Observer-based event-triggered adaptive neural control for time-delay nonlinear systems with input saturation and external disturbances

Abstract

In this article, the issue of adaptive event-triggered tracking control is investigated for time-delay nonlinear systems with input saturation and external disturbances. In the whole process of control design, the radial basis function neural networks are utilized to approximate uncertain nonlinearities. To estimate the unknown states, a neural network-based observer is constructed. Pade approximation method is adopted to eliminate the effect of input delay. A smooth non-affine function is introduced to replace input saturation, and an auxiliary variable is employed to obtain the actual control input. An event-triggered strategy is designed to reduce the utilization of communication and computation resources. Moreover, the command filtering technique is applied to handle the issue of “explosion of complexity” in the conventional backstepping method. The designed controller can assure that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Therefore, the proposed event-triggered neural control scheme can not only save network resources, but also improve the robustness of the system by dealing with input constraints and external disturbances. Finally, two simulation examples are given to verify the availability and feasibility of the designed controller.

Keywords

Input delay command filtering technique state observer event-triggered control input saturation

Introduction

In the past few decades, since nonlinear terms inevitably appeared in practical engineering field, how to solve this problem has attracted extensive attention. In this case, fuzzy logic systems (FLS) (Li et al., 2014; Ling et al., 2019; Liu et al., 2016a; Ma and Ma, 2020) and neural networks (NNs) (Bai et al., 2019; Doudou and Khaber, 2021; Li et al., 2019; Zerari et al., 2018) were presented to approximate uncertain parameters or unknown functions. In practice, limited by the complexity of the system itself and some external conditions, the state variables of the system are often immeasurable or partly measurable only. To solve this problem, the observers are usually combined with the approximators. Therefore, adaptive fuzzy observers (Fu et al., 2018; Zhang et al., 2020) and neural network observers (Chen et al., 2017; Tong et al., 2020) were constructed.

The backstepping method is one of the most powerful tools to solve the numerous problems in the field of nonlinear control (Ba et al., 2019; Liu et al., 2016b; Sun and Guo, 2014; Tong and Li, 2011). In Ba et al. (2019), with the aid of neural networks and backstepping method, a fixed-time adaptive neural control scheme was presented for a class of uncertain nonstrict nonlinear systems. In Tong and Li (2011), an adaptive fuzzy backstepping control approach was considered to handle the problem of unknown dead zones and immeasurable states. Unfortunately, in conventional backstepping method, the multiple differentials of virtual signals will cause the issue of “explosion of complexity” in each step of the controller design process. In Liu et al. (2019), Yangand D and Yue (2018), Shi et al. (2019), the dynamic surface control (DSC) technique was developed to reduce the expansion of the differential terms by introducing the first-order filters in backstepping framework. However, the DSC technology does not address the errors caused by the filters, which may degrade the performance of the system. In Yu et al. (2018), Li (2019), Wang et al. (2021), and Hou and Tong (2017), compared with the DSC technology, the command filtering technology not only employed the command filters to solve the problem of “explosion of complexity,” but also introduced compensation signals to deal with the errors generated by the filters. In this paper, we will further investigate the problem of input constraints in nonlinear systems.

It is a common phenomenon that input constraints exist in many nonlinear systems. Input delay and saturation are two forms of input constraints, which will affect the quality of control or even make the system unstable. In Li et al. (2018) and Han et al. (2021), the Pade approximation method was used to deal with input delay. The advantage of this method is that the input delay can be eliminated by introducing an intermediate variable and constructing a feasible coordinate transformation. In Wang et al. (2020), a novel auxiliary system was constructed to handle the adverse influence caused by input delay. In Chen et al. (2016) and Wang et al. (2019), an anti-saturation function was constructed to compensate for the magnitude and rate of saturation. However, the usage of network resources has not been considered in the above-mentioned references, which may increase the burden of communication bandwidth.

In many practical applications, time-triggered control usually leads to a waste of network resources in Zhang et al. (2021) and Chen et al. (2014). Thus, how to reduce the communication frequency between the controller and actuator has received more and more attention. Event-triggered strategy was proposed as an important method to solve the aforementioned problem in Yu and Li (2021), Xia et al. (2021), Gao et al. (2020), Ma et al. (2019), Liu et al. (2021), and Zhu et al. (2020). Different from conventional time-triggered control, the event-triggered strategy considers the behavior of the system rather than depending on real-time solely, so it has more advantages in saving communication and computation resources. In Yu and Li (2021), with the aid of the DSC technique and simplified barrier Lyapunov function, an adaptive fuzzy event-triggered control (ETC) strategy was constructed to deal with tracking error constrained and unknown dead-zone in a class of non-strict feedback systems. In Xia et al. (2021), an observer-based event-triggered adaptive fuzzy control mechanism was designed for stochastic nonlinear systems to handle the problem of unknown control directions and the unmeasured states.

In the view of issues discussed above, the input constraints and the waste of network resources often occur in practical engineering systems. Therefore, how to handle these adverse factors to maintain the stability of the system is the main purpose of this paper. The motivation of this paper is summarized as follows:

Due to the existence of input constraints, the actual controller cannot be obtained at the current time in the process of the controller design. In Chen et al. (2016) and Wang et al. (2020), an auxiliary system and an anti-saturation function were used to overcome the difficulties of input delay and saturation, respectively. However, to the authors’ best knowledge, the problem of dealing with input delay and saturation simultaneously still needs to be further investigated.

In conventional time-triggered scheme, the transmissions of feedback signals and the updates of control signals are real-time, so it often leads to a waste of communication and computation resources. Therefore, how to incorporate event-triggered mechanism into controller to save network resources is a complex work.

Motivated by the above investigations, this paper is committed to presenting a novel state observer-based event-triggered trajectory tracking controller for time-delay nonlinear systems with input saturation and external disturbances. The highlighted contributions of this article are presented as follows:

In this paper, compared with Li et al. (2019) and Liu et al. (2016b) without considering input constraints, a new coordinate transformation is introduced to solve the difficulties caused by input delay and saturation simultaneously. In addition, with the help of Lyapunov stability theorem, the designed controller can ensure the boundedness of all the signals in the closed-loop system.

