Structure preserving balanced proper orthogonal decomposition for second‐order form systems via shifted Legendre polynomials

Abstract

This study considers structure preserving balanced proper orthogonal decomposition for second-order form systems via shifted Legendre polynomials. The proposed approach is to use time interval empirical Gramians of the first-order representation, which are constructed from impulse responses by solving block tridiagonal linear systems, to generate approximate balanced system for the large-scale second-order form system. The balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states corresponding to the small approximate singular values. Furthermore, in combination with the dominant subspace projection method, the authors modify the reduction procedure to alleviate the shortcoming of the above method, which may unexpectedly lead to unstable systems even though the original one is stable. The stability preservation of the reduced model is briefly discussed. Finally, the effectiveness of the proposed methods is demonstrated by two numerical experiments.