Event-Triggered, Adaptive, Exponentially Asymptotic Tracking Control of Stochastic Nonlinear Systems

Abstract

This paper investigates the problem of event-triggered, adaptive, asymptotic tracking control for a class of non-strict feedback stochastic nonlinear systems with symmetrical structures and sensor faults. Based on the negative exponential function, the event-triggered adaptive tracking control strategy deals with the problem of exponentially asymptotic convergence for the first time. The radial basis function neural network (RBFNN) mechanism addresses uncertain factors and unknown external disturbances in the system. The developed strategy ensures that all the signals of the closed-loop system are semi-globally uniformly bounded in probability, and that the tracking error can exponentially converge to zero. Finally, a simulation example demonstrates the effectiveness of the proposed method.

1. Introduction

Subject to the growing demand from the development of practice and theory, many adaptive tracking control problems for stochastic nonlinear systems [1,2,3,4,5,6,7,8,9,10,11,12,13] are being studied. A lot of scholars combine the backstepping technique and Lyapunov function to design adaptive controllers to learn about the stabilization of stochastic systems [1,2,3]. RBFNN and fuzzy logic systems (FLS) have drawn the attention of numerous scholars since they can handle possible unknown terms in the system [4,5,6,7,8,9,10,11,12,13]. Sui proposes a fuzzy adaptive finite-time control approach for uncertain stochastic nonlinear systems in non-triangular form [4]. Combining backstepping methods and neural network techniques, Niu investigates the adaptive output feedback control problem for a class of non-strict feedback, interconnected stochastic nonlinear systems with unmeasurable states [10]. Li addresses the problem of adaptive tracking control for non-strict feedback stochastic nonlinear systems with dead-zones and output constraints [13]. However, it is worth pointing out that the aforementioned articles only focus on the time-triggered mechanism, which transmits control signals from the controller to the actuator within a fixed time interval, and then easily causes the problem of communication resource waste.

Compared with time-triggered mechanism, event-triggered mechanism makes the system update the control signal of actuator only when the controller signal reaches a certain threshold, which will greatly reduce the communication resource occupancy rate of the system. Therefore, incorporating the event-triggered mechanism into stochastic nonlinear systems has been receiving growing attention [14,15,16,17,18,19,20,21,22,23,24,25]. Ning studies the problem of adaptive fault detection filter design for uncertain stochastic nonlinear systems with missing measurements by using the event-triggered mechanism [15]. Zou addresses the consensus tracking problem for a class of high order uncertain stochastic nonlinear multi-agent system with an event-triggered control strategy [20,21]. Ma presents an adaptive event-triggered output feedback control method for a class of stochastic nonlinear systems; meanwhile, he adopted the fuzzy logic system to approximate the unknown nonlinear functions [22]. However, none of these research results take into account the fact that the data transmitted by the sensors can be biased.

In practical application systems, partial component failure or sudden external interference may cause sensor faults, which will inevitably reduce the performance of the system, or even destroy the whole application system. Therefore, many scholars have been contributing to research on stochastic nonlinear control systems with sensor faults in recent years [26,27,28,29,30,31,32,33]. Xu solves the problem of generalized correntropy-filter-based fault diagnosis and fault tolerant control for stochastic nonlinear systems subject to heavy-tailed distributed noises [28]. Du discusses the problem of cooperative control for second-order, strict-feedback, stochastic, nonlinear, multi-agent systems against sensor faults [29]. Combining event-triggered mechanisms and sliding mode control, Aslam, et al. investigate the complex fault control problems for Markov jump systems [30,31,32]. However, it is important to notice that the tracking performance of the above articles only ensures that the tracking error eventually converges to a neighborhood near the equilibrium point, rather than asymptotically converging to zero.

In order to better achieve zero-error tracking performance, many results have been reported on asymptotic tracking problems for stochastic systems [34,35,36,37,38,39,40,41,42]. For the purpose of addressing the problem of limited transmission resources and achieving asymptotic convergence performance, Liu proposes the event-triggered, fuzzy adaptive compensation control design for uncertain stochastic nonlinear systems [34]. Liu developed a parameter-dependent state and output feedback controller to address the problem of global asymptotic stabilization for a class of stochastic, nonlinear, time-varying delay systems under the weaker condition on nonlinear functions [37]. Furthermore, numerous studies exploring exponential convergence are published in an effort to optimize the speed of convergence of the stochastic nonlinear system to the equilibrium point [39,40,41]. Li developed an exponentially asymptotic, adaptive tracking control method for a class of non-strict feedback nonlinear systems by adding the negative exponential function to the Lyapunov function for the first time [42]. Yet, no consideration has been given in any of the above articles [34,35,36,37,38,39,40,41,42] to the possibility that sensor faults will arise in a physical application system.

Consequently, it is of great relevance to formulate an event-triggered, adaptive, neural tracking control strategy for non-strict, feedback, stochastic, nonlinear systems to ensure exponentially asymptotic tracking performance. The main contributions of this paper are as follows:

(1)

An adaptive neural tracking control scheme for non-strict feedback stochastic nonlinear systems based on the event-triggered mechanism is presented. Compared with the time-triggered mechanism generally used in [26,27,28,29,30,31,36,37,38,39,40,42], this paper employs an event-triggered mechanism that reduces the frequency of control signal transmission in the system to reduce the communication burden.

(2)

In contrast to the normal formulation of the Lyapunov function in [34,35,36,37,38], negative exponential function is applied to the Lyapunov function to obtain exponential asymptotic tracking performance for stochastic nonlinear systems. Meanwhile, all signals within the system are bounded.

(3)

Unlike existing findings, which do not take sensor faults into account [34,35,36,37,38,39,40,41,42], this paper deals with the impact of sensor faults on the control system, which broadens the application of event-triggering mechanisms and exponential asymptotic convergence strategies.

2. Problem Statement and Some Preliminaries

Consider the non-strict feedback nonlinear stochastic system, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}d{x}_i\hfill & =(g_i\left(x\right)+h_i\left(x\right)x_{i+1}+d_i\left(t\right))dt+{\chi }_i^T\left(x\right)d\omega ,i=1,\dots ,n-1,\hfill \\ d{x}_n\hfill & =(g_n\left(x\right)+h_n\left(x\right)u+d_n\left(t\right))dt+{\chi }_n^T\left(x\right)d\omega ,\hfill \\ y\hfill & =x_1,\hfill \end{array}\right.\end{array}$$

(1)

where $x={[x_1,\dots ,x_n]}^T\in R^n$, $u\in R$, and $y\in R$ are the state variable, the control input, and the output of the system, respectively. $g_i(·),h_i(·),{\chi }_i(·)\in R,(i=1,2,\dots ,n)$ represent unknown continue smooth nonlinear functions. $d_i\left(t\right)$ denotes the uncertain bounded disturbance function and satisfies $|d_i\left(t\right)|\le {\overline{d}}_i$, and ${\overline{d}}_i$ is positive constant. $\omega$ is an independent standard Brownian motion.

Consider the analysis we discussed in the introduction—it is inevitable that some errors will occur in the practical application of sensors. Thus, we assume that the measured value of $x_i^E$ and the real value of $x_i$ satisfy the following condition [29]:

$$\begin{array}{c}\hfill x_i^E\left(t\right)={\vartheta }_i\left(t\right)x_i\left(t\right)+{\iota }_i\left(t\right),\end{array}$$

(2)

where ${\vartheta }_i\left(t\right)$ and ${\iota }_i\left(t\right)$ are the multiplicative and additive faults. Moreover, there exist two positive constants, ${\underline{\vartheta }}_i$ and ${\overline{\iota }}_i$, which satisfy $0<{\underline{\vartheta }}_i\le {\vartheta }_i\le 1$ and $|{\dot{\iota }}_i\left(t\right)|\le {\overline{\iota }}_i$.

