System decomposition method-based global stability criteria for T-S fuzzy Clifford-valued delayed neural networks with impulses and leakage term

1.   Introduction

In the past decades, the adaptive consensus control problem of multi-agent systems has attracted increasing attention due to its wide range of applications in industrial and military fields, such as exploration robots, surface vehicles formation and so on ^{[1,2,3,4,5,6]}. Generally, the consensus problem can be divided leader-following consensus problem ^[7] and leaderless consensus problem ^[8]. The leader-following consensus problem similar to tracking control ^[9,10,11]. The purpose of the consensus problem is that these agents can reach synchronization. To achieve this goal, a proper distributed control protocol is designed based on the local information of agents and its neighbors. According to different working conditions, the consensus problem has been widely studied. In ^[12], the consensus problem of discrete-time systems with nonlinear dynamics was studied. The problem of communication noises was studied in ^[13]. In order to realize the consensus control of multi-agent systems, the adaptive control method was applied to the research of multi-agent systems ^[14]. Adaptive consensus control technology refers to the system adjusting itself according to the change of environment so that its behavior can achieve the best or at least allowable characteristics and functions in the new or changed environment. That is to say, it is an online adjustment mode, which cause unnecessary waste of communication resources. To overcome this disadvantage, more and more researchers have begun to research the event-triggered adaptive control technology.

In event-triggered control (ETC) scheme, when the event-triggered condition is satisfied, the information is transmitted. Generally speaking, the event-triggered condition include a predefined threshold and event-triggered error ^{[15,16,17,18,19]}. In general, based on the actual demands, the threshold is set. If the event-triggered error exceeds the threshold, the information is allowed to transmit. In this way, the burden of communication is reduced. In ^[20], the ETC problem was researched for uncertain nonlinear systems, and three different event-triggered conditions were designed of controller update. In ^[21], an ETC technology was studied for a simple single system. In ^[22], the technology in literature ^[21] had been improved, a distributed ETC scheme was proposed, which means that the trigger mechanism was extended to each agent system. In ^[23], the event-triggered tracking consensus control problem of nonlinear systems with unknown disturbances was researched. The fuzzy adaptive distributed ETC protocol was proposed for uncertain systems in ^[24]. With the development of science and technology, there has been a better development on how to use event-triggered mechanism to solve the communication burden. Based on describe the motivation, the methods of algorithmic synthesis, the technical challenges, and their application in distributed control, the development of the event-triggered mechanism of average consensus was introduced, and the event-triggered network system control problem was studied in ^[25]. In ^[26], the event-triggered coverage control problem was studied for asynchronous multi-agent systems, and a completely asynchronous communication sensing solution was proposed by the agent to decide when to push information to others in the networks. Based on ETC scheme, the global stabilization problem of k-valued logical control networks was studied in ^[27]. However, most schemes consider the fault-free system model in the above results.

In many practical systems, various faults may suddenly occur in the process of system operation ^[28,29,30]. Therefore, the effectiveness of components cannot reach the ideal goal. The actual performance of the systems may decline or instability, when the faults occur. In order to guarantee stability and safety of the systems, the some fault-tolerant control (FTC) methods have been developed. Based on fuzzy systems and sliding-mode methods, the problems of fault estimation and fault-tolerant control for stochastic systems with sensor faults were studied in ^[31]. The problem of sensor failures was researched for nonlinear pure-feedback systems, and an adaptive fuzzy fault-tolerant control method was proposed by the parameter separation technology in ^[32]. In ^[33], an adaptive fault-tolerant consensus control method was developed based on local filter to estimate the unmeasurable states for multi-agent systems. In ^[34], Fault tolerant control and event-triggered mechanism were considered at the same time in nonlinear systems. In order to compensate for actuator fault and uncertainty of systems, a robust adaptive decentralized FTC scheme based on neural network was proposed for interconnected systems in ^[35]. An observer-based fault-tolerant controller was designed by the T-S fuzzy and delta operator methods in ^[36]. Hence, it is meaningful that the fault-tolerant control problem is investigated. Moreover, because of the unknown dynamics existed in the actual system models, the neural networks (NNs) and fuzzy control have been widely studied ^{[37,38,39,40,41]}. The Lyapunov stability theorem is often used to prove that the signals of systems are bounded in different control environments, such as discrete-time systems ^[42], impulsive systems ^[43], semi-markov jump systems ^[44], pure-feedback interconnected nonlinear systems ^[45], and so on.

