Adaptive Leader-Following Consensus Tracking Control of Multiple UAVs Subject to Deception Attacks

Abstract

This article is concerned with the problem of leader-following consensus tracking control for multi-unmanned aerial vehicle (UAV) systems under deception attacks and actuator saturation. While guaranteeing the expected control performance, a novel adaptive event-triggered scheme (AETS) is developed to reduce the frequency of data transmission among UAVs and reduce the release of redundant data, where the adaptive threshold can be adjusted online according to the dynamic error instead of giving a default value. With the help of a new Lyapunov function, sufficient conditions and security criteria are developed to ensure that all following UAVs could reach secure consensus asymptotically with a leader UAV. Lastly, an illustrative example is presented to validate the effectiveness of the designed method.

1. Introduction

In recent years, the cooperative control issues of multi-unmanned aerial vehicle (UAV) systems have attracted considerable attention from scientific and engineering fields [1,2,3,4,5,6,7,8], mainly owing to the fact that a group of mobile UAVs come to agreement by utilizing local relative information from adjacent UAVs, which can cooperate to complete some complex tasks that exceed the capability of a single vehicle. Additionally, a group of UAVs has better robustness than a single vehicle when executing tasks. It is noteworthy that cooperative consensus control can be approximately divided into two categories according to the number of leaders, which are leader-following consensus and leaderless consensus [9,10,11,12]. The leader-following problem plays a vital role in consensus control for multi-UAVs, although leaderless consensus is useful in several application fields. Up to the present, a large number of studies and references therein have paid attention to it.

As is well known, due to the limitation of communication and control resources, it is unrealistic that continuous update of the controller and continuous communication between UAVs are required in practical engineering. In order to address this issue, event-triggered scheme (ETS) in multiple-UAV systems has attracted wide attention [13,14,15,16,17,18,19]. Considering the influence of input saturation on a multi-UAV system, the work in [20] probed into the problem of implementing predefined configuration in a distributed finite-time ETS. When a UAV system was suffering time-varying disturbance, the dynamic event-triggered cooperative formation control strategy was investigated in [21]. Despite ensuring the control performance with the aid of the above results, the thresholds are unable to be automatically regulated in the light of system requirements. Hence, it is necessary to design a trigger mechanism to dynamically adjust the trigger conditions on the basis of system performance demand and reduce information transmission rate. Different from the preset constant threshold used in the existing literature, the method of designing an adaptive event-triggered scheme (AETS) for nonlinear interconnected systems was proposed in [22], wherein adaptive threshold was based on the dynamic error of the system. The authors in [23] investigated consensus of multi-agent systems (MASs) on directed graphs through AETS with composite triggering conditions. In the case of switching directional communication topology and security assignments, the issue of cluster tracking cooperative formation control was studied in [6], where the restriction of limited information interchange and the assumption of continuous feedback control gain were relaxed with the help of cluster ETS. Inspired by the above works, the purpose of this paper is to adopt an AETS to realize the tracking control consistency for multi-UAV systems.

As the most common and important nonlinear phenomenon, actuator saturation, due to the limitation of the physical characteristics of the actuator, often results in poor performance so far as to destabilize the UAV system. Over the last decade, a great many scholars have studied the topic whereby UAV systems suffer from actuator saturation [24,25,26,27,28,29]. For instance, comprehensively considering the effects of actuator failure, saturation, and disturbance, the problem of fault-tolerant tracking consensus was discussed in [25]. It was reported in [26], with intermittent communication, that the global consensus problem for MAS subject to actuator saturation was considered. In addition, to achieve global consistency, a distributed feedback controller was designed for the system with switching topology and time-varying delay. How to explore some novel design algorithms for UAV systems with actuator saturation to achieve better performance gives impetus to this work.

Due to the increasing dependence on the network, UAV systems are more vulnerable to malicious cyberattacks. At the same time, the issue of secure consensus control has also attracted extensive attention of scholars [30,31,32,33,34,35,36,37,38,39,40]. Under known and unknown acceleration of targets, in order to address the issue of collaborative siege and simultaneous assault, two collaborative guidance laws based on multiple attackers encircling the target region were proposed in [30]. In addition, with regard to an impulsive system under spoofing attack, the authors in [39] inquired into the topic of secure synchronization and proposed the distributed impulsive control protocol for MASs. In [40], the issue of robust output consensus was studied to guarantee that the consensus goal of the heterogeneous MASs can be achieved, immune from aperiodic sampling and DoS attack. Thus far, the secure consensus of an UAV system, subject to both unknown disturbance and cyber attack, has not been fully discussed, which motivates the current study.

Inspired by the above observations, this paper is concerned with the leader-following consensus tracking control design for multi-UAV systems subject to deception attacks and actuator saturation. The major contributions of this paper lie in three points:

(1) A novel adaptive data transmission scheme is developed for multi-UAV systems. The triggering threshold, introducing measurement error among UAVs, can be adjusted online according to the fluctuation of system state, so as to actively discard the data with little change from the latest released data packet. By utilizing the designed AETS, data-releasing rate can be reduced while the sensitivity of the system to malicious attacks is improved.

(2) A new event-triggered multi-UAV system is constructed, in which the wind disturbance during UAV flight, the uncertainty inside the actuator, actuator saturation, and deception attack injected into the system are considered within a unified framework.

