Enhanced discrete-time Zhang neural network for time-variant matrix inversion in the presence of bias noises

doi:10.1016/j.neucom.2016.05.010

Neurocomputing

Volume 207, 26 September 2016, Pages 220-230

https://doi.org/10.1016/j.neucom.2016.05.010 Get rights and content

Abstract

Inevitable noises and limited computational time are major issues for time-variant matrix inversion in practice. When designing a time-variant matrix inversion algorithm, it is highly demanded to suppress noises without violating the performance of real-time computation. However, most existing algorithms only consider a nominal system in the absence of noises, and may suffer from a great computational error when noises are taken into account. Some other algorithms assume that denoising has been conducted before computation, which may consume extra time and may not be suitable in practice. By considering the above situation, in this paper, an enhanced discrete-time Zhang neural network (EDTZNN) model is proposed, analyzed and investigated for time-variant matrix inversion. For comparison, an original discrete-time Zhang neural network (ODTZNN) model is presented. Note that the EDTZNN model is superior to ODTZNN model in suppressing various kinds of bias noises. Moreover, theoretical analyses show the convergence of the proposed EDTZNN model in the presence of various kinds of bias noises. In addition, numerical experiments including an application to robot motion planning are provided to substantiate the efficacy and superiority of the proposed EDTZNN model for time-variant matrix inversion.

Introduction

Matrix inversion is considered to be one fundamental problem widely encountered in science and engineering fields [1], [2], [3], [4], [5], such as optimization [1], robot control [2] and image processing [3]. In practice, the ability to invert matrix quickly and accurately determines the effectiveness of a computational tool [6]. Therefore, a lot of research efforts have been devoted to such a problem solving [7], [8]. Matrix inversion, especially for large scale cases, has been significantly advanced and its time complexity has been much reduced in past years [6], [7], [8], [9].

As a variant of constant matrix inversion, time-variant matrix inversion is becoming increasingly popular in recent years [10], [11], [12], [13]. Generally speaking, time-variant matrix inversion is more complicated than constant matrix inversion, and the models for time-variant matrix inversion must satisfy the urgent requirement of real-time computation. Note that traditional methods for constant matrix inversion may not satisfy the real-time computational requirement of time-variant matrix inversion [14], [15]. Specifically, the object matrix is varying with time, while time is inevitably consumed by each computational method. After the inverse of a time-variant matrix at a time instant is obtained, the matrix is not the original one.

With the characteristics of high-speed parallel processing and superiority in large scale online processing, neural networks have been widely employed in scientific computation and optimization [16], [17], [18]. Especially, recurrent neural networks (RNNs) have been presented and investigated as powerful alternatives to online scientific problems solving [19], [20], [21]. For solving online time-variant problems, including time-variant matrix inversion, Zhang neural network (ZNN), a special class of RNNs, was proposed [22]. Original Zhang neural network (OZNN) model is able to perfectly track time-variant solution for time-variant matrix inversion with the condition that the solving process is free of noises [10]. However, by considering that noises always exist and denoising may consume extra time, the pre-denoising method may not adapt well for online time-variant matrix inversion due to the urgent requirement of real-time computation in practice.

By considering the above situation, an enhanced Zhang neural network (EZNN) model is presented for time-variant matrix inversion [23]. Note that the EZNN model can perfectly track time-variant solution, and can suppress various kinds of bias noises, such as constant bias noise, bounded random bias noise and even (unbounded) linear-increasing bias noise, in one unified framework. Besides, as discrete-time models are convenient for numerical implementation on digital computers, the EZNN model and the OZNN model, which are continuous-time ZNN models, need to be discretized for practical realization [24], [25], [26]. Different from the OZNN model, the EZNN model contains an integral term. Thus, we may not obtain a convergent model by discretizing the integral term directly. To solve this problem, in this paper, new mathematical manipulations are employed to avoid discretizing the integral term directly. Then an enhanced discrete-time ZNN (EDTZNN) model is obtained for time-variant matrix inversion. Note that the EDTZNN model not only inherits the superiority of original discrete-time ZNN (ODTZNN) model [26] (i.e., the ODTZNN model can predict the inverse of time-variant matrix with high accuracy, or say, the inverse of time-variant matrix at a time instant can be obtained before/at that time instant), but also has the ability of suppressing various kinds of bias noises.