In this paper, the fault-tolerant leaderless consensus control problem is considered for nonlinear multi-agent systems. First, this paper attempts to apply a special event-triggered error definition to reduce the amount of communications in multi-agent systems. Then, a fault-tolerant control scheme is designed by using model reference control method and approximate property of neural networks to unknown functions, which reduces the exact requirements of controller parameters. At last, an event-triggered controller is designed to ensure the stability of multi-agent systems, where adaptive law is updated by event-triggered control method.

The rest of the structure is as follows. The proposes the graph theory and the NNs approximation method are given in Section Ⅱ. Section Ⅲ give the controller design. Stability analysis of multi-agent formations is presented in Section Ⅳ. In Section Ⅴ, it is proved that the ETC scheme avoid zeno behavior. A simulation result is given in Section Ⅵ. Section Ⅶ summarize the conclusion.

2.   Preliminaries

2.1.   Graph theory

The identify matrix is described as $I_N$. Let $\mathcal{G}$ = $\{\mathcal{V}, \mathcal{E}\}$ be an undirected graph, where $\mathcal{V} = \{ 1, 2, ..., N \}$, $N\geq 1$ and $\mathcal{E}$ = $\{(i, j)|i\neq j, i, j\in \mathcal{V}\}$ are described as a node set and an edge set, respectively. The adjacency matrix $\mathcal{A} = [a_{ij}\mathcal]\in{\mathbb{R}}^{N\times N}$ is associated with $\mathcal{G}$, $a_{ij}$ = $1$ if the agent $i$ can receive the information from agent $j$, otherwise $a_{ij}$ = $0$. The degree matrix $\mathcal{D}$ = diag$\{d_1, ..., d_N \}$, where $d_i$ = ${\sum }^{j = N}_{j = 1} a_{ij}$. Define $\mathcal{L}$ = $\mathcal{D}$-$\mathcal{A}$ is Laplacian matrix of $\mathcal{G}$.

For a vector $s = (s_1, ..., s_N)^T$, $\|s\|$ represents the 2-norm of $s$. Let matrix $W = [w_1, w_2;w_3, w_4]$, the vectorization of the matrix $W$ is defined $vec(W) = [w_1, w_2, w_3, w_4] ^T$, and $vec(W)^T vec(W) = tr\{W^T W\}$. For a square matrix $E\in \mathbb{R}^{n\times n}$, the minimum eigenvalue of $E$ is defined as $\lambda_{min}(E)$, and the maximum eigenvalue of $E$ is defined as $\lambda_{max}(E)$. The $Z^+$ is difference of $Z$ in a moment. The $\otimes$ is Kronecker product.

2.2.   Function approximation

In recent years, radial basis function NNs are employed to deal with unknown dynamics of systems ^[46]. Defined a compact set $\Omega_{\alpha}$, the unknown function $F(\alpha) \in \Omega_{\alpha}$, there exists a constant $\omega^{*T}\xi(\alpha)$ satisfying the following form

where $\alpha \in \mathfrak{R}^N$ is the input variable, $\delta$ is arbitrary positive constant, $\omega^{*} \in \mathfrak{R}^N$ is the ideal NNs weight vector, and $\xi(\alpha)$ is a smooth basis vector.

Based on the NNs approximation property, an unknown function $F(\alpha)$ can be represented in the following form

where $\delta(\alpha)$ is the smallest approximation error and $\delta(\alpha)\leq \bar{\delta}$. Throughout this paper, we define $\xi(x) = [\xi_1(x_1)^T, ..., \xi_N(x_N)^T]^T$, and $\delta(x) = [\delta_1(x_1), ..., \delta_N(x_N)]^T$. There exists $\|\xi(x)\| < \bar{\xi}$ and $\|\delta(x)\| < \bar{\delta}$, in which $\bar{\xi}$ and $\bar{\delta}$ are positive constants.

