Event-Triggered Control for SNSs with Distributed Time-Varying Delays and Output Dead Zone

Abstract

This paper addresses the tracking control problem for stochastic nonlinear systems (SNSs) subject to distributed time-varying delays and output dead zones. A novel dynamic event-triggered control scheme is proposed by integrating the backstepping technique with a fuzzy logic system (FLS). The FLS is employed to approximate unknown nonlinear functions, while a Nussbaum-type function is incorporated to mitigate the effects of the output dead zone. The challenges posed by distributed time-varying delays are effectively overcome by constructing novel double-integral Lyapunov–Krasovskii functionals. Furthermore, the introduced dynamic event-triggering mechanism, which features a relative threshold and an adaptive parameter, significantly reduces the network communication burden while maintaining desired system performance. Based on Lyapunov stability theory, it is rigorously proven that all signals in the resulting closed-loop system are semi-globally uniformly ultimately bounded, and the tracking error converges to a small neighborhood of the origin. Simulation results are provided to validate the feasibility and effectiveness of the proposed control approach.

1. Introduction

In practical applications, researchers have found that traditional control methods require extensive computations and the transmission of redundant information. This not only increases communication overhead but also tends to cause network congestion [1]. Such limitations make it difficult for traditional methods to meet the dual requirements of modern control systems for communication efficiency and system performance. Therefore, event-triggered mechanisms (ETMs) have gradually become an effective approach to address this challenge.

Before the emergence of ETMs, time-triggered mechanisms (TTMs) were the mainstream control solutions. Studies have shown [2] that compared with the fixed-period triggering method, event-triggered mechanisms can flexibly execute control operations based on specific system events, significantly improving resource utilization efficiency. Reference [3] provides a comprehensive review of the research progress in DETC and estimation for networked systems. The sliding mode control (SMC) designed under the event-triggered framework also exhibits outstanding performance when dealing with uncertain systems [4]. These advantages have promoted the vigorous development of event-triggered control (ETC) research in various fields [5,6,7,8,9].

Most of the aforementioned studies are extensions of event-triggered control (ETC) in nonlinear systems conducted in recent years. This indicates that compared with linear systems, the research on nonlinear systems has become an important supplementary direction in this field. However, many researchers have pointed out that although the research on nonlinear systems can already explain the dynamic behaviors of complex systems, the research framework of stochastic nonlinear systems (SNSs) describes dynamic characteristics more accurately in practical engineering fields such as robotics, aerospace, and power systems [10,11,12]. Such systems generally exhibit complex characteristics, including stochastic disturbances, nonlinear properties, and time-delay effects [13,14,15,16,17,18], and their multi-factor coupling nature significantly increases the difficulty of control design. Reflecting the vitality of this field, recent research has expanded into diverse advanced methodologies for SNSs, including optimal regulation for systems with specific nonlinearities [19], adaptive fuzzy control with nonmonotonic performance guarantees under unknown control directions [20], and fuzzy covariance control using interval type-2 models to handle uncertainties [21].

For time-delay systems, the primary challenge encountered in the controller design process is how to handle the unknown time-delay terms in the system. Meanwhile, the method based on the Lyapunov–Krasovskii functional exhibits unique advantages in addressing constant or discrete time-varying delays in nonlinear systems [22,23,24,25]. This functional construction approach can effectively analyze the stability of time-varying delay systems. For instance, the core idea of References [26,27] is to propose a Lyapunov-Krasovskii functional suitable for time-delayed T-S fuzzy systems.

However, the non-smooth characteristics commonly found in control engineering further increase system complexity. Phenomena such as saturation [28] and dead zone [29] can severely degrade system performance and even lead to instability. Particularly for the dead zone effect, which is widely present in engineering practice, the dead zone slope modeling method proposed in Reference [30] effectively compensates for the impact of input dead zones. Nevertheless, most existing studies focus on input dead zone compensation, while research on output dead zones is still in its preliminary stage.

Distributed time-varying delays, common in networked and biomedical systems, not only degrade tracking performance but also significantly complicate system modeling and analysis [31]. While Lyapunov–Krasovskii functionals offer a fundamental approach for stability analysis under delays, and contemporary strategies like multilayer neurocontrol [32] and state-filtered disturbance rejection [33] provide potent tools for handling complex uncertainties, their application remains notably absent for stochastic nonlinear systems (SNSs) concurrently plagued by both distributed time-varying delays and output dead zones. This gap necessitates an integrated control framework. It is crucial to clarify that, although tools such as FLS, Nussbaum functions, and Lyapunov–Krasovskii (L-K) functionals are individually prevalent, the novelty of this work stems from their non-trivial synthesis to address this specific compounded challenge. The key technical distinctions include the construction of a suitable double-integral L-K functional to handle distributed delays with unknown bounds, which differs from the standard or single-integral forms commonly used for discrete delays [25,26,27], and the co-design of a dynamic event-triggered mechanism (ETM) featuring a relative threshold, which is intricately coupled with the Nussbaum-based dead-zone compensator within the backstepping stability analysis, ensuring performance under intermittent updates—a complexity not addressed in conventional dead-zone compensation schemes [34,35,36] or standard ETM designs [28].

Based on the analysis of the aforementioned research status, the core contributions of this paper are threefold:

(i) An SNS model incorporating output dead zones and distributed time-varying delays is constructed. Compared to works like [37,38] which consider input dead zones or simpler delay types, this model better reflects practical scenarios where sensor nonlinearities and distributed delays coexist.

(ii) An improved double-integral Lyapunov–Krasovskii functional is adopted to handle distributed time-varying delays with unknown bounds. This novel construction, distinct from the standard forms used for discrete delays in [39,40], eliminates the a priori dependence on the delay upper bound and effectively suppresses phase lag.

(iii) We design an adaptive event-triggered controller based on a relative threshold. Unlike the multi-parameter architectures in [39,40], it requires online adjustment of only a single parameter, significantly reducing overhead. Crucially, it jointly addresses both output dead-zone compensation and stabilization under distributed delays—a co-design challenge not tackled in the aforementioned references.”

2. Preliminaries

2.1. System Design and Preparatory Knowledge

The following SNSs with output dead zone and distributed time-varying delay is investigated:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{x}_1=\left[{\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\left(s\right)\right)\mathrm{d}s+x_2\left(t\right)\right]\mathrm{d}t+h_1\left(x_1\right)\mathrm{d}\omega \hfill \\ \mathrm{d}{x}_j=\left[{\int }_{t-d_j\left(t\right)}^t{f}_j\left({\overline{x}}_j\left(s\right)\right)\mathrm{d}s+x_{j+1}\left(t\right)\right]\mathrm{d}t+h_j\left({\overline{x}}_j\right)\mathrm{d}\omega (2\le j\le n-1)\hfill \\ \mathrm{d}{x}_n=\left[{\int }_{t-d_n\left(t\right)}^t{f}_n\left({\overline{x}}_n\left(s\right)\right)\mathrm{d}s+u\right]\mathrm{d}t+h_n\left({\overline{x}}_n\right)\mathrm{d}\omega \hfill \\ y=\mathrm{\Delta }\left(x_1\right)\hfill \end{array}\right.\end{array}$$

(1)

The system’s state vector is defined as ${\overline{x}}_j={[x_1,\dots ,x_j]}^{\mathsf{T}}\in {\mathbb{R}}^j$, where $y\in \mathbb{R}$ is the measurable output and $u\in \mathbb{R}$ is the measurable control input. The dynamics involve unknown smooth nonlinear mappings $f_j(·)\in \mathbb{R}(j=2,\dots ,n-1)$ and $h_j(·)\in \mathbb{R}(j=2,\dots ,n-1)$ along with time-varying delays $d_j\left(t\right)$ affecting each state $x_j$. The system is subject to stochastic disturbances modeled by a standard Brownian motion $\omega$, and $\omega$ can effectively characterize the ubiquitous Gaussian random disturbance (e.g., grid load fluctuations [41], mechanical vibration [42]) in industrial environments. The nonlinear function $h_j(·)$ describes the physical correlation between the intensity of the perturbation and the state of the system: In power systems, $h_j\left({\overline{x}}_j\right)$ reflects the nonlinear relationship between load fluctuations and current, and in motion control, it characterizes the random component of velocity-dependent friction.

The ultimate goal of this study is to design a dynamic event-triggered controller for the proposed system so that the system output y is able to track a specified reference trajectory $y_d$, make sure that the tracking error probability converges to a small neighborhood, and ensure that Zeno’s behavior doesn’t occur in the closed-loop system. The assumptions, theorems and definitions required for the study are given below.

Assumption 1 

([34]). $y_d$ and ${\dot{y}}_d$ are bounded and known.

Assumption 2. 

When $1\le j\le n$, the function is $0\le \left|f_j\right|\le m_j$.

Assumption 3 

([42]). When $1\le j\le n$, $0\le d_j\left(t\right)\le d$, and the first order derivative of $d_j\left(t\right)$ satisfies ${\dot{d}}_j\left(t\right)\le d^{*}<1$, where both d and $d^{*}$ are unknown positive parameters.

Lemma 1 

([34]). Based on the characterization of the $\tanh (·)$ function, it follows that

$$\begin{array}{c}\hfill 0\le U·sgn\left(U\right)-Utanh\left({\displaystyle \frac{U}{\eta }}\right)\le 0.2785\eta ,\end{array}$$

(2)

when $sgn\left(U\right)$ is the sign function, $U\in R,\eta >0$.

Lemma 2 

([34]). For smooth functions $V\left(t\right)$, $d_0\ge 0$ and $d_1\ge 0$, if the inequality

$$\begin{array}{c}\hfill EV\left(t\right)\le d_0+e^{-d_{1}t}{\int }_0^t(1+N\left(\psi \right)g\left(t\right))\dot{\psi }\left(\tau \right)e^{d_1\tau }d\tau ,\end{array}$$

(3)

is satisfied, then $g\left(t\right)$ is a bounded function, ${\int }_0^t(1+N\left(\psi \right)g\left(t\right))\dot{\psi }d\tau$, $\psi \left(t\right)$ and $V\left(t\right)\in \left[0,t_f\right)$.