Control Objective: For System (1), develop an event-triggered controller, $u\left(t\right)$, such that the semi-global asymptotic convergence of the system states in probability at the origin is achieved, and the semi-global boundedness in probability of all the closed-loop signals is guaranteed.

Remark 1.

For the exponential asymptotic convergence method, an adaptive control scheme proposed in [42] for non-stochastic systems achieves asymptotic zero-error tracking. However, it employs a periodic time-triggered mechanism, greatly occupying communication resources, and does not consider the possibility of sensor faults occurring. To address this problem, this paper designs the controller with the event-triggered mechanism to reduce the communication resource occupation rate, which is more suitable for practical application scenarios. Moreover, the exponential convergence function is extended to stochastic systems, and the sensor faults problem is also considered, which broadens the applicability of the method.

Remark 2.

The main differences between this paper and [25] have been presented as following. Firstly, our work studies the stochastic nonlinear systems in non-strict feedback form with sensor failures, while [25] studies the stochastic nonlinear systems in pure feedback form with the input saturation. Secondly, the controller developed in our paper achieves the asymptotic convergence performance; however, the control scheme in [25] only ensures the boundedness of the closed-loop system.

Remark 3.

The time-triggered mechanism—generally used in [26,27,28,29,30,31,36,37,38,39,40,42]—sends data strictly at fixed time intervals. The event-triggered mechanism used in this paper will only be triggered when the control signal used for the actuator, and the signal generated by the controller, are over a certain threshold. As a result, communication resources can be saved, reducing the system’s communication resource usage.

Definition 1

([23]). Consider the following stochastic system:

$$\begin{array}{c}\hfill dx=\psi \left(x\right)dt+{\chi }^T\left(x\right)d\omega ,x\in R,\end{array}$$

(3)

where x and ω are defined in System (1), $\psi (·)$ and $\chi (·)$ are, locally, Lipschitz functions. For any given Lyapunov function, $V\left(x\right)\in C^2$, the differential operator, $LV$, is defined as follows:

$$\begin{array}{c}\hfill LV=\frac{\partial V}{\partial x}\psi +\frac{1}{2}Tr\{{\chi }^T\frac{{\partial }^{2}V}{\partial x^2}\chi \},\end{array}$$

(4)

where $Tr(·)$ means the trace of matrix.

Lemma 1

([43]). For $\forall x\in R$, and $\sigma >0$, one has

$$\begin{array}{c}\hfill 0\le \left|x\right|<\sigma +\frac{x^2}{\sqrt{x^2+{\sigma }^2}}.\end{array}$$

(5)

Lemma 2

([6]). In this paper, the following $RBFNN$ will be utilized to approximate the unknown continuous functions and uncertain items

$$\begin{array}{c}\hfill g\left(Z\right)=W^{T}S\left(Z\right)+\epsilon \left(Z\right),\end{array}$$

(6)

where $Z\in {\mathsf{\Omega }}_z\subset R^n$ represents the input vector of $NN$, $W\in R^l$ is the optimal weight vector, l means the number of neural network nodes, $S\left(Z\right)={[s_1\left(Z\right),\dots ,s_l\left(Z\right)]}^T$ denotes the basis function vector, and $s_i\left(Z\right)$ is chosen as the Gaussian function, generally, which is described as follows:

$$\begin{array}{c}\hfill s_i\left(Z\right)=exp[-\frac{{(Z-{\sigma }_i)}^T(Z-{\sigma }_i)}{{\varrho }_i^2}],i=1,2,\dots ,l,\end{array}$$

(7)

where ${\sigma }_i={[{\sigma }_{i1},\dots ,{\sigma }_{in}]}^T$ is the center of receptive field, and ${\varrho }_i$ is the width of Gaussian function. $\epsilon \left(Z\right)$ means the approximation error between the unknown function, and $RBFNN$ satisfies $\left|\epsilon \left(Z\right)\right|\le \overline{\epsilon }$ with $\overline{\epsilon }$ as a positive constant.

Lemma 3

([12]). Considering $S_i\left(Z\right)={(s_1\left(Z\right),\dots ,s_i\left(Z\right))}^T$ with $s_i\left(Z\right)$, in the form of Gaussian basic function, and $Z={[z_1,\dots ,z_n]}^T$, the following holds:

$$\begin{array}{c}\hfill \parallel S_i{\left(Z\right)\parallel }^2\le {\parallel S_i\left(Z_i\right)\parallel }^2,\end{array}$$

(8)

where $Z_i={[z_1,\dots ,z_i]}^T$ for $i\le n$.

Assumption 1.

The reference trajectory $y_d\left(t\right)$ and its derivative up to $(n+1)th$ order are continuous, known, and bounded.

Assumption 2.

Generally, it is assumed that $h_i\left(t\right)>0$, and that there exist positive constants, ${\overline{h}}_i$ and ${\underline{h}}_i$, satisfying $0<{\underline{h}}_i\le h_i\left(t\right)\le {\overline{h}}_i$.

3. Controller Design and Stability Analysis

In this section, an event-based adaptive control scheme is designed via the backstepping technique. Considering the sensor error phenomenon in the system, the following controller design will adopt $x^E$ instead of x to represent state variables. The coordinate transformation is constructed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{z}_1=x_1^E-y_d,\hfill \\ z_i=x_i^E-{\alpha }_{i-1},i=2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(9)

where ${\alpha }_{i-1}$ represents the virtual control input, which will be designed later.

Step 1:

Combining Definition 1, substituting (1) into (9), we obtain the following:

$$\begin{array}{cc}\hfill d{z}_1=& ({\dot{\vartheta }}_1{x}_1+{\vartheta }_1(g_1+d_1)+{\breve{h}}_1(z_2+{\alpha }_1-{\iota }_2)+{\dot{\iota }}_1-{\dot{y}}_d)dt+{\chi }_1^{T}d\omega ,\hfill \end{array}$$

(10)

where $z_2=x_2^E-{\alpha }_1$, ${\breve{h}}_1=h_1{\vartheta }_1/{\vartheta }_2$, and ${\underline{\vartheta }}_1{\underline{h}}_1\le \left|{\check{h}}_1\right|\le ({\overline{h}}_1/{\underline{\vartheta }}_1)\triangleq H_1$.

The Lyapunov function candidate is constructed in the following form:

$$\begin{array}{c}\hfill V_1=\frac{1}{4{\underline{h}}_1}{z}_1^4+\frac{1}{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2+\frac{1}{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2,\end{array}$$

(11)

where $\eta >0$, ${\nu }_1>0$ and ${\mu }_1>0$. Furthermore, the derivative of $V_1$ is as follows:

$$\begin{array}{cc}\hfill L{V}_1=& \frac{z_1^3}{{\underline{h}}_1}({\dot{\vartheta }}_1{x}_1+{\vartheta }_1(g_1+d_1)+{\breve{h}}_1(z_2+{\alpha }_1-{\iota }_2)+{\dot{\iota }}_1-{\dot{y}}_d)+\frac{3}{2{\underline{h}}_1}{z}_1^2{\chi }_1^T{\chi }_1\hfill \\ \hfill & -\frac{\eta }{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2-\frac{\eta }{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2-\frac{1}{{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1{\dot{\widehat{\varphi }}}_1-\frac{1}{{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1{\dot{\widehat{\phi }}}_1.\hfill \end{array}$$

(12)

By utilizing Lemma 1 and Young’s inequality, one obtains the following:

$$\begin{array}{cc}\hfill \frac{3}{2{\underline{h}}_1}{z}_1^2{\chi }_1^T{\chi }_1& \le \frac{3}{2{\underline{h}}_1}{z}_1^2{\parallel {\chi }_1\parallel }^2\le \frac{\frac{9}{4{\underline{h}}_1^2}{z}_1^4{\parallel {\chi }_1\parallel }^4}{\sqrt{\frac{9}{4{\underline{h}}_1^2}{z}_1^4{\parallel {\chi }_1\parallel }^4+{\pi }_1^2}}+{\pi }_1,\hfill \\ \hfill {\check{h}}_1{z}_1^3{z}_2& \le \frac{3}{4}{H}_1^{\frac{4}{3}}{z}_1^4+\frac{1}{4}{z}_2^4,\hfill \end{array}$$

(13)

where ${\pi }_1=l_1{e}^{-B_{1}t}$ is a positive parameter.