The remainder of this paper is organized into five sections. Section 2 presents the continuous-time ZNN models for time-variant matrix inversion, including the EZNN model and the OZNN model. In Section 3, by employing a new finite difference formula to discretize the EZNN model, the EDTZNN model is proposed. Moreover, the ODTZNN model is presented for comparison and the stability and convergence of EDTZNN model are analyzed. In Section 4, we analyze and investigate the EDTZNN model for time-variant matrix inversion in the presence of constant bias, bounded random bias noise and even linear-increasing bias noise. In Section 5, an application to robot motion planning is presented for further substantiating the efficacy of the EDTZNN model. Section 6 concludes the paper with final remarks. Before ending this section, it is worth pointing out here that the main contributions of this paper lie in the following facts.

•
The EDTZNN model that is able to handle various kinds of bias noises is firstly proposed for time-variant matrix inversion.
•
Theoretical analyses and results are presented to show the convergence of EDTZNN model.
•
Numerical experiments including an application to robot motion planning are presented to substantiate the efficacy and superiority of the proposed model.

Section snippets

Continuous-time ZNN models

To solve the problem of time-variant matrix inversion, the following definition equation is considered: $A (t) X (t) = I, with t \geq 0,$ where $A (t) \in R^{n \times n}$ is a nonsingular smoothly time-variant matrix, $X (t) \in R^{n \times n}$ is the unknown matrix to be obtained, and $I \in R^{n \times n}$ is the identity matrix. We assume that A(t) and its time derivative are uniformly bounded.

To monitor and control the solving process of (1), we define the following matrix-valued indefinite error function: $E (t) = A (t) X (t) - I \in R^{n \times n} .$ Each element e(t) of E(t)

Discrete-time ZNN models

As aforementioned, the EDTZNN model and ODTZNN model are easier to be developed into numerical algorithms [24]. In this section, a new finite difference formula is presented to approximate the first order time-derivative of f(t) with the truncation error being $O (g^{2})$ , where g denotes sampling period. Then the new finite difference formula is employed to discretize the continuous-time ZNN models [i.e., (4), (7)], and thus two models are obtained for time-variant matrix inversion in the presence

EDTZNN model with bias noises

There always exist noises in practice. In this section, EDTZNN model (10) in the presence of different kinds of discrete-time bias noises is analyzed for time-variant matrix inversion, such as constant bias noise, bounded random bias noise and even linear-increasing bias noise. To further substantiate the efficacy and superiority of model (10) in the presence of bias noises, numerical experiments are conducted. Moreover, the results generated by ODTZNN model (11) in the presence of bias noises

Application to robot motion planning

In this section, EDTZNN model (10) is applied to the robot motion planning in the presence of various kinds of bias noises. For comparison, ODTZNN model (11) is also applied to such a task.

In the simulation, a two-link planar robot manipulator is used, and the following equation is obtained: $ϕ (ϵ (t)) = ζ (t), with t \in [0, T],$ where $ϕ (\cdot)$ is the continuous forward-kinematics mapping function with known structure and parameters for a given manipulator [10], [24], [31]. In addition, $ϵ (t) \in R^{2}$ and $ζ (t) \in R^{2}$ denote

Conclusions

The EDTZNN model (10) has been proposed for the first time, and further analyzed and investigated for time-variant matrix inversion in this paper. For comparison, ODTZNN model (11) generated by discretizing the OZNN model (4) has been presented. Note that model (10) not only inherits the superiority of ODTZNN model (11), but also has the ability of suppressing various kinds of bias noises. For obtaining the model (10), the EZNN model (7) and a new finite difference formula has been presented.