Definition 1 

([34]). If a continuous function $N\left(\psi \right)$ is called a Nussbaum-type function, it must satisfy the following equation

$$\begin{array}{c}\hfill \underset{q\to \infty }{lim}inf{\displaystyle \frac{1}{q}}{\int }_0^{q}N\left(\psi \right)d\psi =-\infty ,\underset{q\to \infty }{lim}sup{\displaystyle \frac{1}{q}}{\int }_0^{q}N\left(\psi \right)d\psi =+\infty ,\end{array}$$

(4)

where $N\left(\psi \right):R\to R$

Lemma 3 

([43]). For any non-negative real numbers $x, y\ge 0$ and exponents $m, n>1$, satisfying ${\displaystyle \frac{1}{m}}+{\displaystyle \frac{1}{n}}=1$:

$$\begin{array}{c}\hfill xy\le {\displaystyle \frac{p^n}{n}}{\left|x\right|}^n+{\displaystyle \frac{1}{m{p}^m}}{\left|y\right|}^m,\end{array}$$

(5)

2.2. Output Dead Zone

The model of the output dead zone required in this paper is as follows [28]:

$$\begin{array}{c}\hfill y=\mathrm{\Delta }\left(x_1\right)=\left\{\begin{array}{cc}{\mathrm{c}}_r\left(x_1-d_r\right),\hfill & x_1>d_r\hfill \\ 0,\hfill & d_l\le x_1\le d_r\hfill \\ c_l\left(x_1-d_l\right),\hfill & x_1<d_l,\hfill \end{array}\right.\end{array}$$

(6)

where $c_r>0$ and $c_l>0$, $d_r>0$ and $d_l<0$ are the transition slope of the dead zone and the width parameters, respectively. The dead zone can also be expressed in the following form [37]:

$$\begin{array}{c}\hfill x_1={\mathrm{\Delta }}^{-1}\left(y\right)={\displaystyle \frac{y+c_r{d}_r}{c_r}}{m}_r\left(y\right)+{\displaystyle \frac{y+c_l{d}_l}{c_l}}{m}_l\left(y\right),\end{array}$$

(7)

where $\kappa >0$, $m_r\left(y\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{arctan\left(\kappa y\right)}{\pi }}$ and $m_l\left(y\right)={\displaystyle \frac{1}{2}}-{\displaystyle \frac{arctan\left(\kappa y\right)}{\pi }}$.

Remark 1 

([35]). Equation (6) is non-differentiable at its sharp corners, which is undesirable for practical controller design. To address this, we replace the dead-zone model with the differentiable Equation (7). This reformulation significantly facilitates the backstepping procedure. Differentiating $x_1$ yields

$$\begin{array}{c}\hfill \mathrm{d}y={\displaystyle \frac{\mathrm{d}y}{\mathrm{d}{x}_1}}\mathrm{d}{x}_1=L\mathrm{d}{x}_1,\end{array}$$

(8)

L is the time-varying coefficient and satisfies $0<L<c_m=max\left(c_r, c_l\right)$.

2.3. FLSs

The following fuzzy rules are used to design an adaptive controller that satisfies the requirements of this paper [36]

$$R^l:\mathrm{if}{x}_1\mathrm{is}{F}_1^l\mathrm{and}.....\mathrm{and}{x}_n\mathrm{is}{F}_n^l,\mathrm{then}y\mathrm{is}{G}^l, l=1,2,\dots ,Q$$

where $x={\left[x_1,\dots ,x_n\right]}^{\mathsf{T}}$ and are the inputs to the fuzzy system y is the output. The affiliation functions of the fuzzy sets $F_j^l$ and $G^l$ are ${\mu }_{F_j^l}\left(x_j\right)$ and ${\mu }_{G^{\prime }}\left(y\right)$, respectively. Q is the number of fuzzy rules. The FLS incorporates four key components: a central average defuzzifier, a product-based inference mechanism, a singleton fuzzifier, and a Gaussian membership function expressed as follows [36]:

$$\begin{array}{c}\hfill y\left(x\right)={\displaystyle \frac{{\sum }_{l=1}^Q{\mathrm{\Phi }}_l{\prod }_{j=1}^n{\mu }_{F_j^{\prime }}\left(x_j\right)}{{\sum }_{l=1}^Q{\prod }_{j=1}^n{\mu }_{F_j^{\prime }}\left(x_j\right)}}\end{array}$$

(9)

where $x=\left[x_1,\dots ,x_n\right]\mathsf{T}$ and $W_l=\arg {\sup }_{y\in R}{\mu }_{F_j^l}\left(x_j\right)$. Define $S\left(x\right)=\left[S_1,\dots ,S_Q\left(x\right)\right]\mathsf{T}$ and $W=\left[W_1,\dots ,W_Q\right]\mathsf{T}$ the fuzzy basis function $S_l$ is defined as $S_l\left(x\right)={\displaystyle \frac{{\prod }_{j=1}^n{\mu }_{F_j^{\prime }}\left(x_j\right)}{{\sum }_{l=1}^Q{\prod }_{j=1}^n{\mu }_{F_j^{\prime }}\left(x_j\right)}}$. Furthermore, the FLS can be expressed in the following form: $y\left(x\right)=W\mathsf{T}S\left(x\right)$. Choose the Gaussian function as the membership function ${\mu }_{F_i^{\prime }}\left(x_i\right)=exp\left(-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x_j-a_j^l}{{\sigma }_j^l}}\right)}^2\right)$ Herein, ${\sigma }_j^l$ and $a_j^l$ represent the width and center of the Gaussian function. Any continuous function defined on a tight set can be approximated by the above FLS with arbitrary accuracy.

Definition 2 

([34]). The FLS applies the following equation when approximating continuous functions:

$$\begin{array}{c}\hfill \phi \left(Z\right)=W\mathsf{T}\left(Z\right)S\left(Z\right)+\delta \left(Z\right),\forall Z\in {\mathrm{\Omega }}_Z\end{array}$$

(10)

where $\delta \left(Z\right)$ is the approximation error, and there exists an unknown positive constant $\overline{\delta }$ that satisfies $\parallel \delta \parallel \le \overline{\delta }$. In this paper, $\theta =max\left\{{∥W_j∥}^2,j=1,\dots ,Q\right\}$, denote the parameter error by $\tilde{\theta }=\widehat{\theta }-\theta$, and $\widehat{\theta }$ is an unknown constant $\overline{\delta }$. $\widehat{\theta }$ is the estimated value of the unknown constant θ.

For the system without modeling the dynamic $f_j(·)$, FLS has the universal approximation characteristic. Compared with traditional radial basis function neural network (RBFNN), fuzzy rules provide interpretable knowledge representation. The product reasoning mechanism of (9) can effectively deal with multivariate coupling.

Remark 2. 

The FLS holds the universal approximation property on compact sets. The approximation error $\delta \left(Z\right)$ in (10) is bounded, and the adaptive law (53) for $\widehat{\theta }$ is designed to accommodate this bounded uncertainty, ensuring system robustness.

2.4. Stochastic Process Theory

Consider the SNSs:

$$\begin{array}{c}\hfill \mathrm{d}x=f\left(x\right)\mathrm{d}t+h\left(x\right)\mathrm{d}\omega \end{array}$$

(11)

where $f\left(x\right)$ and $h\left(x\right)$ are locally Lipschitz functions with $f\left(0\right)=h\left(0\right)=0$, $\omega$ is a standard Brownian motion, and x is state vector.

Definition 3 

([40]). For any given $V\left(x\right)\in C^2$, its differential operator $LV(x,t)\in C^2$ with respect to Equation (11) is defined as follows:

$$\begin{array}{c}\hfill LV(x,t)={\displaystyle \frac{\mathit{\partial }V(x,t)}{\mathit{\partial }t}}+{\displaystyle \frac{\mathit{\partial }V(x,t)}{\mathit{\partial }x}}f\left(x\right)+{\displaystyle \frac{1}{2}}trace\left(h^{\mathsf{T}}\left(x\right){\displaystyle \frac{{\mathit{\partial }}^{2}V(x,t)}{\mathit{\partial }{x}^2}}h\left(x\right)\right).\end{array}$$

(12)

Lemma 4 

([39]). For Equation (11), if there exists a Lyapunov function $V\left(x\right)$, such that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\varphi }_1\left(\right|x\left|\right)\le V(x,t)\le {\varphi }_2\left(\right|x\left|\right),\hfill \\ LV(x,t)\le -CV(x,t)+\rho \hfill \end{array}\right.\end{array}$$

(13)

where $\rho ,C>0$ are constants, ${\varphi }_1$, ${\varphi }_2$ are functions of $k_{\infty }$, and the system Equation (11) possesses a strong solution that satisfies:

$$\begin{array}{c}\hfill E\left(V(x,t)\right)\le V\left(0\right)e^{-Ct}+{\displaystyle \frac{\rho }{C}}\end{array}$$

(14)

3. Controller Design

Relative threshold ETC strategy [44], FLS and backstepping techniques are used to design the controller based on the following coordinate transformations. Firstly, the following error surfaces are constructed:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{z}_1=y-y_d\hfill \\ z_j=x_j-x_{j,d},j=2,\dots ,n\hfill \end{array}\right.\end{array}$$

(15)

where $y_d$ is the reference signal. The DSC technique is defined as follows [44]:

$$\begin{array}{c}\hfill {\zeta }_i{\dot{x}}_{i,d}={\alpha }_{i-1}-x_{i,d},{\alpha }_{i-1}\left(0\right)=x_{i,d}\left(0\right)\end{array}$$

(16)

The intermediate controllers ${\alpha }_{i-1}(i=2,\dots ,n)$ and positive parameters ${\zeta }_i,i=2,\dots ,n$ are defined. Since the filtering error ${\alpha }_{i-1}-x_{i,d}$ directly affects the controller performance, a compensation signal $q_j$ is introduced to eliminate the error. The definition of $q_j$ is given below [34]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{q}}_1=-c_1{q}_1+L\left(x_{2,d}-{\alpha }_1\right)+L{q}_2-l_{1}sign\left(q_1\right)\hfill \\ {\dot{q}}_j=-c_j{q}_j+\left(x_{j+1,d}-{\alpha }_j\right)+q_{j+1}-l_{j}sign\left(q_j\right)\hfill \\ {\dot{q}}_n=-c_n{q}_n-l_{n}sign\left(q_n\right)\hfill \end{array}\right.\end{array}$$

(17)

where $l_j(j=1,\dots ,n)$ are the design parameters, and $q_j(j=1,\dots ,n)$ serves as the state parameter for the compensation mechanism. The compensation tracking error signals are defined as ${\vartheta }_j=z_j-q_j, j=1,\dots ,n$.

Remark 3. 

The compensation signals $q_j$ in (17) are designed to actively cancel the filtering error ${\xi }_j=x_{j,d}-{\alpha }_{j-1}$ induced by the DSC filter (16). The term $-c_j{q}_j$ ensures stable compensation dynamics, while the inclusion of $q_{j+1}$ propagates the compensation effect forward through the backstepping procedure under the event-triggered framework.

