Adaptive Distributed Type-2 Fuzzy Dynamic Event-Triggered Formation Control for Switched Nonlinear Multi-Agent System with Actuator Faults

Abstract

The adaptive distributed type-2 fuzzy dynamic event-triggered (DET) formation control problem of switched nonlinear multi-agent systems (SNMASs) with actuator faults is addressed in this study. Each agent has a switching subsystem and the switching method of each subsystem is heterogeneous. Interval type-2 fuzzy logic systems (T2FLSs) are adopted to handle uncertain nonlinearities. To conserve communication resources (UCRs), a novel distributed DET controller with an event triggering mechanism is proposed. Additionally, Zeno behavior is excluded. Then, the formation objective can be achieved with a designed common Lyapunov function (CLF). Finally, simulation results confirm the validity of the proposed scheme.

1. Introduction

In recent years, the study of multi-agent systems (MASs) has attracted significant attention in various fields, such as robotics [1], unmanned aerial vehicles [2,3,4], smart grids [5,6], and distributed sensor networks [7]. One of the critical challenges in MASs is achieving and maintaining formation control, where agents are required to follow a specific spatial configuration while adapting to dynamic environments and system uncertainties. Formation control is essential for tasks such as cooperative surveillance, search and rescue operations, and collaborative transportation. In [8], the authors studied the adaptive leader–follower formation control problem of second-order nonlinear multi-agent systems (NMASs) with unknown dynamics using a neural network. An adaptive backstepping finite time optimal formation control problem for NMASs was addressed in [9]. The authors in [10] considered the adaptive output consensus problem for heterogeneous NMASs under random link failures, in which the connection of each link may suffer from link failure. Although the above-mentioned studies [8,9,10] promoted the development of the field through methods such as neural networks, finite time control, and stochastic system analysis, their common core lies in the use of adaptive techniques. It can be seen that adaptive techniques are widely used. In [11], the authors provided an adaptive neural network finite time optimal control protocol for a class of nonlinear systems. The observer-based fuzzy adaptive finite time containment problem for a class of NMASs with input delay was studied in [12].

Switched nonlinear multi-agent systems (SNMASs) [13,14,15] introduce significant complexities due to their capability to transition in control design. These systems frequently encounter challenges such as sudden changes in dynamics, external disturbances, and inherent uncertainties. In [13], they used a low-complexity adaptive methodology for consensus tracking for a team with uncertain high order SNMASs which may experience the asynchronous phenomenon. The authors in [14] handled the leader–follower bipartite consensus problem for singular SNMASs in non-uniform time domains. In [15], the study focused on the consensus problem for fully distributed adaptive systems of SNMASs, considering Bouc–Wen hysteresis as an input. Additionally, actuator faults, which are prevalent in real-world applications, can significantly impair system performance and potentially cause instability if not adequately managed. Consequently, it is crucial to address these challenges in NMASs [16,17,18]. The distributed adaptive predefined time bipartite containment problem for NMASs with actuator faults was investigated in [16]. The problem of achieving consensus fault-tolerant control for non-strict feedback NMASs, subject to intermittent actuator faults, is addressed using a distributed adaptive neural network approach in [17]. The author in [18] focused on the consensus tracking issue for second-order NMASs, which were subject to disturbances and actuator faults, using the sliding mode control method. In the past, type-1 fuzzy logic systems (T1FLSs) have achieved many results in dealing with nonlinear terms in the system. For example, ref. [19] addressed distributed adaptive fuzzy control for achieving output consensus in heterogeneous stochastic NMASs. In [20], the fuzzy finite time consensus control problem for a class of strict feedback NMASs with heterogeneous dynamics was investigated. The scholars in [21] investigated the fixed time adaptive fuzzy containment control problem for NMASs with directed topologies. However, T2FLSs have gained prominence as an effective tool for addressing the uncertainties and nonlinearities in control systems. In contrast to conventional T1FLSs, T2FLSs offer enhanced capability in capturing and managing uncertainties, owing to their additional degree of freedom in membership functions. This characteristic renders them particularly well suited for applications in complex and uncertain environments. A fuzzy tracking control scheme with a switching event-based interval type-2 variable universe is introduced for addressing uncertainties in nonlinear systems in [22]. In [23], the discrete time tracking problem of discrete time nonlinear actuator saturated systems via T2FLSs was investigated.

Event-triggered control has become increasingly popular as an efficient approach to reducing the communication and computational demands in NAMSs. Unlike conventional time-triggered control, which relies on continuous communication and updates, event-triggered control initiates control actions only when predefined conditions are satisfied, thereby conserving resources and prolonging system longevity. The event-triggered control problems of NMASs were studied in [24,25,26,27,28] and the authors proposed many effective event triggering mechanisms to conserve UCRs. Nevertheless, dynamic event-triggered mechanisms further refine this strategy by adaptively adjusting the triggering conditions in response to system states, resulting in enhanced performance and minimized unnecessary updates. In [29,30,31], the authors considered the DET control for NMASs.

Thus, an adaptive distributed type-2 fuzzy DET formation control scheme for SNMASs with actuator faults is proposed in this paper. The primary contributions of this study are summarized as follows:

(i) This work addresses the scenarios where each agent operates under arbitrary and asynchronous switching dynamics while also being susceptible to actuator faults, in contrast to prior studies, such as [8]. By selecting a suitable common Lyapounov function, the formation objective of SNMASs can be achieved. Consequently, the designed protocol offers a broader scope of applicability.

(ii) Unlike some previous works [20,21] which use the T1FLSs to approximate the unknown nonlinear terms, the T2FLSs are introduced to address the formation control problem of SNMASs considered in this paper. Compared with traditional T1FLSs, T2FLSs introduce an additional degree of freedom, enabling the more accurate representation and handling of fuzziness and uncertainties in complex environments.

(iii) Different from some adaptive techniques and event-triggered methods which are used in [8,9,24,25], a novel DET controller is proposed for the considered SNMASs with an actuator fault. The dynamic event triggering mechanism further reduces unnecessary communication by adaptively adjusting triggering conditions. Hence, the UCRs can be saved effectively.

The remainder of this paper is arranged as follows: Section 2 includes graph theory, problem formulation and lemmas, and type-2 fuzzy logic systems, which provide the problem formulation and preliminaries. Section 3 consists of adaptive dynamic event-triggered controller design, stability analysis, and the exclusion of the Zeno phenomenon, where the formation control protocol, the DET mechanism, and the type-2 adaptive fuzzy law are derived. Section 4 presents a numerical simulation example to verify the effectiveness of the provided scheme, and Section 5 provides the conclusions.

2. Problem Statement and Preliminaries

2.1. Graph Theory

In this section, an interaction diagram is presented to depict the exchange of information between agents. The communication topology of agents is represented by the graph $H=\left(\mathsf{\Gamma },\mathfrak{I},\mathcal{A}\right)$, where $\mathsf{\Gamma }=\left(1,\dots ,N\right)$ represents the node set, $\mathfrak{I}\subseteq \mathsf{\Gamma }\times \mathsf{\Gamma }$ denotes the set of edge, and the weighted adjacency matrix is denoted as $\mathcal{A}=\left[a_{i,j}\right]\in {\mathbb{R}}^{N\times N}$. An edge from node $i$ to node $j$ is denoted by $\left({\mathsf{\Gamma }}_j,{\mathsf{\Gamma }}_i\right)\in \mathfrak{I}$. The neighbor set of node $i$ is defined as ${\mathcal{N}}_i=\left\{{\mathsf{\Gamma }}_j\mid \left({\mathsf{\Gamma }}_j,{\mathsf{\Gamma }}_i\right)\in \mathfrak{I},i\ne \right.$ $j\}$. In the adjacency matrix $A,a_{i,j}>0$, if the message from node $j$ can be received by node $i$, and otherwise $a_{i,j}=0$. The in-degree of node $i$ is given by $o_i=\sum _{j=1}^N a_{ij}\cdot \mathcal{L}=\mathcal{D}-\mathcal{A}$ is the Laplacian matrix of $H$, where $\mathcal{D}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(d_1,\dots ,d_N\right)$ with $o_i=d_i$. Let $\mathcal{G}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({\mathcal{G}}_1,{\mathcal{G}}_2,\dots ,{\mathcal{G}}_N\right)$, if the $i$ th follower is capable of receiving information from the leader ${\mathcal{G}}_i>0$, else ${\mathcal{G}}_i=0$.

