems: These systems, characterized by energy conservation, are particularly suitable for KRONIC as the Hamiltonian functions qualify as Koopman eigenfunctions.
3. Double-Gyre Model for Ocean Mixing: The paper demonstrates the application to a high-dimensional system, showcasing how KRONIC can manage complex fluid dynamics scenarios.

Implementation and Control

KRONIC allows for a direct formulation of control strategies within the Koopman framework. By using the intrinsic eigenfunction coordinates, control actions can be derived linearly, enabling optimal and robust control techniques traditionally reserved for linear systems. The paper also discusses the practical considerations for implementing KRONIC, including handling computational requirements and extending the approach to handle non-affine control systems.

Conclusion

The paper concludes that leveraging Koopman eigenfunctions provides a promising pathway for reformulating and controlling nonlinear dynamics by utilizing a linear perspective. Future work aims to improve the accuracy of eigenfunction identification, address computational challenges, and explore applications in various domains such as turbulence control and machine interfaces, indicating a broad potential for the KRONIC framework in advancing nonlinear control methodologies.