Zhang Neuro-PID Control for Generalized Bi-Variable Function Projective Synchronization of Nonautonomous Nonlinear Systems with Various Perturbations

Abstract

Nonautonomous nonlinear (NN) systems have broad application prospects and significant research value in nonlinear science. In this paper, a new synchronization type—namely, generalized bi-variable function projective synchronization (GBVFPS)—is proposed. The scaling function matrix of GBVFPS is not one-variable but bi-variable. This indicates that the GBVFPS can be transformed into various synchronization types such as projective synchronization (PS), modified PS, function PS, modified function PS, and generalized function PS. In order to achieve the GBVFPS in two different NN systems with various perturbations, by designing a novel Zhang neuro-PID controller, an effective and anti-perturbation GBVFPS control method is proposed. Rigorous theoretical analyses are presented to prove the convergence performance and anti-perturbation ability of the GBVFPS control method, especially its ability to suppress six different perturbations. Besides, the effectiveness, superiority, and anti-perturbation ability of the proposed GBVFPS control method are further substantiated through two representative numerical simulations, including the synchronization of two NN chaotic systems and the synchronization of two four-dimensional vehicular inverted pendulum systems.

1. Introduction

With the rapid development of modern science, nonlinear science has become one of the important factors driving the overall progress of modern science. Nonlinear science has extensive and significant application value in various fields of science and technology, such as nonlinear optimization [1], nonlinear neural networks [2], nonlinear dynamic systems [3,4,5], fractional order nonlinear systems [6], nonlinear differential equations [7,8,9], and so on. At present, most of the researches are focused on autonomous nonlinear systems. However, lots of real-world systems are highly related to temporally varying coefficient function; thus, nonautonomous nonlinear (NN) systems have great research value and extensive application prospects in various fields. For example, Lancelotti et al. studied the positive solutions of NN critical elliptic problems in unbounded domains [10]. The extinction relationship of two species of NN competitive systems was studied in [11]. Lenka et al. investigated the asymptotic stability of NN fractional-order systems [12].

Synchronization is a typical and important dynamic behavior in nonlinear science. Due to its potential applications in multimotor driving systems [13], chaotic systems [6,14], diagnosis and recognition [15], image encryption [16], chemical reaction [17], and secure communication [18], various types of synchronization have been extensively exploited and investigated in recent years. Specially, many valuable results have been presented, such as complete synchronization (CS) [19,20,21], anti-synchronization (AS), lag synchronization [22], generalized synchronization [23], combination synchronization [24], projective synchronization (PS) [25], modified projective synchronization (MPS) [26], function projective synchronization (FPS) [27], modified function projective synchronization (MFPS) [28], and generalized function projective synchronization (GFPS) [29].

In order to tackle synchronization problems with more complicated relationships between systems and improve the flexibility of the synchronization approach, a generalized bi-variable function projection synchronization (GBVFPS) for two different NN systems is designed and proposed. Unlike the investigated synchronization types in previous researches [25,26,27,28,29], the scaling function matrix of GBVFPS is not one-variable but bi-variable—that is, in the proposed GBVFPS type, the scaling function matrix is bi-variable, which is related to both time variable t and state variable $\mathbf{x}\left(t\right)$ of the driving system. Therefore, the proposed GBVFPS is an improvement and extension of PS, FPS, MPS, GFPS, and MFPS. This not only enriches the theoretical research of synchronization but also enables synchronization to be applied to more complicated situations.

In the hardware-implementation process of many systems, temporally varying perturbations are inevitable. Sometimes, temporally varying perturbations may affect the stability and accuracy of the studied system and even lead to the failure of the target task [30,31]. How to reduce the impact of various temporally varying perturbations has received widespread attention and become an interesting research direction. For instance, in [32], Su et al. provided a practical fixed-time control method for synchronization of chaotic neural networks with perturbations. In [33], Yu et al. constructed a multiple-input multiple-output secure communication method for a four-wing chaotic system in the presence of temporally varying perturbations. In [34], an anti-perturbation distributed network is designed for the cooperative motion control of redundant robot manipulators. Generally, NN systems are affected by internal temporally varying noises and internal temporally varying uncertainties. For simplicity, in this paper, these two types of temporally varying perturbations are collectively termed as mixed temporally varying perturbation (MTVP). However, most researchers study one or two MTVPs without conducting a comprehensive and systematic study of MTVPs. Due to the existence of MTVPs in various forms in NN systems, this paper formally discusses six different types of perturbations, including zero perturbation, bounded constant perturbation, linear-form MTVP, exponential-form MTVP, sine-form MTVP, and exponential-sine-form MTVP.

As a widely applicable and efficient method, the Zhang neural dynamics (ZND) method has been systematically proposed and studied since 2001 [35,36,37,38]. Compared with the direct dynamics (DD) method [39], the ZND method not only makes full use of the time derivative information but also contains error feedback information, which can accurately and efficiently solve temporally varying problems, such as temporally varying quadratic programming [40,41], temporally varying nonlinear optimization [42], temporally varying reciprocal solving [43], temporally varying matrix square root solving [44], temporally varying matrix pseudoinverse solving [45,46], and temporally varying matrix decomposition [47]. Although the ZND method has been widely applied by researchers to solve various temporally varying problems, few researchers have used the ZND method to design a suitable controller for dealing with the synchronization problem of NN systems. In addition, when there are various perturbations in the studied systems, exclusively using the ZND method may make it difficult to obtain the desired results. In view of this, it is necessary, valuable, and challenging to exploit a new and effective method to overcome the knotty perturbation problem.

Inspired by the above discussions, in this paper, a Zhang neuro-proportional integral derivative (neuro-PID) controller is designed and proposed for tackling the GBVFPS problem of two different NN systems with various perturbations. Specifically, the main contributions of this paper are illustrated as follows.

(1)

A new synchronization type, namely GBVFPS, is proposed. The scaling factor of GBVFPS is a bi-variable scaling function matrix, marking a significant progress and breakthrough in the field of synchronization research.

(2)

Unlike existing studies, this paper investigates the synchronization of NN systems with six different perturbations. By designing a novel Zhang neuro-PID controller, an effective and anti-perturbation GBVFPS control method is proposed.

(3)

Rigorous theoretical analyses are provided to prove the convergence performance and anti-perturbation ability of the GBVFPS control method, especially its ability to suppress six different perturbations.

(4)

Two representative numerical simulations and comparisons further substantiate the effectiveness, superiority, and anti-perturbation ability of the proposed GBVFPS control method.

For better readability, the abbreviations are presented in Table 1. The remaining arrangements of this paper are structured as follows. In Section 2, the preliminaries for NN systems are provided. Specifically, the definitions of GBVFPS and exponential convergence are presented. In Section 3, the GBVFPS control and theoretical analyses are provided for two different NN systems with six different perturbations. Numerical simulations on NN chaotic systems and four-dimensional vehicular inverted pendulum (FDVIP) systems are conducted in Section 4. Finally, the conclusion of this paper is given in Section 5.

Table 1. Abbreviations.

NN	Nonautonomous nonlinear
DD	Direct dynamics
ZND	Zhang neural dynamics
MTVP	Mixed temporally varying perturbation
CS	Complete synchronization
AS	Anti-synchronization
PS	Projective synchronization
MPS	Modified projective synchronization
FPS	Function projective synchronization
MFPS	Modified function projective synchronization
GFPS	Generalized function projective synchronization
GBVFPS	Generalized bi-variable function projective synchronization
FDVIP	Four-dimensional vehicular inverted pendulum
Neuro-PID	Neuro-proportional integral derivative

2. Preliminaries

In order to lay the foundation for further discussion and better obtain the main results, some necessary preliminaries for NN systems are presented as follows.

The NN system, which is considered as the driving system, is described as

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{x}}\left(t\right)& =\mathbf{g}\left(\mathbf{x}\right(t),t),\hfill \\ \hfill \mathbf{x}\left(0\right)& ={\mathbf{x}}_0,\hfill \end{array}\right.$$

(1)

where $\mathbf{x}\left(t\right)\in {\mathbb{R}}^m$ denotes the temporally varying state vector, $\dot{\mathbf{x}}\left(t\right)\in {\mathbb{R}}^m$ stands for the first order time derivative of $\mathbf{x}\left(t\right)$, and ${\mathbf{x}}_0\in {\mathbb{R}}^m$ is used to denote the initial state vector. In addition, $\mathbf{g}(\mathbf{x}\left(t\right),t):{\mathbb{R}}^m\times [0,+\infty )\mapsto {\mathbb{R}}^m$ is a temporally varying nonlinear continuous vector-valued function.

