Two-Sided Arnoldi in Order Reduction of Large Scale MIMO Systems

In order reduction of high order linear time invariant systems based on two-sided Krylov subspace methods, the Lanczos algorithm is commonly used to find the bases for input and output Krylov subspaces and to calculate the reduced order model by projection. However, this method can be numerically unstable even for systems with moderate number of states and can only find a limited number of biorthogonal vectors. In this paper, we present another method which we call Two-Sided Arnoldi and which is based on the Arnoldi algorithm. It finds two orthogonal bases for any pair of Krylov subspaces, with one or more starting vectors. This new method is numerically more robust and simpler to implement specially for nonsquare MIMO systems, and it finds a reduced order model with same transfer function as Lanczos. The Two-Sided Arnoldi algorithm can be used for order reduction of the most general case of a linear time invariant Multi-Input-Multi-Output (MIMO) systems. Furthermore, we present some suggestions to improve the method using a column selection procedure and to reduce the computational time using LU-factorization.