2.2. Problem Formulation and Lemmas

In this charter, the following are the considered SNMASs with actuator faults:

$$\left\{\begin{array}{l}{\dot{\chi }}_{i,m} ={\chi }_{i,m+1} +f_{i,m}^{{\sigma }_i\left(t\right)}\left({\bar{\chi }}_{i,m}\right)+d_{i,m}^{{\sigma }_i\left(t\right)}\left(t\right),\\ {\dot{\chi }}_{i,n_i}=v_i\left(t\right)+f_{i,n_i}^{{\sigma }_i\left(t\right)}\left({\bar{\chi }}_{i,n_i}\right)+d_{i,n_i}^{{\sigma }_i\left(t\right)}\left(t\right),\\ {\mathcal{Y}}_i={\chi }_{i,1},\end{array}\right.$$

(1)

where ${\bar{\chi }}_{i,m}={\left[{\chi }_{i,1},{\chi }_{i,2},\dots ,{\chi }_{i,m}\right]}^T\in R^{n_i},m=$ $\mathrm{1,2},\dots ,n_i-1$, is the state vector with $i=\mathrm{1,2},\dots ,N,v_i\left(t\right)$ is the controller of agent which suffer from actuator faults, and ${\mathcal{Y}}_i$ represents the $i$-th agent’s output. The switching signal of the agent is ${\sigma }_i\left(t\right):\left[0,\mathrm{\infty }\right]\to {\mathrm{\wp }}_i=\left\{\mathrm{1,2},\dots ,W_i\right\}$, where $W_i$ is the number of subsystems. $k_i\in {\mathrm{\wp }}_i,f_{i,m}^{k_i}\left(\cdot \right)$ is the unknown smooth nonlinear function, and $d_{i,m}^{k_i}\left(t\right)$ is the unknown external disturbance.

Remark 1. 

In practice, influenced by its own effects or external environment, the NMASs can be modeled as SNMASs. By switching between different subsystems, the performance and adaptability of complex dynamic systems have been significantly improved. In addition, the subsystems of agents can switch asynchronously.

The actuator fault model is described as follows [32]:

$$v_i\left(t\right)={\omega }_i\left(t\right)u_i\left(t\right)+{\nu }_i\left(t\right),$$

(2)

where ${\omega }_i\left(t\right)$ represents a partial loss or degradation in actuator performance due to failure, and $\underset{\_}{{\omega }_i}\le {\omega }_i\left(t\right)<1,\underset{\_}{{\omega }_i}>0$, $u_i\left(t\right)$ is the actual control input. The offset fault ${\nu }_i\left(t\right)$ satisfies $\left|{\bar{\nu }}_i\left(t\right)\right|<{\nu }^{\ast }$ with the unknown constant ${\nu }^{\ast }>0$.

In addition, the above fault model consists of the following four modes:

(1)

${\omega }_i\left(t\right)=1$, ${\nu }_i\left(t\right)=0$, actuators work properly.

(2)

$0<{\omega }_i\left(t\right)<1$, ${\nu }_i\left(t\right)=0$, actuators experience partial loss of effectiveness faults.

(3)

${\omega }_i\left(t\right)=1$, ${\nu }_i\left(t\right)\ne 0$, actuators have bias faults.

(4)

${\omega }_i\left(t\right)=0$, ${\nu }_i\left(t\right)\ne 0$, actuators have stuck faults.

Assumption 1. 

The leader signal is bounded, denoted as ${\mathcal{Y}}_r$, which satisfies the condition that its $n_i$ th-order derivative exists. Furthermore, ${\mathcal{Y}}_r^{\left(\mathcal{P}\right)},\mathcal{P}=1,\dots ,n_i$, are also bounded and continuous.

Assumption 2. 

The actuator failure coefficient ${\omega }_i\left(t\right)$ adheres to the properties of a $C^1$ function. In addition, the first derivative of ${\omega }_i\left(t\right)$ satisfies ${\underset{\_}{\omega }}_i^{*}\le \left|{\omega }_i^{\prime }\left(t\right)\right|\le {\bar{\omega }}_i^{*}$, where ${\underset{\_}{\omega }}_i^{*}$ and ${\bar{\omega }}_i^{*}$ are positive constants.

Assumption 3. 

The unknown external disturbance $d_{i,m}^{{\sigma }_i}\left(t\right)$ is bounded and satisfies $\left|d_{i,m}^{{\sigma }_i}\left(t\right)\right| \le {\bar{d}}_{i,m}$, where ${\bar{d}}_{i,m}$ is a positive constant.

Assumption 4. 

The agents have an undirected and interconnected communication topology, where at least one follower is connected to the leader.

Remark 2. 

Assumption 4 aims to ensure that the communication topology is connected. In other words, the interactive information can traverse all agents.

Lemma 1 

[29]. For $\forall \mathcal{M}>0,\mathcal{N}\in R$, the following inequality holds:

$$0 \le \left|\mathcal{N}\right|-\mathcal{N}\mathit{tanh}\left({\displaystyle \frac{\mathcal{N}}{\mathcal{M}}}\right) \le 0.2785\mathcal{M}.$$

(3)

Lemma 2 

[21]. For any $\gamma >0,\pi >0$ and $\delta >0$, the following inequality holds:

$${\left|{\hslash }_1\right|}^{\gamma }{\left|{\hslash }_2\right|}^{\pi } \le {\displaystyle \frac{\gamma \delta }{\gamma \pi }}{\left|{\hslash }_1\right|}^{\gamma +\pi }{\displaystyle \frac{\gamma {\delta }^{-\frac{\gamma }{\pi }}}{\gamma \pi }}{\left|{\hslash }_2\right|}^{\gamma +\pi },$$

(4)

where ${\hslash }_1$ and ${\hslash }_2$ are real numbers.

2.3. Type-2 Fuzzy Logic Systems

T2FLSs demonstrate superior approximation capabilities and enhanced descriptive power in modeling uncertain nonlinear dynamics when contrasted with T1FLSs. The mechanism map for typical fuzzy membership function is described in Figure 1, where z and μ(z) represent the input and output of the fuzzy membership function, respectively. A is defined as the fuzzy set. Then, the fuzzy membership function adopts a Gaussian function, and its structure is

$${\mu }_A\left(z\right)=\mathit{exp}\left(-{\displaystyle \frac{{\left(z-{\lambda }_0\right)}^2}{{\sigma }^2}}\right),$$

(5)

where σ and ${\lambda }_0$ are denoted as the width and center of the Gaussian function. In contrast to type-1 fuzzy membership functions, type-2 fuzzy membership functions inherently exhibit fuzziness and are characterized by a standardized uncertainty interval with fixed bounds, representing a “deviation” range $\overline{\lambda }$ in $\left[{\lambda }_1,{\lambda }_2\right]$ which can be seen in Figure 2. Unlike type-1 fuzzy membership functions that assign crisp values to a fuzzy set A, the type-2 fuzzy membership functions quantify membership through interval-valued mappings. Different from type-1, the fuzzy membership function of type-2 is defined by two upper and lower boundary fuzzy membership functions of type-1, corresponding to fuzzy sets $\mathcal{A}$ and $\mathcal{B}$ respectively (see Figure 2). Specifically, for any input $z$, the type-2 fuzzy membership function is bounded by two type-1 fuzzy membership functions $\mathcal{A}$ (upper bound) and $\mathcal{B}$ (lower bound). This blurred boundary characteristic enables T2FLSs to inherently encapsulate higher-order uncertainties, thereby achieving superior approximation capabilities for systems with obvious uncertainties.

The definition of T2FLSs are expressed as follows [33]: If ${\kappa }_1$ is ${\mathcal{B}}_1^m,{\kappa }_2$ is ${\mathcal{B}}_2^m,\dots$, and ${\kappa }_{\alpha }$ is ${\mathcal{B}}_{\alpha }^m$, then $y$ is ${\mathcal{U}}^m,m=\mathrm{1,2},\dots ,\mathcal{l}$, where ${\kappa }_1,\dots ,{\kappa }_{\alpha }$ and $y$ are defined as the input and output of T2FLSs. ${\mathcal{B}}_1^m,\cdots ,{\mathcal{B}}_{\alpha }^m$ and ${\mathcal{U}}^m$ are fuzzy sets of T2FLSs. The upper and lower fuzzy membership functions of T2FLSs for fuzzy set ${\mathcal{B}}_{\mathcal{H}}^m,\mathcal{H}=\mathrm{1,2},\cdots ,\alpha$ are denoted as ${\mu }_{{\mathbb{Z}}_{\mathcal{H}}^m}^1\left({\kappa }_{\alpha }\right)$ and ${\mu }_{{\mathbb{Q}}_{\mathcal{H}}^m}^1\left({\kappa }_{\alpha }\right)$, where ${\mathbb{Z}}_{\mathcal{H}}^m$ and ${\mathbb{Q}}_{\mathcal{H}}^m$ represent two fuzzy sets. Then, the output of T2FLSs is indicated as

$$y=\sum _{m=1}^{\mathcal{l}} {\left({\mathsf{\Lambda }}_1^m\right)}^T{\mathsf{\Xi }}_1^m\left(\kappa \right)+\sum _{m=1}^{\mathcal{l}} {\left({\mathsf{\Lambda }}_2^m\right)}^T{\mathsf{\Xi }}_2^m\left(\kappa \right),$$

(6)

where $\kappa ={\left[{\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_{\alpha }\right]}^T,{\mathsf{\Xi }}_1^m\left(\kappa \right)$ and ${\mathsf{\Xi }}_2^m\left(\kappa \right)$ are the fuzzy weighting vectors for upper and lower fuzzy membership function under T2FLSs. Then, the fuzzy basis functions are defined as ${\mathsf{\Xi }}_1^m\left(\kappa \right)={\displaystyle \frac{\prod _{\mathcal{H}=1}^{\alpha } {\mu }_{{\mathbb{Z}}_{\mathcal{H}}^m}^1\left({\kappa }_{\mathcal{H}}\right)}{\sum _{m=1}^{\mathcal{l}} \prod _{\mathcal{H}=1}^{\alpha } {\mu }_{{\mathbb{Z}}_{\mathcal{H}}^m}^1\left({\kappa }_{\mathcal{H}}\right)}}$ and ${\mathsf{\Xi }}_2^m\left(\kappa \right)={\displaystyle \frac{\prod _{\mathcal{H}=1}^{\alpha } {\mu }_{\mathbb{Q}\mathcal{H}}^2\left({\kappa }_{\mathcal{H}}\right)}{\sum _{m=1}^{\mathcal{l}} \prod _{\mathcal{H}=1}^{\alpha } {\mu }_{\mathbb{Q}\mathcal{H}}^2\left({\kappa }_{\mathcal{H}}\right)}}$.

