Event-triggered formation fault-tolerant control for nonlinear multi-agent systems with time-varying state delays

Abstract

This paper presents neural network (NN) event-triggered (ET) formation fault-tolerant control (FTC) problem for nonlinear multi-agent systems (MASs) with unmeasurable states, actuator faults and time-varying state delays. Firstly, a state observer is introduced to estimate the unmeasurable states and prescribed performance (PP) control based on error transformation is presented to maintain system efficiency and reliability. Secondly, to solve the unknown actuator faults, an adaptive law based on the projection opearator is designed and state time-delay is compenstated with an appropriate Lyapunov functional. Thirdly, the biggest innovation in the proposed control approach is that only when ET conditions are violated, the control law and NN weights are updated. Then, the developed control scheme can guarantee MASs are semi-globally uniformly ultimately bounded (SGUUB). Finally, comparative simulations are given to demonstrate the validity of the proposed method.

Introduction

Over the past few years, multi-agent systems (MASs) have attracted much attention due to its broad applications in unmanned surface vehicle (USV), unmanned aerial vehicle (UAV), unmanned autonomous vehicle (UGV) and other fields.^1–3 As a branch of MASs coordinate control, the purpose of formation tracking of MASs is to maintain a desired formation eventually. Thus, many interesting formation control problems are studied in Refs.^4–6 For example, the problem of finite-time formation control for MASs with general linear dynamics is addressed in Refs.^5,6 presents distance-based formation control with prescribed performance in which agents are subjected to unknown external disturbances.

In practical applications, it is difficult to acquire the full states of the system directly due to the restrictions of measuring equipments. To solve these problems, many scholars are committed to studying state estimation^7–9 and Fuzzy/NN approximation.^9–13 Among them, an adaptive NN formation FTC issue is considered for nonlinear MASs in Tong et al., ⁹ where unknown states are estimated by a nonlinear observer. By employing fuzzy logic systems, the unknown nonlinear dynamics and the state-dependent actuator faults are approximated. ¹¹ Moreover, communication delays are inevitable and may be a source of instability or poor performance of MASs, especially when the delay is unknown. Nevertheless, there are not many results in multiagent systems with time delay until in Refs.^14–17 By utilizing the Lyapounov-Krasovskii functional (LKF) to solve state time-delay, the consensus problem is addressed for MASs in Xiao et al. ¹⁴ In Liu et al., ¹⁵ Young’s inequality scaling technique is applied to handle the problem of time delay. In Lai and Chen ¹⁶ , a new sufficient condition of delay dependence for the mean square formation tracking is also proposed. However, the above control schemes all require that the control signals are frequently updated by the time-based sampling schemes, which will inevitably cause a tremendous waste of communication resources.

To reduce unnecessary communication resources, event-triggered control (ETC) is a nice choice without doubt since the controller performs an action only when the event threshold is exceeded. In Xu et al. ¹⁸ dealed with the problem of event-triggered formation tracking in a fixed topology. In Ahmed et al., ¹⁹ the event-triggered consensus control was studied by utilizing a distributed ET estimator with the dynamic threshold mechanisms. In Wang and Li, ²⁰ the control law and NN weights are updated in an aperiodic form, which can reduce data transfer and communication resource. Unfortunately, ETC, which is studied in Refs.,^18–20 is only designed in the controller-to-actuator (C-A) channel, and data transmissions from the sensor-to-controller (S-C) channel are not considered. Therefore, how to design a two-channel ET mechanism and address the formation control problem for MASs with unknown states are a challenging task.

Besides, actuator faults are also inevitable, which may damage system performance or result in the instability of system. Thus, it is essential to learn FTC methods to maintain control performance.^11,13,21,22 For instance, Bounemeur and Chemachema ¹¹ presents an adaptive fuzzy fault-tolerant tracking control for a class of unknown multi-variable nonlinear systems. An active fuzzy fault tolerant tracking control (AFFTTC) scheme is developed to deal with four kinds of faults. ¹³ However, these schemes are only suitable for single agent systems. In order to acheive more effective and accurate formation tasks for MASs, it is important to extend the FTC from single agent systems to MASs. In Liu et al., ²³ the Takagi-Sugeno fuzzy system robust fault estimation and FTCare used to compensate the influence of actuator failure for MASs. In Wu et al. ²⁴ two effective distributed fault-tolerant formation tracking protocols are designed to get the desired formation space positions. The distributed adaptive FTC to solve non-identical matching functions for MASs is studied in Deng and Yang. ²⁵

On the other hand, formation control for nonlinear MASs is known to require high performance. Guaranteeing prescribed performance is critical to maintain system efficiency and reliability. In the work, Li et al. ²⁶ a distributed finite-time consensus controller is proposed to ensure the synchronization of multi-agent systems with specified performance. In the work, Liu et al. ²⁷ a data-driven control strategy is constructed to handle the problem of MASs. Therefore, it is reasonable to use the prescribed performance control to obtain more precise formation tracking tasks. However, how to develop NN formation fault-tolerant control strategies for nonlinear multi-agent systems with unknown states, actuator failure, time-varying state delays and the prescribed performance based on the double-sides ET mechanism are very intersting and difficult, which is the reason inspiring our research.

Motivated by these observations, an adaptive neural network event-triggered formation fault-tolerant control is designed with unmeasurable states, actuator failures, time-varying state delays in this article. Contributions can be summarized as follows.

1. The control schemes proposed in Gao et al. ⁷ and Xu et al. ¹⁸ cannot be applied to systems with unmeasurable states, while the state observer constructed in this paper overcomes the limitation that all states must be measurable. Meanwhile, actuator faults and time-varying state delays are also taken into consideration. Therefore, the control strategy proposed in this paper is more applicable to practical scenarios;

2. Compared with the single channel ET mechanism in Refs.^18–20, a two-channel ET mechanism is proposed to reduce the communication burden in this paper. Moreover, by utilizing the dynamic surface control technique and redesigning virtual controllers and controller, the developed FTC method overcomes the virtual controller nondifferentiability problem by using backstepping control theory;

3. Unlike the results in Refs.^14,28–30, the system stability is guaranteed without prescribed performance. In this paper, the proposed NN event-triggered formation fault-tolerant control method not only ensures the stability of the systems, but also guarantees that the formation control errors are limited within preset boundaries.

Problem statement

Problem formulation of the system

Consider the nonlinear multi-agent systems issue of $N$ followers. Follower $i$ can be presented as

$$\{\begin{array}{c}{\stackrel{·}{x}}_{i,k}=x_{i,k+1}+f_{i,k}\left({\overline{x}}_{i,k}\right)+g_{i,k}\left(x_{i,k}(t-h_{i,k})\right),\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_{i,n_i}=v_i^f+f_{i,n_i}\left({\overline{x}}_{i,n_i}\right)+g_{i,n_i}\left(x_{i,1}(t-h_{i,n_i})\right),}\\ \multicolumn{1}{c}{y=x_{i,1},k=1,\dots ,n_i-1,i=1,2,\cdots ,N,}\end{array}$$

(1)

where ${\overline{x}}_{i,k}={[x_{i,k},\dots ,x_{i,k}]}^{\mathrm{T}}\in R^k$ represents the state. $v_i^f$ and $y_i\in R$ represent input and output of the system, respectively. $f_{i,k}(·)$ denotes the unknown smooth nonlinear function and $g_{i,k}(·)$ denotes the time-delay function. Moreover, we assume that only $y$ can be measured.

In light of Tong et al. ⁹ and Zhang and Tong, ³¹ the loss of effectiveness faults and bias faults are modeled as:

$$v_i^f={\beta }_i\left(t\right)v_i+{\overline{u}}_i\left(t\right),$$

(2)

where ${\beta }_i\left(t\right)\in (0,1]$ represents the time-varying effectiveness factor of the actuator for $i$ th agent and its first-order derivative is bounded such that $\left|{\stackrel{·}{\beta }}_i\left(t\right)\right|\le {\overline{\beta }}_i$ , ${\overline{u}}_i\left(t\right)$ denotes an unknown time-varying bounded function with $\left|{\overline{u}}_i\left(t\right)\right|\le {\overline{u}}_i^{*}$ . From (2), it means that the actuator of $i$ th agent may suffer the loss of effectiveness fault and bias fault.

According to system (1), we can define $y_r\left(t\right)$ as the output of the leader. ⁶ Thus, we define formation control error as

$$\begin{array}{c}{z}_{i,1}=\sum _{j\in M_i}{a}_{i,j}[(y_i-r_i-y_r)-(y_j-r_j-y_r)]\hfill \\ +b_i(y_i-r_i-y_r),\hfill \end{array}$$

(3)

where $y_j$ and $r_i\left(t\right)$ represent neighbor’s signal and the assigned formation variable, respectively.

The control objective of this work is to design a NN event-triggered formation fault-tolerant control scheme for MASs to ensure that:

• (1) In the presence of unmeasurable states, actuator faults, and time-varying delays, all signals of the controlled MASs are semi-globally uniformly ultimately bounded (SGUUB).

• (2)  $N$ followers $y_1,\dots ,y_N$ can follow a leader’s signal $y_r\left(t\right)$ and maintain a desired formation form.

Communication graph

In the networked MASs, for $N$ follower agents encoded from 1 to N, we use a directed graph $G=(H,\epsilon ,\mathfrak{A})$ to present their communication relationship, where $H=\{1,\dots ,N\}$ , $\epsilon =\{(i,j)\in H\times H\}$ and $\mathfrak{A}=\left[a_{\mathrm{ij}}\right]$ are called the node set, the edge set and the weight matrix, respectively. The $(i,j)\in ϵ$ is equivalent to that agent $j$ can obtain the information from $i$ . So, a neighbor set can be represented as $N_i=\{j\mid (i,j)\in \epsilon \}$ , namely, when $j\in N_i$ , $a_{\mathrm{ij}}>0$ , and when $(i,j)\notin N_i$ or $i=j$ , $a_{\mathrm{ij}}=0$ . We assume that $a_{\mathrm{ij}}=0$ and the graph is fixed in this paper. $\mathfrak{L}=\mathfrak{D}-\mathfrak{A}$ represents the laplacian matrix. $\mathfrak{D}=\mathrm{diag}\{q_1,\dots ,q_N\}$ is represented as the in-degree matrix of agent $i$ with $q_i=\sum _{j=1}^N{a}_{\mathrm{ij}}$ . It is important to note that at least one follower can accept a message from a leader.