(3) By introducing an improved Lyapunov functional method, some sufficient conditions for the stability of a multi-UAV system under digraph is proposed to co-design the triggering parameters and controllers in the meantime.

The outline for the rest of this article is as follows. In Section 2, problem formulation including system model description, deception attack model, and AETS is introduced. In Section 3, some sufficient conditions are established to guarantee the desired control performance and the leader-following consensus in security of the multi-UAV system with the AETS and deception attacks. In addition, the design algorithm of controllers is given. Section 4 introduces a simulation example manifesting the effectiveness of the proposed method. Finally, the conclusion is drawn in Section 5.

Notation: ${\mathbb{R}}^n$ and ${\mathbb{R}}^{p\times q}$ denote the n-Euclidean space and the set of all $p\times q$ dimensional real matrices. diag${}_N\left\{S\right\}$ (or diag${}_N\left\{S_i\right\}$) represents an N-block diagonal matrix diag$\{S,\dots ,S\}$ (or diag$\{S_1,\dots ,S_N\}$). Similarly, col$\left\{S\right\}$ (or col${}_N\left\{S_i\right\}$) is used to express column vectors with suitable blocks. ∗ stands for the transpose of block matrix. 0 is the zero matrix of the appropriate dimension. Furthermore, $I_n$ (or $I_N$) and $I_{nN}$ (or $I_{2nN}$) represent the identity matrix of n (or N) order and $nN$ (or $2nN$) order, respectively. ⊗ indicates the Kronecker product. $\parallel •\parallel$ describes Euclidean norm for vectors or matrices. The expectation of X is represented by $\mathbb{E}\left\{X\right\}$.

2. Problem Formulation

2.1. Graph Theory

A weighted digraph $\mathcal{G}=\left(\mathcal{V},\mathcal{E},\mathcal{W}\right)$ without self-loops describes the communication topology among one leader and N followers, where $\mathcal{V}$ denotes index set of N followers, that is, $\mathcal{V}=\left\{v_1,v_2,\cdots ,v_N\right\}$. An edge set is represented by $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$, and $\mathcal{W}=\left[{\omega }_{ij}\right]\in {\mathbb{R}}^{N\times N}$ is defined to represent a weighted adjacency matrix. For any edge $({\mathcal{V}}_j,{\mathcal{V}}_i)\in \mathcal{E}$, ${\omega }_{ij}>0$, and ${\omega }_{ij}=0$ otherwise. ${\mathcal{N}}_i=\left\{j\in \mathcal{V}:\left(j,i\right)\in \mathcal{E}\right\}$ is used to express the set of neighbors of the i-th follower.

Assumption 1.

With regard to a multi-UAV system, the leader UAV is presumed to be the root node of the spanning tree in $\mathcal{G}$, and the followers receive information from the leader.

2.2. System Description

In this paper, a multi-UAV system is made up of N followers and one leader (labeled by 0). For simplicity, assuming that each number of the group has the same mechanical structure and for $i=1,\cdots ,N$, each vehicle is described as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{p}}_{xi}\left(t\right)=V_{i}cos{\phi }_i\left(t\right)\hfill \\ {\dot{p}}_{yi}\left(t\right)=V_{i}sin{\phi }_i\left(t\right)\hfill \\ {\dot{v}}_i\left(t\right)=u_i\left(t\right)\hfill \end{array},\right.\end{array}$$

(1)

where $(p_{xi}\left(t\right),p_{yi}\left(t\right)),v_i\left(t\right)$ represent the position, accelerated velocity of the i-th UAV, respectively. ${\phi }_i\left(t\right)$ is the heading angle, and $V_i$ is flight speed. We denote $u_i\left(t\right)$ as the control input vectors.

The precise feedback linearization is used to linearize the UAV system. We define $s_i\left(t\right)={(p_{xi}\left(t\right),p_{yi}\left(t\right))}^T$, then the system (1) can be transformed as below:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{s}}_i\left(t\right)=v_i\left(t\right)\hfill \\ {\dot{v}}_i\left(t\right)=u_i\left(t\right)\hfill \end{array},\right.\end{array}$$

(2)

where $s_i\left(t\right)\in {\mathbb{R}}^n$ denote position and $v_i\left(t\right)\in {\mathbb{R}}^n$ represent velocity of the i-th UAV. Without loss of generality, the leader vehicle has the following dynamic equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{s}}_0\left(t\right)=v_0\left(t\right)\hfill \\ {\dot{v}}_0\left(t\right)=0\hfill \end{array},\right.\end{array}$$

(3)

Considering the wind disturbance and actuator uncertainty during UAV flight, the second-order integrator model structured above can be described as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\delta }}_i\left(t\right)=A{\delta }_i\left(t\right)+B{u}_i\left(t\right)+f\left({\delta }_i\left(t\right)\right)\hfill \\ {\dot{\delta }}_0\left(t\right)=A{\delta }_0\left(t\right)+f\left({\delta }_0\left(t\right)\right)\hfill \end{array},\right.\end{array}$$

