Optimal interpolation-based model reduction

This dissertation is devoted to the development and study of new techniques in the field of model reduction for large-scale linear time-invariant (LTI) dynamical systems. The behavior of processes in electrical networks, mechanics, aeronautics, civil engineering, micro-electro-mechanical-systems, weather prediction and many others can be described by mathematical models. Such models are usually determined by suitable systems of partial differential equations. Their linearization and discretization by means of finite element or finite difference methods lead to high-dimensional systems of linear ordinary differential or difference equations. The number of resulting equations depends on the quality of the discretization and is typically very large. It can easily reach a few millions. Model reduction methods can then be helpful as they provide automatic processes which construct a reduced order system of the same form but of lower complexity, 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. A standard approach to reduce the order of an unstable system is to divide the system into a completely stable and completely unstable subsystems and then apply model reduction only to the stable part. This approach does not always give a satisfactory result. In contrast, in this thesis new interpolation-based methods are proposed that aim to compute an optimal reduced-order 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, a MIMO (multiple-input-multiple-output) Iterative Rational Interpolation Algorithm (MIRIAm) is proposed which, if it converges, provides a reduced-order system that satisfies the aforementioned first order necessary conditions. For asymptotically stable systems several different sets of the first order necessary optimality conditions have already been developed. Among them, the gramian-based conditions given by Wilson and Hyland-Bernstein are of special interest. In this thesis a generalization of these conditions to unstable continuous-time and discrete-time LTI dynamical systems is introduced and shown to be equivalent to the tangential interpolation-based first order necessary conditions. The major drawback of the gramian-based approach is its high computational cost caused by the need for solving two large Lyapunov equations for continuous-time systems and of two Stein equations for discrete-time systems in each iteration. The complexity of the tangential interpolation-based approach (MIRIAm), introduced in this thesis, is much smaller. In numerical experiments using an example of the CD player, the accuracy of the new method is illustrated and compared with other existing methods. The benefit of the new approach to model order reduction of unstable systems is also demonstrated in several numerical experiments with shallow water test models. In case of systems with large number of small unstable poles, the method is shown to be significantly better than the currently used approach based on the decomposition of the system into stable and unstable part.