Distributed Disturbance Observer-Based Containment Control of Multi-Agent Systems with Event-Triggered Communications

Abstract

This article investigates a class of multi-agent systems (MASs) with known dynamics external disturbances, where the communication graph is directed, and the followers have undirected connections. To eliminate the impacts of external disturbance, the technologies of disturbance observer-based control are introduced into the containment control problems. Additionally, to save communication costs and energy consumption, a distributed disturbance observer-based event-triggered controller is employed to achieve containment control and reject disturbance. Furthermore, designing the event-triggered function using an exponential function is beneficial for a time-dependent term while ensuring the exclusion of Zeno behavior. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis.

1. Introduction

Consensus, as a fundamental problem in the cooperative control of multi-agent systems (MASs), has garnered significant attention over the past few decades due to its potential applications in various areas, such as robotic systems, groups of unmanned aerial vehicles, and power-sharing of DC microgrids (see, for example, [1,2,3,4]). Notably, its application in modeling various system dynamics and engineering scenarios has been conducted on consensus problems, as evidenced by studies such as [5,6,7]. Consensus problems have two categories: leaderless consensus and leader-following tracking consensus.

The works above mainly investigate these two types of consensus problems. However, there are scenarios in practical applications, as well as in social animals and insects. Motivated by practical applications and natural phenomena, the containment control problem with multiple leaders has been widely investigated, that is, all followers tend to the convex hull spanned by all the leaders (see, for example, [6,7,8,9,10,11,12]). Many containment control strategies have been studied for different MASs, including first-order or second-order agent dynamics [9], fractional-order systems [10,11,12], homogeneous high-order MASs [13,14], and heterogeneous high-order MASs [15].

It is worth pointing out that most research on MASs has mainly focused on the ideal models. However, external disturbances in practical MASs are widespread and affect system performance. In practical engineering, systems often operate in environments with various disturbances, impacting control accuracy. Additionally, the cooperative control of MASs has strict requirements for control accuracy. Therefore, it is crucial to study the consensus problem for MASs in the presence of external disturbances. Due to physical disturbances such as external environment disturbance and system constraint disturbance, it is desirable to design distributed disturbance rejection strategies for MASs, such as anti-interference methods, sliding-mode observers, disturbance observers, output regulation, and so on [16,17,18,19,20,21,22,23,24,25,26]. For instance, since each agent cannot access the desired tracking signal and external disturbances, the authors in [16] utilized a dynamic compensator independent of the exogenous signal and introduced a dynamic internal-model-based component. Recently, the authors of [17] utilized the distributed event-based controller and designed a disturbance observer for MASs with matched disturbances to guarantee performance requirements and avoid exogenous disturbances, and similarly for the controllers in [18] under an observer-based dynamic event-triggered scheme. In [22], based on designing disturbance observers, event-triggered control involving controllers and actuator-updating rules are provided. Disturbance observer is designed for MASs under deterministic disturbances in [23]. Recently, significant progress has been made in MASs based on the state or relative state measurements [27,28,29,30].

In the aforementioned studies, participants in MASs must exchange interaction information through their microprocessors. However, highly frequent, continuous communication and heavy computational burden significantly increase the overall system cost. Additionally, to avoid these situations, agents are equipped with limited resources and power. Therefore, a key issue in resource-constrained MASs is ensuring the performance of the closed-loop system while reducing the workload of sensors and actuators. Cooperative control based on event-triggered policies provides an effective solution [31,32,33,34,35,36,37,38,39,40,41]. Distributed event-triggered consensus control usually uses zero-order hold (ZOH) [42,43,44,45]. The works [35,36] make a breakthrough in the estimation method based on model values, whereby the event-triggered protocols help reduce communication frequency. In [33,43,45,46], each agent’s controller is designed based on the most recent event moment and remains unchanged until the next event moment. These control strategies reduce the number of controller or state updates. However, when using inter-agent communication, continuous communication between agents is required to check event-triggered conditions [47,48,49,50,51,52,53]. It is observed that the existence of a positive trigger time cannot be guaranteed in [17], although the Zeno behavior is excluded by incorporating an exponential decay term in the triggering threshold.

Motivated by the aforementioned works, this work considers the distributed disturbance containment problem for linear MASs with event-triggered communications. The communication topology among followers and leaders is directed. To solve this problem, a distributed containment protocol is proposed to reach containment control and simultaneously reject disturbances. In particular, through event-triggered communication, the disturbance estimator is constructed using a disturbance observer control design and non-periodic information from followers. A time-independent threshold is used to design the trigger condition for event-triggered communication.