(4)

where $A=\left[\begin{array}{cc}{0}_{n\times n}& I_n\\ 0_{n\times n}& 0_{n\times n}\end{array}\right],B=\left[\begin{array}{c}{0}_{n\times n}\\ I_n\end{array}\right]$, ${\delta }_i\left(t\right)=col\{s_i\left(t\right),v_i\left(t\right)\},{\delta }_0\left(t\right)=col\{s_0\left(t\right),v_0\left(t\right)\}$ denote the state vector. In addition, in a short time and a small flight space, the wind disturbance and actuator uncertainty of UAV can be regarded as slow variables and expressed as unknown vector parameters, and $f\left({\delta }_i\left(t\right)\right)\in {\mathbb{R}}^{2n},f\left({\delta }_0\left(t\right)\right)\in {\mathbb{R}}^{2n}$ represent the uncertain disturbance of follower and leader UAV, respectively. For all ${\delta }_i\left(t\right),{\delta }_0\left(t\right)\in {\mathbb{R}}^{2n}$, the following inequality holds if there exists an appropriate constant $\beta >0$:

$$\begin{array}{c}\hfill \parallel f\left({\delta }_i\left(t\right)\right)-f\left({\delta }_0\left(t\right)\right)\parallel \le \beta \parallel {\delta }_i\left(t\right)-{\delta }_0\left(t\right)\parallel .\end{array}$$

(5)

The main intention of this study is to design distributed adaptive event-triggered consensus strategies for each UAV such that the consensus problem of the UAV system (1) can be asymptotically solved, i.e., ${lim}_{t\to \infty }∥s_i\left(t\right)-s_0\left(t\right)∥=0$ and ${lim}_{t\to \infty }∥v_i\left(t\right)-v_0\left(t\right)∥=0$ for $i=1,2,\cdots ,N$. For this purpose, we need the following assumptions and lemmas.

2.3. Deception Attack Model

In this paper, actuator saturation is taken into account, which can be described by a saturation function $sat(·):{\mathbb{R}}^m\to {\mathbb{R}}^m$, i.e., $sat\left(u_i\left(t\right)\right):={[sat\left(u_i^1\left(t\right)\right),sat\left(u_i^2\left(t\right)\right),\cdots ,sat\left(u_i^m\left(t\right)\right)]}^T$. For each $s=\{1,2,\cdots ,m\}$, $sat\left(u_i\left(t\right)\right)=sign\left(u_i^s\left(t\right)\right)min\{1,|u_i^s\left(t\right)\left|\right\}$.

In addition, the saturation function resolves into two items, that is $sat\left(u_i\left(t\right)\right)=u_i\left(t\right)-\psi \left(u_i\left(t\right)\right)$, where $\psi \left(u_i\left(t\right)\right)$ is a nonlinear vector-valued function, and $\psi \left(u_i\left(t\right)\right)$ satisfies the following inequality:

$$\begin{array}{c}\hfill \sum _{i=1}^N{\psi }^T\left(\left(u_i\left(t\right)\right)\right)(\psi \left(u_i\left(t\right)\right)-u_i\left(t\right))\le 0.\end{array}$$

(6)

In fact, due to the aging of the actuator or external attacks from the network, the actuators may suffer malicious attacks. The attack signals for transmitted information between follower UAVs are expressed by

$$\begin{array}{c}\hfill {\widehat{\delta }}_j\left(t\right)={\delta }_j\left(t\right)+{\varrho }_j\left(t\right),\end{array}$$

(7)

where the attack function, similar to [41], ${\varrho }_j\left(t\right)\in {\mathbb{R}}^{2n}$, is assumed to satisfy

$$\begin{array}{c}\hfill \parallel {\varrho }_j{\left(t\right)\parallel }_2\le {\parallel E({\delta }_j\left(t\right)-{\delta }_0\left(t\right))\parallel }_2\end{array}$$

(8)

with a known energy constraint matrix E.

2.4. Adaptive Event-Triggered Communication Scheme

To reduce the number of triggering event of data releasing, An AETS is utilized for each vehicle transmission. Under the scheme, the i-th UAV only broadcasts the data packets to its neighbor when the triggering event is generated.

An AETS is proposed to design the following controller for a multiple UAV system:

$$\begin{array}{c}\hfill u_i\left(t\right)=-K{\chi }_i\left(t\right),\end{array}$$

(9)

where K is a feedback matrix to be designed, and ${\chi }_i\left(t\right)={\sum }_{j\in {\mathcal{N}}_i}{\omega }_{ij}[{\delta }_i(t_k^i)-({\delta }_j(t_{k^{\prime }}^j)+{\varrho }_j(t))]+b_i({\delta }_i(t_k^i)-{\delta }_0(t_k^{i}h+lh))$.

Then, we will design the adaptive triggering mechanism to determine the update time of the controller. In order to clearly clarify the mechanism of AETS, we artificially divide the packet release interval into some subsets, shown as $[t_k^{i}h,t_{k+1}^{i}h)={\cup }_{\iota h=t_k^{i}h}^{(t_{k+1}^i-1)h}\left[\iota h,(\iota +1)h\right)$, where $\iota h=t_k^{i}h+lh$. We define measurement error ${\mathit{e}}_i(t_k^{i}h+lh)={\delta }_i\left(t_k^i\right)-{\delta }_0(t_k^{i}h+lh)$ and ${\mathcal{S}}_i=\left\{l|{\mathit{e}}_i^T\left(\iota h\right)\Xi {\mathit{e}}_i\left(\iota h\right)-{\kappa }_i\left(t\right){\chi }_i^T\left(\iota h\right)\Xi {\chi }_i\left(\iota h\right)\le 0\right\}$.