Then, let ${\mathsf{\Xi }}_1={\left[{\mathsf{\Xi }}_1^1\left(\kappa \right),\cdots ,{\mathsf{\Xi }}_1^{\mathcal{l}}\left(\kappa \right)\right]}^T,{\mathsf{\Xi }}_2={\left[{\mathsf{\Xi }}_2^1\left(\kappa \right),\cdots ,{\mathsf{\Xi }}_2^{\mathcal{l}}\left(\kappa \right)\right]}^T,{\mathsf{\Lambda }}_1={\left[{\mathsf{\Lambda }}_1^1\left(\kappa \right),\cdots ,{\mathsf{\Lambda }}_1^{\mathcal{l}}\left(\kappa \right)\right]}^T$, and ${\mathsf{\Lambda }}_2={\left[{\mathsf{\Lambda }}_2^1\left(\kappa \right),\cdots ,{\mathsf{\Lambda }}_2^{\mathcal{l}}\left(\kappa \right)\right]}^T$, the following holds:

$$y={\mathsf{\Lambda }}_1^T{\mathsf{\Xi }}_1\left(\kappa \right)+{\mathsf{\Lambda }}_2^T{\mathsf{\Xi }}_2\left(\kappa \right).$$

(7)

Based on the above content, for a continuous nonlinear function $\mathcal{W}\left(\kappa \right)$, which is defined on the set ${\varpi }_{\kappa }$, there exists a positive constant $\varsigma$. By using the fuzzy logic system, the following inequality holds:

$$\underset{\kappa \in {\varpi }_{\kappa }}{\mathrm{s}\mathrm{u}\mathrm{p}} \left|\mathcal{W}\left(\kappa \right)-{\mathsf{\Lambda }}_1^T{\mathsf{\Xi }}_1\left(\kappa \right)+{\mathsf{\Lambda }}_2^T{\mathsf{\Xi }}_2\left(\kappa \right)\right|\le \varsigma .$$

(8)

Remark 3. 

Compared with T1FLSs, combining Figure 1 and Figure 2, T2FLSs have the following advantages when dealing with the formation control problem of NMASs. At first, it can handle significant heterogeneous uncertainty in the NMASs, such as heterogeneous dynamical modeling, by using member functions to describe the uncertainty. Secondly, T2FLSs have a stronger nonlinear approximation ability, which can reduce the number of rules and thus reduce the computational output of the controller. Combined with event triggering mechanisms, it can further reduce UCRs.

Control objective: In this paper, an adaptive distributed type-2 fuzzy DET formation control scheme is provided for the SNMASs with actuator faults, so that the followers can follow the leader via formation and the UCRs can be saved.

3. Main Results

3.1. Adaptive Dynamic Event-Triggered Controller Design

The formation errors are defined as follows:

$$e_{i,1}=\sum _{j=1}^N a_{ij}\left({\mathcal{Y}}_i-{\mathcal{Y}}_j-{\mathsf{\Phi }}_i+{\mathsf{\Phi }}_j\right)+{\mathcal{G}}_i\left({\mathcal{Y}}_i-{\mathsf{\Phi }}_i-{\mathcal{Y}}_r\right),$$

(9)

where ${\mathsf{\Phi }}_i$ and ${\mathsf{\Phi }}_j$ represent the relative position of agent $i$ and $j$. Meanwhile, ${\mathsf{\Phi }}_i$ and ${\mathsf{\Phi }}_j$ are unknown positive constants.

Next, the coordinate transformation is

$$\begin{array}{c}{z}_{i,1}=e_{i,1}\\ \dots \\ z_{i,m}=x_{i,m}-{\alpha }_{i,m-1},\end{array}$$

(10)

where ${\alpha }_{i,m-1}$ is the virtual controller, which will be given later.

Step 1: The Lyapunov function constructed below is

$$V_{i,1}={\displaystyle \frac{1}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2l_{i,1}}}{\tilde{\theta }}_{i,1}^T{\tilde{\theta }}_{i,1},$$

(11)

where $l_{i,1}$ is a positive design parameter. Then ${\tilde{\theta }}_{i,1}$ will be deigned.

Afterwards, the differentiation of $V_{i,1}$ is

$$\begin{array}{c}{\dot{V}}_{i,1}=\left({\mathcal{G}}_i+o_i\right)z_{i,1}{z}_{i,2}+\left({\mathcal{G}}_i+o_i\right)z_{i,1}{\alpha }_{i,1}-o_i{z}_{i,1}{\chi }_{j,2}+\left({\mathcal{G}}_i+o_i\right)z_{i,1}{f}_{i,1}^{k_i}\left({\chi }_{i,1}\right)\\ +\left({\mathcal{G}}_i+o_i\right)z_{i,1}{d}_{i,1}^{k_i}-o_i{z}_{i,1}{d}_{j,1}^{k_j}-o_i{z}_{i,1}{f}_{j,1}^{k_j}\left({\chi }_{j,1}\right)-z_{i,1}{\mathcal{G}}_i{\dot{\mathcal{Y}}}_r-{\displaystyle \frac{1}{l_{i,1}}}{\tilde{\theta }}_{i,1}^T{\dot{\widehat{\theta }}}_{i,1}.\end{array}$$

(12)

According to Lemma 2, we can get

$$\left({\mathcal{G}}_i+o_i\right)z_{i,1}{z}_{i,2}\le {\displaystyle \frac{1}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}\left({\mathcal{G}}_i+o_i\right)z_{i,2}^2,$$

(13)

$$\left({\mathcal{G}}_i+o_i\right)z_{i,1}{f}_{i,1}^{k_i}\left({\chi }_{i,1}\right)\le {\displaystyle \frac{1}{4}}+\left({\mathcal{G}}_i+o_i\right)z_{i,1}^2{\left(f_{i,1}^{k_i}\left({\chi }_{i,1}\right)\right)}^2,$$

(14)

$$\left({\mathcal{G}}_i+o_i\right)z_{i,1}{d}_{i,1}^{k_i}\left(t\right)\le {\displaystyle \frac{\left({\mathcal{G}}_i+o_i\right)}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}{\left(d_{i,1}^{k_i}\left(t\right)\right)}^2.$$

(15)

Defining $f_{i,1}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{i,1}^{k_i}\left({\chi }_{i,1}\right)\right)}^2\right\},f_{j,1}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{j,1}^{k_j}\left({\chi }_{j,1}\right)\right)}^2\right\}$. Then, substituting Equations (12)–(14) into Equation (11), we obtain

$$\begin{array}{cc}{\dot{V}}_{i,1}\hfill & \le {\displaystyle \frac{1}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}\left({\mathcal{G}}_i+o_i\right)z_{i,2}^2+\left({\mathcal{G}}_i+o_i\right)z_{i,1}{\alpha }_{i,1}+{\left({\mathcal{G}}_i+o_i\right)}^2{z}_{i,1}^2{f}_{i,1}\hfill \\ \hfill & +{\displaystyle \frac{\left({\mathcal{G}}_i+o_i\right)}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}{\overline{d}}_{i,1}^2-o_i{z}_{i,1}{\chi }_{j,2}+z_{i,1}^2{o}_i^2{\displaystyle \sum _{j\in {\aleph }_i}} \underset{j,1}{f}+{\displaystyle \frac{1}{2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}{o}_i^2{\displaystyle \sum _{j\in {\aleph }_i}} {\overline{d}}_{j,1}^2-z_{i,1}{\mathcal{G}}_i{\dot{\mathcal{Y}}}_r-{\displaystyle \frac{1}{l_{i,1}}}{\tilde{\theta }}_{i,1}^T{\dot{\widehat{\theta }}}_{i,1},\hfill \end{array}$$

(16)

where the set ${\mathrm{\aleph }}_i$ is denoted as the set of neighbors for the $i$-th agent.

