Cooperative Control for Multi-Agent Systems with Deception Attack Based on an Attack Detection Mechanism

Abstract

This study highlights the security control challenge for multi-agent systems (MASs) with integrated attack detectors under deception attacks (DAs). We develop an adaptive backstepping security control strategy designed to simultaneously detect DAs and maintain cooperative system performance. First, a DA detection mechanism is proposed using a state observer. The analytical results reveal that observer errors grow unbounded under DAs but converge to zero in attack-free scenarios, enabling effective attack identification. Following detection, we integrate a Nussbaum function into the backstepping control framework to manage unknown time-varying output gains. Additionally, adaptive parameters, dynamically adjusted based on DA signals, are designed to compensate for actuator and sensor deviations induced by attacks. Rigorous Lyapunov-based analysis proves that the proposed controller ensures output tracking under deception attacks, the timely detection of attack signals, and the boundedness of all closed-loop signals. Numerical simulations further confirm the theoretical findings and demonstrate the effectiveness of the proposed method.

1. Introduction

In the past few decades, distributed cooperative control of multi-agent systems (MASs) has received significant attention owing to its wide applications in various fields, for example, smart grids [1,2], transportation networks [3], sensor networks [4], and so on. Most current research on distributed cooperative control problems assumes a secure and non-attacked scenario to guarantee safe and stable data output. However, MASs consisting of N agents, which rely on communication topologies, actuators, and sensor networks to exchange information and transmit data, are vulnerable to attack. As such, in multi-agent systems (MAS), deception attacks may disrupt controllers or actuators, compromising the integrity of their output signals. This can trigger closed-loop stability issues and ultimately prevent the MAS from achieving cooperative control objectives. The authors of [5] proposed an effective, robust distributed frequency controller for multi-area power systems, where network communication between distributed wind energy units is vulnerable to attacks. However, without analyzing the impacts of cyber attacks, the proposed control method may fail to achieve its control objectives in such an adversarial environment. To compensate for the impact of the attack, considering stochastic FDI attacks [6] and sensor attacks [7], Ref. [6] designed adaptive security control protocols considering nonlinear multi-agent systems to realize cooperative control objectives. Consequently, it is crucial to design a secure control protocol for cooperative control of a distributed mass under attack.

Numerous recent studies have focused on addressing denial attacks (DAs) in MASs, proposing various control strategies to counteract them. A distributed impulsive controller is proposed in [8,9] for multi-agent systems with deception attacks under the impulsive control framework to solve the secure synchronization control problem. In [10], a distributed secure control scheme is developed for multi-agent systems under sensor DAs to compensate for the effect of unknown control gains caused by sensor DA. A security control protocol is designed in [11] to ensure secure sampling of data in a communication network for a class of nonlinear multi-agent systems subject to DAs. Zhang et al. [12] used a Bernoulli process to represent the startup success rate of random spoofing attacks and studied the coordinated control problem of networked high-order fully driven multi-agents under random DAs in feedback channels and forward channels. Considering more general uncertain DA scenarios, a neural network-based adaptive algorithm is designed in [13] for uncertain nonlinear multi-agent systems to achieve impulsive consensus control. However, the attack may not be detected under well-designed attack solutions, so it is important to build an effective attack detection mechanism, which motivates our work in this article.

The development of an attack detection mechanism has been the focus of the control community. It enables the system to identify attacks and design solutions to overcome the impact of an attack, which is the foundation for ensuring the safe operation of a system. From the perspective of the direction taken to design attack detection mechanisms, different detection methods have been proposed, see, e.g., [14,15,16,17,18,19,20,21,22,23,24,25,26,27]. The learning method [14,15] can be applied to detect attacks in complex sensor systems. The Kalman filter method [16] applies to attacks in automatic generation control systems, where they can be effectively detected using this type of method. The candidate set construction algorithm [17] is a type of algorithm that can detect attacks by introducing side information. The centralized and distributed monitors method [18] can detect various attacks based on system-theoretic and graph-theoretic processes. Satisfiability modulo theory [19] can detect attacks via the geometric structure of a problem. Existing detection techniques also include adaptive switching mechanisms [20], dynamic watermarking tests [21], distributedrecursivefiltering [22,23], etc. However, none of the attack detection methods mentioned above involve the system’s output information. Therefore, the motivation of this study is to explore how to detect DAs with the help of the output information of a system.

Meanwhile, compared to existing results on the cooperative control of MASs, to realize the cooperative security control of MASs under DAs, we need to overcome three technical barriers. Firstly, the effect of DAs on output information must be fully considered in MASs, but there is currently no detection method based on output information; therefore, a novel approach needs to be developed to design an effective DA detection mechanism. Secondly, the nature of DA makes the actuator and sensor deviation unknown and time-varying, rendering the standard backstepping technique inapplicable due to the lack of prior knowledge of the gain parameter. Thirdly, the indeterminacy of output signals and the presence of deviation terms in the observer under DAs increases the difficulty of designing the system controller using the backstepping method.

This paper constructs a DA detection mechanism via an output feedback observer and further develops an effective control protocol to achieve cooperative control of MASs under DAs. The major contributions of our work are summarized as follows:

(1) In contrast to the attack detection mechanism in reference [28,29], a dynamic detection mechanism, which only utilizes observer error obtained via agent output information as a threshold, is proposed in this paper to identify DAs, thereby overcoming the problem of difficulties in forming a stable mapping relationship between agent outputs.

(2) By adopting the idea of the dynamic filtering technique, as in [30], we propose a first-order filter to avert a direct differentiation of virtual control signals in control design. Meanwhile, introducing two adaptive parameters depends on the DA signal into the Lyapunov function to compensate for the attack deviation still involved in the Lyapunov stability analysis.

(3) A class of adaptive parameters dependent on the signal of the DAs and Nussbaum function are well-integrated into backstepping control to design a control framework with appropriate adaptive laws and a control protocol, which can effectively compensate for the impact caused by DAs and achieve the cooperative control of MASs under DAs.

The remaining part of the paper is organized as follows. Section 2 expresses the preliminaries and problem formulation. The detection method design is given in Section 3. The main results are shown in Section 4, and simulation results are illustrated in Section 5 to show the developed strategy. Finally, Section 6 concludes the paper.

2. Preliminaries and Problem Formulation

2.1. Information Transmission Among Agents

Suppose that the communication topology among the N agents is represented by an undirected graph $\mathsf{\Gamma }=\left\{\mathfrak{V},\mathfrak{F},\mathfrak{D}\right\},$ where $\mathfrak{V}=\left\{1,2,\cdots N\right\},$ $\mathfrak{F}\in \mathfrak{V}\times \mathfrak{V}$ and $\mathfrak{D}=\left[a_{i,j}\right]\in {\mathbb{R}}^{N\times N}$ denote the agent set, the edge set, and the adjacency matrix, respectively. An edge $\left(i,j\right)\in \mathfrak{F}$ indicates that agent i can receive information from agent $j,$ but not necessarily vice versa. The set of neighbors of the i th agent is denoted as $N_i=\left\{j|\left(j,i\right)\in \mathfrak{F}\right\}.$ The weight of the adjacency matrix is defined as $a_{ii}=0,$ $a_{ij}=1$ if $\left(j,i\right)\in \mathfrak{V},$; otherwise, $a_{ij}=0.$ The in-degree matrix is defined as $\Delta =\mathrm{diag}\left({\Delta }_i\right)\in {\mathbb{R}}^{N\times N}$ with ${\Delta }_i={\sum }_{j=1}^N{a}_{i,j}.$ The Laplacian matrix is defined as $\mathcal{L}=\Delta -\mathfrak{D}.$ In addition, we define matrix $H=\mathrm{diag}\left({\mathrm{h}}_1,\cdots ,{\mathrm{h}}_{\mathrm{N}}\right),$ $h_i=1$ to indicate that the i-th follower can receive information from the leader node; otherwise, $h_i=0.$

