Performance Analyses of Four-Instant Discretization Formulas With Application to Generalized-Sylvester-Type Future Matrix Equation

For solving generalized-Sylvester-type future matrix equation (GS-type FME) effectively, the continuous zeroing neural network (ZNN) model should be discretized by appropriate discretization formula. In this paper, theoretical upper bound of truncation errors of four-instant discretization formulas is presented, of which the upper bound is
$O(g^{2})$
. In addition, an inspirational method to decrease the truncation error of general discretization formula is proposed and analysed. Specifically, first of all, on the basis of Taylor expansion, different four-instant discretization formulas are presented by exploiting the higher-order derivative elimination (HODE) methods, which include second-order derivative elimination (SODE) method, third-order derivative elimination (TODE) method and fourth-order derivative elimination (FODE) method. Then, the theoretical upper bound of truncation errors of these discretization formulas is presented, investigated and analysed with the theoretical analyses provided. Secondly, by analyzing the parameter of general discretization formula, we propose and analyse an inspirational method to decrease the truncation error of general discretization formula. Thirdly, by exploiting the general discretization formula to discretize continuous ZNN model for solving GS-type FME, general discrete ZNN model is presented. Finally, an illustrative numerical experiment is presented to substantiate the effectiveness of theoretical results.