To achieve the GBVFPS of the driving system (1) and another different NN system, the corresponding response system is given by

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{y}}\left(t\right)& =\mathbf{f}\left(\mathbf{y}\right(t),\mathbf{u}(t),t),\hfill \\ \hfill \mathbf{y}\left(0\right)& ={\mathbf{y}}_0,\hfill \end{array}\right.$$

(2)

where $\mathbf{y}\left(t\right)\in {\mathbb{R}}^m$ represents the temporally varying state vector, $\dot{\mathbf{y}}\left(t\right)\in {\mathbb{R}}^m$ is the first order time derivative of $\mathbf{y}\left(t\right)$, ${\mathbf{y}}_0\in {\mathbb{R}}^m$ denotes the initial state vector, and $\mathbf{u}\left(t\right)\in {\mathbb{R}}^m$ is the control input vector to be designed. In addition, $\mathbf{f}(\mathbf{y}\left(t\right),\mathbf{u}\left(t\right),t):{\mathbb{R}}^m\times {\mathbb{R}}^m\times [0,+\infty )\mapsto {\mathbb{R}}^m$ is a differentiable temporally varying nonlinear vector-valued function. In this paper, we consider the decomposable function $\mathbf{f}\left(\mathbf{y}\right(t),\mathbf{u}(t),t)=\mathbf{h}\left(\mathbf{y}\right(t),t)+\mathbf{F}\left(\mathbf{y}\right(t),t)\mathbf{u}\left(t\right)$, where $\mathbf{h}(\mathbf{y}\left(t\right),t):{\mathbb{R}}^m\times [0,+\infty )\mapsto {\mathbb{R}}^m$ is a temporally varying nonlinear continuous vector-valued function, $\mathbf{F}(\mathbf{y}\left(t\right),t):{\mathbb{R}}^m\times [0,+\infty )\mapsto {\mathbb{R}}^{m\times m}$ is a differentiable full-row-rank temporally varying matrix-valued function. In this case, the response system (2) is rewritten as

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{y}}\left(t\right)& =\mathbf{h}\left(\mathbf{y}\right(t),t)+\mathbf{F}\left(\mathbf{y}\right(t),t)\mathbf{u}\left(t\right),\hfill \\ \hfill \mathbf{y}\left(0\right)& ={\mathbf{y}}_0.\hfill \end{array}\right.$$

(3)

It is well known that perturbations are not only inevitable and ubiquitous in practical systems but are also temporally varying rather than a fixed constant. In order to make the research results better for application in practice, we study the NN system with MTVP. Therefore, the driving system (1) with MTVP is presented as

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{x}}\left(t\right)& =\mathbf{g}\left(\mathbf{x}\right(t),t)+\mathit{\xi }\left(t\right),\hfill \\ \hfill \mathbf{x}\left(0\right)& ={\mathbf{x}}_0,\hfill \end{array}\right.$$

(4)

where $\mathit{\xi }\left(t\right)\in {\mathbb{R}}^m$ represents the MTVP. Similarly, the response system (3) with MTVP is presented by

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{y}}\left(t\right)& =\mathbf{h}\left(\mathbf{y}\right(t),t)+\mathbf{F}\left(\mathbf{y}\right(t),t)\mathbf{u}\left(t\right)+\mathit{\phi }\left(t\right),\hfill \\ \hfill \mathbf{y}\left(0\right)& ={\mathbf{y}}_0,\hfill \end{array}\right.$$

(5)

where $\mathit{\phi }\left(t\right)\in {\mathbb{R}}^m$ stands for the MTVP. The purpose of designing $\mathbf{u}\left(t\right)$ is to make the driving system (4) and response system (5) to achieve GBVFPS with various MTVPs.

For the convenience of presentation and theoretical analyses, the following two definitions are provided.

Definition 1. 

For the driving system (4) and response system (5), it is said that systems (4) and (5) can achieve generalized bi-variable function projective synchronization (GBVFPS) if there exists a differentiable bi-variable scaling function matrix $\Psi (\mathbf{x}\left(t\right),t):{\mathbb{R}}^m\times [0,+\infty )\mapsto {\mathbb{R}}^{m\times m}$ such that

$$\underset{t\to \infty }{lim}{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=\underset{t\to \infty }{lim}\left|\right|\mathbf{y}\left(t\right)-{\Psi (\mathbf{x}\left(t\right),t)\mathbf{x}\left(t\right)\left|\right|}_2=0,$$

(6)

where $\mathbf{e}\left(t\right)\in {\mathbb{R}}^m$ denotes the GBVFPS error vector.

For Equation (6), it must be particularly pointed out that the bi-variable scaling function matrix $\Psi \left(\mathbf{x}\right(t),t)$ can be transformed into various forms of matrices. Generally speaking, different mathematical expressions of $\Psi \left(\mathbf{x}\right(t),t)$ correspond to different synchronization types. For instance, when $\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11}\left(t\right),{\psi }_{22}\left(t\right),\cdots ,{\psi }_{mm}\left(t\right))$, the GBVFPS becomes MFPS, where $\mathrm{diag}(·)$ denotes a diagonal matrix and ${\psi }_{ii}\left(t\right)$, $i=1,2,\cdots ,m$, represents the ith function on the diagonal of the diagonal matrix. While $\Psi \left(\mathbf{x}\right(t),t)=\mathrm{diag}\left(\psi \right(t),\psi (t),\cdots ,\psi (t\left)\right)$, the GBVFPS turns into FPS. The GBVFPS changes into MPS when $\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11},{\psi }_{22},\cdots ,{\psi }_{mm})$. In particular, when $\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11}\left(\mathbf{x}\left(t\right)\right),{\psi }_{22}\left(\mathbf{x}\left(t\right)\right),\cdots ,{\psi }_{mm}\left(\mathbf{x}\left(t\right)\right))$, the GBVFPS changes into GFPS. All in all, when $\Psi \left(\mathbf{x}\right(t),t)$ turns into a different form of mathematical expression, its corresponding synchronization type subsequently changes. For the convenience of presentation and readability, the relationships between $\Psi \left(\mathbf{x}\right(t),t)$ and various synchronization types are summarized in Table 2.

Table 2. Relationships between $\Psi (\mathbf{x}\left(t\right),t)\in {\mathbb{R}}^{m\times m}$ and various synchronization types.

Bi-Variable Scaling Function Matrix $\mathit{\Psi }\left(\mathit{x}\right(\mathit{t}),\mathit{t})$	Type of Synchronization
$\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11}(\mathbf{x}\left(t\right),t),{\psi }_{22}(\mathbf{x}\left(t\right),t),\cdots ,{\psi }_{mm}(\mathbf{x}\left(t\right),t))$	GBVFPS
$\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11}\left(\mathbf{x}\left(t\right)\right),{\psi }_{22}\left(\mathbf{x}\left(t\right)\right),\cdots ,{\psi }_{mm}\left(\mathbf{x}\left(t\right)\right))$	GFPS
$\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11}\left(t\right),{\psi }_{22}\left(t\right),\cdots ,{\psi }_{mm}\left(t\right))$	MFPS
$\Psi \left(\mathbf{x}\right(t),t)=\mathrm{diag}\left(\psi \right(t),\psi (t),\cdots ,\psi (t\left)\right)$	FPS
$\Psi (\mathbf{x}\left(t\right),t)=\mathrm{diag}({\psi }_{11},{\psi }_{22},\cdots ,{\psi }_{mm})$	MPS
$\Psi \left(\mathbf{x}\right(t),t)=\mathrm{diag}(\psi ,\psi ,\cdots ,\psi )$	PS
$\Psi \left(\mathbf{x}\right(t),t)=\mathrm{diag}(-1,-1,\cdots ,-1)$	AS
$\Psi \left(\mathbf{x}\right(t),t)=\mathrm{diag}(1,1,\cdots ,1)$	CS

Definition 2. 

For the GBVFPS of two different NN systems, starting from an initial vector $\mathbf{e}\left(0\right)$, the GBVFPS error vector $\mathbf{e}\left(t\right)$ is said to be exponentially convergent to zero if it satisfies

$${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2\le {\left|\right|\mathit{\tau }\left|\right|}_{2}exp(-\alpha t),\forall t>0,$$

(7)

where $\mathit{\tau }$ denotes a constant vector and constant $\alpha >0$ denotes the exponential convergence speed.

3. GBVFPS Control and Theoretical Analyses

In this section, a novel Zhang neuro-PID controller is designed to handle the GBVFPS problem of two different NN systems with various perturbations. Six theorems are presented to show the convergence performance and anti-perturbation ability of the proposed GBVFPS control method.

3.1. GBVFPS Control

First, in view of Definition 1, driving system (4), and response system (5), the GBVFPS error system with MTVP is expressed as

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}\left(\mathbf{y}\right(t),t)+\mathbf{F}\left(\mathbf{y}\right(t),t)\mathbf{u}\left(t\right)+\mathit{\varphi }\left(t\right)-\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0,\hfill \end{array}\right.$$

(8)

where $\mathbf{e}\left(0\right)\in {\mathbb{R}}^m$ generally represents the initial error vector, $\Omega \left(\mathbf{x}\right(t),t)$ denotes the derivative of $\Psi \left(\mathbf{x}\right(t),t)$ with respect to t, ${\Psi }_0=\Psi (\mathbf{x}\left(0\right),0)$ is the initial bi-variable scaling function matrix, and $\mathit{\varphi }\left(t\right)=\mathit{\phi }\left(t\right)-\Psi \left(\mathbf{x}\right(t),t)\mathit{\xi }\left(t\right)$ represents the MTVP. For better presentation and understanding, the architecture of GBVFPS for the driving system (4) and response system (5) with MTVP is displayed in Figure 1.

Figure 1. Architecture of GBVFPS for driving system (4) and response system (5) with MTVP.

Then, an effective controller needs to be designed so that driving system (4) and response system (5) with MTVP can achieve GBVFPS. On the basis of the ZND method [48], the first ZND error function containing an integral term is defined as

$$\mathit{\omega }\left(t\right)={\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau .$$

(9)

By utilizing the ZND design formula $\dot{\mathit{\omega }}\left(t\right)=-{\delta }_1\mathit{\omega }\left(t\right)$, with ${\delta }_1>0$ being a flexible and changeable design parameter to control the convergence speed [49], we obtain

$$\mathbf{e}\left(t\right)=-{\delta }_1{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau .$$

(10)

Subsequently, the second ZND error function is constructed as

$$\mathit{\nu }\left(t\right)=\dot{\mathit{\omega }}\left(t\right)+{\delta }_1\mathit{\omega }\left(t\right).$$

(11)

By making use of the ZND design formula $\dot{\mathit{\nu }}\left(t\right)=-{\delta }_2\mathit{\nu }\left(t\right)$, the derivative of $\mathbf{e}\left(t\right)$ is formulated as

$$\dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau .$$

(12)

Substituting Equation (12) into the GBVFPS error system with MTVP (8), the Zhang neuro-PID controller is designed as

$$\begin{array}{cc}\hfill \mathbf{u}\left(t\right)& ={\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau -\mathbf{h}(\mathbf{y}\left(t\right),t)\hfill \\ \hfill & +\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right)).\hfill \end{array}$$

(13)

Reformulating (13), the Zhang neuro-PID controller is rewritten as

$$\begin{array}{cc}\hfill \mathbf{u}\left(t\right)=& -({\delta }_1+{\delta }_2){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t){\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & +{\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-\mathbf{h}(\mathbf{y}\left(t\right),t)+\Psi (\mathbf{x}\left(t\right),t\left)\mathbf{g}\right(\mathbf{x}\left(t\right),t)+\Omega (\mathbf{x}\left(t\right),t\left)\mathbf{x}\right(t\left)\right).\hfill \end{array}$$

(14)

Remark 1. 