2.2. Problem Statement

To better illustrate our developed method of the control protocol analysis and design, we consider a kind of MAS with N$\left(N\ge 2\right)$ agents, and each agent is modeled by

$$\begin{array}{ccc}\hfill {\dot{x}}_{i,h}& =& x_{i,h+1}\hfill \\ \hfill {\dot{x}}_{i,n_i}& =& u_i+{\varsigma }_i\left(t\right)\hfill \\ \hfill y_i& =& {\mathsf{\Xi }}_i\hfill \end{array}$$

(1)

where $i=1,\cdots ,N,$ $h=1,\cdots ,n_i-1,$ $x_{i,h}$ and $u_i\in R^{n_i}$ are the state and control input of the i-th agent, respectively. $y_i$ is the measurable output. ${\varsigma }_i\left(t\right)$ denotes the actuator deviation of the DAs on the i th agent. ${\mathsf{\Xi }}_i$ represents the output model considering DA, and it is represented by

$${\mathsf{\Xi }}_i=\left\{\begin{array}{c}{G}_i\left(x_{i,1}\right), {\varphi }_i\left(t\right)\ne 0\hfill \\ x_{i,1}, \hspace{1em}{\varphi }_i\left(t\right)=0\hfill \end{array}\right.$$

where ${\varphi }_i\left(t\right)={\mathsf{\Xi }}_i-x_{i,1}$ shows the sensor deviation caused by the DAs.

The actuator and sensor deviations are described as follows.

Actuator deviation caused via DA

To provide a more intuitive description of whether the attack is successful, define the actuator deviation ${\varsigma }_i\left(t\right)={\gamma }_i\left(t\right){\breve{\varsigma }}_i\left(t\right)$ caused by the DAs, with ${\breve{\varsigma }}_i\left(t\right)\in R$ being the DA signal generated by the attacker. ${\gamma }_i\left(t\right)$ is an attack variable, and the DA signal ${\breve{\varsigma }}_i\left(t\right)$ satisfies $\left|\right|\breve{\varsigma }{\left|\right|}^2\le \overline{\varsigma },$ in which $\overline{\varsigma }>0$ is a constant and $\breve{\varsigma }={[{\breve{\varsigma }}_i,\cdots ,{\breve{\varsigma }}_N]}^T.$

Sensor deviation caused via DA

Similar to Actuator deviation, we define sensor deviation ${\varphi }_i\left(t\right)={\gamma }_i\left(t\right){\breve{\varphi }}_i\left(t\right)$ with ${\breve{\varphi }}_i\left(t\right)$ being the DA signal dependent on $x_{i,1}$ generated by the attacker. Moreover, ${\breve{\varphi }}_i$ satisfies $\left|\right|\breve{\varphi }{\left|\right|}^2\le \overline{\varphi },$ in which $\overline{\varphi }>0$ is a constant and $\breve{\varphi }={[{\breve{\varphi }}_i,\cdots ,{\breve{\varphi }}_N]}^T.$

Due to the synchronous impact of the DA considered in this paper on both the actuators and sensors, i.e., they are either simultaneously attacked or neither, and there is no case where the actuators and sensors are individually attacked, we employ the same attack variable ${\gamma }_i\left(t\right),$ and it is described by

$${\gamma }_i\left(t\right)=\left\{\begin{array}{c}1, \mathrm{attack} \mathrm{occurrence}\hfill \\ 0, \mathrm{no} \mathrm{attack} \hfill \end{array}\right.$$

(2)

Via (2), we can easily see that if a DA occurs, then $y_i={\mathsf{\Xi }}_i=G_i\left(x_{i,1}\right);$ otherwise, ${\varsigma }_i\left(t\right)=0$ and $y_i={\mathsf{\Xi }}_i=x_{i,1}.$

Remark 1.

Attackers are usually limited by energy, so it is reasonable for the deviation attack signals ${\breve{\varsigma }}_i\left(t\right)$ and ${\breve{\varphi }}_i$ to satisfy the properties mentioned above, and a similar energy-limited constraint can be found in [31] and the references therein. Meanwhile, to better identify the DAs, an attack detection mechanism is designed in Section 3.

2.3. Control Objectives

We develop a class of adaptive cooperative security control protocols for MASs under DA to achieve the following objectives.

(a)

The adaptive cooperative security control is achieved under DA;

(b)

All signals are bounded in the closed-loop system.

In order to achieve the above control objectives, the following assumptions and lemma are required.

Assumption 1.

The communication topology Γ is fixed and connected. The desired trajectory $y_d\left(t\right)$ is available to at least one subsystem in $\mathsf{\Gamma }.$

Assumption 2.

All DAs can be detected via (10).

Assumption 3.

The trajectory $y_d\left(t\right)$ of the leader and its n-th order derivatives are piecewise continuous, bounded, and only known by subsystem i with $h_i=1.$ Moreover, it is known by subsystem i with $h_i=0$ such that $\left|{\dot{y}}_d\right|\le Y_d,$ in which where $Y_d>0$ is an unknown constant.

Lemma 1.

([32]). For any scalars $O\in R$ and $\mu >0,$ the inequality $0\le \left|O\right|-{\displaystyle \frac{O^2}{\sqrt{O^2+{\mu }^2}}}\le \mu$ holds.

3. Detection Method Design

In practical application, the communication network of MASs is always in a dynamic process due to the information exchanges among agents, which brings unexpected attacks to the communication network of MASs. This operating condition of MASs may make it difficult to form a static mapping relationship between the output $y_i$ of an agent, i.e., it is difficult to determine whether MASs achieve cooperative control. Hence, an observer is designed based on the agent’s output information to identify the occurrence of DAs and to develop a reliable cooperative security control protocol for MASs. When DA occurs, the attacked agent is accurately located. Next, we start by designing a DA detection observer, after which an attack detection method and mechanism are developed.

3.1. Attack Detection Observer Design

A DA detection observer was designed to determine whether the agents are attacked or not. The following DA detection observer is designed

$$\begin{array}{ccc}\hfill {\dot{\widehat{x}}}_{i,h}& =& {\widehat{x}}_{i,h+1}+l_{i,h}\left(y_i-{\widehat{x}}_{i,1}\right)\hfill \\ \hfill {\dot{\widehat{x}}}_{i,n_i}& =& u_i+l_{i,n_i}\left(y_i-{\widehat{x}}_{i,1}\right)\hfill \end{array}$$

(3)

with ${\widehat{x}}_{i,h}$ being the estimate of $x_{i,h},$ $h=1,\cdots ,n_i-1.$

Define observer error ${\varkappa }_i=x_i-{\widehat{x}}_i$ with ${\varkappa }_i={[{\varkappa }_{i,1},\cdots ,{\varkappa }_{i,n_i}]}^T,$ $x_i={[y_i,\cdots ,x_{i,n_i}]}^T$ and ${\widehat{x}}_i={[{\widehat{x}}_{i,1},\cdots ,{\widehat{x}}_{i,n_i}]}^T;$ differentiating ${\varkappa }_i$ with respect to time yields

$${\dot{\varkappa }}_i={\mathcal{ϝ}}_i{\varkappa }_i+{\gamma }_i\left(t\right){\breve{\varsigma }}_i\left(t\right)-l_i{\gamma }_i\left(t\right){\breve{\varphi }}_i\left(t\right)$$

(4)

where $l_i={\left[l_{i,1},\cdots ,l_{i,n_i}\right]}^T$ should satisfy the condition that ${\mathcal{ϝ}}_i=\left[\begin{array}{ccc}-l_{i,1}& & \\ ⋮& I_{n_i-1}& \\ -l_{i,n_i}& \cdots & 0\end{array}\right]$ is Hurwitz. This implies that there exists a matrix $P_i=P_i^T$ for $Q_i=Q_i^T$ such that ${\mathcal{ϝ}}_i^T{P}_i+P_i{\mathcal{ϝ}}_i=-Q_i.$

Remark 2.

