TY - INPR A1 - Breiten, Tobias A1 - Burela, Shubhaditya A1 - Schulze, Philipp T1 - Optimal Control for a Class of Linear Transport-Dominated Systems via the Shifted Proper Orthogonal Decomposition N2 - Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. Y1 - 2025 ER - TY - JOUR A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Structure-preserving $H_\infty$ control for port-Hamiltonian systems N2 - We study $H_\infty$ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model. Y1 - 2025 U6 - https://doi.org/10.1016/j.sysconle.2023.105493 ER - TY - JOUR A1 - Karsai, Attila A1 - Breiten, Tobias A1 - Ramme, Justus A1 - Schulze, Philipp T1 - Nonlinear port-Hamiltonian systems and their connection to passivity N2 - Port-Hamiltonian (pH) systems provide a powerful tool for modeling physical systems. Their energy-based perspective allows for the coupling of various subsystems through energy exchange. Another important class of systems, passive systems, are characterized by their inability to generate energy internally. In this paper, we explore first steps towards understanding the equivalence between passivity and the feasibility of port-Hamiltonian realizations in nonlinear systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2409.06256 ER - TY - JOUR A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Passive feedback control for nonlinear systems N2 - 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 a particular choice of the observer gain 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. In addition, we propose a novel passivity preserving discrete gradient scheme for the time discretization of passive systems. To illustrate our results, we numerically realize the controller using the policy iteration and compare it to a controller where the observer gain is given by the extended Kalman filter. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2502.04987 ER -