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Volker Mehrmann

Publications and source records attributed to Volker Mehrmann.

At least 37 records · Page 2Linked to original sources
arXiv2024

Port-Hamiltonian Realizations of Positive Real Descriptor Systems

The relationship between port-Hamiltonian and positive real descriptor systems is investigated. It is well-known that port-Hamiltonian systems are positive real, but the converse implication does not always hold. In [K. Cherifi, H. Gernandt, and D. Hinsen. The difference between port-Hamiltonian, passive and positive real descriptor systems. Math. Control Signals Systems, pages 1-32, 2023] sufficient conditions for the converse are presented. We refine these conditions and present for a completely controllable, completely observable and positive real descriptor system a necessary and sufficient condition as well as an explicit method to compute a port-Hamiltonian realization of a general positive real descriptor system.

math.OC↗
arXiv2024

Port-Hamiltonian Modeling and Control of Electric Vehicle Charging Stations

Electric vehicles (EV) are an important part of future sustainable transportation. The increasing integration of EV charging stations (EVCSs) in the existing power grids require new scaleable control algorithms that maintain the stability and resilience of the grid. Here, we present such a control approach using an averaged port-Hamiltonian model. In this approach, the underlying switching behavior of the power converters is approximated by an averaged non-linear system. The averaged models are used to derive various types of stabilizing controllers, including the typically used PI controllers. The pH modeling is showcased by means of a generic setup of an EVCS, where the battery of the vehicle is connected to an AC grid via power lines, converters, and filters. Finally, the control design methods are compared for the averaged pH system and validated using a simulation model of the switched charging station.

math.OC↗
arXiv2024

Abstract dissipative Hamiltonian differential-algebraic equations are everywhere

In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there are constraints coming from the underlying physics, but many standard PDE models can be seen as an adHDAE on an extended state space. This reflects the fact that models often include closure relations and structural properties. We present a unifying operator theoretic approach to analyze the properties of such operator equations and illustrate this by several applications.

math.FA↗
arXiv2024

On the discrete equivalence of Lagrangian, Hamiltonian and mixed finite element formulations for linear wave phenomena

It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit a Hamiltonian structure when written in mixed form. In this contribution, the discrete equivalence of Lagrangian, symplectic Hamiltonian and mixed formulations is investigated for linear wave propagation phenomena. Under compatibility conditions between the finite elements, the Lagrangian and mixed formulations are indeed equivalent. For the time discretization the leapfrog scheme and the implicit midpoint rule are considered. In mixed methods applied to wave problems the primal variable (e.g. the displacement in mechanics or the magnetic potential in electromagnetism) is not an unknown of the problem and is reconstructed a posteriori from its time derivative. When this reconstruction is performed via the trapezoidal rule, then these time-discretization methods lead to equivalent formulations.

math.NA↗
arXiv2023

Fiedler Linearizations of Rectangular Rational Matrix Functions

Linearization is a standard approach in the computation of eigenvalues, eigenvectors and invariant subspaces of matrix polynomials and rational matrix value functions. An important source of linearizations are the so called Fiedler linearizations, which are generalizations of the classical companion forms. In this paper the concept of Fiedler linearization is extended from square regular to rectangular rational matrix valued functions. The approach is applied to Rosenbrock functions arising in mathematical system theory.

math.CT↗
arXiv2023

Hidden regularity in singular optimal control of port-Hamiltonian systems

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

math.OC↗
arXiv2023

Hypocoercivity in algebraically constrained partial differential equations with application to Oseen equations

The long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models are isotropic Oseen equations where the viscosity acts uniformly in all directions and anisotropic Oseen-type equations with different viscosity directions. The hypocoercivity index is determined (if it exists) and it is shown that similar to the finite dimensional case of ordinary differential equations and differential-algebraic equations it characterizes its decay behavior.

math.AP↗
arXiv2023

Hierarchical modeling for an industrial implementation of a Digital Twin for electrical drives

Digital twins have become popular for their ability to monitor and optimize a process or a machine, ideally through its complete life cycle using simulations and sensor data. In this paper, we focus on the challenge of accurate and real-time simulations for digital twins in the context of electrical machines. To build such a digital twin involves not only computational models for the electromagnetic aspects, but also mechanical and thermal effects need to be taken into account. We address mathematical tools that can be employed to carry out the required simulations based on physical laws as well as surrogate or data-driven models. One of those tools is a model hierarchy of very fine to very coarse models as well as reduced order models for obtaining real-time simulations. The required software tools to carry out simulations in the digital twin are also discussed. The simulation models are implemented in a pipeline that allows for the automatic modeling of new machines and the automatic configuration of new digital twins. Finally, the overall implemented digital twin is tested and implemented in a physical demonstrator.