Radial basis function NNs

RBFNNs can be utilized to approximate any smooth functions $f\left(x\right)$ as Xiao et al. ¹⁴ and Liu et al. ²⁷

$$\begin{array}{c}f\left(x\right)=W^{{*}^{\mathrm{T}}}\mathrm{\Gamma }\left(x\right)+\epsilon \left(x\right),\hfill \end{array}$$

where $\epsilon \left(x\right)$ stands for the approximation error, satisfying $\parallel \epsilon \parallel \le {\epsilon }^{*}$ , ${\epsilon }^{*}>0$ . $W^{*}={[W_1^{*},\dots ,W_l^{*}]}^{\mathrm{T}}$ denotes the ideal weight vector, $l$ is the number of neurons. $\mathrm{\Gamma }\left(x\right)={[{\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_l]}^{\mathrm{T}}$ is the neural network basis function, represented by the Gaussian function. $\phi \left(x\right)=\exp [-{(X-c_l)}^{\mathrm{T}}(X-c_l)/b_l^2]$ . Among them, $b_l$ and $c_l$ represent the width of the Gaussian function and the center of the receptive field, respectively.

To make the control design process more complete and compact, the following assumption and lemma are given.

• Assumption 1 (Yoo ³² ). The reference trajectory vector ${[y_r,{\stackrel{·}{y}}_r,{\stackrel{·}{y}}_r]}^{\mathrm{T}}\in {\mathrm{\Omega }}_b$ is continuous and available, where ${\mathrm{\Omega }}_b:=\{y_r^2+{\stackrel{·}{y}}_r^2+{\stackrel{·}{y}}_r^2\le \mathrm{\Delta }\}$ , and $\mathrm{\Delta }$ is a known positive constant.

• Assumption 2 (Wang et al. ³³ ). At least one follower can access the reference trajectory $y_r\left(t\right)$ directly or through a communication graph.

• Assumption 3 (Li and Tong ³⁴ ). For a nonlinear function $g_{i,k}\left(x_{i,1}(·)\right)$ , there are bounded functions $Q_{i,k}\left(r_{i,1}(·)\right)$ and ${\stackrel{⌣}{g}}_{i,k}\left(y_r(·)\right)$ such that

$$\begin{array}{c}{g}_{i,k}^2\left(x_{i,1}(·)\right)\le r_{i,1}(·)Q_{i,k}\left(r_{i,1}(·)\right)+{\stackrel{⌣}{g}}_{i,k}\left(y_r(·)\right)+{\varrho }_{i,k},\hfill \end{array}$$

where $r_{i,1}=y_i-y_r$ is the tracking error and ${\varrho }_{i,k}>0$ is a positive constant.

• Lemma 1 (Xiao et al. ¹⁴ ). For any variable ${\varsigma }_{i,1}$ and constant ${\gamma }_{i,1}>0$ satisfying $\underset{{\varsigma }_{i,1}\to 0}{\lim }\frac{1}{{\varsigma }_{i,1}}\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)=0$ , a set ${\mho }_a$ is defined suchthat ${\mho }_a=\{{\varsigma }_{i,1}\mid \frac{\left|{\varsigma }_{i,1}\right|}{{\gamma }_{i,1}}<0.8814\}$ . Then, the inequality $1-2\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)\le 0$ is satisfied for ${\varsigma }_{i,1}\notin {\mho }_a$ .

• Remark 1. Assumption 1 addresses that the reference tracking signal and its first and second derivative are bounded, which is a necessary condition for the MASs tracking in DSC. This ensures that the formation tracking errors remain controllable and avoid unbounded control inputs. Assumption 2 ensures that the formation tracking error (3) is well-defined. The requirement for leader-follower connectivity is a common topological constraint in MASs to ensure controllability. Assumption 3 provides the restriction on delay functions. It is obvious to extend the assumption to address the time-delay problem for MASs, where the tracking error $r_{i,1}$ is further considered as formation error $z_{i,1}$ .

Prescribed performance

In this section, to improve the capability of nonlinear systems, the error transformation within a prescribed performance function in Li et al. ²⁶ and Liu et al. ²⁷ can astrict the range of system formation tracking errors $z_{i,1}$ , $i=1,\dots ,N$ . Thereby, we define error transformation function as follows:

$$z_{i,1}={\sigma }_i\left(t\right){\theta }_i\left({\varsigma }_{i,1}\left(t\right)\right),\forall t\ge 0,$$

where $z_{i,1}$ is the formation tracking error, and ${\sigma }_i\left(t\right)=({\sigma }_{i,0}-{\sigma }_{i,\infty })e^{-{\overline{\mu }}_{i,t}}+{\sigma }_{i,\infty }$ . The function ${\sigma }_i\left(t\right)$ : $\underset{t\to \infty }{\lim }{\sigma }_i\left(t\right)={\sigma }_{i,\infty }$ , ${\sigma }_{i,0}={\sigma }_i\left(0\right)>{\sigma }_{i,\infty }$ , ${\theta }_i\left({\varsigma }_{i,1}\left(t\right)\right)$ is an increasing smooth function and ${\varsigma }_{i,1}\left(t\right)$ is the transformed signal. So we have

$${\theta }_i\left({\varsigma }_{i,1}\left(t\right)\right)=\frac{k_{i,2}{e}^{{\varsigma }_{i,1}\left(t\right)}-k_{i,1}{e}^{-{\varsigma }_{i,1}\left(t\right)}}{e^{{\varsigma }_{i,1}\left(t\right)}+e^{-{\varsigma }_{i,1}\left(t\right)}},$$

so

$${\varsigma }_{i,1}\left(t\right)={\theta }^{-1}\left(\frac{z_{i,1}\left(t\right)}{{\sigma }_i\left(t\right)}\right)=\frac{1}{2}\ln \frac{{\theta }_i+k_{i,1}}{k_{i,2}-{\theta }_i},$$

and

$${\stackrel{·}{\varsigma }}_{i,1}\left(t\right)=\frac{1}{2}\frac{d\left(\ln \frac{{\theta }_i+k_{i,1}}{k_{i,2}-{\theta }_i}\right)}{\mathrm{dt}}={\omega }_i({\stackrel{·}{z}}_{i,1}-\frac{z_{i,1}{\stackrel{·}{\sigma }}_i\left(t\right)}{{\sigma }_i\left(t\right)}),$$

(4)

where ${\omega }_i=\frac{1}{2}(\frac{1}{k_{i,1}{\sigma }_i\left(t\right)+z_{i,1}}-\frac{1}{z_{i,1}-k_{i,2}{\sigma }_i\left(t\right)})$ , $k_{i,1}>0$ and $k_{i,2}>0$ . So the prescribed performance can be described as

$$-k_{i,1}{\sigma }_i\left(t\right)<z_{i,1}\left(t\right)<k_{i,2}{\sigma }_i\left(t\right),\forall t\ge 0.$$

State observer design

Rewrite system (1) as the following form:

$$\{\begin{array}{c}{\stackrel{·}{x}}_i=A_i{x}_i+H_i{y}_i+\sum _{k=1}^{n_i}{B}_{i,k}{f}_{i,k}\left({\overline{x}}_{i,k}\right)\hfill \\ \multicolumn{1}{c}{+B_k{v}_i^f+{\overline{g}}_{i,n_i}\left(x_{i,1}(t-h_{i,n_i})\right),}\\ \multicolumn{1}{c}{y_i=C_i{x}_i,}\end{array}$$

(5)

where, $A_i=\left[\begin{array}{cccc}-{\overline{k}}_{i,1}\hfill & 1\hfill & \hfill & \hfill \\ \multicolumn{1}{c}{-{\overline{k}}_{i,2}}& \hfill & \ddots \hfill & \hfill \\ \multicolumn{1}{c}{⋮}& \hfill & \hfill & 1\hfill \\ \multicolumn{1}{c}{-{\overline{k}}_{i,n_i}}& 0\hfill & \cdots \hfill & 0\hfill \end{array}\right]$ , $H_i=\left[\begin{array}{c}{\overline{k}}_{i,1}\hfill \\ \multicolumn{1}{c}{{\overline{k}}_{i,2}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\overline{k}}_{i,n_i}}\end{array}\right]$ , $B_{i,k}={\left[\begin{array}{c}\underset{k}{\underbrace{0,\cdots ,0,1}},0,\dots ,0\hfill \end{array}\right]}_{n_i\times 1}^{\mathrm{T}}$ and $C_i=[1,0,\dots ,0{]}_{1\times n_i}^{\mathrm{T}}$ , $x_i={[x_{i,1},\dots ,x_{i,n_i}]}^{\mathrm{T}}$ , $B_k=[0,0,\dots ,1{]}_{n_i\times 1}^{\mathrm{T}}$ .

Since $f_{i,k}\left({\overline{x}}_{i,k}\right)$ in (5) is unknown, a RBFNN is used to approximate $f_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)$ , then we have

$$\begin{array}{c}{f}_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)={W_{i,k}^{*}}^{\mathrm{T}}{\phi }_{i,k}\left({\overline{x}}_{i,k}\right)+{\epsilon }_{i,k},\hfill \end{array}$$

where ${\epsilon }_{i,k}$ is the approximation error with $\left|{\epsilon }_{i,k}\right|\le {\overline{\epsilon }}_{i,k}$ , and ${\overline{\epsilon }}_{i,k}$ is an unknown constant.