The next releasing instant of the i-th UAV can be expressed by

$$\begin{array}{c}\hfill t_{k+1}^{i}h=t_k^{i}h+(l_M+1)h,\end{array}$$

(10)

where $l_M=\underset{l\in {\mathcal{S}}_i}{max}l$. $t_k^{i}h$ is on behalf of the data releasing instant of ith follower, and immediately sampling packets $t_k^{i}h+lh$ are discarded with $l=1,2,\cdots ,l_M$. $\Xi$ is a triggering matrix to be constructed. In addition, ${\kappa }_i\left(t\right)$ represents the adjustable thresholds satisfying the following adaptive algorithm:

$$\begin{array}{c}\hfill {\dot{\kappa }}_i\left(t\right)=\frac{{\alpha }_i}{{\kappa }_i\left(t\right)}(\frac{1}{{\kappa }_i\left(t\right)}-{\varpi }_i){\mathit{e}}_i^T\left(\iota h\right)\Xi {\mathit{e}}_i\left(\iota h\right),\end{array}$$

(11)

where ${\kappa }_i\left(t\right)\in (0,1],{\alpha }_i>0,\mathrm{and} {\varpi }_i\ge 1$ are used to make the adaptive law obtain better threshold adjustment.

Remark 1.

The next releasing instant is given in Equation (10), and $l_M$ is the maximum number of continuous packet losses allowed, which means that the updated sampling data will be sent when the following trigger condition are violated; otherwise, the data packet will be discarded,

$$\begin{array}{c}\hfill {\mathit{e}}_i^T\left(\iota h\right)\Xi {\mathit{e}}_i\left(\iota h\right)-{\kappa }_i\left(t\right){\chi }_i^T\left(\iota h\right)\Xi {\chi }_i\left(\iota h\right)\le 0.\end{array}$$

(12)

Consequently, the amount of transmitted packets from the i-th UAV are greatly reduced.

Remark 2.

If one sets ${\alpha }_i=0$, then the proposed AETS will deduce to a conventional ETS

$$\begin{array}{c}\hfill {\mathit{e}}_i^T\left(\iota h\right)\Xi {\mathit{e}}_i\left(\iota h\right)-{\overline{\kappa }}_i{\chi }_i^T\left(\iota h\right)\Xi {\chi }_i\left(\iota h\right)\le 0,\end{array}$$

(13)

where ${\overline{\kappa }}_i$ are constant triggering thresholds.

Through the above discussion, for $t\in [\iota h,(\iota +1\left)h\right)$, we define the tracking error as ${\xi }_i\left(\iota h\right)={\delta }_i\left(\iota h\right)-{\delta }_0\left(\iota h\right),i=1,2,\cdots ,N$, and $d_i\left(t\right)=t-\iota h$, $d_i\left(t\right)$ satisfies $d_i\left(t\right)\in [0,d_M)$ with the derivative ${\dot{d}}_i\left(t\right)=1$ at $t\ne \iota h$. Then, the leader-following consensus tracking error system is given as follows:

$$\begin{array}{c}\hfill {\dot{\xi }}_i\left(t\right)=A{\xi }_i\left(t\right)+B{u}_i\left(t\right)+f\left({\delta }_i\left(t\right)\right)-f\left({\delta }_0\left(t\right)\right).\end{array}$$

(14)

Combining the measurement error ${\mathit{e}}_i(t-d_i\left(t\right))$ and tracking error ${\xi }_i(t-d_i\left(t\right))$, the controller $u_i\left(t\right)$ can be described as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_i\left(t\right)=& -K[\sum _{j=1}^N{l}_{ij}({\xi }_j(t-d_j\left(t\right))+{\mathit{e}}_j(t-d_j\left(t\right)))+b_i({\xi }_i(t-d_i\left(t\right))\hfill \\ \hfill & +{\mathit{e}}_i(t-d_i\left(t\right)))-\sum _{j=1}^N{\omega }_{ij}{\varrho }_j\left(t\right)],\hfill \end{array}\end{array}$$

(15)

where $L={\left[l_{ij}\right]}_{N\times N},l_{ii}={\sum }_{j\in {\mathcal{N}}_i}{\omega }_{ij},l_{ij}=-{\omega }_{ij}$, and $\mathcal{B}=dia{g}_N\left\{b_i\right\}$.