Compared with DSC technique in Ling et al. (2019) and Liu et al. (2019), the designed controller based on the command filtering technique not only overcomes the drawbacks of “explosion of complexity” in conventional backstepping method, but also effectively compensates for the errors caused by the filters.

In this paper, compared with Chen et al. (2014) and Zhang et al. (2021) by using time-triggered method to handle with relevant problems, an event-triggered relative threshold mechanism is incorporated into controller to alleviate the burden of network resources greatly. Moreover, a neural network-based observer is constructed to estimate the unknown states, which is more general for practical systems.

This article is organized as follows. In the second section, the problem statement and preliminaries are given. In the third section, control scheme design and stability analysis are presented. In the fourth section, two simulation examples demonstrate effectiveness and superiority of the proposed control strategy. The conclusion of this paper is presented in the fifth section.

Problem statement and preliminaries

Model construction

Consider a class of strict-feedback time-delay nonlinear systems with input saturation and external disturbances as

{\begin{matrix} {\overset{\cdot}{x}}_{j} = x_{j + 1} + f_{j} ({\bar{x}}_{j}) + d_{j} (t), j = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} = u (υ (t - τ)) + f_{n} (x) + d_{n} (t) \\ y = x_{1} \end{matrix}

(1)

where ${\bar{x}}_{j} = [x_{1}, x_{2}, \dots, x_{j}]^{T} \in R^{j}$ are the system state vectors and $x = {\bar{x}}_{n}$ . $y \in R$ is the system output. $f_{j} (\cdot)$ , $(j = 1, 2, \dots, n)$ are unknown smooth functions. $d_{j} (t)$ , $(j = 1, 2, \dots, n)$ are external disturbances. $u (\cdot)$ denotes the control signal with input saturation and $τ$ is a known non-negative delay. $υ (\cdot)$ is the controller input, which is to be designed.

Remark 1. There are many practical systems that can be expressed as the systems (1), such as robot arm system (Chen and Ge, 2015), pendulum system (Gao et al., 2016), and so on. Input constraints and external disturbances always occur in practical engineering, which greatly degrade the performance of the system. Therefore, they deserve our study.

Assumption 1. (Bai et al., 2019). $\exists D_{j} > 0, (j = 1, 2, \dots, n)$ , the unknown continuous disturbances $d_{j} (t)$ can be bounded by $| d_{j} (t) | \leq D_{j}$ .

Lemma 1. ( Gao et al., 2020 ). The hyperbolic function satisfies the following condition

0 \leq | ϑ | - ϑ \tanh (\frac{ϑ}{c}) \leq 0.2785 c

(2)

where $c > 0$ and $ϑ \in R$ .

Lemma 2. (Zhang et al., 2020). The Young’s inequality is described as

ab \leq \frac{ϕ^{x}}{x} | a |^{x} + \frac{1}{y ϕ^{y}} | b |^{y}, \forall (a, b) \in R^{2}

(3)

where $ϕ > 0$ , $x > 1$ , $y > 1$ , and $(x - 1) (y - 1) = 1$ .

Lemma 3. (Sun et al., 2018). There exist a $C^{2}$ function $V$ : $R^{n} \to R_{+}$ , two constants $C > 0$ and $D > 0$ , two functions $\underline{Δ}$ and $\bar{Δ}$ , if

{\begin{matrix} \underline{Δ} (| x |) \leq V (x) \leq \bar{Δ} (| x |) \\ \overset{\cdot}{V} (x) \leq - CV (x) + D \end{matrix}

(4)

$\forall x \in R^{n}$ and $t \geq 0$ , one has

V (t) \leq V (0) e^{- Ct} + \frac{D}{C}

(5)

In addition, if inequality (5) is satisfied, the signals of the system (1) are semi-globally uniformly ultimately bounded (SGUUB).

Assumption 2. (Chen et al., 2017). For the nonlinear functions $f_{j} (Y), (j = 1, 2, \dots, n)$ , there exist known positive constants $ι_{j}$ such that

| f_{j} (Y_{1}) - f_{j} (Y_{2}) | \leq ι_{j} | Y_{1} - Y_{2} |, \forall Y_{1}, Y_{2} \in R^{l}

(6)

Radial basis function neural network (RBFNN)

For any smooth function $f (Z)$ defined on a compact set $Ω_{Z}$ , there exists a RBFNN $ω^{* T} S (Z)$ that can be employed to approximate $f (Z)$ , which can be described as

f (Z) = ω^{* T} S (Z) + δ (Z)

(7)

where $Z \in Ω_{Z} \subset R^{n}$ stands for the input vector. $δ (Z)$ denotes the approximation error that satisfies the inequality $| δ (Z) | \leq ε$ .

The basis function vector $S (Z)$ has the following form

S_{j} (Z) = \exp [- \frac{{(Z - θ_{j})}^{T} (Z - θ_{j})}{{η_{j}}^{2}}], 1 \leq j \leq N

(8)

where $θ_{j} = [θ_{j 1}, θ_{j 2}, \dots, θ_{jn}]^{T}$ and $η_{j}$ denote the center and the width of the Gaussian function, respectively. $N > 1$ is the NN node number. $ω^{*} = [{ω_{1}}^{*}, {ω_{2}}^{*}, \dots, {ω_{N}}^{*}]$ is the weight vector, which can be expressed as

ω^{*} : = \arg min_{ω \in R^{N}} {sup_{Z \in Ω_{z}} | f (Z) - ω^{T} S (Z) |}

(9)

Input delay and saturation

We introduce Pade approximation approach to tackle the problem of input delay. One has

\begin{matrix} ℓ {u (υ (t - τ))} = e^{- τ s} ℓ {u (υ (t))} = \frac{e^{- τ s / 2}}{e^{τ s / 2}} ℓ {u (υ (t))} \\ \approx \frac{1 - τ s / 2}{1 + τ s / 2} ℓ {u (υ (t))} \end{matrix}

(10)

where $s$ stands for the Laplace variable, $ℓ (\cdot)$ represents the Laplace operator.