Note that the first and second parts of the Zhang neuro-PID controller (14) contain the proportional and integral information of the GBVFPS error vector $\mathbf{e}\left(t\right)$, respectively. The remaining part of the Zhang neuro-PID controller (14) includes the derivatives of system and bi-variable scaling function matrix. Referring to the PID [41], the designed Zhang neuro-PID controller (14) has the PID characteristics, effectively suppressing the impact of various perturbations on NN systems. From controller (13), we know that the controller involves matrix inversion operation, leading to a computational complexity of $O\left(m^3\right)$. In addition, from the design process, the parameters ${\delta }_1$ and ${\delta }_2$ have physical meaning. In a practical physical sense, the parameters ${\delta }_1$ and ${\delta }_2$ generally correspond to the reciprocal of the capacitance parameter [35]. Therefore, the parameters ${\delta }_1$ and ${\delta }_2$ should be set to be large enough for hardware allowances or appropriately large for actual control purposes.

Remark 2. 

It is worth noting that this paper studies the GBVFPS problem of two different NN systems. On one hand, NN systems are temporally varying nonlinear systems with complicated nonlinear dynamic characteristics. On the other hand, the scaling factor of the GBVFPS is a bi-variable scaling function matrix. Both of these aspects make the GBVFPS problem pretty troublesome and interesting. Therefore, by designing two ZND error functions and utilizing the ZND design formula twice, the GBVFPS control method with a Zhang neuro-PID controller is proposed to handle the GBVFPS problem.

3.2. Theoretical Analyses

In this subsection, the effectiveness of the proposed GBVFPS control method is theoretically analyzed. Specifically, this subsection investigates the convergence performance and anti-perturbation ability of the GBVFPS error system (8) with six different perturbations under Zhang neuro-PID controller (13), where six different types of perturbations are zero perturbation, bounded constant perturbation, linear-form MTVP, exponential-form MTVP, sine-form MTVP, and exponential-sine-form MTVP. The specific six different types of perturbations and their characteristics are exhibited in Table 3.

Table 3. Six different types of perturbations and their characteristics.

No.	Perturbation	Expression	Characteristic
1	Zero perturbation	$\mathit{\varphi }\left(t\right)=\mathbf{0}\in {\mathbb{R}}^m$	Ideal environment
2	Bounded constant perturbation	$\mathit{\varphi }\left(t\right)=\mathbf{c}\in {\mathbb{R}}^m$	Stabilization
3	Linear-form MTVP	$\mathit{\varphi }\left(t\right)=\mathbf{a}t+\mathbf{c}\in {\mathbb{R}}^m$	Linear change
4	Exponential-form MTVP	$\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}+\mathbf{c}\in {\mathbb{R}}^m$	Exponential decay change
5	Sine-form MTVP	$\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{d}+\mathbf{c}\in {\mathbb{R}}^m$	Sinusoidal periodic change
6	Exponential-sine-form MTVP	$\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}\in {\mathbb{R}}^m$	Composite change of exponential and sinusoidal

where $\mathbf{c}$ and $\mathbf{a}$ are constant vectors, $\mathbf{b}={[exp(-b_{1}t),exp(-b_{2}t),\cdots ,exp(-b_{m}t)]}^{\mathrm{T}}$ with $b_j$$(j\in \{1,2,\cdots ,m\left\}\right)$ being positive constant, $\mathbf{d}={[sin\left(d_{1}t\right),sin\left(d_{2}t\right),\cdots ,sin\left(d_{m}t\right)]}^{\mathrm{T}}$ with $d_j$ being constant, and operator ⊙ denotes Hadamard product between two vectors.

Theorem 1. 

Consider the zero perturbation and the GBVFPS problem of NN systems (4) and (5) starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the perturbation $\mathit{\varphi }\left(t\right)=\mathbf{0}\in {\mathbb{R}}^m$ and the design of the Zhang neuro-PID controller is shown in controller (13), the GBVFPS error system (8) converges to zero globally and exponentially.

Proof. 

First, we prove the global convergence performance of the GBVFPS error function $\mathbf{e}\left(t\right)$. Substituting the designed Zhang neuro-PID controller (13) into the GBVFPS error system (8) with zero perturbation $\mathit{\varphi }\left(t\right)=\mathbf{0}$, we obtain

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}(\mathbf{y}\left(t\right),t)+\mathbf{F}(\mathbf{y}\left(t\right),t){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & -\mathbf{h}\left(\mathbf{y}\right(t),t)+\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right))\hfill \\ \hfill & -\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0.\hfill \end{array}\right.$$

(15)

Thus, the derivative of the GBVFPS error vector $\mathbf{e}\left(t\right)$ is simplified as

$$\dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau .$$

(16)

The Lyapunov function candidate for the ith subsystem is constructed as

$$l_i\left(t\right)={\displaystyle \frac{1}{2}}{e}_i^2\left(t\right)+{\displaystyle \frac{{\delta }_1{\delta }_2}{2}}{\left({\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau \right)}^2.$$

Generally, the Lyapunov function candidate $l_i\left(t\right)$ is guaranteed to be positive definite—in other words, $l_i\left(t\right)>0$ for any $e_i\left(t\right)\ne 0$ or ${\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau \ne 0$, and $l_i\left(t\right)=0$ for $e_i\left(t\right)={\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau =0$. Besides, when $e_i\left(t\right)\to \infty$, $l_i\left(t\right)\to \infty$. More importantly, the derivative of $l_i\left(t\right)$ with respect to t is

$$\begin{array}{cc}\hfill {\dot{l}}_i\left(t\right)& =e_i\left(t\right)\left(-({\delta }_1+{\delta }_2)e_i\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau \right)+{\delta }_1{\delta }_2{e}_i\left(t\right){\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & =-({\delta }_1+{\delta }_2)e_i^2\left(t\right)\le 0.\hfill \end{array}$$

According to the Lyapunov theory [50], $l_i\left(t\right)$ of the ith subsystem globally converges to zero for any $i=1,2,\cdots ,m$. Therefore, the GBVFPS error system (8) converges to zero globally.

Next, the detailed theoretical proof of the exponential convergence performance of GBVFPS error system (8) is presented. On the basis of the ZND error function, we have $\mathit{\omega }\left(t\right)={\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau$. Let $e_i\left(t\right)$, ${\omega }_i\left(t\right)$, ${\dot{\omega }}_i\left(t\right)$, and ${\ddot{\omega }}_i\left(t\right)$ be the ith element of $\mathbf{e}\left(t\right)$, $\mathit{\omega }\left(t\right)$, $\dot{\mathit{\omega }}\left(t\right)$, and $\ddot{\mathit{\omega }}\left(t\right)$, respectively. Then, $\dot{\mathit{\omega }}\left(t\right)=\mathbf{e}\left(t\right)$, ${\omega }_i\left(0\right)=0$, and ${\dot{\omega }}_i\left(0\right)=e_i\left(0\right)$. Thereby, the ith subsystem of Equation (16) is rewritten as

$${\ddot{\omega }}_i\left(t\right)+({\delta }_1+{\delta }_2){\dot{\omega }}_i\left(t\right)+{\delta }_1{\delta }_2{\omega }_i\left(t\right)=0.$$

(17)

In view of the differential equation theory [50], the characteristic equation of Equation (17) is

$${\zeta }^2+({\delta }_1+{\delta }_2)\zeta +{\delta }_1{\delta }_2=0.$$

(18)

Evidently, the two roots of characteristic Equation (18) are ${\zeta }_1=-{\delta }_2$ and ${\zeta }_2=-{\delta }_1$. From the previous parameter settings, parameters ${\delta }_1$ and ${\delta }_2$ can be changed according to the actual situation. Therefore, we roughly divide parameters ${\delta }_1$ and ${\delta }_1$ into the following two situations for discussion.

