Koopman Representations of Dynamic Systems with Control
This work provides an analysis framework for evaluating simplifications in Koopman representations for controlled dynamical systems, which is incremental as it builds on existing methods to address a known bottleneck in control theory.
The paper tackles the challenge of designing optimal control for nonlinear dynamical systems by developing consistent Koopman representations that linearize dynamics, resulting in a hybrid formulation that accommodates a larger space of systems while keeping the Koopman operator independent of state and control inputs.
The design and analysis of optimal control policies for dynamical systems can be complicated by nonlinear dependence in the state variables. Koopman operators have been used to simplify the analysis of dynamical systems by mapping the flow of the system onto a space of observables where the dynamics are linear (and possibly infinte). This paper focuses on the development of consistent Koopman representations for controlled dynamical system. We introduce the concept of dynamical consistency for Koopman representations and analyze several existing and proposed representations deriving necessary constraints on the dynamical system, observables, and Koopman operators. Our main result is a hybrid formulation which independently and jointly observes the state and control inputs. This formulation admits a relatively large space of dynamical systems compared to earlier formulations while keeping the Koopman operator independent of the state and control inputs. More generally, this work provides an analysis framework to evaluate and rank proposed simplifications to the general Koopman representation for controlled dynamical systems.