Abstract |
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The modelling of physical processes gives rise to mathematical systems of increasing complexity. Good mathematical models have to reproduce the physical process as precisely as possible while the computing time and the storage resources needed to simulate the mathematical model are limited. As a consequence, there must be a tradeoff between accuracy and computational constraints. At the present time, one is often faced with systems that have an unacceptably high level of complexity. It is then desirable to approximate such systems by systems of lower complexity. This is the Model Reduction Problem. This thesis focuses on the study of new model reduction techniques for linear systems.
Our objective is twofold. First, there is a need for a better understanding of Krylov techniques. With such techniques, one can construct a reduced order transfer function that satisfies a set of interpolation conditions with respect to the original transfer function. A study of the generality of such techniques and their extension for MIMO systems via the concept of tangential interpolation constitutes the first part of this thesis. This also led us to study the generality of the projection technique for model reduction.
Most large scale systems have a particular structure. They can be modelled as a set of subsystems that interconnect to each other. It then makes sense to develop model reduction techniques that preserve the structure of the original system. Both interpolation-based and gramian-based structure preserving model reduction techniques are developed in a unified way. Second order systems that appear in many branches of engineering deserve a special attention. This constitutes the second part of this thesis. |