Subsequently, we provide the detailed procedure for designing a controller on the basis of the backstepping technique:

• Step 1: (Purpose: define the tracking error $z_1$ and filter the virtual control to prevent peaking caused by fast dynamics and noisy derivatives. The compensation state $q_1$ cancels the filtering mismatch ${\alpha }_1-x_{1,d}$.)
• According to ${\vartheta }_1=z_1-q_1$, $z_1=y-y_d$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{d}{\vartheta }_1& =\mathrm{d}{z}_1-\mathrm{d}{q}_1=\mathrm{d}y-\mathrm{d}{y}_{\mathrm{d}}-\mathrm{d}{q}_1=L\mathrm{d}{x}_1-\mathrm{d}{y}_{\mathrm{d}}-d{q}_1\hfill \\ & =L\left[{\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\left(s\right)\right)\mathrm{d}s\right]+L{x}_2\left(t\right)\mathrm{d}t-{\dot{y}}_d\mathrm{d}t-{\dot{q}}_1\mathrm{d}t+L{h}_1^{\mathsf{T}}\left(x_1\right)\mathrm{d}\omega \hfill \end{array}\end{array}$$

(18)

Then, on the basis of $z_2=x_2-x_{2,d}$, ${\vartheta }_2=z_2-q_2$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathrm{d}{\vartheta }_1& =\left[L\left({\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\left(s\right)\right)\mathrm{d}s\right)+L{\vartheta }_2+L{q}_2+L{x}_{2,d}-{\dot{y}}_d-{\dot{q}}_1\right]\mathrm{d}t\hfill \\ & +L{h}_1^{\mathsf{T}}\left(x_1\right)\mathrm{d}\omega \hfill \end{array}\end{array}$$

(19)

Considering Lyapunov functions

$$\begin{array}{c}\hfill V_{{\vartheta }_1}={\displaystyle \frac{1}{4}}{\vartheta }_1^4+{\mathrm{\Lambda }}_1\end{array}$$

(20)

where ${\mathrm{\Lambda }}_1={\displaystyle \frac{{\lambda }_1{e}^{-\gamma (t-d)}}{2\left(1-d^{*}\right)}}d{\int }_{t-d_1\left(t\right)}^t{\int }_{\tau }^t{e}^{rs}{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s\mathrm{d}\tau$, ${\lambda }_1>0$ is a design parameterand, and in this study, ${\mathrm{\Lambda }}_1>0$ always holds. Differentiate $V_{g_1}$ and combine (17) and (20) to obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& ={\vartheta }_1^3\left[L\left({\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\left(s\right)\right)\mathrm{d}s\right)+L{\vartheta }_2+L{q}_2+L{x}_{2,d}-{\dot{y}}_d-{\dot{q}}_1\right]\hfill \\ & -{\displaystyle \frac{{\lambda }_1^2\left(1-{\dot{d}}_1\left(t\right)\right)}{2\left(1-d^{*}\right)}}·d{\int }_{t-d_1\left(t\right)}^t{e}^{-r(t-d-s)}{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{3}{2}}{\vartheta }_1^{2}L{h}_1^{\mathsf{T}}{h}_1\hfill \\ & +{\displaystyle \frac{{\lambda }_1^2{d}_1\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}·f_1^2\left(x_1\right)-\gamma {\mathrm{\Lambda }}_1\hfill \end{array}\end{array}$$

(21)

Substituting the compensation signal Equation (17) (−${\dot{q}}_1=c_1{q}_1-L\left(x_{2,d}-{\alpha }_1\right)-L{q}_2+l_{1}sign\left(q_1\right)$) into Equation (21) ($-{\dot{q}}_1$) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& ={\vartheta }_1^3\left[L\left({\int }_{t-d_1\left(t\right)}^t{f}_1\left({\overline{x}}_1\left(s\right)\right)\mathrm{d}s\right)+c_1{q}_1+L{\alpha }_1+l_{1}sign\left(q_1\right)-{\dot{y}}_d\right]\hfill \\ & -{\displaystyle \frac{{\lambda }_1^2\left(1-d_1\left(t\right)\right)}{2\left(1-d^{*}\right)}}·d{\int }_{t-d_1\left(t\right)}^t{e}^{-r(t-d-s)}\left.f_1^2(x_1\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{3}{2}}{\vartheta }^{2}L{h}_1^{\top }{h}_1\hfill \\ & +L{\vartheta }_1^3{\vartheta }_2+{\displaystyle \frac{{\lambda }_1^2{d}_1\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}{f}_1^2\left(x_1\right)-\gamma {\mathrm{\Lambda }}_1\hfill \end{array}\end{array}$$

(22)

By using Young’s inequality, one has

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{{\lambda }_1^2{d}_1\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}{f}_1^2\left(x_1\right)\le {\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}_1\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}} ,{\vartheta }_1^3{l}_{1}sign\left(q_1\right)\le {\displaystyle \frac{3}{4}}{\vartheta }_1^4+{\displaystyle \frac{1}{4}}{l}_1^4,\hfill \\ & -{\displaystyle \frac{{\lambda }_1^2\left(1-{\dot{d}}_1\left(t\right)\right)}{2\left(1-d^{*}\right)}}d{\int }_{t-d_1\left(t\right)}^t{e}^{-r(t-d-s)}{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s\hfill \\ & \le -{\displaystyle \frac{{\lambda }_1^{2}d}{2}}{\int }_{t-d_1\left(t\right)}^t{e}^{-r(t-d-s)}{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s,\hfill \\ & {\vartheta }_1^3{\vartheta }_{2}L\le {\displaystyle \frac{3}{4}}{c}_m^{{\displaystyle \frac{4}{3}}}{\vartheta }_1^4+{\displaystyle \frac{1}{4}}{\vartheta }_2^4,{\displaystyle \frac{3}{2}}{\vartheta }_1^{2}L{h}_1^{\mathsf{T}}{h}_1\le {\displaystyle \frac{3}{4}}{\displaystyle \frac{c_m^2{\vartheta }^4{∥h_1∥}^4}{{\lambda }_1^2}}+{\displaystyle \frac{3}{4}}{\lambda }_1^2,\hfill \\ & {\vartheta }_1^{3}L\left({\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\left(s\right)\right)\mathrm{d}s\right)\le {\displaystyle \frac{{\lambda }_1^2}{2}}{\int }_{t-d_1\left(t\right)}^t{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{{\vartheta }_1^6{c}_m^2{d}_1\left(t\right)}{2{\lambda }_1^2}},\hfill \\ & -{\displaystyle \frac{{\lambda }_1^{2}d}{2}}{\int }_{t-d_1\left(t\right)}^t{e}^{-r(t-d-s)}{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s\le -{\displaystyle \frac{{\lambda }_1^2}{2}}{\int }_{t-d_1\left(t\right)}^t{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s,\hfill \\ & d-d_1\left(t\right)\ge 0,{\displaystyle \frac{1-d_1\left(t\right)}{1-d^{*}}}\ge 1,{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}_1\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}\le {\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}},\hfill \end{array}\end{array}$$

(23)

Then, substituting Young’s inequality into Equation (22) yields that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& \le {\displaystyle \frac{{\lambda }_1^2}{2}}{\int }_{t-d_1\left(t\right)}^t{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{{\vartheta }_1^6{c}_m^2{d}_1\left(t\right)}{2{\lambda }_1^2}}+{\vartheta }_1^3\left(c_1{q}_1+L{\alpha }_1-{\dot{y}}_d\right)\hfill \\ & +{\displaystyle \frac{3}{4}}{\vartheta }_1^4+{\displaystyle \frac{1}{4}}{l}_1^4-{\displaystyle \frac{{\lambda }_1^2}{2}}{\int }_{t-d_1\left(t\right)}^t{f}_1^2\left(x_1\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{1}{4}}{\vartheta }_2^4-\gamma {\mathrm{\Lambda }}_1+{\displaystyle \frac{3}{4}}{\lambda }_1^2\hfill \\ & +{\displaystyle \frac{3}{4}}{\displaystyle \frac{c_m^2{\vartheta }_1^4{∥h_1∥}^4}{{\lambda }_1^2}}+{\displaystyle \frac{3}{4}}{c}_m^{{\displaystyle \frac{4}{3}}}{\vartheta }_1^4+{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}\hfill \\ & ={\vartheta }_1^3\left(c_1{q}_1+L{\alpha }_1+{\phi }_1\left(Z_1\right)\right)+{\displaystyle \frac{1}{4}}{\vartheta }_2^4+{\displaystyle \frac{1}{4}}{l}_1^4+{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_1,\hfill \end{array}\end{array}$$

(24)

where ${\phi }_1\left(Z_1\right)={\displaystyle \frac{3}{4}}{\displaystyle \frac{c_m^2{\vartheta }_1{∥h_1∥}^4}{{\lambda }_1^2}}+{\displaystyle \frac{{\vartheta }_1^3{c}_m^2{d}_1\left(t\right)}{2{\lambda }_1^2}}+{\displaystyle \frac{3}{4}}{c}_m^{{\displaystyle \frac{4}{3}}}{\vartheta }_1+{\displaystyle \frac{3}{4}}{\vartheta }_1-{\dot{y}}_d$ is continuous, $Z_1={\left[x_1,{\dot{y}}_d\right]}^{\mathsf{T}}$. Thus, the unknown continuous function ${\phi }_1\left(Z_1\right)$ can be approximated by the FLS $W_1^{\mathsf{T}}{S}_1\left(Z_1\right)$. One gets

$$\begin{array}{c}\hfill {\phi }_1\left(Z_1\right)=W_1^{\mathsf{T}}{S}_1\left(Z_1\right)+{\delta }_1\left(Z_1\right)\end{array}$$

(25)

By using Young’s inequality, one has

$$\begin{array}{c}\hfill \left.{\vartheta }_1^3\left(W_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)\right)+{\delta }_1\left(Z_1\right)\right)\le {\displaystyle \frac{{\vartheta }_1^6\theta S_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{{\vartheta }_1^6}{2}}+{\displaystyle \frac{1}{2}}{a}_1^2+{\displaystyle \frac{1}{2}}{\delta }_1^2\end{array}$$

(26)

where $a_1$, ${\overline{\delta }}_1$ are all positive constants. Substituting Equations (26) into (24), one gets

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& \le {\vartheta }_1^3\left(c_1{q}_1+L{\alpha }_1+{\displaystyle \frac{{\vartheta }_1^3}{2}}\right)+{\displaystyle \frac{{\vartheta }_1^6\theta S_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{1}{4}}{\vartheta }_2^4+{\displaystyle \frac{3}{4}}{\lambda }_1^2\hfill \\ & +{\displaystyle \frac{1}{2}}{\overline{\delta }}_1^2+{\displaystyle \frac{1}{4}}{l}_1^4+{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_1\hfill \end{array}\end{array}$$