Then, the intermediate variable $x_{n + 1}$ holds the following equation

\frac{1 - τ s / 2}{1 + τ s / 2} ℓ {u (υ (t))} = ℓ {x_{n + 1}} - ℓ {u (υ (t))}

(11)

Based on the inverse Laplace transformation, (11) can be rewritten as

{\overset{\cdot}{x}}_{n + 1} = 2 λ u (υ (t)) - λ x_{n + 1}

(12)

where $λ = \frac{2}{τ}$

Thus, the system (1) can be transformed as follows

{\begin{matrix} {\overset{\cdot}{x}}_{j} = x_{j + 1} + f_{j} ({\bar{x}}_{j}) + d_{j} (t), j = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} = x_{n + 1} + f_{n} (x) - u (υ (t)) + d_{n} (t) \\ {\overset{\cdot}{x}}_{n + 1} = 2 λ u (υ (t)) - λ x_{n + 1} \\ y = x_{1} . \end{matrix}

(13)

Taking the saturation restriction into consideration, the system input $u (υ (t))$ can be described as

u (υ (t)) = sat (υ (t)) = {\begin{matrix} sign (υ (t)) u_{N}, | υ (t) | \geq u_{N} \\ υ (t), | υ (t) | < u_{N} \end{matrix}

(14)

where $u_{N}$ is the bound of $sat (υ (t))$ .

From Zerari et al. (2018), it is shown that there exists a sharp corner when $| υ (t) | \geq u_{N}$ . Therefore, a smooth non-affine function is utilized to approximate the saturation function as

g (υ (t)) = u_{N} \times \tanh (\frac{υ (t)}{u_{N}}) = u_{N} \frac{e^{υ (t) / u_{N}} - e^{- υ (t) / u_{N}}}{e^{υ (t) / u_{N}} + e^{- υ (t) / u_{N}}}

(15)

Then, $sat (υ (t))$ in (14) can be expressed as

sat (υ (t)) = g (υ (t)) + μ (υ) = u_{N} \times \tanh (\frac{υ (t)}{u_{N}}) + μ (υ)

(16)

where $μ (υ) = sat (υ (t)) - g (υ (t))$ and it holds the following condition

| μ (υ) | = | sat (υ (t)) - g (υ (t)) | \leq u_{N} (1 - \tanh (1))

(17)

The auxiliary variable $ϵ$ is introduced to obtain the actual control input $υ$ , which is defined as

\overset{\cdot}{ϵ} = - ϵ + (g (υ) - υ)

(18)

Finally, the nonlinear system (13) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{j} = x_{j + 1} + f_{j} ({\bar{x}}_{j}) + d_{j} (t), j = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} = x_{n + 1} + f_{n} (x) - g (υ) - μ + d_{n} (t) \\ {\overset{\cdot}{x}}_{n + 1} = - λ x_{n + 1} + 2 λ g (υ) + 2 λ μ \\ y = x_{1} . \end{matrix}

(19)

Remark 2. Compared with Zhang et al. (2021) and Li et al. (2014), this paper not merely handles the problems caused by input constraints of the system, but also constructs state observer and adopts event-triggered mechanism to estimate the unknown states and lighten the burden of communication bandwidth. Therefore, the control strategy proposed in this paper can be widely used in engineering practice.

Remark 3. The Pade approximation method is adopted to handle the adverse influence caused by input delay in this paper. However, this method has certain limitations in dealing with time-varying delays. Therefore, how to solve the problem of time-varying delays is worthy of our further investigation.

Control scheme design and stability analysis

In this section, an observer-based event-triggered adaptive neural control scheme for nonlinear systems with input constraints and external disturbances is presented. To understand the proposed control scheme, a block diagram of event-triggered adaptive neural control scheme is illustrated in Figure 1.

Neural observer design

In this section, a neural observer will be designed to estimate the unknown state of system (1). By using the formula (7), the unknown continuous nonlinear function $f_{j} ({\hat{\bar{x}}}_{j})$ can be represented. Therefore, system (1) can be written as

{\begin{matrix} {\overset{\cdot}{x}}_{j} = x_{j + 1} + {ω_{j}}^{* T} S_{j} ({\hat{\bar{x}}}_{j}) + δ_{j} ({\hat{\bar{x}}}_{j}) + Δ f_{j} + d_{j} (t), j = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{x}}_{n} = u (υ (t - τ)) + {ω_{n}}^{* T} S_{n} ({\hat{x}}_{n}) + δ_{n} ({\hat{x}}_{n}) + Δ f_{n} + d_{n} (t) \\ y = x_{1} \end{matrix}

(20)

where $Δ f_{j} = f_{j} ({\bar{x}}_{j}) - f_{j} ({\hat{\bar{x}}}_{j})$ and ${\hat{\bar{x}}}_{j}$ are the estimates of ${\bar{x}}_{j}$ .

Then, a neural observer is constructed as follows

{\begin{matrix} {\overset{\cdot}{\hat{x}}}_{j} = {\hat{x}}_{j + 1} + {\hat{ω}}_{j}^{T} S_{j} ({\hat{\bar{x}}}_{j}) + ς_{j} (y - {\hat{x}}_{1}), j = 1, 2, \dots, n - 1 \\ {\overset{\cdot}{\hat{x}}}_{n} = u (υ (t - τ)) + {\hat{ω}}_{n}^{T} S_{n} ({\hat{x}}_{n}) + ς_{n} (y - {\hat{x}}_{1}) \end{matrix}

(21)

where $ς_{j}, (j = 1, 2, \dots, n)$ are the parameters of the controller design.

