Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems

Abstract

A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Laguerre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states with small approximate Hankel singular values (HSVs). The above method has a disadvantage that it may unexpectedly result in unstable systems although the original one is stable. Therefore, modified reduction procedure combined with the dominant subspace projection method is presented to alleviate the limitation. Finally, two numerical experiments are provided to demonstrate the effectiveness of the algorithms.