Inspired by the above discussion, we propose a disturbance observer-based containment control algorithm. The main contributions of this paper are inthree aspects:

(1)

In contrast to the existing disturbance containment control algorithms that only focus on traditional MASs [23], this paper targets a class of external disturbances with known dynamics. We propose a disturbance observer-based containment control algorithm, which introduces a disturbance compensation term to offset the external disturbance.

(2)

Compared with the existing containment control methods developed in [23,47] for MASs with continuous communication and actuation, the proposed event-triggered containment control method has the advantage of reducing the burden of communication and actuation.

(3)

Different from the control methods in [40,45], the triggering functions designed in this paper are independent of the number of nodes (which represent the scale of the network) and do not require continuous communication. Hence, the event-triggered strategies are expected to be scalable with low communication costs. The design of the event-triggered function using an exponential function is beneficial for a time-dependent term and also eliminates Zeno behavior. By selecting appropriate coefficients for each exponential function, we can balance the communication frequency and control performance.

2. Preliminaries and Problem Formulation

2.1. Graph Theroy

The communication topology among the N agents is represented by a directed graph $\mathcal{G}=(𝒱,\mathcal{E})$, where $𝒱=\{1,2,\cdots ,N\}$ and $\mathcal{E}\subset 𝒱\times 𝒱$ are the node set and the edge set, respectively. The adjacency matrix $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ is given by $a_{ij}=0$, $a_{ij}>0$ if $(i,j)\in \mathcal{E}$. The Laplacian matrix of $\mathcal{G}$ is defined as $L=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$, where $l_{ii}={\sum }_{j\ne i}{a}_{ij}$ and $l_{ij}=-a_{ij}$, where $i\ne j$. The in-degree of agent i is defined as $r_i=l_{ii}$. In this article, a group of N agents composed of M followers and $N-M$ leaders is represented by a directed graph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$. The set of followers and leaders is denoted by $\mathfrak{L}\triangleq \{M+1,\dots ,N\}$ and $\mathfrak{F}\triangleq \{1,\dots ,M\}$, respectively. The leaders have no neighbors among themselves. Thus, the Laplacian matrix of ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$ can be partitioned as

$$L\triangleq \left[\begin{array}{cc}{L}_{\mathfrak{F}}& L_{\mathfrak{L}}\\ {\mathbf{0}}_{(N-M)\times M}& {\mathbf{0}}_{(N-M)\times (N-M)}\end{array}\right],$$

where $L_{\mathfrak{F}}\in {\mathbb{R}}^{M\times M}$ and $L_{\mathfrak{L}}\in {\mathbb{R}}^{M\times (N-M)}$.

2.2. Notations

Throughout this article, the following notations are used. $\parallel ·\parallel$ denotes the Euclidean norm of matrices. For matrices A and B, $A\otimes B$ denotes the matrix Kronecker product of A and B. For a square real matrix, $Z>0(Z\ge 0)$ means that Z is a positive definite (semi-definite), and $\lambda \left(Z\right)$ represents its eigenvalues. $I_N$ denotes the $N\times N$ identity matrix and ${\mathbf{1}}_N$ denotes an $N\times 1$ column vector of all 1.

2.3. Problem Statement

Consider N agents of a linear MAS with a directed graph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$. The dynamics of followers $i\in \mathfrak{F}\cup \mathfrak{L}$ are described by

$$\begin{array}{cc}\hfill {\dot{x}}_i& =A{x}_i+B{u}_i+D{d}_i,i\in \mathfrak{F},\hfill \end{array}$$

(1a)

$$\begin{array}{cc}\hfill {\dot{x}}_i& =A{x}_i,i\in \mathfrak{L}.\hfill \end{array}$$

(1b)

where $x_i\in {\mathbb{R}}^n$ and $u_i\in {\mathbb{R}}^m$ are the ith agent’s state and output state. A, B, and D are known constant matrices of appropriate dimensions. $d_i\in {\mathbb{R}}^n$ is a disturbance whose dynamics are given as

$${\dot{d}}_i=S{d}_i,i\in \mathfrak{F},$$

(2)

with S being a known constant matrix.

To proceed, we also need the following assumptions and Lemma.

Assumption 1

([17]). $(A,B)$ is stabilizable.

Assumption 2

([17]). The digraph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$ is connected. For each follower $i\in \mathfrak{F}$, at least one leader $k\in \mathfrak{L}$ has a directed path to its follower.

Assumption 3

([17]). The disturbance is matched, i.e., there exists a matrix F, such that $D=BF$.