Based on (20) and (21), the observer error equation can be obtained as follows

\overset{\cdot}{π} = A π + ψ + Δ f + δ + d

(22)

where $π = x - \hat{x}$ , ${\tilde{ω}}_{j} = ω_{j} - {\hat{ω}}_{j}$ , $ψ = [\begin{matrix} {\tilde{ω}}_{1}^{T} S_{1} (x_{1}) \\ {\tilde{ω}}_{2}^{T} S_{2} (x_{2}) \\ ⋮ \\ {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) \end{matrix}]$ , $Δ f = [\begin{matrix} Δ f_{1} \\ Δ f_{2} \\ ⋮ \\ Δ f_{n} \end{matrix}],$ and $A = [\begin{matrix} - ς_{1} \\ - ς_{2} & I \\ ⋮ \\ - ς_{n} & 0 & \dots & 0 \end{matrix}] .$

To evaluate the performance of the state observer (21), the following Lyapunov function is chosen as

V_{0} = π^{T} P π .

(23)

The derivation of $V_{0}$ is

{\overset{\cdot}{V}}_{0} = π^{T} (A^{T} P + PA) π + 2 π^{T} P (ψ + Δ f + δ + d)

(24)

With the aid of Lemma 2 and Assumption 2, we can get the following inequalities

\begin{matrix} 2 π^{T} P Δ f \leq | | π | |^{2} + | | P | |^{2} | | Δ f | |^{2} \\ \leq | | π | |^{2} + | | P | |^{2} \sum_{j = 1}^{n} ι_{j}^{2} | | π | |^{2} = (ι + 1) | | π | |^{2} \end{matrix}

(25)

2 π^{T} P ψ \leq | | π | |^{2} + | | P | |^{2} \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j}

(26)

2 π^{T} P (δ + d) \leq 2 | | π | |^{2} + | | P | |^{2} (\sum_{j = 1}^{n} {ε_{j}}^{2} + \sum_{j = 1}^{n} {D_{j}}^{2})

(27)

From (24) to (27), we have

{\overset{\cdot}{V}}_{0} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} + χ_{0} + | | P | |^{2} \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j}

(28)

where $ι = | | P | |^{2} \sum_{j = 1}^{n} {ι_{j}}^{2}$ , $χ_{0} = | | P | |^{2} (\sum_{j = 1}^{n} {ε_{j}}^{2} + \sum_{j = 1}^{n} {D_{j}}^{2})$ .

Remark 4. The coefficient vector $ς$ is chosen, such that A is a strict Hurwitz matrix. Then, for any given positive definite matrix $Q = Q^{T} > 0$ , the equation $A^{T} P + PA = - Q$ can be obtained.

Controller design

At first, a new coordinate transformation is given

{\begin{matrix} z_{1} = {\hat{x}}_{1} - y_{r} \\ z_{j} = {\hat{x}}_{j} - β_{j}, j = 2, \dots, n - 1 \\ z_{n} = {\hat{x}}_{n} - β_{n} + \frac{x_{n + 1}}{λ} - ϵ \end{matrix}

(29)

where $β_{j}$ are the outputs of the first-order command filter, which satisfy the following condition

a_{j} {\overset{\cdot}{β}}_{j} + β_{j} = α_{j}, β_{j} (0) = α_{j} (0), j = 2, \dots, n

(30)

where $α_{j}$ are the visual controllers to be designed and the inputs of the above filters, and $a_{j} > 0$ are the design parameters.

In order to eliminate the filter errors, we introduce the following auxiliary system

{\begin{matrix} {\overset{\cdot}{ζ}}_{1} = - k_{1} ζ_{1} + (β_{2} - α_{2}) + ζ_{2} \\ {\overset{\cdot}{ζ}}_{j} = - k_{j} ζ_{j} + (β_{j + 1} - α_{j + 1}) - ζ_{j - 1} + ζ_{j + 1}, j = 2, \dots, n - 1 \\ {\overset{\cdot}{ζ}}_{n} = - k_{n} ζ_{n} - ζ_{n - 1} \end{matrix}

(31)

where $k_{j} > 0$ are design parameters.

The compensated error signals are defined as

e_{j} = z_{j} - ζ_{j}, j = 1, 2, \dots, n .

(32)

Step 1: Differentiating $e_{1}$ produces

\begin{matrix} {\overset{\cdot}{e}}_{1} = {\overset{\cdot}{z}}_{1} - {\overset{\cdot}{ζ}}_{1} \\ = {\hat{x}}_{2} + {\hat{ω}}_{1}^{T} S_{1} (x_{1}) + ς_{1} π_{1} + k_{1} ζ_{1} - (β_{2} - α_{2}) - ζ_{2} - {\overset{\cdot}{y}}_{r} \\ = {\hat{ω}}_{1}^{T} S_{1} (x_{1}) + ς_{1} π_{1} + e_{2} + k_{1} ζ_{1} + α_{2} - {\overset{\cdot}{y}}_{r} \end{matrix}

(33)

Select the Lyapunov function as follows

V_{1} = V_{0} + \frac{1}{2} {e_{1}}^{2} + \frac{1}{2 r_{1}} {\tilde{ω}}_{1}^{T} {\tilde{ω}}_{1}

(34)

where the design parameter $r_{1} > 0$ .

Based on (28) and (33), the derivation of $V_{1}$ is

\begin{matrix} {\overset{\cdot}{V}}_{1} = {\overset{\cdot}{V}}_{0} + e_{1} {\overset{\cdot}{e}}_{1} - \frac{1}{r_{1}} {\tilde{ω}}_{1}^{T} {\overset{\cdot}{\hat{ω}}}_{1} \\ = {\overset{\cdot}{V}}_{0} + e_{1} [{\hat{ω}}_{1}^{T} S_{1} (x_{1}) - {\tilde{ω}}_{1}^{T} S_{1} (x_{1}) + ς_{1} π_{1} + e_{2} \\ + k_{1} ζ_{1} + α_{2} - {\overset{\cdot}{y}}_{r}] \\ - \frac{1}{r_{1}} {\tilde{ω}}_{1}^{T} [{\overset{\cdot}{\hat{ω}}}_{1} - r_{1} e_{1} S_{1} (x_{1})] \end{matrix}

(35)