It is worth noting that due to the existence of DAs, only $y_i$ rather than $x_{i,1}$ can be used in (3), which leads to the extra terms ${\gamma }_i\left(t\right){\breve{\varsigma }}_i\left(t\right)-l_i{\gamma }_i\left(t\right){\breve{\varphi }}_i\left(t\right)$ in (4). As a result, the actuator and sensor deviation appearing in (27) and (29) and in each step of the backstepping design procedure poses an additional challenge for the stability analysis. To handle the actuator and sensor deviation caused by the DAs, two adaptive parameters related to DA signal ${\breve{\varsigma }}_i\left(t\right)$ and ${\breve{\varphi }}_i\left(t\right),$ that is, ${\xi }_i=max\left\{{sup}_{t\ge 0}\left|{\breve{\varsigma }}_i\left(t\right)\right|\right\}$ and ${\beta }_{i,h}=max\{{sup}_{t\ge 0}{l}_{i,h}|{\breve{\varphi }}_i\left(t\right)\left|\right\},$ are defined, and adaptive parameters ${\xi }_i$ and ${\beta }_{i,h}$ are introduced into the Lyapunov function. Then, we perform an online estimation of ${\xi }_i$ and ${\beta }_{i,h}$ by designing suitable adaptive update laws.

To better identify whether a DA has occurred, we propose the following lemma.

Lemma 2.

Consider MAS (1) with DAs under observer (3); the following properties are satisfied:

$$\begin{array}{ccc}\hfill \underset{t\to \infty }{lim}{\varkappa }_{i,1}& =& 0, \mathit{no} \mathit{attack}\hfill \\ \hfill \underset{t\to \infty }{lim}{\varkappa }_{i,1}& \ne & 0, \mathit{attack}\hfill \end{array}$$

Proof. 

Consider a Lyapunov function candidate as follows:

$$V_0=\sum _{i=1}^N{\varkappa }_i^T{P}_i{\varkappa }_i$$

Based on (4), take the derivative of $V_0$ and use Young’s inequality; one has

$$\begin{array}{ccc}\hfill {\dot{V}}_0& \le & \sum _{i=1}^N\left\{-{\varkappa }_i^T{\breve{Q}}_i{\varkappa }_i+{\left({\gamma }_i\left(t\right)\right)}^2\left|\right|\breve{\varsigma }{\left|\right|}^2+l_i^2{\left({\gamma }_i\left(t\right)\right)}^2\left|\right|\breve{\varphi }{\left|\right|}^2\right\}\hfill \\ & \le & \sum _{i=1}^N\left\{-{\varkappa }_i^T{\breve{Q}}_i{\varkappa }_i+{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varsigma }+l_i^2{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varphi }\right\}\hfill \end{array}$$

(5)

where ${\breve{Q}}_i={\lambda }_{min}\left(Q_i\right)-2{∥P_i∥}^2$ with ${\lambda }_{min}\left(·\right)$ denotes the minimum eigenvalue of $\left(·\right).$

Further, one can obtain

$${\dot{V}}_0\le -a_i{V}_0+b_i$$

(6)

where $a_i=min\left\{{\sum }_{i=1}^N{\displaystyle \frac{{\breve{Q}}_i}{{\lambda }_{max}\left(p_i\right)}}\right\}>0,$ $b_i={\sum }_{i=1}^N\{{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varsigma }+l_i^2{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varphi }\}.$

For the potential DAs, we proceed with the following discussion based on the DA variable model (2).

$Case$ 1: No DAs occur, i.e., ${\gamma }_i\left(t\right)=0,$ where (6) is rewritten as

$${\dot{V}}_0\le -a_i{V}_0$$

(7)

We can easily obtain ${lim}_{t\to \infty }{\varkappa }_i=0.$

$Case$ 2: DAs occur, i.e., ${\gamma }_i\left(t\right)=1;$ one has

$${\dot{V}}_0\le -a_i{V}_0+b_i$$

(8)

By integrating both sides of (8), one has

$$0\le V_0\left(t\right)\le e^{-a_{i}t}{V}_0\left(0\right)+{\displaystyle \frac{b_i}{a_i}}\left(1-e^{-a_{i}t}\right)$$

(9)

In theory, we can easily obtain ${lim}_{t\to \infty }{\varkappa }_i={\displaystyle \frac{b_i}{a_i}}\ne 0.$

Hence, based on the above analysis, Lemma 2 is proved.

□

3.2. Attack Detection Mechanism

Based on a DA detection observer and Lemma 2, a dynamic detection method for DAs is presented, which is expressed as follows:

$$\left\{\begin{array}{c}\mathrm{no} \mathrm{attack}, \mathrm{if} {\gamma }_i\left(t\right)=0,\hfill \\ \mathrm{attack}, \mathrm{if} {\gamma }_i\left(t\right)=1,\hfill \end{array}\right.$$

(10)

Hence, according to (2), we can easily know $y_i=G\left(x_{i,1}\right)$ when a DA occurs and satisfies equation ${\dot{y}}_i={\omega }_i\left(t\right){\dot{x}}_{i,1};$ otherwise, ${\dot{y}}_i={\dot{x}}_{i,1}$ and ${\varsigma }_i\left(t\right)=0,$ in which ${\omega }_i\left(t\right)$ is an unknown time-varying function, $\left|{\omega }_i\left(t\right)\right|\in \left[{\omega }_i^{min},{\omega }_i^{max}\right]$, with ${\omega }_i^{min}$ and ${\omega }_i^{max}$ being unknown upper and lower bounds and $0\notin \left[{\omega }_i^{min},{\omega }_i^{max}\right];$ the attack detection threshold ${\varkappa }_i$ is designed, and the update rule logic is represented as follows:

$$\left\{\begin{array}{c}\underset{t\to \infty }{lim}{\varkappa }_{i,1}=0, \mathrm{if} {\gamma }_i\left(t\right)=0,\hfill \\ \underset{t\to \infty }{lim}{\varkappa }_{i,1}\ne 0, \mathrm{if} {\gamma }_i\left(t\right)=1,\hfill \end{array}\right.$$

(11)

Remark 3.

It can be observed from (11) that ${lim}_{t\to \infty }{\varkappa }_{i,1}=0$ indicates no DAs occur; otherwise, DAs occur. Meanwhile, the threshold ${\varkappa }_i$ is chosen and updated based on the observer error obtained from the real-time operational output dynamics of MASs.

4. Main Results

In this section, according to a backstepping technique, a class of distributed adaptive cooperative security control protocols is developed for MASs with DAs, and a block diagram of adaptive cooperative control for MASs is given in Figure 1.

