Model reduction based on parallelisation and Krawtchouk polynomials for discrete time systems with delays

Discrete time-delay systems arise widely in engineering and scientific applications. These systems are often modelled as high-dimensional dynamical systems, and the presence of delays increases the computational complexity and cost of both simulation and analysis. To alleviate this challenge, this paper proposes a new model reduction method in a parallel manner tailored for discrete time-delay systems. The proposed method begins with the explicit difference relations of Krawtchouk polynomials and a structural analysis of the shift-transformation matrices. We expand the original system over a Krawtchouk polynomial basis, which yields a system of linear algebraic equations characterised by a block lower triangular Toeplitz matrix. Further, applying the block discrete Fourier transform to the involved block α-circulant matrices, we design a parallel strategy to efficiently construct the projection basis used for reducing discrete time-delay systems. This is the main contribution of this paper. We provide rigorous theoretical results on the invertibility of the block α-circulant matrices and present error estimation bounds to ensure the validity and stability of the reduced model. Numerical examples demonstrate that our method achieves accurate approximation while offering substantial computational speed-ups, especially for large-scale time-delay systems.