(1) If ${\delta }_1\ne {\delta }_2$, the general solution of differential Equation (17) is obtained by

$${\omega }_i\left(t\right)=a_{i}exp(-{\delta }_{1}t)+b_{i}exp(-{\delta }_{2}t),$$

(19)

where $a_i=e_i\left(0\right)/({\delta }_1-{\delta }_2)$ and $b_i=-e_i\left(0\right)/({\delta }_1-{\delta }_2)$. For reading convenience as well as for completeness, Equation (19) is rewritten as

$${\omega }_i\left(t\right)=e_i\left(0\right)(exp(-{\delta }_{1}t)-exp(-{\delta }_{2}t))/({\delta }_1-{\delta }_2),$$

and, further, we have

$$\begin{array}{c}\hfill e_i\left(t\right)=e_i\left(0\right)(-{\delta }_{1}exp(-{\delta }_{1}t)+{\delta }_{2}exp(-{\delta }_{2}t))/({\delta }_1-{\delta }_2).\end{array}$$

Let $\delta =min\{{\delta }_1,{\delta }_2\}$, we obtain

$$|e_i\left(t\right)|\le |e_i\left(0\right)|{\displaystyle \frac{{\delta }_1+{\delta }_2}{|{\delta }_1-{\delta }_2|}}exp(-\delta t).$$

Thus, the result is obtained as

$$|e_i\left(t\right)|\le |{\tau }_i|exp(-\delta t).$$

(2) If ${\delta }_1={\delta }_2$, the general solution of differential Equation (17) is obtained as

$${\omega }_i\left(t\right)=(a_i+b_{i}t)exp(-{\delta }_{1}t),$$

where $a_i=0$ and $b_i=e_i\left(0\right)$. Therefore, we further obtain

$${\omega }_i\left(t\right)=e_i\left(0\right)texp(-{\delta }_{1}t),$$

and

$$e_i\left(t\right)=e_i\left(0\right)(\exp (-{\delta }_{1}t)-{\delta }_{1}texp(-{\delta }_{1}t)).$$

Finally, the following inequality is obtained:

$$|e_i\left(t\right)|\le |e_i\left(0\right)|(1+{\delta }_{1}t)exp(-{\delta }_{1}t).$$

In view of Lemma 1 in [37], we have

$$|e_i\left(t\right)|\le |{\tau }_i|exp(-\alpha t),$$

where constant $\alpha >0$ denotes the exponential convergence speed. Thus, the result is obtained as

$${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2\le {\left|\right|\mathit{\tau }\left|\right|}_{2}exp(-\alpha t).$$

According to Definitions 1 and 2, the GBVFPS error system (8) without perturbations converges to zero globally and exponentially. The proof is thus completed. □

Theorem 2. 

Consider the bounded constant perturbation and the GBVFPS problem of NN systems (4) and (5). Starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the perturbation $\mathit{\varphi }\left(t\right)=\mathbf{c}\in {\mathbb{R}}^m$ is a constant vector and the design of the Zhang neuro-PID controller is shown in controller (13), the GBVFPS error system (8) converges to zero as time t goes.

Proof. 

Substituting the designed Zhang neuro-PID controller (13) into the GBVFPS error system (8) with bounded constant perturbation $\mathit{\varphi }\left(t\right)=\mathbf{c}$, we obtain

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}(\mathbf{y}\left(t\right),t)+\mathbf{F}(\mathbf{y}\left(t\right),t){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & -\mathbf{h}\left(\mathbf{y}\right(t),t)+\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right))+\mathbf{c}\hfill \\ \hfill & -\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0.\hfill \end{array}\right.$$

(20)

Hence, the derivative of the GBVFPS error vector $\mathbf{e}\left(t\right)$ is simplified as

$$\dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau +\mathbf{c}.$$

(21)

The ith subsystem of Equation (21) is reformulated as

$${\dot{e}}_i\left(t\right)=-({\delta }_1+{\delta }_2)e_i\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau +c_i.$$

By applying Laplace transform [51] to the above equation, we have

$$s{e}_i\left(s\right)-e_i\left(0\right)=-({\delta }_1+{\delta }_2)e_i\left(s\right)-{\delta }_1{\delta }_2{e}_i\left(s\right)/s+c_i/s.$$

By rearranging the above equation, we further obtain

$$s^2{e}_i\left(s\right)-s{e}_i\left(0\right)=-({\delta }_1+{\delta }_2)s{e}_i\left(s\right)-{\delta }_1{\delta }_2{e}_i\left(s\right)+c_i.$$

Then, the above equation is rewritten as

$$e_i\left(s\right)={\displaystyle \frac{s{e}_i\left(0\right)+c_i}{s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2}}.$$

By virtue of the final value theorem [50], we obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{e}_i\left(t\right)=\underset{s\to 0}{lim}s{e}_i\left(s\right)=\underset{s\to 0}{lim}{\displaystyle \frac{s(s{e}_i\left(0\right)+c_i)}{s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2}}=0.\end{array}$$

Thus, the result is obtained as ${lim}_{t\to \infty }{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=0.$ That is, the GBVFPS error system (8) with bounded constant perturbation converges to zero. The proof is completed. □

Theorem 3. 

Consider the linear-form MTVP and the GBVFPS problem of NN systems (4) and (5) starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}t+\mathbf{c}\in {\mathbb{R}}^m$ is a temporally varying linear function vector with $\mathbf{a}$ being a constant vector, and the Zhang neuro-PID controller is shown in controller (13), then the GBVFPS error system (8) is bounded with steady-state GBVFPS error ${lim}_{t\to \infty }{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=\sqrt{{\sum }_{i=1}^m{\left(a_i\right)}^2}/\left({\delta }_1{\delta }_2\right)$.

Proof. 

Substituting the designed Zhang neuro-PID controller (13) into the GBVFPS error system (8) with linear-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}t+\mathbf{c}$, we have

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}(\mathbf{y}\left(t\right),t)+\mathbf{F}(\mathbf{y}\left(t\right),t){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & -\mathbf{h}\left(\mathbf{y}\right(t),t)+\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right))+\mathbf{a}t+\mathbf{c}\hfill \\ \hfill & -\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0.\hfill \end{array}\right.$$

(22)

Thus, the derivative of the GBVFPS error vector $\mathbf{e}\left(t\right)$ is simplified as

$$\dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau +\mathbf{a}t+\mathbf{c}.$$

(23)

Then, the ith subsystem of Equation (23) is rewritten as

$${\dot{e}}_i\left(t\right)=-({\delta }_1+{\delta }_2)e_i\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau +a_{i}t+c_i.$$

By employing Laplace transform to the above equation, the following equation is derived:

$$\begin{array}{cc}\hfill s{e}_i\left(s\right)-e_i\left(0\right)=& -({\delta }_1+{\delta }_2)e_i\left(s\right)-{\delta }_1{\delta }_2{\displaystyle \frac{e_i\left(s\right)}{s}}+{\displaystyle \frac{a_i}{s^2}}+{\displaystyle \frac{c_i}{s}}.\hfill \end{array}$$

By rearranging the above equation, we further have

$$\begin{array}{cc}\hfill s^3{e}_i\left(s\right)-s^2{e}_i\left(0\right)=& -({\delta }_1+{\delta }_2)s^2{e}_i\left(s\right)-{\delta }_1{\delta }_{2}s{e}_i\left(s\right)+a_i+c_{i}s.\hfill \end{array}$$

Then, the following equation is further derived:

$$e_i\left(s\right)={\displaystyle \frac{s^2{e}_i\left(0\right)+a_i+c_{i}s}{s^3+({\delta }_1+{\delta }_2)s^2+{\delta }_1{\delta }_{2}s}}.$$

According to the final value theorem [8], we obtain

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{e}_i\left(t\right)=\underset{s\to 0}{lim}s{e}_i\left(s\right)=\underset{s\to 0}{lim}{\displaystyle \frac{s^2{e}_i\left(0\right)+a_i+c_{i}s}{s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2}}={\displaystyle \frac{a_i}{{\delta }_1{\delta }_2}}.\end{array}$$

Therefore, the GBVFPS error system (8) with linear-form MTVP is bounded with steady-state GBVFPS error ${lim}_{t\to \infty }{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=\sqrt{{\sum }_{i=1}^m{\left(a_i\right)}^2}/\left({\delta }_1{\delta }_2\right)$. The proof is completed. □

Theorem 4. 

Consider the exponential-form MTVP and the GBVFPS problem of NN systems (4) and (5) starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}+\mathbf{c}\in {\mathbb{R}}^m$ is a function vector in temporally varying exponential-form, where operator ⊙ is the Hadamard product between two vectors and $\mathbf{b}={[exp(-b_{1}t),exp(-b_{2}t),\cdots ,exp(-b_{m}t)]}^{\mathrm{T}}$ with $b_j$$(j\in \{1,2,\cdots ,m\left\}\right)$ being positive constant, and the Zhang neuro-PID controller is presented in controller (13), then the GBVFPS error system (8) converges to zero.

Proof. 

Substituting the designed Zhang neuro-PID controller (13) into the GBVFPS error system (8) with exponential-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}+\mathbf{c}$, we obtain

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}(\mathbf{y}\left(t\right),t)+\mathbf{F}(\mathbf{y}\left(t\right),t){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & -\mathbf{h}\left(\mathbf{y}\right(t),t)+\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right))+\mathbf{a}\odot \mathbf{b}+\mathbf{c}\hfill \\ \hfill & -\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0.\hfill \end{array}\right.$$

(24)

Hence, the derivative of the GBVFPS error vector $\mathbf{e}\left(t\right)$ is simplified as

$$\dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau +\mathbf{a}\odot \mathbf{b}+\mathbf{c}.$$

(25)

Then, the ith subsystem of Equation (25) is rewritten as

$$\begin{array}{cc}\hfill {\dot{e}}_i\left(t\right)& =-({\delta }_1+{\delta }_2)e_i\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau +a_{i}exp(-b_{i}t)+c_i.\hfill \end{array}$$

Similarly, we obtain the following explicit form:

$$\begin{array}{cc}\hfill s{e}_i\left(s\right)-e_i\left(0\right)& =-({\delta }_1+{\delta }_2)e_i\left(s\right)-{\delta }_1{\delta }_2{\displaystyle \frac{e_i\left(s\right)}{s}}+{\displaystyle \frac{a_i}{s+b_i}}+{\displaystyle \frac{c_i}{s}}.\hfill \end{array}$$

Next, by adjusting the above explicit expression, we obtain

$$\begin{array}{cc}\hfill (s+b_i)s^2& e_i\left(s\right)-s(s+b_i)e_i\left(0\right)\hfill \\ \hfill =& -({\delta }_1+{\delta }_2)s(s+b_i)e_i\left(s\right)-{\delta }_1{\delta }_2(s+b_i)e_i\left(s\right)+a_{i}s+c_i(s+b_i).\hfill \end{array}$$

The detailed mathematical expression of $e_i\left(s\right)$ is expressed as

$$\begin{array}{c}\hfill e_i\left(s\right)={\displaystyle \frac{s(s+b_i)e_i\left(0\right)+a_{i}s+c_i(s+b_i)}{(s+b_i)(s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2)}}={\displaystyle \frac{(s+b_i)s{e}_i\left(0\right)+(a_i+c_i)s+c_i{b}_i}{(s+b_i)(s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2)}}.\end{array}$$

Finally, as time t goes, $e_i\left(t\right)$ tends to zero—that is,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}{e}_i\left(t\right)=\underset{s\to 0}{lim}s{e}_i\left(s\right)=\underset{s\to 0}{lim}{\displaystyle \frac{s((s+b_i)s{e}_i\left(0\right)+(a_i+c_i)s+c_i{b}_i)}{(s+b_i)(s^2+({\delta }_1+{\delta }_2)s+{\delta }_1{\delta }_2)}}=0.\end{array}$$

Thus, the result is obtained as ${lim}_{t\to \infty }{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=0.$ Therefore, the GBVFPS error system (8) with exponential-form MTVP converges to zero. The proof is completed. □

Theorem 5. 