(27)

Then, the virtual controller ${\alpha }_1$ is designed as

$$\begin{array}{c}\hfill {\alpha }_1=N\left(\psi \right)\overline{{\alpha }_1},\psi ={\vartheta }_1^3\overline{{\alpha }_1},\overline{{\alpha }_1}=k_1{\vartheta }_1+{\displaystyle \frac{{\vartheta }_1^3\widehat{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{{\vartheta }_1^3}{2}}+c_1{q}_1\end{array}$$

(28)

substituting Equations (28) into (27) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& \le {\vartheta }_1^3\left(c_1{q}_1+LN\left(\psi \right)\left[-k_1{\vartheta }_1-{\displaystyle \frac{{\vartheta }_1^3\widehat{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}-{\displaystyle \frac{{\vartheta }_1^3}{2}}-c_1{q}_1\right]\right)\hfill \\ & +{\displaystyle \frac{{\vartheta }_1^6}{2}}+{\displaystyle \frac{{\vartheta }_1^6\theta S_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{1}{4}}{\vartheta }_2^4+{\displaystyle \frac{3}{4}}{\lambda }_1^2+{\displaystyle \frac{1}{4}}{l}_1^4+{\displaystyle \frac{1}{2}}{a}_1^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_1^2\hfill \\ & +{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_1\hfill \end{array}\end{array}$$

(29)

Through simplification, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_1}& \le (LN\left(\psi \right)+1)\dot{\psi }-k_1{\vartheta }_1^4-{\displaystyle \frac{{\vartheta }_1^6\widehat{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}-{\displaystyle \frac{{\vartheta }_1^6}{2}}-{\vartheta }_1^3{c}_1{q}_1\hfill \\ & +{\displaystyle \frac{{\vartheta }_1^6}{2}}+{\displaystyle \frac{{\vartheta }_1^6\theta S_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\vartheta }_1^3{c}_1{q}_1+{\displaystyle \frac{1}{4}}{\vartheta }_2^4+c_1-\gamma {\mathrm{\Lambda }}_1\hfill \end{array}\end{array}$$

(30)

Continuing simplification, we have

$$\begin{array}{c}\hfill L{V}_{{\vartheta }_1}\le -k_1{\vartheta }_1^4-{\displaystyle \frac{{\vartheta }_1^6\tilde{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+(LN\left(\psi \right)+1)\dot{\psi }+{\displaystyle \frac{1}{4}}{\vartheta }_2^4+c_1-\gamma {\mathrm{\Lambda }}_1\end{array}$$

(31)

where $c_1={\displaystyle \frac{3}{4}}{\lambda }_1^2+{\displaystyle \frac{1}{4}}{l}_1^4+{\displaystyle \frac{1}{2}}{a}_1^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_1^2+{\displaystyle \frac{{\lambda }_1^2{m}_1^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}$. In the subsequent step, ${\displaystyle \frac{1}{4}}{\vartheta }_2^4$ is eliminated.

• Step $j(2\le j\le n-1)$: Considering Lyapunov functions

$$\begin{array}{c}\hfill V_{{\vartheta }_j}={\displaystyle \frac{1}{4}}{\vartheta }_j^4+{\mathrm{\Lambda }}_j\end{array}$$

(32)

where ${\mathrm{\Lambda }}_j={\displaystyle \frac{{\lambda }_j{e}^{-\gamma (t-d)}}{2\left(1-d^{*}\right)}}d{\int }_{t-d_j\left(t\right)}^t{\int }_{\tau }^t{e}^{rs}{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}sd\tau$ and where ${\lambda }_j>0$ is a design parameter. Then, similar to Equations (18)–(20), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_j}& ={\vartheta }_j^3\left[{\int }_{t-d_j\left(t\right)}^t{f}_j\left(\overline{x_j}\left(s\right)\right)\mathrm{d}s+{\vartheta }_{j+1}+q_{j+1}+x_{j+1,d}-{\dot{q}}_j-{\dot{x}}_{j,d}\right]\hfill \\ & +{\displaystyle \frac{3}{2}}{\vartheta }_j^2{h}_j^{\mathsf{T}}{h}_j+{\vartheta }_j^3{\vartheta }_{j+1}+{\displaystyle \frac{{\lambda }_j^2{d}_j\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}{f}_j^2\left(\overline{x_j}\right)-\gamma {\mathrm{\Lambda }}_j\hfill \\ & -{\displaystyle \frac{{\lambda }_j^2\left(1-{\dot{d}}_j\left(t\right)\right)}{2\left(1-d^{*}\right)}}d{\int }_{t-d_j\left(t\right)}^t{e}^{-r(t-d-s)}{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}s\hfill \end{array}\end{array}$$

(33)

Substituting the compensation signal Equations (17) into (33) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_j}& ={\vartheta }_j^3\left[{\int }_{t-d_j\left(t\right)}^t{f}_j\left(\overline{x_j}\left(s\right)\right)\mathrm{d}s+c_j{q}_j+L{\alpha }_j+l_{j}sign\left(q_j\right)-{\dot{x}}_{j,d}\right]\hfill \\ & +{\displaystyle \frac{3}{2}}{\vartheta }_j^2{h}_j^{\mathsf{T}}{h}_j+{\vartheta }_j^3{\vartheta }_{j+1}+{\displaystyle \frac{{\lambda }_j^2{d}_j\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}{f}_j^2\left(\overline{x_j}\right)\hfill \\ & -{\displaystyle \frac{{\lambda }_j^2\left(1-{\dot{d}}_j\left(t\right)\right)}{2\left(1-d^{*}\right)}}d{\int }_{t-d_j\left(t\right)}^t{e}^{-r(t-d-s)}{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}s-\gamma {\mathrm{\Lambda }}_j\hfill \end{array}\end{array}$$

(34)

Based on Young’s inequality, it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}& d-d_j\left(t\right)\ge 0,{\displaystyle \frac{1-d_j\left(t\right)}{1-d^{*}}}\ge 1,{\vartheta }_j^3{l}_{j}sign\left(q_j\right)\le {\displaystyle \frac{3}{4}}{\vartheta }_j^4+{\displaystyle \frac{1}{4}}{l}_j^4,\hfill \\ & {\vartheta }_j^3{\vartheta }_{j+1}\le {\displaystyle \frac{3}{4}}{\vartheta }_j^4+{\displaystyle \frac{1}{4}}{\vartheta }_{j+1}^4,{\displaystyle \frac{3}{2}}{\vartheta }_j^2{h}_j^{\mathsf{T}}{h}_j\le {\displaystyle \frac{3}{4}}{\displaystyle \frac{{\vartheta }_j^4{∥h_j∥}^4}{{\lambda }_j^2}}+{\displaystyle \frac{3}{4}}{\lambda }_j^2,\hfill \\ & {\vartheta }_j^3\left({\int }_{t-d_j\left(t\right)}^t{f}_j\left(x_j\left(s\right)\right)\mathrm{d}s\right)\le {\displaystyle \frac{{\lambda }_j^2}{2}}{\int }_{t-d_j\left(t\right)}^t{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}s+{\displaystyle \frac{{\vartheta }_j^6{d}_j\left(t\right)}{2{\lambda }_j^2}},\hfill \\ & -{\displaystyle \frac{{\lambda }_j^2\left(1-{\dot{d}}_j\left(t\right)\right)}{2\left(1-d^{*}\right)}}d{\int }_{t-d_j\left(t\right)}^t{e}^{-r(t-d-s)}{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}s\le -{\displaystyle \frac{{\lambda }_j^2}{2}}{\int }_{t-d_j\left(t\right)}^t{f}_j^2\left(x_j\left(s\right)\right)\mathrm{d}s,\hfill \end{array}\end{array}$$

(35)

Substituting Equations (35) into (34) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_j}& \le {\vartheta }_j^3\left(c_j{q}_j+{\alpha }_j+{\phi }_j\left(Z_j\right)\right)+{\displaystyle \frac{3}{4}}{\lambda }_j^2+{\displaystyle \frac{1}{4}}{\vartheta }_{j+1}^4-{\displaystyle \frac{1}{4}}{\vartheta }_j^4+{\displaystyle \frac{1}{4}}{l}_j^4+{\displaystyle \frac{{\lambda }_j^2{m}_j^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_j\hfill \end{array}\end{array}$$

(36)

where ${\phi }_j\left(Z_j\right)={\displaystyle \frac{3}{4}}{\displaystyle \frac{{\vartheta }_j{∥h_j∥}^4}{{\lambda }_j^2}}+{\displaystyle \frac{{\vartheta }_j^3{d}_j\left(t\right)}{2{\lambda }_j^2}}+{\displaystyle \frac{3}{2}}{\vartheta }_j-{\dot{x}}_{j,d}$ is continuous, $Z_j={\left[x_1,x_2,\dots ,x_j\right]}^{\mathsf{T}}\in {\mathrm{\Omega }}_Z$. Thus, ${\phi }_j\left(Z_j\right)$ can be approximated by the FLS $W_j^{\mathsf{T}}{S}_j\left(Z_j\right)$. One gets

$$\begin{array}{c}\hfill {\phi }_j\left(Z_j\right)=W_j^{\mathsf{T}}{S}_j\left(Z_j\right)+{\delta }_j\left(Z_j\right)\end{array}$$

(37)

Using Young’s inequality it follows that

$$\begin{array}{c}\hfill \left.{\vartheta }_j^3\left(W_j^{\mathsf{T}}\left(Z_j\right)S_j\left(Z_j\right)\right)+{\delta }_j\left(Z_j\right)\right)\le {\displaystyle \frac{{\vartheta }_j^6\theta S_j^{\mathsf{T}}\left(Z_j\right)S_j\left(Z_j\right)}{2a_j^2}}+{\displaystyle \frac{{\vartheta }_j^6}{2}}+{\displaystyle \frac{1}{2}}{a}_j^2+{\displaystyle \frac{1}{2}}{\delta }_j^2\end{array}$$