Substituting (13) into (12), we obtain the following:

$$\begin{array}{c}\hfill L{V}_1\le \frac{z_1^3}{{\underline{h}}_1}(G_1+{\breve{h}}_1{\alpha }_1)+{\pi }_1+\frac{1}{4{\underline{h}}_1}{z}_2^4-\frac{\eta }{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2-\frac{\eta }{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2-\frac{1}{{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1{\dot{\widehat{\varphi }}}_1-\frac{1}{{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1{\dot{\widehat{\phi }}}_1,\end{array}$$

(14)

where $G_1\left(Z_1\right)={\dot{\vartheta }}_1{x}_1+{\vartheta }_1(g_1+{\overline{d}}_1)-{\breve{h}}_1{\iota }_2+{\overline{\iota }}_1-{\dot{y}}_d+\frac{\frac{9}{4{\underline{h}}_1}{z}_1{\parallel {\chi }_1\parallel }^4}{\sqrt{\frac{9}{4{\underline{h}}_1^2}{z}_1^4{\parallel {\chi }_1\parallel }^4+{\pi }_1^2}}+\frac{3}{4}{H}_1^{\frac{4}{3}}{z}_1=W_1^T{S}_1\left(Z_1\right)+{\epsilon }_1$, with $Z_1={[x_1,\dots ,x_n,y_d,{\dot{y}}_d]}^T$. Let ${\varphi }_1=\frac{\parallel W_1\parallel }{{\underline{h}}_1}$, and ${\tilde{\varphi }}_i={\varphi }_i-{\widehat{\varphi }}_i$, i = 1, …, n, represents the estimation error, and ${\widehat{\varphi }}_i$ is the estimation of $\varphi$. Based on Lemmas 1 and 3, and $S_1^T{S}_1\le 1$, we obtain the following:

$$\begin{array}{c}\hfill \frac{z_1^3}{{\underline{h}}_1}{W}_1^T{S}_1\left(Z_1\right)\le |z_1{|}^3{\varphi }_1\parallel S_1\left(Z_1\right)\parallel \le |z_1{|}^3{\varphi }_1\parallel S_1\left({\overline{Z}}_1\right)\parallel \\ \hfill \le \frac{z_1^6{S}_1^T{S}_1{\varphi }_1}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}+{\pi }_1{\varphi }_1\le \frac{z_1^6{\varphi }_1}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}+{\pi }_1{\varphi }_1,\end{array}$$

(15)

where ${\overline{Z}}_1={[x_1,y_d,{\dot{y}}_d]}^T$. Let ${\phi }_1=\frac{{\epsilon }_1}{{\underline{h}}_1}$, we have as follows:

$$\begin{array}{c}\hfill \frac{1}{{\underline{h}}_1}{z}_1^3{\epsilon }_1\le {\left|z_1\right|}^3{\phi }_1\le \frac{z_1^6{\phi }_1}{\sqrt{z_1^6+{\pi }_1^2}}+{\pi }_1{\phi }_1.\end{array}$$

(16)

Along with (15) and (16), (14) can be rewritten as follows:

$$\begin{array}{cc}\hfill L{V}_1\le & \frac{1}{4{\underline{h}}_1}{z}_2^4+\frac{{\breve{h}}_1}{{\underline{h}}_1}{z}_1^3{\alpha }_1+{\pi }_1(1+{\varphi }_1+{\phi }_1)+\frac{z_1^6{\varphi }_1}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}+\frac{z_1^6{\phi }_1}{\sqrt{z_1^6+{\pi }_1^2}}\hfill \\ \hfill & -\frac{\eta }{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2-\frac{\eta }{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2-\frac{1}{{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1{\dot{\widehat{\varphi }}}_1-\frac{1}{{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1{\dot{\widehat{\phi }}}_1\hfill \\ \hfill \le & -c_1{z}_1^4-\frac{\eta }{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2-\frac{\eta }{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2+\frac{1}{4{\underline{h}}_1}{z}_2^4+\frac{{\breve{h}}_1}{{\underline{h}}_1}{z}_1^3{\alpha }_1+z_1^3{\lambda }_1+{\pi }_1(1+{\varphi }_1+{\phi }_1)\hfill \\ \hfill & +\frac{1}{{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1({\mu }_1{e}^{\eta t}\frac{z_1^6}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}-{\dot{\widehat{\varphi }}}_1)+\frac{1}{{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1({\nu }_1{e}^{\eta t}\frac{z_1^6}{\sqrt{z_1^6+{\pi }_1^2}}-{\dot{\widehat{\phi }}}_1),\hfill \end{array}$$

(17)

where ${\lambda }_1=c_1{z}_1+\frac{z_1^3{\widehat{\varphi }}_1}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}+\frac{z_1^3{\widehat{\phi }}_1}{\sqrt{z_1^6+{\pi }_1^2}}$, $c_1>0$.

The virtual controller and adaptive laws are constructed in the following form:

$$\begin{array}{cc}\hfill {\alpha }_1=& -\frac{z_1^3{\lambda }_1^2}{\sqrt{z_1^6{\lambda }_1^2+{\pi }_1^2}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\dot{\widehat{\varphi }}}_1=& {\mu }_1{e}^{\eta t}(\frac{z_1^6}{\sqrt{z_1^6{S}_1^T{S}_1+{\pi }_1^2}}-{\pi }_1{\widehat{\varphi }}_1),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\dot{\widehat{\phi }}}_1=& {\nu }_1{e}^{\eta t}(\frac{z_1^6}{\sqrt{z_1^6+{\pi }_1^2}}-{\pi }_1{\widehat{\phi }}_1),\hfill \end{array}$$

(20)

Considering the following:

$$\begin{array}{c}\hfill \frac{{\breve{h}}_1}{{\underline{h}}_1}{z}_1^3{\alpha }_1+z_1^3{\lambda }_1\le -\frac{h_1{\vartheta }_1}{{\underline{h}}_1{\vartheta }_2}\frac{z_1^6{\lambda }_1^2}{\sqrt{z_1^6{\lambda }_1^2+{\pi }_1^2}}+z_1^3{\lambda }_1\le -\frac{z_1^6{\lambda }_1^2}{\sqrt{z_1^6{\lambda }_1^2+{\pi }_1^2}}+z_1^3{\lambda }_1\le {\pi }_1.\end{array}$$

(21)

According to (18)–(21), we obtain the following:

$$\begin{array}{cc}\hfill L{V}_1\le & -c_1{z}_1^4-\frac{\eta }{2{\mu }_1}{e}^{-\eta t}{\tilde{\varphi }}_1^2-\frac{\eta }{2{\nu }_1}{e}^{-\eta t}{\tilde{\phi }}_1^2+\frac{1}{4{\underline{h}}_1}{z}_2^4\hfill \\ \hfill & +{\pi }_1(2+{\varphi }_1+{\phi }_1)+{\pi }_1({\tilde{\varphi }}_1{\widehat{\varphi }}_1+{\tilde{\phi }}_1{\widehat{\phi }}_1).\hfill \end{array}$$

(22)