From (7), (14), and (15), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{\xi }}_i\left(t\right)& =A{\xi }_i\left(t\right)-BK\sum _{j=1}^N{l}_{ij}({\xi }_j(t-d_j\left(t\right))+{\mathit{e}}_j(t-d_j\left(t\right)))-BK{b}_i{\xi }_i(t-d_i\left(t\right))\hfill \\ \hfill & -BK{b}_i{\mathit{e}}_i(t-d_i\left(t\right))+BK\sum _{j=1}^N{c}_{ij}{\varrho }_j\left(t\right)-B\psi \left(u_i\left(t\right)\right)+f\left({\delta }_i\left(t\right)\right)-f\left({\delta }_0\left(t\right)\right).\hfill \end{array}\end{array}$$

(16)

By using Kronecker product, for $t\in [\iota h,(\iota +1\left)h\right),$ the overall error dynamic can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\xi }\left(t\right)& =(I_N\otimes A)\xi \left(t\right)-(H\otimes BK)ϝ\left(t\right)-(H\otimes BK)\varkappa \left(t\right)+(C\otimes BK)\mathrm{Y}\left(t\right)\hfill \\ \hfill & -(I_N\otimes B)\psi \left(u\left(t\right)\right)+\mathbb{F}\left(\xi \left(t\right)\right),\hfill \end{array}\end{array}$$

(17)

where

$$\begin{array}{cc}\hfill & H=L+\mathcal{B},\xi \left(t\right)=col\{{\xi }_1\left(t\right),{\xi }_2\left(t\right),\cdots ,{\xi }_N\left(t\right)\},\hfill \\ \hfill & ϝ\left(t\right)=col\{{\xi }_1(t-d_1\left(t\right)),{\xi }_2(t-d_2\left(t\right)),\cdots ,{\xi }_N(t-d_N\left(t\right))\},\hfill \\ \hfill & \varkappa \left(t\right)=col\{{\mathit{e}}_1(t-d_1\left(t\right)),{\mathit{e}}_2(t-d_2\left(t\right)),\cdots ,{\mathit{e}}_N(t-d_N\left(t\right))\},\hfill \\ \hfill & \mathrm{Y}\left(t\right)=col\{{\varrho }_1\left(t\right),{\varrho }_2\left(t\right),\cdots ,{\varrho }_N\left(t\right)\},\hfill \\ \hfill & \psi \left(u\left(t\right)\right)=col\{\psi \left(u_1\left(t\right)\right),\psi \left(u_2\left(t\right)\right),\cdots ,\psi \left(u_N\left(t\right)\right)\},\hfill \\ \hfill & \mathbb{F}\left(\xi \left(t\right)\right)=col\{f\left({\delta }_1\left(t\right)\right)-f\left({\delta }_0\left(t\right)\right),\cdots ,f\left({\delta }_N\left(t\right)\right)-f\left({\delta }_0\left(t\right)\right)\}.\hfill \end{array}$$

The objective of this paper is to design a consensus controller for the UAV system by applying the AETS, such that the system subject to the deception attacks and actuator saturation may maintain the performance at a certain level.

3. Main Results

Before further derivation, we will give some useful definitions and lemmas.

Definition 1

([42]). The leader-following consensus of multi-UAV system (2) with the designed controller (9) is achieved if the following criterion

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}∥{\delta }_i\left(t\right)-{\delta }_0\left(t\right)∥=0,i=1,2,\cdots ,N,\end{array}$$

(18)

holds for any initial conditions ${\delta }_i\left(0\right)$.

Lemma 1

([43]). For given vector-valued function $\rho \left(t\right)$ satisfying $\rho \left(t\right)\in [0,\overline{\rho }]$, and any scalar $\dot{x}\left(t\right):[-\overline{\rho },0]\to {\mathbb{R}}^n$, if any symmetric constant matrix $R\in {\mathbb{R}}^{n\times n}>0$, then the following inequality holds:

$$-\overline{\rho }{\int }_{t-\overline{\rho }}^t{\dot{x}}^T\left(t\right)R\dot{x}\left(t\right)dt\le {\left[\begin{array}{c}x\left(t\right)\\ x(t-\rho (t\left)\right)\\ x(t-\overline{\rho })\end{array}\right]}^T\left[\begin{array}{ccc}-R& *& *\\ R& -2R& *\\ 0& R& -R\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x(t-\rho (t\left)\right)\\ x(t-\overline{\rho })\end{array}\right].$$

(19)

Proof.

Defining $0={\rho }_1\le \rho \left(t\right)\le {\rho }_2=\overline{\rho }$, from Jensen’s integral inequality and Leibnitz–Newton formula, one can obtain that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -({\rho }_2-{\rho }_1){\int }_{t-{\rho }_2}^{t-{\rho }_1}{\dot{x}}^T\left(v\right)R\dot{x}\left(v\right)dv\hfill \\ \hfill & =-\overline{\rho }\left[{\int }_{t-\rho \left(t\right)}^t{\dot{x}}^T\left(v\right)R\dot{x}\left(v\right)dv+{\int }_{t-\overline{\rho }}^{t-\rho \left(t\right)}{\dot{x}}^T\left(v\right)R\dot{x}\left(v\right)dv\right]\hfill \\ \hfill & \le -\rho \left(t\right){\int }_{t-\rho \left(t\right)}^t{\dot{x}}^T\left(v\right)R\dot{x}\left(v\right)dv-(\overline{\rho }-\rho \left(t\right)){\int }_{t-\overline{\rho }}^{t-\rho \left(t\right)}{\dot{x}}^T\left(v\right)R\dot{x}\left(v\right)dv\hfill \\ \hfill & \le -{\int }_{t-\rho \left(t\right)}^t{\dot{x}}^T\left(v\right)dvR{\int }_{t-\rho \left(t\right)}^t\dot{x}\left(v\right)dv-{\int }_{t-\overline{\rho }}^{t-\rho \left(t\right)}{\dot{x}}^T\left(v\right)dvR{\int }_{t-\overline{\rho }}^{t-\rho \left(t\right)}\dot{x}\left(v\right)dv\hfill \\ \hfill & =-{[x\left(t\right)-x(t-\rho \left(t\right))]}^{T}R[x\left(t\right)-x(t-\rho \left(t\right))]\hfill \\ \hfill & -{[x(t-\rho \left(t\right))-x(t-\overline{\rho })]}^{T}R[x(t-\rho \left(t\right))-x(t-\overline{\rho })].\hfill \end{array}\end{array}$$

(20)

Rearranging some terms of (20) yields the result (1). This completes the proof.    □

Theorem 1.

