Two-sided dimension reduction for linear control systems of kth-order form

In this article, a two-sided structure-preserving dimension reduction method for linear control systems of kth-order form is discussed. We first recall the Petrov–Galerkin framework of dimension reduction for the kth linear systems. Then the moments of systems at a given frequency point s₀ are discussed. According to the recurrence form of the moments, we obtain the input Krylov subspace and the projection matrix V. Via dually designing output Krylov subspace, we obtain the output Krylov subspace basis matrix W. Hence, the two-sided dimension reduction method is obtained via combination V with W. Afterwards one can obtain the result that the number of moments matched is twice as much as the column dimension of the projection matrix. Finally, one numerical experiment is performed to verify the proposed dimension reduction method.