Step $i(2\le i\le n-1)$: Similar to (10), it is by using Definition 1 that we obtain the following:

$$\begin{array}{cc}\hfill d{z}_i=& ({\dot{\vartheta }}_i{x}_i+{\vartheta }_i(g_i+d_i)+{\breve{h}}_i(z_{i+1}+{\alpha }_i-{\iota }_{i+1})+{\dot{\iota }}_i-L{\alpha }_{i-1})dt+{\overline{\chi }}_i^{T}d\omega ,\hfill \end{array}$$

(23)

where $z_{i+1}=x_{i+1}^E-{\alpha }_i$, ${\breve{h}}_i=h_i{\vartheta }_i/{\vartheta }_{i+1}$, ${\underline{\vartheta }}_i{\underline{h}}_i\le \left|{\check{h}}_i\right|\le ({\overline{h}}_i/{\underline{\vartheta }}_i)\triangleq H_i$, ${\overline{\chi }}_i={\chi }_i-{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}{\chi }_j$, $L{\alpha }_{i-1}={\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}(h_j{x}_{j+1}+g_j+d_j)+{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\widehat{\varphi }}_j}{\dot{\widehat{\varphi }}}_j+{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\widehat{\phi }}_j}{\dot{\widehat{\phi }}}_j+{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\vartheta }_j}{\dot{\vartheta }}_j+{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\iota }_j}{\dot{\iota }}_j$ $+{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\pi }_j}{\dot{\pi }}_j+{\sum }_{j=0}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial y_d^{\left(j\right)}}{y}_d^{(j+1)}+\frac{1}{2}{\sum }_{p,q=1}^{i-1}\frac{\partial {\alpha }_{i-1}^2}{\partial x_p{x}_q}{\chi }_p^T{\chi }_q$.

Considering the Lyapunov function candidate, as follows:

$$\begin{array}{c}\hfill V_i=V_{i-1}+\frac{1}{4{\underline{h}}_i}{z}_i^4+\frac{1}{2{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i^2+\frac{1}{2{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i^2,\end{array}$$

(24)

where ${\nu }_i>0$ and ${\mu }_i>0$. Furthermore, the derivative of $V_i$ is as follows:

$$\begin{array}{cc}\hfill L{V}_i=& L{V}_{i-1}+\frac{z_i^3}{{\underline{h}}_i}({\dot{\vartheta }}_i{x}_i+{\vartheta }_i(g_i+d_i)+{\breve{h}}_i(z_{i+1}+{\alpha }_i-{\iota }_{i+1})+{\dot{\iota }}_i-L{\alpha }_{i-1})\hfill \\ \hfill & -\frac{\eta }{2{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i^2-\frac{\eta }{2{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i^2+\frac{3}{2{\underline{h}}_i}{z}_i^2{\overline{\chi }}_i^T{\overline{\chi }}_i-\frac{1}{{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i{\dot{\widehat{\varphi }}}_i-\frac{1}{{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i{\dot{\widehat{\phi }}}_i.\hfill \end{array}$$

(25)

Similar to (13), one obtains the following:

$$\begin{array}{cc}\hfill \frac{3}{2{\underline{h}}_i}{z}_i^2{\overline{\chi }}_i^T{\overline{\chi }}_i& \le \frac{\frac{9}{4{\underline{h}}_i^2}{z}_i^4{\parallel {\overline{\chi }}_i\parallel }^4}{\sqrt{\frac{9}{4{\underline{h}}_i^2}{z}_i^4{\parallel {\overline{\chi }}_i\parallel }^4+{\pi }_i^2}}+{\pi }_i,\hfill \\ \hfill {\check{h}}_i{z}_i^3{z}_{i+1}& \le \frac{3}{4}{H}_i^{\frac{4}{3}}{z}_i^4+\frac{1}{4}{z}_{i+1}^4,\hfill \end{array}$$

(26)

where ${\pi }_i=l_i{e}^{-B_{i}t}$ is a positive parameter.

Substituting (26) into (25), we obtain the following:

$$\begin{array}{cc}\hfill L{V}_i\le & -\sum _{j=1}^{i-1}{c}_j{z}_j^4-\sum _{j=1}^i\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^i\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^{i-1}{\pi }_j(2+{\varphi }_j+{\phi }_j)\hfill \\ \hfill & +\sum _{j=1}^{i-1}{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)+\frac{z_i^3}{{\underline{h}}_i}(G_i+{\breve{h}}_i{\alpha }_i-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}{d}_j-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\pi }_j}{\dot{\pi }}_j)\hfill \\ \hfill & +{\pi }_i+\frac{1}{4{\underline{h}}_i}{z}_{i+1}^4-\frac{1}{{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i{\dot{\widehat{\varphi }}}_i-\frac{1}{{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i{\dot{\widehat{\phi }}}_i,\hfill \end{array}$$

(27)

where $$\begin{gathered} G_i\left(Z_i\right)={\dot{\vartheta }}_i{x}_i+{\vartheta }_i(g_i+{\overline{d}}_i)-{\breve{h}}_i{\iota }_{i+1}+{\overline{\iota }}_i-{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}(h_j{x}_{j+1}+g_j)+\frac{3}{4}{H}_i^{\frac{4}{3}}{z}_i \\ -{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\widehat{\varphi }}_j}{\dot{\widehat{\varphi }}}_j-{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\widehat{\phi }}_j}{\dot{\widehat{\phi }}}_j-{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\vartheta }_j}{\dot{\vartheta }}_j-{\sum }_{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\iota }_j}{\dot{\iota }}_j-{\sum }_{j=0}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial y_d^{\left(j\right)}}{y}_d^{j+1}+\frac{{\underline{h}}_i}{4{\underline{h}}_{i-1}}{z}_i \\ -\frac{1}{2}{\sum }_{p,q=1}^{i-1}\frac{\partial {\alpha }_{i-1}^2}{\partial x_p{x}_q}{\chi }_p^T{\chi }_q+\frac{9}{4{\underline{h}}_i}{z}_i\parallel {\overline{\chi }}_i{\parallel }^4(\frac{9}{4{\underline{h}}_i^2}{z}_i^4\parallel {\overline{\chi }}_i{{\parallel }^4+{\pi }_i^2)}^{-\frac{1}{2}}=W_i^T{S}_i\left(Z_i\right)+{\epsilon }_i, \end{gathered}$$

with $Z_i={[x_1,\dots ,x_n,{\widehat{\varphi }}_1,\dots ,{\widehat{\varphi }}_i,{\widehat{\phi }}_1,\dots ,{\widehat{\phi }}_i,y_d,\dots ,y_d^{\left(i\right)}]}^T$.

