Double Features Zeroing Neural Network Model for Solving the Pseudoninverse of a Complex-Valued Time-Varying Matrix

Abstract

The solution of a complex-valued matrix pseudoinverse is one of the key steps in various science and engineering fields. Owing to its important roles, researchers had put forward many related algorithms. With the development of research, a time-varying matrix pseudoinverse received more attention than a time-invarying one, as we know that a zeroing neural network (ZNN) is an efficient method to calculate a pseudoinverse of a complex-valued time-varying matrix. Due to the initial ZNN (IZNN) and its extensions lacking a mechanism to deal with both convergence and robustness, that is, most existing research on ZNN models only studied the convergence and robustness, respectively. In order to simultaneously improve the double features (i.e., convergence and robustness) of ZNN in solving a complex-valued time-varying pseudoinverse, this paper puts forward a double features ZNN (DFZNN) model by adopting a specially designed time-varying parameter and a novel nonlinear activation function. Moreover, two nonlinear activation types of complex number are investigated. The global convergence, predefined time convergence, and robustness are proven in theory, and the upper bound of the predefined convergence time is formulated exactly. The results of the numerical simulation verify the theoretical proof, in contrast to the existing complex-valued ZNN models, the DFZNN model has shorter predefined convergence time in a zero noise state, and enhances robustness in different noise states. Both the theoretical and the empirical results show that the DFZNN model has better ability in solving a time-varying complex-valued matrix pseudoinverse. Finally, the proposed DFZNN model is used to track the trajectory of a manipulator, which further verifies the reliability of the model.

1. Introduction

As an extend form of matrix inverse, the pseudoinverse is also known as the Moore–Penrose inverse (or generalized inverses), which appeared in mathematics and engineering fields frequently, e.g., stochastic systems estimation [1], optimization [2], hyperspectral image processing [3], robotics [4], and forecasting [5]. Due to the important role of pseudoinverse in mentioned fields, for calculating a matrix pseudoinverse fast and accurately, considerable effort has been exerted toward the numerical algorithms. For instance, for obtaining the Drazin and Moore–Penrose inverses, a third order iterative method was presented in [6]. Some direct algorithms were established for computing a pseudoinverse [7,8]. Santiago Artidiello et al. designed a secant-type method for approximating generalized inverses of complex matrices [9]. In general, most existing methods for solving real-valued and time-invariant matrix pseudoinverse are iterative algorithms. However, these iterative algorithms are not an optimal scheme when developed to solve the time-variant issue, and they will sometimes even appear to be powerless or fail to complete the calculation.

It has been proven that the neural network was an effective solution of real-time computing. In the past two decades, recurrent neural network (RNN) methods were popularly applied in science and engineering fields [10,11,12]. For example, Xiao et al. proposed RNN for computing quadratic minimization [12]. As a classical RNN, the gradient-based RNN (GNN) and its extensions were designed for handling scientific issues [13,14,15]. GNN took the Frobenius-norm of the error matrix as the performance index. In the time-invariant case, GNN ran along the negative gradient direction to monitor the error-norm converging to zero in infinite time. In the time-variant case, as it did not utilize time derivative information, GNN produced a lagging error between the theoretical solution and the neural output solution. Thus, the error norm of GNN failed in vanishing to zero, which cannot meet the requirements of dynamic real-time computing. For overcoming the defect of GNN, Zhang et al. proposed a zeroing neural network (ZNN) method [16]. By adopting a time derivative to trace the theory solution, ZNN made the error exponential decrease to zero as time goes on. Therefore, ZNNs were developed to solve the time-variant problem as required [17,18,19,20,21,22]. However, in implementations of the ZNN models mentioned, it is assumed that they were free of noises. As we known that the neural network systems are inevitably interfered with by various noise, and that the invading noise has a great effect on the accuracy of resolving a time-variant issue [23,24]. Therefore, improving the robustness of ZNN in a noise environment became a hot topic. Thus, researchers developed some noise-resistant ZNN models. For instance, in [25], an integration-enhanced ZNN was first proposed for solving real-time-varying matrix inversion in the presence of different noises. A noise-tolerant ZNN was proposed to solve a time-dependent matrix inverse [26]. The proposal of these noise-resistant ZNN models greatly extends the application of ZNN in practice.

It is worth pointing out that the initial ZNN and noise-resistant ZNN models take infinite time to converge, which cannot satisfy real-time application. As time is precious for handling a time-varying problem in practice, and the convergence speed is a vital performance indicator of a neural system. Thus, there has been a wave of research on the limited time convergence ZNN (LTCZNN) model [27,28]. In [27], Li embedded a delicately designed nonlinear function in the ZNN model for accelerating the convergence speed; thus, the ZNN model achieved limited time convergence, which is seminal in the ZNN field. Xiao proposed two novel activation functions that were better structured and more convenient to implement [28]. These LTCZNN models, by embedding a certain nonlinear activation function, greatly improve the convergence speed. For these LTCZNN models, a fly in the ointment is that initial values and noise have significant impact on the convergence speed. To overcome the disadvantages of LTCZNN, in [29], a predefined time convergence ZNN (PTCZNN) with an improved nonlinear activation function was proposed. However, the design formula of the PTCZNN only controlled the influence of initial values within a particular scope, and the PTCZNN still did not improve the robustness.

On the other hand, the complex-valued neural networks have been widely used in numerous scientific and engineering issues, such as signal processing [30], imaging [31], automatic control [32], and optimization [33]. Due to the high storage capacity of stable equilibrium points, the stability analysis of complex-valued neural networks has received considerable attention, and many interesting results have been obtained [34,35,36,37]. In contrast to the real-valued neural networks, some complex-valued ZNN models have highlighted the competitive advantage in compatibility and processing power [38,39,40]. Inspired by the capability of these complex-valued ZNN models, in this paper, for solving pseudoinverse of complex-valued time-varying matrix, we explore a complex-valued ZNN with better performance.

In [41], an RNN model with time-varying parameters was first designed for solving a time-varying Sylvester equation, compared to the RNN with fixed parameters, the robustness of RNN with time-varying parameter is improved significantly. The result indicates that time-varying parameters in ZNN have crucial implications. Recently, increasing attention has been focused on time-varying parameters in a neural network, for example, a varying-parameter ZNN model is designed for approximating outer inverses [42], Xiao and He also designed a flexible variable parameter ZNN model with noise tolerance ability [43]. In order to simultaneously improve the robustness and convergence of ZNN for solving the pseudoinverse of a complex-valued time-varying matrix, inspired by literature [29,41], we explore a double features ZNN (DFZNN) model, which embedded a novel delicately designed time-varying parameter and nonlinear activation function. Moreover, in this paper, we find two ways (called type I and type II) to activate a complex-valued number. Both activated types are proven to insure the predefine-time convergence and good robustness of the DFZNN for solving a pseudoninverse of a complex-valued time-varying matrix in different noise circumstances. The research further enhances the scope of compatibility and processing power of ZNN in real-time computation. The innovations and contributions of this paper can be listed below:

• For solving complex-valued time-varying issues, the design of the DFZNN model constructs a mechanism to deal with both convergence and robustness, which enriches the usage of complex-valued ZNN;
• The convergence rate is improved; specifically, the DFZNN model can converge in the predefine time. Moreover, the convergence rate is not affected by the initial value;
• Robustness is better, compared with the existing ZNNs, and the DFZNN model has stronger stability in noise situation;
• The simulation example and an application further illustrate the reliability of the proposed DFZNN model.

