On krylov matrices and controllability of n‐dimensional linear time‐invariant state equations

Abstract

The classical Rosenbrock's algorithm (based on the Gauss elimination method) for n-dimensional linear time invariant state equation matrices is analysed and modernized. The method of orthogonal similarity reduction to block Hessenberg form is used to assure numerical stability. The updated version of the Rosenbrock's algorithm is then justified in a very easy way using properties of Krylov matrices. Additionally, this algorithm can be used to determine an equivalence transformation which converts an n-dimensional linear state equation into a controllable form (or a time invariant one into an equivalent observable form). It is advantageous for big and medium size problems and can be easily parallelized. Numerical examples are presented.