Let ${\varphi }_i=\frac{\parallel W_i\parallel }{{\underline{h}}_i}$. Similar to (15), one obtains the following:

$$\begin{array}{c}\hfill \frac{z_i^3}{{\underline{h}}_i}{W}_i^T{S}_i\left(Z_i\right)\le |z_i{|}^3{\varphi }_i\parallel S_i\left(Z_i\right)\parallel \le |z_i{|}^3{\varphi }_i\parallel S_i\left({\overline{Z}}_i\right)\parallel \\ \hfill \le \frac{z_i^6{S}_i^T{S}_i{\varphi }_i}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}+{\pi }_i{\varphi }_i\le \frac{z_i^6{\varphi }_i}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}+{\pi }_i{\varphi }_i,\end{array}$$

(28)

where ${\overline{Z}}_i={[x_1,\dots ,x_i,{\widehat{\varphi }}_1,\dots ,{\widehat{\varphi }}_i,{\widehat{\phi }}_1,\dots ,{\widehat{\phi }}_i,y_d,\dots ,y_d^{\left(i\right)}]}^T$. Denoting that ${\phi }_i={sup}_{t\ge 0}\parallel {\vartheta }_i\parallel$, we obtain the following:

$$\begin{array}{cc}\hfill \frac{z_i^3}{{\underline{h}}_i}({\epsilon }_i-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial x_j}{d}_j-\sum _{j=1}^{i-1}\frac{\partial {\alpha }_{i-1}}{\partial {\pi }_j}{\dot{\pi }}_j)& \le |z_i{|}^3\parallel {\beta }_i\parallel \parallel {\vartheta }_i\parallel \hfill \\ \hfill & \le |z_i{|}^3\parallel {\beta }_i\parallel {\phi }_i\le \frac{z_i^6{\parallel {\beta }_i\parallel }^2{\phi }_i}{\sqrt{z_i^6{\parallel {\beta }_i\parallel }^2+{\pi }_i^2}}+{\pi }_i{\phi }_i,\hfill \end{array}$$

(29)

where ${\beta }_i={[1,\frac{\partial {\alpha }_{i-1}}{\partial x_1},\dots ,\frac{\partial {\alpha }_{i-1}}{\partial x_{i-1}},\frac{\partial {\alpha }_{i-1}}{\partial {\pi }_1},\dots ,\frac{\partial {\alpha }_{i-1}}{\partial {\pi }_{i-1}}]}^T$ and ${\vartheta }_i=\frac{1}{{\underline{h}}_i}[{\epsilon }_i,d_1,\dots ,d_{i-1},{\dot{\pi }}_1,\dots ,{\dot{\pi }}_{i-1}]$.

Substituting (28) and (29) into (27), the following holds:

$$\begin{array}{cc}\hfill L{V}_i\le & -\sum _{j=1}^{i-1}{c}_j{z}_j^4-\sum _{j=1}^i\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^i\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^{i-1}{\pi }_j(2+{\varphi }_j+{\phi }_j)+\sum _{j=1}^{i-1}{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)\hfill \\ \hfill & +\frac{1}{4{\underline{h}}_i}{z}_{i+1}^4+\frac{{\breve{h}}_i}{{\underline{h}}_i}{z}_i^3{\alpha }_i+{\pi }_i(1+{\varphi }_i+{\phi }_i)+\frac{z_i^6{\varphi }_i}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}+\frac{z_i^6{\parallel {\beta }_i\parallel }^2{\phi }_i}{\sqrt{z_i^6{\parallel {\beta }_i\parallel }^2+{\pi }_i^2}}\hfill \\ \hfill & -\frac{1}{{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i{\dot{\widehat{\varphi }}}_i-\frac{1}{{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i{\dot{\widehat{\phi }}}_i\hfill \\ \hfill \le & -\sum _{j=1}^i{c}_j{z}_j^4-\sum _{j=1}^i\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^i\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^{i-1}{\pi }_j(2+{\varphi }_j+{\phi }_j)+\sum _{j=1}^{i-1}{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)\hfill \\ \hfill & +\frac{1}{{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i({\nu }_i{e}^{\eta t}\frac{z_i^6{\parallel {\beta }_i\parallel }^2}{\sqrt{z_i^6{\parallel {\beta }_i\parallel }^2+{\pi }_i^2}}-{\dot{\widehat{\phi }}}_i)+\frac{1}{{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i({\mu }_i{e}^{\eta t}\frac{z_i^6}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}-{\dot{\widehat{\varphi }}}_i)\hfill \\ \hfill & +\frac{1}{4{\underline{h}}_i}{z}_{i+1}^4+\frac{{\breve{h}}_i}{{\underline{h}}_i}{z}_i^3{\alpha }_i+z_i^3{\lambda }_i+{\pi }_i(1+{\varphi }_i+{\phi }_i),\hfill \end{array}$$

(30)

where ${\lambda }_i=c_i{z}_i+\frac{z_i^3{\widehat{\varphi }}_i}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}+\frac{z_i^3{\parallel {\beta }_i\parallel }^2{\widehat{\phi }}_i}{\sqrt{z_i^6{\parallel {\beta }_i\parallel }^2+{\pi }_i^2}}$, $c_i>0$.

The virtual controller and adaptive laws are constructed as follows:

$$\begin{array}{cc}\hfill {\alpha }_i=& -\frac{z_i^3{\lambda }_i^2}{\sqrt{z_i^6{\lambda }_i^2+{\pi }_i^2}},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\dot{\widehat{\varphi }}}_i=& {\mu }_i{e}^{\eta t}(\frac{z_i^6}{\sqrt{z_i^6{S}_i^T{S}_i+{\pi }_i^2}}-{\pi }_i{\widehat{\varphi }}_i),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\dot{\widehat{\phi }}}_i=& {\nu }_i{e}^{\eta t}(\frac{z_i^6{\parallel {\beta }_i\parallel }^2}{\sqrt{z_i^6{\parallel {\beta }_i\parallel }^2+{\pi }_i^2}}-{\pi }_i{\widehat{\phi }}_i),\hfill \end{array}$$

(33)

through a process similar to (21), and then combining the virtual controller and adaptive law with (30), it follows that:

$$\begin{array}{cc}\hfill L{V}_i\le & -\sum _{j=1}^i{c}_j{z}_j^4-\sum _{j=1}^i\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^i\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2\hfill \\ \hfill & +\sum _{j=1}^i{\pi }_j(2+{\varphi }_j+{\phi }_j)+\sum _{j=1}^i{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)+\frac{1}{4{\underline{h}}_i}{z}_{i+1}^4.\hfill \end{array}$$

(34)

Step n: Because the event-triggered mechanism is introduced to reduce the communication burden, as follows:

$$\begin{array}{cc}\hfill u\left(t\right)=& v\left(t_s\right),\forall t\in [t_s,t_{s+1}),\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill t_{s+1}=& inf\{t\in R|\left|\xi \left(t\right)\right|\ge {\rho }_0\left|u\left(t\right)\right|+{\rho }_1\},\hfill \end{array}$$

(36)

where $0<{\rho }_0<1$ and ${\rho }_1>0$ are the parameters to be designed. $v\left(t\right)$ is the output of actual controller, and $\xi \left(t\right)=v\left(t\right)-u\left(t\right)$. Thus, for the interval, $[t_s,t_{s+1})$, we obtain the following:

$$\begin{array}{c}\hfill v\left(t\right)=(1+{\zeta }_0{\rho }_0)u\left(t\right)+{\zeta }_1{\rho }_1,\forall t\ge 0,\end{array}$$

(37)

where $|{\zeta }_0|\le 1$ and $|{\zeta }_1|\le 1$ are time-varying constants. Then, the following can be derived:

$$\begin{array}{c}\hfill u\left(t\right)=\frac{v-{\zeta }_1{\rho }_1}{1+{\zeta }_0{\rho }_0}.\end{array}$$

(38)

Similar to (23), one obtains the following:

$$\begin{array}{cc}\hfill d{z}_n=& ({\dot{\vartheta }}_n{x}_n+{\vartheta }_n(h_{n}u+g_n+d_n)+{\dot{\iota }}_n-L{\alpha }_{n-1})dt+{({\chi }_n-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}{\chi }_j)}^{T}d\omega ,\hfill \end{array}$$

(39)

where $$\begin{gathered} L{\alpha }_{n-1}={\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}(h_j{x}_{j+1}+g_j+d_j)+{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\widehat{\varphi }}_j}{\dot{\widehat{\varphi }}}_j+{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\widehat{\phi }}_j}{\dot{\widehat{\phi }}}_j+{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\vartheta }_j}{\dot{\vartheta }}_j \\ +{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\iota }_j}{\dot{\iota }}_j+{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\pi }_j}{\dot{\pi }}_j+{\sum }_{j=0}^{n-1}\frac{\partial {\alpha }_{i-1}}{\partial y_d^{\left(j\right)}}{y}_d^{(j+1)}+\frac{1}{2}{\sum }_{p,q=1}^{n-1}\frac{\partial {\alpha }_{n-1}^2}{\partial x_p{x}_q}{\chi }_p^T{\chi }_q. \end{gathered}$$

