arXiv subjects

Volker Mehrmann

Publications and source records attributed to Volker Mehrmann.

At least 73 records · Page 4Linked to original sources

arXiv2019

Optimal robustness of port-Hamiltonian systems

We construct optimally robust port-Hamiltonian realizations of a given rational transfer function that represents a passive system. We show that the realization with a maximal passivity radius is a normalized port-Hamiltonian one. Its computation is linked to a particular solution of a linear matrix inequality that defines passivity of the transfer function, and we provide an algorithm to construct this optimal solution. We also consider the problem of finding the nearest passive system to a given non-passive one and provide a simple but suboptimal solution.

math.OC↗

arXiv2019

Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis

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.

math.OC↗

arXiv2019

Structure-preserving discretization for port-Hamiltonian descriptor systems

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.

math.OC↗

arXiv2019

Numerical methods to compute a minimal realization of a port-Hamiltonian system

Port-Hamiltonian (pH) systems are a very important modeling tool in almost all areas of systems and control, in particular in network based model of multi-physics multi-scale systems. They lead to remarkably robust models that can be easily interconnected. This paper discusses the derivation of pH models from time-domain input-output data. While a direct construction of pH models is still an open problem, we present three different indirect numerical methods for the realization of pH systems. The algorithms are implemented in MATLAB and their performance is illustrated via several numerical examples.

math.OC↗

arXiv2019

Low rank perturbation of regular matrix pencils with symmetry structures

The generic change of the Weierstrass Canonical Form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues $0$ and $\infty$, respectively $+1$ and $-1$. The differences arise in those cases where the parity of the partial multiplicities in the perturbed pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-$1$ and rank-$2$ cases that were obtained in [L. Batzke, Generic Low-Rank Perturbations of Structured Regular Matrix Pencils and Structured Matrices, PhD Thesis, TU Berlin, Berlin, Germany, 2015] and [L. Batzke, Generic rank-two perturbations of structured regular matrix pencils, Oper. Matrices,10:83-112, 2016] for the case of special structure-preserving perturbations. As the main tool, we use decompositions of matrix pencils with symmetry structure into sums of rank-one pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.

math.SP↗

arXiv2019

Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems

Port-based network modeling of multi-physics problems leads naturally to a formulation as port-Hamiltonian differential-algebraic system. In this way, the physical properties are directly encoded in the structure of the model. Since the state space dimension of such systems may be very large, in particular when the model is a space-discretized partial differential-algebraic system, in optimization and control there is a need for model reduction methods that preserve the port-Hamiltonian structure while keeping the (explicit and implicit) algebraic constraints unchanged. To combine model reduction for differential-algebraic equations with port-Hamiltonian structure preservation, we adapt two classes of techniques (reduction of the Dirac structure and moment matching) to handle port-Hamiltonian differential-algebraic equations. The performance of the methods is investigated for benchmark examples originating from semi-discretized flow problems and mechanical multibody systems.

math.OC↗

arXiv2019

A Robust Iterative Scheme for Symmetric Indefinite Systems

We propose a two-level nested preconditioned iterative scheme for solving sparse linear systems of equations in which the coefficient matrix is symmetric and indefinite with relatively small number of negative eigenvalues. The proposed scheme consists of an outer Minimum Residual (MINRES) iteration, preconditioned by an inner Conjugate Gradient (CG) iteration in which CG can be further preconditioned. The robustness of the proposed scheme is illustrated by solving indefinite linear systems that arise in the solution of quadratic eigenvalue problems in the context of model reduction methods for finite element models of disk brakes as well as on other problems that arise in a variety of applications.

math.NA↗

arXiv2018

Modeling and discretization methods for the numerical simulation of elastic stents

A new model description for the numerical simulation of elastic stents is proposed. Based on the new formulation an inf-sup inequality for the finite element discretization is proved and the proof of the inf-sup inequality for the continuous problem is simplified. The new formulation also leads to faster simulation times despite an increased number of variables. The techniques also simplify the analysis and numerical solution of the evolution problem describing the movement of the stent under external forces. The results are illustrated via numerical examples.