2. Problem Formulation and ZNN Models

In this section, the corresponding preliminaries and the formulations of the complex-valued time-varying matrix pseudoinverse are introduced first. Secondly, we provide the design process of the ZNN model. Different evolutions of IZNN, LTCZNN, PTCZNN, and DFZNN are introduced. Finally, based on two complex-valued activation types, we construct the DFZNN model.

2.1. Problem Formulation

Definition 1.

Given a complex-valued time-varying matrix $A\left(t\right)\in {\mathbb{C}}^{m\times n}$, if $Y\left(t\right)\in {\mathbb{C}}^{n\times m}$ satisfies all the Penrose equations as below:

$$\begin{array}{cc}\hfill & A\left(t\right)Y\left(t\right)A\left(t\right)=A\left(t\right),Y\left(t\right)A\left(t\right)Y\left(t\right)=Y\left(t\right),\hfill \\ \hfill & {\left(A\left(t\right)Y\left(t\right)\right)}^H=A\left(t\right)Y\left(t\right),{\left(Y\left(t\right)A\left(t\right)\right)}^H=Y\left(t\right)A\left(t\right),\hfill \end{array}$$

then $Y\left(t\right)$ is named the pseudoinverse of matrix $A\left(t\right)$, denoted by $A^{+}\left(t\right)$, which exists and is unique; here, $A^H\left(t\right)$ denotes the conjugate transpose of the matrix $A\left(t\right)$.

Moreover, if the complex-valued time-varying matrix $A\left(t\right)\in {\mathbb{C}}^{m\times n}$ is full rank, which means rank($A\left(t\right)$) = min($m,n$) for any t, we can obtain the pseudoinverse of $A\left(t\right)$ according to the following lemma.

Lemma 1.

If the complex-valued time-varying matrix $A\left(t\right)\in {\mathbb{C}}^{m\times n}$ is full rank, then $A\left(t\right)A^H\left(t\right)$ or $A^H\left(t\right)A\left(t\right)$ is nonsingular. Hence, the pseudoinverse of $A\left(t\right)$ can be obtained as

$$\begin{array}{c}\hfill A^{+}\left(t\right)=\left\{\begin{array}{c}{A}^H\left(t\right){\left(A\left(t\right)A^H\left(t\right)\right)}^{-1},m>n,\hfill \\ A^{-1}\left(t\right),m=n,\hfill \\ {\left(A^H\left(t\right)A\left(t\right)\right)}^{-1}{A}^H\left(t\right),m<n.\hfill \end{array}\right.\end{array}$$

For the situation $m>n$ and $m<n$, the procedure of computing pseudoinverse is identical. Thus, this paper investigates the full rank complex-valued time-varying matrix $A\left(t\right)\in {\mathbb{C}}^{m\times n}$ with $m<n$. The complex-valued time-varying pseudoinverse problem can be presented as

$$Y\left(t\right)A\left(t\right)A^H\left(t\right)=A^H\left(t\right).$$

$Y\left(t\right)$ is an unknown complex-valued time-varying matrix, and the purpose of this work is to design a DFZNN model for solving the unknown matrix $Y\left(t\right)$ within a predefined time under additive noises.

2.2. Design Process of ZNN Model

The design process of the ZNN model for solving a complex-valued time-varying pseudoinverse is the same as widely described in the previously mentioned literature [16,17,18,19,20,21,22], which can be listed as the following procedure.

Firstly, defining an error-monitoring function:

$$\begin{array}{c}\hfill \mathbb{D}\left(t\right)=Y\left(t\right)A\left(t\right)A^H\left(t\right)-A^H\left(t\right).\end{array}$$

(1)

Secondly, defining an evolution for $\mathbb{D}\left(t\right)$:

$$\begin{array}{c}\hfill \dot{\mathbb{D}}\left(t\right)=\frac{\mathrm{d}\mathbb{D}\left(t\right)}{\mathrm{d}t}=-\alpha \mathcal{W}\left(\mathbb{D}\left(t\right)\right),\end{array}$$

(2)

where $\alpha >0$ is a scaling parameter, $\mathcal{W}(·)$ is an activation method which is defined in a complex-value element-wise. Thirdly, substituting (2) into (1), one can obtain a ZNN model:

$$\begin{array}{c}\hfill \dot{Y}\left(t\right)A\left(t\right)A^H\left(t\right)={\dot{A}}^H\left(t\right)-Y\left(t\right)\left(\dot{A}\left(t\right)A^H\left(t\right)-A\left(t\right){\dot{A}}^H\left(t\right)\right)-\alpha \mathcal{W}\left(\mathbb{D}\left(t\right)\right).\end{array}$$

(3)

For complex-value element-wise, the activation way operates in two types in our work: one is that the activation function operates the real part and the imaginary part separately, another is that the activation function operates the modulus and the argument separately. The definitions of two types are given as follows:

Definition 2.

The activation type I

$$\begin{array}{c}\hfill {\mathcal{W}}_1(P+iQ)=\mathfrak{F}\left(P\right)+i\mathfrak{F}\left(Q\right).\end{array}$$

(4)

The activation type II

$$\begin{array}{c}\hfill {\mathcal{W}}_2(P+iQ)=\mathfrak{F}\left(\tau \right)\circ exp(i\Theta ),\end{array}$$

(5)

where $P\in {\mathbb{R}}^{m\times n}$, $Q\in {\mathbb{R}}^{m\times n}$, i is the imaginary unit, $\tau \in {\mathbb{R}}^{m\times n}$, and $\mathsf{\Theta }\in {[-\pi ,\pi )}^{m\times n}$ respectively denote the modulus and the argument of the complex matrix $P+iQ$. $\mathfrak{F}(·)$ denotes an activation function, and ∘ denotes matrix opposite element multiplication.

In initial ZNN (IZNN), $\mathfrak{F}(·)$ is a linear activation function, i.e., $\mathfrak{F}\left(x\right)=x$; according to (3), the IZNN for solving complex-valued time-varying pseudoinverse can be expressed as:

$$\begin{array}{c}\hfill \dot{Y}\left(t\right)A\left(t\right)A^H\left(t\right)={\dot{A}}^H\left(t\right)-Y\left(t\right)\left(\dot{A}\left(t\right)A^H\left(t\right)-A\left(t\right){\dot{A}}^H\left(t\right)\right)-\alpha \mathbb{D}\left(t\right),\end{array}$$

(6)

where $\dot{Y}\left(t\right),{\dot{A}}^H\left(t\right),\dot{A}\left(t\right)$ respectively denote the time derivatives of $Y\left(t\right),A^H\left(t\right),A\left(t\right)$.