4.1. Nussbaum Functions Method

The Nussbaum technique is employed to overcome the unknown time-varying gain problem caused by a DA. Based on this, the following type of Nussbaum function is considered:

$$N_i\left({\varrho }_i\right)=-\chi exp\left({\displaystyle \frac{{\varrho }_i^2}{2}}\right)\left({\varrho }_i^2+2\right)sin\left({\varrho }_i\right)$$

with $\chi >0$ being a constant and ${\varrho }_i$ being a real variable.

Lemma 3.

([33]). Let $V\left(•\right)$ be a smooth function defined on $\left[0,t_{\omega }\right)$ with $V\left(t\right)\ge 0$ for any $t\in \left[0,t_{\omega }\right),$ and ${\varrho }_i\left(s\right)$ $\left(i=1,\cdots ,N\right)$ are designed as smooth functions with their initials ${\varrho }_i\left(0\right)$ being bounded. If the following inequality holds

$$V\left(t\right)\le W+{\int }_0^t\sum _{i=1}^N\left({\omega }_i\left(s\right)N_i\left({\varrho }_i\left(s\right)\right)+\overline{W}\right){\dot{\varrho }}_i\left(s\right)ds$$

where W and $\overline{W}$ are any constants, ${\omega }_i\left(s\right)$ expresses unknown time-varying functions which take values in the unknown closed intervals $Q_i:\left[{\omega }_i^{min},{\omega }_i^{max}\right],$ $V\left(t\right),$ ${\varrho }_i\left(t\right)$ and ${\int }_0^t{\sum }_{i=1}^N\left({\omega }_i\left(s\right)N_i\left({\varrho }_i\right)+\overline{W}\right){\dot{\varrho }}_{i}ds$ must be bounded on $t\in \left[0,t_{\omega }\right),$ for $i=1,\cdots ,N.$

4.2. Controller Design and Stability Analysis

Based on MAS control theory, the cooperative security control protocol is developed via the backstepping technique. The following coordinate transformations are given for system (1)

$$\begin{array}{ccc}\hfill e_{i,1}& =& \sum _{j=1}^N{a}_{i,j}\left(y_i-y_j\right)+h_i(y_i-y_d)\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill e_{i,h}& =& {\widehat{x}}_{i,h}-c_{i,h}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\vartheta }_{i,h}& =& c_{i,h}-{\mathsf{\Psi }}_{i,h-1}, h\ge 2\hfill \end{array}$$

(14)

where ${\mathsf{\Psi }}_{i,h-1}$ is a virtual controller that is designed later. Via defining $e_1={[e_{1,1},\cdots e_{N,1}]}^T,$ one has

$$e_1=\left(\mathcal{L}+H\right)z$$

where $z={[y_1-y_d,\cdots y_N-y_d]}^T.$ The introduced state variable $c_{i,h}$ is obtained by the following first-order low-pass filter:

$${\varpi }_{i,h}{\dot{c}}_{i,h}+c_{i,h}={\mathsf{\Psi }}_{i,h-1}, c_{i,h}\left(0\right)={\mathsf{\Psi }}_{i,h-1}\left(0\right)$$

(15)

with ${\varpi }_{i,h}>0$ being a design constant.

To simplify the notation, ${\omega }_i(•)$ is denoted as ${\omega }_i$ hereafter.

The following sections show the virtual/real controllers and adaptive update laws from two aspects:

$Case$ 1: Agents are detected via (10) in relation to occurring DAs, where one has ${\dot{y}}_i={\omega }_i{\dot{x}}_{i,1}.$

By choosing appropriate Lyapunov functions, the virtual/real controllers are designed as follows:

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{i,1}& =& f_i{N}_i\left({\varrho }_i\right){\alpha }_{i,1}\hfill \\ \hfill {\mathsf{\Psi }}_{i,h}& =& -b_{i,h}{e}_{i,h}-{\displaystyle \frac{3}{2}}{e}_{i,h}-l_{i,h}{\varkappa }_{i,1}\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & -{\displaystyle \frac{{\widehat{\beta }}_{i,h}{e}_{i,h}}{\sqrt{e_{i,h}^2+{\mu }_i^2\left(t\right)}}}+{\dot{c}}_{i,h}\hfill \\ \hfill u_i& =& -b_{i,n_i}{e}_{i,n_i}-{\displaystyle \frac{1}{2}}{e}_{i,n_i}-l_{i,n_i}{\varkappa }_{i,1}+{\dot{c}}_{i,n_i}\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & -{\displaystyle \frac{{\widehat{\beta }}_{i,n_i}{e}_{i,n_i}}{\sqrt{e_{i,n_i}^2+{\mu }_i^2\left(t\right)}}}-{\displaystyle \frac{{\widehat{\xi }}_i{e}_{i,n_i}}{\sqrt{e_{i,n_i}^2+{\mu }_i^2\left(t\right)}}}\hfill \end{array}$$

(18)

where ${\alpha }_{i,1}=b_{i,1}{e}_{i,1}+{\displaystyle \frac{3}{4}}{e}_{i,1}^3+{\displaystyle \frac{{\widehat{Y}}_i{e}_{i,1}}{\sqrt{e_{i,1}^2+{\mu }_i^2\left(t\right)}}}-h_i{\dot{y}}_d,$ ${\widehat{Y}}_i$ is the estimation of $Y_i=\left(h_i-1\right)Y_d, f_i>0$ and $b_{i,h}>0$ $(h=1,\cdots ,n_i)$ are design parameters. ${\mu }_i\left(t\right)=s_{i1}{e}^{-s_{i2}t}$ with $s_{i1}$ and $s_{i2}$ being positive constants. ${\widehat{\beta }}_{i,h}$ and ${\widehat{\xi }}_i$ are the estimations of ${\beta }_{i,h}$ and ${\xi }_i,$ respectively, and ${\beta }_{i,h}$ and ${\xi }_i$ have been defined in Remark 2.

The adaptive update laws are defined as

$$\begin{array}{ccc}\hfill {\dot{\varrho }}_i& =& r_i{f}_i{e}_{i,1}{\alpha }_{i.1}\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill {\dot{\widehat{Y}}}_i& =& -{\nu }_i{\widehat{Y}}_i+{\displaystyle \frac{{\psi }_i{e}_{i,1}^2}{\sqrt{e_{i,1}^2+{\mu }_i^2\left(t\right)}}}\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\beta }}}_{i,h}& =& -{\kappa }_{i,m}{\widehat{\beta }}_{i,h}+{\displaystyle \frac{d_{i,h}{e}_{i,h}^2}{\sqrt{e_{i,h}^2+{\mu }_i^2\left(t\right)}}}\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill {\dot{\widehat{\xi }}}_i& =& -{\phi }_i{\widehat{\xi }}_i+{\displaystyle \frac{g_i{e}_{i,n_i}^2}{\sqrt{e_{i,n_i}^2+{\mu }_i^2\left(t\right)}}}\hfill \end{array}$$

(22)

where $h=2,\cdots ,n_i,$ $r_i,$ ${\nu }_i, {\kappa }_i, {\phi }_i, {\psi }_i,$ $d_{i,h}$ and $g_i$ are positive design parameters.