Consider the sine-form MTVP and the GBVFPS problem of NN systems (4) and (5) starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{d}+\mathbf{c}\in {\mathbb{R}}^m$ is a function vector in temporally varying sine-form, where $\mathbf{d}={[sin\left(d_{1}t\right),sin\left(d_{2}t\right),\cdots ,sin\left(d_{m}t\right)]}^{\mathrm{T}}$ with $d_j$$(j\in \{1,2,\cdots ,m\left\}\right)$ being constant, and the Zhang neuro-PID controller is shown in controller (13), then the GBVFPS error system (8) converges to zero.

Proof. 

The proof is omitted due to the similarity with Theorem 4. □

In some complicated actual environments, the temporally varying perturbations to the NN system may change along with environments. In order to make the results of our research better apply to the actual situation, the exponential-sine-form MTVP is investigated.

Theorem 6. 

Consider the exponential-sine-form MTVP and the GBVFPS problem of NN systems (4) and (5) starting with random generated initial vectors ${\mathbf{y}}_0$, ${\mathbf{x}}_0$ and initial bi-variable scaling function matrix ${\Psi }_0$. When the MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}\in {\mathbb{R}}^m$ is a function vector with a combination of temporally varying exponential and sine forms, and the Zhang neuro-PID controller is presented in controller (13), then the GBVFPS error system (8) converges to zero as time t goes.

Proof. 

Substituting the designed Zhang neuro-PID controller (13) into the GBVFPS error system (8) with exponential-sine-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}$, we obtain

$$\left\{\begin{array}{cc}\hfill \dot{\mathbf{e}}\left(t\right)& =\mathbf{h}(\mathbf{y}\left(t\right),t)+\mathbf{F}(\mathbf{y}\left(t\right),t){\mathbf{F}}^{-1}(\mathbf{y}\left(t\right),t)(-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & -\mathbf{h}\left(\mathbf{y}\right(t),t)+\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)+\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right))+\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}\hfill \\ \hfill & -\Psi \left(\mathbf{x}\right(t),t)\mathbf{g}\left(\mathbf{x}\right(t),t)-\Omega \left(\mathbf{x}\right(t),t)\mathbf{x}\left(t\right),\hfill \\ \hfill \mathbf{e}\left(0\right)& ={\mathbf{y}}_0-{\Psi }_0{\mathbf{x}}_0.\hfill \end{array}\right.$$

(26)

Thus, the derivative of the GBVFPS error vector $\mathbf{e}\left(t\right)$ is simplified as

$$\begin{array}{c}\hfill \dot{\mathbf{e}}\left(t\right)=-({\delta }_1+{\delta }_2)\mathbf{e}\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t\mathbf{e}\left(\tau \right)\mathrm{d}\tau +\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}.\end{array}$$

(27)

Thus, the ith subsystem of Equation (27) is rewritten as

$$\begin{array}{cc}\hfill {\dot{e}}_i\left(t\right)& =-({\delta }_1+{\delta }_2)e_i\left(t\right)-{\delta }_1{\delta }_2{\int }_0^t{e}_i\left(\tau \right)\mathrm{d}\tau +a_{i}exp(-b_{i}t)sin\left(d_{i}t\right)+c_i.\hfill \end{array}$$

Similarly, the following equation is obtained:

$$\begin{array}{cc}\hfill s{e}_i\left(s\right)-e_i\left(0\right)& =-({\delta }_1+{\delta }_2)e_i\left(s\right)-{\delta }_1{\delta }_2{\displaystyle \frac{e_i\left(s\right)}{s}}+{\displaystyle \frac{a_i{d}_i}{{(s+b_i)}^2+d_i^2}}+{\displaystyle \frac{c_i}{s}}.\hfill \end{array}$$

Then, multiplying both sides of the above equation by $s({(s+b_i)}^2+d_i^2)$ simultaneously, the following expression is obtained:

$$\begin{array}{cc}\hfill s\left(\right(s+& b_i{)}^2+d_i^2\left)\right(s{e}_i\left(s\right)-e_i\left(0\right))\hfill \\ \hfill =& -({\delta }_1+{\delta }_2)s({(s+b_i)}^2+d_i^2)e_i\left(s\right)+a_i{d}_{i}s-{\delta }_1{\delta }_2({(s+b_i)}^2+d_i^2)e_i\left(s\right)\hfill \\ \hfill & +c_i({(s+b_i)}^2+d_i^2).\hfill \end{array}$$

By rearranging the above equation, we further obtain

$$\begin{array}{cc}\hfill e_i\left(s\right)=& {\displaystyle \frac{s({(s+b_i)}^2+d_i^2)e_i\left(0\right)+a_i{d}_{i}s}{(s^2+({\delta }_1+{\delta }_2)s-{\delta }_1{\delta }_2)({(s+b_i)}^2+d_i^2)}}\hfill \\ \hfill & +{\displaystyle \frac{c_i({(s+b_i)}^2+d_i^2)}{(s^2+({\delta }_1+{\delta }_2)s-{\delta }_1{\delta }_2)({(s+b_i)}^2+d_i^2)}}.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}{e}_i\left(t\right)=& \underset{s\to 0}{lim}s{e}_i\left(s\right)\hfill \\ \hfill =& \underset{s\to 0}{lim}({\displaystyle \frac{s^2\left(\right({(s+b_i)}^2+d_i^2)e_i\left(0\right)+a_i{d}_i)}{(s^2+({\delta }_1+{\delta }_2)s-{\delta }_1{\delta }_2)({(s+b_i)}^2+d_i^2)}}\hfill \\ \hfill & +{\displaystyle \frac{c_{i}s({(s+b_i)}^2+d_i^2)}{(s^2+({\delta }_1+{\delta }_2)s-{\delta }_1{\delta }_2)({(s+b_i)}^2+d_i^2)}})=0.\hfill \end{array}$$

Therefore, the result is derived as ${lim}_{t\to \infty }{\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2=0.$ That is, the GBVFPS error system (8) with exponential-sine-form MTVP converges to zero. The proof is completed. □

Remark 3. 

According to [34,42,52], the integration of error function is critical to minimize the steady-state error, ensuring that the system eventually reaches the desired state. Specifically, the integral term within a controller progressively sums the error. This cumulative effect prompts the controller to make appropriate adjustments to effectively minimize the steady-state error and ensure a stable response of the system. Generalized from [34,42,52], in the rigorous proofs of Theorems 1–6, integrating the error function can effectively suppress the impact of various perturbations, thus improving the anti-perturbation ability of the proposed GBVFPS control method.

4. Numerical Simulations and Comparisons

In this section, two numerical simulations, including the synchronization of two NN chaotic systems and the synchronization of two FDVIP systems, are provided to verify the effectiveness of the proposed GBVFPS control method. In addition, comparative experiments are conducted to further illustrate the anti-perturbation ability and superiority of the proposed GBVFPS control method.

4.1. Chaotic System Synchronization Example

In this example, the designed Zhang neuro-PID controller is used to synchronize two different three-dimensional NN chaotic systems with six different perturbations (i.e., zero perturbation, bounded constant perturbation, linear-form MTVP, exponential-form MTVP, sine-form MTVP, and exponential-sine-form MTVP).

Consider that the driving system is a three-dimensional NN chaotic system with MTVP, which is described by

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =x_2\left(t\right)+{\xi }_1\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =x_3\left(t\right)+sin\left(5t\right)+{\xi }_2\left(t\right),\hfill \\ \hfill {\dot{x}}_3\left(t\right)& =-p{x}_1\left(t\right)-q{x}_2\left(t\right)-r{x}_3\left(t\right)+x_1^2\left(t\right)+{\xi }_3\left(t\right),\hfill \end{array}\right.$$

(28)

where $x_i\left(t\right)$ is the ith state variable with the index $i=1,2,3$ and ${\xi }_i\left(t\right)$ represents the ith MTVP. Besides, $p=6$, $q=2.92$, and $r=1.2$. When the driving system (28) is not affected by various perturbations (i.e., ${\xi }_i\left(t\right)=0$), the driving system (28) is chaotic, and its chaotic attractor is exhibited in Figure 2a. Besides, Figure 2b depicts the trajectories of state variables $x_i\left(t\right)$.