2.3.   Systems description

The nonlinear multi-agent systems are considered, and its mathematical model can be expressed as

where $x_{i}\in\mathbb{R}^{m}$ is state of the $i$-th agent, and $x = [x_1^T, x_2^T, ..., x_N^T]^T$. $A_0$ is a known constant matrix with compatible dimension. The $f_{0i}\in f_{0} = [f_{01}^T, ..., f_{0N}^T]^T$ is smooth continuous nonlinear function. $u_{i}$ is the controller of each agent, and $u$ = $[u_1^T, u_2^T, ..., u_N^T]^T$. $g(x_i)\in g(x) = [g(x_1)^T, ..., g(x_N)^T]^T$ is gain coefficient. For the multi-agent systems (2.3) have following assumption.

Assumption 1. System (2.3) is controllable, and the nonlinear function $f_2(x)$ can be linearizable. $g(x)$ is the control gain matrix, and $g(x)$ is bounded for all $x$, satisfying $\|g(x)\| \leq g_{max}$ and $g_{max} > 0$.

In this paper, the actuator fault is considered, the multi-agent systems model can be rewritten as

where $p_{i}$ = diag$(\rho_1, ..., \rho_N)$ is the actuator effectiveness factor of each agent with $\rho_i\in(0, 1)$, and $p$ = diag$(p_{1}, ..., p_{N})$. $A_1$ is a known constant matrix with compatible dimension. The $f_{1i}\in f_{1} = [f_{11}^T, ..., f_{1N}^T]^T$ is smooth continuous nonlinear function. It is assumed that the ideal controller $u^*_i = Q(x)$ enables the multi-agent systems to achieve the desired performances. If the controller $u^*_i = Q(x)$ is added to the fault multi-agent systems, the systems model can be rewritten as

where $A_2$ is a known constant matrix with compatible dimension, and the $f_{2i}\in f_{2} = [f_{21}^T, ..., f_{2N}^T]^T$ is smooth continuous nonlinear function. There are the following assumptions for equality (2.2) and system (2.5).

Assumption 2. For the multi-agent systems (2.5), the positive matrices $P$ and $Q$ satisfy the following form

where $\sigma$ represents an appropriate positive constant, $I$ is defined as the identity matrix.

Assumption 3. For nonlinear smooth bounded function functions $f_2(x)$ and $\xi(x)$, there exist constant $l_f$ and $l_{\xi}$ satisfying

where $l_{f}$ and $l_{\xi}$ are Lipschitz constants.

2.4.   Stability theoretics

An impulsive dynamical system is considered ^[47], it is defined as

where $X\in \mathcal{I}$ is the state vector of the system, and $\mathcal{I}$ is an open set with $0\in \mathcal{I}$. $\mathcal{F}$ and $\mathcal{J}$ are the flow and the jump sets, respectively. $\Delta X(t^+) = \underset{a\rightarrow 0}{lim}X(t+a)$. The functions $F_{c}(X)$ represents continuous dynamics of the impulsive dynamical system, and the functions $F_{d}(X)$ is reset dynamics of the impulsive dynamical system.

Remark 1. In this paper, because the event-triggered sampling mechanism is used, the influence of sampling impulse should be considered when discussing the stability. And, the various impulsive theories have been widely used ^[48].

Lemma 1. ^[49] It is assumed that the function $V(X)$ is continuously differentiable. $M(*)$ and $O(*)$ are continuous functions with initial value being 0, such that $V(X)$, $M(*)$ and $O(*)$ satisfy the following forms

where $\chi$ is a positive constant. If the above equations are satisfied, the system state is locally ultimately bounded.