$Case$ 2: No DA is detected via (10); we have ${\varsigma }_i\left(t\right)=0$ and ${\dot{y}}_i={\dot{x}}_{i,1}.$

By selecting appropriate Lyapunov functions, we define the following virtual/real controllers:

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{i,1}& =& -{\overline{b}}_{i,1}{e}_{i,1}-{\displaystyle \frac{3}{2}}{e}_{i,1}-h_i{\dot{y}}_d-{\displaystyle \frac{{\widehat{Y}}_i{e}_{i,1}}{\sqrt{e_{i,1}^2+{\mu }_i^2\left(t\right)}}}\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_{i,h}& =& -{\overline{b}}_{i,h}{e}_{i,h}-{\rho }_{i,h}{e}_{i,h}-l_{i,h}{\varkappa }_{i,1}+{\dot{c}}_{i,h}\hfill \end{array}$$

(24)

where ${\rho }_{i,h}={\displaystyle \frac{3}{2}},$ $(h=2,\cdots ,n_i-1), {\rho }_{i,n_i}={\displaystyle \frac{1}{2}}$ and $u_i={\mathsf{\Psi }}_{i,n_i}.$ ${\overline{b}}_{i,h}>0$ $(h=1,\cdots ,n_i)$ are design parameters.

The adaptive update law is defined as

$${\dot{\widehat{Y}}}_i=-{\overline{\nu }}_i{\widehat{Y}}_i+{\displaystyle \frac{{\overline{\psi }}_i{e}_{i,h}^2}{\sqrt{e_{i,h}^2+{\mu }_i^2\left(t\right)}}}$$

(25)

where ${\overline{\nu }}_i$ and ${\overline{\psi }}_i$ are positive design parameters.

The main results are formally stated by the following theorem.

$Theorem$ 1: Consider the MASs (1) subject to DAs satisfying Assumptions 1–3 under $Case$ 1 and $2.$ If the cooperative control protocol associated with the virtual control signals and adaptive laws are applied, then for any initial condition, there exist positive design parameters $b_{i,h},$ $r_i,$ ${\nu }_i,$ ${\kappa }_i,$ ${\phi }_i,$ $f_i,$ $d_{i,h},$ $g_i,$ ${\psi }_i,$ ${\overline{b}}_{i,h}, {\overline{\nu }}_i$ and ${\overline{\psi }}_i$ such that the control objectives can be achieved.

Proof. 

$Case$ 1: According to (2), we know that ${\gamma }_i\left(t\right)=1$ when a DA occurs.

Step 1: Choose the Lyapunov function candidate as

$$V_{i,1}^{a1}=V_0+{\displaystyle \frac{1}{2}}{z}^T\left(\mathcal{L}+H\right)z+{\displaystyle \frac{1}{2}}\sum _{i=1}^N{\displaystyle \frac{1}{{\psi }_i}}{\tilde{Y}}_i^2$$

where ${\tilde{Y}}_i={\widehat{Y}}_i-Y_i$ denote the estimation errors.

Based on Young’s inequation, from (1), (5), and (12)–(14), one has

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,1}^{a1}& =& V_0+\sum _{i=1}^N{e}_{i,1}[{\omega }_i({\varkappa }_{i,2}+e_{i,2}+{\vartheta }_{i,2}+{\Psi }_{i,1})\hfill \\ & & -{\dot{y}}_d]+\sum _{i=1}^N{\displaystyle \frac{1}{{\psi }_i}}{\tilde{Y}}_i{\dot{\widehat{Y}}}_i\hfill \\ & \le & \sum _{i=1}^N\{-{\breve{Q}}_i{∥{\varkappa }_i∥}^2+l_i^2{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varphi }+{\left({\gamma }_i\left(t\right)\right)}^2\overline{\varsigma }\hfill \\ & & +e_{i,1}[{\displaystyle \frac{3}{4}}{e}_{i,1}^3+{\omega }_i{\mathsf{\Psi }}_{i,1}+\left(h_i-1\right){\dot{y}}_d\hfill \\ & & -h_i{\dot{y}}_d]+{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4}+{\displaystyle \frac{1}{2}}{e}_{i,2}^2+{\displaystyle \frac{1}{2}}{∥{\varkappa }_i∥}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\vartheta }_{i,2}^2+\sum _{i=1}^N{\displaystyle \frac{1}{{\psi }_i}}{\tilde{Y}}_i{\dot{\widehat{Y}}}_i\}\hfill \end{array}$$

(26)

Then, via using Lemma 1, substituting (16), (19), and (20) into (26), ${\dot{V}}_{i,1}^{a1}$ can be rewritten as

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,1}^{a1}& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2\right.+\overline{\varsigma }+l_i^2\overline{\varphi }-b_{i,1}{e}_{i,1}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\vartheta }_{i,2}^2+{\displaystyle \frac{1}{2}}{e}_{i,2}^2+{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4}+{\mu }_i{Y}_i\hfill \\ & & -{\displaystyle \frac{{\nu }_i}{{\psi }_i}}{\tilde{Y}}_i{\widehat{Y}}_i\left.+{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i\right\}\hfill \end{array}$$

where $\mathsf{\Phi }={\breve{Q}}_i-{\displaystyle \frac{1}{2}}.$

Step $h\left(2\le h\le n_i-1\right)$: Via (3), (13), and (14), the derivative of $e_{i,h}$ can be computed as

$$\begin{array}{ccc}\hfill {\dot{e}}_{i,h}& =& e_{i,h+1}+{\mathsf{\Psi }}_{i,h}+{\vartheta }_{i,h+1}+l_{i,h}{\varkappa }_{i,1}\hfill \\ & & +l_{i,h}{\varphi }_i\left(t\right)-{\dot{c}}_{i,h}\hfill \end{array}$$

(27)

Choose the Lyapunov function candidate as

$$V_{i,h}^{a1}=V_{i,h-1}^{a1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^N\left(e_{i,h}^2+{\vartheta }_{i,h}^2+{\displaystyle \frac{1}{d_{i,h}}}{\tilde{\beta }}_{i,h}^2\right)$$

where ${\tilde{\beta }}_{i,h}={\widehat{\beta }}_{i,h}-{\beta }_{i,h}.$

By applying Young’s inequation, the definition of ${\beta }_{i,h}$ and ${\gamma }_i\left(t\right),$ and the resulting ${\dot{V}}_{i,h}^{a1}$ is

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,h}^{a1}& \le & {\dot{V}}_{i,h-1}^{a1}+\sum _{i=1}^N\left\{e_{i,h}\right(e_{i,h}+{\mathsf{\Psi }}_{i,h}+l_{i,h}{\varkappa }_{i,1}\hfill \\ & & +{\beta }_{i,h}-{\dot{c}}_{i,h})+{\displaystyle \frac{1}{2}}{e}_{i,h+1}^2+{\displaystyle \frac{1}{2}}{\vartheta }_{i,h+1}^2\hfill \\ & & +{\vartheta }_{i,h}{\dot{\vartheta }}_{i,h}+{\displaystyle \frac{1}{d_{i,h}}}{\tilde{\beta }}_{i,h}{\dot{\widehat{\beta }}}_{i,h}\}\hfill \end{array}$$

(28)

Then, according to Lemma 1, virtual controller (17) and adaptive update law (21), we arrive at

