H₂ Optimal Linear Model Order Reduction Methods Based On The Cross Gramian

In recent years,the simulation,optimization and control of large-scale dynamical systems have been extensively applied.These systems are generally be described by differential equations.Since the simulation of these systems required large computational cost.Then some engineering and mathematical researchers engage in how to reduce the computational complexity and data storage in the simulation.Model order reduction is an effective way to deal with the of approximation large-scale systems.It generates a smaller system to approximation the large-scale system,sufficiently reduces the computation cost,and save the simulation time.Meanwhile,some properties are preserved in the reduced order system.For the single input and single output?SISO?continuous system,the H₂optimal model order reduction method based on the cross Gramian and the Riemannian manifold is proposed.First,the H₂norm of the error system is expressed by the cross Gramian,and regarded as the cost function of transformation matrices.Next,by introducing the Riemannian manifold,the cost function is viewed as a nonnegative real function defined on the Riemannian manifold.Then some optimization techniques are employed to perform the linear search on the Riemannian manifold and seek a couple of transformation matrices,which make the cost function as small as possible.Applying this method to the linear system,a more accurate reduced order system is obtained.This paper also investigates the H₂optimal model order reduction method for the large-scale multiple-input and multiple-output?MIMO?discrete dynamical systems is also discussed.First,the MIMO discrete system is resolved into several SISO subsystems and the H₂norm of the subsystem is derived by the cross Gramian.At the same time,the H₂norm of the original system is represented by the cross Gramian according to the relation between the subsystem and the original system.Moreover,the vector transport and the retraction on the Riemannian manifold are introduced,and the geometric conjugate gradient method for the H₂optimal model reduction of the subsystem is proposed.Finally,the proposed method is individually applied to these subsystems,and the reduced order system of the original system is reformulated by all the reduced order subsystems.