Assumption 4

([17]). The eigenvalues of the matrix S are on the imaginary axis, and the pair $(S,D)$ is observable.

Remark 1.

Based on the same system model object, we use the same assumptions as in [17]. These assumptions meet the requirements of system stability and observable, and satisfy the matched disturbance-related conditions.

Definition 1

(Containment control problem [6]). Given the MASs (1) and a directed graph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$, find a distributed controller so that the followers converge to the convex hull spanned by the states of the leaders, that is, ${lim}_{t\to \infty }\parallel (L_{\mathfrak{F}}\otimes I_n)x_{\mathfrak{F}}+(L_{\mathfrak{L}}\otimes I_n)x_{\mathfrak{L}}\parallel =0$.

Lemma 1

([6]). Under Assumption 2, all the eigenvalues of $L_{\mathfrak{F}}$ have positive real parts, $-L_{\mathfrak{F}}^{-1}{L}_{\mathfrak{L}}$ is non-negative, and $-L_{\mathfrak{F}}^{-1}{L}_{\mathfrak{L}}{\mathbf{1}}_{N-M}={\mathbf{1}}_M$.

3. Main Results

In this section, we design a distributed disturbance observer-based containment controller for systems (1) and (2) as follows:

$$\begin{array}{cc}\hfill & u_i=c{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)-F{\hat{d}}_i,\hfill \\ \hfill & {\hat{d}}_i=z_i+H{x}_i,\hfill \\ \hfill & {\dot{z}}_i=(S-HD){\hat{d}}_i-H(A{x}_i+B{u}_i),i\in \mathfrak{F},t\in [t_k^i,t_{k+1}^i),\hfill \end{array}$$

(3)

where ${\hat{d}}_i\in {\mathbb{R}}^s$ and $z_i\in {\mathbb{R}}^m$ are the estimates of the disturbance and the internal variable of the observer, respectively. $K_1,F,H$ and D are gain matrices to be determined. And ${\tilde{x}}_i=e^{A\left(t-t_k^i\right)}{x}_i\left(t_k^i\right)$, where $t_k^i$ is the kth event-triggered instant of agent $i\in \mathfrak{F}$. For agent $i\in \mathfrak{F}\cup \mathfrak{L}$, we define the measurement error $e_i\left(t\right)$ as

$$e_i\left(t\right)={\tilde{x}}_i-x_i,t\in [t_k^i,t_{k+1}^i).$$

The event-triggered function for each agent i is given by

$$\begin{array}{c}\hfill f_i\left(t\right)=4r_i\parallel K_1{e}_i{\parallel }^2-\sum _{j=1}^N{a}_{ij}{\parallel K_1({\tilde{x}}_i-{\tilde{x}}_j)\parallel }^2-\mu e^{-\nu t}\end{array}$$

(4)

where $r_i$ is the in-degree of agent i, and $\mu$ and $\nu$ are positive constants. As long as the triggering condition $f_i\left(t\right)\ge 0$ is fulfilled, an event is triggered for agent i. At this time instant, agent i updates its observer-based controller (3) using its current state of the observer and broadcasts the current state of the observer to its out-neighbors. Meanwhile, $e_i\left(t\right)$ is reset to zero. When agent i receives new information from its neighbors, the controller for agent i will also be updated immediately. If $f_i\left(t\right)<0$ for agent $i\in \mathfrak{F}$, there is no communication occurring until the next event is triggered.

Remark 2.

Compared with [40], we deal with the MASs with disturbance, and the applicability of the control algorithm is better. Compared with reference [41], our event-triggered controller adopts the time-independent exponential function trigger condition, which is more general than the trigger condition of reference [41].

Define for $i\in \mathfrak{F}$ as follows:

$${\epsilon }_i={\hat{d}}_i-d_i.$$

From (1)–(3), we have

$$\begin{array}{cc}\hfill {\dot{x}}_i& =A{x}_i+B{u}_i+D{d}_i\hfill \\ & =A{x}_i-BF{\hat{d}}_i+cB{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)+D{d}_i\hfill \\ & =A{x}_i-BF{\epsilon }_i+cB{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j).\hfill \end{array}$$

(5)

Let $x_{\mathfrak{F}}\triangleq {[x_1^T,x_2^T,\dots ,x_M^T]}^T\in {\mathbb{R}}^{nM}$, $x_{\mathfrak{L}}\triangleq {[x_{M+1}^T,x_{M+2}^T,\dots ,x_N^T]}^T\in {\mathbb{R}}^{n(N-M)}$ and $\chi \triangleq {[{\chi }_1^T,{\chi }_2^T,\dots ,{\chi }_M^T]}^T\in {\mathbb{R}}^{nM}$. Define the relative input measurements of the ith follower as follows:

$${\chi }_i\left(t\right)=\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}(x_i\left(t\right)-x_j\left(t\right)).$$

