Krylov Subspace Methods in Linear Model Order Reduction: Introduction and Invariance Properties

In recent years, Krylov subspace methods have become popular tools for computing reduced order models of high order linear time invariant systems. The reduction can be done by applying a projection from high order to lower order space using bases of some subspaces called input and output Krylov subspaces. One aim of this paper is describing the invariancies of reduced order models using these methods for MIMO systems: The effects of changing the starting vectors of Krylov subspaces or its bases and changing the representation and the realization of original state space model on the input-output behaviour of reduced order system are discussed. The differences between one-sided Krylov methods (like Arnoldi algorithm) and two-sided methods (like Lanczos algorithm) with respect to invariancies are pointed out. Furthermore, it is shown that how a matching of the moments and Markov parameters of original and reduced order models can be achieved at the same time. Finally, a new two-sided Arnoldi algorithm is suggested.