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A method to preserve stability in parametric model order reduction by matrix interpolation for the whole parameter range is proposed for high-order linear timeinvariant systems. In the first step, system matrices of the highdimensional parameter-dependent system are computed for a discrete set of parameter vectors. The local high-order systems are reduced by a projection-based reduction method. Secondly, the reduced models are made contractive by solving lowdimensional Lyapunov equations. Thirdly, they are transformed into a consistent set of generalized coordinates for accurate interpolation results. These three steps are done offline and the matrices of the local systems are stored. Finally, a stable reduced order model for a new parameter vector can be calculated online by interpolating the precomputed matrices of the local low-dimensional models. We show that this approach works without any limiting conditions concerning the structure of the large-scale model and is suitable for real-time applications.

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This thesis deals with model order reduction of parameter-dependent systems based on interpolation of locally reduced system matrices. A Black-Box method is proposed that automatically determines the optimal design parameters and delivers a reduced system with desired accuracy. In addition, the method is extended to stability preservation and interpolation for high-dimensional parameter spaces.

Dynamic systems with time-varying parameters arise in numerous industrial applications, e.g. in structural dynamics or systems with moving loads. A spatial discretization of such systems often leads to high-dimensional linear parameter-varying models, which need to be reduced in order to enable a fast simulation. In the following we present time-varying parametric model order reduction (p(t)MOR) based on matrix interpolation and apply this novel framework to a system with moving load. Model Reduction:• p(t)MOR by Matrix Interpolation applied• Order of locally reduced systems:Further study:• Interpretation of the new matrix Projective pMOR: Choose appropriate projection matrices to approximate the state-vector by . Application for Systems with Moving Loads Projective p(t)MOR:Analogously, we aim to approximate the state-vector by Systems with Moving Loads:• position of the acting load varies with time

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A black-box method for parametric model order reduction is presented that includes method selection, model refinement and error prediction using a cross-validation-based error indicator. The method is demonstrated for the interpolation of reduced system matrices.

In this paper controllers and observers are designed for high-dimensional parameter-dependent LTI systems. The application of parametric model order reduction by matrix interpolation is proposed in order to use common methods of control. In the offline phase, the parameter space is sampled and a set of locally reduced systems is obtained using projection-based model order reduction, which results in a database of system matrices. In the online phase, a reduced system can be calculated for a desired parameter vector by interpolating the system matrices from the database. Controllers and observers can be obtained for the interpolated system using common methods of control design. The approach is demonstrated through a practical example on a test rig for a gantry crane operating with different loads. Zusammenfassung: In diesem Beitrag werden Regler und Beobachter für hochdimensionale, parameterabhängige Systeme bestimmt. Um gängige Verfahren zur Regler-bzw. Beobachterauslegung nutzen zu können, wird die Anwendung von parametrischer Modellreduktion basierend auf Matrixinterpolation vorgeschlagen. In der Offline-Phase wird für ein Raster des Parameterraums eine Menge von lokal reduzierten Systemen mittels projektionsbasierter Modellreduktion berechnet, was eine Datenbasis von Systemmatrizen ergibt. In der Online-Phase erhält man das reduzierte Modell für einen gewünschten Parametervektor durch Interpolation der Systemmatrizen der Datenbasis. Für das interpolierte System können dann mit den gängi-gen Verfahren der Regelungstechnik Regler bzw. Beobachter ausgelegt werden. Die Methode wird durch ein praktisches Beispiel an einem Prüfstand, der das Verhalten eines Brückenkrans mit unterschiedlichen Lastmassen nachbilden soll, unterstützt. Schlüsselwörter: Parametrische Modellreduktion, Matrixinterpolation, modellprädiktive Regelung, flachheitsbasierte Folgeregelung, Brückenkran.