Figure 2. Simulation results of driving system (28) and response system (29) with ${\xi }_i\left(t\right)={\phi }_i\left(t\right)=0$, $x_i\left(0\right)=$ rand$(0,1)$, and $y_i\left(0\right)=$ rand$(0,1)$, $i=1,2,3$. (a) Chaotic attractor of driving system (28). (b) Trajectories of state variables $x_i\left(t\right)$. (c) Chaotic attractor of response system (29). (d) Trajectories of state variables $y_i\left(t\right)$.

The controlled response system, which is a three-dimensional NN chaotic system with MTVP, is expressed as

$$\left\{\begin{array}{cc}\hfill {\dot{y}}_1\left(t\right)& =-y_2\left(t\right)-y_3\left(t\right)+cos\left(10t\right)+{\phi }_1\left(t\right)+f_{11}(\mathbf{y}\left(t\right),t)u_1\left(t\right),\hfill \\ \hfill {\dot{y}}_2\left(t\right)& =y_1\left(t\right)+\alpha y_2\left(t\right)+sin(cos\left(t\right))+{\phi }_2\left(t\right)+f_{22}(\mathbf{y}\left(t\right),t)u_2\left(t\right),\hfill \\ \hfill {\dot{y}}_3\left(t\right)& =\beta +y_3\left(t\right)(y_1\left(t\right)-\gamma )+{\phi }_3\left(t\right)+f_{33}(\mathbf{y}\left(t\right),t)u_3\left(t\right),\hfill \end{array}\right.$$

(29)

where $y_i\left(t\right)$ is the ith state variable, ${\phi }_i\left(t\right)$ stands for the ith MTVP, $u_i\left(t\right)$ is the ith designed controller, and $f_{ii}(\mathbf{y}\left(t\right),t)$ represents the ith differentiable temporally varying nonlinear function. Let $f_{11}(\mathbf{y}\left(t\right),t)=1/(5+{sin}^2\left(t\right))$, $f_{22}(\mathbf{y}\left(t\right),t)=1/(3+cos\left(y_2\left(t\right)\right)sin\left(t\right))$, $f_{33}(\mathbf{y}\left(t\right),t)=1/(4+sin\left(t\right)sin\left(y_3\left(t\right)\right))$, $\alpha =0.2$, $\beta =0.2$, and $\gamma =5.7$. When the response system (29) is free from MTVP—i.e., ${\phi }_i\left(t\right)=0$—the response system (29) is chaotic, and its chaotic attractor is presented in Figure 2c. Besides, Figure 2d depicts the trajectories of state variables $y_i\left(t\right)$.

In the process of numerical simulation, both NN chaotic systems (28) and (29) start with randomly generated initial values, the parameters ${\delta }_1=10$ and ${\delta }_2=15$. ${\varphi }_i\left(t\right)={\phi }_i\left(t\right)-{\psi }_{ii}(\mathbf{x}\left(t\right),t){\xi }_i\left(t\right)$ is MTVP, where ${\phi }_i\left(t\right)$ and ${\xi }_i\left(t\right)$ represent the ith MTVP of NN chaotic systems (28) and (29), respectively. Furthermore, ${\psi }_{ii}(\mathbf{x}\left(t\right),t)$ stands for the ith bi-variable scaling function factor, which is constructed as

$$\left\{\begin{array}{cc}\hfill & {\psi }_{11}(\mathbf{x}\left(t\right),t)=x_1\left(t\right)+cos\left(t\right)+2,\hfill \\ \hfill & {\psi }_{22}(\mathbf{x}\left(t\right),t)=x_2\left(t\right)-2cos\left(t\right)+3,\hfill \\ \hfill & {\psi }_{33}(\mathbf{x}\left(t\right),t)=x_3\left(t\right)-sin\left(t\right)cos\left(t\right)+2.\hfill \end{array}\right.$$

(30)

Next, numerical experiments are conducted on the following six different perturbations to show the validity and anti-perturbation ability of the proposed GBVFPS control method.

(1)

Consider the zero perturbation, i.e., ${\xi }_i\left(t\right)=0$ and ${\phi }_i\left(t\right)=0$. The corresponding numerical simulation results are shown in Figure 3 and Figure 4.

Figure 3. Simulation results of GBVFPS for NN chaotic systems (28) and (29) with zero perturbation based on Zhang neuro-PID controller. (a) ${\left|\right|}_k\mathbf{e}\left(t\right){\left|\right|}_2$ starting from six different sets of random initial vectors with $k\in \{1,2,\cdots ,6\}$ and ${\delta }_1={\delta }_2=10$. (b) ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ with six different sets of parameter values.

Figure 4. Simulation results of GBVFPS for NN chaotic systems (28) and (29) with zero perturbation based on Zhang neuro-PID controller and DD controller. (a) Control inputs $u_{i_{\mathrm{Z}}}$. (b) Control inputs $u_{i_{\mathrm{D}}}$. (c) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$. (d) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$.

(2)

Consider the bounded constant perturbation, i.e., ${\xi }_1\left(t\right)=0.05$, ${\xi }_2\left(t\right)=0.02$, ${\xi }_3\left(t\right)=0.06$, ${\phi }_1\left(t\right)=0.07$, ${\phi }_2\left(t\right)=0.01$, and ${\phi }_3\left(t\right)=0.02$. The corresponding numerical simulation results are displayed in Figure 5a,b.

Figure 5. Simulation results of GBVFPS for NN chaotic systems (28) and (29) with different perturbations based on Zhang neuro-PID controller and DD controller. (a) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$ with bounded constant perturbations. (b) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$ with bounded constant perturbations. (c) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$ with linear-form MTVP. (d) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$ with linear-form MTVP.

(3)

Consider the linear-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}t+\mathbf{c}$, where $\mathbf{a}$ and $\mathbf{c}$ stand for constant vectors. In this case, the linear-form MTVP is expressed as

$$\left\{\begin{array}{cc}\hfill {\xi }_1\left(t\right)& =0.05t+0.02,\hfill \\ \hfill {\xi }_2\left(t\right)& =0.02t+0.09,\hfill \\ \hfill {\xi }_3\left(t\right)& =0.06t+0.3,\hfill \\ \hfill {\phi }_1\left(t\right)& =0.07t+0.01,\hfill \\ \hfill {\phi }_2\left(t\right)& =0.01t+0.05,\hfill \\ \hfill {\phi }_3\left(t\right)& =0.02t+0.3.\hfill \end{array}\right.$$

(31)

The simulation results are shown in Figure 5c,d.

(4)

Consider the exponential-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}+\mathbf{c}$, where $\mathbf{a}$ and $\mathbf{c}$ are constant vectors and $\mathbf{b}={[exp(-b_{1}t),exp(-b_{2}t),exp(-b_{3}t)]}^{\mathrm{T}}$. In this case, the exponential-form MTVP is considered as

$$\left\{\begin{array}{cc}\hfill {\xi }_1\left(t\right)& =0.05exp(-2t)+0.02,\hfill \\ \hfill {\xi }_2\left(t\right)& =0.02exp(-0.5t)+0.09,\hfill \\ \hfill {\xi }_3\left(t\right)& =0.06exp(-2t)+0.3,\hfill \\ \hfill {\phi }_1\left(t\right)& =0.07exp(-2t)+0.01,\hfill \\ \hfill {\phi }_2\left(t\right)& =0.01exp(-0.5t)+0.05,\hfill \\ \hfill {\phi }_3\left(t\right)& =0.02exp(-2t)+0.3.\hfill \end{array}\right.$$

(32)

The simulation results are displayed in Figure 6a,b.

Figure 6. Simulation results of GBVFPS for NN chaotic systems (28) and (29) with different perturbations based on Zhang neuro-PID controller and DD controller. (a) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$ with exponential-form MTVP. (b) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$ with exponential-form MTVP. (c) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$ with sine-form MTVP. (d) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$ with sine-form MTVP.

(5)

Consider the sine-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{d}+\mathbf{c}$, where $\mathbf{d}={[sin\left(d_{1}t\right),sin\left(d_{2}t\right),sin\left(d_{3}t\right)]}^{\mathrm{T}}$. In this case, the sine-form MTVP is considered as

$$\left\{\begin{array}{cc}\hfill {\xi }_1\left(t\right)& =0.05sin\left(2t\right)+0.02,\hfill \\ \hfill {\xi }_2\left(t\right)& =0.02sin\left(0.5t\right)+0.09,\hfill \\ \hfill {\xi }_3\left(t\right)& =0.06sin\left(2t\right)+0.3,\hfill \\ \hfill {\phi }_1\left(t\right)& =0.07sin\left(2t\right)+0.01,\hfill \\ \hfill {\phi }_2\left(t\right)& =0.01sin\left(0.5t\right)+0.05,\hfill \\ \hfill {\phi }_3\left(t\right)& =0.02sin\left(2t\right)+0.3.\hfill \end{array}\right.$$

(33)

The simulation results are exhibited in Figure 6c,d.

(6)

Consider the exponential-sine-form MTVP $\mathit{\varphi }\left(t\right)=\mathbf{a}\odot \mathbf{b}\odot \mathbf{d}+\mathbf{c}$. In this case, the exponential-sine-form MTVP is considered as

$$\left\{\begin{array}{cc}\hfill {\xi }_1\left(t\right)& =0.05exp(-0.6t)sin\left(2t\right)+0.02,\hfill \\ \hfill {\xi }_2\left(t\right)& =0.02exp(-2t)sin\left(0.5t\right)+0.09,\hfill \\ \hfill {\xi }_3\left(t\right)& =0.06exp(-0.05t)sin\left(2t\right)+0.3,\hfill \\ \hfill {\phi }_1\left(t\right)& =0.07exp(-0.3t)sin\left(2t\right)+0.01,\hfill \\ \hfill {\phi }_2\left(t\right)& =0.01exp(-0.5t)sin\left(0.5t\right)+0.05,\hfill \\ \hfill {\phi }_3\left(t\right)& =0.02exp(-2t)sin\left(2t\right)+0.3.\hfill \end{array}\right.$$

(34)

The simulation results are shown in Figure 7.