Choose appropriate vector ${\overline{k}}_{i,k}$ so that matrix $A_i$ is Hurwitz. Choose a symmetric matrix $R_i={R_i}^{\mathrm{T}}>0$ to satisfy the following formula

$${A_i}^{\mathrm{T}}{R}_i+R_i{A}_i=-2P_i,$$

(6)

To reduce the data transmission, an ET mechanism is constructed in the sensor-to-observer channel as follows:

$$\begin{array}{c}\begin{array}{c}{\tilde{y}}_i=y_i\left(t_a\right),t\in [t_a,t_{a+1})\hfill \end{array}\hfill \\ \begin{array}{c}{t}_{a+1}=\inf \{t>t_a:|y_i-{\tilde{y}}_i|\ge {\lambda }_{i,y}\}.\hfill \end{array}\hfill \end{array}$$

(7)

Taking into account the output of each follwer is triggered, the NN stste observer is constructed as

$$\begin{array}{c}\{\begin{array}{c}{\stackrel{.}{\hat{x}}}_{i,k}={\hat{x}}_{i,k+1}+{\hat{W}}_{i,k}^{\mathrm{T}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)+{\overline{K}}_{i,k}({\tilde{y}}_i-{\hat{y}}_i),\hfill \\ \multicolumn{1}{c}{{\stackrel{.}{\hat{x}}}_{i,n_i}=g_i{\hat{\beta }}_i{v}_i+{\hat{W}}_{i,n_i}^{\mathrm{T}}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)+{\overline{K}}_{i,n_i}({\tilde{y}}_i-{\hat{y}}_i),}\\ \multicolumn{1}{c}{{\hat{y}}_i={\hat{x}}_{i,1}}\end{array}\hfill \end{array}$$

(8)

where ${\hat{\overline{x}}}_{i,k}^c=[{\tilde{y}}_i,{\hat{\overline{x}}}_{i,2},\dots ,{\hat{\overline{x}}}_{i,n_i}]$ .

• Remark 2. It is worth noting that the NN or fuzzy state observer designs in Refs.^18–20 are only using the output $y_i$ instead of triggered output ${\tilde{y}}_i$ . While the state observer (8) designed in this paper utilizes triggered output ${\tilde{y}}_i$ . Therefore, the proposed state observer (8) has advantages in estimating the unmeasurable states, unknown actuator effectiveness factor and reducing the unnecessary data transmissions in the S-C channel simultaneously.

• Remark 3. The triggered output ${\tilde{y}}_i$ is not continuous, if the backstepping control design^18–20 is adopted directly, the nondifferentiable problem of virtual controllers will happen. Thus, the control design of this paper is more difficult than Refs.^18–20 In this work, by utilizing the dynamic surface control technique (32) and redesigning virtual controllers (29), (36), (39) and (44), the developed FTC method overcomes the virtual controller nondifferentiability problem by using backstepping control theory.

The observer error is defined as $e_i=x_i-{\hat{x}}_i$ . From (5) and (8), we consider the observer error dynamic as

$$\begin{array}{c}{\stackrel{·}{e}}_i=A_i{e}_i+{\overline{\epsilon }}_{i,n_i}+H_i({\tilde{y}}_i-{\hat{y}}_i)+\sum _{k=1}^{n_i}{B}_{i,k}{\tilde{W}}_{i,k}^{\mathrm{T}}\phi \left({\hat{\overline{x}}}_{i,k}^c\right)\hfill \\ \multicolumn{1}{c}{+{\overline{\xi }}_{i,n_i}+B_k({\tilde{\beta }}_i{v}_i+{\overline{u}}_i)+{\overline{g}}_{i,n_i}\left(x_{i,1}(t-h_{i,n_i})\right),}\end{array}$$

(9)

where ${\overline{\epsilon }}_{i,n_i}={[{\epsilon }_{i,1},\dots ,{\epsilon }_{i,n_i}]}^{\mathrm{T}}$ , ${\overline{\xi }}_{i,n_i}={[{\xi }_{i,1},\dots ,{\xi }_{i,n_i}]}^{\mathrm{T}}$ and ${\xi }_{i,k}=f_{i,k}\left({\overline{x}}_{i,k}\right)-f_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)$ .

To verify the validity of the designed state observer, we construct Lyapounov-Krasovskii candidate functional as

$$\begin{array}{c}{V}_0=V_{a0}+V_{b0},\hfill \end{array}$$

(10)

where

$$\begin{array}{c}\multicolumn{1}{c}{V_{a0}=\sum _{i=1}^N{V}_{i,a0}=\sum _{i=1}^N\{\frac{1}{2}{e_i}^{\mathrm{T}}{R}_i{e}_i+\frac{1}{3{\mu }_{i,o}}{\left|{\tilde{\beta }}_i\right|}^3\},}\end{array}$$

(11)

$$\begin{array}{c}{V}_{b0}=\sum _{i=1}^N{V}_{i,b0}=\sum _{i=1}^N\frac{1}{2b}{e}^{-r(t-h_i)}\parallel R_i{\parallel }_F^2*,\hfill \end{array}$$

(12)

where $*=\sum _{k=1}^{n_i}{\int }_{t-h_{i,k}}^t{e}^{r\varphi }{z}_{i,1}\left(\varphi \right)Q_{i,k}\left(z_{i,1}\left(\varphi \right)\right)d\varphi$ ,

${\mu }_{i,o}>0$ , $h_i=max\left\{h_{i,k}\right\}$ and $r$ is a constant.

By using ${\phi }_{i,k}^{\mathrm{T}}\left({\hat{\overline{x}}}_{i,k}^c\right){\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)\le {\iota }_i$ , we get

$$\begin{array}{c}{e_i}^{\mathrm{T}}{R}_i\sum _{k=1}^{n_i}{B}_{i,k}{\tilde{W}}_{i,k}^{\mathrm{T}}\phi \left({\hat{\overline{x}}}_{i,k}^c\right)\hfill \\ \le \frac{n}{2}\parallel R_i{\parallel }_F^2\parallel e_i{\parallel }^2+\frac{1}{2}\sum _{k=1}^{n_i}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k},\hfill \end{array}$$

(13)

$$\begin{array}{c}{e_i}^{\mathrm{T}}{R}_i({\overline{\epsilon }}_{i,n_i}+{\overline{\xi }}_{i,n_i}+H_i({\tilde{y}}_i-y_i)+B_k{\overline{u}}_i)\hfill \\ \le \frac{n}{2}\parallel R_i{\parallel }_F^2+\frac{1}{2}\parallel R_i{\parallel }_F^2\parallel {\overline{\epsilon }}_{i,N}{\parallel }^2+\frac{1}{2}\parallel R_i{\parallel }_F^2\parallel H_i{\parallel }^2{\lambda }_{i,y}^2\hfill \\ +\frac{1}{2}\parallel R_i{\parallel }_F^2\sum _{k=1}^{n_i}{W_{i,k}^{*}}^{\mathrm{T}}{W_{i,k}^{*}}^{\mathrm{T}}+\frac{1}{2}\parallel R_i{\parallel }_F^2{\overline{u}}_i^{*2}+2\parallel e_i{\parallel }^2,\hfill \end{array}$$

(14)

where ${\overline{\epsilon }}_{i,N}={[{\epsilon }_{1,n_i},\dots ,{\epsilon }_{N,n_i}]}^{\mathrm{T}}$ and $n=n_i{\iota }_i$ . According to Assumption 3, we have

$$\begin{array}{c}{e_i}^{\mathrm{T}}{R}_i{\overline{g}}_{i,n_i}\left(x_{i,1}(t-h_{i,n_i})\right)\hfill \\ \multicolumn{1}{c}{\le \frac{1}{2b}\parallel e_i{\parallel }^2+{\varrho }_{i,k}+\sum _{k=1}^{n_i}\frac{1}{2b}\parallel R_i{\parallel }_F^2{g}_{i,k}\left(y_r(t-h_{i,k})\right)}\\ \multicolumn{1}{c}{+\sum _{k=1}^{n_i}\frac{1}{2b}\parallel R_i{\parallel }_F^2{z}_{i,1}(t-h_{i,k})Q_{i,k}\left(z_{i,1}(t-h_{i,k})\right).}\end{array}$$

(15)

To fully design the ${\stackrel{·}{\hat{\beta }}}_i$ , and we consider the state observer error as

$$e_i={[{\tilde{e}}_{i,1},0,\dots ,0]}^{\mathrm{T}}+{[0,e_{i,2},\dots ,e_{i,n_i}]}^{\mathrm{T}}+{[{\mathrm{\Delta }}_{i,1},0,\dots ,0]}^{\mathrm{T}}$$

where ${\tilde{e}}_{i,1}={\tilde{y}}_i-{\hat{y}}_i$ and ${\mathrm{\Delta }}_{i,1}=y_i-{\tilde{y}}_i$ .

Thus, we can obtain

$$\begin{array}{c}{e_i}^{\mathrm{T}}{R}_i{B}_k{\tilde{\beta }}_i{v}_i\le \frac{1}{2}\parallel R_i{\parallel }_F^2\parallel e_i{\parallel }^2+(\frac{{ϵ}_i}{2}{\tilde{e}}_{i,1}+1){\tilde{\beta }}_i^2{v}_i^2\hfill \\ +\frac{1}{2}\parallel R_i{\parallel }_F^2(\frac{1}{{\epsilon }_i}+{\lambda }_{i,y}^2).\hfill \end{array}$$

(16)

The adaptive law can be designed as

$${\stackrel{·}{\hat{\beta }}}_i=\mathrm{pro}{j}_{[\underset{,}{\underset{¯}{{\beta }_i}}1]}\{-{\mu }_{i,o}(\frac{{ϵ}_i}{2}{\tilde{e}}_{i,1}+1)v_i^2-{\delta }_{i,o}{\hat{\beta }}_i\},$$

(17)

where $\underset{¯}{{\beta }_i}>0$ is a constant. The $\mathrm{pro}{j}_{[\underset{¯}{{\beta }_i},1]}$ operator is designed as

$${\stackrel{·}{\hat{\beta }}}_i=\mathrm{pro}{j}_{[\underset{,}{\underset{¯}{{\beta }_i}}1]}\left\{\mathrm{\Gamma }\right\}=\{\begin{array}{c}\mathrm{\Gamma },\mathrm{if}\mathrm{\Gamma }\le 0\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}$$

(18)

Because of the definition in Yoo, ³² the ${\stackrel{·}{\hat{\beta }}}_i\le 0$ is always ensure. So, utilizing Young’s inequality, one gets

$$\begin{array}{c}\frac{1}{{\mu }_{i,o}}{\tilde{\beta }}_i^2{\stackrel{·}{\beta }}_i-\frac{{\delta }_{i,o}}{{\mu }_{i,o}}{\tilde{\beta }}_i^2{\hat{\beta }}_i\le -\frac{{\delta }_{i,o}-2}{3{\mu }_{i,o}}{\left|{\tilde{\beta }}_i\right|}^3+\frac{{\delta }_{i,o}}{3{\mu }_{i,o}}{\beta }_i^3+\frac{1}{3{\mu }_{i,o}}{\overline{\beta }}_i^3.\hfill \end{array}$$