3.   Controller and event-triggered mechanism design

3.1.   Controller design

Based on the NNs approximate property (2.2), the ideal controller $u_{i}^*$ can be shown as

According to the above analysis, the actual controller can be designed in the following form

where $H(t) = \exp({-\tau(t-t_k)})$. The $\hat{\omega}_i$ is estimation of ${\omega}_i$, and $\hat{\omega}_i \in \hat{\omega}^{T} = [\hat{\omega}^{T}_1, ..., \hat{\omega}^{T}_N]^T$, ${\omega}^{T} = [{\omega}^{T}_1, ..., {\omega}^{T}_N]^T$.

Remark 2. Because the multi-agent systems exist interconnections among agents, the structure of the controller (3.1) is different from the one in ^[34,47]. In this way, the technology is further introduced into the field of multi-agent.

At the time $t = t_k$, the event-triggered condition is triggered, so the $\hat{\omega}_i$ is updated. The next triggered time is defined as $t_{k+1}$. Until the next triggered time $t_{k+1}$, the $\hat{\omega}_i$ is held. The time interval $(t_k, t_{k+1}]$ represents the occurrence of an event triggered. Therefore, the adaptive update law is designed as

where $0 < \nu < 1$, $a_{e} > 0$, $a_{\tilde{x}} > 0$, $b_{e}$ and $b_{\tilde{x}}$ are small positive constants, $B_{1i}\in B_1 =$diag$[B_{11}, ..., B_{1N}]$ and $B_{2i}\in B_2 =$diag$[B_{21}, ..., B_{2N}]$ are nonzero matrices with appropriate dimension. $e_i$ is event-triggered error.

Remark 3. The $b_{e}$ and $b_{\tilde{x}}$ are defined as small positive constants. They exist to avoid the denominator of equality (3.3) equal to $0$.

Let $\tilde{x}_i = x_i-\hat{x}_i$, where $\hat{x}_i$ is reference dynamics with the following form

Next, the $\tilde{\omega}_i = \omega_i- \hat{\omega}_i$ denotes the estimation error, and $\tilde{\omega} = [\tilde{\omega}_1^T, ..., \tilde{\omega}_N^T]^T$. The estimation error dynamics are obtained as

where $\Delta\tilde{\omega}_i = (1-\nu)\hat{\omega}_i+{W}_{{e}_i}\xi(x_i)e^{T}_i{B}_{1i}+{W}_{\tilde{x}_i}\xi(x_i)\tilde{x}^{T}_i{B}_{2i}, {W}_{{e}_i} = \frac{a_{e}}{b_{e}+\| e_i \|^2}, {W}_{\tilde{x}_i} = \frac{a_{\tilde{x}}}{b_{\tilde{x}}+\| \tilde{x}_i \|^2}$, ${W}_{{e}_i}\in{W}_{e} = [{W}_{{e}_1}, ..., {W}_{{e}_N}]^T$ and ${W}_{\tilde{x}_i}\in {W}_{\tilde{x}} = [{W}_{\tilde{x}_1}, ..., {W}_{\tilde{x}_N}]^T$.

3.2.   Event-triggered mechanism design

In this section, the event-triggered mechanism is introduced ^[34]. First, $e_i$ represents event-triggered error. Define the $e_i = x_i(t)-x_{ik}$, where $e_i \in e = [e_1^T, ..., e_N^T]^T$, $x_{ik} = x_i(t_k)\exp({-\tau(t-t_k)})$. Then, considering the stability of the multi-agent systems, the design of event-triggered condition is as follow

where $\bar{{B}}_{X}$ is a small positive constant. The $\bar{{B}}_{X}$ can avoid the frequent occurrence of events, when the $\|x\|$ is too small. In order to ensure the stability of the multi-agent systems, $\zeta_{e}$ is designed as follows

where $K_\zeta$ is a constant, and $0 < K_\zeta < \frac{1}{\zeta} \lambda_{min}(Q)$. The constant $\zeta > 0$.

Remark 4. The event-triggered error $e_i$ has been designed, and a negative exponential function is added to the event-triggered error. According to the characteristics of the negative exponential function, the event-triggered mechanism avoided no triggering after a long period of system stability. Because $\xi(x)$ is a smooth continuous, $l_{\xi}$ donot equal to zero. In multi-agent systems, the design of event-triggered scheme needs to consider the interconnections among agents. Although $\|\hat{{\omega}}\|$ is equal to zero, it is guaranteed that the denominator of equality (3.7) is not equal to zero. Differ in ^[34], the parameter selection needs to consider another form when the 2-norm of weight vector estimation equal to zero. In other words, the denominator of $\zeta_{e}$ does not appear to be equal to zero in this paper.