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,h}^{a1}& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2\right.+\overline{\varsigma }+l_i^2\overline{\varphi }+{\displaystyle \frac{1}{2}}{e}_{i,h+1}^2+{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4}\hfill \\ & & +{\displaystyle \frac{1}{2}}\sum _{m=2}^{h+1}{\vartheta }_{i,m}^2+\sum _{m=2}^h{\vartheta }_{i,m}{\dot{\vartheta }}_{i,m}-\sum _{m=1}^h{b}_{i,m}{e}_{i,m}^2\hfill \\ & & +\sum _{m=2}^h{\beta }_{i,m}{\mu }_i-\sum _{m=2}^h{\displaystyle \frac{{\kappa }_{i,m}}{d_{i,m}}}{\tilde{\beta }}_{i,m}{\widehat{\beta }}_{i,h}+{\mu }_i{Y}_i\hfill \\ & & -{\displaystyle \frac{{\nu }_i}{{\psi }_i}}{\tilde{Y}}_i{\widehat{Y}}_i\left.+{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i\right\}\hfill \end{array}$$

Step $n_i$: Choose the Lyapunov function candidate as

$$V_{i,n_i}^{a1}=V_{i,n_i-1}^{a1}+{\displaystyle \frac{1}{2}}\sum _{i=1}^N\left(e_{i,n_i}^2+{\vartheta }_{i,n_i}^2+{\displaystyle \frac{1}{d_{i,n_i}}}{\tilde{\beta }}_{i,n_i}^2+{\displaystyle \frac{1}{g_i}}{\tilde{\xi }}_i^2\right)$$

where ${\tilde{\xi }}_i={\widehat{\xi }}_i-{\xi }_i.$

Using (3) and (13), the derivative of $V_{i,n_i}^{a1}$ is computed as

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,n_i}^{a1}& \le & {\dot{V}}_{i,n_i-1}^{a1}+\sum _{i=1}^N\left\{e_{i,n_i}(u_i+{\varsigma }_i\left(t\right)+l_{i,n_i}{\varkappa }_{i,1}\right.\hfill \\ & & +l_{i,n_i}{\varphi }_i\left(t\right)-{\dot{c}}_{i,n_i})+{\vartheta }_{i,n_i}{\dot{\vartheta }}_{i,n_i}\hfill \\ & & +{\displaystyle \frac{1}{d_{i,h}}}{\tilde{\beta }}_{i,h}{\dot{\widehat{\beta }}}_{i,h}+{\displaystyle \frac{1}{g_i}}\left.{\tilde{\xi }}_i{\dot{\widehat{\xi }}}_i\right\}\hfill \end{array}$$

(29)

In light of Lemma 1 and the definition of ${\beta }_{i,n_i},$ ${\xi }_i$ and ${\gamma }_i\left(t\right),$ by inserting (18), (21), and (22) into (29), we can readily deduce that

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,n_i}^{a1}& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2\right.+\overline{\varsigma }+l_i^2\overline{\varphi }+{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4}+{\mu }_i{Y}_i\hfill \\ & & +{\displaystyle \frac{1}{2}}\sum _{m=2}^{n_i}\left({\vartheta }_{i,m}^2+{\vartheta }_{i,m}{\dot{\vartheta }}_{i,m}\right)-\sum _{m=1}^{n_i}{b}_{i,m}{e}_{i,m}^2\hfill \\ & & +\sum _{m=2}^{n_i}{\beta }_{i,m}{\mu }_i-\sum _{m=2}^{n_i}{\displaystyle \frac{{\kappa }_{i,m}}{d_{i,m}}}{\tilde{\beta }}_{i,m}{\widehat{\beta }}_{i,m}-{\displaystyle \frac{{\nu }_i}{{\psi }_i}}{\tilde{Y}}_i{\widehat{Y}}_i\hfill \\ & & +{\mu }_i{\xi }_i-{\displaystyle \frac{{\phi }_i}{g_i}}{\tilde{\xi }}_i{\widehat{\xi }}_i\left.+{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i\right\}\hfill \end{array}$$

(30)

According to properties of ${\mu }_i,$ one can obtain

$$\begin{array}{ccc}\hfill {\overline{k}}_i& =& (Y_i+{\xi }_i+\sum _{m=2}^{n_i}{\beta }_{i,m}){\int }_0^{+\infty }{\mu }_i\left(\tau \right)d\left(\tau \right)\hfill \\ & >& (Y_i+{\xi }_i+\sum _{m=2}^{n_i}{\beta }_{i,m}){\int }_0^t{\mu }_i\left(\tau \right)d\left(\tau \right)\hfill \end{array}$$

(31)

such that ${\overline{k}}_i>0$ is a finite constant.

Meanwhile, by defining ${\overline{p}}_{i,m}=max\left\{\right|{\dot{\Psi }}_{i,m-1}\left|\right\},$ from (14) and (15), we obtain

$$\begin{array}{ccc}\hfill {\vartheta }_{i,m}{\dot{\vartheta }}_{i,m}& \le & {\vartheta }_{i,m}\left(-{\displaystyle \frac{{\vartheta }_{i,m}}{{\varpi }_{i,m}}}+p_{i,m}\right)\hfill \\ & \le & -\left({\displaystyle \frac{1}{{\varpi }_{i,m}}}-{\displaystyle \frac{p_{i,m}^2}{2{\sigma }_{i,m}}}\right){\vartheta }_{i,m}^2+{\displaystyle \frac{{\sigma }_{i,m}}{2}}\hfill \end{array}$$

(32)

with ${\overline{\sigma }}_{i,m}>0$ being a constant.

Based on Young’s inequality, substituting (31)–(32) into (30) gives

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,n_i}^{a1}& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2-{\displaystyle \frac{{\phi }_i}{2g_i}}{\tilde{\xi }}_i^2-\sum _{m=2}^{n_i}{\displaystyle \frac{{\kappa }_{i,m}}{2d_{i,m}}}{\tilde{\beta }}_{i,m}^2\right.\hfill \\ & & -{\displaystyle \frac{{\nu }_i}{2{\psi }_i}}{\tilde{Y}}_i^2+\sum _{m=2}^{n_i}\left({\displaystyle \frac{1}{{\varpi }_{i,m}}}-{\displaystyle \frac{p_{i,m}^2}{2{\sigma }_{i,m}}}-{\displaystyle \frac{1}{2}}\right){\vartheta }_{i,m}^2\hfill \\ & & +\overline{\varsigma }+l_i^2\overline{\varphi }-\sum _{m=1}^{n_i}{b}_{i,m}{e}_{i,m}^2+{\displaystyle \frac{{\phi }_i}{2g_i}}{\xi }_i^2+{\displaystyle \frac{{\nu }_i}{2{\psi }_i}}{Y}_i^2\hfill \\ & & +{\overline{k}}_i+\sum _{m=2}^{n_i}\left({\displaystyle \frac{{\kappa }_{i,m}}{2d_{i,m}}}{\beta }_{i,m}^2+{\displaystyle \frac{{\sigma }_{i,m}}{2}}\right)\hfill \\ & & +{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4}\left.+{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i\right\}\hfill \\ & \le & -\overline{\eta }{V}^{a1}+\overline{\upsilon }+{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i\hfill \end{array}$$

(33)

where $\overline{\eta }=min\{{\displaystyle \frac{\Phi }{{\lambda }_{max}\left(P\right)}},$ $b_{i,m},$ ${\phi }_i,$ ${\kappa }_{i,m},$ ${\nu }_i,$ $2({\displaystyle \frac{1}{{\varpi }_{i,h}}}-{\displaystyle \frac{p_{i,h}^2}{2{\sigma }_{i,h}}}-{\displaystyle \frac{1}{2}})\}>0,$ $\overline{\upsilon }={\sum }_{i=1}^N(\overline{\varsigma }+l_i^2\overline{\varphi }+{\displaystyle \frac{{\phi }_i}{2g_i}}{\xi }_i^2+{\displaystyle \frac{{\nu }_i}{2{\psi }_i}}{Y}_i^2+{\overline{k}}_i+{\sum }_{m=2}^{n_i}({\displaystyle \frac{{\kappa }_{i,m}}{2d_{i,m}}}{\beta }_{i,m}^2+{\displaystyle \frac{{\sigma }_{i,m}}{2}})+{\displaystyle \frac{3}{4}}{\omega }_i^{{max}^4})$.

