Output Regulation of Nonlinear Systems in a Koopman Operator Framework

Abstract

This thesis considers the problem of nonlinear output regulation in a Koopman operator framework. The goal of output regulation is to asymptotically track a reference and/or simultaneously reject a disturbance signal, both generated by some external autonomous system called the exosystem. The nonlinear output regulation problem is solvable if and only if a set of partial differential equations (PDE) are satisfied. From the solution, a feedback law can be obtained that achieves output regulation. However, solving the PDE is difficult. In this thesis, we instead aim to construct a feedback law by utilizing the Koopman operator instead.
The Koopman operator associated with a state-space model of a (nonlinear) dynamical system describes the evolution of functions of the states, called observable functions, by propagating the state forward in time according to the flow of the system, and evaluating this at each possible observable function. The space of observables is an infinite-dimensional vector field. Therefore, the Koopman operator is infinite-dimensional and linear. The Koopman operator of an autonomous system associated with a nonlinear control system provides a bilinear description of the control system instead. The use of the Koopman operator to tackle the output regulation problem has not been done before in the literature. We identify conditions under which the Koopman operator can be used to rephrase the nonlinear output regulation problem as a bilinear output regulation problem. We then show when the bilinear output regulation problem is solved using linear dynamic error feedback. In particular, a Lyapunov-based approach is used to characterize a set of initial conditions for which the output is regulated. Finally, to verify the results, a numerical example is presented.