For given positive constants $\beta ,d_M,{\alpha }_i,{\varpi }_i$, and matrix K, the multi-UAV system (4) subject to deception attacks can achieve consensus if there exist symmetrical matrices $P>0,$$Q>0,R>0,\Xi >0$, such that

$$\Pi =\left[\begin{array}{ccc}{\Pi }_1& *& *\\ d_M\mathcal{A}& -R^{-1}& *\\ {\Pi }_3& 0& -I_{2nN}& *\\ {\Pi }_4& 0& 0& -{(I_N\otimes P)}^{-1}\end{array}\right]<0,$$

(21)

where

$$\begin{array}{cc}\hfill & {\Pi }_1=\left[\begin{array}{ccccccc}{\Pi }_{11}& *& *& *& *& *& *\\ {\Pi }_{21}& {\Pi }_{22}& *& *& *& *& *\\ 0& R& -Q-R& *& *& *& *\\ {\Pi }_{41}& {\Pi }_{42}& 0& {\Pi }_{44}& *& *& *\\ {\Pi }_{51}& {\Pi }_{52}& 0& {\Pi }_{54}& {\Pi }_{55}& *& *\\ {\Pi }_{61}& {\Pi }_{62}& 0& {\Pi }_{64}& C\otimes K& -I_{nN}& *\\ {\Pi }_{71}& 0& 0& 0& 0& 0& -I_{2nN}\end{array}\right],\hfill \\ \hfill & {\Pi }_{11}={(I_N\otimes A)}^T(I_N\otimes P)+(I_N\otimes P)(I_N\otimes A)+Q-R,\hfill \\ \hfill & {\Pi }_{21}=-{(H\otimes BK)}^T(I_N\otimes P)+R,\hfill \\ \hfill & {\Pi }_{41}=-{(H\otimes BK)}^T(I_N\otimes P),{\Pi }_{51}={(C\otimes BK)}^T(I_N\otimes P),\hfill \\ \hfill & {\Pi }_{61}=-{(I_N\otimes B)}^T(I_N\otimes P),{\Pi }_{71}=I_N\otimes P,\hfill \\ \hfill & {\Pi }_{22}=-2R+M{H}^{T}H\otimes \Xi ,{\Pi }_{42}=M{H}^{T}H\otimes \Xi ,\hfill \\ \hfill & {\Pi }_{52}=-M{C}^{T}H\otimes \Xi ,{\Pi }_{62}=-H\otimes K,\hfill \\ \hfill & {\Pi }_{44}=M{H}^{T}H\otimes \Xi -\Theta \otimes \Xi ,{\Pi }_{54}=-M{C}^{T}H\otimes \Xi ,\hfill \\ \hfill & {\Pi }_{55}=M{C}^{T}C\otimes \Xi -I_N\otimes P,{\Pi }_{64}=-H\otimes K,\hfill \\ \hfill & M=dia{g}_N\left\{{\alpha }_i\right\},\Theta =dia{g}_N\left\{{\alpha }_i{\varpi }_i\right\},\hfill \\ \hfill & \mathcal{A}=[I_N\otimes A-H\otimes BK0-H\otimes BKC\otimes BK-I_N\otimes B{I}_{2nN}],\hfill \\ \hfill & {\Pi }_3=\left[\begin{array}{ccccccc}\beta I_{2nN}& 0& 0& 0& 0& 0& 0\end{array}\right],{\Pi }_4=\left[\begin{array}{ccccccc}{I}_N\otimes E& 0& 0& 0& 0& 0& 0\end{array}\right].\hfill \end{array}$$

Proof.

Construct the following Lyapunov–Krasovskii functional candidate for the multi-UAV system

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{U=1}^3{V}_U\left(t\right),\end{array}$$

(22)

where

$$\begin{array}{cc}\hfill & V_1\left(t\right)={\xi }^T\left(t\right)(I_N\otimes P)\xi \left(t\right),\hfill \\ \hfill & V_2\left(t\right)={\int }_{t-d_M}^t{\xi }^T\left(s\right)Q\xi \left(s\right)ds+d_M{\int }_{-d_M}^0{\int }_{t+\alpha }^t{\dot{\xi }}^T\left(s\right)R\dot{\xi }\left(s\right)dsd\alpha ,\hfill \\ \hfill & V_3\left(t\right)=\frac{1}{2}\sum _{i=1}^N{\kappa }_i^2\left(t\right).\hfill \end{array}$$

