Elsevier

Information Sciences

Volume 524, July 2020, Pages 216-228
Information Sciences

A parallel computing method based on zeroing neural networks for time-varying complex-valued matrix Moore-Penrose inversion

https://doi.org/10.1016/j.ins.2020.03.043Get rights and content

Highlights

  • •

    The complex-valued zeroing neural network (CVZNN) model, activated by the complex-valued saturation-allowed activation functions, is proposed for solving the dynamic complex-valued matrix Moore-Penrose inverse.

  • •

    The convergence performance of the proposed CVZNN model for solving the dynamic complex-valued matrix Moore-Penrose inverse problem is analyzed by using Lyapunov stability theory.

  • •

    The robustness of the CVZNN model under noise interference, i.e., the time-varying bounded noises and constant noises, is investigated, separately.

  • •

    Simulation results and theoretical analysis verify the validity and superiority of the proposed CVZNN model for solving the dynamic complex-valued Moore-Penrose inverse problem.

Abstract

This paper analyzes the existing zeroing neural network (ZNN) models from the perspective of control theory. It proposes an exclusive ZNN model for solving the dynamic complex-valued matrix Moore-Penrose inverse problem: the complex-valued zeroing neural network (CVZNN). Then, a method of constructing a special type of saturation-allowed activation function is defined, which relaxes the convex constraint on the activation function when constructing the ZNN model. The convergence of the CVZNN model activated by proposed saturation-allowed functions is analyzed. Besides, the robustness of the CVZNN model under different types of noise interference is investigated based on the perspective of the control theory. Finally, the effectiveness and superiority of the CVZNN model are illustrated by simulation experiments.

Introduction

The dynamic complex-valued matrix Moore-Penrose inverse problem is an essential branch of the matrix theory and is universally used in scientific research [18] and engineering applications, such as robotic kinematics [38], control theory [14], digital image processing [2], [3], and communication engineering [7]. Since it is applied to many fields, the issue of accurately solving the complex-valued matrix Moore-Penrose inverse with dynamic parameters is received more considerable critical attention [9], [10], [11]. In recent researches and investigations, some numerical algorithms are presented to solve the above problem [8], [24]. For example, previous researchers have reported that the Newton iteration method can be used to calculate the matrix Moore-Penrose inverse. Courrieu et al. [4] presents an algorithm based on the full rank Cholesky decomposition to calculate the matrix Moore-Penrose inverse problems.

Newton iteration method is a classical numerical algorithm for solving discrete-time problems [23]. From the control theory, the Newton iteration method is a proportional feedback controller. Obviously, according to the control theory, a controller only using the proportional feedback cannot control a system with time-varying parameters in a predictive manner, thus leading to lagging errors. Recurrent neural network (RNN) is usually designed as an ordinary differential function when solving the zero-finding problem [20], [22], [27]. The RNN model needs to start from an arbitrary initial value, continue to evolve along a given direction, and recursively calculate the estimated value of each point until it converges to the required precision. Accordingly, the evolution direction of algorithm needs to be modified according to the input state, forcing the residual error to decrease to zero as time goes by [40]. Zeroing neural network (ZNN) as a parallel computing method [5], [17], [30] is an essential component in RNNs and plays a vital role in the linear computation problems [29], [31], [34] or optimization problems [6], [15], [25]. For example, Wei et al. proposes and investigates three kinds of ZNN models to solve the real domain Moore-Penrose inverse problem of non-full rank matrices [28]. Jin et al. proposes a systematic method based on control theory to study the ZNN model and gives a plan to analyze the stability and convergence of ZNN model [13]. It is worth further attention that as what we have mentioned above, at the present stage, most researches aim at the real-valued issues. There are few literatures devoted to solving the complex-valued matrix Moore-Penrose inverse problems, not to mention the dynamic complex-valued matrix Moore-Penrose inverse. Thus, it is urgent and necessary to design a new robust solving framework for this issue.

Based on the discussions proposed above, a modified complex-valued ZNN (CVZNN) model with saturation-allowed activation functions for solving the Moore-Penrose inverse of the dynamic complex-valued matrix is proposed [16], [33], [36]. This parallel computing model transforms the dynamic complex-valued matrix Moore-Penrose inverse problem into a complex-valued zero-finding problem of a linear equation. Then, this paper introduces two methods to extend the real-valued activation functions into the complex-valued ones, and defines the saturation-allowed activation functions mentioned above based on the F-norm. The CVZNN model activated by saturation-allowed activation function proposed in this paper fixes the defect that the original ZNN model converges when time approaches infinity, that is to say, the modified ZNN model converge in finite time [1], [37], [41].

In this paper, from the perspective of control theory, we construct a Lyapunov function candidate and prove the real solution to the proposed CVZNN model globally converges to its theoretical solution in finite time by mathematical derivation using the above created Lyapunov function [26], [32]. Whereafter, the robustness of the proposed CVZNN model is analyzed [35]. Further, we analyze and discuss the convergence properties of the CVZNN model perturbed by constant noises [21] and time-varying noises, respectively. Finally, this paper gives several examples for finding the complex-valued dynamic matrix Moore-Penrose inverse and analyzes the convergence performance of the proposed CVZNN model under the condition of using different activation functions or different regulation parameters. At the end of the introduction, the main contributions of the paper are listed as follows.