(19)

From (11) to (19), the derivative of $V_0$ is

$$\begin{array}{c}{\stackrel{·}{V}}_0\le \sum _{i=1}^N(-R_0\parallel e_i{\parallel }^2+\frac{1}{2}\sum _{k=1}^{n_i}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k}-\frac{{\delta }_{i,o}-2}{3{\mu }_{i,o}}{\left|{\tilde{\beta }}_i\right|}^3\hfill \\ -r{V}_{i,b0}+{\pi }_{i,0}+\sum _{k=1}^{n_i}\frac{1}{2b}{e}^{r{h}_i}\parallel R_i{\parallel }_F^2{z}_{i,1}\left(t\right)Q_{i,k}\left(z_{i,1}\left(t\right)\right)),\hfill \end{array}$$

(20)

where $R_0=\min \{{\lambda }_{\min }\left(P_i\right)-\frac{n}{2}\parallel R_i{\parallel }_F^2-2-\frac{1}{2}\parallel R_i{\parallel }_F^2\}$ , ${\pi }_{i,0}=\frac{1}{2}\parallel R_i{\parallel }_F^2(\parallel {\overline{\epsilon }}_{i,N}{\parallel }^2+\sum _{k=1}^{n_i}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k}+(\parallel H_i{\parallel }^2+1){\lambda }_{i,y}^2+\frac{1}{{ϵ}_i})+\frac{{\delta }_{i,o}}{3{\mu }_{i,o}}{\beta }_i^3+\frac{1}{3{\mu }_{i,o}}{\overline{\beta }}_i^3+h_o$ , $h_o=\sum _{i=1}^N{\wp }_{i0}$ and ${\wp }_{i0}>\sum _{k=1}^{n_i}(\frac{1}{2b}\parallel R_i{\parallel }_F^2{\stackrel{⌣}{g}}_{i,k}\left(y_r(t-h_{i,k})\right)+{\varrho }_{i,k})$ .

Controller design

In this section, we utilize a recurisve backstepping-based dynamic surface control (DSC) to design formation fault-tolerant controller. The specific steps are as follows.

• Step 1. In light of (4), the derivatives of ${\varsigma }_{i,1}$ is written as

$$\begin{array}{c}{\stackrel{·}{\varsigma }}_{i,1}={\omega }_i({\stackrel{·}{z}}_{i,1}-\frac{z_{i,1}{\stackrel{·}{\sigma }}_i\left(t\right)}{{\sigma }_i\left(t\right)}).\hfill \end{array}$$

(21)

Meanwhile, according to (3), we get

$$\begin{array}{c}{\stackrel{·}{z}}_{i,1}=q_i\left(g_{i,1}(x_{i,1}(t-h_{i,1})+x_{i,2}+f_{i,1}\left({\overline{x}}_{i,1}\right))\right)-b_i{\stackrel{·}{y}}_r\hfill \\ \multicolumn{1}{c}{+{\overline{r}}_i-\sum _{j\in N_i}{a}_{\mathrm{ij}}\left(g_{j,1}(x_{j,1}(t-h_{j,1})+x_{j,2}+f_{j,1}\left({\overline{x}}_{j,1}\right))\right),}\end{array}$$

(22)

where,

$$q_i=(\sum _{j\in N_i}{a}_{\mathrm{ij}}+b_i)\mathrm{and}{\overline{r}}_i=-q_i{\stackrel{·}{r}}_i+\sum _{j\in N_i}{a}_{\mathrm{ij}}{\stackrel{·}{r}}_j.$$

Define nonlinear function as ${\overline{f}}_{j,1}\left(x_{j,1}\right)=-(1/q_i)\sum _{j\in N_i}{a}_{\mathrm{ij}}(x_{j,2}+f_{j,1}\left({\overline{x}}_{j,1}\right))$ and by using a RBFNN to approximate it, then we have

$$\begin{array}{c}{\overline{f}}_{j,1}\left({\overline{x}}_{j,1}\right)={W_{j,1}^{*}}^{\mathrm{T}}{\phi }_{j,1}\left({\overline{x}}_{j,1}\right)+{\epsilon }_{j,1},\hfill \end{array}$$

(23)

where $x_{i,2}=e_{i,2}+{\hat{x}}_{i,2}$ , ${\hat{x}}_{i,2}=z_{i,2}+s_{i,2}+{\alpha }_{i,1}$ .

Putting (23) into (22) and we get

$$\begin{array}{l}{z}_{i,1}\left(t\right)=q_i(e_{i,2}+z_{i,2}+s_{i,2}+g_{i,1}\left(x_{i,1}\left(t-h_{i,1}\right)\right).\\ +{\epsilon }_{j,1}+{\epsilon }_{i,1}+W_{i,1}^{{\ast }^{\mathrm{T}}}{\phi }_{i,1}\left(y_i\right)+W_{j,1}^{{\ast }^{\mathrm{T}}}{\phi }_{j,1}\left(y_j\right)\\ +{\alpha }_{i,1}+\frac{{\overline{r}}_i}{q_i}-\frac{b_i{\stackrel{.}{y}}_r}{q_i})-\sum _{j\in N_i}{g}_{j,1}\left(x_{j,1}\left(t-h_{j,1}\right)\right).\end{array}$$

(24)

The Lyapunov-Krasovskii candidate functional is considered as

$$\begin{array}{c}{V}_{i,1}={\overline{V}}_{i,1}+V_{i,d}+V_{j,f},\hfill \end{array}$$

(25)

Where

$$\begin{array}{c}{\overline{V}}_{i,1}=\frac{1}{2}({\varsigma }_{i,1}^2+\sum _{j\in N_i}\frac{a_{\mathrm{ij}}}{{\eta }_{j,1}}{\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}^{\mathrm{T}}+\frac{1}{{\eta }_{i,1}}{\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1}^{\mathrm{T}}),\hfill \end{array}$$

(26)

$$\begin{array}{c}{V}_{i,d}=\frac{1}{d}{e}^{-r(t-h_{i,1})}{\int }_{t-h_{i,1}}^t{e}^{r\varphi }{z}_{i,1}\left(\varphi \right)Q_{i,1}\left(z_{i,1}\left(\varphi \right)\right)d\varphi ,\hfill \end{array}$$

(27)

$$\begin{array}{c}{V}_{j,f}=\sum _{j\in N_i}\frac{1}{2f}{e}^{-r(t-h_{j,1})}{\int }_{t-h_{j,1}}^t{e}^{r\varphi }{z}_{j,1}\left(\varphi \right)Q_{j,1}\left(z_{j,1}\left(\varphi \right)\right)d\varphi ,\hfill \end{array}$$

(28)

where $d$ , $f$ , ${\eta }_{i,1}$ and ${\eta }_{j,1}$ are positive constants.

In light of the derivative of $V_{i,1}$ , the virtual control law and adaptive laws are constructed as follows:

$$\begin{array}{c}{\tilde{\alpha }}_{i,1}=-({\rho }_{i,1}+\frac{9}{2}+\frac{d}{4}{q}_i+\frac{f{N}_i}{2}){\varsigma }_{i,1}{\omega }_{i,1}+\frac{{\tilde{z}}_{i,1}{\stackrel{·}{\sigma }}_i\left(t\right)}{{\sigma }_i\left(t\right)}\hfill \\ -\frac{{\overline{r}}_i}{q_i}+\frac{b_i{\stackrel{·}{y}}_r}{q_i}-{\hat{W}}_{i,1}^{\mathrm{T}}{\phi }_{i,1}\left(y_i\right)-{\hat{W}}_{j,1}^{\mathrm{T}}{\phi }_{j,1}\left(y_j\right),\hfill \end{array}$$

(29)

$$\begin{array}{c}{\stackrel{.}{\hat{W}}}_{i,1}={\eta }_{i,1}({\varsigma }_{i,1}{\omega }_i{q}_i{\phi }_{i,1}\left({\tilde{y}}_i\right)-{\sigma }_{i,1}{\hat{W}}_{i,1}),\hfill \end{array}$$

(30)

$$\begin{array}{c}{\stackrel{.}{\hat{W}}}_{j,1}={\eta }_{j,1}({\varsigma }_{i,1}{\omega }_i{q}_i{\phi }_{i,1}\left({\tilde{y}}_j\right)-{\sigma }_{j,1}{\hat{W}}_{j,1}).\hfill \end{array}$$

(31)

Step k ( $k=2,\dots ,n_i-1$ ). To solve the problem of “explosion of complexity,” we use DSC technique.

$$\begin{array}{c}{z}_{i,k}={\hat{x}}_{i,k}-{\alpha }_{i,k}^f,\hfill \\ \multicolumn{1}{c}{s_{i,k}={\alpha }_{i,k}^f-{\alpha }_{i,k-1},}\end{array}$$

(32)

where $z_{i,k}$ represents the error surface. ${\alpha }_{i,k}^f$ is a new variable, which satisfies the following first-order filter

$$\begin{array}{c}{r}_{i,k}{\stackrel{·}{\alpha }}_{i,k}^f+{\alpha }_{i,k}^f={\alpha }_{i,k-1}.\hfill \end{array}.$$

(33)

In view of the designed observer (5) and (32), one can get

$$\begin{array}{c}{\stackrel{·}{z}}_{i,k}={\stackrel{.}{\hat{x}}}_{i,k}-{\stackrel{·}{\alpha }}_{i,k}^f\hfill \\ =z_{i,k+1}+s_{i,k+1}+{\alpha }_{i,k}+W_{i,k}^{{*}^{\mathrm{T}}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)\hfill \\ -{\tilde{W}}_{i,k}^{\mathrm{T}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)-W_{i,k}^{{*}^{\mathrm{T}}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)\hfill \\ +W_{i,k}^{{*}^{\mathrm{T}}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)+{\overline{k}}_{i,k}({\tilde{y}}_i-{\hat{y}}_i)-{\stackrel{·}{\alpha }}_{i,k}^f.\hfill \end{array}.$$

(34)

The Lyapunov candidate function is considered as:

$$\begin{array}{c}{V}_{i,k}=\frac{1}{2}(z_{i,k}^2+s_{i,k}^2+\frac{1}{{\eta }_{i,k}}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k})+V_{i,k-1},\hfill \end{array}$$

(35)

where ${\eta }_{i,k}$ is a parameter to be designed.