(38)

where $a_j$, ${\overline{\delta }}_j$ is a positive constant. Then, substituting Equations (38) into (36), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_j}& \le {\vartheta }_j^3\left(c_j{q}_j+{\alpha }_j+{\displaystyle \frac{{\vartheta }_j^3}{2}}\right)+{\displaystyle \frac{{\vartheta }_j^6\theta S_j^{\mathsf{T}}\left(Z_j\right)S_j\left(Z_j\right)}{2a_j^2}}+{\displaystyle \frac{3}{4}}{\lambda }_j^2+{\displaystyle \frac{1}{4}}{\vartheta }_{j+1}^4\hfill \\ & -{\displaystyle \frac{1}{4}}{\vartheta }_j^4+{\displaystyle \frac{1}{2}}{a}_j^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_j^2+{\displaystyle \frac{1}{4}}{l}_j^4+{\displaystyle \frac{{\lambda }_j^2{m}_j^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_j\hfill \end{array}\end{array}$$

(39)

Then, the virtual controller ${\alpha }_j$ is designed as

$$\begin{array}{c}\hfill {\alpha }_j=-k_j{\vartheta }_j-{\displaystyle \frac{{\vartheta }_j^3\widehat{\theta }{S}_j^{\mathsf{T}}\left(Z_j\right)S_j\left(Z_j\right)}{2a_j^2}}-{\displaystyle \frac{{\vartheta }_j^3}{2}}-c_j{q}_j\end{array}$$

(40)

substituting Equations (40) into (39) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_j}& \le -k_1{\vartheta }_1^4-{\displaystyle \frac{{\vartheta }_1^6\tilde{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{1}{4}}{\vartheta }_{j+1}^4-{\displaystyle \frac{1}{4}}{\vartheta }_j^4+c_j-\gamma {\mathrm{\Lambda }}_j\hfill \end{array}\end{array}$$

(41)

where $c_j={\displaystyle \frac{3}{4}}{\lambda }_j^2+{\displaystyle \frac{1}{2}}{a}_j^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_j^2+{\displaystyle \frac{1}{4}}{l}_j^4+{\displaystyle \frac{{\lambda }_j^2{m}_j^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}$.

• Step n: In this step, the following event-triggered controller is designed.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u\left(t\right)=\varpi \left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right]\hfill \\ \varpi \left(t\right)=-2\left({\alpha }_{n}tanh{\displaystyle \frac{{\vartheta }_n^3{\alpha }_n}{\eta }}+{\mathsf{ı}}_{1}tanh{\displaystyle \frac{{\vartheta }_n^3{\mathsf{ı}}_1}{\eta }}\right)\hfill \\ t_{k+1}=inf\left\{t\in R|e\left(t\right)\ge tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)\left|u\left(t\right)\right|+\mathsf{ı}\right\}\hfill \end{array}\right.\end{array}$$

(42)

where $\epsilon$, $\eta$, ı, and ${\mathsf{ı}}_1>{\displaystyle \frac{\mathsf{ı}}{1-tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)}}$ are positive constants, $e\left(t\right)=\varpi \left(t\right)-u\left(t\right)$ are the trigger errors and where $\varpi \left(t\right)$ is the actual controller and ${\alpha }_n$ virtual controller. When $t\in \left[t_k,t_{k+1}\right)$, the system output $u\left(t\right)$ remains $\varpi \left(t_k\right)$, and when $t_{k+1}$ is the update time of $\varpi \left(t\right)$, $u\left(t\right)$ will change to $\varpi \left(t_{k+1}\right)$. Therefore, for $t\in \left[t_k,t_{k+1}\right)$, we have $|\varpi \left(t\right)-u\left(t\right)|\le tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)\left|u\left(t\right)\right|+\mathsf{ı}$.

Based on the discussion of the classification of $u\left(t\right)$ in different cases, it is possible to obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\varpi \left(t\right)=u\left(t\right)+tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)s_1\left(t\right)u\left(t\right)+\mathsf{ı}{s}_2\left(t\right),\hfill \\ s_1\left(t\right)=s_2\left(t\right)=s\left(t\right),u\left(t\right)>0,\hfill \\ s_1\left(t\right)=s\left(t\right),s_2\left(t\right)=-s\left(t\right),u\left(t\right)<0,\hfill \end{array}\right.\end{array}$$

(43)

where $\left|s_1\left(t\right)\right|\le 1$, $\left|s_2\left(t\right)\right|\le 1$. Then, one obtains

$$\begin{array}{c}\hfill u\left(t\right)={\displaystyle \frac{\varpi \left(t\right)-l{s}_2\left(t\right)}{1+s_1\left(t\right)tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)}}\end{array}$$

(44)

According to ${\vartheta }_{\mathrm{n}}=z_n-q_n,z_n=x_n-x_{n,d}$, one has

$$\begin{array}{c}\hfill \mathrm{d}{\vartheta }_n=\left[\left({\int }_{t-d_n\left(t\right)}^t{f}_n\left(x_n\left(s\right)\right)\mathrm{d}s\right)+u+{\dot{x}}_{n,d}-{\dot{q}}_n\right]\mathrm{d}t+h_n^{\mathsf{T}}\left(x_n\right)\mathrm{d}\omega \end{array}$$

(45)

Considering Lyapunov functions

$$\begin{array}{c}\hfill V_{{\vartheta }_n}={\displaystyle \frac{1}{4}}{\vartheta }_n^4+{\mathrm{\Lambda }}_n\end{array}$$

(46)

where ${\mathrm{\Lambda }}_n={\displaystyle \frac{{\lambda }_n{e}^{-\gamma (t-d)}}{2\left(1-d^{*}\right)}}d{\int }_{t-d_n\left(t\right)}^t{\int }_{\tau }^t{e}^{rs}{f}_n^2\left(x_n\left(s\right)\right)\mathrm{d}s\mathrm{d}\tau$ and ${\lambda }_n>0$ is a design parameter. Then, similar to Equations (18)–(20), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_n}& ={\vartheta }_n^3\left[{\int }_{t-d_n\left(t\right)}^t{f}_n\left(\overline{x_n}\left(s\right)\right)\mathrm{d}s+c_n{q}_n+u+l_{n}sign\left(q_n\right)-{\dot{x}}_{n,d}\right]\hfill \\ & +{\displaystyle \frac{3}{2}}{\vartheta }_n^2{h}_n^{\mathsf{T}}{h}_n-{\displaystyle \frac{{\lambda }_n^2\left(1-{\dot{d}}_n\left(t\right)\right)}{2\left(1-d^{*}\right)}}d{\int }_{t-d_n\left(t\right)}^t{e}^{-r(t-d-s)}{f}_n^2\left(x_n\left(s\right)\right)\mathrm{d}s\hfill \\ & +{\displaystyle \frac{{\lambda }_n^2{d}_n\left(t\right)d{e}^{rd}}{2\left(1-d^{*}\right)}}{f}_n^2\left(\overline{x_n}\right)-\gamma {\mathrm{\Lambda }}_n\hfill \end{array}\end{array}$$

(47)

By using Young’s inequality, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_{{\vartheta }_n}& \le {\vartheta }_n^3\left(c_n{q}_n+{\alpha }_n-{\alpha }_n+u+{\phi }_n\left(Z_n\right)\right)+{\displaystyle \frac{3}{4}}{\lambda }_n^2-{\displaystyle \frac{1}{4}}{\vartheta }_n^4+{\displaystyle \frac{1}{4}}{l}_n^4+{\displaystyle \frac{{\lambda }_n^2{m}_n^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}-\gamma {\mathrm{\Lambda }}_n\hfill \end{array}\end{array}$$

(48)

where ${\phi }_n\left(Z_n\right)={\displaystyle \frac{3}{4}}{\displaystyle \frac{{\vartheta }_n{∥h_n∥}^4}{{\lambda }_n^2}}+{\vartheta }_n-{\dot{x}}_{n,d}$ is continuous, $Z_n={\left[x_1,x_2,\dots ,x_n\right]}^{\mathsf{T}}\in {\mathrm{\Omega }}_{Z_n}$. Thus, the unknown continuous function ${\phi }_n\left(Z_n\right)$ can be approximated by the FLS $W_n^{\mathsf{T}}{S}_n\left(Z_n\right)$. One gets

$$\begin{array}{c}\hfill {\phi }_n\left(Z_n\right)=W_n^{\mathsf{T}}{S}_n\left(Z_n\right)+{\delta }_n\left(Z_n\right)\end{array}$$

(49)

Using Young’s inequality, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& \left.{\vartheta }_n^3\left(W_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)\right)+{\delta }_n\left(Z_n\right)\right)\le {\displaystyle \frac{{\vartheta }_n^6\theta S_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)}{2a_n^2}}+{\displaystyle \frac{{\vartheta }_n^6}{2}}+{\displaystyle \frac{1}{2}}{a}_n^2+{\displaystyle \frac{1}{2}}{\delta }_n^2\hfill \end{array}\end{array}$$

(50)

where $a_n$, ${\overline{\delta }}_n$ are all positive constants. Substitute Equations (50) into (48) to get it:

$$\begin{array}{c}\hfill \begin{array}{cc}& L{V}_{{\vartheta }_n}\le {\vartheta }_n^3\left(c_n{q}_n+{\alpha }_n-{\alpha }_n+u+{\displaystyle \frac{{\vartheta }_n^3}{2}}\right)+{\displaystyle \frac{{\vartheta }_n^6\theta S_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)}{2a_n^2}}+{\displaystyle \frac{3}{4}}{\lambda }_n^2\hfill \\ & -{\displaystyle \frac{1}{4}}{\vartheta }_n^4+{\displaystyle \frac{1}{2}}{a}_n^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_n^2+{\displaystyle \frac{{\lambda }_n^2{m}_n^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}+{\displaystyle \frac{1}{4}}{l}_n^4-\gamma {\mathrm{\Lambda }}_n\hfill \end{array}\end{array}$$

(51)

Then, the virtual controller ${\alpha }_n$ is designed as

$$\begin{array}{c}\hfill {\alpha }_n=-k_n{\vartheta }_n-{\displaystyle \frac{{\vartheta }_n^3\widehat{\theta }{S}_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)}{2a_n^2}}-{\displaystyle \frac{{\vartheta }_n^3}{2}}-c_n{q}_n\end{array}$$

(52)

$$\begin{array}{c}\hfill \dot{\widehat{\theta }}=\sum _{i=1}^n{\displaystyle \frac{o{\vartheta }_i^6{S}_i^{\mathsf{T}}\left(Z_i\right)S_i\left(Z_i\right)}{2a_i^2}}-\sigma \widehat{\theta }\end{array}$$

(53)

where $\sigma$ is a design parameter. Substituting Equations (52) for (51) yields

$$\begin{array}{c}\hfill L{V}_{{\vartheta }_n}\le -k_n{\vartheta }_n^4-{\displaystyle \frac{{\vartheta }_n^6\theta S_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)}{2}}+{\vartheta }_n^3\left(u-{\alpha }_n\right)+c_n-{\displaystyle \frac{1}{4}}{\vartheta }_n^4-\gamma {\mathrm{\Lambda }}_n\end{array}$$

