Model Reduction in Parallelization Based on Equivalent Transformation of Block Bi-Diagonal Toeplitz Matrices for Two-Dimensional Discrete-Time Systems

This study proposes a parallel model reduction method for two-dimensional discrete-time systems, utilizing Krawtchouk moments and equivalent transformation. This work makes two significant contributions. First, we introduce a projection subspace that is independent of the input as well as of the Krawtchouk parameters, thus ensuring robustness. Second, we propose an efficient parallel algorithm for computing the basis of the projection subspace. With the difference relation of Krawtchouk polynomials and the analytic identity theorem, we obtain the explicit formula for the Krawtchouk moments of the state, which is input-dependent and Krawtchouk-parameter-dependent. We derive a projection subspace that is independent of both input and Krawtchouk parameter, such that it is equivalent to the subspace spanned by the Krawtchouk moments. Further, we propose a parallel strategy based on the equivalent transformation of the block bi-diagonal Toeplitz matrices with bi-diagonal blocks to compute the basis of the projection subspace, facilitating acceleration of the model reduction process on high-performance computers. Moreover, we analyze the Krawtchouk moment invariants of the proposed parallel method. Finally, the effectiveness of the proposed method is illustrated by two numerical examples.