Figure 7. Simulation results of GBVFPS for NN chaotic systems (28) and (29) with exponential-sine-form MTVP based on Zhang neuro-PID controller and DD controller. (a) Control inputs $u_{i_{\mathrm{Z}}}$. (b) Control inputs $u_{i_{\mathrm{D}}}$. (c) GBVFPS errors $e_{i_{\mathrm{Z}}}$ and $e_{i_{\mathrm{D}}}$. (d) Residual errors $|e_{i_{\mathrm{Z}}}|$ and $|e_{i_{\mathrm{D}}}|$.

Specifically, with zero perturbation, the trajectories of GBVFPS errors ${\left|\right|}_k\mathbf{e}\left(t\right){\left|\right|}_2$ starting from six different sets of random initial vectors with $k\in \{1,2,\cdots ,6\}$ and ${\delta }_1={\delta }_2=10$ are displayed in Figure 3a. Besides, the trajectories of GBVFPS errors ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ with six different sets of parameter values are shown in Figure 3b. As the parameter values increase, the convergence rate of ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ increases. Figure 3a,b illustrate that the GBVFPS control method is insensitive to initial values and sensitive to parameter variations. In addition, the trajectories of control inputs $u_{i_{\mathrm{Z}}}$ and $u_{i_{\mathrm{D}}}$ with zero perturbation are displayed in Figure 4a,b, respectively. The trajectories of control inputs $u_{i_{\mathrm{Z}}}$ and $u_{i_{\mathrm{D}}}$ with exponential-sine-form MTVP are exhibited in Figure 7a,b, respectively. Evidently, both the Zhang neuro-PID controller and DD controller are feasible during execution.

Furthermore, in the case of zero perturbation, bounded constant perturbation, linear-form MTVP, exponential-form MTVP, sine-form MTVP, and exponential-sine-form MTVP, the GBVFPS errors are exhibited in Figure 4c, Figure 5a,c, Figure 6a,c and Figure 7c, respectively. The corresponding residual errors are illustrated in Figure 4d, Figure 5b,d, Figure 6b,d and Figure 7d, respectively. From the figures, the GBVFPS errors $e_{i_{\mathrm{Z}}}$ generated by the Zhang neuro-PID controller converge to zero quickly, and the residual errors $|e_{i_{\mathrm{Z}}}|$ are mostly between orders ${10}^{-2}$ and ${10}^{-5}$. Especially, with zero perturbation, the residual errors $|e_{i_{\mathrm{Z}}}|$ are mostly about order ${10}^{-5}$. However, the GBVFPS errors $e_{i_{\mathrm{D}}}$ generated by DD controller cannot converge to zero, and residual errors $|e_{i_{\mathrm{D}}}|$ are greater than order 10.

In summary, the DD controller designed by DD method fails to solve the GBVFPS problem of two different NN chaotic systems with various perturbations. In contrast, the Zhang neuro-PID controller effectively and accurately solves the GBVFPS problem of NN chaotic systems (28) and (29) with various perturbations. Thus, the numerical simulation results substantiate the superiority of the designed Zhang neuro-PID controller as well as the effectiveness and anti-perturbation ability of the proposed GBVFPS control method.

4.2. FDVIP System Synchronization Example

To illustrate that the proposed GBVFPS type is an important breakthrough in the field of synchronization research, in this example, four representative synchronization simulations of two different FDVIP systems with exponential-sine-form MTVP are considered. Besides, the proposed Zhang neuro-PID controller and the existing DD controller are applied to tackle this challenging perturbation problem.

The driving system is considered as the following FDVIP system with exponential-sine-form MTVP:

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =x_2\left(t\right)+{\xi }_1\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)& ={\displaystyle \frac{m_2(l_1{x}_4^2\left(t\right)-g_{1}cos\left(x_3\left(t\right)\right))sin\left(x_3\left(t\right)\right)-b_1{x}_2\left(t\right)}{m_1+m_2{sin}^2\left(x_3\left(t\right)\right)}}+0.1sin\left(t\right)+{\xi }_2\left(t\right),\hfill \\ \hfill {\dot{x}}_3\left(t\right)& =x_4\left(t\right)+{\xi }_3\left(t\right),\hfill \\ \hfill {\dot{x}}_4\left(t\right)& =-{\displaystyle \frac{(m_2{l}_1{x}_4^2\left(t\right)sin\left(x_3\left(t\right)\right)b_1{x}_2\left(t\right))cos\left(x_3\left(t\right)\right)+(m_1+m_2)g_{1}sin\left(x_3\left(t\right)\right)}{l_1(m_1+m_2{sin}^2\left(x_3\left(t\right)\right))}}\hfill \\ \hfill & +0.5cos\left(t\right)+{\xi }_4\left(t\right),\hfill \end{array}\right.$$

(35)

where ${\xi }_j\left(t\right)$ denotes the jth exponential-sine-form MTVP with the index $j=1,2,3,4$, and the exponential-sine-form MTVP is designed as

$$\left\{\begin{array}{cc}\hfill {\xi }_1\left(t\right)& =0.005exp(-0.6t)sin\left(2t\right)+0.002,\hfill \\ \hfill {\xi }_2\left(t\right)& =0.002exp(-2t)sin\left(0.5t\right)+0.009,\hfill \\ \hfill {\xi }_3\left(t\right)& =0.006exp(-0.05t)sin\left(2t\right)+0.003,\hfill \\ \hfill {\xi }_4\left(t\right)& =0.008exp(-3t)sin\left(3t\right)+0.007.\hfill \end{array}\right.$$

The response system is considered as the following FDVIP system with exponential-sine-form MTVP:

$$\left\{\begin{array}{cc}\hfill {\dot{y}}_1\left(t\right)& =y_2\left(t\right)+{\phi }_1\left(t\right)+f_{11}(\mathbf{y}\left(t\right),t)u_1\left(t\right),\hfill \\ \hfill {\dot{y}}_2\left(t\right)& ={\displaystyle \frac{m_4(l_2{y}_4^2\left(t\right)-g_{2}cos\left(y_3\left(t\right)\right))sin\left(y_3\left(t\right)\right)-b_2{y}_2\left(t\right)}{m_3+m_4{sin}^2\left(y_3\left(t\right)\right)}}\hfill \\ \hfill & +0.01sin\left(t\right)+{\phi }_2\left(t\right)+f_{22}(\mathbf{y}\left(t\right),t)u_2\left(t\right),\hfill \\ \hfill {\dot{y}}_3\left(t\right)& =y_4\left(t\right)+{\phi }_3\left(t\right)+f_{33}(\mathbf{y}\left(t\right),t)u_3\left(t\right),\hfill \\ \hfill {\dot{y}}_4\left(t\right)& ={\displaystyle \frac{(m_3+m_4)g_{2}sin\left(y_3\left(t\right)\right)-m_4{l}_2{y}_4^2\left(t\right)sin\left(y_3\left(t\right)\right)b_2{y}_2\left(t\right)cos\left(y_3\left(t\right)\right)}{l_2(m_3+m_4{sin}^2\left(y_3\left(t\right)\right))}}\hfill \\ \hfill & +0.2cos\left(0.5t\right)+{\phi }_4\left(t\right)+f_{44}(\mathbf{y}\left(t\right),t)u_4\left(t\right),\hfill \end{array}\right.$$

(36)

where $u_j\left(t\right)$ represents the jth controller and ${\phi }_j\left(t\right)$ represents the jth exponential-sine-form MTVP, which is described as

$$\left\{\begin{array}{cc}\hfill {\phi }_1\left(t\right)& =0.007exp(-0.3t)sin\left(2t\right)+0.001,\hfill \\ \hfill {\phi }_2\left(t\right)& =0.001exp(-0.5t)sin\left(0.5t\right)+0.005,\hfill \\ \hfill {\phi }_3\left(t\right)& =0.002exp(-2t)sin\left(2t\right)+0.003,\hfill \\ \hfill {\phi }_4\left(t\right)& =0.005exp(-0.4t)sin\left(3t\right)+0.002.\hfill \end{array}\right.$$

Let ${\delta }_1=20$, ${\delta }_2=25$, $f_{11}(\mathbf{y}\left(t\right),t)=1$, $f_{22}(\mathbf{y}\left(t\right),t)=1/(m_3+m_4{sin}^2\left(y_3\left(t\right)\right))$, $f_{33}(\mathbf{y}\left(t\right),t)=1$, and $f_{44}(\mathbf{y}\left(t\right),t)=-1/\left(l_2(m_3+m_4{sin}^2\left(y_3\left(t\right)\right))\right)$. For better understanding and presentation, two mathematical models of FDVIP systems (35) and (36) with exponential-sine-form MTVP are presented in Figure 8. In addition, some relevant parameters of the FDVIP systems (35) and (36) are exhibited in Table 4.

Figure 8. Schematic of FDVIP systems with different MTVP. (a) Schematic of FDVIP system (35) without control input. (b) Schematic of FDVIP system (36) with control input $\mathbf{u}\left(t\right)$.

Table 4. Parameter values of FDVIP systems (35) and (36).