(54)

where $c_n={\displaystyle \frac{3}{4}}{\lambda }_n^2+{\displaystyle \frac{1}{2}}{a}_n^2+{\displaystyle \frac{1}{2}}{\overline{\delta }}_n^2+{\displaystyle \frac{1}{4}}{l}_n^4+{\displaystyle \frac{{\lambda }_n^2{m}_n^2{d}^2{e}^{rd}}{2\left(1-d^{*}\right)}}$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u-{\alpha }_n=& -{\displaystyle \frac{2}{1+tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)s_1\left(t\right)}}\left({\alpha }_{n}tanh{\displaystyle \frac{{\vartheta }_n^3{\alpha }_n}{\eta }}+{\mathsf{ı}}_{1}tanh{\displaystyle \frac{{\vartheta }_n^3{\mathsf{ı}}_1}{\eta }}\right)\hfill \\ & -{\displaystyle \frac{\mathsf{ı}{s}_2\left(t\right)}{1+tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)s_1\left(t\right)}}-{\alpha }_n\hfill \end{array}\end{array}$$

(55)

Then on the basis of $\left|s_1\left(t\right)\right|\le 1$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\vartheta }_n^3\left(u-{\alpha }_n\right)& \le \left|{\vartheta }_n^3{\alpha }_n\right|-{\vartheta }_n^3{\alpha }_{n}tanh{\displaystyle \frac{{\vartheta }_n^3{\alpha }_n}{\eta }}+\left|{\vartheta }_n^3{l}_1\right|-{\vartheta }_n^3{l}_{1}tanh{\displaystyle \frac{{\vartheta }_n^3{l}_1}{\eta }}\hfill \\ & -\left|{\vartheta }_n^3{l}_1\right|-{\displaystyle \frac{{\vartheta }_n^{3}l{s}_2\left(t\right)}{1+tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)s_1\left(t\right)}}\hfill \end{array}\end{array}$$

(56)

According to $-{\displaystyle \frac{{\vartheta }_n^3\mathsf{ı}{s}_2\left(t\right)}{1+tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)s_1\left(t\right)}}\le \left|{\displaystyle \frac{{\vartheta }_n^3\mathsf{ı}}{1-tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)}}\right|$,${\mathsf{ı}}_1>{\displaystyle \frac{\mathsf{ı}}{1-tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)}}$ and Lemma 1, one obtains

$$\begin{array}{c}\hfill {\vartheta }_n^3\left(u-{\alpha }_n\right)\le 0.557\eta \end{array}$$

(57)

Then, substituting Equations (57) into (54) yields

$$\begin{array}{c}\hfill L{V}_{g_n}\le -k_n{g}_n^4-{\displaystyle \frac{{\vartheta }_n^6\theta S_n^{\mathsf{T}}\left(Z_n\right)S_n\left(Z_n\right)}{2}}+0.557\eta -{\displaystyle \frac{1}{4}}{g}_n^4+c_n-\gamma {\mathrm{\Lambda }}_n\end{array}$$

(58)

4. Stability Analysis

Theorem 1. 

Applying the system controllers (22, 31, 42) and the parameter adaptive law (43) to the stochastic nonlinear system (1) with state time-varying delays and output dead zone, in combination with the error compensation mechanism (12) and the ETM (34), subject to Assumptions 1–3, ensures the following: (i) All signals in the closed-loop system are ultimately bounded in accordance with probabilistic semi-global agreement; (ii) The tracking error converges to a small neighbourhood of the origin [39]; (iii) Zeno behaviour can be effectively avoided [39].

Proof. 

Considering the Lyapunov functions

$$\begin{array}{c}\hfill V=\sum _{i=1}^n{V}_{{\theta }_i}+{\displaystyle \frac{{\tilde{\theta }}^2}{2o}}\end{array}$$

(59)

Differentiating V with respect to using Equations (31), (41) and (58) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill LV\le & -\sum _{i=1}^n{k}_i{\vartheta }_i^4-\sum _{i=1}^n{\displaystyle \frac{{\vartheta }_i^6\tilde{\theta }{S}_i^{\mathsf{T}}\left(Z_i\right)S_i\left(Z_i\right)}{2a_i^2}}+\left(L(1+N\left(\psi \right))\right)\dot{\psi }+0.557\eta \hfill \\ & +\sum _{i=1}^n{c}_i+{\displaystyle \frac{\tilde{\theta }\dot{\widehat{\theta }}}{o}}-\gamma \sum _{i=1}^n{\mathrm{\Lambda }}_i\hfill \end{array}\end{array}$$

(60)

Substituting (43) into (50) yields

$$\begin{array}{c}\hfill LV\le -\sum _{i=1}^n{k}_i{\vartheta }_i^4+(L(1+N\left(\psi \right))\psi +0.557\eta +\sum _{i=1}^n{c}_i-{\displaystyle \frac{\sigma \overline{\theta }\dot{\theta }}{o}}-\gamma \sum _{i=1}^n{\mathrm{\Lambda }}_i\end{array}$$

(61)

According to $-{\displaystyle \frac{-\sigma \tilde{\theta }\widehat{\theta }}{o}}\le -{\displaystyle \frac{\sigma {\tilde{\theta }}^2}{o}}+{\displaystyle \frac{\sigma {\theta }^2}{o}}$ yields

$$\begin{array}{c}\hfill LV\le -CV+(L(1+N\left(\psi \right))\dot{\psi }+{\rho }_0\end{array}$$

(62)

where $C=min\left\{4k_i,2\sigma ,\gamma ,i=1,\dots ,n\right\},{\rho }_0=0.557\eta +{\sum }_{i=1}^n{c}_i+{\displaystyle \frac{\sigma {\theta }^2}{o}}$. Multiply both sides of Equation (52) by ${\mathrm{e}}^{Ct}$ and differentiate it, we obtain

$$\begin{array}{c}\hfill 0\le EV\left(t\right)\le V\left(0\right)+{\displaystyle \frac{{\rho }_0}{C}}+e^{-Ct}{\int }_0^t(1+N\left(\psi \right)g\left(t\right))\dot{\psi }{e}^{C\tau }\mathrm{d}\tau \end{array}$$

(63)

According to Lemma 3 and Equation (63), we know that ${\int }_0^t(L(1+N\left(\psi \right))\dot{\psi }{e}^{C\tau }\mathrm{d}\tau$ is bounded. Therefore, we define $(L(1+N\left(\psi \right))\psi \le {\rho }_1$.We can further conclude that

$$\begin{array}{c}\hfill LV\le -CV+\rho ,\end{array}$$

(64)

where $\rho ={\rho }_0+{\rho }_1$. Thus, all signals in this system are ultimately bounded in accordance with the probabilistic semi-global consistency. $E\left(V\left(t\right)\right)\le V\left(0\right)e^{-Ct}+{\displaystyle \frac{\rho }{C}}$ holds; then,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{4}}E\sum _{i=1}^n{\vartheta }_i^4\le E\left(V\left(t\right)\right)\le V\left(0\right)e^{-Ct}+{\displaystyle \frac{\rho }{C}}\end{array}$$

(65)

We can obtain $E{\sum }_{i=1}^n{\vartheta }_i^4\le 4\mathrm{\Omega }$ where $\mathrm{\Omega }=V\left(0\right)e^{-Ct}+{\displaystyle \frac{\rho }{C}}$. Therefore, ${\vartheta }_i$ converges to the closed set ${\mathrm{\Omega }}_{{\vartheta }_i}=\left\{{\vartheta }_i\in R\mid E{\sum }_{i=1}^n{\vartheta }_i^4<4U\right\}$. Since ${\vartheta }_i=z_i-q_i$, when the compensation signal $qi$ converges, the error signal $zi$ converges to a small neighborhood of the origin. Considering Lyapunov functions $V_q={\sum }_{i=1}^n{\displaystyle \frac{1}{2}}{q}_i^2$. Then, combine the differentials of $\left|x_{i+1,d}-{\alpha }_i\right|\le {\epsilon }_i(i=1,\dots ,n-1)$ and $V_q$, yielding

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_q& =q_1\left(-c_1{q}_1+L\left(x_{2,d}-{\alpha }_1\right)+L{q}_2-l_{1}sign\left(q_1\right)\right)\hfill \\ & +\sum _{i=2}^{n-1}{q}_i\left(-c_i{q}_i+\left(x_{i+1,d-{\alpha }_i}-{\alpha }_i\right)+q_{i+1}-l_{i}sign\left(q_i\right)\right)\hfill \\ & +q_n\left(-c_n{q}_n-l_{n}sign\left(q_n\right)\right)\hfill \\ & \le \sum _{i=1}^n-c_i{q}_i^2+L{q}_1{q}_2+\sum _{i=2}^{n-1}{q}_i{q}_{i+1}+q_1\left(L{\epsilon }_1-l_1\right)+\sum _{i=2}^n{q}_i\left({\epsilon }_i-l_i\right)\hfill \end{array}\end{array}$$

(66)

where ${\epsilon }_n=0$ and ${\epsilon }_i>0$. By using Young’s inequality, it yields that

$$\begin{array}{c}\hfill \begin{array}{cc}& L{q}_1{q}_2\le {\displaystyle \frac{c_M^2}{2r_1^2}}{q}_1^2+{\displaystyle \frac{1}{2}}{q}_2^2,q_i{q}_{i+1}\le {\displaystyle \frac{1}{2r_i^2}}{q}_i^2+{\displaystyle \frac{r_i^2}{2}}{q}_{i+1}^2(i=2,\dots ,n-1),\hfill \\ & q_1\left(L{\epsilon }_1-l_1\right)\le {\displaystyle \frac{1}{2}}{q}_1^2+{\displaystyle \frac{1}{2}}{\left(c_M{\epsilon }_1-l_1\right)}^2,q_i\left({\epsilon }_i-l_i\right)\le {\displaystyle \frac{1}{2}}{q}_i^2+{\displaystyle \frac{1}{2}}{\left({\epsilon }_i-l_i\right)}^2\hfill \end{array}\end{array}$$

(67)

