A mixed model reduction method based on the symmetrizer for two classes of dynamical systems

Abstract

In regard of the cross Gramian and the block Arnoldi process, this paper focuses on mixed model reduction for two classes of dynamical systems. By studying the correlation between the cross Gramian and the Hankel singular values, model reduction via the cross Gramian is investigated for symmetric regular linear systems. Considering the non-symmetric regular linear system, associated with the symmetrizer, a convex optimization problem is established to obtain a symmetric system. Then, the non-symmetric regular linear system is divided into two parts, in which one is determined by the symmetric system and another is a non-symmetric counterpart. Model reduction via the cross Gramian is employed to reduce the first part. According to the structure of the non-symmetric counterpart, a modified version of the block Arnoldi process is developed to reduce the non-symmetric counterpart. The H2 error bound between the original system and the reduced system is discussed. This model reduction approach avoids partitioning the non-symmetric system into several single-input and single-output systems. Moreover, the proposed model reduction method can also be applied to the non-symmetric linear systems. Finally, the proposed model reduction approach is illustrated by a 2D convective thermal flow benchmark and an optimal cooling problem of steel profiles.