IZNN is an effective approach for dealing with time-varying problems, but the convergence speed is restricted for IZNN adopting the linear activation function. To improve the convergence rate, Li proposed an LTCZNN embedded with a delicately designed nonlinear activation function, which achieved limited time convergence [27]. The designed nonlinear activation function is expressed as:

$$\begin{array}{c}\hfill {\mathfrak{F}}_1\left(x\right)=\mathrm{sign}\left(x\right)\frac{{\left|x\right|}^{\mu }+{\left|x\right|}^{\frac{1}{\mu }}}{2},\end{array}$$

(7)

where $x\in \mathbb{R},0<\mu <1$. $\mathrm{sign}(·)$ is defined as:

$$\begin{array}{c}\hfill \mathrm{sign}\left(x\right)=\left\{\begin{array}{c}1,x>0,\hfill \\ 0,x=0,\hfill \\ -1,x<0.\hfill \end{array}\right.\end{array}$$

In terms of convergence, the LTCZNN achieved a significant improvement. However, the initial value will greatly impact the convergence effect in LTCZNN. In order to reduce the effect of the initial value, Li et al. proposed a PTCZNN with a modified nonlinear activation function as:

$$\begin{array}{c}\hfill {\mathfrak{F}}_2\left(x\right)=\mathrm{sign}\left(x\right)(l_1{\left|x\right|}^{\mu }+l_2{\left|x\right|}^{\nu }+l_3)+l_{4}x,\end{array}$$

(8)

where the parameter $\mu$ is the same definition as (7), and $\nu >1,l_1,l_2>0,l_3,l_4\ge 0$.

In all of the above ZNN models, the parameters in (2) are fixed. To improve the performance of the models, the parameters often need to be adjusted frequently to find the approximate optimal value, which is inefficient in practical application. Thus, Zhang et al. first proposed a VPZNN; in contrast to the fixed parameter ZNN, VPZNN has better robustness, which is beneficial for online processing [41]. In VPZNN, the evolution of $\mathbb{D}\left(t\right)$ was defined as

$$\dot{\mathbb{D}}\left(t\right)=-(t^{\theta }+\theta ){\mathfrak{F}}_1\left(\mathbb{D}\left(t\right)\right),$$

where $\theta >0$, $t^{\theta }+\theta >0$ is a time-varying scaling parameter, and ${\mathfrak{F}}_1$ denotes the activation function defined in (7).

The proposal of VPZNN makes a breakthrough in the design of time-varying parameters; however, the initial value still impacts the solving task of the VPZNN model. Furthermore, when we increase the convergence rate of VPZNN, efficiency will be sacrificed. In addition, the PTCZNN achieves predefined time convergence by controlling the influence of the initial value in a limited scope. This inspires us to explore a new time-varying parameter model which is not disturbed by the initial value and accelerate convergence rate without remarkable efficiency loss. The novel evolution of $\mathbb{D}\left(t\right)$ was designed as:

$$\begin{array}{c}\hfill \dot{\mathbb{D}}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)\mathcal{W}\left(\mathbb{D}\left(t\right)\right).\end{array}$$

(9)

where $\alpha ,{\zeta }_1,{\zeta }_2>0$, the exponential part of time varying parameter contains attenuation factor ${\zeta }_1\mathrm{arccot}\left(t\right)$ and noise suppression factor ${\zeta }_{2}t$, and the attenuation factor design is based on the properties of the arccotangent function, which can guarantee that the model converges quickly and possesses short-term noise suppression capability. The noise suppression factor ensures that the the model possesses long-term noise suppression capability. The nonlinear activation function $\mathfrak{F}(·)$ in $\mathcal{W}(·)$ is designed as:

$$\begin{array}{c}\hfill {\mathfrak{F}}_3\left(x\right)=\left\{\begin{array}{c}{l}_1{\left|x\right|}^{\mu }\mathrm{sign}\left(x\right)+l_{5}x,\mathrm{if}\left|x\right|\le 1,\hfill \\ l_2{\left|x\right|}^{\nu }\mathrm{sign}\left(x\right)+l_{5}x,\mathrm{if}\left|x\right|>1,\hfill \end{array}\right.\end{array}$$

(10)

where $l_1,l_2$ are defined the same as (8) and $l_5>0$. Then, the DFZNN model is constructed as:

$$\begin{array}{cc}\hfill \dot{Y}\left(t\right)A\left(t\right)A^H\left(t\right)=& {\dot{A}}^H\left(t\right)-Y\left(t\right)\left(\dot{A}\left(t\right)A^H\left(t\right)-A\left(t\right){\dot{A}}^H\left(t\right)\right)\hfill \\ \hfill & -\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)\mathcal{W}\left(\mathbb{D}\left(t\right)\right).\hfill \end{array}$$

(11)

In addition, the DFZNN model with noise intervention can be described as:

$$\begin{array}{cc}\hfill \dot{Y}\left(t\right)A\left(t\right)A^H\left(t\right)=& {\dot{A}}^H\left(t\right)-Y\left(t\right)\left(\dot{A}\left(t\right)A^H\left(t\right)-A\left(t\right){\dot{A}}^H\left(t\right)\right)\hfill \\ \hfill & -\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)\mathcal{W}\left(\mathbb{D}\left(t\right)\right)+\Delta \left(t\right),\hfill \end{array}$$

(12)

where $\Delta \left(t\right)$ denotes additive complex-valued noise.

3. Theoretical Analysis

In this section, we mainly present the theoretical results of the DFZNN model. At first, the global stability and predefined time convergence of the DFZNN model are proven in theory. Then, the robustness of the DFZNN model with unknown additive noises is discussed.

3.1. Global Stability

Theorem 1.

Given a complex-valued time-varying matrix $A\left(t\right)$ of full rank, the state matrix $Y\left(t\right)$ synthesized by the DFZNN with the activated type I, starting from stochastic initial $Y\left(0\right)$, always converges to theoretical solution, i.e., the error matrix $\mathbb{D}\left(t\right)$ globally converges to 0.

Proof. 

Defining the error matrix $\mathbb{D}\left(t\right)$, according to Equation (9), we obtain

$$\dot{\mathbb{D}}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_1\left(\mathbb{D}\left(t\right)\right).$$

Based on the definition of ${\mathcal{W}}_1$ in (4), element-wise,

$$\begin{array}{cc}\hfill {\dot{\mathfrak{D}}}_{ij}\left(t\right)& =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_1\left({\mathfrak{D}}_{ij}\left(t\right)\right)\hfill \\ \hfill & =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)({\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)+i{\mathfrak{F}}_3\left(q_{ij}\left(t\right)\right)),\hfill \end{array}$$

where ${\mathfrak{D}}_{ij}\left(t\right)\in \mathbb{C}$ is the $ij$th item of the complex matrix $\mathbb{D}\left(t\right)$, and $p_{ij}\left(t\right)$ and $q_{ij}\left(t\right)$ respectively denote the real and imaginary parts of ${\mathfrak{D}}_{ij}\left(t\right)$. We study the real and imaginary parts of ${\dot{\mathfrak{D}}}_{ij}\left(t\right)$ separately, and two equations can be obtained as follows:

$$\begin{array}{c}\hfill {\dot{p}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill {\dot{q}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(q_{ij}\left(t\right)\right).\end{array}$$

(14)

For the variable $p_{ij}\left(t\right)$, we choose the Lyapunov function ${\upsilon }_{ij}\left(t\right)=p_{ij}^2\left(t\right)$. Then, the derivative can be obtained as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& =2p_{ij}\left(t\right){\dot{p}}_{ij}\left(t\right)\hfill \\ \hfill & =-2\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)p_{ij}\left(t\right){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)\hfill \\ \hfill & \le -2\alpha exp\left({\zeta }_{2}t\right)p_{ij}\left(t\right){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right).\hfill \end{array}$$

(15)

According to the magnitude of $p_{ij}\left(t\right)$, i.e., $|p_{ij}\left(t\right)|>1$ and $|p_{ij}\left(t\right)|\le 1$, two situations are discussed as below.