Let $F_{i,1}={\left({\mathcal{G}}_i+o_i\right)}^2{z}_{i,1}{f}_{i,1}-o_i{\chi }_{j,2}+z_{i,1}{o}_i^2\sum _{j\in {\mathrm{\aleph }}_i} f_{j,1}-$ ${\mathcal{G}}_i{\dot{\mathcal{Y}}}_r$. Based on the type-2 fuzzy logic systems, $F_{i,1}$ can be approximated as $F_{i,1}={\vartheta }_{i,1}^T{\varphi }_{i,1}\left({\mathsf{\Omega }}_{i,1}\right)+{\mathsf{\Theta }}_{i,1}^T{\phi }_{i,1}\left({\mathsf{\Omega }}_{i,1}\right)+$ ${\zeta }_{i,1}\left({\mathsf{\Omega }}_{i,1}\right)$, where ${\mathsf{\Omega }}_{i,1}={\left[{\chi }_{i,1},{\chi }_{j,1},{\chi }_{j,2},{\mathcal{Y}}_r,{\dot{\mathcal{Y}}}_r\right]}^T,j\in {\mathrm{\aleph }}_i$. Then, the following inequality holds:

$$z_{i,1}{F}_{i,1}\le {\displaystyle \frac{1}{2}}{\theta }_{i,1}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{z_{i,1}^2}{2}}+{\displaystyle \frac{{\bar{\zeta }}_{i,1}^2}{2}},$$

(17)

where ${\theta }_{i,1}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{‖{\mathsf{\Theta }}_{i,1}^{k_i}‖}^2+{‖{\vartheta }_{i,1}^{k_i}‖}^2\right\}>0,{\hat{\theta }}_{i,1}$ is the estimation of ${\theta }_{i,1}$ and ${\tilde{\theta }}_{i,1}={\theta }_{i,1}-{\hat{\theta }}_{i,1}$. In addition, ${\left|{\zeta }_{i,1}\right|}^2\le {\bar{\zeta }}_{i,1}^2$, and ${\bar{\zeta }}_{i,1}$ are positive constants.

Substituting Equation (17) into Equation (16), we can get

$$\begin{array}{cc}{\dot{V}}_{i,1}\hfill & \le {\displaystyle \frac{3}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}\left({\mathcal{G}}_i+o_i\right)z_{i,2}^2+{\displaystyle \frac{\left({\mathcal{G}}_i+o_i\right)}{2}}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}{\theta }_{i,1}{z}_{i,1}^2+\left({\mathcal{G}}_i+o_i\right)z_{i,1}{\alpha }_{i,1}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\overline{d}}_{i,1}^2+1+{\displaystyle \frac{1}{2}}{o}_i^2{\displaystyle \sum _{j\in {\aleph }_i}} {\overline{d}}_{j,1}^2+{\displaystyle \frac{{\overline{\epsilon }}_{i,1}^2}{2}}-{\displaystyle \frac{1}{l_{i,1}}}{\tilde{\theta }}_{i,1}^T{\widehat{\theta }}_{i,1}.\hfill \end{array}$$

(18)

Subsequently, the virtual control law and adaptive law are designed as follows:

$${\alpha }_{i,1} =-{\displaystyle \frac{1}{\left({\mathcal{G}}_i+o_i\right)}}\left({\beta }_{i,1}+{\displaystyle \frac{3}{2}}+{\displaystyle \frac{\left({\mathcal{G}}_i+o_i\right)}{2}}\right)z_{i,1}-{\displaystyle \frac{1}{\left({\mathcal{G}}_i+o_i\right)}}{z}_{i,1}{\widehat{\theta }}_{i,1},$$

(19)

$${\dot{\hat{\theta }}}_{i,1}=l_{i,1}{z}_{i,1}-{\eta }_{i,1}{\hat{\theta }}_{i,1},$$

(20)

where ${\eta }_{i,1}>0$ and ${\beta }_{i,1}>0$ are design parameters.

At last, Equation (17) is transformed into

$${\dot{V}}_{i,1}\le -{\beta }_{i,1}{z}_{i,1}^2+{\displaystyle \frac{1}{2}}\left({\mathcal{G}}_i+o_i\right)z_{i,2}^2-{\displaystyle \frac{{\eta }_{i,1}{\tilde{\theta }}_{i,1}^T{\tilde{\theta }}_{i,1}}{2l_{i,1}}}+{\mathcal{Q}}_{i,1},$$

(21)

where ${\mathcal{Q}}_{i,1}={\displaystyle \frac{1}{2}}{\bar{d}}_{i,1}^2+{\displaystyle \frac{1}{2}}{o}_i^2\sum _{j\in {\mathrm{\aleph }}_i} {\bar{d}}_{j,1}^2+1+{\displaystyle \frac{1}{2}}{\bar{\zeta }}_{i,1}^2+{\displaystyle \frac{{\eta }_{i,1}{‖{\theta }_{i,1}‖}^2}{2l_{i,1}}}$.

Step $\mathcal{S}\left(2<\mathcal{S}<n_i-1\right)$: The Lyapunov function is constructed as follows:

$$V_{i,\mathcal{S}}=V_{i,\mathcal{S}-1}+{\displaystyle \frac{1}{2}}{z}_{i,\mathcal{S}}^2+{\displaystyle \frac{1}{2l_{i,\mathcal{S}}}}{\tilde{\theta }}_{i,\mathcal{S}}^T{\tilde{\theta }}_{i,\mathcal{S}},$$

(22)

with a positive design parameter $l_{i,\mathcal{S}}$.

Derivatizing $V_{i,s}$ results in

$$\begin{array}{c}{\dot{V}}_{i,\mathcal{S}}={\dot{V}}_{i,\mathcal{S}-1}+z_{i,\mathcal{S}}{z}_{i,\mathcal{S}+1}+z_{i,\mathcal{S}}{\alpha }_{i,\mathcal{S}}+z_{i,\mathcal{S}}{f}_{i,\mathcal{S}}^{k_i}\left({\overline{\chi }}_{i,\mathcal{S}}\right)\hfill \\ +z_{i,\mathcal{S}}{d}_{i,\mathcal{S}}^{k_i}-z_{i,\mathcal{S}}{\dot{\alpha }}_{i,\mathcal{S}-1}-{\displaystyle \frac{1}{l_{i,\mathcal{S}}}}{\tilde{\theta }}_{i,\mathcal{S}}^T{\dot{\widehat{\theta }}}_{i,\mathcal{S}}.\hfill \end{array}$$

(23)

The differentiation of ${\alpha }_{i,\mathcal{S}-1}$ is

$$\begin{array}{c}{\dot{\alpha }}_{i,\mathcal{S}-1}={\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}\left({\chi }_{i,\mathcal{P}+1}+f_{i,\mathcal{P}}^{k_i}\left({\overline{\chi }}_{i,\mathcal{P}}\right)+d_{i,\mathcal{P}}^{k_i}\right)\hfill \\ +{\displaystyle \sum _{j\in {\aleph }_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}\left({\chi }_{j,\mathcal{P}+1}+f_{j,\mathcal{P}}^{k_j}\left({\overline{\chi }}_{j,\mathcal{P}}\right)+d_{j,\mathcal{P}}^{k_j}\right)\hfill \\ +{\displaystyle \sum _{\mathcal{P}=0}^{s-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\mathcal{Y}}_r^{\left(\mathcal{P}\right)}}}{\mathcal{Y}}_r^{\left(\mathcal{P}+1\right)}+{\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\widehat{\theta }}_{i,\mathcal{P}}}}{\dot{\widehat{\theta }}}_{i,\mathcal{P}}.\hfill \end{array}$$

(24)

Similar to step 1, defining $f_{i,\mathcal{S}}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{i,s}^{k_i}\left({\bar{\chi }}_{i,s}\right)\right)}^2\right\}$, $f_{i,\mathcal{P}}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{i,\mathcal{P}}^{k_i}\left({\bar{\chi }}_{i,\mathcal{P}}\right)\right)}^2\right\},f_{j,\mathcal{P}}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}}$ $\left\{{\left(f_{j,\mathcal{P}}^{k_j}\left({\bar{\chi }}_{j,\mathcal{P}}\right)\right)}^2\right\}$ and according to Lemma 2, we have