Choose the following Lyapunov function, as follows:

$$\begin{array}{c}\hfill V_n=V_{n-1}+\frac{1}{4{\underline{h}}_n}{z}_n^4+\frac{1}{2{\mu }_n}{e}^{-\eta t}{\tilde{\varphi }}_n^2+\frac{1}{2{\nu }_n}{e}^{-\eta t}{\tilde{\phi }}_n^2,\end{array}$$

(40)

where ${\nu }_n>0$ and ${\mu }_n>0$. Furthermore, the derivative of $V_n$ is as follows:

$$\begin{array}{cc}\hfill L{V}_n=& L{V}_{n-1}+\frac{z_n^3}{{\underline{h}}_n}({\dot{\vartheta }}_n{x}_n+{\vartheta }_n(h_n\frac{v-{\zeta }_1{\rho }_1}{1+{\zeta }_0{\rho }_0}+g_n+d_n)+{\dot{\iota }}_n-L{\alpha }_{n-1})\hfill \\ \hfill & -\frac{\eta }{2{\mu }_n}{e}^{-\eta t}{\tilde{\varphi }}_n^2-\frac{\eta }{2{\nu }_n}{e}^{-\eta t}{\tilde{\phi }}_n^2+\frac{3}{2{\underline{h}}_n}{z}_n^2{\overline{\chi }}_n^T{\overline{\chi }}_n-\frac{1}{{\mu }_n}{e}^{-\eta t}{\tilde{\varphi }}_n{\dot{\widehat{\varphi }}}_n-\frac{1}{{\nu }_n}{e}^{-\eta t}{\tilde{\phi }}_n{\dot{\widehat{\phi }}}_n,\hfill \end{array}$$

(41)

where ${\overline{\chi }}_n={\chi }_n-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}{\chi }_j$. Due to $|{\zeta }_0|\le 1$ and $|{\zeta }_1|\le 1$, and by applying Lemma 1, we obtain the following:

$$\begin{array}{cc}\hfill \frac{h_n}{{\underline{h}}_n}{\vartheta }_n{z}_n^3|\frac{{\zeta }_1{\rho }_1}{1+{\zeta }_0{\rho }_0}|& \le \frac{h_n}{{\underline{h}}_n}{\left|z_n\right|}^3\frac{{\rho }_1}{1-{\rho }_0}\le \frac{\frac{h_n^2}{{\underline{h}}_n^2}{z}_n^6\frac{{\rho }_1^2}{{(1-{\rho }_0)}^2}}{\sqrt{\frac{h_n^2}{{\underline{h}}_n^2}{z}_n^6\frac{{\rho }_1^2}{{(1-{\rho }_0)}^2}}+{\pi }_n^2}+{\pi }_n,\hfill \\ \hfill \frac{3}{2{\underline{h}}_n}{z}_n^2{\overline{\chi }}_n^T{\overline{\chi }}_n& \le \frac{\frac{9}{4{\underline{h}}_n^2}{z}_n^4{\parallel {\overline{\chi }}_n\parallel }^4}{\sqrt{\frac{9}{4{\underline{h}}_n^2}{z}_n^4{\parallel {\overline{\chi }}_n\parallel }^4+{\pi }_n^2}}+{\pi }_n,\hfill \end{array}$$

(42)

where ${\pi }_n=l_n{e}^{-B_{n}t}$ is a positive parameter.

Substituting (42) into (41), gives the following:

$$\begin{array}{cc}\hfill L{V}_n\le & -\sum _{j=1}^{n-1}{c}_j{z}_j^4-\sum _{j=1}^n\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^n\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^{n-1}{\pi }_j(2+{\varphi }_j+{\phi }_j)\hfill \\ \hfill & +\sum _{j=1}^{n-1}{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)+\frac{z_n^3}{{\underline{h}}_n}(\frac{{\vartheta }_n{h}_{n}v}{1+{\zeta }_0{\rho }_0}+G_n-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}{d}_j-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\pi }_j}{\dot{\pi }}_j)\hfill \\ \hfill & +2{\pi }_n-\frac{1}{{\mu }_n}{e}^{-\eta t}{\tilde{\varphi }}_n{\dot{\widehat{\varphi }}}_n-\frac{1}{{\nu }_n}{e}^{-\eta t}{\tilde{\phi }}_n{\dot{\widehat{\phi }}}_n,\hfill \end{array}$$

(43)

where $G_n\left(Z_n\right)={\dot{\vartheta }}_n{x}_n+{\vartheta }_n(g_n+{\overline{d}}_n)+{\overline{\iota }}_n-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}(h_j{x}_{j+1}+g_j)-{\sum }_{j=1}^{n-1}$ $\frac{\partial {\alpha }_{n-1}}{\partial {\widehat{\varphi }}_j}{\dot{\widehat{\varphi }}}_j-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\widehat{\phi }}_j}{\dot{\widehat{\phi }}}_j-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\vartheta }_j}{\dot{\vartheta }}_j-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\iota }_j}{\dot{\iota }}_j-{\sum }_{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial y_d^{\left(j\right)}}{y}_d^{j+1}-\frac{1}{2}{\sum }_{p,q=1}^{n-1}$ $\frac{\partial {\alpha }_{n-1}^2}{\partial x_p{x}_q}{\chi }_p^T{\chi }_q+\frac{{\underline{h}}_n}{4{\underline{h}}_{n-1}}{z}_n+\frac{\frac{9}{4{\underline{h}}_n}{z}_n{\parallel {\overline{\chi }}_n\parallel }^4}{\sqrt{\frac{9}{4{\underline{h}}_n^2}{z}_n^4{\parallel {\overline{\chi }}_n\parallel }^4+{\pi }_n^2}}+\frac{\frac{h_n^2}{{\underline{h}}_n}{z}_n^3\frac{{\rho }_1^2}{{(1-{\rho }_0)}^2}}{\sqrt{\frac{h_n^2}{{\underline{h}}_n^2}{z}_n^6\frac{{\rho }_1^2}{{(1-{\rho }_0)}^2}}+{\pi }_n^2}=W_n^T{S}_n\left(Z_n\right)+{\epsilon }_n$, with $Z_n={[x_1,\dots ,x_n,{\widehat{\varphi }}_1,\dots ,{\widehat{\varphi }}_n,{\widehat{\phi }}_1,\dots ,{\widehat{\phi }}_n,y_d,\dots ,y_d^{\left(n\right)}]}^T$. Let ${\varphi }_n=\frac{\parallel W_n\parallel }{{\underline{h}}_n}$, because of Lemma 1, as follows:

$$\begin{array}{c}\hfill \frac{z_n^3}{{\underline{h}}_n}{W}_n^T{S}_n\left(Z_n\right)\le |z_n{|}^3{\varphi }_n\parallel S_n\left(Z_n\right)\parallel \le \frac{z_n^6{S}_n^T{S}_n{\varphi }_n}{\sqrt{z_n^6{S}_n^T{S}_n+{\pi }_n^2}}+{\pi }_n{\varphi }_n\le \frac{z_n^6{\varphi }_n}{\sqrt{z_n^6{S}_n^T{s}_n+{\pi }_n^2}}+{\pi }_n{\varphi }_n,\end{array}$$