By using Young’s inequality, one has

- e_{1} {\tilde{ω}}_{1}^{T} S_{1} (x_{1}) \leq \frac{1}{2} {e_{1}}^{2} + \frac{1}{2} {\tilde{ω}}_{1}^{T} {\tilde{ω}}_{1}

(36)

Then, the virtual controller $α_{2}$ and the adaptation law ${\overset{\cdot}{\hat{ω}}}_{1}$ are given as

α_{2} = - k_{1} z_{1} - {\hat{ω}}_{1}^{T} S_{1} (x_{1}) + {\overset{\cdot}{y}}_{r} - \frac{1}{2} e_{1} - ς_{1} π_{1}

(37)

{\overset{\cdot}{\hat{ω}}}_{1} = r_{1} [e_{1} S_{1} (x_{1}) - σ_{1} {\hat{ω}}_{1}]

(38)

From (35) to (38), one yields

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} - k_{1} {e_{1}}^{2} + e_{1} e_{2} + | | P | |^{2} \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} \\ + \frac{1}{2} {\tilde{ω}}_{1}^{T} {\tilde{ω}}_{1} + χ_{0} + σ_{1} {\tilde{ω}}_{1}^{T} {\hat{ω}}_{1} \end{matrix}

(39)

Step j $(j = 2, \dots, n - 1)$ : Taking the derivation of $e_{j}$ , one has

\begin{matrix} {\overset{\cdot}{e}}_{j} = {\overset{\cdot}{z}}_{j} - {\overset{\cdot}{ζ}}_{j} \\ = {\hat{x}}_{j + 1} + {\hat{ω}}_{j}^{T} S_{j} (x_{j}) + ς_{j} π_{1} - {\overset{\cdot}{β}}_{j} + k_{j} ζ_{j} - (β_{j + 1} - α_{j + 1}) \\ + ζ_{j - 1} - ζ_{j + 1} \\ = e_{j + 1} + {\hat{ω}}_{j}^{T} S_{j} (x_{j}) + ς_{j} π_{1} - {\overset{\cdot}{β}}_{j} + k_{j} ζ_{j} + α_{j + 1} + ζ_{j - 1} \end{matrix}

(40)

Select the Lyapunov function as follows

V_{j} = V_{j - 1} + \frac{1}{2} {e_{j}}^{2} + \frac{1}{2 r_{j}} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j}

(41)

where the design parameters $r_{j} > 0$ .

Based on (40), the derivation of $V_{j}$ is

\begin{matrix} {\overset{\cdot}{V}}_{j} = {\overset{\cdot}{V}}_{j - 1} + e_{j} {\overset{\cdot}{e}}_{j} - \frac{1}{r_{j}} {\tilde{ω}}_{j}^{T} {\overset{\cdot}{\hat{ω}}}_{j} \\ = {\overset{\cdot}{V}}_{j - 1} + e_{j} [e_{j + 1} + {\hat{ω}}_{j}^{T} S (x_{j}) - {\tilde{ω}}_{j}^{T} S_{j} (x_{j}) + ς_{j} π_{1} \\ - {\overset{\cdot}{β}}_{j} + k_{j} ζ_{j} + α_{j + 1} + ζ_{j - 1}] \\ - \frac{1}{r_{j}} {\tilde{ω}}_{j}^{T} [{\overset{\cdot}{\hat{ω}}}_{j} - r_{j} e_{j} S_{j} (x_{j})] \end{matrix}

(42)

By using Young’s inequality, one has

- e_{j} {\tilde{ω}}_{j}^{T} S_{j} (x_{j}) \leq \frac{1}{2} {e_{j}}^{2} + \frac{1}{2} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j}

(43)

Then, the virtual controller $α_{j + 1}$ and the adaptation law ${\overset{\cdot}{\hat{ω}}}_{j}$ are given as

α_{j + 1} = - k_{j} z_{j} - z_{j - 1} - {\hat{ω}}_{j}^{T} S_{j} (x_{j}) + {\overset{\cdot}{β}}_{j} - \frac{1}{2} e_{j} - ς_{j} π_{1}

(44)

{\overset{\cdot}{\hat{ω}}}_{j} = r_{j} [e_{j} S_{j} (x_{j}) - σ_{j} {\hat{ω}}_{j}]

(45)

From (42) to (45), one yields

\begin{matrix} {\overset{\cdot}{V}}_{j} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} - \sum_{i = 1}^{j} k_{i} {e_{i}}^{2} + e_{j} e_{j + 1} \\ + | | P | |^{2} \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} \\ + \frac{1}{2} \sum_{i = 1}^{j} {\tilde{ω}}_{i}^{T} {\tilde{ω}}_{i} + χ_{0} + \sum_{i = 1}^{j} σ_{i} {\tilde{ω}}_{i}^{T} {\hat{ω}}_{i} \end{matrix}

(46)

Step n: Considering the network resources constraint, an event-triggered relative threshold strategy is designed to mitigate the unnecessary waste of communication resources. The ETC conditions are defined as

{\begin{matrix} υ (t) = ϖ (t), \forall t \in [t_{m}, t_{m + 1}] \\ t_{m + 1} = inf {t \in R | | κ (t) | \geq ϱ | υ (t) | + Θ} \end{matrix}

(47)

where $ϖ (t)$ is transition control signal, $t_{m}$ is the controller update time, and $κ (t) = ϖ (t) - υ (t)$ represents the measurement error. $0 < ϱ < 1$ , $Θ > 0$ , $ℏ > \frac{Θ}{1 - ϱ}$ are design parameters.