$$\begin{array}{c}{\dot{V}}_{i,\mathcal{S}}\le {\dot{V}}_{i,\mathcal{S}-1}+{\displaystyle \frac{1}{2}}{z}_{i,\mathcal{S}}^2+{\displaystyle \frac{1}{2}}{z}_{i,\mathcal{S}+1}^2+z_{i,\mathcal{S}}{\alpha }_{i,\mathcal{S}}+z_{i,\mathcal{S}}^2{f}_{i,\mathcal{S}}\hfill \\ +{\displaystyle \frac{1}{2}}{z}_{i,\mathcal{S}}^2+{\displaystyle \frac{1}{2}}{\overline{d}}_{i,S}^2-z_{i,\mathcal{S}}{\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial x_{i,\mathcal{P}}}}{\chi }_{i,\mathcal{P}+1}\hfill \\ +z_{i,\mathcal{S}}^2{\left({\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2{\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} f_{i,\mathcal{P}}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\overline{d}}_{i,p}^2\hfill \\ +{\displaystyle \frac{1}{2}}{\left({\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2{z}_{i,\mathcal{S}}^2-z_{i,\mathcal{S}}{\displaystyle \sum _{j\in {\mathfrak{N}}_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}{\chi }_{j,\mathcal{P}+1}\hfill \\ +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j\in B_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\overline{d}}_{j,p}^2+{\displaystyle \frac{1}{2}}{\left({\displaystyle \sum _{j\in {\aleph }_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}\right)}^2{z}_{i,\mathcal{S}}^2\hfill \\ +z_{i,\mathcal{S}}^2{\left({\displaystyle \sum _{j\in {\aleph }_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}\right)}^2{\displaystyle \sum _{j\in B_i}} {\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} f_{j,\mathcal{P}}\hfill \\ -z_{i,\mathcal{S}}{\displaystyle \sum _{\mathcal{P}=0}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\mathcal{Y}}_r^{\left(\mathcal{P}\right)}}}{\mathcal{Y}}_r^{\left(\mathcal{P}+1\right)}-z_{i,\mathcal{S}}{\displaystyle \sum _{\mathcal{P}=1}^{\mathcal{S}-1}} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\widehat{\theta }}_{i,\mathcal{P}}}}{\dot{\widehat{\theta }}}_{i,\mathcal{P}}\hfill \\ +{\displaystyle \frac{3}{4}}-{\displaystyle \frac{1}{l_{i,\mathcal{S}}}}{\tilde{\theta }}_{i,\mathcal{S}}^T{\dot{\widehat{\theta }}}_{i,\mathcal{S}}. \hfill \end{array}$$

(25)

$\mathrm{L}\mathrm{e}\mathrm{t} F_{i,\mathcal{S}}=z_{i,\mathcal{S}}{f}_{i,\mathcal{S}}-\sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}{\chi }_{i,\mathcal{P}+1}+z_{i,\mathcal{S}}{\left(\sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2\sum _{\mathcal{P}=1}^{\mathcal{S}-1} f_{i,\mathcal{P}} +$   ${\displaystyle \frac{1}{2}}{\left(\sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2{z}_{i,\mathcal{S}}{z}_{i,\mathcal{S}}{\left(\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}\right)}^2\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{\mathcal{S}-1} f_{j,\mathcal{P}}\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}{\chi }_{j,\mathcal{P}+1} +$   ${\displaystyle \frac{1}{2}}{\left(\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\chi }_{j,\mathcal{P}}}}\right)}^2$t $z_{i,\mathcal{S}}-\sum _{\mathcal{P}=0}^{\mathcal{S}-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\mathcal{Y}}_r^{\left(h\right)}}}{\mathcal{Y}}_r^{\left(\mathcal{P}+1\right)}-\sum _{\mathcal{P}=1}^{k-1} {\displaystyle \frac{\partial {\alpha }_{i,\mathcal{S}-1}}{\partial {\hat{\theta }}_{i,\mathcal{P}}}}{\dot{\hat{\theta }}}_{i,\mathcal{P}}$. According to T2FLS, $F_{i,\mathcal{S}}={\vartheta }_{i,\mathcal{S}}^T{\varphi }_{i,\mathcal{S}}\left({\mathsf{\Omega }}_{i,\mathcal{S}}\right)+{\mathsf{\Theta }}_{i,\mathcal{S}}^T{\phi }_{i,\mathcal{S}}\left({\mathsf{\Omega }}_{i,\mathcal{S}}\right)+$${\zeta }_{i,\mathcal{S}}\left({\mathsf{\Omega }}_{i,\mathcal{S}}\right)$, where $\left|{\zeta }_{i,\mathcal{S}}\left({\mathsf{\Omega }}_{i,\mathcal{S}}\right)\right|<{\bar{\zeta }}_{i,\mathcal{S}},{\bar{\zeta }}_{i,\mathcal{S}}>0$ and ${\mathsf{\Omega }}_{i,\mathcal{S}}={\left[{\chi }_{i,1},\dots ,{\chi }_{i,\mathcal{S}},{\chi }_{j,1},\dots ,{\chi }_{j,\mathcal{S}},{\mathcal{Y}}_r,\dots ,{\mathcal{Y}}_r^{\left(\mathcal{S}\right)},{\hat{\theta }}_{i,1},\dots ,{\hat{\theta }}_{i,\mathcal{S}-1}\right]}^T$.

Then, we can deduce that

$$z_{i,\mathcal{S}}{F}_{i,\mathcal{S}}\le {\displaystyle \frac{1}{2}}{\theta }_{i,\mathcal{S}}{z}_{i,\mathcal{S}}^2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{z_{i,\mathcal{S}}^2}{2}}+{\displaystyle \frac{{\bar{\zeta }}_{i,\mathcal{S}}^2}{2}}.$$

(26)

The virtual control law and adaptive law are designed as follows:

$${\alpha }_{i,\delta }=-\left({\beta }_{i,\delta }+{\displaystyle \frac{3}{2}}\right)z_{i,\delta }-z_{i,\delta }{\widehat{\theta }}_{i,\delta },$$

(27)

$${\dot{\hat{\theta }}}_{i,\mathcal{S}}=l_{i,\mathcal{S}}{z}_{i,\mathcal{S}}-{\eta }_{i,\mathcal{S}}{\hat{\theta }}_{i,\mathcal{S}},$$

(28)

with the positive design parameters ${\beta }_{i,\mathcal{S}}$ and ${\eta }_{i,\mathcal{S}}$.

From Equations (25)–(27), we can obtain

$$\begin{array}{c}{\dot{V}}_{i,\mathcal{S}}\le -{\beta }_{i,1}{z}_{i,1}^2-\cdots -{\beta }_{i,\mathcal{S}}{z}_{i,\mathcal{S}}^2-{\displaystyle \frac{{\eta }_{i,1}{\theta }_{i,1}^T{\theta }_{i,1}}{2l_{i,1}}}-\cdots \hfill \\ -{\displaystyle \frac{{\eta }_{i,\mathcal{S}}{\tilde{\theta }}_{i,\mathcal{S}}^T{\tilde{\theta }}_{i,\mathcal{S}}}{2l_{i,\mathcal{S}}}}+{\mathcal{Q}}_{i,1}+\cdots +{\mathcal{Q}}_{i,\mathcal{S}}+{\displaystyle \frac{1}{2}}{z}_{i,\mathcal{S}+1}^2,\hfill \end{array}$$

(29)

where ${\mathcal{Q}}_{i,\mathcal{S}}={\displaystyle \frac{1}{2}}{\bar{d}}_{i,\mathcal{S}}^2+{\displaystyle \frac{1}{2}}\sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\bar{d}}_{i,\mathcal{P}}^2+{\displaystyle \frac{1}{2}}\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{\mathcal{S}-1} {\bar{d}}_{j,\mathcal{P}}^2+{\displaystyle \frac{1}{2}}{\bar{\zeta }}_{i,\mathcal{S}}^2+$ ${\displaystyle \frac{5}{4}}+{\displaystyle \frac{{\eta }_{i,\mathcal{S}}{‖{\theta }_{i,\mathcal{S}}‖}^2}{2l_{i,\mathcal{S}}}}$.

Step $n_i$: Select the following Lyapunov function:

$$V_{i,n_i}=V_{i,n_i-1}+{\displaystyle \frac{1}{2{\omega }_i\left(t\right)}}{z}_{i,n_i}^2+{\displaystyle \frac{1}{2l_{i,n_i}}}{\tilde{\theta }}_{i,n_i}^T{\tilde{\theta }}_{i,n_i},$$

(30)

where $l_{i,n_i}$ is a positive design parameter.

Differentiating $V_{i,n_i}$ yields

$${\dot{V}}_{i,n_i}={\dot{V}}_{i,n_i-1}+{\displaystyle \frac{1}{{\omega }_i\left(t\right)}}{z}_{i,n_i}\left({\dot{x}}_{i,n_i}-{\dot{\alpha }}_{i,n_i-1}\right)-{\displaystyle \frac{{\omega }_i^{\prime }\left(t\right)}{2{\omega }_i^2\left(t\right)}}{z}_{i,n_i}^2-{\displaystyle \frac{1}{l_{i,n_i}}}{\tilde{\theta }}_{i,n_i}^T{\dot{\widehat{\theta }}}_{i,n_i}.$$

(31)

Then, the differentiation of ${\alpha }_{i,n_i-1}$ is

$$\begin{array}{cc}{\dot{\alpha }}_{i,n_i-1}\hfill & ={\displaystyle \sum _{\mathcal{P}=1}^{n_i-1}} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{i,p}}}\left({\chi }_{i,p+1}+f_{i,h}^{k_i}\left({\overline{\chi }}_{i,p}\right)+d_{i,p}^{k_i}\right)+{\displaystyle \sum _{\mathcal{P}=0}^{n_i-1}} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\mathcal{Y}}_r^{\left(\mathcal{P}\right)}}}{\mathcal{Y}}_r^{\left(\mathcal{P}+1\right)}\hfill \\ \hfill & +{\displaystyle \sum _{\mathcal{P}=1}^{n_i-1}} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\widehat{\theta }}_{i,p}}}{\dot{\widehat{\theta }}}_{i,p}+{\displaystyle \sum _{j\in {\aleph }_i}} {\displaystyle \sum _{p=1}^{n_i-1}} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{j,p}}}\left({\chi }_{j,\mathcal{P}+1}+f_{j,\mathcal{P}}^{k_j}\left({\overline{\chi }}_{j,h}\right)+d_{j,\mathcal{P}}^{k_j}\right).\hfill \end{array}$$