$Case$ 2: According to (2), we know that ${\gamma }_i\left(t\right)=0$ when no DA occurs.

Step 1: Choose the Lyapunov function candidate as

$$V_{i,1}^s=V_0+{\displaystyle \frac{1}{2}}{z}^T\left(\mathcal{L}+H\right)z+{\displaystyle \frac{1}{2}}\sum _{i=1}^N{\displaystyle \frac{1}{{\overline{\psi }}_i}}{\tilde{Y}}_i^2$$

Based on Young’s inequation, from (1), (5), and (12)–(14), one has

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,1}^s& \le & \sum _{i=1}^N\{-{\breve{Q}}_i{∥{\varkappa }_i∥}^2+e_{i,1}[{\displaystyle \frac{3}{2}}{e}_{i,1}+{\Psi }_{i,1}-h_i{\dot{y}}_d\hfill \\ & & +\left(h_i-1\right){\dot{y}}_d]+{\displaystyle \frac{1}{2}}{e}_{i,2}^2+{\displaystyle \frac{1}{2}}{∥{\varkappa }_i∥}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{\vartheta }_{i,2}^2+\sum _{i=1}^N{\displaystyle \frac{1}{{\overline{\psi }}_i}}{\tilde{Y}}_i{\dot{\widehat{Y}}}_i\}\hfill \end{array}$$

(34)

Then, via using Lemma 1, substituting (23) and (25) into (34), ${\dot{V}}_{i,1}^s$ can be rewritten as

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,1}^s& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2\right.-{\overline{b}}_{i,1}{e}_{i,1}^2+{\displaystyle \frac{1}{2}}{\vartheta }_{i,2}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}{e}_{i,2}^2+{\mu }_i{Y}_i-{\displaystyle \frac{{\overline{\nu }}_i}{{\overline{\psi }}_i}}\left.{\tilde{Y}}_i{\widehat{Y}}_i\right\}\hfill \end{array}$$

with $\mathsf{\Phi }={\breve{Q}}_i-{\displaystyle \frac{1}{2}}.$

Step $h\left(2\le h\le n_i\right)$: Choose the Lyapunov function candidate as $V_{i,h}^s=V_{i,h-1}^s+{\displaystyle \frac{1}{2}}{\sum }_{i=1}^N\left(e_{i,h}^2+{\vartheta }_{i,h}^2\right).$ Apply the same method as (32), define ${\overline{p}}_{i,h}=max\left\{\right|{\dot{\Psi }}_{i,h-1}\left|\right\},$ based on (1), (13), (14), and (24), where the derivative of $V_{i,h}^s$ can be calculated as

$$\begin{array}{ccc}\hfill {\dot{V}}_{i,n_i}^s& \le & \sum _{i=1}^N\left\{-\mathsf{\Phi }{∥{\varkappa }_i∥}^2\right.-{\displaystyle \frac{{\nu }_i}{2{\psi }_i}}{\tilde{Y}}_i^2-\sum _{m=1}^{n_i}{\overline{b}}_{i,m}{e}_{i,m}^2\hfill \\ & & +\sum _{m=2}^{n_i}\left({\displaystyle \frac{1}{{\varpi }_{i,m}}}-{\displaystyle \frac{{\overline{p}}_{i,m}^2}{2{\overline{\sigma }}_{i,m}}}-{\displaystyle \frac{1}{2}}\right){\vartheta }_{i,m}^2\hfill \\ & & +{\displaystyle \frac{{\overline{\nu }}_i}{2{\overline{\psi }}_i}}{Y}_i^2+{\overline{k}}_1+\sum _{m=2}^{n_i}\left.{\displaystyle \frac{{\overline{\sigma }}_{i,m}}{2}}\right\}\hfill \\ & \le & -\breve{\eta }{V}^s+\breve{\upsilon }\hfill \end{array}$$

(35)

where ${\overline{\sigma }}_{i,m}>0$ is a constant, and $\breve{\eta }=min\{{\displaystyle \frac{\mathsf{\Phi }}{{\lambda }_{max}\left(P\right)}},$ ${\overline{b}}_{i,m},$ ${\overline{\nu }}_i,$ $2({\displaystyle \frac{1}{{\varpi }_{i,h}}}-{\displaystyle \frac{{\overline{p}}_{i,m}^2}{2{\overline{\sigma }}_{i,m}}}-{\displaystyle \frac{1}{2}})\}>0,$ $\breve{\upsilon }={\sum }_{i=1}^N({\displaystyle \frac{{\overline{\nu }}_i}{2{\overline{\psi }}_i}}{Y}_i^2+{\overline{k}}_i+{\sum }_{m=2}^{n_i}{\displaystyle \frac{{\overline{\sigma }}_{i,m}}{2}})$, with ${\overline{k}}_1={\sum }_{i=1}^N{Y}_i{\int }_0^{+\infty }{\mu }_i\left(\tau \right)d\left(\tau \right)>{\sum }_{i=1}^N{Y}_i{\int }_0^t{\mu }_i\left(\tau \right)d\left(\tau \right)>0$ being a finite constant.

Then, considering $Case$ 1 and $Case$ 2, construct a total Lyapunov function $V=V^{a1}+V^s.$ Based on (33) and (35), the derivative of V can be expressed as

$$V\le -\eta V+\upsilon +{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f_i^{-1}}{r_i}}{\dot{\varrho }}_i$$

(36)

where $\eta =min\{\overline{\eta },\breve{\eta }\}$ and $\upsilon =max\{\overline{\upsilon },\breve{\upsilon }\}.$

By integrating both sides of (36), one has

$$\begin{array}{ccc}\hfill 0& \le & V\left(t\right)\le e^{-\eta t}V\left(0\right)+{\displaystyle \frac{\upsilon }{\eta }}\left(1-e^{-\eta t}\right)\hfill \\ & & +e^{-\eta t}\sum _{i=1}^N{\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)e^{\eta \tau }d\tau \hfill \end{array}$$

Then, according to Lemma 3, we know that $V\left(t\right),$ ${\varrho }_i\left(t\right)$ and ${\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)d\tau$ are bounded on $t\in \left[0,t_{\omega }\right).$ Moreover, we can easily obtain

$$\begin{array}{ccc}& & {\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)e^{-\eta \left(t-\tau \right)}d\tau \hfill \\ & \le & {\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)d\tau \hfill \end{array}$$

Hence, ${\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)e^{-\eta \left(t-\tau \right)}d\tau$ is bounded on $t\in \left[0,t_{\omega }\right).$

Then, define

$$\delta =\underset{t\in \left[0,t_{\omega }\right)}{max}\left|\sum _{i=1}^N{\int }_0^t{\displaystyle \frac{{\omega }_i{N}_i\left({\varrho }_i\right)+f^{-1}}{r}}{\dot{\varrho }}_i\left(\tau \right)e^{-\eta \left(t-\tau \right)}d\tau \right|,$$

where we can obtain

$$0\le V\left(t\right)\le e^{-\eta t}V\left(0\right)+{\displaystyle \frac{\upsilon }{\eta }}\left(1-e^{-\eta t}\right)+\delta$$