Calculating the derivative of $V\left(t\right)$, one obtains

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\dot{V}}_1\left(t\right)={\dot{\xi }}^T\left(t\right)(I_N\otimes P)\xi \left(t\right)+{\xi }^T\left(t\right)(I_N\otimes P)\dot{\xi }\left(t\right),\hfill \\ \hfill & {\dot{V}}_2\left(t\right)={\xi }^T\left(t\right)Q\xi \left(t\right)-{\xi }^T(t-d_M)Q\xi (t-d_M)+d_M^2{\zeta }^T\left(t\right){\mathcal{A}}^{T}R\mathcal{A}\zeta \left(t\right)\hfill \\ \hfill & -d_M{\int }_{t-d_M}^t{\dot{\xi }}^T\left(s\right)R\dot{\xi }\left(s\right)ds,\hfill \\ \hfill & {\dot{V}}_3\left(t\right)=\sum _{i=1}^N[\frac{1}{{\kappa }_i\left(t\right)}{\mathit{e}}_i^T(t-d_i\left(t\right))\Xi {\mathit{e}}_i(t-d_i\left(t\right))-{\varpi }_i{\mathit{e}}_i^T(t-d_i\left(t\right))\Xi {\mathit{e}}_i(t-d_i\left(t\right))],\hfill \end{array}\end{array}$$

(23)

where $\zeta \left(t\right)=col\left\{\xi \left(t\right)ϝ\left(t\right)\xi (t-d_M)\varkappa \left(t\right)\psi \left(u\left(t\right)\right)\mathbb{F}\left(\xi \left(t\right)\right)\right\}$.

Considering (5), one can see that the nonlinear functions $F^T\left(\xi \left(t\right)\right)$ satisfies

$$\begin{array}{c}\hfill F^T\left(\xi \left(t\right)\right)F\left(\xi \left(t\right)\right)\le {\beta }^2\parallel {\delta }_i\left(t\right)-{\delta }_0\left(t\right)\parallel ={\beta }^2{\xi }^T\left(t\right)\xi \left(t\right).\end{array}$$

(24)

Applying the adaptive algorithm (11) and the trigger condition (12), it yields that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V_3\left(t\right)}& \le \sum _{i=1}^N({\alpha }_i{\chi }_i^T(t-d_i\left(t\right))\Xi {\chi }_i(t-d_i\left(t\right))-{\alpha }_i{\varpi }_i{\mathit{e}}_i^T(t-d_i\left(t\right))\Xi {\mathit{e}}_i(t-d_i\left(t\right)))\hfill \\ \hfill & =M{[(H\otimes I_{2n})(ϝ\left(t\right)+\varkappa \left(t\right))-(C\otimes I_{2n})\mathrm{Y}\left(t\right)]}^T(I_N\otimes \Xi )\left[(H\otimes I_{2n})\right(ϝ\left(t\right)\hfill \\ \hfill & +\varkappa \left(t\right))-(C\otimes I_{2n})\mathrm{Y}\left(t\right)]-{\varkappa }^T\left(t\right)(\Theta \otimes \Xi )\varkappa \left(t\right).\hfill \end{array}\end{array}$$

(25)

According to Lemma 1, we have

$$\begin{array}{c}\hfill -d_M{\int }_{t-d_M}^t\dot{\xi }{\left(s\right)}^{T}R\dot{\xi }\left(s\right)ds\le {\zeta }^T\left(t\right)\Delta \zeta \left(t\right),\end{array}$$

(26)

where

$$\begin{array}{c}\hfill \Delta =\left[\begin{array}{cccccc}-R& *& *& *& *& *\\ R& -2R& *& *& *& *\\ 0& R& -R& *& *& *\\ 0& 0& 0& 0& *& *\\ 0& 0& 0& 0& 0& *\\ 0& 0& 0& 0& 0& 0\end{array}\right].\end{array}$$

According to (6)

$$\begin{array}{c}\hfill {\psi }^T\left(u\left(t\right)\right)(\psi \left(u\left(t\right)\right)-u\left(t\right))\le 0,\end{array}$$

(27)

which can be expressed as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\psi }^T\left(u\left(t\right)\right)[-(H\otimes K)(ϝ\left(t\right)+\varkappa \left(t\right))+(C\otimes K)\mathrm{Y}\left(t\right)]-{\psi }^T\left(u\left(t\right)\right)\psi \left(u\left(t\right)\right)\ge 0.\end{array}\end{array}$$

In addition, recalling Equation (8), the following inequality holds for a given symmetric matrix P

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\xi }^T\left(t\right){(I_N\otimes E)}^T(I_N\otimes P)(I_N\otimes E)\xi \left(t\right)\ge {\mathrm{Y}}^T\left(t\right)(I_N\otimes P)\mathrm{Y}\left(t\right).\end{array}\end{array}$$

(28)