(32)

Let $f_{i,n_i}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{i,n_i}^{k_i}\left({\bar{\chi }}_{i,n_i}\right)\right)}^2\right\},f_{i,\mathcal{P}}=$ $\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{i,\mathcal{P}}^{k_i}\left({\bar{\chi }}_{i,n_i}\right)\right)}^2\right\},f_{j,\mathcal{P}}=\underset{k_i\in {\mathrm{\wp }}_i}{\mathrm{m}\mathrm{a}\mathrm{x}} \left\{{\left(f_{j_2\mathcal{P}}^{k_j}\left({\bar{\chi }}_{j,n_i}\right)\right)}^2\right\}$   $\mathrm{a}\mathrm{n}\mathrm{d} F_{i,n_i}={\displaystyle \frac{z_{i,n_i}}{{\underset{\_}{\omega }}^2}}\left[f_{i,n}+\left(\sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{i,\mathcal{P}}}}{\chi }_{i,\mathcal{P}+1}\right)\right.+{\left(\sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2\sum _{\mathcal{P}=1}^{n_i-1} f_{i,\mathcal{P}}+$   ${\displaystyle \frac{1}{2}}{\left(\sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{i,\mathcal{P}}}}\right)}^2+{\left(\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{j,\mathcal{P}}}}{\chi }_{j,\mathcal{P}+1}\right)}^2+{\left(\sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\hat{\theta }}_{i,\mathcal{P}}}}{\dot{\hat{\theta }}}_{i,\mathcal{P}}\right)}^2+$   ${\displaystyle \frac{1}{2}}{\left(\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{j,P}}}\right)}^2+{\left(\sum _{\mathcal{P}=0}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\mathcal{Y}}_r^{\left(\mathcal{P}\right)}}}{\mathcal{Y}}_r^{\left(\mathcal{P}+1\right)}\right)}^2+{\left(\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{n_i-1} {\displaystyle \frac{\partial {\alpha }_{i,n_i-1}}{\partial {\chi }_{j,P}}}\right)}^2\left.\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{n_i-1} f_{j,\mathcal{P}}\right]$ $+{\displaystyle \frac{{\left({\bar{\omega }}^{\ast }\right)}^2}{2{\underset{\_}{\omega }}^2}}{z}_{i,n_i}$ Then, the $F_{i,n_i}$ can be approximated as $F_{i,n_i}\left({\mathsf{\Omega }}_{i,n_i}\right)={\vartheta }_{i,n_i}^T{\varphi }_{i,n_i}\left({\mathsf{\Omega }}_{i,n_i}\right)+{\mathsf{\Theta }}_{i,n_i}^T{\phi }_{i,n_i}\left({\mathsf{\Omega }}_{i,n_i}\right)+{\zeta }_{i,n_i}\left({\mathsf{\Omega }}_{i,n_i}\right)$ where ${\mathsf{\Omega }}_{i,n_i}=\left[{\chi }_{i,1},\dots ,{\chi }_{i,n_i},{\chi }_{j,1},\dots ,{\chi }_{j,n_i},{\mathcal{Y}}_r,\dots ,{\mathcal{Y}}_r^{\left(n_i\right)},{\hat{\theta }}_{i,1},\dots \right.$, ${\left.{\hat{\theta }}_{i,n_i-1}\right]}^T$ and $\left|{\zeta }_{i,n_i}\left({\mathsf{\Omega }}_{i,n_i}\right)\right|<{\bar{\zeta }}_{i,n_i},{\bar{\zeta }}_{i,n_i}>0$. Then, we can get

$$z_{i,n_i}{F}_{i,n_i}\le {\displaystyle \frac{1}{2}}{\theta }_{i,n_i}{z}_{i,n_i}^2+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{z_{i,n_i}^2}{2}}+{\displaystyle \frac{{\bar{\zeta }}_{i,n_i}^2}{2}}.$$

(33)

Subsequently, we have

$$\begin{array}{cc}{\dot{V}}_{i,n_i}\hfill & \le {\dot{V}}_{i,n_i-1}+z_{i,n_i}{u}_i+{\displaystyle \frac{v^{*2}}{2{\underset{\_}{\omega }}^2}}+{\displaystyle \frac{3}{2}}{z}_{i,n_i}^2+{\displaystyle \frac{1}{2}}{\theta }_{i,n_i}{z}_{i,n_i}^2+{\displaystyle \frac{{\overline{\zeta }}_{i,n_i}^2}{2}}+{\displaystyle \frac{9}{4}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\overline{d}}_{i,n_i}^2}{{\underset{\_}{\omega }}^2}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\displaystyle \sum _{p=1}^{n_i-1}} {\overline{d}}_{i,p}^2+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j\in {\mathbb{N}}_i}} {\displaystyle \sum _{p=1}^{n_i-1}} {\overline{d}}_{j,p}^2+{\displaystyle \frac{z_{i,n_i}^2}{2{\left({\underset{\_}{\omega }}^{*}\right)}^2}}-{\displaystyle \frac{1}{l_{i,n_i}}}{\tilde{\theta }}_{i,n_i}^T{\dot{\widehat{\theta }}}_{i,n_i}.\hfill \end{array}$$

(34)

Afterwards, the virtual control law and adaptive laws can be designed as

$${\alpha }_{i,n_i}=-\left({\beta }_{i,n_i}+2\right)z_{i,n_i}-z_{i,n_i}{\hat{\theta }}_{i,n_i},$$

(35)

$${\dot{\hat{\theta }}}_{i,n_i}=l_{i,n}{z}_{i,n_i}-{\eta }_{i,n_i}{\hat{\theta }}_{i,n_i},$$

(36)

where ${\beta }_{i,n_i}>0$ and ${\eta }_{i,n_i}>0$ are design parameters.

Remark 4. 

In this paper, the backstepping method is used to achieve the objective of formation. It recursively constructs virtual control laws and Lyapunov functions, starting from the unstable part of the innermost layer of the system, gradually expanding outward, and ultimately designing actual control laws that ensure the global stability of the entire closed-loop system. Its core advantages lie in its structured decomposition and strict stability assurance.

Further, we can deduce that

$$\begin{array}{cc}{\dot{V}}_{i,n_i}\hfill & \le -{\beta }_{i,1}{z}_{i,1}^2-\cdots -{\beta }_{i,n}{z}_{i,n_i}^2-{\displaystyle \frac{{\eta }_{i,1}{\tilde{\theta }}_{i,1}^T{\tilde{\theta }}_{i,1}}{2l_{i,1}}}-\cdots -{\displaystyle \frac{{\eta }_{i,n_i}{\tilde{\theta }}_{i,n_i}^T{\tilde{\theta }}_{i,n_i}}{2l_{i,n}}}+{\mathcal{Q}}_{i,1}\hfill \\ \hfill & +\cdots +{\mathcal{Q}}_{i,n_i-1}+z_{i,n}\left(u_i\left(t\right)-{\alpha }_{i,n_i}\right)+{\Delta }_{i,n_i},\hfill \end{array}$$

(37)

where ${\Delta }_{i,n_i}={\displaystyle \frac{9}{4}}+{\displaystyle \frac{1}{2}}{\bar{\zeta }}_{i,n_i}^2+{\displaystyle \frac{{\nu }^{\ast 2}}{2{\underset{\_}{\omega }}^2}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\bar{d}}_{i,n_i}^2}{{\underset{\_}{\omega }}^2}}+{\displaystyle \frac{1}{2}}\sum _{\mathcal{P}=1}^{n_i-1} {\bar{d}}_{i,\mathcal{P}}^2+$${\displaystyle \frac{1}{2}}\sum _{j\in {\mathrm{\aleph }}_i} \sum _{\mathcal{P}=1}^{n_i-1} {\bar{d}}_{j,\mathcal{P}}^2+{\displaystyle \frac{{\eta }_{i,n_i}{‖{\theta }_{i,n_i}‖}^2}{2l_{i,n_i}}}$.

To ensure that the formation objective can be achieved, the DET controller can be designed as follows:

$$u_i^{*}\left(t\right)=-\left(1+{\rho }_i\right)\left({\alpha }_{i,n_i}\mathit{tanh}{\displaystyle \frac{z_{i,n_i}{\alpha }_{i,n_i}}{q_i}}+{\overline{w}}_i\mathit{tanh}{\displaystyle \frac{z_{i,n_i}{\overline{w}}_i}{q_i}}\right),$$

(38)

and the following is the event triggering mechanism:

$$\begin{array}{c}{t}_i^{\sigma \left(t_{k+1}\right)}=inf\left\{t>t_i^{\sigma \left(t_k\right)}|E_i(t)|\ge {\rho }_i|u_i\mid +w_i(t)\right\},\hfill \\ {\dot{w}}_i\left(t\right)=-a{w}_i\left(t\right)+b{E}^{-h{w}_i(t-T)}{w}_i(t-T)+c\left|E_i\left(t\right)\right|,\hfill \\ u_i=u_i^{*}\left(t_i^{\sigma \left(t_k\right)}\right),\hfill \end{array}$$

(39)

where $E_i\left(t\right)= u_i-u_i^{*}\left(t\right)$ represents the event-triggered error, $0<{\rho }_i<1,a,b,c,\hslash >0$ are design parameters satisfies $a>b+{\displaystyle \frac{c}{2}},{\bar{w}}_i\left(t\right)={\displaystyle \frac{w_i\left(t\right)}{\left(1-{\rho }_i\right)}}$, and $T>0$ is the memory span.

