Publikationen der Gruppe

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained systems with the solution of certain saddle point problems in order to meet the constraints throughout the integration process. The result is a novel class of semi-explicit time integration schemes. We prove the expected convergence rates and illustrate the performance on two numerical examples including a parabolic equation with nonlinear dynamic boundary conditions.

We extend the modeling framework of port-Hamiltonian descriptor systems to include under- and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Numerical examples are provided.

We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in [Peherstorfer, Gugercin, Willcox, SIAM J. Sci. Comput., 39(5):2152–2178, 2017]. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.

In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous- and discrete-time systems. Numerical methods are derived to solve these equations via steepest ascent and Newton-like methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.

In this paper we derive a formulation for Model Predictive Control (MPC) of linear time-invariant systems based on H-infinity loop-shaping. The design provides an optimized stability margin for problems that require state estimation. Input and output weights are designed in the frequency domain to satisfy steady-state and transient performance requirements, in lieu of conventional MPC plant model augmentations. The H-infinity loop-shaping synthesis results in an observer-based state feedback structure. Using the linear state feedback law, an inverse optimal control problem is solved to design the MPC cost function, and the H-infinity state estimator is used to initialize the prediction model at each time step. The MPC inherits the closed-loop performance and stability margin of the loop-shaped design when constraints are inactive. We apply the methodology to a multi-zone heat pump system in simulation. The design rejects constant unmeasured disturbances and tracks constant references with zero steady-state error, has good transient performance, provides an excellent stability margin, and enforces input and output constraints.

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model with inertia. The usual formulation in the form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model with inertia as a polynomial port-Hamiltonian system of differential-algebraic equations, with a quadratic Hamiltonian function including a generalized order parameter. This leads to a robust representation of the system with respect to disturbances: it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, and it explicitly displays all possible constraints and allows for robust simulation methods. The model is immersed into a system of model hierarchies that will be helpful for applying adaptive simulations in future works. We illustrate the advantages of the modified modeling approach with analytics and numerical results.

Metriplectic systems are state space formulations that have become well-known under the acronym GENERIC. In this work we present a GENERIC based state space formulation in an operator setting that encodes a weak-formulation of the field equations describing the dynamics of a homogeneous mixture of compressible heat-conducting Newtonian fluids consisting of reactive constituents. We discuss the mathematical model of the fluid mixture formulated in the framework of continuum thermodynamics. The fluid mixture is considered an open thermodynamic system that moves free of external body forces. As closure relations we use the linear constitutive equations of the phenomenological theory known as Thermodynamics of Irreversible Processes (TIP). The phenomenological coefficients of these linear constitutive equations satisfy the Onsager-Casimir reciprocal relations. We present the state space representation of the fluid mixture, formulated in the extended GENERIC framework for open systems, specified by a symmetric, mixture related dissipation bracket and a mixture related Poisson-bracket for which we prove the Jacobi-identity.

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kinds of systems. It extends the proper orthogonal decomposition (POD) by introducing time-dependent shifts of the snapshot matrix. The approach, called shifted proper orthogonal decomposition (sPOD), features a determination of the multiple transport velocities and a separation of these. One- and two-dimensional test examples reveal the good performance of the sPOD for transport-dominated phenomena and its superiority in comparison to the POD.

A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (operator DAEs), is presented. The given procedure works for semi-explicit and semi-linear operator DAEs of first order including the Navier-Stokes and other flow equations. The proposed reformulation is consistent, i.e., the solution of the PDE remains untouched. Its main advantage is that it regularizes the operator DAE in the sense that a semi-discretization in space leads to a DAE of lower index. Furthermore, a stability analysis is presented for the linear case, which shows that the regularization provides benefits also for the application of the Rothe method. For this, the influence of perturbations is analyzed for the different formulations. The results are verified by means of a numerical example with an adaptive space discretization.

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and eciency. The existing package Guptri is very elegant but may sometimes be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specic rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.

A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system.

A first order perturbation theory for eigenvalues of real or complex J-symplectic matrices under structure- preserving perturbations is developed. As main tools structured canonical forms and Lidskii-like formulas for eigenvalues of multiplicative perturbations are used. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues 1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions.

The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

The dynamics of elastic media, constrained by Dirichlet boundary conditions, can be modeled as an operator DAE of semi-explicit structure. These models include flexible multibody systems as well as applications with boundary control. In order to use adaptive methods in space, we analyse the properties of the Rothe method concerning stability and convergence for this kind of systems. We consider a regularization of the operator DAE and prove the weak convergence of the implicit Euler scheme. Furthermore, we consider perturbations in the semi-discrete systems which correspond to additional errors such as spatial discretization errors.