Mingzhi Mao received the B.S., M.S., and Ph.D. degrees in the Department of Computer Science from Sun Yat-sen University, Guangzhou, China, 1988, 1998, and 2008, respectively. He is currently a professor at the School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China. His main research interests include intelligence algorithm, software engineering, and management information system.

References (32)

S. Gigola et al.
The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem
J. Comput. Appl. Math.
(2016)
Y. Fan et al.
An efficient computational approach for multiframe blind deconvolution
J. Comput. Appl. Math.
(2012)
L. Lu
Gram matrix of Bernstein basisproperties and applications
J. Comput. Appl. Math.
(2015)
X. Ma
Two matrix inversions associated with the Hagen–Rothe formula, their q-analogues and applications
J. Combin. Theory A
(2011)
G. Sharma et al.
A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA
Comput. Struct.
(2013)
D. Coppersmith et al.
Matrix multiplication via arithmetic progressions
J. Symbol. Comput.
(1990)
D. Guo et al.
Zhang neural network, Getz–Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots׳ kinematic control
Neurocomputing
(2012)
Y. Zhang et al.
Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion
Appl. Math. Comput.
(2011)
Y. Bai et al.
New stability criteria for recurrent neural networks with interval time-varying delay
Neurocomputing
(2013)
S. Wen et al.
Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays
Neurocomputing
(2012)

L. Li et al.

Exponential convergence and Lagrange stability for impulsive Cohen–Grossberg neural networks with time-varying delays

J. Comput. Appl. Math.

(2015)

S. Qin et al.

A new one-layer recurrent neural network for nonsmooth pseudoconvex optimization

Neurocomputing

(2013)

X. Wang et al.

A hybrid optimization-based recurrent neural network for real-time data prediction

Neurocomputing

(2013)

L. Jin et al.

Discrete-time Zhang neural network of $O (τ^{3})$ pattern for time-varying matrix pseudoinversion with application to manipulator motion generation

Neurocomputing

(2014)

N.H. Getz et al.

Dynamical methods for polar decomposition and inversion of matrices

Linear Algebra Appl.

(1997)

J. Chen et al.

Dual quaternion-based inverse kinematics of dexterous finger

J. Vibroeng.

(2014)

Cited by (52)