Remark 5. 

Different from some existing event-triggered mechanisms in [24,26,27], $w_i\left(t-T\right)$ is introduced to avoid $w_i\left(t\right)$ from approaching zero too quickly, and the performance of saving communication resources is improved by upgrading the event-triggered threshold ${\rho }_i\left|u_i\right|+w_i\left(t\right)$. In addition, a compromise can be made between the parameters ${\rho }_i$, $w_i\left(0\right)$ and the designed constant $a$ to avoid excessive event-triggered errors which are caused by wide time intervals between trigger events.

Remark 6. 

Compared with the fixed-threshold event triggering mechanism, the DET mechanism has the following advantages: 1. It can adaptively adjust and reduce unnecessary frequent triggering, further saving the UCRs, 2. Online parameter adjustment provides greater flexibility and can balance performance and resource consumption.

Based on Equation (38), there exists $\left|{\mathcal{R}}_{1,i}\right|\le 1$ and $\left|{\mathcal{R}}_{2,i}\right|\le 1$, then $u_i\left(t\right)$ can be rewritten as

$$u_i={\displaystyle \frac{u_i^{\ast }\left(t\right)}{1+{\mathcal{R}}_{1,i}{\rho }_i}}-{\displaystyle \frac{{\mathcal{R}}_{2,i}{w}_i\left(t\right)}{1+{\mathcal{R}}_{1,i}{\rho }_i}}.$$

(40)

From Equations (37)–(39), it can be obtained that

$$\begin{array}{cc}{\dot{V}}_{i,n_i}\hfill & \le -{\beta }_{i,1}{z}_{i,1}^2-\cdots -{\beta }_{i,n}{z}_{i,n_i}^2-{\displaystyle \frac{{\eta }_{i,1}{\tilde{\theta }}_{i,1}^T{\tilde{\theta }}_{i,1}}{2l_{i,1}}}-\cdots -{\displaystyle \frac{{\eta }_{i,n_i}{\tilde{\theta }}_{i,n_i}^T{\tilde{\theta }}_{i,n_i}}{2l_{i,n}}}+{\mathcal{Q}}_{i,1}+\cdots +{\mathcal{Q}}_{i,n_i-1}+{\Delta }_{i,n_i}\hfill \\ \hfill & -z_{i,n}{\displaystyle \frac{\left(1+{\rho }_i\right)\left({\alpha }_{i,n_i}\mathit{tanh}\frac{z_{i,n_i}{\alpha }_{i,n_i}}{q_i}+{\overline{w}}_i\mathit{tanh}\frac{z_{i,n_i}{\overline{w}}_i}{q_i}\right)}{1+{\mathcal{R}}_{1,i}{\rho }_i}}-z_{i,n}{\displaystyle \frac{{\mathcal{R}}_{2,i}{w}_i\left(t\right)}{1+{\mathcal{R}}_{1,i}{\rho }_i}}-z_{i,n}{\alpha }_{i,n_i}.\hfill \end{array}$$

(41)

According to Lemma 1, the following inequalities hold

$$-{\displaystyle \frac{1+{\rho }_i}{1+{\mathcal{R}}_{1,i}{\rho }_i}}{z}_{i,n_i}{\overline{w}}_i\mathit{tanh}{\displaystyle \frac{z_{i,n_i}{\overline{w}}_i}{q_i}}\le 0.2785q_i-\left|z_{i,n_i}{\overline{w}}_i\right|,$$

(42)

$$-{\displaystyle \frac{1+{\rho }_i}{1+{\mathcal{R}}_{1,i}{\rho }_i}}{z}_{i,n_i}{\alpha }_{i,n_i}\mathit{tanh}{\displaystyle \frac{z_{i,n_i}{\alpha }_{i,n_i}}{q_i}}\le 0.2785q_i-\left|z_{i,n_i}{\alpha }_{i,n_i}\right|.$$

(43)

Then, it yields

$${\dot{V}}_{i,n_i}\le -{\beta }_{i,1}{z}_{i,1}^2-\cdots -{\beta }_{i,n}{z}_{i,n_i}^2-{\displaystyle \frac{{\eta }_{i,1}{\tilde{\theta }}_{i,1}^T{\tilde{\theta }}_{i,1}}{2l_{i,1}}}-\cdots -{\displaystyle \frac{{\eta }_{i,n_i}{\tilde{\theta }}_{i,n_i}^T{\tilde{\theta }}_{i,n_i}}{2l_{i,n}}}+{\mathcal{Q}}_{i,1}+\cdots +{\mathcal{Q}}_{i,n_i},$$

(44)

where ${\mathcal{Q}}_{i,n_i}={\Delta }_{i,n_i}+0.557q_i$.

3.2. Stability Analysis

Theorem 1. 

For the considered SNMAS with actuator faults, with Assumptions 1–4 held, the formation objective can be achieved and the UCRs can be saved under the proposed event-triggered controller Equation (37) with dynamic event triggering mechanism Equation (38). In addition, the Zeno phenomenon can be excluded.

Proof. 

The following is the selected CLF for the studied SNMASs in this paper:

$$V=\sum _{i=1}^N V_{i,n_i}.$$

(45)

Let ${\mathcal{C}}_i=\mathrm{m}\mathrm{i}\mathrm{n}\left\{2{\beta }_{i,1},..,2{\underset{\_}{\omega }}_i{\beta }_{i,n_i},{\eta }_{i,1},\dots ,{\eta }_{i,n_i}\right\}$, we can get the following inequality:

$$\dot{V}\le \sum _{i=1}^N \left(-{\mathcal{C}}_i{V}_{i,n_i}+{\mathcal{Q}}_{i,n_i}\right).$$

(46)

Then, denote $\mathcal{C}=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathcal{C}}_1,{\mathcal{C}}_2,\dots ,{\mathcal{C}}_N\right\}$, and $\mathcal{Q}=\sum _{j=1}^{n_i} \sum _{i=1}^N {\mathcal{Q}}_{i,j},i=\mathrm{1,2},\dots .N$. Thus, it yields

$$\dot{V}\le -\mathcal{C}V+\mathcal{Q}.$$

(47)

Next, we can obtain that

$$V\le e^{-\mathcal{C}t}\left(V\left(0\right)-{\displaystyle \frac{\mathcal{Q}}{\mathcal{C}}}\right)+{\displaystyle \frac{\mathcal{Q}}{\mathcal{C}}}.$$

(48)

When $t\to \mathrm{\infty },\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}} V={\displaystyle \frac{\mathcal{Q}}{\mathcal{C}}}$. So, the formation errors are bounded and formation objective can be achieved. □

Remark 7. 

In this paper, an adaptive distributed type-2 fuzzy DET formation protocol for a class of SNMASs with actuator faults has been provided. However, the scheme we designed still has shortcomings, such as the calculating explosion problem. By learning the command filters method and convex optimization-based method in [34,35], we will deal with the calculating explosion problem in our future work.

3.3. The Exclusion of Zeno Phenomenon

The derivative of $E_i$ is

$${\dot{E}}_i\left(t\right)={\dot{\hat{u}}}_i(t)\le \left|{\dot{\hat{u}}}_i\left(t\right)\right|\le {\mathcal{K}}_i,$$

(49)

with $\forall t\in \left[t_{i,j},t_{i,j+1}\right)$ and the positive constant ${\mathcal{K}}_i$.

Since the state variables of ${\dot{\hat{u}}}_i\left(t\right)$ are bounded, we have

$${\int }_{t_{i,j}}^{t_{i,j+1}} {\dot{E}}_i\left(\tau \right)d\tau \le {\int }_{t_{i,j}}^{t_{i,j+1}} {\mathcal{K}}_{i}dt.$$

(50)

When $t\to t_{i,j+1}$, we can deduce

$$\left|E_i\left(t_{i,j+1}\right)\right|={\mathcal{R}}_{1,i}\left|{\hat{u}}_i\left(t_{i,j+1}\right)\right|+{\mathcal{R}}_{2,i}\ge {\mathcal{W}}_i,$$

(51)

where ${\mathcal{W}}_i$ is a positive constant.

Hence, we get ${\mathcal{W}}_i\le {\int }_{t_{i,j}}^{t_{i,j+1}} \left|{\dot{E}}_i\left(\tau \right)\right|d\tau =\left|E_i\left(t_{i,j+1}\right)\right|\le$$\left(t_{i,j+1}-t_{i,j}\right){\mathcal{K}}_i$. Then, the lower bound of the interval between two triggering instants is

$$t_{i,j+1}-t_{i,j}\ge {\displaystyle \frac{{\mathcal{W}}_i}{{\mathcal{K}}_i}}>0.$$

(52)

There is a lower bound between the two triggers. Hence, the Zeno phenomenon can be excluded effectively.