math.NA↗

arXiv2018

Computation of Stability Radii for Large-Scale Dissipative Hamiltonian Systems

A linear time-invariant dissipative Hamiltonian (DH) system x' = (J-R)Q x, with a skew-Hermitian J, an Hermitian positive semi-definite R, and an Hermitian positive definite Q, is always Lyapunov stable and under weak further conditions even asymptotically stable. In various applications there is uncertainty on the system matrices J, R, Q, and it is desirable to know whether the system remains asymptotically stable uniformly against all possible uncertainties within a given perturbation set. Such robust stability considerations motivate the concept of stability radius for DH systems, i.e., what is the maximal perturbation permissible to the coefficients J, R, Q, while preserving the asymptotic stability. We consider two stability radii, the unstructured one where J, R, Q are subject to unstructured perturbation, and the structured one where the perturbations preserve the DH structure. We employ characterizations for these radii that have been derived recently in [SIAM J. Matrix Anal. Appl., 37, pp. 1625-1654, 2016] and propose new algorithms to compute these stability radii for large scale problems by tailoring subspace frameworks that are interpolatory and guaranteed to converge at a super-linear rate in theory. At every iteration, they first solve a reduced problem and then expand the subspaces in order to attain certain Hermite interpolation properties between the full and reduced problems. The reduced problems are solved by means of the adaptations of existing level-set algorithms for H-infinity norm computation in the unstructured case, while, for the structured radii, we benefit from algorithms that approximate the objective eigenvalue function with a piece-wise quadratic global underestimator. The performance of the new approaches is illustrated with several examples including a system that arises from a finite-element modeling of an industrial disk brake.

math.NA↗

arXiv2018

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert--Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.

math.AP↗

arXiv2018

Model Reduction for a Pulsed Detonation Combuster via Shifted Proper Orthogonal Decomposition

We propose a new algorithm to compute a shifted proper orthogonal decomposition (sPOD) for systems dominated by multiple transport velocities. The sPOD is a recently proposed mode decomposition technique which overcomes the poor performance of classical methods like the proper orthogonal decomposition (POD) for transport-dominated phenomena. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. Our new algorithm carries out a residual minimization in which the main computational cost arises from solving a nonlinear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detonation combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

math.NA↗

arXiv2018

The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kind 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 {\it 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.

math.NA↗

arXiv2018

Robust port-Hamiltonian representations of passive systems

We discuss the problem of robust representations of stable and passive transfer functions in particular coordinate systems, and focus in particular on the so-called port-Hamiltonian representations. Such representations are typically far from unique and the degrees of freedom are related to the solution set of the so-called Kalman-Yakubovich-Popov linear matrix inequality (LMI). In this paper we analyze robustness measures for the different possible representations and relate it to quality functions defined in terms of the eigenvalues of the matrix associated with the LMI. In particular, we look at the analytic center of this LMI. From this, we then derive inequalities for the passivity radius of the given model representation.

math.OC↗

arXiv2018

Linear algebra properties of dissipative Hamiltonian descriptor systems

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.

math.NA↗

arXiv2017

Qualitative Stability and Synchronicity Analysis of Power Network Models in Port-Hamiltonian Form

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies such as wind and solar power, 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. Although based on basic physical principles, the usual formulation in form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model as port-Hamiltonian system of differential-algebraic equations. This leads to a very 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, it explicitly displays all possible constraints and allows for robust simulation methods. Due to its systematic energy based formulation the model class allows easy extension, when further effects have to be considered, higher fidelity is needed for qualitative analysis, or the system needs to be coupled in a robust way to other networks. We demonstrate the advantages of the modified modeling approach with analytic results and numerical experiments.

cond-mat.dis-nn↗

arXiv2017

Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control

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.

math.NA↗

arXiv2017

Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization

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.

math.OC↗

arXiv2017

Port-Hamiltonian descriptor systems

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.

math.OC↗