Model order reduction of linear time-varying systems via shifted Legendre polynomials

In this paper, we present a model order reduction algorithm for linear time-varying systems via shifted Legendre polynomials. The main idea of the approach is to use approximate low-rank factors of the finite-time empirical Gramians, which are constructed from the the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, to generate approximate balanced system for the large-scale system. Then, the reduced-order model is constructed by truncating the states corresponding to the small approximate Hankel singular values (HSVs). Finally, a numerical experiment is given to demonstrate the effectiveness of the proposed methods.