(44)

denoting that ${\phi }_n={sup}_{t\ge 0}\parallel {\vartheta }_n\parallel$, we obtain the following:

$$\begin{array}{cc}\hfill \frac{z_n^3}{{\underline{h}}_n}({\epsilon }_n-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial x_j}{d}_j-\sum _{j=1}^{n-1}\frac{\partial {\alpha }_{n-1}}{\partial {\pi }_j}{\dot{\pi }}_j)& \le |z_n{|}^3\parallel {\beta }_n\parallel \parallel {\vartheta }_n\parallel \hfill \\ \hfill & \le |z_n{|}^3\parallel {\beta }_n\parallel {\phi }_n\le \frac{z_n^6{\parallel {\beta }_n\parallel }^2{\phi }_n}{\sqrt{z_n^6{\parallel {\beta }_n\parallel }^2+{\pi }_n^2}}+{\pi }_n{\phi }_n,\hfill \end{array}$$

(45)

where ${\beta }_n={[1,\frac{\partial {\alpha }_{n-1}}{\partial x_1},\dots ,\frac{\partial {\alpha }_{n-1}}{\partial x_{n-1}},\frac{\partial {\alpha }_{n-1}}{\partial {\pi }_1},\dots ,\frac{\partial {\alpha }_{n-1}}{\partial {\pi }_{n-1}}]}^T$ and ${\vartheta }_n=\frac{1}{{\underline{h}}_n}[{\epsilon }_n,d_1,\dots ,d_{n-1},{\dot{\pi }}_1,\dots ,{\dot{\pi }}_{n-1}]$.

Substituting (44) and (45) into (43), the following holds:

$$\begin{array}{cc}\hfill L{V}_n\le & -\sum _{j=1}^n{c}_j{z}_j^4-\sum _{j=1}^n\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^n\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^n{\pi }_j(2+{\varphi }_j+{\phi }_j)+\sum _{j=1}^{n-1}{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)\hfill \\ \hfill & +\frac{h_n}{{\underline{h}}_n}\frac{{\vartheta }_n{z}_n^{3}v}{1+{\zeta }_0{\rho }_0}+z_n^3{\lambda }_n+\frac{1}{{\mu }_n}{e}^{-\eta t}{\tilde{\varphi }}_n({\mu }_n{e}^{\eta t}\frac{z_n^6}{\sqrt{z_n^6{S}_n^T{S}_n+{\pi }_n^2}}-{\dot{\widehat{\varphi }}}_n)\hfill \\ \hfill & +\frac{1}{{\nu }_n}{e}^{-\eta t}{\tilde{\phi }}_n({\nu }_n{e}^{\eta t}\frac{z_n^6{\parallel {\beta }_n\parallel }^2}{\sqrt{z_n^6{\parallel {\beta }_n\parallel }^2+{\pi }_n^2}}-{\dot{\widehat{\phi }}}_n),\hfill \end{array}$$

(46)

where ${\lambda }_n=c_n{z}_n+\frac{z_n^3{\widehat{\varphi }}_n}{\sqrt{z_n^6{S}_n^T{S}_n+{\pi }_n^2}}+\frac{z_n^3{\parallel {\beta }_n\parallel }^2{\widehat{\phi }}_n}{\sqrt{z_n^6{\parallel {\beta }_n\parallel }^2+{\pi }_n^2}}$, $c_n>0$.

Then, we design the virtual controller and adaptive laws as follows:

$$\begin{array}{cc}\hfill v=& -(1+{\rho }_0)\frac{z_n^3{\lambda }_n^2}{{\underline{\vartheta }}_n\sqrt{z_n^6{\lambda }_n^2+{\pi }_n^2}},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\dot{\widehat{\varphi }}}_n=& {\mu }_n{e}^{\eta t}(\frac{z_n^6}{\sqrt{z_n^6{S}_n^T{S}_n+{\pi }_n^2}}-{\pi }_n{\widehat{\varphi }}_n),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\dot{\widehat{\phi }}}_n=& {\nu }_n{e}^{\eta t}(\frac{z_n^6{\parallel {\beta }_n\parallel }^2}{\sqrt{z_n^6{\parallel {\beta }_n\parallel }^2+{\pi }_n^2}}-{\pi }_n{\widehat{\phi }}_n),\hfill \end{array}$$

(49)

according to (47), we can obtain $z_n^{3}v<0$, furthermore satisfying the following:

$$\begin{array}{c}\hfill \frac{z_n^{3}v}{1+{\zeta }_0{\rho }_0}\le \frac{z_n^{3}v}{1+{\rho }_0}.\end{array}$$

(50)

Then, combining the virtual controller and adaptive laws with (46), the following inequality holds:

$$\begin{array}{cc}\hfill L{V}_n\le & -\sum _{j=1}^n{c}_j{z}_j^4-\sum _{j=1}^n\frac{\eta }{2{\mu }_j}{e}^{-\eta t}{\tilde{\varphi }}_j^2-\sum _{j=1}^n\frac{\eta }{2{\nu }_j}{e}^{-\eta t}{\tilde{\phi }}_j^2+\sum _{j=1}^n{\pi }_j(\kappa +{\varphi }_j+{\phi }_j)+\sum _{j=1}^n{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j)\hfill \\ \hfill \le & -k{V}_n+\sum _{j=1}^n{\pi }_j(\kappa +{\varphi }_j+{\phi }_j)+\sum _{j=1}^n{\pi }_j({\tilde{\varphi }}_j{\widehat{\varphi }}_j+{\tilde{\phi }}_j{\widehat{\phi }}_j),\hfill \end{array}$$

(51)

where $k=min\{4c_j,\eta \}$ for $j=1,2,\dots ,n$, and

$$\begin{array}{c}\hfill \kappa =\left\{\begin{array}{c}2,j=1,2,\dots ,n-1,\hfill \\ 3,j=n.\hfill \end{array}\right.\end{array}$$

(52)

Due to the following:

$$\begin{array}{cc}\hfill {\tilde{\varphi }}_j{\widehat{\varphi }}_j& =-{\tilde{\varphi }}_j^2+{\tilde{\varphi }}_j{\varphi }_j\le \frac{{\varphi }_j^2}{4},\hfill \\ \hfill {\tilde{\phi }}_j{\widehat{\phi }}_j& =-{\tilde{\phi }}_j^2+{\tilde{\phi }}_j{\phi }_j\le \frac{{\phi }_j^2}{4},\hfill \end{array}$$

(53)

furthermore, one obtains:

$$\begin{array}{cc}\hfill L{V}_n\le & -k{V}_n+\sum _{j=1}^n{\pi }_j(\kappa +{\varphi }_j+{\phi }_j)+\sum _{j=1}^n\frac{{\pi }_j}{4}({\varphi }_j^2+{\phi }_j^2).\hfill \end{array}$$

(54)

Theorem 1.

Under Assumptions 1 and 2, consider the non-strict, feedback, stochastic, nonlinear System (1) with the virtual controllers, (18), (31) and (47), and the adaptive laws, (19), (20), (32), (33), (48) and (49). By selecting the appropriate design parameters, all the closed-loop signals of System (1) are bounded in probability, and the asymptotic convergence is exponentially achieved.

Proof. 