$ϖ (t)$ satisfies the following condition

ϖ (t) = - (1 + ϱ) [α_{n + 1} \tanh (\frac{e_{n} α_{n + 1}}{γ}) + ℏ \tanh (\frac{e_{n} ℏ}{γ})]

(48)

Differentiating $e_{n}$ produces

\begin{matrix} {\overset{\cdot}{e}}_{n} = {\overset{\cdot}{z}}_{n} - {\overset{\cdot}{ζ}}_{n} \\ = x_{n + 1} - g (υ) - μ + {\hat{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} - {\overset{\cdot}{β}}_{n} \\ + \frac{1}{λ} [- λ x_{n + 1} + 2 λ g (υ) + 2 λ μ] \\ + ϵ - (g (υ) - υ) + k_{n} ζ_{n} + ζ_{n - 1} \\ = {\hat{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} + υ + μ + ϵ - {\overset{\cdot}{β}}_{n} + k_{n} ζ_{n} + ζ_{n - 1} \end{matrix}

(49)

Select the Lyapunov function as follows

V_{n} = V_{n - 1} + \frac{1}{2} {e_{n}}^{2} + \frac{1}{2 r_{n}} {\tilde{ω}}_{n}^{T} {\tilde{ω}}_{n}

(50)

where the design parameter $r_{n} > 0$ .

Based on (50), the derivation of $V_{n}$ is

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + e_{n} {\overset{\cdot}{e}}_{n} - \frac{1}{r_{n}} {\tilde{ω}}_{n}^{T} {\overset{\cdot}{\hat{ω}}}_{n} \\ = {\overset{\cdot}{V}}_{n - 1} + e_{n} [{\hat{ω}}_{n}^{T} S_{n} (x_{n}) - {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} + υ \\ + μ + ϵ - {\overset{\cdot}{β}}_{n} + k_{n} ζ_{n} + ζ_{n - 1}] \\ - \frac{1}{r_{n}} {\tilde{ω}}_{n}^{T} [{\overset{\cdot}{\hat{ω}}}_{n} - r_{n} e_{n} S_{n} (x_{n})] \end{matrix}

(51)

From (47), we can obtain $ϖ (t) = (1 + Ξ_{1} ϱ) υ (t) + Ξ_{2} Θ, \forall t \in [t_{m}, t_{m + 1}]$ , where $Ξ_{1} (t) \in [- 1, 1], Ξ_{2} (t) \in [- 1, 1]$ . Then, it is not difficult to come up with the following equation

υ (t) = \frac{ϖ (t)}{1 + Ξ_{1} ϱ} - \frac{Ξ_{2} Θ}{1 + Ξ_{1} ϱ}

(52)

Substituting (52) to (51), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + \frac{e_{n} ϖ (t)}{1 + Ξ_{1} ϱ} - \frac{e_{n} Ξ_{2} Θ}{1 + Ξ_{1} ϱ} + e_{n} \\ [{\hat{ω}}_{n}^{T} S_{n} (x_{n}) - {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} + μ + ϵ - {\overset{\cdot}{β}}_{n} \\ + k_{n} ζ_{n} + ζ_{n - 1}] - \frac{1}{r_{n}} {\tilde{ω}}_{n}^{T} [{\overset{\cdot}{\hat{ω}}}_{n} - r_{n} e_{n} S_{n} (x_{n})] \end{matrix}

(53)

As $e_{n} \frac{ϖ (t)}{1 + Ξ_{1} ϱ} \leq e_{n} \frac{ϖ (t)}{1 + ϱ}, e_{n} | \frac{Ξ_{2} Θ}{1 + Ξ_{1} ϱ} | \leq e_{n} \frac{Θ}{1 - ϱ}$ and inserting (2) and (48) into (53), we can get

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq {\overset{\cdot}{V}}_{n - 1} + \frac{e_{n} ϖ (t)}{1 + ϱ} + \frac{e_{n} Θ}{1 - ϱ} + e_{n} \\ [{\hat{ω}}_{n}^{T} S_{n} (x_{n}) - {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} + μ + ϵ - {\overset{\cdot}{β}}_{n} \\ + k_{n} ζ_{n} + ζ_{n - 1}] - \frac{1}{r_{n}} {\tilde{ω}}_{n}^{T} [{\overset{\cdot}{\hat{ω}}}_{n} - r_{n} e_{n} S_{n} (x_{n})] \\ \leq {\overset{\cdot}{V}}_{n - 1} + e_{n} [{\hat{ω}}_{n}^{T} S_{n} (x_{n}) - {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) + ς_{n} π_{1} + μ \\ + ϵ - {\overset{\cdot}{β}}_{n} + k_{n} ζ_{n} + ζ_{n - 1}] + e_{n} α_{n + 1} \\ - | e_{n} ℏ | + | \frac{e_{n} Θ}{1 - ϱ} | + 0.557 γ - \frac{1}{r_{n}} {\tilde{ω}}_{n}^{T} [{\overset{\cdot}{\hat{ω}}}_{n} - r_{n} e_{n} S_{n} (x_{n})] \end{matrix}

(54)

By using Young’s inequality, one has

- e_{n} {\tilde{ω}}_{n}^{T} S_{n} (x_{n}) \leq \frac{1}{2} {e_{n}}^{2} + \frac{1}{2} {\tilde{ω}}_{n}^{T} {\tilde{ω}}_{n}

(55)

Then, the virtual controller $α_{n + 1}$ and the adaptation ${\overset{\cdot}{\hat{ω}}}_{n}$ are given as

α_{n + 1} = - k_{n} z_{n} - z_{n - 1} - {\hat{ω}}_{n}^{T} S_{n} (x_{n}) - ϵ - μ + {\overset{\cdot}{β}}_{n} - \frac{1}{2} e_{n} - ς_{n} π_{1}

(56)

{\overset{\cdot}{\hat{ω}}}_{n} = r_{n} [e_{n} S_{n} (x_{n}) - σ_{n} {\hat{ω}}_{n}]

(57)

Substituting (55)–(57) into (54), one has

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} - \sum_{i = 1}^{n} k_{i} {e_{i}}^{2} \\ + (| | P | |^{2} + \frac{1}{2}) \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} + 0.557 γ - | e_{n} ℏ | \\ + | \frac{e_{n} Θ}{1 - ϱ} | + χ_{0} + \sum_{i = 1}^{n} σ_{i} {\tilde{ω}}_{i}^{T} {\hat{ω}}_{i} \end{matrix}

(58)

Then, according to $ℏ > \frac{Θ}{1 - ϱ}$ , (58) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} - \sum_{i = 1}^{n} k_{i} {e_{i}}^{2} + (| | P | |^{2} + \frac{1}{2}) \\ \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} \\ + χ_{0} + \sum_{i = 1}^{n} σ_{i} {\tilde{ω}}_{i}^{T} {\hat{ω}}_{i} + 0.557 γ \end{matrix}

(59)

Remark 5. The input constraints and external disturbances are addressed in backstepping framework. The command filtering technology is adopted to avoid the adverse effects of “explosion of complexity.” In addition, the neural networks are utilized to approximate uncertain nonlinearities. Therefore, the computational complexity of the designed controller is greatly reduced.

