A truncated model reduction method via cross Gramian and optimization

This paper explores a truncated model reduction (TMR) approach for generalized linear systems and linear systems, leveraging cross Gramian. For generalized linear systems, an optimization problem is carefully constructed, factoring in the symmetrizer and the system matrices. The convexity characteristics of the objective function with respect to each variable are analyzed, followed by the rigorous derivation of its gradient. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is then employed to efficiently solve the optimization problem, facilitating the construction of an approximate generalized linear system. Subsequently, the cross-Gramian-based TMR approach is applied to obtaining the reduced-order model (ROM), and an investigation is conducted into the H_∞ error bound between the full-order model and the ROM. Moreover, the proposed model reduction (MR) technique is adapted for linear systems, with a series of detailed discussions presented. This paper successfully bridges the gap between non-symmetric and symmetric systems in reduction, rendering the cross-Gramian-based TMR method applicable to non-symmetric systems. Finally, several numerical examples are provided, which effectively illustrate the efficiency of the proposed MR method.