Parallel Input-Independent Model Order Reduction for Discrete-Time Parametric Systems

Abstract

In this paper, based on discrete orthogonal polynomials and the block <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -circulant matrix, we explore a parallel input-independent model order reduction (MOR) method, which is suitable for the single-input discrete-time systems characterizing non-affine uncertainty about a scalar parameter. With the explicit difference relations of Charlier polynomials, Meixner polynomials, and Krawtchouk polynomials, the expansion coefficients of the state variable are obtained. Further, we derive an input-independent projection subspace, such that it is equivalent to the subspace spanned by the expansion coefficients for arbitrary input. Based on the block discrete Fourier transform (DFT) of the block <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -circulant matrix, a parallel strategy is proposed to compute the basis of the equivalent projection subspace. Then, the projection matrix is constructed and used to reduce discrete-time parametric systems. Moreover, we analyze the feasibility of the parallel strategy by presenting the invertibility of the block <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -circulant matrices and the corresponding error. Finally, the efficiency of the proposed method is illustrated by the numerical experiment.