TY - JOUR A1 - Breiten, Tobias A1 - Morandin, Riccardo A1 - Schulze, Philipp T1 - Error bounds for port-Hamiltonian model and controller reduction based on system balancing N2 - Linear quadratic Gaussian (LQG) control design for port-Hamiltonian systems is studied. The recently proposed method from [42] is reviewed and modified such that the resulting controllers have a port-Hamiltonian (pH) realization. Based on this new modification, a reduced-order controller is obtained by truncation of a balanced system. The approach is shown to be closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. With regard to this error bound, a theoretically optimal pH-representation is derived. Consequences for pH-preserving balanced truncation model reduction are discussed and shown to yield two different classical H∞ -error bounds. Numerical examples illustrate the main theoretical findings. KW - port-Hamiltonian systems KW - model order reduction KW - LQG control design KW - error bound Y1 - 2020 U6 - https://doi.org/10.1016/j.camwa.2021.07.022 CY - Computers & Mathematics with Applications ER - TY - INPR A1 - Breiten, Tobias A1 - Unger, Benjamin T1 - Passivity preserving model reduction via spectral factorization N2 - We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2 -optimal reduced-order models. KW - passivity KW - port-Hamiltonian systems KW - structure-preserving model order reduction KW - spectral factorization KW - H2-optimal Y1 - 2021 ER - 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 - TY - INPR A1 - Breiten, Tobias A1 - Schulze, Philipp T1 - Structure-Preserving Linear Quadratic Gaussian Balanced Truncation for Port-Hamiltonian Descriptor Systems N2 - We present a new balancing-based structure-preserving model reduc- tion technique for linear port-Hamiltonian descriptor systems. The pro- posed method relies on a modification of a set of two dual generalized algebraic Riccati equations that arise in the context of linear quadratic Gaussian balanced truncation for differential algebraic systems. We de- rive an a priori error bound with respect to a right coprime factorization of the underlying transfer function thereby allowing for an estimate with respect to the gap metric. We further theoretically and numerically ana- lyze the influence of the Hamiltonian and a change thereof, respectively. With regard to this change of the Hamiltonian, we provide a novel proce- dure that is based on a recently introduced Kalman–Yakubovich–Popov inequality for descriptor systems. Numerical examples demonstrate how the quality of reduced-order models can significantly be improved by first computing an extremal solution to this inequality. Y1 - 2021 ER - TY - INPR A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Breiten, Tobias A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Hiller, Benjamin A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview (Extended English Version) N2 - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. Y1 - 2023 ER - TY - INPR A1 - Breiten, Tobias A1 - Karsai, Attila A1 - Mehrmann, Volker A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Tscherpel, Tabea A1 - Hiller, Benjamin A1 - Morandin, Riccardo A1 - Tischendorf, Caren T1 - A Catalog of Gas Network Models: PDEs, Coupling Conditions, and Numerical Schemes N2 - This document aims to provide a concise and clear introduction to the topic of gas flow modeling. We present several models for gas flow, organized into hierarchies based on complexity. We discuss in detail the modeling of individual components such as valves and compressors. Network model classes based on purely algebraic relations and energy-based port-Hamiltonian models are included, along with a brief overview of basic numerical methods for hyperbolic balance laws and port-Hamiltonian systems. We do not claim completeness and refer in many places to the existing literature. Y1 - N1 - This is an updated version of [P. Domschke, B. Hiller, J. Lang, V. Mehrmann, R. Morandin, and C. Tischendorf. Gas Network Modeling: An Overview. Preprint, TRR 154, 2021], available at: https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/411 ER -