First situation: if $|p_{ij}\left(t\right)|>1$. Based on the definition of ${\mathfrak{F}}_3(·)$ in (10), (15) can be divided into two steps.

(1) For $|p_{ij}\left(t\right)|>1$, Equation (15) can be represented as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& \le -2\alpha exp\left({\zeta }_{2}t\right)p_{ij}\left(t\right)(l_2|p_{ij}\left(t\right){|}^{\nu }\mathrm{sign}\left(p_{ij}\left(t\right)\right)+l_5{p}_{ij}\left(t\right))\hfill \\ \hfill & =-2\alpha exp\left({\zeta }_{2}t\right)(l_2|p_{ij}\left(t\right){|}^{\nu +1}+l_5|p_{ij}\left(t\right){|}^2).\hfill \end{array}$$

Thus, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite, which means that $p_{ij}\left(t\right)$ will decrease to $p_{ij}\left(t\right)\le 1$, then enter the second step as:

(2) For $|p_{ij}\left(t\right)|\le 1$. Equation (15) can be represented as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& \le -2\alpha exp\left({\zeta }_{2}t\right)p_{ij}\left(t\right)(l_1|p_{ij}\left(t\right){|}^{\mu }\mathrm{sign}\left(p_{ij}\left(t\right)\right)+l_5{p}_{ij}\left(t\right))\hfill \\ \hfill & =-2\alpha exp\left({\zeta }_{2}t\right)(l_2|p_{ij}\left(t\right){|}^{\mu +1}+l_5|p_{ij}\left(t\right){|}^2)\le 0,\hfill \end{array}$$

clearly, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite.

Second situation: if $|p_{ij}\left(t\right)|\le 1$, which is just like the second step in the first situation. Thus, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite. In general, based on the Lyapunov stability theory, it can be concluded that $p_{ij}\left(t\right)$ globally converges to 0. Similarly, it can be proven that $q_{ij}\left(t\right)$ globally converges to 0. Then, we can conclude that ${\mathfrak{D}}_{ij}\left(t\right)$ globally converges to 0. In summary, the error matrix $\mathbb{D}\left(t\right)$ globally converging to 0 is proven. □

Theorem 2.

Given a complex-valued time-varying matrix $A\left(t\right)$ of full rank, the state matrix $Y\left(t\right)$ synthesized by the DFZNN with the activated type II, starting from stochastic initial $Y\left(0\right)$, always converges to a theoretical solution, i.e., the error matrix $\mathbb{D}\left(t\right)$ globally converges to 0.

Proof. 

According to the activated type II, we obtain the derivative of $\mathbb{D}\left(t\right)$ as

$$\dot{\mathbb{D}}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_2\left(\mathbb{D}\left(t\right)\right).$$

Element-wise,

$${\dot{\mathfrak{D}}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_2\left({\mathfrak{D}}_{ij}\left(t\right)\right).$$

where ${\mathfrak{D}}_{ij}\left(t\right)\in \mathbb{C}$ is the $ij$th item of the complex matrix $\mathbb{D}\left(t\right)$; note that ${\mathfrak{D}}_{ij}\left(t\right)$ is a complex scalar, and we construct a Lyapunov function

$${\upsilon }_{ij}\left(t\right)={\left|{\mathfrak{D}}_{ij}\left(t\right)\right|}^2={\mathfrak{D}}_{ij}\left(t\right)\overline{{\mathfrak{D}}_{ij}\left(t\right)},$$

where $|{\mathfrak{D}}_{ij}\left(t\right)|$ denotes the modulus of ${\mathfrak{D}}_{ij}\left(t\right)$. The derivative of ${\upsilon }_{ij}\left(t\right)$ can be obtained as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& ={\dot{\mathfrak{D}}}_{ij}\left(t\right)\overline{{\mathfrak{D}}_{ij}\left(t\right)}+{\mathfrak{D}}_{ij}\left(t\right)\dot{\overline{{\mathfrak{D}}_{ij}\left(t\right)}}.\hfill \\ \hfill & =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)\left({\mathcal{W}}_2\left({\mathfrak{D}}_{ij}\left(t\right)\right)\overline{{\mathfrak{D}}_{ij}\left(t\right)}+{\mathfrak{D}}_{ij}\left(t\right)\overline{{\mathcal{W}}_2\left({\mathfrak{D}}_{ij}\left(t\right)\right)}\right).\hfill \end{array}$$

(16)

According to the definition of ${\mathcal{W}}_2(·)$ in (5), we have that

$${\mathcal{W}}_2\left({\mathfrak{D}}_{ij}\left(t\right)\right)={\mathfrak{F}}_3\left({\mathfrak{D}}_{ij}\left(t\right)\right)exp\left(\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right),$$

where $\sqrt{-1}$ represents the imaginary unit; therefore,

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)(\overline{{\mathfrak{D}}_{ij}\left(t\right)}{\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right)exp\left(\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right)\hfill \\ \hfill & +{\mathfrak{D}}_{ij}\left(t\right){\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right)exp\left(-\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right))\hfill \\ \hfill & =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)\hfill \\ \hfill & \left(\right|{\mathfrak{D}}_{ij}\left(t\right)|exp\left(-\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right){\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right)exp\left(\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right)\hfill \\ \hfill & +|{\mathfrak{D}}_{ij}\left(t\right)|exp\left(\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right){\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right)exp\left(-\sqrt{-1}·arg\left({\mathfrak{D}}_{ij}\left(t\right)\right)\right))\hfill \\ \hfill & =-2\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)|{\mathfrak{D}}_{ij}\left(t\right)|{\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right),\hfill \end{array}$$

(17)

considering similarly the magnitude of $|{\mathfrak{D}}_{ij}\left(t\right)|$, i.e., $|{\mathfrak{D}}_{ij}\left(t\right)|>1$ and $|{\mathfrak{D}}_{ij}\left(t\right)|\le 1$, two situations are discussed as below.

First situation: if $|{\mathfrak{D}}_{ij}\left(t\right)|>1$. Based on the definition of ${\mathfrak{F}}_3(·)$ in (10), Equation (17) can be divided into two steps.