Firstly, decide that $\eta \le 4c_j$, $k=\eta$. Applying $It{\widehat{o}}^{\prime }s$ formula to (54), we obtain the following:

$$\begin{array}{cc}\hfill d\left(V_n{e}^{\eta t}\right)\le & e^{\eta t}(\sum _{j=1}^n{\pi }_j(\kappa +{\varphi }_j+{\phi }_j)+\sum _{j=1}^n\frac{{\pi }_j}{4}({\varphi }_j^2+{\phi }_j^2))dt+e^{\eta t}\sum _{j=1}^n\frac{\partial V}{\partial x_j}{\chi }_{j}d\omega .\hfill \end{array}$$

(55)

let ${\pi }_j=l_j{e}^{-B_{j}t}$, integrating (55) over $(0,t)$, we obtain the following:

$$\begin{array}{cc}\hfill E\left[V_n\left(t\right)\right]\le & e^{-\eta t}[V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2}{4}{\int }_0^t{e}^{-(B_j-\eta )\tau }d\tau \hfill \\ \hfill & +\sum _{j=1}^n(\kappa +{\varphi }_j+{\phi }_j){\int }_0^t{e}^{-(B_j-\eta )\tau }d\tau \hfill \\ \hfill & +\sum _{j=1}^n\frac{\partial V}{\partial x_j}{\chi }_j{\int }_0^t{e}^{-\eta (t-\tau )}d\omega \left(\tau \right)],\hfill \end{array}$$

(56)

beacause the stochastic integral is local martingale, we obtain the following:

$$\begin{array}{cc}\hfill E\left[V_n\left(t\right)\right]\le & e^{-\eta t}[V_n\left(0\right)+\sum _{j=1}^n\frac{1}{4}\frac{{\varphi }_j^2+{\phi }_j^2}{B_j-\eta }(1-e^{-(B_j-\eta )t})\hfill \\ \hfill & +\sum _{j=1}^n(\kappa +{\varphi }_j+{\phi }_j)\frac{1}{B_j-\eta }(1-e^{-(B_j-\eta )t})],\hfill \end{array}$$

(57)

then, we define $b_j>\eta$, and obtain the following:

$$\begin{array}{cc}\hfill E\left[V_n\left(t\right)\right]\le & e^{-\eta t}(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )}).\hfill \end{array}$$

(58)

Furthermore, we obtain the following:

$$\begin{array}{c}\hfill E[\frac{z_i^4}{4{\underline{h}}_i}]\le e^{-\eta t}(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )}),\\ \hfill \frac{1}{2{\mu }_i}{e}^{-\eta t}{\tilde{\varphi }}_i^2\le e^{-\eta t}(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )}),\\ \hfill \frac{1}{2{\nu }_i}{e}^{-\eta t}{\tilde{\phi }}_i^2\le e^{-\eta t}(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )}),\end{array}$$

(59)

moreover, we obtain the following:

$$\begin{array}{cc}\hfill E\left[\right|z_i\left|\right]\le & e^{-\frac{\eta t}{4}}{[4{\underline{h}}_i(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )})]}^{\frac{1}{4}},\hfill \\ \hfill |{\tilde{\varphi }}_i|\le & {[2{\mu }_i(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )})]}^{\frac{1}{2}},\hfill \\ \hfill |{\tilde{\phi }}_i|\le & {[2{\nu }_i(V_n\left(0\right)+\sum _{j=1}^n\frac{{\varphi }_j^2+{\phi }_j^2+4(\kappa +{\varphi }_j+{\phi }_j)}{4(B_j-\eta )})]}^{\frac{1}{2}}.\hfill \end{array}$$

(60)

Thus, we can know that $z_i$ can exponentially, asymptotically converge to zero, and ${\tilde{\varphi }}_i$ and ${\tilde{\phi }}_i$ are bounded. Furthermore we can obtain that ${\widehat{\varphi }}_j$ is bounded from ${\widehat{\varphi }}_i={\varphi }_i-{\tilde{\varphi }}_i$, and ${\varphi }_i$ is a positive constant. In the same way, the boundedness of ${\phi }_i$ can be obtained. Then, we can ensure that ${\alpha }_i$ is bounded. In addition, according to (9), it can also know that $x_i$ is bounded. □

4. Simulation Example

Consider the following second-order, non-strict feedback, stochastic nonlinear system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}d{x}_1=(x_1^{2}sin\left(x_2\right)+(1.5+0.5sin\left(x_1\right))x_2+d_1\left(t\right))dt+0.5cos\left(x_1\right)d\omega ,\hfill \\ d{x}_2=((1.5+sin\left(x_1{x}_2\right))u+x_1^2{x}_2{e}^{x_2}+x_{1}cos\left(x_1{x}_2\right)+d_2\left(t\right))dt+0.2sin\left(x_2\right)d\omega ,\hfill \end{array}\right.\end{array}$$

(61)

where $d_1=0.1sin\left(t\right)$, $d_2=0.01cos\left(t\right)$ are the external disturbance.

We decide that $y_d=sin\left(2t\right)+cos\left(2t\right)$, $x_1^E=x_1+e^{-2t}$ and $x_2^E=x_2+e^{-2t}$. The design parameters of controllers and adaptive laws are selected as $c_1=20$, $c_2=10$, ${\mu }_1=5$, ${\mu }_2=1$, ${\nu }_1=1$, ${\nu }_2=2$, $\eta =0.2$, ${\pi }_1=30e^{-2.2t}$, ${\pi }_2=10e^{-0.5t}$, and ${\underline{\vartheta }}_2=0.5$. Select the initial system status as $x_1\left(0\right)=0.8$, $x_2\left(0\right)=-0.3$, ${\widehat{\varphi }}_1\left(0\right)=9$, ${\widehat{\varphi }}_2\left(0\right)=4$, and ${\widehat{\phi }}_1\left(0\right)={\widehat{\phi }}_2\left(0\right)=5$. The design parameters of the event-triggered mechanism were selected as ${\rho }_0=0.5$, ${\rho }_1=1$.

The simulation results are shown in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5, which demonstrate the statements in Theorem 1 have been verified. Figure 1 plots the system output signal, y, and the reference signal, $y_d$, from which we can see that the output, y, will track $y_d$ accurately at 9 s. Figure 2 verifies that the tracking error can exponentially, asymptotically converge to zero. The control law v and the actual input signal u generated after the event-triggered mechanism are given in Figure 3. Figure 4 shows the adaptive laws. The boundedness of the closed-loop signals can be seen from Figure 1, Figure 2, Figure 3 and Figure 4. Figure 5 shows the event trigger interval diagrams under the event-triggered control (ETC) mechanism. Setting a fixed step size of 0.001 s under the time-triggered control (TTC) mechanism, we obtain that the number of triggers is 10,000 in 10 s. It can be seen that compared to a fixed period TTC, the ETC mechanism used in this paper allows for a longer trigger period and reduces the communication burden on the system.

In order to demonstrate the superiority of the event triggering mechanism, we present the quantitative analysis of the data from the experimental results in

Table 1. Quantitative comparison between the literature [42] and this paper.

Control Method	Trigger Num	IAE	ITAE
TTC method	10,000	1.9659	3.5466
ETC method	6087	1.9468	3.4717

, where the integrated absolute error (IAE) and integrated time absolute error (ITAE), defined as IAE $={\int }_0^t\left|z_1\left(s\right)\right|ds$ and ITAE $={\int }_0^{t}t\left|z_1\left(s\right)\right|ds$, respectively, are used to judge the tracking performance. As evident from Table 1 (with a fixed step size of 0.001 s), the number of triggers using the ETC method is considerably less than using the TTC method when achieving the same system performance metrics IAE and ITAE.

5. Conclusions

This paper has investigated an adaptive neural tracking control strategy, combining the event-triggered mechanism and sensor fault to study the problem of exponentially asymptotic tracking for a class of uncertain, non-strict feedback systems. A new Lyapunov function with negative exponential function is designed to address the problem of asymptotic tracking. Event-triggered mechanism is introduced to reduce the communication resource usage in the system. The proposed strategy ensures that the tracking error converges exponentially to zero, and the signals in the closed-loop system are bounded. Finally, a simulation example is used to verify the feasibility of the proposed control strategy. In future research, to obtain a better control performance, some novel event-triggered mechanisms can be employed to solve the exponential, asymptotic tracking control problem of the stochastic nonlinear system, such as the dynamic event-triggered mechanism and the memory event-triggered mechanism.