Substituting Equations (67) into (66), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill L{V}_q\le & -\left(c_1-{\displaystyle \frac{c_M^2}{2r_1^2}}-{\displaystyle \frac{1}{2}}\right)q_1^2-\left(c_2-{\displaystyle \frac{1}{2r_2^2}}-{\displaystyle \frac{r_1^2}{2}}-{\displaystyle \frac{1}{2}}\right)q_2^2+{\displaystyle \frac{1}{2}}{\left(c_M{\epsilon }_1-l_1\right)}^2\hfill \\ & +\sum _{i=2}^n{\displaystyle \frac{1}{2}}{\left({\epsilon }_i-l_i\right)}^2-\cdots -\left(c_{n-1}-{\displaystyle \frac{1}{2r_{n-1}^2}}-{\displaystyle \frac{r_{n-2}^2}{2}}-{\displaystyle \frac{1}{2}}\right)q_{n-1}^2\hfill \\ & -\left(c_n-{\displaystyle \frac{r_{n-1}^2}{2}}-{\displaystyle \frac{1}{2}}\right)q_n^2\hfill \\ \hfill \le & -A{V}_q+B\hfill \end{array}\end{array}$$

(68)

where

$A=min\left\{c_1-{\displaystyle \frac{c_M^2}{2r_1^2}}-{\displaystyle \frac{1}{2}},c_i-{\displaystyle \frac{1}{2r_i^2}}-{\displaystyle \frac{r_{i-1}^2}{2}}-{\displaystyle \frac{1}{2}}(i=2,\dots ,n-1),c_n-{\displaystyle \frac{r_{n-1}^2}{2}}-{\displaystyle \frac{1}{2}}\right\}>0,$$B={\displaystyle \frac{1}{2}}{\left(c_M{\epsilon }_1-l_1\right)}^2+{\sum }_{i=2}^n{\displaystyle \frac{1}{2}}{\left({\epsilon }_i-l_i\right)}^2,$ Therefore, from (54) and (55), we have $E{\sum }_{i=1}^n{q}_i^2<2U_q$, where $U_q=V_q\left(0\right)e^{-At}+{\displaystyle \frac{B}{A}}$. □

It is further concluded that $q_i$ converges to a compact set A ${\mathrm{\Omega }}_{q_i}=\{q_i\in R\mid E{\sum }_{i=1}^n$$q_i^2<2U_q\}$. Thus, the error signal $z_i$ converges to a small neighbourhood of the origin. The following proof shows that this control strategy can prevent Zeno behavior from occurring, derived from $\forall t\in \left[t_k,t_{k+1}\right),e\left(t\right)=\varpi \left(t\right)-u\left(t\right)$, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{\mathrm{d}t}}\left|e\left(t\right)\right|\le {\displaystyle \frac{d}{\mathrm{d}t}}{\left(e\left(t\right)e\left(t\right)\right)}^{{\displaystyle \frac{1}{2}}}=sign\left(e\left(t\right)\right)\dot{e}\left(t\right)\le \left|\dot{\omega }\left(t\right)\right|\end{array}$$

(69)

Because $\dot{\omega }\left(t\right)$ is bounded, there exists a positive normal number b such that $|\dot{\varpi }\left(t\right)|\le b$. By the Lagrange’s mean value theorem, we have

$$\begin{array}{c}\hfill e\left(t_{k+1}\right)-e\left(t_k\right)\le \dot{e}\left(t_{\mathrm{\Delta }}\right)\left(t_{k+1}-t_k\right),t_{\mathrm{\Delta }}\in \left[t_k,t_{k+1}\right),\end{array}$$

(70)

From Equations (69) and (73), $e\left(t_k\right)=0$ and

$$\begin{array}{c}\hfill \underset{t\to t_{k+1}}{lim}e\left(t\right)=tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)\left|u\left(t\right)\right|+\mathsf{ı}\end{array}$$

(71)

it yields that

$$\begin{array}{c}\hfill tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)\left|u\left(t\right)\right|+\mathsf{ı}\le b\left(t_{k+1}-t_k\right)\end{array}$$

(72)

Then, from Equation (72), one gets $t^{*}=\left|t_{k+1}-t_k\right|\ge {\displaystyle \frac{tanh\left(\frac{1}{u\left(t\right)+\epsilon }\right)\left|u\left(t\right)\right|+t}{b}}$, it can be concluded that the event-triggered mechanism designed in this paper can effectively avoid the occurrence of Zeno behavior. To numerically verify the exclusion of Zeno behavior, we compute the minimum inter-event time based on the theoretical lower bound derived in (72):

$$\begin{array}{c}\hfill t^{*}={\displaystyle \frac{tanh\left(\frac{1}{\left|u\right(t\left)\right|+\epsilon }\right)\left|u\left(t\right)\right|+\mathsf{\ell }}{b}}\end{array}$$

(73)

From the simulation data, we observe $\left|u\right(t\left)\right|\le 5.3,b\approx 12.6$, and with $\epsilon =0.8$ and $\mathsf{\ell }=0.4$, the minimum inter-event time is bounded below by $t^{*}\ge 0.062\mathrm{s}$. The actual triggering intervals shown in Figure 11 remain strictly positive and are consistently greater than this bound, which provides clear numerical confirmation that Zeno behavior is practically avoided.

Remark 4. 

The design parameters are selected according to standard Lyapunov-based synthesis. Gains $k_i,c_i>0$ are chosen sufficiently large to ensure dominance of negative definite terms in the Lyapunov derivative. Parameters $a_i>0$ balance approximation accuracy and control effort. The adaptive gain $\sigma >0$ influences the convergence rate of $\widehat{\theta }$, while $\epsilon =0.8$ and $\mathsf{\ell }=0.4$ (in Example 1) tune the trade-off between communication frequency and tracking performance.

5. Simulation

To verify the effectiveness of the proposed dynamic event-triggered adaptive tracking control scheme for SNSs with distributed time-varying delays and output dead zone, two simulation examples are provided below.

Example 1. 

Consider the stochastic nonlinear system with output dead zone and distributed time delay as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{x}_1=\left({\int }_{t-d_1\left(t\right)}^t{f}_1\left(x_1\right)\mathrm{d}s+x_2\right)\mathrm{d}t+h_1\left(x_1\right)\mathrm{d}\omega \hfill \\ \mathrm{d}{x}_2=\left({\int }_{t-d_2\left(t\right)}^t{f}_2\left(\overline{x_2}\right)\mathrm{d}s+u\right)\mathrm{d}t+h_2\left(x_1,x_2\right)\mathrm{d}\omega \hfill \\ y=\mathrm{\Delta }\left(x_1\right)\hfill \end{array}\right.\end{array}$$

(74)

In the aforementioned system, the tracked signal is $y_d=sin\left(t\right)$, the distributed time-varying delay is selected as $d_1\left(t\right)=0.01sin\left(t\right)+0.1$, $d_2\left(t\right)=0.02cos\left(t\right)+0.2$, the initial values are $x_1\left(0\right)=0$, $x_2\left(0\right)=0$ and the specific expression of the involved function is $f_1\left(x_1\right)=2sin\left(x_1\right)$, $h_1\left(x_1\right)=1-cos\left(x_1\right)$, $f_2\left({\overline{x}}_2\right)=sin\left(x_1^2\right)+10sin\left(x_2\right)$, $h_2\left({\overline{x}}_2\right)=0.01x_2^{2}cos\left(x_1\right)$.

The design of the simulated compensation signal is as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{q}}_1=-c_1{q}_1+L\left(x_{2,d}-{\alpha }_1\right)+L{q}_2-l_{1}sign\left(q_1\right),\hfill \\ {\dot{q}}_2=-c_2{q}_2-l_{2}sign\left(q_2\right).\hfill \end{array}\right.\end{array}$$

(75)

The design of the virtual controllers ${\alpha }_1$, ${\alpha }_2$ and $\dot{\widehat{\theta }}$ is presented as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& {\alpha }_1=N\left(\psi \right)\overline{{\alpha }_1},\dot{\psi }={\vartheta }_1^3\overline{{\alpha }_1},\hfill \\ & \overline{{\alpha }_1}=k_1{\vartheta }_1+{\displaystyle \frac{{\vartheta }_1^3\widehat{\theta }{S}_1^{\mathsf{T}}\left(Z_1\right)S_1\left(Z_1\right)}{2a_1^2}}+{\displaystyle \frac{{\vartheta }_1^3}{2}}+c_1{q}_1\hfill \\ & {\alpha }_2=-k_2{\vartheta }_2-{\displaystyle \frac{{\vartheta }_2^3\widehat{\theta }{S}_2^{\mathsf{T}}\left(Z_2\right)S_2\left(Z_2\right)}{2a_2^2}}-{\displaystyle \frac{{\vartheta }_2^3}{2}}-c_2{q}_2,\hfill \\ & \dot{\widehat{\theta }}=\sum _{i=1}^2{\displaystyle \frac{o{\vartheta }_i^6{S}_i^{\mathsf{T}}\left(Z_i\right)S_i\left(Z_i\right)}{2a_i^2}}-\sigma \widehat{\theta }.\hfill \end{array}\end{array}$$

(76)

The design of the ETC is as follows [34]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\varpi \left(t\right)=-2\left({\alpha }_{2}tanh{\displaystyle \frac{{\vartheta }_2^3{\alpha }_2}{\eta }}+{\mathsf{ı}}_{1}tanh{\displaystyle \frac{{\vartheta }_2^3{\mathsf{ı}}_1}{\eta }}\right)\hfill \\ u\left(t\right)=\varpi \left(t_k\right),\forall t\in \left[t_k,t_{k+1}\right]\hfill \\ t_{k+1}=inf\left\{t\in R\parallel e\left(t\right)\ge tanh\left({\displaystyle \frac{1}{\left|u\right(t\left)\right|+\epsilon }}\right)+\mathsf{ı}\right\}\hfill \end{array}\right.\end{array}$$

(77)

The selected design parameters are $c_1=1, c_2=1, l_1=0.01, l_2=0.01, k_1=98,$$k_2=58, a_1=a_2=0.1, o=1, \sigma =1.6, \eta =0.02, \epsilon =0.8, l_1=5.1, l=0.4, \zeta ={\displaystyle \frac{1}{6.8}}$, $N\left(\psi \right)=e^{{\psi }^2}cos\left(0.5\pi \psi \right)$, Output dead zone parameter selection as $d_r=d_l=1,$$c_r=0.02, c_l=-0.03$, Initial value is $\widehat{\theta }\left(0\right)=1, \psi \left(0\right)=0$.

Figure 1 shows the reference signal $y_d$ and the system output y, demonstrating good tracking performance. Figure 2 and Figure 3 show the response of the state variables $x_1$ and $x_2$. Figure 4 depicts the curves of $N\left(\psi \right)$, while Figure 5 illustrates the adaptive law $\dot{\widehat{\theta }}$. Figure 6 and Figure 7 show the virtual controllers ${\alpha }_1$, ${\alpha }_2$.