4.   System stability analysis

4.1.   Impulsive dynamical model

According to the multi-agent systems (2.4) and controller (3.2), the multi-agent systems can be rewritten as

Furthermore, by the definition of Laplacian matrix, one has

Add and subtracting ${g}(x){p}u^{*}$ yields

Using the ideal controller (8) and system (4), we have

According to $\tilde{\omega} = \omega- \hat{\omega}$, one obtains

Define the state error $\tilde{x} = x-\hat{x}$. According to the above equation, the following equation can be obtained

Define a sign $\psi = [x^T, x^T_k, \tilde{x}^T, vec(\hat{\omega})^T]^T\in \mathcal{I}$, where $\psi$ is an augmented vector. Then, the closed-loop impulsive dynamical is obtained as

where ${\Gamma} = vec((1-\nu)\hat{\omega}+{W}_{{e}}\xi(x)e^{T}{B}_{1}+{W}_{\tilde{x}}\xi(x)\tilde{x}^{T}{B}_{2})$.

4.2.   Stability analysis

In this section, the stability is established via Lyapunov function. Firstly, The estimation error $\tilde{\omega}$ needs to be bounded, so the following Lemma is given.

Lemma 2. Consider the multi-agent systems (2.3) and the controller (3.1) expressed as the the impulsive system (4.6), and the adaptive update law is (3.5). Let Assumptions 1-3 be satisfied, and the initial $\hat{\omega}(0)$ in a compact set. Therefore, the estimated error $\tilde{\omega}$ is bounded by selecting the appropriate constant.

Proof : In impulsive systems, the stability of continuous dynamics and stability of jump dynamics need to be considered, respectively. The function $V_w = tr\{ \tilde{\omega}^{T}\tilde{\omega}\}$ is defined as Lyapunov function for the impulsive dynamics (3.5). The $\hat{\omega}_i$ is held in the continuous part of impulsive dynamics, so $\dot{V}_w = 0$ for $t \in (t_k, t_{k+1}]$. Further, the adaptive estimation error $\tilde{\omega}$ is bounded.

Then, the $\hat{\omega}_i$ is updated when the event-triggered condition is triggered. The stability of jump part is considered. According to the above analysis, we know that

Using the equality (3.5), we can get

According to the equality (3.3), the $0\leq{W}_{\tilde{x}}\|\tilde{x}\| < 1$ and $0\leq{W}_{e}\|e\| < 1$ is obtained. We get

Combined with the above inequalities and $0 < \nu < 1$, we can get

where

And, inequality (4.11) can be rewritten as follows

where $\eta_4 = \eta_3 + ({\eta_2}^2 / 2 \eta_1)$. It is found that $\Delta V_w$ is negative when $\| \tilde{\omega}\|^2 > {2\eta_4}/{ \eta_1}$. In the triggered instant $t_k$, the estimated error $\tilde{\omega}$ is ultimately bounded.

According to the above analysis, it is proved that the error $\tilde{\omega}$ is locally ultimately bounded. Then, the stability is illustrated in the following theorem.

Theorem 1. Consider the multi-agent systems (2.4), the controller (3.1), the event-triggered condition (3.6), and adaptive update law (3.3), suppose that the initial augmentation state $\psi(0)\in \mathcal{I}$. If the Assumptions 1–3 are satisfied. Then, the augmented state $\psi$ is locally ultimately bounded.

Proof: In order to prove the stability of system (4.7), the Lyapunov function is constructed as

where $V_x = x^TPx$, $V_{x_k} = {x_{k}}^Tx_{k}$, $V_{\tilde{x}} = {\tilde{x}}^TP{\tilde{x}}$ and $V_w = tr\{ \tilde{\omega}^{T}\tilde{\omega}\}$. $P$ is a positive matrix satisfying the Assumption 3.