According to (7), the ET virtual control law is reconstructed as

$$\begin{array}{l}{\tilde{\alpha }}_{i,k}=-\left({\rho }_{i,k}+3\right){\tilde{z}}_{i,k}-{\overline{K}}_{i,k}\left({\tilde{y}}_i-{\widehat{y}}_i\right)\\ -{\widehat{W}}_{i,k}^{\mathrm{T}}{\phi }_{i,k}\left({\widehat{\overline{x}}}_{i,k}^c\right)+\frac{{\tilde{\alpha }}_{i,k-1}-{\tilde{\alpha }}_{i,k}^f}{r_{i,k}},\end{array}$$

(36)

and the adaptive law is constructed as follows:

$$\begin{array}{c}{\stackrel{.}{\hat{W}}}_{i,k}={\eta }_{i,k}({\tilde{z}}_{i,k}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)-{\sigma }_{i,k}{\hat{W}}_{i,k}).\hfill \end{array}$$

(37)

Step $n_i$ . Similar to Step $k$ , we have

$$\begin{array}{c}{\stackrel{·}{z}}_{i,n_i}={\stackrel{.}{\hat{x}}}_{i,n_i}-{\stackrel{·}{\alpha }}_{i,n_i}^f\hfill \\ ={\hat{\beta }}_i{v}_i+{\hat{W}}_{i,n_i}^{\mathrm{T}}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)+{\overline{k}}_{i,k}({\tilde{y}}_i-{\hat{y}}_i)-{\stackrel{·}{\alpha }}_{i,k}^f.\hfill \end{array}$$

(38)

The ET mechanism in the controller-to-actuator channel is designed as

$$\begin{array}{c}{\tilde{\alpha }}_{i,k}^f\left(t\right)={\alpha }_{i,k}^f\left(t_{m,a}\right),t\in [t_{m,a},t_{m,a+1}),\hfill \\ t_{m,a+1}=\inf \{t>t_{m,a},|{\alpha }_{i,k}^f-{\tilde{\alpha }}_{i,k}^f|\ge {\gamma }_{\omega }\},\hfill \end{array}$$

(39)

$$\begin{array}{c}{v}_i={\vartheta }_i\left(t_{n,a}\right),t\in [t_{n,a},t_{n,a+1}),\hfill \\ t_{n,a+1}=\inf \{t>t_{n,a},|{\vartheta }_i-v_i|\ge h_{i,v}\left|v_i\right|+{\lambda }_{i,v}\},\hfill \end{array}$$

(40)

where $0<h_{i,v}<1$ , ${\lambda }_{i,v}>0$ and ${\gamma }_{\omega }>0$ are design parameters.

From (40), we have

$$\begin{array}{c}{v}_i\left(t\right)=\frac{{\vartheta }_i\left(t\right)}{1+{\overline{\lambda }}_{i,1}{h}_{i,v}}-\frac{{\overline{\lambda }}_{i,2}{\lambda }_{i,v}}{1+{\overline{\lambda }}_{i,1}{h}_{i,v}},\hfill \end{array}$$

(41)

where $0<{\overline{\lambda }}_{i,1}<1$ is a parameter and ${\vartheta }_i\left(t\right)$ is similar to the definition in Tong et al. ⁹

From (2) and (40), one has

$$\begin{array}{c}{z}_{i,n_i}{\hat{\beta }}_i{v}_i=z_{i,n_i}{\hat{\beta }}_i(\frac{{\vartheta }_i\left(t\right)}{1+{\overline{\lambda }}_{i,1}{h}_{i,v}}-\frac{{\overline{\lambda }}_{i,2}{\lambda }_{i,v}}{1+{\overline{\lambda }}_{i,1}{h}_{i,v}}).\hfill \end{array}$$

(42)

Consider the Lyapunov candidate function as follows:

$$\begin{array}{c}\multicolumn{1}{c}{V_{i,n_i}=\frac{1}{2}(z_{i,n_i}^2+s_{i,n_i}^2+\frac{1}{{\eta }_{i,n_i}}{\tilde{W}}_{i,n_i}^{\mathrm{T}}{\tilde{W}}_{i,n_i})+V_{i,n_i-1},}\end{array}$$

(43)

according to (7), (39) and (40), the ET virtual control law is reconstructed as

$$\begin{array}{c}{\tilde{\alpha }}_{i,n_i}=-({\rho }_{i,n_i}+3){\tilde{z}}_{i,n_i}-{\overline{k}}_{i,n_i}({\tilde{y}}_i-{\hat{y}}_i)\hfill \\ \multicolumn{1}{c}{-{\hat{W}}_{i,n_i}^{\mathrm{T}}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)+\frac{{\tilde{\alpha }}_{i,n_i-1}-{\tilde{\alpha }}_{i,n_i}^f}{r_{i,n_i}},}\end{array}$$

(44)

and the adaptive law is constructed as follows:

$$\begin{array}{c}{\stackrel{.}{\hat{W}}}_{i,n_i}={\eta }_{i,n_i}({\tilde{z}}_{i,n_i}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)-{\sigma }_{i,n_i}{\hat{W}}_{i,n_i}).\hfill \end{array}$$

(45)

The designed FTC control algorithm can be seen in Figure 1

Figure 1.

The designed FTC control algorithm.

Stability analysis

In the section, we give a detailed proof that the proposed control algorithm can ensure the stability of nonlinear MASs (1). The proof process is described as follow.

• Theorem 1. Taking into account the nonlinear multi-agent systems (1) under Assumption 1-3. If the observer (8), the virtual controllers (29), (36) and (44), NN weight adaptive laws (30), (31), (37) and (45) are adopted. The proposed NN event-triggered formation fault-tolerant control algorithm can ensure the following three conclusions:

• (1) All signals are SGUUB;

• (2) All agents can maintain a desired formation form;

• (3) Zeno phenomenon caused by the ET mechanism can be avoided.

Proof. To make the explanation clearer, we divide the proof process into the following three parts.

First, according to Lyapunov stability theory, Lyapunov-Krasovskii functional $V$ is selected as all followers

$$\begin{array}{c}V=V_0+\sum _{i=1}^N{V}_{i,n_i},\hfill \end{array}$$

(46)

according to the time derivative ${\overline{V}}_{i,1}$ , under Young’s inequality, one gets

$$\begin{array}{c}{\varsigma }_{i,1}{\omega }_i{q}_i(e_{i,2}+{\epsilon }_{i,1}+{\epsilon }_{j,1})\hfill \\ \le \frac{3}{2}{q}_i{\varsigma }_{i,1}^2{\omega }_i^2+\frac{q_i}{2}{\epsilon }_{i,1}^{*2}+\frac{q_i}{2}{\epsilon }_{j,1}^{*2}+\frac{q_i}{2}\parallel e_i{\parallel }^2,\hfill \end{array}$$

(47)

$$\begin{array}{c}{\varsigma }_{i,1}{\omega }_i{q}_i({\tilde{W}}_{i,1}^{\mathrm{T}}{\phi }_{i,1}\left(y_i\right)-{\tilde{W}}_{i,1}^{\mathrm{T}}{\phi }_{i,1}\left({\tilde{y}}_i\right))\hfill \\ \le q_i{\varsigma }_{i,1}^2{\omega }_i^2+q_i{\iota }_i{\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1},\hfill \end{array}$$

(48)

$$\begin{array}{c}{\varsigma }_{i,1}{\omega }_i{q}_i({\tilde{W}}_{j,1}^{\mathrm{T}}{\phi }_{j,1}\left(y_j\right)-{\tilde{W}}_{j,1}^{\mathrm{T}}{\phi }_{j,1}\left({\tilde{y}}_j\right))\hfill \\ \le q_i{\varsigma }_{i,1}^2{\omega }_i^2+q_i{\iota }_i{\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}.\hfill \end{array}$$

(49)

According to Assumption 3, it can be obtained that

$$\begin{array}{c}{\varsigma }_{i,1}{\omega }_i{q}_i{g}_{i,1}\left(x_{i,1}(t-h_{i,1})\right)-{\varsigma }_{i,1}{\omega }_i\sum _{j\in N_i}{g}_{j,1}\left(x_{j,1}(t-h_{j,1})\right)\hfill \\ \le \frac{1}{d}(z_{i,1}(t-h_{i,1})Q_{i,1}\left(z_{i,1}(t-h_{i,1})\right)+{\stackrel{⌣}{g}}_{i,1}\left(y_r(t-h_{i,1})\right)+{\sigma }_{i,1})\hfill \\ +\frac{1}{2f}\sum _{j\in N_i}(z_{j,1}(t-h_{j,1})Q_{j,1}\left(z_{j,1}(t-h_{j,1})\right)+{\stackrel{⌣}{g}}_{j,1}\left(y_r(t-h_{j,1})\right))\hfill \\ +\frac{d}{4}{q}_i^2{\varsigma }_{i,1}^2{\omega }_i^2+\frac{f{N}_i}{2}{\varsigma }_{i,1}^2{\omega }_i^2,\hfill \end{array}$$

(50)

where,

$$\begin{array}{c}{\mathrm{\Xi }}_{i,1}=\sum _{j\in N_i}\frac{e^{r{h}_{i,1}}}{d}{z}_{i,1}\left(t\right)Q_{i,1}\left(z_{i,1}\left(t\right)\right)\hfill \\ +\sum _{j\in N_i}\frac{e^{r{h}_{j,1}}}{2f}{z}_{j,1}\left(t\right)Q_{j,1}\left(z_{j,1}\left(t\right)\right)\hfill \\ +\sum _{k=1}^{n_i}\frac{1}{2b}{e}^{r{h}_i}\parallel R_i{\parallel }_F^2{z}_{i,1}\left(t\right)Q_{i,k}\left(z_{i,1}\left(t\right)\right).\hfill \end{array}$$

Combining RBFNNs with Young’s inequality, we have

$$\begin{array}{c}{\varsigma }_{i,1}{\omega }_i{q}_i\frac{2}{{\varsigma }_{i,1}{\omega }_i{q}_i}\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right){\mathrm{\Xi }}_{i,1}\hfill \\ \le {\varsigma }_{i,1}{\omega }_i{q}_i(W_{i,l_1}^{{*}^{\mathrm{T}}}{\phi }_{i,l_1}\left({\hat{\overline{x}}}_{i,l_1}\right)+{\epsilon }_{i,l_1})\hfill \\ \le q_i{\varsigma }_{i,1}^2{\omega }_i^2+\frac{q_i{\iota }_i}{2}{W}_{i,l_1}^{{*}^{\mathrm{T}}}{W}_{i,l_1}^{*}+\frac{q_i}{2}{\epsilon }_{i,l_1}^{*2},\hfill \end{array}$$

(51)

where ${\epsilon }_{i,l_1}$ is the approximation error.

