Nonlinear systems and passivity: feedback control, model reduction, and time discretization

Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this paper, we show that the combination of an optimal feedback law characterized by the Hamilton-Jacobi-Bellman equation and output feedback gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. To illustrate our results, we numerically realize the controller using the policy iteration and computationally verify passivity via a custom passivity-preserving discrete gradient scheme suitable for a wide class of passive systems.