(1) For $|{\mathfrak{D}}_{ij}\left(t\right)|>1$. Equation (17) can be represented as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& \le -2\alpha exp\left({\zeta }_{2}t\right){\mathfrak{D}}_{ij}\left(t\right)(l_2|{\mathfrak{D}}_{ij}\left(t\right){|}^{\nu }\mathrm{sign}\left({\mathfrak{D}}_{ij}\left(t\right)\right)+l_5{\mathfrak{D}}_{ij}\left(t\right))\hfill \\ \hfill & =-2\alpha exp\left({\zeta }_{2}t\right)(l_2|{\mathfrak{D}}_{ij}\left(t\right){|}^{\nu +1}+l_5|{\mathfrak{D}}_{ij}\left(t\right){|}^2).\hfill \end{array}$$

Thus, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite, which means that ${\mathfrak{D}}_{ij}\left(t\right)$ will decrease to ${\mathfrak{D}}_{ij}\left(t\right)\le 1$, then enter the second step as:

(2) For $|{\mathfrak{D}}_{ij}\left(t\right)|\le 1$. Equation (17) can be represented as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& \le -2\alpha exp\left({\zeta }_{2}t\right){\mathfrak{D}}_{ij}\left(t\right)(l_1|{\mathfrak{D}}_{ij}\left(t\right){|}^{\mu }\mathrm{sign}\left({\mathfrak{D}}_{ij}\left(t\right)\right)+l_5{\mathfrak{D}}_{ij}\left(t\right))\hfill \\ \hfill & =-2\alpha exp\left({\zeta }_{2}t\right)(l_2|{\mathfrak{D}}_{ij}\left(t\right){|}^{\mu +1}+l_5|{\mathfrak{D}}_{ij}\left(t\right){|}^2)\le 0,\hfill \end{array}$$

clearly, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite.

Second situation: if $|{\mathfrak{D}}_{ij}\left(t\right)|\le 1$, which is just like the second step in the first situation. Thus, ${\dot{\upsilon }}_{ij}\left(t\right)$ is negative definite. In general, based on the Lyapunov stability theory, it can be concluded that ${\mathfrak{D}}_{ij}\left(t\right)$ globally converges to 0. Therefore, the error matrix $\mathbb{D}\left(t\right)$ globally converging to 0 is proven. □

3.2. Predefined Time Convergence

Theorem 3.

Given the complex-valued time-varying matrix $A\left(t\right)$ of full rank, the state matrix $Y\left(t\right)$ synthesized by the DFZNN with the activated type I(or type II), starting from stochastic initial $Y\left(0\right)$, always converges to theoretical solution in predefine time $t_{top}$:

$$\begin{array}{c}\hfill t_{\mathrm{top}}\le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}+\frac{{\zeta }_2}{\alpha l_2(\nu -1)}\right).\end{array}$$

Proof. 

In this proof, we first define the same error matrix $\mathbb{D}\left(t\right)$ as Theorem 1, then consider the case with the activated type I and the activated type II, respectively.

When choosing the activated type I, the same as Theorem 1, we study the real and imaginary parts of ${\dot{\mathfrak{D}}}_{ij}\left(t\right)$ separately; two equations can be obtained as follows:

$$\begin{array}{c}\hfill {\dot{p}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right),\end{array}$$

$$\begin{array}{c}\hfill {\dot{q}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(q_{ij}\left(t\right)\right),\end{array}$$

where $p_{ij}\left(t\right)$ and $q_{ij}\left(t\right)$ respectively denote the real and imaginary parts of ${\mathfrak{D}}_{ij}\left(t\right)$; for the real variable $p_{ij}\left(t\right)$, we choose the Lyapunov function ${\upsilon }_{ij}\left(t\right)=\left|p_{ij}\left(t\right)\right|$. Then, the derivative can be obtained as

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\upsilon }_{ij}\left(t\right)}{\mathrm{d}t}& ={\dot{p}}_{ij}\left(t\right)\mathrm{sign}\left(p_{ij}\left(t\right)\right)\hfill \\ \hfill & =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)\mathrm{sign}\left(p_{ij}\left(t\right)\right)\hfill \\ \hfill & \le -\alpha exp\left({\zeta }_{2}t\right){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)\mathrm{sign}\left(p_{ij}\left(t\right)\right).\hfill \end{array}$$

(18)

Similarly, according to $|p_{ij}\left(t\right)|>1$ and $|p_{ij}\left(t\right)|\le 1$, two steps are divided for the whole process.

First step, the initial value $|p_{ij}\left(0\right)|>1$, the elapsed time $t_1$ notes $|p_{ij}\left(0\right)|>1$ decreases to $|p_{ij}\left(t_1\right)|=1$, then inequality (18) yields

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\upsilon }_{ij}\left(t\right)}{\mathrm{d}t}& \le -\alpha exp\left({\zeta }_{2}t\right){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)\hfill \\ \hfill & =-\alpha exp\left({\zeta }_{2}t\right)\left(l_2{\left|p_{ij}\left(t\right)\right|}^{\nu }+l_5{p}_{ij}\left(t\right)\right)\hfill \\ \hfill & \le -\alpha l_{2}exp\left({\zeta }_{2}t\right){\left|p_{ij}\left(t\right)\right|}^{\nu },\hfill \end{array}$$

then we have

$$\begin{array}{c}\hfill |p_{ij}{\left(t\right)|}^{-\nu }\mathrm{d}{\upsilon }_{ij}\left(t\right)\le -\alpha l_{2}exp\left({\zeta }_{2}t\right)\mathrm{d}t,\end{array}$$

that is,

$$\begin{array}{c}\hfill {\upsilon }_{ij}{\left(t\right)}^{-\nu }\mathrm{d}{\upsilon }_{ij}\left(t\right)\le -\alpha l_{2}exp\left({\zeta }_{2}t\right)\mathrm{d}t.\end{array}$$

(19)

Integrating on both sides of (19), we obtain

$$\begin{array}{c}\hfill {\int }_{{\upsilon }_{ij}\left(0\right)}^{{\upsilon }_{ij}\left(t_1\right)}{\upsilon }_{ij}{\left(t\right)}^{-\nu }\mathrm{d}{\upsilon }_{ij}\left(t\right)\le {\int }_0^{t_1}-\alpha l_{2}exp\left({\zeta }_{2}t\right)\mathrm{d}t\\ \hfill \frac{1}{1-\nu }\left({\upsilon }_{ij}{\left(t\right)}^{1-\nu }\right){{|}_{{\upsilon }_{ij}\left(0\right)}^1\le -\frac{\alpha l_2}{{\zeta }_2}exp\left({\zeta }_{2}t\right)|}_0^{t_1}\\ \hfill \frac{1}{1-\nu }(1-{\upsilon }_{ij}{\left(0\right)}^{1-\nu })\le -\frac{\alpha l_2}{{\zeta }_2}(exp\left({\zeta }_2{t}_1\right)-1).\end{array}$$

(20)

Then, we have

$$\begin{array}{c}\hfill t_1\le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2(1-{\upsilon }_{ij}{\left(0\right)}^{1-\nu })}{\alpha l_2(\nu -1)}\right)\le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2}{\alpha l_2(\nu -1)}\right).\end{array}$$

(21)

Second step, the elapsed time $t_2$ notes $|p_{ij}\left(0\right)|=1$ decreases to $|p_{ij}\left(t_1\right)|=0$, according to (10), the inequality (18) yields

