UC Santa Barbara

The Koopman operator, which describes a dynamical system via a linear representation that can be approximately learned from data, allows application of linear control techniques to nonlinear systems. One may consider these aspects of system representation and controller design separately, but successful application will ultimately require a fusion of both. This thesis studies contributions to both aspects. The predictive accuracy of a Koopman-based representation is heavily dependent on the chosen set of lifting functions, termed observables. Identifying an appropriate lifting is a challenging domain-specific problem. We focus our study on a class of systems that are "entity-based systems" that we introduce and formally define. The states of such systems may be subdivided into a variable number of components, which we refer to as "entities". In the Koopman operator framework, we introduce a type of observable that describes a notion of the density of these entities. We discuss limitations of these types of observables, and further consider how products of these densities allow us to capture a richer class of interactions between entities. Given a Koopman representation for a dynamical system, we may study a nonlinear optimal control problem via a linear problem on a lifted state space. We show that the optimal cost-to-go to this linear problem has a piecewise linear form with respect to the lifted state. A robust formulation, in which we minimize the worst-case cost with respect to bounded errors in the initial state and Koopman dynamics, retains a piecewise affine structure. Due to the combinatorial nature of the problem, the number of possible input sequences grows exponentially in the horizon, and so we additionally provide a heuristic pruning algorithm that aims to approximate the cost-to-go by restricting the search space to a much smaller subset.