In light of $V_1=\sum _{i=1}^M{V}_{i,1}$ , substituting (46)–(51) into the derivative of $V_{i,1}$ results in

$$\begin{array}{c}{\stackrel{·}{V}}_{i,1}\le -{\rho }_{i,1}{q}_i{\varsigma }_{i,1}^2{\omega }_i^2+q_i{\iota }_i{\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1}+q_i{\iota }_i{\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}\hfill \\ -r{V}_{i,d}-\sum _{j\in N_i}r{V}_{j,f}+{\varsigma }_{i,1}{\omega }_i{q}_i(s_{i,2}+z_{i,2})+{\pi }_{i,1}\hfill \\ +{\sigma }_{i,1}{\tilde{W}}_{i,1}^{\mathrm{T}}{\hat{W}}_{i,1}+{\sigma }_{j,1}{\tilde{W}}_{j,1}^{\mathrm{T}}{\hat{W}}_{j,1}+\frac{q_i}{2}\parallel e_i{\parallel }^2\hfill \\ +(1-\frac{2}{{\varsigma }_{i,1}{\omega }_i{q}_i}\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\mathrm{\Xi }}_{i,1},\hfill \end{array}$$

(52)

where ${\pi }_{i,1}=\frac{q_i{\iota }_i}{2}{W}_{i,l_1}^{{*}^{\mathrm{T}}}{W}_{i,l_1}^{*}+\frac{q_i}{2}{\epsilon }_{i,1}^{*2}+\frac{q_i}{2}{\epsilon }_{j,1}^{*2}+\frac{q_i}{2}{\epsilon }_{i,l_1}^{*2}$ .

Then, utilizing Young’s inequality, we have

$$\begin{array}{c}{z}_{i,k}{\overline{k}}_{i,k}({\tilde{y}}_i-y_i)\le \frac{1}{2}{z}_{i,k}^2+\frac{{\overline{k}}_{i,k}^2}{2}{\lambda }_{i,y}^2,\hfill \end{array}$$

(53)

$$\begin{array}{c}-z_{i,k}{\tilde{W}}_{i,k}^{\mathrm{T}}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)\le \frac{1}{2}{z}_{i,k}^2+\frac{{\iota }_i}{2}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k},\hfill \end{array}$$

(54)

$$\begin{array}{c}{z}_{i,k}{W}_{i,k}^{{*}^{\mathrm{T}}}({\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right)-{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}\right))\hfill \\ \le z_{i,k}^2+{\iota }_i{W}_{i,k}^{{*}^{\mathrm{T}}}{W}_{i,k}^{*}.\hfill \end{array}$$

(55)

$$\begin{array}{c}{\tilde{W}}_{i,k}^{\mathrm{T}}(z_{i,k}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}\right)-{\tilde{z}}_{i,k}{\phi }_{i,k}\left({\hat{\overline{x}}}_{i,k}^c\right))\hfill \\ \le z_{i,k}^2+\frac{3{\iota }_i}{2}{\tilde{W}}_{i,k}^{\mathrm{T}}{\tilde{W}}_{i,k}+\frac{1}{2}{\gamma }_{\omega }^2.\hfill \end{array}$$

(56)

Substituting (53)–(56) into the derivative of $V_{i,k}$ results in

$$\begin{array}{c}{\stackrel{·}{V}}_{i,k}\le -\sum _{q=2}^k{\rho }_{i,q}{z}_{i,q}^2+\sum _{q=2}^k{s}_{i,q}(-\frac{s_{i,q}}{r_{i,q}}+E_{i,q}(·))\hfill \\ +\sum _{q=2}^k{z}_{i,q}(z_{i,q+1}+s_{i,q})+\sum _{q=2}^{k}2{\iota }_i{\tilde{W}}_{i,q}^{\mathrm{T}}{\tilde{W}}_{i,q}\hfill \\ +\sum _{q=1}^k{\sigma }_{i,q}{\tilde{W}}_{i,q+}^{\mathrm{T}}{\hat{W}}_{i,q}+{\sigma }_{j,1}{\tilde{W}}_{j,1}^{\mathrm{T}}{\hat{W}}_{j,1}-{\rho }_{i,1}{q}_i{\varsigma }_{i,1}^2{\omega }_i^2\hfill \\ +q_i{\iota }_i{\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1}+q_i{\iota }_i{\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}-r{V}_{i,d}-\sum _{j\in N_i}r{V}_{j,f}\hfill \\ +{\varsigma }_{i,1}{\omega }_i{q}_i(s_{i,2}+z_{i,2})+\frac{q_i}{2}\parallel e_i{\parallel }^2\hfill \\ +(1-2\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\mathrm{\Xi }}_{i,1}+{\pi }_{i,q},\hfill \end{array}$$

(57)

where ${\pi }_{i,q}={\pi }_{i,q-1}+{\iota }_i{W}_{i,k}^{{*}^{\mathrm{T}}}{W}_{i,k}^{*}+\frac{1}{2}{\gamma }_{\omega }^2+\frac{{\overline{k}}_{i,k}^2}{2}{\lambda }_{i,y}^2$ .

Similarly, one can obtain

$$\begin{array}{c}\multicolumn{1}{c}{z_{i,n_i}{\overline{k}}_{i,n_i}({\tilde{y}}_i-y_i)\le \frac{1}{2}{z}_{i,n_i}^2+\frac{{\overline{k}}_{i,n_i}^2}{2}{\lambda }_{i,y}^2,}\end{array}$$

(58)

$$\begin{array}{c}-z_{i,n_i}{\tilde{W}}_{i,n_i}^{\mathrm{T}}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)\le \frac{1}{2}{z}_{i,n_i}^2+\frac{{\iota }_i}{2}{\tilde{W}}_{i,n_i}^{\mathrm{T}}{\tilde{W}}_{i,n_i},\hfill \end{array}$$

(59)

$$\begin{array}{c}{z}_{i,n_i}(W_{i,n_i}^{{*}^{\mathrm{T}}}({\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)-{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}\right))\hfill \\ +{\tilde{W}}_{i,n_i}^{\mathrm{T}}({\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}\right)-{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)))\hfill \\ \le 2z_{i,n_i}^2+{\iota }_i{W}_{i,n_i}^{{*}^{\mathrm{T}}}{W}_{i,n_i}^{*}+{\iota }_i{\tilde{W}}_{i,n_i}^{\mathrm{T}}{\tilde{W}}_{i,n_i}.\hfill \end{array}$$

(60)

$$\begin{array}{c}{\tilde{W}}_{i,n_i}^{\mathrm{T}}(z_{i,n_i}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right)-{\tilde{z}}_{i,n_i}{\phi }_{i,n_i}\left({\hat{\overline{x}}}_{i,n_i}^c\right))\hfill \\ \le \frac{{\iota }_i}{2}{\tilde{W}}_{i,n_i}^{\mathrm{T}}{\tilde{W}}_{i,n_i}+\frac{1}{2}{\gamma }_{\omega }^2.\hfill \end{array}$$

(61)

In light of (42) and according to $0\le \left|\mathrm{\Theta }\right|-\mathrm{\Theta }\tanh (\mathrm{\Theta }/\varrho )\le 0.2785\varrho$ , ( $\mathrm{\Theta }\in R$ , $\varrho >0$ ), we have:

$$\begin{array}{c}{\tilde{z}}_{i,n_i}{\hat{\beta }}_i{v}_i\le -\left|{\tilde{z}}_{i,n_i}{\tilde{\alpha }}_{i,n_i}\right|-\left|{\tilde{z}}_{i,n_i}{\overline{\lambda }}_{i,v}\right|\hfill \\ +\left|\frac{{\tilde{z}}_{i,n_i}{\lambda }_{i,v}}{1+{\overline{\lambda }}_{i,1}{h}_{i,v}}\right|+0.557o_i\hfill \\ \le {\tilde{z}}_{i,n_i}{\tilde{\alpha }}_{i,n_i}+0.557o_i.\hfill \end{array}$$

(62)

Utilizing Young’s inequality, we can get:

$$\begin{array}{c}{\tilde{W}}_{i,q}^{\mathrm{T}}{\hat{W}}_{i,q}\le \frac{1}{2}{W}_{i,q}^{{*}^{\mathrm{T}}}{W}_{i,q}^{*}-\frac{1}{2}{\tilde{W}}_{i,q}^{\mathrm{T}}{\tilde{W}}_{i,q},\hfill \end{array}$$

(63)

$$\begin{array}{c}\multicolumn{1}{c}{{\varsigma }_{i,1}{\omega }_i{q}_i(s_{i,2}+z_{i,2})\le \frac{1}{2}{\varsigma }_{i,1}^2{\omega }_i^2{q}_i^2+\frac{1}{2}{s}_{i,2}^2+\frac{1}{2}{z}_{i,2}^2,}\end{array}$$

(64)

$$\begin{array}{c}{z}_{i,q}(z_{i,q+1}+s_{i,q+1})+s_{i,q}{E}_{i,k}(·)\hfill \\ \le \frac{1}{2}{z}_{i,q+1}^2+\frac{1}{2}{s}_{i,q+1}^2+z_{i,q}^2+\frac{1}{2}{s}_{i,k}^2+\frac{1}{2}{\overline{E}}_{i,q}^2.\hfill \end{array}$$

