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

Zhangqiang Yin,Yuanhang Chen,Rui Xue,Kang Liu,Jian Li,Yang Shi

doi:10.1109/access.2019.2948082

Zhangqiang Yin, Yuanhang Chen + Show 4 more

Open Access

https://doi.org/10.1109/access.2019.2948082

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Journal: IEEE Access	Publication Date: Jan 1, 2019
Citations: 11	License type: CC BY 4.0

Affiliation: Yangzhou University, Xinyang Normal University

Abstract

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(g^{2})$ </tex-math></inline-formula> . 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.

Highlights

In recent decades, matrix equation solving problem, which is one of fundamental and significant problems in some science and engineering areas, e.g., control system design [1], [2], robotics [3], [4] and image-processing [5], has been investigated by many researchers
Theorem 4: The general discrete zeroing neural network (ZNN) model (9) is consistent and convergent, which converges with the order of its truncation error being O(g3)
In this paper, theoretical upper bound of truncation errors of four-instant discretization formulas has been presented, of which the upper bound is O(g2)

Summary

INTRODUCTION

Matrix equation solving problem, which is one of fundamental and significant problems in some science and engineering areas, e.g., control system design [1], [2], robotics [3], [4] and image-processing [5], has been investigated by many researchers. Based on the above analyses on different discretization formulas, the theoretical upper bound of their truncation errors is proposed. Based on the above analyses, for solving generalized-Sylvestertype future matrix equation (GS-type FME), we propose a general discrete ZNN model. 1) In this paper, theoretical upper bound of truncation errors of four-instant discretization formulas is presented and analysed. Theoretical results present the facts that the presented general discrete ZNN model converges with the order of its truncation error being O(g3) with the effective domain of parameter a being a ∈ (−∞, −1) ∪ (1/3, +∞). FOUR-INSTANT DISCRETIZATION FORMULAS a series of theoretical analyses of four-instant discretization formulas are proposed

THEORETICAL UPPER BOUND

NUMERICAL EXPERIMENT

CONCLUSION

Result