A new ZNN model for finding discrete time-variant matrix square root: From model design to parameter analysis
2023, Journal of Computational and Applied Mathematics
In recent years, artificial neural network technology has developed very rapidly. More and more experts and scholars use neural network technology to solve the related problems of applied mathematics. As a representative of the Hopfield neural network, zeroing neural network (ZNN) which can deal with the time-variant problem that may not be solved by traditional methods effectively. In this paper, to solve the problem of time-variant matrix square root, the continuous-time ZNN (CT-ZNN) model is presented based on the ZNN design formula. Next, to meet the requirements of computer simulation, the general square-pattern discretization (SPD) formula is applied to obtain the general discrete-time ZNN (G-DT-ZNN) model. Moreover, by changing the value of the selected parameter in the G-DT-ZNN model, it is presented that the effect of different values of the selected parameter on the precision of the G-DT-ZNN model. Finally, in the numerical experiments, three discrete time-variant matrix square root problems are presented as examples for verifying such theoretical results of the proposed G-DT-ZNN model.
General ELLRFS-DAZN algorithm for solving future linear equation system under various noises
2023, Neurocomputing
Citation Excerpt :
Compared with continuous models, discrete algorithms are more easily implemented on digital computers[12,16–18]. Generally speaking, the discrete advanced ZN (DAZN) algorithms can solve discrete time-dependent problems with future unknownness (or termed, future problems) to better satisfy the needs of strict real-time computing [5,19–21]. For example, in [5], two novel DAZN algorithms were proposed to solve future LES and future nonlinear equation system in noisy environments.
This work investigates the problem of future linear equation system under various noises. Firstly, a continuous advanced zeroing neurodynamic (CAZN) model under noise is developed to solve continuous linear equation system. Subsequently, by combining a general explicit linear left–right four-step (ELLRFS) formula with the CAZN model, a general ELLRFS discrete advanced zeroing neurodynamic (ELLRFS-DAZN) algorithm under noise is proposed to solve the future linear equation system. Theoretical analyses and results manifest the convergence performance of the general ELLRFS-DAZN algorithm under various noises. Moreover, numerical experimental results, including those based on a UR5 manipulator, validate the effectiveness and robustness of the general ELLRFS-DAZN algorithm under various noises. Numerical experimental results based on mobile acoustic source localization further substantiate the superiority of the general ELLRFS-DAZN algorithm under constant noise. Finally, physical experimental results based on a Kinova JACO² manipulator substantiate the practicability of the general ELLRFS-DAZN algorithm under bounded random noise.
Discrete-time ZNN-based noise-handling ten-instant algorithm solving Yang-Baxter-like matrix equation with disturbances
2022, Neurocomputing
Time-variant Yang-Baxter-like matrix equation (YBLME) with the disturbances of noises is the hotspot in various scientific disciplines. The existing research, either can not handle the noise, or rely on the build-in numerical algorithm provided by MATLAB. Therefore, it is necessary to further study the digital-computer discrete solution of this problem. In this paper, the authors propose a noise-handling ten-instant discrete solution for the time-variant YBLME. By defining the matrix-form error function, Zhang neural network (ZNN) is exploited to design the continuous-time noise-handling ZNN model. For potential digital hardware (i.e., digital computer) realization, a discrete-time ZNN-based noise-handling ten-instant (DTZNH10i) algorithm, which is capable of handling three types of noises, is proposed on the basis of a Zhang time discretization (ZTD) formula. The convergence and precision of the proposed DTZNH10i algorithm are investigated and discussed. Theoretical analyses show the correctness and effectiveness of the proposed algorithm. The paper offers three examples of time-variant YBLME with disturbances of constant, linear, and bounded random noises to illustrate the efficacy and convergence of the proposed algorithm.
Real-domain QR decomposition models employing zeroing neural network and time-discretization formulas for time-varying matrices
2021, Neurocomputing
This study investigated the problem of QR decomposition for time-varying matrices. We transform the original QR decomposition problem into an equation system using its constraints. Then, we propose a continuous-time QR decomposition (CTQRD) model by applying zeroing neural network method, equivalent transformations, Kronecker product, and vectorization techniques. Subsequently, a high-precision ten-instant Zhang et al. discretization (ZeaD) formula is proposed. A ten-instant discrete-time QR decomposition model is also proposed by using the ten-instant ZeaD formula to discretize the CTQRD model. Moreover, three discrete-time QR decomposition models are proposed by applying three other ZeaD formulas, and three examples of QR decomposition are presented. The experimental results confirm the effectiveness and correctness of the proposed models for the QR decomposition of time-varying matrices.
Two neural dynamics approaches for computing system of time-varying nonlinear equations
2020, Neurocomputing
Citation Excerpt :
To better determine the evolution direction in a predictive manner, Zhang et al. develop a type of ZND model by taking advantage of the time-derivative information of the time-varying parameters, which becomes more efficient for solving time-varying problems [26,27]. As a development, several ZND models are presented for solving the time-varying matrix inversion, dynamics optimization and so on [38–40]. Furthermore, considering the fact that the ZND is difficult to be implemented directly on digital computers, the ZND model is further extended to the discrete version termed discrete-time zeroing neural dynamics (DTZND).
The problem of solving time-varying nonlinear equations has received much attention in many fields of science and engineering. In this paper, firstly, considering that the classical gradient-based neural dynamics (GND) might result in nonnegligible residual error in handling time-varying nonlinear equations, an adaptive coefficient GND (AGND) model is constructed as an improvement. Besides, the secondly new designed model is the projected zeroing neural dynamics (PZND) to relieve the limitation on the available activation function, which can be of saturation and non-convexity different from that should be unbounded and convex described in the traditional zeroing neural dynamics (ZND) approach. Moreover, theoretical analyses on the AGND model and PZND model are provided to guarantee their effectiveness. Furthermore, computer simulations are conducted and analyzed to illustrate the efficacy and superiority of the two new neural dynamics models designed for online solving time-varying nonlinear equations. Finally, applications to robot manipulator motion generation and Lorenz system are provided to show the feasibility and practicability of the proposed approaches.
Online singular value decomposition of time-varying matrix via zeroing neural dynamics
2020, Neurocomputing
In this paper, the problem of online singular value decomposition (SVD) for time-varying matrix is proposed, analyzed and investigated. In order to solve this complex and difficult problem in real time, we consider to transform it into an equation system firstly. Then, by applying zeroing neural dynamics (ZND) method and a dimensional reduction technique, a continuous-time SVD (CTSVD) model is proposed. Besides, a high-precision eight-instant Zhang et al discretization (ZeaD) formula with theoretical analysis is proposed and studied. Furthermore, by using this new ZeaD formula to discretize the CTSVD model, an eight-instant discrete-time SVD (EIDTSVD) model is thus proposed. Moreover, three other discrete-time SVD (DTSVD) models termed Euler-type DTSVD (ETDTSVD) model, four-instant DTSVD (FIDTSVD) model and six-instant DTSVD (SIDTSVD) model are derived and proposed, respectively, for the purpose of comparison. Finally, numerical experiments and results further substantiate the great effectiveness, accuracy and superiority of the proposed CTSVD and EIDTSVD models.

