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Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples., 19 pages, 2 figures, 5 tables

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We show and compare the effectiveness of all considered methods by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems., 31 pages, 18 figures

Output-based controllers are known to be fragile with respect to model uncertainties. The standard $\mathcal{H}_{\infty}$-control theory provides a general approach to robust controller design based on the solution of the $\mathcal{H}_{\infty}$-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples., Comment: 20 pages, 4 figures, 3 tables

In this paper, we extend the structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples., 26 pages, 9 figures

The MORLAB (Model Order Reduction Laboratory) toolbox is a software solution in MATLAB for the model reduction of linear time-invariant continuous-time systems. It contains several methods to deal with unstructured medium-scale descriptor systems. In order to illustrate the use of MORLAB in this context, in this paper, a damped mass-spring system with a holonomic constraint is reduced by the generalized balanced truncation subroutine from the MORLAB toolbox. The basic ideas of the MORLAB implementation are shown by introducing the used spectral projection methods based on the matrix sign and matrix disk function. To show the numerical behavior, the MORLAB routine and all the corresponding subroutines are used for computations applied to the introduced example. An overview about further model reduction methods for descriptor systems in MORLAB is given, on which similar implementational ideas as for the generalized balanced truncation method are used.

We consider the stabilization of incompressible fluid flow using linearized and spatially discretized models. In order to potentially work in applications, the designed controller must stabilize the discrete model with a robustness margin that covers linearization, discretization, and modeling errors. We expand on previous results that a linearization error in the infinite-dimensional model amounts to a coprime factor uncertainty and show that H∞-robust controllers can compensate this in the discrete approximation. In numerical experiments, we quantify the robustness margins and show that the H∞-robust controller, unlike the LQG-controller, is capable of stabilizing nonlinear incompressible Navier-Stokes equations with an inexact linearization.

In order to apply control theory in small autonomous vehicles, mathematical models with small numbers of states are required for using the limited computational power in embedded programming. In this paper, we consider an artificial fishtail as an example for a complex mechanical system with a second-order large-scale model, which is derived by using the finite element method. To meet the above limitations, the several hundreds of thousands of degrees of freedom need to be reduced to merely a handful of surrogate degrees of freedom. We seek to achieve this task by various second-order model order reduction methods. All methods are applied on the fishtail’s matrices and their results are evaluated and compared in the frequency domain as well as in the time domain.

A case of sarcoidosis is reported, which began 25 years ago. As a late complication the patient developed compression syndrome of the superior caval vein, which was caused by swelling of the lymph nodes. After surgical treatment, which was carried out by removing the regional lymph nodes and implantation of a dacron prosthesis, stenosis occurred within the proximal part of the prosthesis, which was treated by the PTA technique according to the Dotter method.