(6)

Then, it follows that

$$\chi \left(t\right)=(L_{\mathfrak{F}}\otimes I_n)x_{\mathfrak{F}}+(L_{\mathfrak{L}}\otimes I_n)x_{\mathfrak{L}}.$$

(7)

And we also define the relative observer measurements of the ith follower as follows:

$$\tilde{\chi }\left(t\right)=(L_{\mathfrak{F}}\otimes I_n){\tilde{x}}_{\mathfrak{F}}+(L_{\mathfrak{L}}\otimes I_n){\tilde{x}}_{\mathfrak{L}}.$$

(8)

where ${\tilde{x}}_{\mathfrak{F}}\triangleq {[{\tilde{x}}_1^T,{\tilde{x}}_2^T,\dots ,{\tilde{x}}_M^T]}^T\in {\mathbb{R}}^{nM}$, ${\tilde{x}}_{\mathfrak{L}}\triangleq {[{\tilde{x}}_{M+1}^T,{\tilde{x}}_{M+2}^T,\dots ,{\tilde{x}}_N^T]}^T\in {\mathbb{R}}^{n(N-M)}$ and $\tilde{\chi }\triangleq {[{\tilde{\chi }}_1^T,{\tilde{\chi }}_2^T,\dots ,{\tilde{\chi }}_M^T]}^T\in {\mathbb{R}}^{nM}$. For $i\in \mathfrak{F}\cup \mathfrak{L}$, it follows from (1)–(8) that

$$\begin{array}{cc}\hfill {\dot{x}}_{\mathfrak{F}}=& (I_{\mathfrak{F}}\otimes A)x_{\mathfrak{F}}-(I_{\mathfrak{F}}\otimes BF)\epsilon +(I_{\mathfrak{F}}\otimes cB{K}_1)\tilde{\chi }\hfill \\ \hfill {\dot{x}}_{\mathfrak{L}}=& (I_{\mathfrak{L}}\otimes A)x_{\mathfrak{L}},\hfill \end{array}$$

(9)

Using (9) for (7), one can obtain that

$$\begin{array}{cc}\hfill \dot{\chi }=& (L_{\mathfrak{F}}\otimes I_n){\dot{x}}_{\mathfrak{F}}+(L_{\mathfrak{L}}\otimes I_n){\dot{x}}_{\mathfrak{L}}\hfill \\ \hfill =& (L_{\mathfrak{F}}\otimes I_n)[(I_{\mathfrak{F}}\otimes A)x_{\mathfrak{F}}-(I_{\mathfrak{F}}\otimes BF)\epsilon \hfill \\ & +(I_{\mathfrak{F}}\otimes cB{K}_1)\tilde{\chi }]+(L_{\mathfrak{L}}\otimes I_n)(I_{\mathfrak{L}}\otimes A)x_{\mathfrak{L}}\hfill \\ \hfill =& (I_{\mathfrak{F}}\otimes A)\chi -(L_{\mathfrak{F}}\otimes BF)\epsilon +(L_{\mathfrak{F}}\otimes cB{K}_1)\tilde{\chi }.\hfill \end{array}$$

(10)

Using (2) and (3), one can obtain that

$$\begin{array}{cc}\hfill {\dot{\epsilon }}_i=& {\dot{\hat{d}}}_i-{\dot{d}}_i\hfill \\ \hfill =& (S-HD){\hat{d}}_i-H(A{x}_i+B{u}_i)+H{\dot{x}}_i-S{d}_i\hfill \\ \hfill =& (S-HD){\epsilon }_i.\hfill \end{array}$$

(11)

Next, Algorithm 1 is presented with the procedure of the controller’s implementation.

Theorem 1.

Under Assumption 1–5, consider the MAS (1) and disturbance signals (2) with the distributed disturbance observer-based event-triggered controller (3) using Algorithm 1. Then, the protocol (3) solves the containment control problem.