In the first step, the continuous part of the dynamical (4.7) is considered for $t \in\left(t_{k}, t_{k+1}\right]$. The time derivative of the Lyapunov function (4.14) can be expressed in the following four parts.

We know that the $\dot{V}_x$ can be described as

Using the Assumption 2 and Young's inequality, one has

Using Assumption 3, it is rewritten as

It is fact that $0 < \|p\| < 1$, Lipschitz condition (6) of Assumption 2, and the $g(x)$ is bounded, we can get

Consider the event-triggered condition (3.6), the $\|e\|\leq \zeta_{e}(\|x\|+\bar{{B}}_{X})$ is given. Therefore, we can obtained

The derivative of ${\dot{V}_{x_k}}$ is given as

Next, the derivative of ${V_{\tilde{x}}}$ is given as follows

Similar to (4.16)-(4.18), we can obtain

Based on the event-triggered condition (3.6) and Young's inequality, the following inequality holds

where $\beta$ is a positive constant, and the equality $\beta^2 = \beta+1$ is satisfied.

Because the $\hat{\omega}_i$ is held in the event-triggered time intervals, we can get $\dot{V}_w = 0$. Therefore, one has

According to $\beta$ is solution of the $\beta^2 = \beta+1$ and $\beta > 0$, we can get

where $\bar{X} = [\|x\|, \|\tilde{x}\|]^T$, $\theta_1 = \lambda_{min}(Q)-\beta K_{\zeta}$ and $\theta_2 = 2\sqrt{2}({g}_{max}\|P\|(\|{\tilde{\omega}}\|\bar{\xi}+\bar{\delta})+K_\zeta{\bar{{B}}_{X}})$. According to Lemma 2, we know that $\tilde{\omega}$ is bounded, we can conclude that the $\theta_2$ is bounded. It is fact that $\dot{V}_f (\psi)$ is less than zero when $\|\bar{X}\| > \bar{G}_{\bar{X}} = {\theta_2}/{\theta_1}$ is satisfied, and it means that the multi-agent systems state $x(t)$ and the state error $\tilde{x}$ are locally ultimately bounded for $t\in(t_k, t_{k+1}]$.

Consider the jump part of the dynamical (4.7) in $t = t_{k}$, the Lyapunov function is defined as (4.14). The difference is obtained as follow form

Using $\Delta \psi = \psi^+ - \psi$ and the dynamics (4.7), we have

For any $t\in(t_k, t_{k+1}]$, it is fact that the $x(t)$ is bounded. Consider the inequality (4.11), one obtains

where $\eta_x$ is bounded for $\|x\|^2$ and $\eta_x < \|\bar{X}\|^2$.

Consider the above analysis, $\Delta V_f (\psi)$ is negative when $\|{x_{k}}\| > G_{x_k}$ or $\|\tilde{\omega}\| > G_{\tilde{\omega}}$, where

Further, the state $\psi$ is locally ultimately bounded in $t = t_{k}$. Because of $x^+ = x$ and $\|\tilde{x}^+\| \leq\|\tilde{x}\|$ for $t = t_k$, the $\|\bar{X}\|\leq \bar{G}_{\bar{X}}$ in $t = t_k$. we can conclude that $G_{x_k}$ and $G_{\tilde{\omega}}$ converge to $\bar{G}_{x_k}$ and $\bar{G}_{\tilde{\omega}}$, where

The $\hat{\omega}$ remains constant, and $\|x_k\|$ is monotone decreasing function on $t\in(t_k, t_{k+1}]$. If $\|\psi\|\geq \sqrt{\bar{{G}}_{\bar{{X}}}^{2}+\bar{{G}}_{x_{k}}^{2}+\bar{{G}}_{\tilde{\omega}}^{2}}$, the Lemma 1 is satisfied. Then, it can be proven that all signals of the multi-agent systems are locally ultimately bounded.

5.   The event-triggered time interval

In this section, it is proven that the event-triggered time interval is a nonzero positive constant in the following theorem. Therefore, the event-triggered time interval is a lower bound.

