Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics (2010.09961v3)

Abstract: Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This paper describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm are constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

• The paper introduces bilinear Koopman realizations to transform nonlinear control-affine dynamics into an efficient bilinear framework.
• It presents analytical conditions and simulation results showing that bilinear models outmatch linear ones in prediction accuracy as the number of basis functions increases.
• The findings highlight the potential for improved real-time control in applications like robotic trajectory tracking with reduced computational complexity.

Overview of Bilinear Koopman Realizations for Modeling and Control

The research paper "Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics," authored by Daniel Bruder, Xun Fu, and Ram Vasudevan, explores leveraging the Koopman operator for constructing bilinear realizations from data to model and control nonlinear systems with unknown dynamics. The paper provides a comprehensive theoretical background, presents analytical conditions for realizations, and demonstrates empirical results through simulations.

Theoretical Foundation

Koopman operator theory facilitates converting nonlinear systems into linear systems by projecting the system dynamics into higher-dimensional spaces of observable functions. While linear systems theory is well developed, it does not directly apply to nonlinear systems without approximation. The paper introduces bilinear Koopman realizations as a method to bridge this gap, providing a more adaptable representation compared to purely linear realizations in certain contexts.

The authors present necessary and sufficient conditions for a dynamical system to be represented bilinearly over a set of observables. It establishes the potential of bilinear realizations in capturing the dynamics of control-affine systems wherein each control-affine system admits a well-defined (possibly infinite-dimensional) bilinear realization. This potential corrects a limitation of linear realizations, which might not always exist.

Numerical Methods and Empirical Validation

This paper includes developing multiple Koopman-based realizations, including linear, bilinear, and nonlinear types, each formed from data using Extended Dynamic Mode Decomposition (EDMD). The efficacy of these methods was validated through simulations with a robot arm tasked with trajectory following.

Empirical results showed bilinear realizations to outperform linear models regarding prediction accuracy as the number of basis functions increased. Among the three realization types, bilinear models struck the best compromise between prediction accuracy and computation efficiency, highlighting the inadequacy of linear models in specific applications with large basis dimensions.

Implications and Applications

The advantages of bilinear realizations manifest in their broader applicability in control scenarios, particularly where computational complexity of nonlinear MPC may be prohibitive. Bilinear models offered both computational efficiency and improved prediction accuracy in simulations of the robotic arm control tasks. This positions bilinear Koopman realizations as a significant tool for the control of systems where model complexity and real-time performance are crucial.

Speculation on Future Developments

As control systems increasingly demand sophistication, future research could focus on refining bilinear model predictive control frameworks. This might include exploring optimization algorithms tailored for bilinear systems, providing guarantees regarding solution fidelity, and addressing computational challenges inherent in real-time applications.

Conclusion

The paper makes a sound argument that bilinear Koopman realizations present a valuable approach for modeling and controlling systems with unknown dynamics. By successfully navigating the limitations associated with linear realizations, bilinear models highlight a promising area of research for computationally efficient and accurate representations of nonlinear dynamical systems. For scholars and practitioners working in control theory and nonlinear dynamics, this represents a substantial advance in understanding and applying Koopman operator theory.