Open Access Publications from the University of California

## Koopman Representations in Control

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Abstract

The Koopman operator describes the time-evolution of scalar-valued functions under the action of a dynamical system. These functions are called observables, and their evolution is always linear, even if the underlying dynamical system is nonlinear. The linearity of the Koopman operator framework is attractive to both dynamical systems theorists who study the spectral properties of these operators as well as to control theorists who leverage linearity to simplify control design. Recently, the theory of Koopman representations has emerged, with researchers gradually exploring the benefits of alternate, potentially nonlinear ways of representing these systems. In this thesis, we explore three distinct ways of representing the Koopman operator and explore their application to control design.

In the first part of this dissertation, we develop the mathematical underpinning of Koopman representation theory. The evolution from the state-space representation of a dynamical system to the Koopman operator is described, and its spectral content is explored. Next, nonlinear representations of the Koopman operator and its extension to systems with input is described. Finally, we introduce some of the numerical approximation schemes for the representations that are used in this paper. This chapter is meant to give the reader the mathematical background necessary to appreciate the results presented in the remaining chapters.

In the second part of this dissertation, we demonstrate a linear representation of the Koopman operator which fully leverages spectral objects such as eigenfunctions and eigenvalues. The eigenfunctions are special observables which evolve under the action of the Koopman operator via multiplication by a complex scalar, the eigenvalues. This is analogous to the eigenvectors and eigenvalues of a linear transformation. A collection of eigenfunctions forms a finite-dimensional, linear representation of a dynamical system, and their evolution spans a Koopman-invariant subspace. Finding this finite-dimensional representation allows for the application of well-developed linear systems methodologies to nonlinear systems such as spectral analysis and linear optimal control methods. In this work, we introduce a deep learning architecture that learns the Koopman eigenfunctions of a dynamical system from data and constructs the resulting finite-dimensional, linear representation of the Koopman operator. In numerical examples, the eigenfunctions learned using this framework exhibit a predictive performance superior to popular fixed-basis methods such as Extended Dynamic Mode Decomposition (EDMD). Finally, we extend the architecture to controlled dynamical systems by simultaneously learning the eigenfunctions of the natural dynamics with special system-decoupling observables on the inputs. Numerical examples show that the linear predictors obtained in this way can be readily used to design controllers that act directly on the Koopman modes of the system.

In the third part of this dissertation, we introduce our first use of the static Koopman operator in control. In our application, this is a linear operator which maps the space of functions of static poses of a soft robotic arm to the space of functions of the pressures in the arm's actuating muscles. We use static Koopman operator as a pregain term in our optimal control implementation alongside a traditional dynamic Koopman operator. Using both Koopman representations, we advance the modeling and control of soft robots into the inertial, nonlinear regime. We control motions of a soft, continuum arm with velocities 10x larger and accelerations 40x larger than those of previous work, and do so for high-deflection shapes with over 110 degrees of curvature. This work advances rapid modeling and control for soft robots from the realm of quasi-static to inertial, laying the groundwork for the next generation of compliant and highly dynamic robots.

Lastly, in the fourth chapter, we introduce a nonlinear Koopman representation which leverages so-called input-parameterized Koopman eigenfunctions. In the control of systems with multiple fixed points, it is typical to use piecewise control methods and local Koopman models. In contrast, our input-parameterized eigenfunction representation is accurate globally and enables a finite dimensional model which can handle these control problems without ad-hoc piecewise methods. We illustrate this on the control between the basins of attraction of the Duffing oscillator with dissipation.