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{\upsilon }_{ij}\left(t\right)}{\mathrm{d}t}& \le -\alpha exp\left({\zeta }_{2}t\right)\left(l_1{\left|p_{ij}\left(t\right)\right|}^{\mu }+l_5{p}_{ij}\left(t\right)\right)\hfill \\ \hfill & \le -\alpha l_{1}exp\left({\zeta }_{2}t\right){\left|p_{ij}\left(t\right)\right|}^{\left(\mu \right)},\hfill \end{array}$$

that is,

$$\begin{array}{c}\hfill {\upsilon }_{ij}{\left(t\right)}^{-\mu }\mathrm{d}{\upsilon }_{ij}\left(t\right)\le -\alpha l_{1}exp\left({\zeta }_{2}t\right)\mathrm{d}t.\end{array}$$

(22)

Integrating on both sides of (22), we obtain

$$\begin{array}{c}\hfill {\int }_{{\upsilon }_{ij}\left(t_1\right)}^{{\upsilon }_{ij}(t_1+t_2)}{\upsilon }_{ij}{\left(t\right)}^{-\mu }\mathrm{d}{\upsilon }_{ij}\left(t\right)\le {\int }_{t_1}^{t_1+t_2}-\alpha l_{1}exp\left({\zeta }_{2}t\right)\mathrm{d}t\\ \hfill \frac{1}{1-\mu }{\upsilon }_{ij}{\left(t\right)}^{1-\mu }{{|}_1^0\le -\frac{\alpha l_1}{{\zeta }_2}exp\left({\zeta }_2{t}_1\right)|}_{t_1}^{t_1+t_2}\\ \hfill exp\left({\zeta }_2(t_1+t_2)\right)\le exp\left({\zeta }_2{t}_1\right)+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}.\end{array}$$

Combining with (20) and (21), the predefined convergence time $t_{\mathrm{top}}$ of the real variable $p_{ij}\left(t\right)$ is obtained as

$$\begin{array}{cc}\hfill t_{\mathrm{top}}=t_1+t_2& \le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2(1-{\upsilon }_{ij}{\left(0\right)}^{1-\nu })}{\alpha l_2(\nu -1)}+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}\right)\hfill \\ \hfill & \le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}+\frac{{\zeta }_2}{\alpha l_2(\nu -1)}\right).\hfill \end{array}$$

Identically, the same predefined convergence time $t_{\mathrm{top}}$ of the imaginary variable $q_{ij}\left(t\right)$ is obtained.

When choosing the activated type II, for ${\dot{\mathfrak{D}}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_2\left({\mathfrak{D}}_{ij}\left(t\right)\right)$, we also define a Lyapunov function ${\upsilon }_{ij}\left(t\right)=\left|{\mathfrak{D}}_{ij}\left(t\right)\right|$. Then, the derivative of ${\upsilon }_{ij}\left(t\right)$ can be obtained as

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& =-\frac{1}{2}\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_2\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right)\hfill \\ \hfill & =-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(\right|{\mathfrak{D}}_{ij}\left(t\right)\left|\right).\hfill \end{array}$$

(23)

Similarly to the activated type I, from $|{\mathfrak{D}}_{ij}\left(t\right)|>1$ to $|{\mathfrak{D}}_{ij}\left(t\right)|=0$, two steps are divided for the whole process, the predefined convergence time $t_{\mathrm{top}}$ of ${\mathfrak{D}}_{ij}\left(t\right)$ can be obtained as

$$\begin{array}{cc}\hfill t_{\mathrm{top}}=t_1+t_2& \le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2(1-{\upsilon }_{ij}{\left(0\right)}^{1-\nu })}{\alpha l_2(\nu -1)}+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}\right)\hfill \\ \hfill & \le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}+\frac{{\zeta }_2}{\alpha l_2(\nu -1)}\right).\hfill \end{array}$$

This completes the proof. □

3.3. Robustness

Theorem 4.

Given a complex-valued time-varying matrix $A\left(t\right)$ of full rank, the state matrix $Y\left(t\right)$ synthesized by the DFZNN with an unknown bound noise $\Delta \left(t\right)$, starting from stochastic initial $Y\left(0\right)$, always converges to a theoretical solution, i.e., the error matrix $\mathbb{D}\left(t\right)$ globally converges to 0.

Proof. 

According to the error matrix $\mathbb{D}\left(t\right)$, when choosing the activated type I, we obtain

$$\dot{\mathbb{D}}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_1\left(\mathbb{D}\left(t\right)\right)+\Delta \left(t\right).$$

Element-wise,

$${\dot{\mathfrak{D}}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathcal{W}}_1\left({\mathfrak{D}}_{ij}\left(t\right)\right)+{\delta }_{ij}\left(t\right).$$

then, two equations of the real part and imaginary part can be obtained as follows:

$$\begin{array}{c}\hfill {\dot{p}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)+{\delta }_a\left(t\right),\end{array}$$

$$\begin{array}{c}\hfill {\dot{q}}_{ij}\left(t\right)=-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(q_{ij}\left(t\right)\right)+{\delta }_b\left(t\right),\end{array}$$

where ${\delta }_a\left(t\right),{\delta }_b\left(t\right)$ respectively denote the real part and imaginary part of ${\delta }_{ij}\left(t\right)$. For real variable $p_{ij}\left(t\right)$, we also choose the Lyapunov function ${\upsilon }_{ij}\left(t\right)=\frac{1}{2}{p}_{ij}^2\left(t\right)$; then,

$$\begin{array}{cc}\hfill {\dot{\upsilon }}_{ij}\left(t\right)& =p_{ij}\left(t\right){\dot{p}}_{ij}\left(t\right)\hfill \\ \hfill & =p_{ij}\left(t\right)(-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)+{\delta }_a\left(t\right)).\hfill \end{array}$$

(24)

If ${\dot{\upsilon }}_{ij}\left(t\right)<0$, $|p_{ij}\left(t\right)|$ will approach zero. If ${\dot{\upsilon }}_{ij}\left(t\right)>0$, $|p_{ij}\left(t\right)|$ will increase, i.e., $p_{ij}\left(t\right)(-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)+{\delta }_a\left(t\right))>0$. Furthermore, we have

$$\begin{array}{c}\hfill |{\delta }_a\left(t\right)|>|\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)|.\end{array}$$

(25)

$|p_{ij}\left(t\right)|$ will not increase until $-\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)+{\delta }_a\left(t\right))$ decreases to zero. According to Equation (25), we have

$$|{\mathfrak{F}}_3\left(p_{ij}\left(t\right)\right)|=|\frac{{\delta }_a\left(t\right)}{\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)}|.$$

Therefore, as the time goes, $|p_{ij}\left(t\right)|$ is bounded by

$$\begin{array}{c}\hfill 0<|p_{ij}\left(t\right)|<|{\mathfrak{F}}^{-1}\left(\frac{{\delta }_a\left(t\right)}{\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)}\right)|,\end{array}$$

(26)

where ${\mathfrak{F}}_3^{-1}$ notes the inverse function of ${\mathfrak{F}}_3$. According to (10) and $|{\mathfrak{F}}_3\left(x\right)|\ge |l_{5}x|$, we have ${\mathfrak{F}}_3^{-1}\left(x\right)\le |x/l_5|$. Inequality (26) can be expressed as

$$\begin{array}{c}\hfill 0<|p_{ij}\left(t\right)|<|\frac{{\delta }_a\left(t\right)}{\alpha l_{5}exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)}|.\end{array}$$