Figure 8 shows that when it is stabilized, the tracking error $z_1$ and compensation tracking error signal ${\vartheta }_1$ converge to a small region around the origin. Figure 9 shows the ETC signal $u\left(t\right)$ and the TTC signal $\varpi \left(t\right)$. The trajectory of the input signal $u\left(t\right)$ in a random system indicates that the next signal will only be collected when the triggering conditions are met. Therefore, the event-triggered mechanism can reduce communication burdens, thereby conserving resources.

To quantitatively evaluate the advantage of the proposed dynamic event-triggering mechanism (DETM), a direct comparison with a conventional fixed-threshold method is provided. Figure 10 and Figure 11 show the dynamic ETM and the fixed threshold time interval $t_{k+1}-t_k$, respectively [34]. The proposed dynamic ETM demonstrates superior effectiveness in conserving communication resources for systems with distributed time-varying delays. Crucially, the mechanism ensures a finite number of triggering events, thereby excluding Zeno behavior.

Table 1. Quantitative performance comparison for Example 1.

Control Method	RMSE of ${\mathit{z}}_1$	AATE	TCE ($\times {10}^2$)	Trigger Count	Comm. Reduction
Proposed (DETC)	0.042	0.032	0.106	1171	76.58%
Fixed-Threshold ETC	0.046	0.037	8.34	2598	48.04%
Time-Triggered (TTC)	0.042	0.032	8.52	5000	–

presents a quantitative performance comparison of the proposed DETC against a Fixed-Threshold ETC and Time-Triggered Control (TTC), evaluating tracking accuracy via RMSE (0.042) and AATE (0.032), control energy via TCE, and communication load via trigger counts. The results confirm that the proposed DETC maintains tracking precision identical to TTC while reducing communication by 76.58% (1171 vs. 5000 triggers), albeit with a moderate increase in control energy (TCE of $1.06\times {10}^1$ vs. $8.52\times {10}^2$ for TTC). Moreover, compared to Fixed-Threshold ETC, DETC achieves superior tracking accuracy, comparable control energy, and a higher communication reduction (76.58% vs. 48.04%), demonstrating its efficacy in balancing performance and resource efficiency for communication-constrained applications.

In addition to the aforementioned numerical simulation examples, a simulation case in a practical application scenario is provided below to further demonstrate the effectiveness of the proposed method.

Example 2. 

Considering a single-chain manipulator system, where the dynamic equations are described as follows in [39].

$$\begin{array}{c}\hfill \left\{\begin{array}{c}D\ddot{p}+F\dot{p}+Nsin\left(p\right)=\tau +{\tau }_d,\hfill \\ M\dot{\tau }+H\tau =u-J_m\dot{p}+u_d,\hfill \end{array}\right.\end{array}$$

(78)

In this context, p is the position of the link, $\dot{p}$ is the velocity of the link and $\ddot{p}$ is the acceleration of the link. $D=1{\mathrm{kgm}}^2$ is the mechanical inertia, $F=1\mathrm{Nm}·\mathrm{s}/\mathrm{rad}$ is the viscous friction coefficient at the joint, $N=10$ is a parameter related to the load mass and gravitational coefficient, $M=0.1H$ is the armature inductance, ${\tau }_d=p^{2}cos\left(\dot{p}\right)$ and $u_d=0.01cos\left(\dot{p}\right){\left(0.1p\right)}^2$ are the random disturbance torques, $H=1\mathrm{\Omega }$ is the armature resistance, and $J_m=0.2\mathrm{Nm}/\mathrm{A}$ is the back electromotive force coefficient.

By defining $x_1=p$, $x_2=\dot{p}$, and $x_3=\tau$. From the first equation of (77), $D\ddot{p}=\tau +{\tau }_d-F\dot{p}-Nsin\left(p\right)$. Substituting the states yields $D{\dot{x}}_2=x_3+{\tau }_d-F{x}_2-Nsin\left(x_1\right)$. Rearranging and incorporating the distributed delay and stochastic disturbance as per our problem formulation leads to ${\dot{x}}_2={\displaystyle \frac{1}{D}}\left[x_3-F{x}_2-Nsin\left(x_1\right)\right]+\left[{\int }_{t-d_1\left(t\right)}^t{f}_2\left(x_1,x_2\right)ds\right]+h_2\left(x_1,x_2\right)\dot{\omega }$. Here, the term ${\displaystyle \frac{{\tau }_d}{D}}$ and part of the deterministic dynamics are absorbed into the unknown function $f_2(·)$ approximated by the FLS. From the second equation of (77) $M\dot{\tau }=u-J_m\dot{p}+u_d-H\tau$. Substituting states gives $M{\dot{x}}_3=u-J_m{x}_2+u_d-H{x}_3$. Similarly, this is rearranged into the form ${\dot{x}}_3={\displaystyle \frac{1}{M}}\left[u-H{x}_3-J_m{x}_2\right]+\left[{\int }_{t-d_2\left(t\right)}^t{f}_3\left(x_2,x_3\right)ds\right]+h_3\left(x_1,x_2\right)\dot{\omega }$, where ${\displaystyle \frac{u_d}{M}}$ and other terms are incorporated into $f_3(·)$ and the stochastic coefficient $h_3(·)$; then, the dynamic Equation (78) can be rewritten as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{x}_1=x_2\mathrm{d}t\hfill \\ \mathrm{d}{x}_2=\left(x_3+{\int }_{t-d_1\left(t\right)}^t{f}_2\left(x_1,x_2\right)\mathrm{d}s\right)\mathrm{d}t+h_2\left(x_1,x_2\right)\mathrm{d}w\hfill \\ \mathrm{d}{x}_3=\left(10u+{\int }_{t-d_2\left(t\right)}^t{f}_3\left(x_2,x_3\right)\mathrm{d}s\right)\mathrm{d}t+h_3\left(x_1,x_2\right)\mathrm{d}w\hfill \\ y=x_1\hfill \end{array}\right.\end{array}$$

(79)

where $d_1\left(t\right)=0.2+0.01sin\left(t\right)$, $f_2\left(x_1,x_2\right)=-x_2-2sin\left(x_1\right)$, $h_2\left(x_1,x_2\right)=x_1^{2}cos\left(x_2\right)$, $d_2\left(t\right)=0.2cos\left(0.2t\right)$, $f_3\left(x_2,x_3\right)=2x_2+2.5sin\left(x_3\right)$, $h_3\left(x_1,x_2\right)=0.01x_1^{2}cos\left(x_2\right)$, the tracking signal is given as $y_d=sin\left(t\right)$. The initial values are $x_1\left(0\right)=0.5,x_2\left(0\right)=-0.2,x_3\left(0\right)=0.1$. The design parameters are selected as $k_1=98$, $k_2=88$, $k_3=58$, $c_1=2.1$, $c_2=88$, $a_1=a_2=a_3=0.1$, $c_3=98$, $l_1=0.01$, $l_2=0.2$, $l_3=0.1$, $\sigma =2.2$, $o=1$, $\eta =0.9$, ${\tau }_1=2.6$, ${\varsigma }_2={\displaystyle \frac{1}{56}}$, ${\varsigma }_3={\displaystyle \frac{1}{52}}$.

Figure 12 shows the convergence of the tracking error $z_1$ after stabilization. Figure 13 shows the response of the state variables $x_1$ and $x_2$. Figure 14 shows the trajectory of the output signal y and the tracking signal $y_d$. Figure 15 shows the response of the state variable $x_3$.

Figure 16 shows the curve of the adaptive law $\widehat{\theta }$. Figure 17 shows the TTC signal $\varpi \left(t\right)$ and the ETC signal $u\left(t\right)$.

Figure 18 shows the time interval $t_{k+1}-t_k$ for the dynamic ETM. Obviously, real-world application cases can better demonstrate the effectiveness of the dynamic ETM discussed in this paper in saving communication resources, and also the fact that the mechanism is triggered a finite number of times, which means that the Zeno phenomenon does not occur. The simulation parameters and initial conditions used in these examples are summarized in

Table 2. Simulation parameters and initial conditions for Examples 1 and 2.

Parameter	Example 1 Value	Example 2 Value	Description
$c_1,c_2,c_3$	$1,1,--$	$2.1,88,98$	Compensation gains
$l_1,l_2,l_3$	$0.01,0.01,--$	$0.01,0.2,0.1$	Sign function gains
$k_1,k_2,k_3$	$98,58,--$	$98,88,58$	Controller gains
$a_1,a_2,a_3$	$0.1,0.1,--$	$0.1,0.1,0.1$	Fuzzy approximation parameters
$\sigma$	$1.6$	$2.2$	Adaptive law gain
o	1	1	Normalization constant
$\eta$	$0.02$	$0.9$	Smoothing parameter for $tanh(·)$
$\epsilon$	$0.8$	$0.8$	Trigger threshold offset
$t_1$	$5.1$	$2.6$	Trigger threshold constant
l	$0.4$	$0.6$	Trigger error bound
${\zeta }_2,{\zeta }_3$	$1/54,1/58$	$1/56,1/52$	Filter time constants
$d_r,d_l$	$1,1$	−−	Dead-zone width parameters
$c_r,c_l$	$0.02,-0.03$	$0.02,-0.03$	Dead-zone slopes
$x_1\left(0\right),x_2\left(0\right),x_3\left(0\right)$	$0,0,--$	$0.5,-0.2,0.1$	Initial states
$\widehat{\theta }\left(0\right),\psi \left(0\right)$	$1,0$	$1,0$	Adaptive parameter initial values
$N\left(\psi \right)$	$e^{{\psi }^2}cos\left(0.5\pi \psi \right)$	$e^{{\psi }^2}cos\left(0.5\pi \psi \right)$	Nussbaum function

.

6. Conclusions

To solve the problem of SNSs with distributed time-varying delays and output dead zone, a control scheme combining backstepping and adaptive control is proposed in this paper. The scheme employs an FLS to estimate the unknown continuous function, handles the distributed time-varying delay with an unknown upper limit by introducing the Lyapunov–Krasovskii function in double-integral form, and compensates for the output dead zone effect in the system by applying the Nussbaum function. The adaptive tracking controller designed using the backstepping method ensures the boundedness of all signals in the closed-loop system and converges the tracking error to a small region near the origin in a probabilistic manner. Simulation results validate the effectiveness of the proposed method.

Although the proposed scheme offers a feasible solution for the considered class of systems, several open problems remain for future research. First, extending the control design from strict-feedback to more general non-strict-feedback or pure-feedback stochastic nonlinear systems with distributed delays would broaden the applicability of the method. Second, further improvements in communication efficiency could be pursued by integrating the dynamic event-triggered mechanism with signal quantization or network transmission protocols under limited bandwidth.