(37)

By applying (37) and the definition of V, we arrive at ${∥e∥}^2\le 2e^{-\eta t}V\left(0\right)+{\displaystyle \frac{2\upsilon }{\eta }}\left(1-e^{-\eta t}\right)+\delta .$ Then, theoretically, for any $ϵ>0,$ on account of the definitions of $\eta ,$ $\upsilon$ and $\delta ,$ and selecting the design parameters appropriately makes the inequality ${\displaystyle \frac{2\upsilon }{\eta }}+\delta \le {ϵ}^2{\left(\iota \left(\mathcal{L}+H\right)\right)}^2$ hold, where $\iota \left(\mathcal{L}+H\right)$ is the minimum singular value of $\left(\mathcal{L}+H\right).$ Further, based on Lemma 2 in [34], we have ${lim}_{t\to \infty }∥y_i-y_r∥\le ϵ,$ i.e., the tracking error $z_i=y_i-y_r$ is proved to be bounded. Distinctly, (37) means that all variables are bounded in system (1). Further, the control objectives of MASs are achieved. This completes the proof.

□

Remark 4.

Although the study presented in [35] employs a nonlinear backstepping control strategy optimized by metaheuristic algorithms to improve trajectory tracking for a quadrotor-slung load system, it primarily focuses on enhancing performance in a deterministic and attack-free environment. In contrast, our work investigates a fundamentally different and more challenging problem: achieving resilient cooperative control in MASs subjected to DAs. Instead of relying on metaheuristic optimization, we construct an output-feedback-based detection mechanism and integrate it into a Lyapunov-guided adaptive backstepping framework, which simultaneously ensures timely attack identification and boundedness of all closed-loop signals. Our approach addresses both the detection and mitigation of malicious attacks, which are not considered in [35], thereby contributing to the security and robustness of MASs under adversarial conditions.

Remark 5.

Generally speaking, most research on adaptive cooperative control has only focused on ensuring system stability in the presence of attacks. There are few studies that explore joint methods for detecting and mitigating attacks. Moreover, current DA detection methods mainly adopt a class of detection mechanisms with a constant threshold and linear feedback controllers to deal with DAs, which leads to constraints and complex DA detection mechanism designs. However, in this paper, we propose an output-based DA detection method, which only requires the output information of agents and does not depend on any additional conditions. With this method, the predictability of DAs is achieved in a system. Based on this, a control protocol is developed to mitigate the effects of DAs and ensure the performance of MASs under DAs. Compared to previous works, this approach provides a more efficient and effective DA detection and mitigation method for MASs.

5. Simulation Results

To illustrate the effectiveness of the developed control scheme, the following MASs are considered

$$\begin{array}{ccc}\hfill {\dot{x}}_{i,1}& =& x_{i,2}\hfill \\ \hfill {\dot{x}}_{i,2}& =& u_i+{\varsigma }_i\left(t\right)\hfill \\ \hfill y_i& =& {\mathsf{\Xi }}_i\hfill \end{array}$$

where ${\varsigma }_i\left(t\right)=5+2sin\left(t\right),$ and if $\varphi \left(t\right)\ne 0;$ then, ${\mathsf{\Xi }}_i=sin\left(x_{i,1}\right)+x_{i,1};$ otherwise, ${\mathsf{\Xi }}_i=x_{i,1}.$ Meanwhile, the communication digraph is depicted in Figure 2, where the leader node represents a reference unit that provides the desired trajectory or consensus target for the system, and the agents are individual dynamic nodes that aim to follow or synchronize with the leader based on local interactions.

In the simulation, all the states and adaptive update law initials are selected as zero, except that ${\widehat{x}}_{12}\left(0\right)=0.1,$ ${\widehat{x}}_{22}\left(0\right)=0.12,$ ${\widehat{x}}_{32}\left(0\right)=0.13,$ ${\widehat{x}}_{42}\left(0\right)=0.14,$ ${\widehat{x}}_{52}\left(0\right)=0.25.$ The controller coefficients and adaptive update laws parameters are chosen as $b_{i,1}=120,$ $b_{i,2}=150,$ $l_{i,1}=15,$ $l_{i,2}=20,$ $f_i=0.05,$ $\chi =5,$ $r_i=0.005,$ ${\nu }_i=0.5,$ ${\psi }_i=0.3,$ ${\kappa }_{i,h}=0.2,$ $d_{i,h}=25,$ ${\phi }_i=0.045,$ $g_i=0.015,$ $s_{i1}={\overline{s}}_{i1}={\breve{s}}_{i1}=3,$ $s_{i2}={\overline{s}}_{i2}={\breve{s}}_{i2}=0.01.$ In addition, the leader trajectory is selected as $y_d=0.5sin\left(0.1t\right)+0.5sin\left(0.05t\right).$

The simulation results are presented in Figure 3, Figure 4 and Figure 5 to verify the effectiveness of the proposed detection and control framework. As illustrated in Figure 3, the behavior of the observer errors under different conditions is examined under the developed control protocol (24) for MASs with or without DA; (a) this represents the fact that the observer errors can converge to 0 when t tends to infinity (i.e., ${lim}_{t\to \infty }{\varkappa }_{i,1}=0)$ in the absence of a DA; (b) here, the observer errors cannot converge to 0 when t tends to infinity (i.e., ${lim}_{t\to \infty }{\varkappa }_{i,1}\ne 0)$ in the presence of a DA, i.e., the design sees that the attack detection observer can effectively detect the DA via Lemma 2. Under a DA, the tracking performance of each agent and the evolution of control input are shown in Figure 4 and Figure 5, which further illustrate the performance of the multi-agent system under a DA. In both figures, two different controllers are compared: (a) this shows the results using a controller (18), which does not incorporate DA-resilient features, while (b) employs controller (24), which is specifically designed to handle DAs. Figure 4 demonstrates the tracking performance of each agent under these controllers, and Figure 5 presents the evolution of the control inputs. From the comparative analysis, it is evident that controller (24) not only ensures the effective detection of DAs but also maintains satisfactory tracking performance and bounded control inputs despite the presence of attacks. This confirms the robustness and resilience of the proposed control strategy in adversarial environments. The combined simulation results strongly support the theoretical developments and show that the proposed method achieves both secure state estimation and cooperative tracking control in the presence of DAs.

6. Conclusions

In this article, a DA detection mechanism based on the output information of an agent has been developed in MASs. The adverse effects generated by a DA have been resolved through the design of the Nussbaum function and adaptive update laws dependent on the DA signal. Moreover, an adaptive cooperative security control protocol has been proposed for MASs under a DA via backstepping technology to achieve cooperative control and ensure that all the closed-loop signals are bounded. The proposed framework effectively integrates detection, mitigation, and control strategies to enhance the resilience of MASs against a DA. The introduction of the Nussbaum function addresses unknown control directions caused by DA, while the adaptive laws allow the system to dynamically compensate for disturbances without requiring prior knowledge of attack models. Furthermore, the backstepping-based design ensures stability and cooperation in a fully distributed manner. These key findings demonstrate the potential of the proposed approach in real-world applications where security and cooperation are critical. It is worth noting that this article only considers the distributed resilient cooperative control problem under a single-leader case. Thus, generalizing this result to the multiple-leader case is an interesting direction for future research.