Reduced-order modeling methods via bivariate discrete orthogonal polynomials for two-dimensional discrete state-delayed systems

Model order reduction (MOR) techniques via bivariate discrete orthogonal polynomials are developed for two-dimensional (2-D) discrete state-delayed systems. The mathematical model of 2-D discrete systems is established on the basis of the Fornasini-Marchesini local state-space model. First, the forward shift transformation matrix and the backward shift transformation matrix of classical discrete orthogonal polynomials of one variable are algebraically deduced. Moreover, MOR is investigated for 2-D discrete state-delayed systems with single-delay. This system is expanded in terms of bivariate discrete orthogonal polynomials, then the coefficient matrix is calculated by a linear matrix equation. The resulting coefficient matrix is exploited to define the orthogonal projection matrix, so the reduced-order system is obtained. Theoretically, the output of the reduced-order system can match a certain number of the expansion coefficients of the output of the original system. Meanwhile, the MOR methods are extended to 2-D discrete state-delayed systems with multiple-delay as well. Finally, one illustrative example is provided to verify the feasibility of the proposed methods.