Stability analysis

Theorem 1. Considering time-delay nonlinear system (1) with input saturation and external disturbances, under the designed state observer (21), the adaptive controller (56), and the adaptive law (57), the stability of all the signals in the closed-loop system can be guaranteed.

Proof: By using Young’s inequality, one has

{\tilde{ω}}_{j}^{T} {\hat{ω}}_{j} \leq - \frac{1}{2} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} + \frac{1}{2} | | ω_{j} | |^{2}

(60)

Inserting (60) into (59) yields

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - (λ_{\min} (Q) - ι - 4) | | π | |^{2} - \sum_{j = 1}^{n} k_{j} {e_{j}}^{2} \\ - (\frac{σ_{j}}{2} - | | P | |^{2} - \frac{1}{2}) \sum_{j = 1}^{n} {\tilde{ω}}_{j}^{T} {\tilde{ω}}_{j} \\ + \frac{σ_{j}}{2} \sum_{j = 1}^{n} | | ω_{j} | |^{2} + χ_{0} + 0.557 γ \\ \leq - a_{n} V_{n} + b_{n} \end{matrix}

(61)

where $a_{n} = min_{j = 1, \dots, n} {λ_{\min} (Q) - ι - 4, 2 k_{j}, σ_{j} - 2 | | P | |^{2} - 1}, b_{n} = \frac{σ_{j}}{2} \sum_{j = 1}^{n} | | ω_{j} | |^{2} + χ_{0} + 0.557 γ .$

We can deduce from (61) that the states $π$ , $e_{j}$ and ${\tilde{ω}}_{j}, (j = 1, 2, \dots, n)$ are bounded. As $ω_{j}$ are constants, the signals ${\hat{ω}}_{j}$ are also bounded. Since $e_{j} = z_{j} - ζ_{j}$ , if we want to ensure the stability of $z_{j}$ , then we have to prove $ζ_{j}$ are bounded.

For the error compensation system (31), we select the Lyapunov function as follows

V_{ζ} = \sum_{j = 1}^{n} \frac{1}{2} {ζ_{j}}^{2}

(62)

The derivation of (62) is calculated as

\begin{matrix} {\overset{\cdot}{V}}_{ζ} = - k_{1} {ζ_{1}}^{2} + ζ_{1} (β_{2} - α_{2}) + ζ_{1} ζ_{2} + \dots - k_{j} {ζ_{j}}^{2} + ζ_{j} \\ (β_{j + 1} - α_{j + 1}) \\ - ζ_{j} ζ_{j - 1} + ζ_{j} ζ_{j + 1} + \dots - k_{n} {ζ_{n}}^{2} - ζ_{n} ζ_{n - 1} \\ = - \sum_{j = 1}^{n} k_{j} {ζ_{j}}^{2} + \sum_{j = 1}^{n - 1} ζ_{j} (β_{j + 1} - α_{j + 1}) \end{matrix}

(63)

According to Lemma in (Li, 2019), the output $β_{j + 1}$ satisfy the condition $| β_{j + 1} - α_{j + 1} | \leq q_{j}$ , where $q_{j} > 0$ are the positive constants.

By using Young’s inequality, one has

ζ_{i} q_{j} \leq \frac{1}{2} {ζ_{j}}^{2} + \frac{1}{2} {q_{j}}^{2}

(64)

Inserting the inequality $| β_{j + 1} - α_{j + 1} | \leq q_{j}$ and (61) into (60), one has

\begin{matrix} {\overset{\cdot}{V}}_{ζ} \leq - \sum_{j = 1}^{n} k_{j} {ζ_{j}}^{2} + \sum_{j = 1}^{n - 1} ζ_{j} q_{j} \\ \leq - \sum_{j = 1}^{n} k_{j} {ζ_{j}}^{2} + \frac{1}{2} \sum_{j = 1}^{n - 1} {ζ_{j}}^{2} + \frac{1}{2} \sum_{j = 1}^{n - 1} {q_{j}}^{2} \\ \leq - \sum_{j = 1}^{n} (k_{j} - \frac{1}{2}) {ζ_{j}}^{2} - \frac{1}{2} {ζ_{n}}^{2} + \frac{1}{2} \sum_{j = 1}^{n - 1} {q_{j}}^{2} \\ \leq - a_{s} V_{ζ} + b_{ζ} \end{matrix}

(65)

where $a_{ζ} = min_{j = 1, \dots, n} {2 k_{j} - 1, 1}, b_{ζ} = \frac{1}{2} \sum_{j = 1}^{n - 1} {q_{j}}^{2}$ .

From (65), the conclusion can be drawn that the states $ζ_{j}$ are bounded. Thus, all closed-loop signals are SGUUB. This completes the proof.

Remark 6. According to (61) and (65), we can adjust the values of design parameters $ς_{j}$ , $k_{j}$ , $r_{j}$ and $σ_{j}$ to achieve the desired control performance of the system.

Simulation examples

In this section, two simulation examples are given to demonstrate the applicability of the presented control scheme.