Remark 8. 

Unlike previous works, a novel adaptive type-2 DET formation protocol is proposed to conserve communication resources. To validate the feasibility of the proposed method, this study rigorously demonstrates that the Zeno phenomenon does not occur. Consequently, the proposed protocol is both practical and effective.

Remark 9. 

According to the above analysis, there is a certain influence relationship between the controller coefficients and adaptive parameters and the UCRs. Increasing the controller coefficients and adaptive parameters may make the system’s response faster, thereby achieving the formation goals more quickly, but it may also increase the UCRs. Therefore, a suitable choice needs to be made between the selection of parameters and the UCRs.

4. Simulation Example

In this paper, the effectiveness of the proposed protocol can be verified by the simulation results. Consider the SNMASs as consisting of one leader and four followers.

The considered SNMASs are described as follows:

$$\left\{\begin{array}{l}{\dot{\chi }}_{i,1}={\chi }_{i,2}+f_{i,1}^{{\sigma }_i\left(t\right)}\left({\chi }_{i,1}\right)+d_{i,1}^{{\sigma }_i\left(t\right)}\left(t\right),\\ {\dot{\chi }}_{i,2}=v_i\left(t\right)+f_{i,2}^{{\sigma }_i\left(t\right)}\left({\chi }_{i,1},{\chi }_{i,2}\right)+d_{i,2}^{{\sigma }_i\left(t\right)}\left(t\right)\\ {\mathcal{Y}}_i={\chi }_{i,1},\end{array}\right.,$$

(53)

where ${\sigma }_i\left(t\right):\left[0,\mathrm{\infty }\right)\to M=\left\{\mathrm{1,2}\right\},f_{\mathrm{1,1}}^1=\sin \left({\chi }_{\mathrm{1,1}}^2\right),f_{\mathrm{1,1}}^2=\cos \left({\chi }_{\mathrm{1,1}}^2\right),f_{\mathrm{1,2}}^1=\cos \left({\chi }_{\mathrm{1,1}}{\chi }_{\mathrm{1,2}}\right),$  $f_{\mathrm{1,2}}^2=\sin \left({\chi }_{\mathrm{1,1}}{\chi }_{\mathrm{1,2}}\right),f_{\mathrm{2,1}}^1=\sin \left({\chi }_{\mathrm{2,1}}\right),f_{\mathrm{2,1}}^2= {\chi }_{\mathrm{2,1}}^2,f_{\mathrm{2,2}}^1 =\sin \left({\chi }_{\mathrm{2,1}}{\chi }_{\mathrm{2,2}}\right),f_{\mathrm{2,2}}^2=$   $\cos \left({\chi }_{\mathrm{2,1}}{\chi }_{\mathrm{2,2}}\right), d_{\mathrm{1,1}}^1=0.1\sin \left(t\right),d_{\mathrm{1,1}}^2=0.13\cos \left(t\right),d_{\mathrm{1,2}}^1=0.15\cos \left(t\right),d_{\mathrm{1,2}}^2=$   $0.11\sin \left(t\right),d_{\mathrm{2,1}}^1=0.13 \cos \left(0.8t\right),d_{\mathrm{2,1}}^2=0.25\sin \left(t\right),d_{\mathrm{2,2} }^1=0.18\sin \left(t\right),d_{\mathrm{2,2}}^2=$   $0.19\cos \left(0.5t\right) \mathrm{and} v_i\left(t\right)=\left(0.86-0.35\sin \left(t\right)\right)u_i\left(t\right)+0.08.$

${\mathcal{Y}}_r=2.5\sin \left(0.05t\right)$ is selected as the leader signal in this paper. Choosing the parameters of the SNMASs as follows: ${\beta }_{\mathrm{1,1}}=91,{\beta }_{\mathrm{1,2}}=98,{\beta }_{\mathrm{2,1}}=95,{\beta }_{\mathrm{2,2}}=96,{\beta }_{\mathrm{3,1}}=92$, ${\beta }_{\mathrm{3,2}}=91,{\beta }_{\mathrm{4,1}}=93,{\beta }_{\mathrm{4,2}}=95,l_{\mathrm{1,1}}=10,l_{\mathrm{1,2}}=10$, $l_{\mathrm{2,1}}=10,l_{\mathrm{2,2}}=10,l_{\mathrm{3,1}}=100,l_{\mathrm{3,2}}=5,l_{\mathrm{4,1}}=100$, $l_{\mathrm{4,2}}=5,{\eta }_{\mathrm{1,1}}=15,{\eta }_{\mathrm{1,2}}=0.1,{\eta }_{\mathrm{2,1}}=15,{\eta }_{\mathrm{2,2}}=0.1$, ${\eta }_{\mathrm{3,1}}=15,{\eta }_{\mathrm{3,2}}=0.1,{\eta }_{\mathrm{4,1}}=15,{\eta }_{\mathrm{4,2}}=0.5$ and $a=1.2$, $b=0.5,c=1,\hslash =60,T=0.5,{\rho }_i=0.05,q_i=0.05$.

The relative positions of the four agents are ${\mathsf{\Phi }}_1=-0.5$, ${\mathsf{\Phi }}_2=-0.25,{\mathsf{\Phi }}_3=0.25,{\mathsf{\Phi }}_4=0.5$. The initial conditions of the SNMASs are selected as ${\chi }_{\mathrm{1,1}}\left(0\right)=-0.5,{\chi }_{\mathrm{1,2}}\left(0\right)=0.15,{\chi }_{\mathrm{2,1}}\left(0\right)=-0.3,{\chi }_{\mathrm{2,2}}\left(0\right)=0.08,{\chi }_{\mathrm{3,1}}\left(0\right)=0.3,{\chi }_{\mathrm{3,2}}\left(0\right)=$   $0.12,{\chi }_{\mathrm{4,1}}\left(0\right)=0.5,{\chi }_{\mathrm{4,2}}\left(0\right)=0.1,\mathrm{a}\mathrm{n}\mathrm{d}{\hat{\theta }}_{\mathrm{1,1}}\left(0\right)=0.01,{\hat{\theta }}_{\mathrm{1,2}}\left(0\right)=0.01,{\hat{\theta }}_{\mathrm{2,1}}\left(0\right)=$   $0.02,{\hat{\theta }}_{\mathrm{2,2}}\left(0\right)=0.02,{\hat{\theta }}_{\mathrm{3,1}}\left(0\right)=0,{\hat{\theta }}_{\mathrm{3,2}}\left(0\right)=0,{\widehat{\theta }}_{\mathrm{4,1}}\left(0\right)=0.015,{\widehat{\theta }}_{\mathrm{4,2}}\left(0\right)=0.015$.

Figure 3 shows the communication topology of the considered SNMASs. The simulation results for the adaptive distributed type-2 DET formation control scheme are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 below. Figure 4 depicts the formation of agents, showing that the formation objective can be achieved. Additionally, Figure 5 displays the relative position trajectories of the four agents and one leader. The control inputs of four agents which suffer from actuator faults are shown in Figure 6. Figure 7, Figure 8, Figure 9 and Figure 10 exhibit the event triggering intervals (ETIs) of agents 1–4. Under the proposed continuous time strategies, the controllers of agents require 10,000 updates. Nevertheless, based on the given method, the number of updates for the agents 1–4 controllers are 1908, 2115, 1755, and 1587 separately. Hence, the proposed protocol can save UCRs effectively. Figure 11 and Figure 12 illustrate the asynchronous switching signals. Additionally, the effectiveness of the proposed protocol can be verified through simulation results.

On the premise that all formation objectives can be achieved, comparing the results of the designed protocol with the T1FLS-based fixed-threshold event triggering protocol which is shown in Figure 13, Figure 14, Figure 15 and Figure 16 as shown in

Table 1. Comparison of event trigger instants table.

	Agent 1	Agent 2	Agent 3	Agent 4
T1FLS-based fixed-threshold event triggering protocol	4397	4787	3697	2239
T2FLS-based DET protocol	1819	1923	1836	1727

, it can be concluded that the provided scheme in this paper has a better ability to save UCRs.

Remark 10. 

It is noted that the proposed method has some limitations. The relative position between two agents is an unknown positive constant which is not time varying. The consideration of formation error is simple, which makes the formation form unchanged throughout the entire protocol design process and unable to adapt to more practical scenarios. In future work, we will improve this limitation.

5. Conclusions

This article investigated an adaptive type-2 fuzzy fault-tolerant DET formation control problem for a specific class of SNMASs, characterized by heterogeneous agent dynamics. A distributed event-triggered controller with a dynamic event triggering mechanism has been designed to minimize the UCRs. It has been rigorously demonstrated that the intended formation control objective can be accomplished, while effectively eliminating the Zeno phenomenon. Future work may extend the DET formation scheme to SNMASs with jointly connected topologies.