A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.

Abstract We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

In the simulation and optimization of gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the simulation of one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques. By means of the well-known n-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.

We are concerned with the computation of the $\mathcal{L}_\infty$-norm for an $\mathcal{L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$, where the middle factor is the inverse of a meromorphic matrix-valued function, and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large scale. We propose a subspace projection method to obtain approximations of the function $H$ where the middle factor is of much smaller dimension. The $\mathcal{L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the $\mathcal{L}_\infty$-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.

Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-k perturbation the k largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest $k$, but the largest 2k Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank k. Special emphasis is also given to Hermitian and skew-Hermitian quaternion matrices and generic low rank perturbations that are structure-preserving.

The present work deals with the inverse dynamics simulation of underactuated mechanical systems relying on servo constraints. The servo-constraint problem of discrete mechanical systems is governed by differential–algebraic equations (DAEs) with high index. We propose a new index reduction approach, which makes possible the stable numerical integration of the DAEs. The new method is developed in the framework of a specific crane formulation and facilitates a reduction from index five to index three and even to index one. Particular attention is placed on the special case in which the reduced index-1 formulation is purely algebraic. In this case the system at hand can be classified as differentially flat system. Both redundant coordinates and minimal coordinates can be employed within the newly developed approach. The success of the proposed method is demonstrated with two representative numerical examples.

In this master's thesis we adapt recent results on optimal control of continuous-time linear differential-algebraic equations to the discrete-time case of implicit difference equations. First, we adapt equivalent characterizations of solvability of the so-called Kalman- Yakubovich-Popov inequality for differential-algebraic equations to the case of implicit difference equations. That is, we relate the solvability of a certain matrix inequality to the positivity of the Popov function on the unit circle. An essential difference between the continuous-time and the discrete-time linear-quadratic optimal control problem is due to different structures occurring during the analysis in the form of even or palindromic matrix pencils, respectively. Therefore, with the help of certain structured Kronecker canonical forms, we adapt characterizations of inertia of even matrix pencils to palindromic matrix pencils. To this end, we first introduce a suitable notion of inertia for palindromic matrix pencils. These results are used – analogously to the continuous-time case – to characterize solvability of Lur'e equations equivalently by the existence of certain deflating subspaces of the palindromic matrix pencil. Then we use these findings to describe feasibility and the structure of solutions of the linear-quadratic control problem with infinite time horizon. Finally, these results are illustrated by means of an example.

In this paper a moment-matching procedure for descriptor systems is presented. Based on a time-domain notion of moments, parametrized families of reduced order models matching the moments of the original system are derived. An example of exploiting the flexibility of choosing the free matrix parameters is demonstrated by achieving two-sided moment matching, i. e., obtaining reduced order models of order $\nu$ matching $2\nu$ moments. In this context, a connection with the Loewner framework is shown which is a special case of the framework presented in this paper.

In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential-algebraic equations. Motivated by the index reduction technique of minimal extension, a remodelling of the flow equations is proposed. It is shown how this reformulation can be realized for standard finite elements via a decomposition of the discrete spaces and that it ensures stable and accurate approximations. The presented decomposition preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated.

The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline. The statistical analysis is performed using both a Monte Carlo Simulation and the Univariate Reduced Quadrature Method and is used to illustrate and confirm the obtained theoretical results.

Numerous research projects in automotive engineering focus on the industrialization of the thermoelectric generator. The development and the implementation of thermoelectric systems into the vehicle environment are commonly supported by virtual designing activities. In this paper a customized simulation architecture [1] for thermoelectric generators, developed in research project TEG2020, is used to compare two very different TEG concepts: a plate and a tube bundle geometry. The installation space serves as a restriction and therefore is the same for both systems. There are slight advantages in maximum power output, nominal back pressure and overall weight of the tube bundle geometry compared to the lightweight and cost-effective planar thermoelectric generator presented in [1] and [2]. The efficiency of both systems is nearly the same. However there are some disadvantages of the tube bundle geometry in required mass of thermoelectric material, efficiency based on the mass of thermoelectric material, complexity and necessary cooling load.

In space semi-discretized equations of elastodynamics with weakly enforced Dirichlet boundary conditions lead to differential algebraic equations (DAE) of index 3. We rewrite the continuous model as operator DAE and present an index reduction technique on operator level. This means that a semi-discretization leads directly to an index-1 system. We present existence results for the operator DAE with nonlinear damping term and show that the reformulated operator DAE is equivalent to the original equations of elastodynamics. Furthermore, we show that index reduction and semi-discretization in space commute if the discretization schemes are chosen in an appropriate way.