Port-Hamiltonian modeling, discretization and feedback control of a circular water tank

This work presents the development of the nonlinear 2D Shallow Water Equations (SWE) in polar coordinates as a boundary port controlled Hamiltonian system. A geometric reduction by symmetry is obtained, simplifying the system to one-dimension. The recently developed Partitioned Finite Element Method is applied to semi-discretize the equations, preserving the boundary power-product of both the original 2D and the reduced 1D system. The main advantage of this power-preserving semi-discretization method is that it can be applied using well-established finite element software. In this work, we use FEniCS to solve the variational formulation, including the nonlinearity provided by the non-quadratic Hamiltonian of the SWE. A passive output-feedback controller using damping injection is used to dissipate the water waves

Twenty years of distributed port-Hamiltonian systems:A literature review

The port-Hamiltonian (pH) theory for distributed parameter systems has developed greatly in the past two decades. The theory has been successfully extended from finite-dimensional to infinite-dimensional systems through a lot of research efforts. This article collects the different research studies carried out for distributed pH systems. We classify over a hundred and fifty studies based on different research focuses ranging from modeling, discretization, control and theoretical foundations. This literature review highlights the wide applicability of the pH systems theory to complex systems with multi-physical domains using the same tools and language. We also supplement this article with a bibliographical database including all papers reviewed in this paper classified in their respective groups

Port-Hamiltonian Neural Networks with State-Dependent Ports

Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We show that port-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the port-Hamiltonian structure that facilitates the separation of internal and external forces. Numerical results are provided for a forced and damped mass-spring system and a tank system of higher complexity, and a symmetric fourth-order integration scheme is introduced for improved training on sparse and noisy data.Comment: 21 pages, 12 figures; v3: restructured the paper for more clarity, major changes to the text, updated plot

Pseudo-Hamiltonian neural networks with state-dependent external forces

Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a generalization of the Hamiltonian formulation via the port-Hamiltonian formulation, and show that pseudo-Hamiltonian neural network models can be used to learn external forces acting on a system. We argue that this property is particularly useful when the external forces are state dependent, in which case it is the pseudo-Hamiltonian structure that facilitates the separation of internal and external forces. Numerical results are provided for a forced and damped mass–spring system and a tank system of higher complexity, and a symmetric fourth-order integration scheme is introduced for improved training on sparse and noisy data.publishedVersio

Dissipative Shallow Water Equations: a port-Hamiltonian formulation

The dissipative Shallow Water Equations (DSWEs) are investigated as port-Hamiltonian systems. Dissipation models of different types are considered: either as nonlinear bounded operators, or as linear unbounded operators involving a classical diffusion term in 1D, or the vectorial Laplacian in 2D. In order to recast the dissipative SWE into the framework of pHs with dissipation, a physically meaningful factorization of the vectorial Laplacian is being used, which nicely separates the divergent and the rotational components of the velocity field. Finally, the structure-preserving numerical scheme provided by the Partitioned Finite Element Method (PFEM) is applied to the nonlinear bounded dissipative fluid models. For the linear unbounded cases, a change of variables is highlighted, to transform the DSWEs into a new pHs with a polynomial structure, which proves more suitable for numerics

PFEM: a mixed structure-preserving discretization method for port-Hamiltonian systems

PFEM: a mixed structure-preserving discretization method for port-Hamiltonian systems

A Partitioned Finite Element Method (PFEM) for power-preserving discretization of port-Hamiltonian systems (pHs) with polynomial nonlinearity

The Partitioned Finite Element Method introduced in [IMA J. MCIControl and Information, 2021]. provides a structure-preserving discretization for the solution of systems of boundary controlled and observed Partial Differential Equations (PDEs), formulated as distributed-parameter port-Hamiltonian systems (pHs). In particular, the energy balance is preserved at the discrete level. This method, already well-developped for linear systems, is also suitable for nonlinear systems with polynomial nonlinearity, such as the 2D Shallow Water Equations, or the full von-Kármán plate equations

Numerical analysis of a structure-preserving space-discretization for an anisotropic and heterogeneous boundary controlled N-dimensional wave equation as port-Hamiltonian system

The anisotropic and heterogeneous N-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.Comment: 36 pages, 1 figure, submitte

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008