Algorithm 1 Distributed Disturbance Observer-Based Containment Control

Under Assumptions 1–5, for disturbance signals in (2), distributed disturbance observer-based event-triggered controller (3) can be constructed by the following form: (I) Solve the following LMI: $$A^{T}P+PA-PB{B}^{T}P+{ϵ}_{1}I<0.$$ (12) to get one solution $P>0$. (II) Let the feedback matrix $K_1=-B^{T}P$. (III) Take a symmetric matrix $\hat{P}\in {\mathbb{R}}^{s\times s}>0$, $$S^T\hat{P}+\hat{P}S-2D^{T}D+{ϵ}_2{I}_n<0.$$ (13) (IV) Let the observer gain $H={\hat{P}}^{-1}{D}^T$. (V) Select positive constants ${ϵ}_1,{ϵ}_2$, the gains be designed in the proof of Theorem 1. (VI) Select the parameters $\mu$ and $\nu$ to be any positive constants.

Proof of Theorem 1.

Choose a Lyapunov function candidate $V=V_1+V_2$, and let

$$V_1={\displaystyle \frac{1}{2}}{\chi }^T(I_{\mathfrak{F}}\otimes P)\chi ,$$

(14)

and

$$V_2={\displaystyle \frac{1}{2}}{\epsilon }^T(I_{\mathfrak{F}}\otimes \hat{P})\epsilon ,$$

(15)

where P and $\hat{P}$ are positive definite. Evidently, V is also positive definite.

Its derivative is obtained as

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)\hfill \\ \hfill =& {\chi }^T(I_{\mathfrak{F}}\otimes P)\dot{\chi }+{\epsilon }^T(\omega I_{\mathfrak{F}}\otimes \hat{P})\dot{\epsilon }\hfill \\ \hfill =& {\chi }^T[I_{\mathfrak{F}}\otimes PA]\chi +{\chi }^T\left[c(L_{\mathfrak{F}}\otimes PB{K}_1)\right]\tilde{\chi }-{\chi }^T(L_{\mathfrak{F}}\otimes PD)\epsilon \hfill \\ & +{\displaystyle \frac{1}{2}}{\epsilon }^T\left(I_{\mathfrak{F}}\otimes (S^T\hat{P}+\hat{P}S-\hat{P}HD-D^T{H}^T\hat{P})\right)\epsilon .\hfill \end{array}$$

(16)

Note that

$$\begin{array}{cc}\hfill & -{\chi }^T(L_{\mathfrak{F}}\otimes PD)\epsilon \hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\lambda }_{max}\left(D^{T}PPD\right){\chi }^T\chi +{\displaystyle \frac{1}{2}}{\lambda }_{max}^2\left(L_{\mathfrak{F}}\right){\epsilon }^T\epsilon .\hfill \end{array}$$

(17)

Note that $K_1=-B^{T}P$, we have

$$\begin{array}{cc}\hfill & -{\chi }^T(c{L}_{\mathfrak{F}}\otimes PB{B}^{T}P)\tilde{\chi }\hfill \\ \hfill =& -{\displaystyle \frac{1}{2}}{\chi }^T(c{L}_{\mathfrak{F}}\otimes PB{B}^{T}P)\chi +{\displaystyle \frac{c}{2}}\{e^T(L_{\mathfrak{F}}\otimes PB{B}^{T}P)e\hfill \\ & -{\tilde{x}}^T(L_{\mathfrak{F}}\otimes PB{B}^{T}P)\tilde{x}\}.\hfill \end{array}$$

(18)

Because $a_{ij}=a_{ji}$, by using Young’s inequality, we have

$$e^T(L_{\mathfrak{F}}\otimes PB{B}^{T}P)e\le 2\sum _{i=1}^N{r}_i{\parallel K_1{e}_i\parallel }^2.$$

(19)

and

$${\tilde{x}}^T(L_{\mathfrak{F}}\otimes PB{B}^{T}P)\tilde{x}={\displaystyle \frac{1}{2}}\sum _{i=1}^N\sum _{j=1}^N{a}_{ij}{\parallel K_1({\tilde{x}}_i-{\tilde{x}}_j)\parallel }^2.$$

(20)

Then, it follows from (16) that

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\chi }^T\left[I_{\mathfrak{F}}\otimes (A^{T}P+PA+{\lambda }_{max}\left(D^{T}PPD\right)I_n)-c{L}_{\mathfrak{F}}\otimes PB{B}^{T}P\right]\chi \hfill \\ & +{\displaystyle \frac{1}{2}}{\epsilon }^T\left(I_{\mathfrak{F}}\otimes (S^T\hat{P}+\hat{P}S-\hat{P}HD-D^T{H}^T\hat{P}+{\lambda }_{max}^2\left(L_{\mathfrak{F}}\right)I_n)\right)\epsilon \hfill \\ & +{\displaystyle \frac{c}{4}}\sum _{i=1}^N\left(4r_i\parallel K{e}_i{\parallel }^2-\sum _{j=1}^N{a}_{ij}{\parallel K({\tilde{x}}_i-{\tilde{x}}_j)\parallel }^2\right).\hfill \end{array}$$

