A Black-Box method for parametric model order reduction based on matrix interpolation with application to simulation and control

Accurate modeling of dynamical systems can lead to large-scale systems of ordinary differential equations. For many applications, these systems additionally depend on parameters, which, for example, describe geometric attributes. In order to be able to perform tasks such as optimization, simulation or control, it is necessary to reduce the computational effort of solving the system. For that reason, methods of parametric model order reduction have been developed which reduce the order of the large-scale system and at the same time preserve its parametric dependencies. This thesis deals with a novel approach for model order reduction of parameterdependent systems. The method calculates a set of reduced systems with individual projection matrices for a number of grid points. Subsequently, the bases of the local systems are adjusted. In order to obtain a model for a new parameter value, the reduced system matrices are interpolated. A general framework is proposed which can be applied by the user as a construction kit. It illustrates the necessary steps and presents the different options for each step. In particular, three approaches for adjusting the bases are proposed and it is pointed out that the user can choose between a large number of candidate interpolation methods and manifolds. In many cases, the user lacks insight into the physics of the model in order to decide on the options leading to the most accurate reduced model. Hence, a Black-Box method is proposed that automatically determines the optimal interpolation method and grid points and delivers a reduced system with the desired accuracy. In addition, the general framework is extended to stability preservation. For this, a low-order optimization problem based on semidefinite programming is solved for every grid point. Moreover, as common gridbased approaches lead to costs that grow exponentially with the number of parameters, sparse-grid-based interpolation is introduced for the interpolation of system matrices. This allows the user to apply the framework to high-dimensional parameter spaces. The general framework is also extended to the interpolation of differently-sized reduced models. For this, a resizing procedure, which is based on pseudoinverses, is presented. The effectiveness of the proposed methods is shown for examples from microsystems technology, mechatronics and structural mechanics. In addition, the application of the interpolation-based framework to control is demonstrated for a system on a test rig.