Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control (1510.03007v2)

Abstract: In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace. The Koopman operator is an infinite-dimensional linear operator that evolves observable functions of the state-space of a dynamical system [Koopman 1931, PNAS]. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing nonlinear observable functions to form an invariant subspace where it is possible to obtain linear models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis using a new algorithm to determine terms in a dynamical system by sparse regression of the data in a nonlinear function space [Brunton et al. 2015, arxiv]; we show how this algorithm is related to DMD. Finally, we demonstrate how to design optimal control laws for nonlinear systems using techniques from linear optimal control on Koopman invariant subspaces.

• The paper introduces a finite-dimensional linear representation of nonlinear systems by restricting the Koopman operator to a carefully chosen invariant subspace of observables.
• It critiques conventional DMD methods and proposes using SINDy to select optimal observable functions that facilitate effective linear control techniques.
• The study demonstrates that employing Koopman Optimal Control can outperform traditional control approaches when the invariant subspace accurately captures the system's dynamics.

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

The paper provides an exploration of finite-dimensional linear representations of nonlinear dynamical systems by employing the Koopman operator framework. Central to this work is the restriction of the infinite-dimensional Koopman operator to an invariant subspace spanned by strategically chosen observable functions. The paper chiefly investigates how these subspaces can facilitate the application of optimal linear control techniques to nonlinear problems.

Key Findings and Methodologies

• Koopman Operator Framework: The Koopman operator offers an operator-theoretic lens to paper dynamical systems by evolving observable functions of the system's state. Traditionally, it is an infinite-dimensional linear operator, but this work focuses on finding finite-dimensional linear representations.
• Dynamic Mode Decomposition (DMD): Dominant terms in Koopman expansions are generally computed using DMD, which relies on linear measurements. The paper critiques this approach as potentially restrictive for nonlinear systems, suggesting a need for more sophisticated selection of observable functions to define invariant subspaces suitable for control applications.
• Observable Functions and Control: A significant portion of the paper explores selecting the right observable functions for Koopman analysis to enable effective control strategies like those used in a Linear Quadratic Regulator (LQR). The authors reveal that incorporating states in the observable subspace is practicable only when there exists a single isolated fixed point. In cases with multiple fixed points or complicated attractors, obtaining a finite-dimensional linear representation that includes the original states is generally infeasible.
• Sparse Identification of Nonlinear Dynamics (SINDy): The paper introduces a data-driven strategy leveraging SINDy to identify relevant observable functions for Koopman analysis. SINDy uses $\ell_1$-regularized regression in a nonlinear function space to discover essential terms in a system’s dynamics, thereby aiding in forming a Koopman-invariant subspace.
• Koopman Optimal Control (KOOC): The paper presents an illustration of using linear control frameworks within the Koopman operator setting, particularly for nonlinear systems. By employing linear optimal control techniques on the Koopman linear system, the authors derive control laws that outperform traditional strategies by effectively inducing nonlinear control laws beneficial for the state dynamics.

Implications and Future Directions

The implications of this paper are both practical and theoretical. Practically, the application of optimal control methods on truncations of the Koopman operator could revolutionize control strategies for nonlinear systems, extending beyond the limitations of traditional linearization approaches. Theoretically, the work advances understanding of the structure and identification of Koopman invariant subspaces that include state variables—an area ripe for further exploration.

Moreover, the paper acknowledges unresolved challenges and potential extensions. Notably, it raises questions about identifying systems that allow for finite-dimensional Koopman-invariant subspaces explicitly spanning state variables. It also points to the challenge of selecting proper observables when dealing with systems featuring more complex attractors.

Conclusion

This paper provides a detailed examination of the use of Koopman operators in finite-dimensional spaces for modeling and controlling nonlinear dynamical systems. By illustrating both the potential and limitations of these methods, it offers a pathway for developing more effective control strategies in nonlinear dynamics, while also setting the stage for further research into identifying and leveraging Koopman eigenfunctions and invariant subspaces. As this domain continues to evolve, such analyses will be crucial for the advancement of control theory and its applications across diverse complex systems.