FDVIP System (35)		FDVIP System (36)
Parameter	Value	Parameter	Value
$m_1$	0.378 (kg)	$m_3$	1.12 (kg)
$m_2$	0.037 (kg)	$m_4$	0.15 (kg)
$l_1$	0.125 (m)	$l_2$	0.1 (m)
$g_1$	9.81 (N/kg)	$g_2$	9.81 (N/kg)
$b_1$	0.001 (kg/s)	$b_2$	0.44 (kg/s)

With zero perturbation, the trajectories of GBVFPS errors ${\left|\right|}_k\mathbf{e}\left(t\right){\left|\right|}_2$ starting from six different sets of random initial vectors with $k\in \{1,2,\cdots ,6\}$ and ${\delta }_1={\delta }_2=10$ are displayed in Figure 9a. Besides, the trajectories of GBVFPS errors ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ with six different sets of parameter values are shown in Figure 9b. As the parameter values increase, the convergence rate of ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ increases. Figure 9a,b illustrate that the GBVFPS control method is insensitive to initial values and sensitive to the parameter variations.

Figure 9. Simulation results of GBVFPS for FDVIP systems (35) and (36) with zero perturbation based on Zhang neuro-PID controller. (a) ${\left|\right|}_k\mathbf{e}\left(t\right){\left|\right|}_2$ starting from six different sets of random initial vectors with $k\in \{1,2,\cdots ,6\}$ and ${\delta }_1={\delta }_2=10$. (b) ${\left|\right|\mathbf{e}\left(t\right)\left|\right|}_2$ with six different sets of parameter values.

Next, by changing the mathematical expression of the bi-variable scaling function factors ${\psi }_{jj}(\mathbf{x}\left(t\right),t)$, the GBVFPS becomes a different synchronization type. For the sake of simplicity, we only simulate four different situations, namely, MPS, MFPS, GFPS, and GBVFPS.

(1)

Considering the bi-variable scaling function factors being constant factors—that is, ${\psi }_{11}(\mathbf{x}\left(t\right),t)=2$, ${\psi }_{22}(\mathbf{x}\left(t\right),t)=4$, ${\psi }_{33}(\mathbf{x}\left(t\right),t)=3$, and ${\psi }_{44}(\mathbf{x}\left(t\right),t)=1$. In this situation, the GBVFPS turns into MPS and the corresponding simulation results are presented in Figure 10.

Figure 10. Simulation results during MPS of FDVIP systems (35) and (36) with exponential-sine-form MTVP. (a) Control inputs $u_{j_{\mathrm{Z}}}$. (b) Control inputs $u_{j_{\mathrm{D}}}$. (c) MPS errors $e_{j_{\mathrm{Z}}}$ and $e_{j_{\mathrm{D}}}$. (d) Residual errors $|e_{j_{\mathrm{Z}}}|$ and $|e_{j_{\mathrm{D}}}|$.

(2)

Consider the bi-variable scaling function factors being related to t, which are expressed as

$$\left\{\begin{array}{cc}\hfill {\psi }_{11}(\mathbf{x}\left(t\right),t)& =cos\left(t\right)+2,\hfill \\ \hfill {\psi }_{22}(\mathbf{x}\left(t\right),t)& =-2cos\left(t\right)+3,\hfill \\ \hfill {\psi }_{33}(\mathbf{x}\left(t\right),t)& =-sin\left(t\right)cos\left(t\right)+2,\hfill \\ \hfill {\psi }_{44}(\mathbf{x}\left(t\right),t)& =-sin\left(t\right)+4.\hfill \end{array}\right.$$

(37)

In this situation, the GBVFPS becomes MFPS and the corresponding simulation results are given in Figure 11.

Figure 11. Simulation results during MFPS of FDVIP systems (35) and (36) with exponential-sine-form MTVP. (a) Control inputs $u_{j_{\mathrm{Z}}}$. (b) Control inputs $u_{j_{\mathrm{D}}}$. (c) MFPS errors $e_{j_{\mathrm{Z}}}$ and $e_{j_{\mathrm{D}}}$. (d) Residual errors $|e_{j_{\mathrm{Z}}}|$ and $|e_{j_{\mathrm{D}}}|$.

(3)

Consider the bi-variable scaling function factors being related to $\mathbf{x}\left(t\right)$, which are constructed as

$$\left\{\begin{array}{cc}\hfill {\psi }_{11}(\mathbf{x}\left(t\right),t)& =x_1\left(t\right)+2,\hfill \\ \hfill {\psi }_{22}(\mathbf{x}\left(t\right),t)& =x_2\left(t\right)+3,\hfill \\ \hfill {\psi }_{33}(\mathbf{x}\left(t\right),t)& =x_3\left(t\right)+2,\hfill \\ \hfill {\psi }_{44}(\mathbf{x}\left(t\right),t)& =x_4\left(t\right)+4.\hfill \end{array}\right.$$

(38)

In this situation, the GBVFPS is changed into GFPS and the corresponding simulation results are displayed in Figure 12.

Figure 12. Simulation results during GFPS of FDVIP systems (35) and (36) with exponential-sine-form MTVP. (a) Control inputs $u_{j_{\mathrm{Z}}}$. (b) Control inputs $u_{j_{\mathrm{D}}}$. (c) GFPS errors $e_{j_{\mathrm{Z}}}$ and $e_{j_{\mathrm{D}}}$. (d) Residual errors $|e_{j_{\mathrm{Z}}}|$ and $|e_{j_{\mathrm{D}}}|$.

(4)

Considering the bi-variable scaling function factors being related to t and $\mathbf{x}\left(t\right)$, which are formulated as

$$\left\{\begin{array}{cc}\hfill {\psi }_{11}(\mathbf{x}\left(t\right),t)& =x_1\left(t\right)+cos\left(t\right)+2,\hfill \\ \hfill {\psi }_{22}(\mathbf{x}\left(t\right),t)& =x_2\left(t\right)-2cos\left(t\right)+3,\hfill \\ \hfill {\psi }_{33}(\mathbf{x}\left(t\right),t)& =x_3\left(t\right)-sin\left(t\right)cos\left(t\right)+2,\hfill \\ \hfill {\psi }_{44}(\mathbf{x}\left(t\right),t)& =x_4\left(t\right)-sin\left(t\right)+4.\hfill \end{array}\right.$$

(39)

In this situation, the synchronization type becomes GBVFPS. By utilizing the proposed GBVFPS control, the corresponding simulation results are exhibited in Figure 13.

Figure 13. Simulation results during GBVFPS of FDVIP systems (35) and (36) with exponential-sine-form MTVP. (a) Control inputs $u_{j_{\mathrm{Z}}}$. (b) Control inputs $u_{j_{\mathrm{D}}}$. (c) GBVFPS errors $e_{j_{\mathrm{Z}}}$ and $e_{j_{\mathrm{D}}}$. (d) Residual errors $|e_{j_{\mathrm{Z}}}|$ and $|e_{j_{\mathrm{D}}}|$.

Specifically, the trajectories of control inputs $u_{j_{\mathrm{Z}}}$ generated by Zhang neuro-PID controller are displayed in Figure 10a, Figure 11a, Figure 12a and Figure 13a. In addition, for comparison, the trajectories of control inputs $u_{j_{\mathrm{D}}}$ generated by DD method are illustrated in Figure 10b, Figure 11b, Figure 12b and Figure 13b. Evidently, both the Zhang neuro-PID controller and DD controller are feasible during execution.

Furthermore, the trajectories of the MPS errors, MFPS errors, GFPS errors, and GBVFPS errors are illustrated in Figure 10c, Figure 11c, Figure 12c, and Figure 13c, respectively. Besides, the trajectories of the corresponding residual errors are displayed in Figure 10d, Figure 11d, Figure 12d and Figure 13d, respectively. From the figures, we see that the MPS errors, MFPS errors, GFPS errors, and GBVFPS errors $e_{j_{\mathrm{Z}}}$ generated by the Zhang neuro-PID controller converge to zero quickly, and the corresponding residual errors $|e_{j_{\mathrm{Z}}}|$ are mostly about order ${10}^{-2}$. However, the MPS errors, MFPS errors, GFPS errors, and GBVFPS errors $e_{j_{\mathrm{D}}}$ generated by DD controller no longer converge to zero, and the corresponding residual errors $|e_{j_{\mathrm{D}}}|$ are larger than order 1.

In summary, the designed Zhang neuro-PID controller is superior to the DD controller and has the ability to suppress various perturbations. Besides, from the simulation results, the proposed GBVFPS type can be transformed into various types of synchronization, such as MPS, MFPS, and GFPS, to meet the requirements of different synchronization types for driving system and response system in different situations.

5. Conclusions

In this paper, a new GBVFPS type for NN systems synchronization was proposed. The scaling function matrix of GBVFPS is bi-variable, which is related to time t and the state variable $\mathbf{x}\left(t\right)$ of the driving system. So, the GBVFPS is an improvement and promotion of existing synchronization types, such as PS, MPS, FPS, MFPS, and GFPS. This indicates that the proposed GBVFPS has a significant driving and breakthrough effect on synchronization research. In addition, the NN systems with six different perturbations were investigated and presented, including zero perturbation, bounded constant perturbation, linear-form MTVP, exponential-form MTVP, sine-form MTVP, and exponential-sine-form MTVP. To achieve GBVFPS for two different NN systems with six different perturbations, the Zhang neuro-PID controller was designed. The convergence properties and anti-perturbation ability of the proposed GBVFPS control method were proved via rigorous theoretical analyses. Moreover, numerical simulations on NN chaotic systems and FDVIP systems were performed to further verify the effectiveness, superiority, and anti-perturbation ability of the proposed GBVFPS control method. Finally, some meaningful and promising future research directions are given as follows.

(1)

Designing a different synchronization control method that does not require matrix inversion operation will significantly improve the real-time processing capability.

(2)

Conducting more systematic synchronization research and experimental validation in ships, helicopters, robotics, and so on.

(3)

Exploring control methods for NN systems with unknown parameters.

(4)

Conducting discrete-time synchronization research for better application to computer control.