Example 1: Consider the strict-feedback time-delay nonlinear system with input saturation and external disturbances as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} - 0.02 x_{1} - x_{1} / (1 + x_{1}^{2}) + 0.5 \cos (x_{2}) \\ {\overset{\cdot}{x}}_{2} = 0.08 e^{\sin (x_{1})} + x_{1} x_{2} + \sin (x_{1}^{2}) + u (υ (t - τ)) \\ y = x_{1} \end{matrix}

(66)

where $x_{1}$ and $x_{2}$ are the system states. The initial condition is $x_{1} (0) = 0.8$ , $x_{2} (0) = 1$ and the other initial values are selected as zero. The system output is denoted by $y$ . The input delay is chosen as $τ = 0.01$ . The input saturation $u (υ (t))$ is represented in (14), where $u_{N} = 15$ . The reference signal is given as $y_{r} = \sin (t)$ .

In the simulation, the parameters are given as $k_{1} = 10$ , $k_{2} = 100$ , $ς_{1} = 10$ , $ς_{2} = 50$ , $r_{1} = r_{2} = 0.5$ , $σ_{1} = σ_{2} = 1$ , $a_{2} = 0.1$ , $ϱ = 0.5$ , $ℏ = 3$ , $γ = 10$ , $Θ = 1$ . The center vector and the width of each Gaussian basic function are designed as $θ = 0.5 \times$ [-2 -1 0 1 2] and $η = 3$ , respectively.

The simulation results are shown in Figures 2 –8. Figure 2 describes the trajectories of $y$ and $y_{r}$ , which shows the desired tracking performance. Figures 3–4 display the tracking trajectories of state variables $x_{1}$ , $x_{2}$ and their estimated values ${\hat{x}}_{1}$ , ${\hat{x}}_{2}$ . Obviously, it can be seen that the system states $x_{1}$ and $x_{2}$ can be well estimated by the constructed observer. The trajectories of the control input $υ$ and input saturation $sat (υ)$ are plotted in Figure 5. It is clear that the input saturation occurs during the process of operation. The evolution of the adaptive parameters is shown in Figure 6. The time intervals of the triggering events are plotted in Figure 7. To further illustrate the contributions of this paper, the numbers of transmissions for the proposed ETC method and the conventional time-triggered control method are shown in Figure 8. It can be seen that the transmission numbers of the above methods are 96 and 4000, respectively, which implies that the proposed method can greatly reduce utilization of communication and computation resources compared with the time-triggered method in Zhang et al. (2021).

Example 2: To verity the applicability of the presented scheme, we consider a single-link robot arm system as depicted in Figure 9.

Its dynamics satisfy the following equation form

J \overset{\cdot\cdot}{q} + H \overset{\cdot}{q} + MGL \sin (q) = ξ, y = q

(67)

where $ξ$ denotes the control force of the link. $J$ , $H$ , $M$ , $G$ , and $L$ denote the inertia of motor, the damping coefficient, the mass of link, the acceleration due to gravity, and the location of mass center, respectively. The parameter values $J$ , $H$ , $M$ , $G$ , and $L$ are chosen the same as in Ma and Ma (2020) $q$ denotes the link angle.

Define $x_{1} = q$ , $x_{2} = \overset{\cdot}{q} + \frac{H}{J} q = \overset{\cdot}{q} + 2 q$ , and $u (υ (t - τ)) = ξ$ , one gets

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} - 2 x_{1} \\ {\overset{\cdot}{x}}_{2} = u (υ (t - τ)) - 9.8 \sin (x_{1}) + φ (t) \\ y = x_{1} \end{matrix}

(68)

where the input delay is chosen as $τ = 0.01$ . $u (υ (t))$ indicates the saturation limit of the input, where $u_{N} = 15$ . $φ (t) = 0.2 \cos (t)$ . The initial condition is $x_{1} (0) = 1.1$ , $x_{2} (0) = 1.4$ and the other initial values are selected as zero. The reference signal is $y_{r} = \sin (t)$ .

The parameters are given as $k_{1} = 10$ , $k_{2} = 60$ , $ς_{1} = 10$ , $ς_{2} = 52$ , $r_{1} = r_{2} = 0.5$ , $σ_{1} = 1$ , $σ_{2} = 2$ , $a_{2} = 0.1$ , $ϱ = 0.5$ , $ℏ = 3$ , $γ = 8$ , $Θ = 1$ . The center vector and the width of each Gaussian basic function are designed as $θ = 0.5 \times$ [-2 -1 0 1 2] and $η = 3$ , respectively.

The simulation results are plotted in Figures 10 –16. The output tracking performance is displayed in Figure 10. The trajectories of states $x_{i}$ and their estimates ${\hat{x}}_{i}, (i = 1, 2)$ are plotted in Figures 11 and 12. The curves of control input $υ$ and $sat (υ)$ are shown in Figure 13. The adaptive parameters are displayed in Figure 14. The relevant time intervals of triggering are plotted in Figure 15. The triggering numbers of the proposed ETC method and the conventional time-triggered control method in Zhang et al. (2021) are shown in Figure 16, which are 114 and 4000, respectively. Therefore, we can get that our proposed control strategy can also alleviate the burden of network resources greatly in single-link robot arm system.

Conclusion

An observer-based event-triggered adaptive neural tracking control approach has been presented for time-delay nonlinear systems with input saturation and external disturbances in this research. The nonlinear terms have been approximated by applying the RBFNNs. The ETC mechanism has been proposed to alleviate the burden of network resource. With the help of the command filter technology, the adverse effects of “explosion of complexity” in the conventional backstepping method have been eliminated. The Pade approximation method has been introduced to eliminate the influence of input delay. In addition, a smooth non-affine function and an auxiliary variable have been applied to deal with the problem caused by input saturation. The main features of this article are summarized as follows: (1) the burden of network resources is alleviated; (2) the problem of “explosion of complexity” is addressed; (3) the closed-loop system is SGUUB. Finally, two simulation examples have demonstrated the effectiveness and superiority of the proposed control strategy. In the future work, the finite-time adaptive event-triggered tracking control problem for switched nonlinear systems with input constraints and time-varying delays will be further considered.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Natural Science Research Project of Jiangsu Higher Education Institutions under Grant 22KJB510028.

ORCID iD

Peng Wu

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