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Edited by: Shuai Li, Hong Kong Polytechnic University, Hong Kong

Reviewed by: Weibing Li, University of Leeds, United Kingdom; Yinyan Zhang, Hong Kong Polytechnic University, Hong Kong; Dechao Chen, Sun Yat-sen University, China; Ke Chen, Tampere University of Technology, Finland

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

To obtain the online solution of complex-valued systems of linear equation in complex domain with higher precision and higher convergence rate, a new neural network based on Zhang neural network (ZNN) is investigated in this paper. First, this new neural network for complex-valued systems of linear equation in complex domain is proposed and theoretically proved to be convergent within finite time. Then, the illustrative results show that the new neural network model has the higher precision and the higher convergence rate, as compared with the gradient neural network (GNN) model and the ZNN model. Finally, the application for controlling the robot using the proposed method for the complex-valued systems of linear equation is realized, and the simulation results verify the effectiveness and superiorness of the new neural network for the complex-valued systems of linear equation.

Today, the complex-valued systems of linear equation has been applied into many fields (Duran-Diaz et al.,

Considering that a complex variable can be written as the combination of its real and imaginary parts, we have _{re} + _{im}, _{re} + _{im}, and _{re}(_{im}(

Thus, we can express the equation (

We can write the equation (

In recent years, the research on robot has become a hot spot (Khan et al.,

The rest of this paper is organized into four sections. Section

The research object focuses on a complex-valued system of linear equation in complex domain, which is quite different from the previously investigated real-valued system of linear equation in real domain.

A new finite-time recurrent neural network is proposed and investigated for solving complex-valued systems of linear equation in complex domain. In addition, it is theoretically proved to be convergent within finite time.

Theoretical analyses and simulative results are presented to show the effectiveness of the proposed finite-time recurrent neural network. In addition, a five-link planar manipulator is used to validate the applicability of the finite-time recurrent neural network.

Considering that the complex-valued system of linear equation can be computed in real domain, the error function

Then, according to the design formula _{1} and _{2} satisfy _{1} > 0, _{2} > 0, and

To simplify the formula, Φ(·) uses the linear activation function. Then we have

_{u} regardless of its randomly generated initial error E_{M}

P

To deal with the dynamic response of the equation (

Now let us define ^{(}^{f}^{–}^{j}^{)/}^{f}

Thus, the differential equation (

This is a typical first order differential equation, and we have

So we have

From the equation (_{u}

Considering each element of the matrix _{ik}_{ik}_{M}_{ik}

According to the above analysis, we can draw a conclusion that the error matrix _{u}

_{u} regardless of its randomly generated initial state x_{M}_{L}

P_{(}_{FT}_{)}(_{(}_{org}_{)}(_{(}_{FT}_{)}(_{(}_{org}_{)}(

The equation (

Substitutes the above equation into FTRNN model (12), we have

Considering _{(}_{org}_{)}(

Furthermore, considering _{(}_{org}_{)}(

Let

So according to the equation (

Let us define

The above equation shows that the state matrix _{(FT)}(_{(}_{org}_{)}(

In this section, a digital example will be carried out to show the superiority of FTRNN model (12) to GNN model (6) and ZNN model (8). We can choose the design parameters

So we have

Now the randomly generated vector ^{T} in Xiao et al. (_{(}_{org}_{)} = [−0.4683−0.2545^{T}. So the theoretical solution of the complex-valued linear equation system can be rewritten as _{(}_{org}_{)} = [−0.4683, 1.2425, −0.6126, 1.5082, −0.2545, 0.3239, 0.0112, 0.4683]^{T}.

First, a zero initial complex-valued state

Now GNN model (6), ZNN model (8), and FTRNN model (12) are applied to solve this complex-valued systems of linear equation problem. The output trajectories of the corresponding neural-state solutions are displayed in Figures

Output trajectories of neural states

Output trajectories of neural states

Output trajectories of neural states

To directly show the solution process of such three neural-network models, the evolution of the corresponding residual errors, measured by the norm ||_{2}, is plotted in Figure

Output trajectories of residual functions ||_{2} synthesized by different neural-network models with

Now we can draw a conclusion that, as compared with GNN model (6) and ZNN model (8), FTRNN model (12) has the most superiority for solving the complex-valued system of linear equation problem.

In this section, a five-link planar manipulator is used to validate the applicability of the finite-time recurrent neural network (FTRNN) (Zhang et al., ^{m}^{×}^{n}

To realize the motion tacking of this five-link planar manipulator, the inverse kinematic equation has been solved. Especially, equation (

In the simulation experiment, a square path (with the radius being 1 m) is allocated for the five-link planar manipulator to track. Besides, initial state of the mobile manipulator is set as ^{T}, γ = 10^{3} and task duration is 20 s. The experiment results are shown in Figures

Simulative results synthesized by FTRNN model (12) when the end-effector of five-link planar manipulator tracking the square path.

Motion trajectories of joint angle and joint velocity synthesized by FTRNN model (12) when the end-effector of five-link planar manipulator tracking the square path.

In this paper, a finite-time recurrent neural network (FTRNN) for the complex-valued system of linear equation in complex domain is proposed and investigated. This is the first time to propose such a neural network model, which can convergence within finite time to online deal with the complex-valued system of linear equation in complex domain, and the first time to apply this FTRNN model for robotic path tracking by solving the system of linear equation. The simulation experiments show that the proposed FTRNN model has better effectiveness, as compared to the GNN model and the ZNN model for the complex-valued system of linear equation in complex domain.

LD: experiment preparation, publication writing; LX: experiment preparation, data processing, publication writing; BL: technology support, data acquisition, publication review; RL: supervision of data processing, publication review; HP revised the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer, YZ, and handling editor declared their shared affiliation.