(65)

Thus, substituting (58)–(65) into the derivative of $V_{i,n_i}$ results in

$$\begin{array}{c}{\stackrel{·}{V}}_{i,n_i}\le -\sum _{q=2}^{n_i}({\rho }_{i,q}-\frac{3}{2})z_{i,q}^2+\sum _{q=2}^{n_i}{s}_{i,q}^2(\frac{1}{r_{i,q}}-1)\hfill \\ -\sum _{q=2}^{n_i}(\frac{{\sigma }_{i,q}}{2}-2{\iota }_i){\tilde{W}}_{i,q}^{\mathrm{T}}{\tilde{W}}_{i,q}-({\rho }_{i,1}-\frac{q_i}{2})q_i{\varsigma }_{i,1}^2{\omega }_i^2\hfill \\ -(\frac{{\sigma }_{i,1}}{2}-q_i{\iota }_i){\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1}-(\frac{{\sigma }_{j,1}}{2}-q_i{\iota }_i){\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}\hfill \\ -r{V}_{i,d}-\sum _{j\in N_i}r{V}_{j,f}+\frac{q_i}{2}\parallel e_i{\parallel }^2+{\overline{\pi }}_{i,n_i}\hfill \\ +(1-2\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\mathrm{\Xi }}_{i,1},\hfill \end{array}$$

(66)

where ${\overline{\pi }}_{i,n_i}={\pi }_{i,n_i-1}+{\iota }_i{W}_{i,k}^{{*}^{\mathrm{T}}}{W}_{i,k}^{*}+\frac{1}{2}{\gamma }_{\omega }^2+\frac{{\overline{k}}_{i,n_i}^2}{2}{\lambda }_{i,y}^2+0.557o_i+\frac{{\sigma }_{i,q}}{2}{W}_{i,q}^{{*}^{\mathrm{T}}}{W}_{i,q}^{*}$ .

Finally, from (52), (57) and (66), one has

$$\begin{array}{c}\stackrel{·}{V}\le \sum _{i=1}^N(-l_0\parallel e_i{\parallel }^2-\frac{{\delta }_{i,o}-2}{3{\mu }_{i,o}}{\left|{\tilde{\beta }}_i\right|}^3-\sum _{q=2}^{n_i}({\rho }_{i,q}-\frac{3}{2})z_{i,q}^2\hfill \\ -\sum _{q=2}^{n_i}{s}_{i,q}^2(\frac{1}{r_{i,q}}-1)-\sum _{q=2}^{n_i}(\frac{{\sigma }_{i,q}}{2}-2{\iota }_i-\frac{1}{2}){\tilde{W}}_{i,q}^{\mathrm{T}}{\tilde{W}}_{i,q}\hfill \\ -\sum _{j\in N_i}r{V}_{j,f}-({\rho }_{i,1}-\frac{3}{2})q_i{\varsigma }_{i,1}^2{\omega }_i^2-r{V}_{i,b_0}-r{V}_{i,d}\hfill \\ -(\frac{{\sigma }_{i,1}}{2}-q_i{\iota }_i-1){\tilde{W}}_{i,1}^{\mathrm{T}}{\tilde{W}}_{i,1}-(\frac{{\sigma }_{j,1}}{2}-q_i{\iota }_i){\tilde{W}}_{j,1}^{\mathrm{T}}{\tilde{W}}_{j,1}\hfill \\ +(1-2\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\mathrm{\Xi }}_{i,1})+\mathrm{\Gamma },\hfill \end{array}$$

(67)

where $l_0=R_0-\frac{q_i}{2}$ , ${\chi }_{i,q}={\rho }_{i,q}-\frac{3}{2}$ and $\mathrm{\Gamma }=\sum _{i=1}^N({\pi }_{i,0}+{\overline{\pi }}_{i,n_i})$ , we choose the parameter as $C_i=min\{\frac{2l_0}{{\lambda }_{\min }\left(R_i\right)},{\delta }_{i,o}-2,2{\chi }_{i,q},\frac{2}{r_i}-2,{\sigma }_{i,q}-4{\iota }_i-1,2{\chi }_{i,1}{q}_i{\omega }_i^2,{\sigma }_{i,1}-2q_i{\iota }_i-1,{\sigma }_{j,1}-2q_i{\iota }_i\}$ and $\mathrm{\Psi }=\min \{C_1,\dots ,C_N\}$ , $i=1,\dots ,N$ , $q=2,\dots ,n_i-1$ .

Thus, one obtains

$$\begin{array}{c}\stackrel{·}{V}\le -\mathrm{\Psi }V\left(t\right)+\mathrm{\Gamma }+\sum _{i=1}^N(1-2\tan h^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\mathrm{\Xi }}_{i,1},\hfill \end{array}$$

(68)

$\mathrm{\Psi }$ and $\mathrm{\Gamma }$ will become positive constants by adjusting constantly relevant parameters. However, $(1-2tan{h}^2\left(\frac{{\varsigma }_{i,1}}{{\gamma }_{i,1}}\right)){\Xi }_{i,1}$ in (68) can be positive or it can be negative, which depends on the size of ${\varsigma }_{i,1}$ . Therefore, it is necessary to consider the following two situations.

According to Lemma 1, we can get

$$\begin{array}{c}\stackrel{·}{V}\le -\mathrm{\Psi }V\left(t\right)+\stackrel{⌣}{\mathrm{\Gamma }},\hfill \end{array}$$

(69)

where $\stackrel{⌣}{\mathrm{\Gamma }}=\mathrm{\Gamma }$ , for ${\varsigma }_{i,1}\notin {\mho }_a$ , and where $\stackrel{⌣}{\mathrm{\Gamma }}={\mathrm{\Gamma }}^{*}$ , for ${\varsigma }_{i,1}\in {\mho }_a$ .

Solving (69), we get

$$\begin{array}{c}0\le V\le V\left(0\right)e^{-\mathrm{\Psi }t}+\frac{\stackrel{⌣}{\mathrm{\Gamma }}}{\mathrm{\Psi }}.\hfill \end{array}$$

(70)

From (10), (46) and (70), we get $e_i$ , ${\tilde{\beta }}_i$ , ${\varsigma }_{i,1}$ , ${\tilde{W}}_{j,1}$ , ${\tilde{W}}_{i,k}$ , $z_{i,k}$ and $s_{i,k}$ are bounded. According to Lemma 1, it shows that ${\tilde{z}}_{i,k}$ is bounded. Since ${\beta }_i$ and $W_{i,k}$ are bounded, ${\hat{\beta }}_i$ and ${\hat{w}}_{i,k}$ are bounded. ${\vartheta }_i$ is a function of ${\hat{\beta }}_i$ and ${\tilde{z}}_{i,k}$ , thus, ${\vartheta }_i$ and $v_i$ are also bounded. Therefore, conclusion (1) has been proven.

Second, we define $z_i={[z_{1,1},\dots ,z_{N,1}]}^{\mathrm{T}}$ . Then we can get $z_i$ and $e_i$ converge to

$$\begin{array}{c}\parallel z_i\parallel \le \sqrt{2V\left(0\right)e^{-\mathrm{\Psi }t}+\stackrel{⌣}{\mathrm{\Gamma }}/\mathrm{\Psi }}\hfill \end{array}$$

(71)

$$\begin{array}{c}\parallel e_i\parallel \le \sqrt{(2V\left(0\right)e^{-\mathrm{\Psi }t}+\stackrel{⌣}{\mathrm{\Gamma }}/\mathrm{\Psi })/{\lambda }_{\min }\left(R_i\right)}.\hfill \end{array}$$

• Remark 4. Note that the parameter selections are only sufficient conditions to guarantee the stability of the controlled system. However, from (67) and (71), it is clearly that tracking errors $z_i$ are related to $c_{i,q}$ , ${\sigma }_{i,q}$ and $r_{i,q}$ . The tracking errors $z_i$ can be adjusted to be small parameter $r_{i,q}$ , but the large parameters $c_{i,q}$ and ${\sigma }_{i,q}$ . Therefore, we can make tracking error $z_i$ be smaller, and thus the conclusion (2) is proven.

Third, we define $t_{a+1}-t_a\ge t_{i,1}$ , $t_{m,a+1}-t_{m,a}\ge t_{i,2}$ and $t_{n,a+1}-t_{n,a}\ge t_{i,3}$ , where $t_{i,1}$ , $t_{i,2}$ and $t_{i,3}$ are positive constants. Let $e_y=y_i\left(t\right)-{\tilde{y}}_i\left(t\right)$ for $t\in [t_a,t_{a+1})$ , $e_{\alpha }={\alpha }_{i,k}^f-{\tilde{\alpha }}_{i,k}^f$ for $t\in [t_{m,a},t_{m,a+1})$ and ${\hslash }_3$ . $e_v={\vartheta }_i\left(t\right)-v_i\left(t\right)$ for $t\in [t_{n,a},t_{n,a+1})$ . Besides, from (70), it follows ${\stackrel{·}{y}}_i$ , ${\stackrel{·}{\alpha }}_{i,k}^f$ and ${\stackrel{·}{\vartheta }}_i$ are bounded. Then, there are ${\hslash }_1$ , ${\hslash }_2$ and satisfy $\left|{\stackrel{·}{y}}_i\right|\le {\hslash }_1$ , $\left|{\stackrel{·}{\alpha }}_{i,k}^f\right|\le {\hslash }_2$ and $\left|{\stackrel{·}{\vartheta }}_i\right|\le {\hslash }_3$ . Then, the time derivative of $\left|e_y\right|$ , $\left|e_{\alpha }\right|$ and $\left|e_v\right|$ are as follows:

$$\begin{array}{c}\frac{d}{\mathrm{dt}}\left|e_y\right|={\stackrel{·}{e}}_y\mathrm{sign}\left(e_y\right)\le \left|{\stackrel{·}{y}}_i\right|\le {\hslash }_1,\hfill \\ \multicolumn{1}{c}{\frac{d}{\mathrm{dt}}\left|e_{\alpha }\right|={\stackrel{·}{e}}_{\alpha }\mathrm{sign}\left(e_{\alpha }\right)\le \left|{\stackrel{·}{\alpha }}_{i,k}^f\right|\le {\hslash }_2,}\\ \multicolumn{1}{c}{\frac{d}{\mathrm{dt}}\left|e_v\right|={\stackrel{·}{e}}_v\mathrm{sign}\left(e_v\right)\le \left|{\stackrel{·}{\vartheta }}_i\right|\le {\hslash }_3.}\end{array}$$