(21)

Let $\Theta =I_{\mathfrak{F}}\otimes (A^{T}P+PA+{\lambda }_{max}\left(D^{T}PPD\right)I_n)-c\Lambda \otimes PB{B}^{T}P$ and $\Xi =S^T\hat{P}+\hat{P}S-\hat{P}HD-D^T{H}^T\hat{P}+{\lambda }_{max}^2\left(L_{\mathfrak{F}}\right)I_n$. Under Assumptions 2 and 3 and Lemma 1, $L_{\mathfrak{F}}$ is a symmetric positive definite, and there exists a unitary matrix $U\in {\mathbb{C}}^{M\times M}$ satisfying $U^H{L}_{\mathfrak{F}}U=\Lambda$, where $\Lambda$ is an upper-triangular matrix with ${\lambda }_i,i=1,...,M$ as its diagonal entries.

$$\begin{array}{cc}\hfill & A^{T}P+PA+{\lambda }_{max}\left(D^{T}PPD\right)I_n-c{\lambda }_{i}PB{B}^{T}P\hfill \\ \hfill \le & A^{T}P+PA-PB{B}^{T}P+{ϵ}_1{I}_n\le 0.\hfill \end{array}$$

(22)

where ${ϵ}_1>{\lambda }_{max}\left(D^{T}PPD\right)$ and $c\ge {\displaystyle \frac{1}{{\lambda }_{min}\left(L_{\mathfrak{F}}\right)}}$.

Noting that $H={\hat{P}}^{-1}{D}^T$, we have

$$\begin{array}{cc}\hfill & S^T\hat{P}+\hat{P}S-\hat{P}HD-D^T{H}^T\hat{P}+{\lambda }_{max}^2\left(L_{\mathfrak{F}}\right)I_n\hfill \\ \hfill =& S^T\hat{P}+\hat{P}S-2D^{T}D+{ϵ}_2{I}_n\le 0.\hfill \end{array}$$

(23)

where ${ϵ}_2={\lambda }_{max}^2\left(L_{\mathfrak{F}}\right)$.

Substituting (22), (23), and the event-triggered function (4) into (21) yields

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\chi }^T\left[I_N\otimes (A^{T}P+PA-PB{B}^{T}P+{ϵ}_1{I}_n)\right]\chi \hfill \\ & +{\displaystyle \frac{1}{2}}{\epsilon }^T\left(I_{\mathfrak{F}}\otimes (S^T\hat{P}+\hat{P}S-2D^{T}D+{ϵ}_2{I}_n)\right)\epsilon +{\displaystyle \frac{c}{2}}N\mu e^{-\nu t}\hfill \\ \hfill \le & -{\theta }_{1}V\left(t\right)+{\displaystyle \frac{c}{2}}N\mu e^{-\nu t}.\hfill \end{array}$$

(24)

where ${\theta }_1=min\{{\displaystyle \frac{{ϵ}_1}{{\lambda }_{max}(P)}},{\displaystyle \frac{{ϵ}_2}{{\lambda }_{max}(\hat{P})}}\}$.

According to the comparison lemma [17], we have

$$\begin{array}{cc}\hfill V\left(t\right)& \le V\left(0\right)e^{-{\theta }_{1}t}+{\displaystyle \frac{c}{2}}N\psi (t,\nu )\hfill \end{array}$$

where $\psi (t,\beta )$ is defined as follows:

$$\begin{array}{c}\hfill \psi \left(t\right)=\left\{\begin{array}{cc}t{e}^{-{\theta }_{1}t},\hfill & \mathrm{if}{\theta }_1=\nu \hfill \\ {\displaystyle \frac{1}{{\theta }_1-\nu }}(e^{-\nu t}-e^{-{\theta }_{1}t}),\hfill & \mathrm{if}{\theta }_1\ne \nu \hfill \end{array}\right..\end{array}$$

with $\beta$ being a parameter. It is not difficult to verify that ${lim}_{t\to \infty }\psi (t,\nu )=0$. We conclude that ${lim}_{t\to \infty }\parallel x_{\mathfrak{F}}\left(t\right)+(L_{\mathfrak{F}}^{-1}{L}_{\mathfrak{L}}\otimes I_n)x_{\mathfrak{L}}\left(t\right)\parallel =0$ and ${lim}_{t\to \infty }\epsilon =0$. Therefore, the containment control problem stated in Definition 1 is solved. □

Feasibility Analysis

In this section, we show the development method to analyze the feasibility of the proposed controller (3). The result is summarized in the following theorem.

