A geometric analysis of model reduction of linear systems
In this thesis we study the model reduction problem in terms of the geometric concepts of linear system theory. By appropriate selection of reducing subspaces, useful lower-order system models can be achieved. The reducing subspaces can be chosen as parts of a system which are "most" and "least" controllable or observable; retaining, of course, the most controllable/observable subspace for model reduction. We review results showing how several measures of controllability and observability can provide this information. Balanced, Jordan canonical form, and dual GHR representations are shown to be state space realizations which naturally identify the reducing subspaces based on these measures. Several results unifying these methods are given.
In another approach, we show that the reducing subspaces can be chosen such that after completing model reduction, a number of Markov parameters and time moments of the full system are retained by the reduced order model. We show how the dual GHR can be used as a tool which identifies these subspaces and state space realizations which naturally display them. Along these lines, a connection between model reduction in the state space and second-order systems is established, particularly the reduction of structures via the Lanczos algorithm.