Optimale interpolations-basierte Modellreduktion

This dissertation is devoted to the development and study of new techniques for model reduction of large-scale linear time-invariant dynamical systems. The behavior of processes in electrical networks, mechanics, weather prediction and many others can be described by high-dimensional systems of linear ordinary differential or difference equations. Model reduction methods can then be helpful as they provide automatic processes which construct a reduced-order system whose input-output behavior approximates the behavior of the original system. Most of the current methods are designed for approximating asymptotically stable systems. However, some processes such as weather development are unstable. In this thesis new interpolation-based methods are proposed that aim to compute an optimal reduced model for stable as well as for unstable systems. For these optimization problems tangential interpolation-based first order necessary optimality conditions are derived. On the basis of the established theory an iterative algorithm is proposed which, if it converges, provides a reduced system that satisfies the aforementioned first order conditions. In numerical experiments, the accuracy of the new method is illustrated and compared with other existing techniques. The benefit of the new approach to model reduction of unstable systems is also demonstrated on example of shallow water models. In case of systems with large number of unstable poles, the method is shown to be significantly better than other approaches