Theorem 2.

Consider the linear MAS (1), controller (3), and triggering condition (4). The Zeno behavior is excluded.

Proof of Theorem 2.

We adopt the method of contradiction. Suppose there is at least one agent exhibiting Zeno behavior. Without loss of generality, assume that agent i exhibits Zeno behavior.

Then, for agent i, there exists a finite time $T_i$, such that $t_k^i\le T_i$ and ${lim}_{t\to \infty }{t}_k^i=T_i$. For any ${\varsigma }_i>0$, $\exists \mathrm{\Omega }\ge 0$, such that $T_i-{\varsigma }_i<t_m^i\le T_i$, for any $m\ge \mathrm{\Omega }$.

Consider the evolution of $\parallel e_i\left(t\right)\parallel$, we have ${\dot{e}}_i\left(t\right)=-{\dot{x}}_i\left(t\right)$; then, the upper right-hand Dini derivative of $e_i\left(t\right)$ over $[t_k^i,t_{k+1}^i)$ can be written as

$$D^{+}{e}_i=A{e}_i-cB{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)+BF{\epsilon }_i.$$

It follows that

$$\begin{array}{cc}\hfill & D^{+}\parallel e_i\left(t\right)\parallel \hfill \\ \hfill \le & \parallel A{e}_i-cB{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)+BF{\epsilon }_i\parallel \hfill \\ \hfill \le & \parallel A\parallel \parallel e_i\left(t\right)\parallel +\parallel BF\parallel \parallel {\epsilon }_i\left(t\right)\parallel +\parallel cB{K}_1\sum _{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)\parallel .\hfill \end{array}$$

(25)

As shown in Theorem 1, $\chi$ is bounded, which implies that $x_i\left(t\right)-x_j\left(t\right),\forall i,j\in \mathfrak{F}\cup \mathfrak{L}$ is bounded. Therefore, we can determine that for any $t\in [t_k^i,t_{k+1}^i)$, ${\tilde{x}}_i-{\tilde{x}}_j=e^{A\left(t-t_k^i\right)}{x}_i\left(t_k^i\right)-e^{A\left(t-t_k^j\right)}{x}_j\left(t_{k^{\prime }}^j\right)$ is also bounded, where $t_{k^{\prime }}^j$ denotes the latest event-triggering instant of agent j.

From (25), we can obtain that $\parallel e_i\left(t\right)\parallel$ will not approach zero unless $\parallel {\epsilon }_i\left(t\right)\parallel$ approaches zero, which implies that there exists $0<R<\infty$, such that ${\displaystyle \frac{\parallel {\epsilon }_i\left(t\right)\parallel }{\parallel e_i\left(t\right)\parallel }}<R$. Substituting (5) and (9) into (25), one has

$$D^{+}\parallel e_i\left(t\right)\parallel \le {\zeta }_i\parallel e_i\left(t\right)\parallel +{\varphi }_k^i,$$

(26)

where ${\zeta }_i=\parallel A\parallel$ and ${\varphi }_k^i=\parallel BF\parallel R+{max}_{t\in [t_k^i,t_{k+1}^i]}\parallel cB{K}_1{\sum }_{j\in \mathfrak{F}\cup \mathfrak{L}}{a}_{ij}({\tilde{x}}_i-{\tilde{x}}_j)\parallel$.

Consider a non-negative function $\phi :[0,\infty )⟶\Re \ge 0$ satisfying the following:

$$\begin{array}{c}\hfill \dot{\phi }={\zeta }_i\phi +{\varphi }_k^i,\phi \left(0\right)=\parallel e_i\left(t_k^i\right)\parallel =0.\end{array}$$

(27)

Then, we have that $\parallel e_i\left(t\right)\parallel \le \phi (t-t_k^i)$, where $\phi$ is the analytical solution to (27). Then, it follows that

$$\phi \le {\displaystyle \frac{{\varphi }_k^i}{{\zeta }_i}}\left[exp\left({\zeta }_i(t-t_k^i)\right)-1\right].$$

(28)

It is not difficult to see that the triggering function (4) satisfies $f_i\left(t\right)\le 0$ if we have the following condition:

$$\begin{array}{c}\hfill \parallel e_i{\parallel }^2\le {\displaystyle \frac{\mu e^{-\nu t}+{\varphi }_k^i}{4r_i{\parallel K\parallel }^2}}.\end{array}$$