  • •

    The complex-valued zeroing neural network (CVZNN) model, activated by the complex-valued saturation-allowed activation functions, is proposed for solving the dynamic complex-valued matrix Moore-Penrose inverse.

  • •

    The convergence performance of the CVZNN model for solving the dynamic complex-valued matrix Moore-Penrose inverse problem is analyzed by using the Lyapunov stability theory.

  • •

    In this paper, the robustness of the CVZNN model under noise interference, i.e., the time-varying bounded noises and constant noises, is investigated separately.

  • •

    Simulation results and theoretical analysis verify the validity and superiority of the proposed CVZNN model for solving the dynamic complex-valued Moore-Penrose inverse problem.

Section snippets

Problem formulation and solving models

The main purpose of this paper is to design and construct a complex-valued zeroing neural network (CVZNN) for solving the dynamic complex-valued matrix Moore-Penrose inverse (DCMPI) problem. In this section, some basic definitions are given, and formulas for the CVZNN model are derived.

Complex-valued saturation-allowed activation function

In this section, we will define a designing method for the saturation-allowed activation function. In this paper, we prove that CVZNN model (4) using a saturation-allowed activation function can globally converge to the theoretical solution to DCMPI (1). This article defines a projection based on the Frobenius norm: PΛ(K)=argminG∈Λ∥G−K∥F with 0 ∈ Λ. Therefore, E˙(t)=−βΨ(E(t)) can be rewritten asE˙(t)=−β(PΛ(Er(t))+iPΛ(Ei(t))),E˙(t)=−βPΛ(∥E(t)∥F)exp(i·arg(E(t))),where i denotes the imaginary

Stability on the CVZNN model

This section investigates and discusses the robustness of the proposed CVZNN model (5) under the interference of noises. It involves complex-valued constant noises ζ and bounded random noises ζ(t), respectively. Above all, CVZNN model (5) activated by a saturation-allowed activation function and under bounded random noises interference ζ(t) is analyzed. For further discussion, the following theory is given.

Illustrative examples

In this section, several examples of solving the DCMPI problem will be given to illustrate the performance of CVZNN model (5) for the noise-free case and cases with different kinds of noises. Meanwhile, the experimental results and the superiority of the proposed approach will be analyzed and dissertated in detail.

Application to inverse kinematic motion generation

The robot manipulator kinematics is widely researched and used in many engineering applications and scientific fields. In this section, in order to prove the validity and superiority of the CVZNN model (5) proposed in this paper, the corresponding simulation experiments are constructed based on the PUMA 560 manipulator. The relationship between the end-effector r∈Cm of the PUMA 560 manipulator and the joint angle ϖ(t)∈Cn satisfies the formula Υ(ϖ)=r, of which Υ(·) stores the mechanical and

Conclusions

In this paper, a complex-valued dynamic zeroing neural network (CVZNN) model for solving the problem of dynamic complex-valued Moore-Penrose inverse has been proposed. To relax the convex constraint on the activation function, a kind of special saturation-allowed activation function has been offered, and three concrete examples for constructing these activation functions have been given. Then, the global convergence of the CVZNN model has been proved with the theoretical analysis. Furthermore,

Declaration of Competing Interest

We wish to draw the attention of the Editor to the following facts which may be considered as potential conflicts of interest and to significant financial contributions to this work. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the other of authors listed in the manuscript has been approved by all of us and that the second author prepared

CRediT authorship contribution statement

Xiuchun Xiao: Investigation, Software, Methodology, Conceptualization, Writing - original draft. Chengze Jiang: Validation, Formal analysis, Visualization, Software. Huiyan Lu: Resources, Writing - review & editing, Supervision, Data curation. Long Jin: Resources, Conceptualization, Methodology, Writing - review & editing. Dazhao Liu: Conceptualization, Validation, Formal analysis, Visualization, Software. Haoen Huang: Writing - review & editing. Yi Pan: Writing - review & editing.

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    This work is supported by the Fund of Southern Marine Science and Engineering Guangdong Laboratory of Zhanjiang, China (with no. ZJW-2019-08), by the Innovation and Strength Project in Guangdong Province, China (Natural Science) (with no. 230419065), by the Key Lab of Digital Signal and Image Processing of Guangdong Province, China (with no. 2019GDDSIPL-01), by the Industry-University-Research Cooperation Education Project of Ministry of Education (with no. 201801328005), by the Guangdong Graduate Education Innovation Project, Graduate Summer School (with no. 2020SQXX19), by the Guangdong Graduate Education Innovation Project, Graduate Academic Forum (with no. 2020XSLT27), by the Doctoral Initiating Project of Guangdong Ocean University (with no. E13428), and also by the Special Project in Key Fields of Universities in Department of Education of Guangdong Province, China (with no. 2019033).

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