eess.SY↗
arXiv2023

On discrete-time dissipative port-Hamiltonian (descriptor) systems

Port-Hamiltonian (pH) systems have been studied extensively for linear continuous-time dynamical systems. This manuscript presents a discrete-time pH descriptor formulation for linear, completely causal, scattering passive dynamical systems based on the system coefficients. The relation of this formulation to positive and bounded real systems and the characterization via positive semidefinite solutions of Kalman-Yakubovich-Popov inequalities is also studied.

math.OC↗
arXiv2023

A geometric framework for discrete time port-Hamiltonian systems

Port-Hamiltonian systems provide an energy-based formulation with a model class that is closed under structure preserving interconnection. For continuous-time systems these interconnections are constructed by geometric objects called Dirac structures. In this paper, we derive this geometric formulation and the interconnection properties for scattering passive discrete-time port-Hamiltonian systems.

math.OC↗
arXiv2023

Adaptive Nonlinear Optimization of District Heating Networks Based on Model and Discretization Catalogs

We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.

math.OC↗
arXiv2023

Hypocoercivity and hypocontractivity concepts for linear dynamical systems

For linear dynamical systems (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.

math.DS↗
arXiv2022

Linear port-Hamiltonian DAE systems revisited

Port-Hamiltonian systems theory provides a systematic methodology for the modeling, simulation and control of multi-physics systems. The incorporation of algebraic constraints has led to a multitude of definitions of port-Hamiltonian differential-algebraic equations (DAE) systems. This paper presents extensions of results in Gernandt, Haller & Reis (2021) and Mehrmann & Van der Schaft (2022) in the context of maximally monotone structures and shows that any such space can be written as composition of a Dirac and a resistive structure. Furthermore, appropriate coordinate representations are presented as well as explicit expressions for the associated transfer functions.

math.OC↗
arXiv2022

On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with "singular part", while the "positive definite part" can still be treated with the three-term recurrence methods.

math.NA↗
arXiv2022

Structure-Preserving Model Order Reduction for Index Two Port-Hamiltonian Descriptor Systems

We present a new optimization-based structure-preserving model order reduction (MOR) method for port-Hamiltonian descriptor systems (pH-DAEs) with differentiation index two. Our method is based on a novel parameterization that allows us to represent any linear time-invariant pH-DAE with a minimal number of parameters, which makes it well-suited to model reduction. We propose two algorithms which directly optimize the parameters of a reduced model to approximate a given large-scale model with respect to either the H-infinity or the H-2 norm. This approach has several benefits. Our parameterization ensures that the reduced model is again a pH-DAE system and enables a compact representation of the algebraic part of the large-scale model, which in projection-based methods often requires a more involved treatment. The direct optimization is entirely based on transfer function evaluations of the large-scale model and is therefore independent of the system matrices' structure. Numerical experiments are conducted to illustrate the high accuracy and small reduced model orders in comparison to other structure-preserving MOR methods.

eess.SY↗
arXiv2022

Structure-Preserving Model Order Reduction for Index One Port-Hamiltonian Descriptor Systems

We develop optimization-based structure-preserving model order reduction (MOR) methods for port-Hamiltonian (pH) descriptor systems of differentiation index one. Descriptor systems in pH form permit energy-based modeling and intuitive coupling of physical systems across different physical domains, scales, and accuracies. This makes pH models well-suited building-blocks for component-wise modeling of large system networks. In this context, it is often necessary to preserve the pH structure during MOR. We discuss current projection-based and structure-preserving MOR algorithms for pH systems and present a new optimization-based framework for that task. The benefits of our method include a simplified treatment of algebraic constraints and often a higher accuracy of the resulting reduced-order model, which is demonstrated by several numerical examples.

math.OC↗
arXiv2022

Discretization of inherent ODEs and the geometric integration of DAEs with symmetries

Discretization methods for differential-algebraic equations (DAEs) are considered that are based on the integration of an associated inherent ordinary differential equation (ODE). This allows to make use of any discretization scheme suitable for the numerical integration of ODEs. For DAEs with symmetries it is shown that the inherent ODE can be constructed in such a way that it inherits the symmetry properties of the given DAE and geometric properties of its flow. This in particular allows the use of geometric integration schemes with a numerical flow that has analogous geometric properties.

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