(29)

Therefore, a lower bound $t_{k+1}^i-t_k^i$ can be obtained by solving the following inequality:

$${\displaystyle \frac{\mu e^{-\nu t_{k+1}^i}+{\varphi }_k^i}{4r_i{\parallel K\parallel }^2}}\le {\displaystyle \frac{{\varphi }_k^i}{{\zeta }_i}}\left[exp\left({\zeta }_i(t_{k+1}^i-t_k^i)\right)-1\right].$$

Then, we can determine that

$$\begin{array}{cc}\hfill t_{k+1}^i-t_k^i& \ge {\tau }_k^i\hfill \\ & \ge {\displaystyle \frac{1}{{\zeta }_i}}ln\left(1+{\displaystyle \frac{{\zeta }_i}{2{\varphi }_k^i\parallel K\parallel }}\sqrt{{\displaystyle \frac{\mu e^{-\nu (t_k^i+{\tau }_k^i)}+{\varphi }_k^i}{r_i}}}\right).\hfill \end{array}$$

(30)

where ${\tau }_k^i$ is a lower bound $t_{k+1}^i-t_k^i$.

Note that ${\tau }_k^i$ is strictly positive for any finite time. One has from (30) that $t_{k+1}^i-t_k^i\ge 2{\varsigma }_i$, which implies that $t_{k+1}^i\ge t_k^i+2{\varsigma }_i\ge T_i+{\varsigma }_i$. This contradicts the fact that $T_i-{\varsigma }_i<t_m^i\le T_i$. Therefore, the Zeno behavior is excluded for all the agents for any finite time. □

4. Simulation

Consider MASs with the communication graph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$, where six followers $\{1,\dots ,6\}\in \mathfrak{F}$ and three leaders $\{7,\dots ,9\}\in \mathfrak{L}$. Assume that the dynamics matrices of (1) are

$$\begin{array}{cc}\hfill & A=\left[\begin{array}{cc}0& 1\\ -0.5& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],\hfill \\ & D=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\hfill \\ & S=\left[\begin{array}{cc}0& -1\\ 0& -2\end{array}\right],F=\left[\begin{array}{cc}0& 1\end{array}\right].\hfill \end{array}$$

(31)

By solving the LMI (12) and (13) in Algorithm 1, the feedback gain matrices $K_1$ and H are

$$\begin{array}{cc}\hfill & K_1=\left[\begin{array}{cc}-2.5& -1.5\end{array}\right],H=\left[\begin{array}{cc}2& 0\\ 0& -1\end{array}\right].\hfill \end{array}$$

(32)

The communication graph ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$ is given in Figure 1, in which ${\mathcal{G}}_{\mathfrak{F}\cup \mathfrak{L}}$ satisfies Assumption 1. Then, the matrices $L_{\mathfrak{F}}$ and $L_{\mathfrak{L}}$ are as follows:

$$L_{\mathfrak{F}}=\left[\begin{array}{cccccc}5& -1& -1& -1& -1& -1\\ -1& 2& -1& 0& 0& 0\\ -1& -1& 2& 0& 0& 0\\ -1& 0& 0& 2& -1& 0\\ -1& 0& 0& -1& 3& -1\\ -1& 0& 0& 0& -1& 2\end{array}\right],L_{\mathfrak{L}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right].$$

The initial conditions of the closed-loop system are randomly chosen. The other parameters are set as follows for all six followers $\{1,\dots ,6\}\in \mathfrak{F}$: $c=1.6$, $\mu =3$, and $\nu =0.5$.

Through Figure 2 and Figure 3, we can obtain the convex hull of all followers’ states converging to the respective leader’s states, where the real line represents the follower’s trajectory, the imaginary line represents the leader’s trajectory, and the black rough line represents the convex hull of the leader’s trajectory.

The three-dimensional effect diagram in Figure 4 clearly shows the movement trajectories of six agents and three leaders over time.

Moreover, the triggering instants of six followers can be found in Figure 5, which shows that the communication among agents is only carried out in some special instants.

5. Conclusions

This paper targets a class of external disturbances with a known matrix system S. We propose a disturbance observer-based containment control algorithm that introduces a disturbance compensation term to offset the external disturbances. Additionally, combined with an event-triggered control strategy, the proposed method can effectively reduce the burden of communication and actuation. In particular, the design of the event-triggered function using an exponential function is independent of the number of nodes (which represents the scale of the network), does not require continuous communication, and eliminates Zeno behavior.