Three variants of the Lanczos algorithm for generating Krylov sequences and three examples of linear control theory applications of those were given. The systems considered are linearized systems arising from physical models with up to thousand states. Let us remind that the Kalman controllability matrix is a slight generalization of the Krylov sequence. The block Arnoldi algorithm is used; it is based on the QR factorization algorithm of Kublanovskaa and Francis. Model reductions based on balanced realization and parameter matching are considered.