Because $\Delta \left(t\right)$ is an unknown bounded noise, there must be a constant $\xi$ that satisfies ${\delta }_a\left(t\right)\le \xi$. Therefore, we have

$$\begin{array}{c}\hfill 0<|p_{ij}\left(t\right)|<|\frac{\xi }{\alpha l_{5}exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)}|.\end{array}$$

It means that $|p_{ij}\left(t\right)|$ will be close to zero over time, i.e.,

$$\begin{array}{c}\hfill 0<\underset{t\to \infty }{lim}\left|p_{ij}\left(t\right)\right|<\underset{t\to \infty }{lim}\left|\frac{\xi }{\alpha l_{5}exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t)}\right|\le 0.\end{array}$$

Similarly, $|q_{ij}\left(t\right)|$ will be close to zero over time, so, when $t\to \infty$, $|{\mathfrak{D}}_{ij}\left(t\right)|\to 0$. Hence,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\parallel \mathbb{D}\left(t\right)\parallel }_{\mathrm{F}}=0.\end{array}$$

When choosing the activated type II, we also define a Lyapunov function ${\upsilon }_{ij}\left(t\right)=\frac{1}{2}{\mathfrak{D}}_{ij}^2\left(t\right)$. According to the modulus, $|{\mathfrak{D}}_{ij}\left(t\right)|$ is bounded and Equation (17), ${\upsilon }_{ij}\left(t\right)$ is negative definite. It can be concluded that $|{\mathfrak{D}}_{ij}\left(t\right)|$ globally converges to 0. Hence,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{\parallel \mathbb{D}\left(t\right)\parallel }_{\mathrm{F}}=0.\end{array}$$

The proof completes. □

4. Numerical Experiments

In this section, we employ an example to verify the superiority of the proposed DFZNN in solving the pseudoinverse of the complex-valued time-varying matrix.

Example 1.

For the purpose of comparative and demonstrative, the following complex-valued time-dependent matrix $A\left(t\right)$ of full rank is exhibited:

$$A\left(t\right)=\left[\begin{array}{ccc}{e}^{it}& e^{i(t+0.5\pi )}& 2e^{it}\\ e^{i(t-0.5\pi )}& e^{it}& e^{i(t+0.5\pi )}\end{array}\right],$$

the theoretical pseudoinverse of $A\left(t\right)$ is

$$A^{+}\left(t\right)=\left[\begin{array}{cc}-0.5e^{-it}& e^{i(-t+0.5\pi )}\\ 0.5e^{i(-t+0.5\pi )}& e^{-it}\\ e^{-it}& e^{i(-t-0.5\pi )}\end{array}\right].$$

For comparing the performance of the models, the relevant parameters in the four models (DFZNN, PTCZNN, LTCZNN, IZNN) are set with the same value. Without loss of generality, in this paper, $\alpha =3,\mu =0.5,\nu =2,l_1=l_2=l_4=0.5,l_3=l_5=4,{\zeta }_1=1,{\zeta }_2=0.8$.

Firstly, for computing a pseudoinverse of a complex-valued time-varying matrix in a noiseless situation, the experimental results generated by the proposed DFZNN model with two different activation types are shown in Figure 1. In Figure 1a, beginning from a randomly initial state $Y\left(0\right)\in {\mathbb{C}}^{m\times n}$, the dynamic trajectories $Y\left(t\right)$ generated by the proposed DFZNN with activation type I converge to the theoretical solution precisely and rapidly. Figure 1b depicts the profile of dynamic trajectories generated by the proposed DFZNN with activation type II; it also can be observed that the neural network output $Y\left(t\right)$ arrives to the theoretical solution precisely and rapidly. The estimation error is measured by Frobenius norm ${\parallel \mathbb{D}\left(t\right)\parallel }_{\mathrm{F}}$, and the estimation error of the DFZNN is plotted in Figure 2 for the activation type I and activation type II, respectively. The simulation results indicate that the DFZNN with two different activation types have the same stability and convergence. The results verify the performance proven in Theorems 1–3.

Secondly, we compare the convergence of the four ZNN models in different noise situations in Figure 3. Figure 3a depicts that the estimation error declines to zero with ${\delta }_{ij}\left(t\right)=0$. The convergence time of DFZNN, PTCZNN, LTCZNN, and IZNN is about 0.15 s, 0.3 s, 0.8 s, and 2 s, respectively. According to Theorem 3, for the DFZNN, the theoretical upper bound of the convergence time can be calculated as:

$$t_{\mathrm{top}}\le \frac{1}{{\zeta }_2}ln\left(1+\frac{{\zeta }_2}{\alpha l_1(1-\mu )}+\frac{{\zeta }_2}{\alpha l_2(\nu -1)}\right)=1.47\mathrm{s}.$$

For PTCZNN and LTZCNN, from Refs. [27,41], the theoretical upper bound of the convergence time can be obtained respectively as:

$$t_{\mathrm{top}}\le \frac{1}{\alpha l_1(1-\mu )}+\frac{1}{\alpha l_2(\nu -1)}=2\mathrm{s},$$

and

$$t_{\mathrm{top}}\le \frac{d_0^{1-\mu }}{\alpha (1-\mu )}=1.73\mathrm{s},$$

where $d_0={\max }_{\forall i,j}\left|{\mathfrak{D}}_{ij}\left(0\right)\right|$, the convergence of LTCZNN is affected by the initial value. At this point, it is obvious that the DFZNN has an advantage in convergence. In order to confirm the robustness in Theorem 4, four complex-valued noise situations are considered, including constant noise ${\delta }_{ij}\left(t\right)=5+5i$, bounded random noise ${\delta }_{ij}\left(t\right)=[2,4]+[2,4]i$, bounded time varying noise ${\delta }_{ij}\left(t\right)=0.5cos\left(2t\right)+0.5cos\left(2t\right)i$, unbounded time varying noise ${\delta }_{ij}\left(t\right)=0.3exp\left(0.2t\right)+0.3exp\left(0.2t\right)i$. The rest of the results demonstrated in Figure 4 show that the proposed DFZNN has stronger robustness than others.

Finally, we discuss the convergence property of the DFZNN model with different values of parameters for the complex-valued time-varying pseudoinverse. From the designed evolution of $\mathbb{D}\left(t\right)$ in (9), the time varying parameters contain attenuation factor ${\zeta }_1\mathrm{arccot}\left(t\right)$ and noise suppression factor ${\zeta }_{2}t$. The convergence and robustness of the DFZNN mainly depend on the time varying parameter in the exponential part. Therefore, we show convergence time of the DFZNN with the different values of ${\zeta }_1$ and ${\zeta }_2$. Furthermore, we discuss convergence time $T_c$ in terms of zero noise and time varying noise; sixteen groups of experiments are summarized in

Table 1. The convergence time $T_c$ (in seconds) of the DFZNN model under different parameters.