Theorem 2. Consider the controller (3.1) and the event-triggered condition (3.6) are proposed in this paper. Let Assumptions 1-3 be satisfied and the initial $\hat{\omega}(0)$ is in a compact set. Then, the event-triggered time interval $T_{k}$ is a positive scalar with nonzero lower bound.

In the event-triggered time interval $t \in(t_k, t_{k+1}]$, the following equalities hold

Based on the system dynamics (4.4) and (4.6), one has

Therefore, one obtains

where $\Gamma_e = (\|{\tau}I_n+{A}_{2}\otimes{I}_{n}\|+l_f)\|x_{k}\| +{g}_{max}(2\|{\hat{\omega}}\|\bar{\xi}+\|{\tilde{\omega}}\|\bar{\xi}+\bar{\delta})$.

Consider the comparison lemma in ^[50] and $e(t_k) = 0$, one has

where $\Gamma_1 = \|({A}_{2}\otimes{I}_{n})\|+{g}_{max}\|L\otimes I_n\|+l_{f}$

Based on the event-triggered condition (3.6), the $\|e\|\geq\zeta_{e}\bar{{B}}_{x}$ in the trigger instant $t = t_k$. The $T_{k} = t_{k+1}-t_k$ represents the $k$-th event-triggered time interval. Further, we can get

which means that

Therefore, the event-triggered time interval is a lower bound. In other words, there is no Zeno behavior.

6.   Simulation

In this section, we prove the theoretical results by numerical examples. We consider the leaderless multi-agent systems with four follower nodes in Figure 1. The each agent's dynamic is given as follows

where ${x}_{i} = [{x}_{i1}, {x}_{i2}]^T$, $u_{i} = [u_{i1}, u_{i2}]^T$, $i = 1, \ldots, 4$. The matrices $A_1$, $f_{1i}(x_{i})$, ${g}(x_{i})$, and $p_{i}$ are chosen as follows

We assume the ideal controller $u^*_i = Q(x)$, so the each agent's dynamic is rewritten as

where $A_2 =$diag$\{-20, -25\}$ and $f_{2i} = [4x_{i1}, 5x_{xi2}]^T$.

The constants are chosen as $P =$diag$(1/60, 1/60)$, $l_{f} = 2.4$, $l_{\xi} = 2.4$, $\sigma = 1$, $\nu = 0.0015$, $a_{e} = 5$, $a_{\tilde{x}} = 5$, $b_{\tilde{x}} = 1$, $b_{e} = 1$, $K_\zeta = 0.09$, $\tau = 0.81$, $B_{1i} =$diag$(1, 1)$ and $B_{2i} =$diag$(1, 1)$. We choose the initial values as $x_1 = [5, 2.5]^T$, $x_2 = [3,2]^T$, $x_3 = [1, 1.21]^T$, $x_4 = [4,3]^T$. Figures 2–7 show the simulation results. The Figure 2 depicts the responses of the system state vector $x_{11}$, $x_{21}$, $x_{31}$ and $x_{41}$. The Figure 3 depicts the responses of the system state vector $x_{12}$, $x_{22}$, $x_{32}$ and $x_{42}$. The above figures show the leaderless consensus. The event-triggered time intervals are presented in Figures 4–7.

7.   Conclusions

In this paper, an event-triggered fault-tolerant consensus control strategy for nonlinear multi-agent systems has been proposed. Based on the approximate property of neural networks and the model reference control method, the fault-tolerant method has been designed to ensure security for leaderless multi-agent systems, which reduces the exact requirements of control parameters. The ETC scheme based on a relative threshold method has been proposed to reduce communications. The event-triggered scheme has been proved that there is no Zeno behavior. By the Lyapunov stability theory, we have obtained that all signals are bounded. The simulation result has confirmed the validity of proposed approach. In the future, we will continue to study the issue of event-triggered and fault-tolerant control. These problems are of great significance in various practical systems.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61703051), and the Project of Liaoning Province Science and Technology Program under Grant (2019-KF-03-13)

Conflict of interest

The authors declare no conflict of interest in this paper.