View all citing articles on Scopus

Jian Li received the B.E. degree in Computer Science and Technology from Hainan University, Haikou, China, in 2014. He is currently pursuing the Ph.D. degree in Communication and Information Systems at the School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China. His main research interests include neural networks, numerical computation and intelligent information processing.

Long Jin received the B.S. degree in Automation from Sun Yat-sen University, Guangzhou, China, in 2011. He is currently pursuing the Ph.D. degree in Communication and Information Systems at the School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China. His main research interests include neural networks, robotics and intelligent information processing.

Shuai Li (M׳14) received the B.E. degree in Precision Mechanical Engineering from Hefei University of Technology, China, in 2005, the M.E. degree in Automatic Control Engineering from University of Science and Technology of China, China, in 2008, and the Ph.D. degree in Electrical and Computer Engineering from Stevens Institute of Technology, USA, in 2014. He is currently a research assistant professor with Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. He is on the editorial board of the International Journal of Distributed Sensor Networks. His current research interests include dynamic neural networks, wireless sensor networks, robotic networks, machine learning, and other dynamic problems defined on a graph.

Yunong Zhang received the B.S. degree from Huazhong University of Science and Technology, Wuhan, China, in 1996, the M.S. degree from South China University of Technology, Guangzhou, China, in 1999, and the Ph.D. degree from Chinese University of Hong Kong, Shatin, Hong Kong, China, in 2003. He is currently a professor with the School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China. Before joining SYSU in 2006, he had been with the National University of Ireland, Maynooth, the University of Strathclyde, Glasgow, UK, and the National University of Singapore, Singapore, since 2003. His main research interests include neural networks, robotics, computation and optimization.

^☆: This work is supported by the National Natural Science Foundation of China (with numbers 61473323 and 61401385), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), by the Science and Technology Program of Guangzhou, China (with number 2014J4100057), by Hong Kong Research Grants Council Early Career Scheme (with number 25214015), and also by Departmental General Research Fund of Hong Kong Polytechnic University (with number G.61.37.UA7L). Besides, kindly note that all authors of the paper are jointly of the first authorship.

View full text

Enhanced discrete-time Zhang neural network for time-variant matrix inversion in the presence of bias noises☆

Abstract

Introduction

Section snippets

Continuous-time ZNN models

Discrete-time ZNN models

EDTZNN model with bias noises

Application to robot motion planning

Conclusions

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Combin. Theory A

Comput. Struct.

J. Symbol. Comput.

Neurocomputing

Appl. Math. Comput.

Neurocomputing

Neurocomputing

J. Comput. Appl. Math.

Neurocomputing

Neurocomputing

Neurocomputing

Linear Algebra Appl.

Dual quaternion-based inverse kinematics of dexterous finger

J. Vibroeng.