Number	Influencing Parameter		${\mathit{T}}_{\mathit{c}}$
	${\mathit{\zeta }}_1$	${\mathit{\zeta }}_2$	Zero Noise	Time-Varying Noise
1	0.5	0.5	0.1694	1.2766
2	0.5	1	0.1664	0.6436
3	0.5	1.5	0.1640	0.6388
4	0.5	2	0.1592	0.6338
5	1	0.5	0.0822	0.6384
6	1	1	0.0774	0.6326
7	1	1.5	0.0758	0.6284
8	1	2	0.0750	0.6284
9	0.5	0.5	0.1694	1.2766
10	1	0.5	0.0822	0.6384
11	1.5	0.5	0.0374	0.6286
12	2	0.5	0.0170	0.0168
13	0.5	1	0.1640	0.6436
14	1	1	0.0774	0.6326
15	1.5	1	0.0368	0.6256
16	2	1	0.0168	0.0168

. Without loss of generality, here we set a bounded time varying noise $10sin\left(5t\right)+10sin\left(5t\right)i$. From the table, we conclude that larger ${\zeta }_1$ (or ${\zeta }_2$) has a shorter convergence time. In terms of robustness, parameter ${\zeta }_2$ has the role of stabilizing the neural system. For example, from numbers 1–4 and 5–8, the convergence time is relatively stable. In particular, from numbers 12 and 16, the convergence time of the DFZNN has no difference even if it is identical in zero noise and time varying noise. While, in terms of convergence speed, the effect of ${\zeta }_1$ is more significant, from numbers 9–12 and 13–16, the convergence time changes significantly when ${\zeta }_1$ increases.

5. Application

Referring to the mechanical structure of the manipulator in [44], this section introduces a mobile manipulator trajectory control task to verify the effectiveness of the DFZNN model. Therefore, the kinematic model of the mobile manipulator considering only the position of the end effector can be obtained as follows:

$${\mathcal{P}}_a\left(t\right)=\mathcal{F}\left(\mathsf{\Theta }\left(t\right)\right)\in {\mathbb{R}}^m,$$

where $\mathsf{\Theta }\left(t\right)={[{\psi }^{\mathrm{T}}\left(t\right),{\theta }^{\mathrm{T}}\left(t\right)]}^{\mathrm{T}}\in {\mathbb{R}}^{n+2}$ denotes the angle vector, including the mobile platform angle vector $\psi \left(t\right)={[{\psi }_L\left(t\right),{\psi }_R\left(t\right)]}^{\mathrm{T}}$ and the manipulator angle vector $\theta \left(t\right)={[{\theta }_1\left(t\right),\dots ,{\theta }_n\left(t\right)]}^{\mathrm{T}}$. In addition, ${\mathcal{P}}_a\left(t\right)={[r_x\left(t\right),r_y\left(t\right),r_z\left(t\right)]}^{\mathrm{T}}$ denotes the actual position of end-effector in space coordinates, and $\mathcal{F}(·)$ represents a known nonlinear mapping from $\mathsf{\Theta }\left(t\right)$ to ${\mathcal{P}}_a\left(t\right)$.

Similarly, redefine a new error function $\mathbb{E}\left(t\right)$:

$$\mathbb{E}\left(t\right)={\mathcal{P}}_d\left(t\right)-{\mathcal{P}}_a\left(t\right),$$

(27)

where ${\mathcal{P}}_d\left(t\right)$ stands for the desired trajectory. Then, the derivative of $\mathbb{E}\left(t\right)$ can be obtained:

$$\dot{\mathbb{E}}\left(t\right)={\dot{\mathcal{P}}}_d\left(t\right)-J\left(\mathsf{\Theta }\left(t\right)\right)\dot{\mathsf{\Theta }}\left(t\right),$$

(28)

where $J\left(\mathsf{\Theta }\left(t\right)\right)=\partial \mathcal{F}\left(\mathsf{\Theta }\left(t\right)\right)/\partial \mathsf{\Theta }\in {\mathbb{R}}^{m\times (n+2)}$. Based on the DFZNN model design method, the dynamic equation of the mobile manipulator can be obtained to achieve the task:

$$J\left(\mathsf{\Theta }\left(t\right)\right)\dot{\mathsf{\Theta }}\left(t\right)={\dot{\mathcal{P}}}_d\left(t\right)+\alpha exp({\zeta }_1\mathrm{arccot}\left(t\right)+{\zeta }_{2}t){\mathfrak{F}}_3\left(\mathbb{E}\left(t\right)\right),$$

(29)

in which $\mathbb{E}\left(t\right)={\mathcal{P}}_d\left(t\right)-\mathcal{F}\left(\mathsf{\Theta }\left(t\right)\right)$. In a similar way, based on the LTCZNN model design method, another dynamic equation can be obtained to achieve the task:

$$J\left(\mathsf{\Theta }\left(t\right)\right)\dot{\mathsf{\Theta }}\left(t\right)={\dot{\mathcal{P}}}_d\left(t\right)+\alpha {\mathcal{W}}_1\left(\mathbb{E}\left(t\right)\right),$$

(30)

In this simulation, we make use of the manipulator to track an infinite symbol trajectory, and the corresponding results are shown in Figure 5 and Figure 6. Among them, the parameters set are: $\alpha =2$; $\mathsf{\Theta }\left(0\right)=[0,0,\pi /6,\pi /2,\pi /3,\pi /3,\pi /4,\pi /3]$.

Figure 5 shows the whole tracking process of the mobile manipulator controlled by the DFZNN model and LTCZNN model. Thereinto, Figure 5a,c represent the entire space trajectory of the DFZNN model and LTCZNN model. Figure 5b,d are the desired trajectory and the actual manipulator trajectory. In a noisy environment (${\delta }_i\left(t\right)=0.3sin\left(5t\right)$, $i=1,2,\dots ,m$), we can observe that only the manipulator controlled by the DFZNN can complete the desired trajectory task well. Figure 6a,b show the errors of tracking the butterfly trajectory of the mobile manipulator controlled by DFZNN and LTCZNN. After two seconds, the upper bound of the trajectory error controlled by LTCZNN is about $5\times {10}^{-2}$, while the upper bound of the trajectory error controlled by DFZNN is only about $1.344\times {10}^{-3}$. Obviously, the mobile manipulator controlled by DFZNN can satisfactorily complete the infinite symbol trajectory tracking task in the noise ${\delta }_i\left(t\right)$.

6. Conclusions

In this paper, a double feature ZNN with a time varying parameter is proposed for solving a complex-valued time-varying matrix pseudoinverse. A new nonlinear activation function is designed to accelerate convergence speed, the function of a time varying parameter is to further improve the convergence speed and enhance the robustness in additional noises. Moreover, two complex-valued activation types are investigated in our work. The stability and robustness of the proposed DFZNN are proven theoretically. In the DFZNN, the upper bound of the predefine time convergence is deduced, which is smaller than PTCZNN and is independent of the initial value. Furthermore, the validity and superiority of the DFZNN are elaborated by comparing with other zeroing neural network models in different noise environments. An application on controlling manipulator movement illustrates the utility value of the DFZNN model in artificial intelligence. Compared with complex-valued ZNN, quaternion-valued ZNN has superiority in storage capacity and dealing with color image problems. In the future, extending the DFZNN to solve quaternion-valued time-varying issue is a worthy direction.