Nonlinear energy-based systems: modeling, control and numerical realization

Dynamical systems consist of ordinary and partial differential equations and are among the most prominent approaches to model physical processes. They describe the evolution of the system in terms of the current system state and external control inputs. To improve model accuracy, it can be beneficial to explicitly include properties such as energy conservation or energy dissipation, which are usually present in the real-world phenomena, in the mathematical problem description. Ideally, these properties should be kept in mind whenever one interacts with the model. The focus of this thesis is on two kinds of interactions: the discretization of such models, and their control. Discretization techniques are necessary whenever the system evolution is to be approximated using computational methods. Similarly fundamental is the numerical realization of control inputs that lead to desired outcomes. Both aspects require special attention to retain an energy-based perspective. For this, the first step is usually to encode the energy properties in the algebraic description of the model. This description must strike a balance between general applicability and its corresponding benefits. The next step is to leverage the algebraic description in further analysis. While the field is well-developed for linear systems, the nonlinear case often poses additional difficulties. First, there seems to be no clear consensus about what model class to use to describe nonlinear physical phenomena. Second, many discretization methods that preserve the energy-based viewpoint in the linear case are not trivial to generalize to nonlinear systems. Third, studying the behavior of energy-optimal controls and finding control laws that can be realized via energy-based models is usually more difficult. In this thesis, these points are addressed. We give an overview of energy-based model classes used for nonlinear phenomena in both finite and infinite dimensions and investigate their relationship. Moreover, using a modified Petrov--Galerkin method and discrete gradients, we present multiple structure-preserving discretization schemes. Furthermore, we show that similar to the linear case, energy-optimal controls steer the associated trajectories to the submanifold of the state space where no dissipation is present. Finally, we combine an optimal feedback law characterized by the Hamilton--Jacobi--Bellman equation with output feedback to state an energy-based feedback controller. Our theoretical results are illustrated using numerical experiments.