(72)

According to (7), (39) and (40), event occurs when $\left|e_y\right|\ge {\lambda }_{i,y}$ , $\left|e_{\alpha }\right|\ge {\gamma }_{\omega }$ and $\left|e_v\right|\ge h_{i,v}\left|v_i\right|+{\lambda }_{i,v}$ . After integrating both sides of formula (72) with respect to $t\in [t_a,t_{a+1}]$ , $t\in [t_{m,a},t_{m,a+1}]$ and $t\in [t_{n,a},t_{n,a+1}]$ , we have

$$\begin{array}{c}{\lambda }_{i,y}\le {\hslash }_1(t_{a+1}-t_a),\hfill \\ {\gamma }_{\omega }\le {\hslash }_2(t_{m,a+1}-t_{m,a}),\hfill \\ h_{i,v}\left|v_i\right|+{\lambda }_{i,v}\le {\hslash }_3(t_{n,a+1}-t_{n,a}{t}_{n,a}).\hfill \end{array}$$

(73)

Further simplification of (73) can be obtained

$$\begin{array}{c}{t}_{a+1}-t_a\ge \frac{{\lambda }_{i,y}}{{\hslash }_1}=t_{i,1},\hfill \\ t_{m,a+1}-t_{m,a}\ge \frac{{\gamma }_{\omega }}{{\hslash }_2}=t_{i,2},\hfill \\ t_{n,a+1}-t_{n,a}\ge \frac{h_{i,v}\left|v_i\right|+{\lambda }_{i,v}}{{\hslash }_3}=t_{i,3}.\hfill \end{array}$$

(74)

Obviously, $t_{a+1}-t_a$ , $t_{m,a+1}-t_{m,a}$ and $t_{n,a+1}-t_{n,a}$ have positive lower bound. Therefore, zeno phenomenon can be successfully excluded. As far, the proof of Theorem 1 has been completed.

Simulation results

The validity of the proposed formation control algorithm can be illustrated by the example below.

$$\begin{array}{c}\{\begin{array}{c}{\stackrel{·}{x}}_{i,1}=x_{i,2}-x_{i,1}{e}^{-0.5x_{i,1}}+0.1\sin x_{i,1},\hfill \\ {\stackrel{·}{x}}_{i,2}=v_i^f+\frac{x_{i,1}(t-h_1)}{1+x_{i,1}^2(t-h_2)}\hfill \\ +x_{i,1}{\sin }_{i,2}^2+0.1\cos x_{i,1}\sin x_{i,1},\hfill \\ y_i=x_{i,1}.\hfill \end{array}\hfill \end{array}$$

The $\mathfrak{L}$ is described as $\mathfrak{L}=[0,0,0,0;-1,1,0,0;0,-1,1,0;0,0,-1,1]$ . The functions are selected as ${\sigma }_{i,1}\left(t\right)=0.3e^{-0.6t}+0.05$ , ${\sigma }_{i,2}\left(t\right)=-0.32e^{-0.6t}-0.06$ .

The parameters are designed as follows: $c=1$ , $M_i=0.2$ , $b=2$ , ${\lambda }_{i,y}=0.02$ , ${\gamma }_{\omega }=0.001$ , $h_{i,v}=0.3$ , ${\lambda }_{i,v}=1.5$ , $r_{i,1}=r_{i,2}=0.08$ , ${\rho }_{i,1}={\rho }_{i,2}=30$ , ${\overline{K}}_{i,2}=300$ . $h_{i,1}=1$ , $h_{i,2}=1.2$ , $k_{i,1}=0.8$ , $k_{i,2}=1$ , ${\overline{K}}_{i,1}=100$ .

The initial state can be defined as $x_1\left(0\right)=[0.01,0.01]$ , $x_2\left(0\right)=[-0.02,-0.01]$ , $x_3\left(0\right)=[0.02,0.01]$ , $x_4\left(0\right)=[-0.02,0.01]$ , $f_{i,k}\left(0\right)=0$ , $g_{i,k}\left(0\right)=0,{\hat{x}}_{i,k}\left(0\right)=0.1$ , ${\hat{w}}_{i,k}\left(0\right)=0.1$ , $i$ =1, 2, 3, 4, $k$ =1, 2. $h_i=[0.10,0.53,0.5,0.08]$ .

To demonstrate, the intermittent fault is described as follows:

$$f_i\left(t\right)=\{\begin{array}{c}-0.4\sin \left(4t\right),\mathrm{if}3<t<7,\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}$$

The following two cases are employed to validate the effectiveness of the method proposed in this paper.

• Case 1 (Chain Topology with Intermittent Faults) In this case, a linear chain topology is implemented where agents connect sequentially. The connectivity graph is shown in Figure 2.

Figure 2.

Communication topology.

The simulation results are presented in Figures 3 to 9. Figure 3 shows the curves of followers, illustrating that the proposed algorithm achieves the formation control objective. The unmeasurable states $x_{i,1}$ estimated by the observer are shown in Figure 4, it is easy to show that the designed observer is effective. Figure 5 shows that the formation tracking errors $z_{i,1}$ decay within the PP bounds. Figure 6 indicates that the formation error of the proposed method remains constrained within PP bounds, whereas the approach in Xiao et al. ¹⁴ violates these bounds and exhibits inferior stability. The triggering times of $y_i$ and $v_i$ are shown in Figures 7 and 8. Figure 9 shows that the proposed FTC can keep the system stable even after the intermittent faults occur.

• Case 2 (Ring Topology with Intermittent Faults) In this case, a ring topology connects all agents symmetrically.

Figure 3.

Curves of followers for case 1.

Figure 4.

Curves of $x_{k,1}$ and ${\hat{x}}_{k,1}$ , $k$  = 1,2,3,4: (a) curves of $x_{1,1}$ and ${\hat{x}}_{1,1}$ , (b) curves of $x_{2,1}$ and ${\hat{x}}_{2,1}$ , (c) curves of $x_{3,1}$ and ${\hat{x}}_{3,1}$ , and (d) curves of $x_{4,1}$ and ${\hat{x}}_{4,1}$ .

Figure 5.

The trajectories of errors $z_{i,1}$ , $i$  = 1,2,3,4: (a) the trajectories of errors $z_{1,1}$ , (b) the trajectories of errors $z_{2,1}$ , (c) the trajectories of errors $z_{3,1}$ , and (d) the trajectories of errors $z_{4,1}$ .

Figure 6.

Comparison results in formation tracking error.

Figure 7.

Triggering times of $y_i$ for case 1, $i$  = 1,2,3,4.

Figure 8.

Triggering times of $v_i$ for case 1, $i$  = 1,2,3,4.

Figure 9.

Trajectories of $v_i^f$ for case 1, $i$  = 1,2,3,4.

Compared with Figure 3, the tracking curve in Figure 10 fluctuates more relative to the reference curve, indicating poor tracking performance. Figures 11 and 12 result in greater wastage of communication resources to some extent compared to that in Figures 7 and 8. Figure 13 suggests that its fault tolerance performance may not be as effective as that shown in Figure 9 when the intermittent faults occur.

Figure 10.

Curves of followers for case 2.

Figure 11.

Triggering times of $y_i$ for case 2, $i$  = 1,2,3,4.

Figure 12.

Triggering times of $v_i$ for case 2, $i$  = 1,2,3,4.

Figure 13.

Trajectories of $v_i^f$ for case 2, $i$  = 1,2,3,4.

To demonstrate that the proposed scheme can decrease the convergence time of the formation tracking error, we compared with the control scheme in Wang and Li. ²⁰ The original Table 1 in Wang and Li ²⁰ presented convergence times from a single simulation run. Now, we have conducted 10 additional independent simulation trials under identical conditions. Table 1 now includes the mean ± variance deviation of convergence times for both our proposed method and the method from Wang and Li. ²⁰ From Table 1, we can know that the proposed method consistently achieves faster convergence with smaller variances, demonstrating its superior stability compared to method in Wang and Li. ²⁰

Table 1.

Convergence time.

Error threshold	Proposed method (mean ± variance)	Method in Wang and Li 20 (mean ± variance)
$\sum _{i=1}^4\left|z_{i,1}\right|\le 0.50$	0.052 ± 0.003 s	0.178 ± 0.008 s
$\sum _{i=1}^4\left|z_{i,1}\right|\le 0.25$	0.088 ± 0.005 s	0.243 ± 0.012 s
$\sum _{i=1}^4\left|z_{i,1}\right|\le 0.10$	0.137 ± 0.007 s	0.405 ± 0.015 s

Moreover, to further proof the superiority of the double-sides ET mechanism, the single channel event-triggered control method in Gao et al. ⁷ is proposed. The results are exhibited in Table 2. According to Table 2, we get control method in this paper can abundantly decrease the unnecessary data updates.

Table 2.

Triggering times in this paper and Gao et al. ⁷

Control methods	$y_i$ (Gao et al. 7 )	$v_i$ (Gao et al. 7 )	$y_i$ (this paper)	$v_i$ (this paper)
Agent 1	6000	492	169	368
Agent 2	6000	687	175	566
Agent 3	6000	651	175	518
Agent 4	6000	827	177	794
Total times	24,000	2657	696	2148

Conclusion

In this note, NN event-triggered formation fault-tolerant control is designed with the unmeasurable states, actuator faults and time-varying state delays. The nonlinear observer and NNs have been introduced to estimate the unmeasurable states by using the triggered output signal and approximate the unknown dynamic, respectively. Time-varying state delays have been solved by using appropriate LKF. In addition, DSC technique and backstepping control theory have been utilized to avoid the problem of “explosion of complexity.” The proposed control method not only ensures the stability of the system but also guarantees that the formation control errors are limited within preset bounds. In the future, there remain several issues to be investigated, for example, how to use the methods to MASs with interactive multi-model and complex input nonlinearities.