Combining (24)–(28) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathbb{E}\left\{\dot{V}\left(t\right)\right\}& \le 2{\xi }^T\left(t\right)(I_N\otimes P)[(I_N\otimes A)\xi \left(t\right)-(H\otimes BK)ϝ\left(t\right)-(H\otimes BK)\varkappa \left(t\right)\hfill \\ \hfill & +(C\otimes BK)\mathrm{Y}\left(t\right)-(I_N\otimes B)\psi \left(u\left(t\right)\right)+\mathbb{F}\left(\xi \left(t\right)\right)]+{\xi }^T\left(t\right)Q\xi \left(t\right)\hfill \\ \hfill & -{\xi }^T(t-d_M)Q\xi (t-d_M)+{\zeta }^T\left(t\right)[d_M^2{\mathcal{A}}^{T}R\mathcal{A}+\Delta ]\zeta \left(t\right)-{\varkappa }^T\left(t\right)(\Theta \otimes \Xi )\varkappa \left(t\right)\hfill \\ \hfill & +M{[(H\otimes I_{2n})(ϝ\left(t\right)+\varkappa \left(t\right))-(C\otimes I_{2n}\mathrm{Y}\left(t\right))]}^T(I_N\otimes \Xi )\left[(H\otimes I_{2n})\right(ϝ\left(t\right)\hfill \\ \hfill & +\varkappa \left(t\right))-(C\otimes I_{2n}\mathrm{Y}\left(t\right))]+{\beta }^2{\xi }^T\left(t\right)\xi \left(t\right)-{\mathbb{F}}^T\left(\xi \left(t\right)\right)\mathbb{F}\left(\xi \left(t\right)\right)\hfill \\ \hfill & +{\psi }^T\left(u\left(t\right)\right)[-(H\otimes K)(ϝ\left(t\right)+\varkappa \left(t\right))+(C\otimes K)\mathrm{Y}\left(t\right)]-{\psi }^T\left(u\left(t\right)\right)\psi \left(u\left(t\right)\right)\hfill \\ \hfill & +{\xi }^T\left(t\right){(I_N\otimes E)}^T(I_N\otimes P)(I_N\otimes E)\xi \left(t\right)-{\mathrm{Y}}^T\left(t\right)(I_N\otimes P)\mathrm{Y}\left(t\right).\hfill \end{array}\end{array}$$

(29)

The inequality (29) can be expressed as

$$\begin{array}{c}\hfill \mathbb{E}\left\{\dot{V}\left(t\right)\right\}\le {\zeta }^T\left(t\right)({\Pi }_1+d_M^2{\mathcal{A}}^{T}R\mathcal{A}+{\Pi }_3^T{\Pi }_3+{\Pi }_4^T{\Pi }_4)\zeta \left(t\right).\end{array}$$

(30)

Applying Schur complement, one can see that (21) is a sufficient condition to guarantee (30) and $\mathbb{E}\{\dot{V}\left(t\right)\}<0$.

Then, the tracking error closed-loop system (14) is asymptotically stable on the basis of condition (21), and the consensus of leader-following is achieved. This completes the proof.    □

In the following text, the co-design of controller gains with the assistance of the proposed AETS for the system (4) subject to deception attacks is given.

Theorem 2.

Under the proposed AETS (12), for some given positive constants $\beta ,d_M,\nu ,\alpha ,{\varpi }_i$, the leader-following consensus of UAV system (4) can be achieved if there exist matrices $\breve{Q}>0,\breve{R}>0,\breve{\Xi }>0,Y$ and W with appropriate dimensions, such that

$$\begin{array}{c}\hfill \breve{\Pi }=\left[\begin{array}{ccc}{\breve{\Pi }}_1& *& *\\ {\breve{\Pi }}_2& {\nu }^2\breve{R}-2\nu (I_N\otimes Y)& *\\ {\breve{\Pi }}_3& 0& -I_{2nN}& *\\ {\breve{\Pi }}_4& 0& 0& -I_N\otimes Y\end{array}\right]<0,\end{array}$$

(31)

where

$$\begin{array}{cc}\hfill & {\breve{\Pi }}_1=\left[\begin{array}{ccccccc}{\breve{\Pi }}_{11}& *& *& *& *& *& *\\ {\breve{\Pi }}_{21}& {\breve{\Pi }}_{22}& *& *& *& *& *\\ 0& \breve{R}& -\breve{Q}-\breve{R}& *& *& *& *\\ {\breve{\Pi }}_{41}& {\breve{\Pi }}_{42}& 0& {\breve{\Pi }}_{44}& *& *& *\\ {\breve{\Pi }}_{51}& {\breve{\Pi }}_{52}& 0& {\breve{\Pi }}_{54}& {\breve{\Pi }}_{55}& *& *\\ {\breve{\Pi }}_{61}& {\breve{\Pi }}_{62}& 0& {\breve{\Pi }}_{64}& {\breve{\Pi }}_{65}& -I_{nN}& *\\ I_{2nN}& 0& 0& 0& 0& 0& -I_{2nN}\end{array}\right],\hfill \\ \hfill & {\breve{\Pi }}_{11}=I_N\otimes Y{A}^T+I_N\otimes AY+\breve{Q}-\breve{R},{\breve{\Pi }}_{21}=-{(H\otimes BW)}^T+\breve{R},\hfill \\ \hfill & {\breve{\Pi }}_{41}=-{(H\otimes BW)}^T,{\breve{\Pi }}_{51}={(C\otimes BW)}^T,{\breve{\Pi }}_{61}=-I_N\otimes B^T,\hfill \\ \hfill & {\breve{\Pi }}_{22}=-2\breve{R}+M{H}^{T}H\otimes \breve{\Xi },{\breve{\Pi }}_{42}=M{H}^{T}H\otimes \breve{\Xi },{\breve{\Pi }}_{52}=-M{C}^{T}H\otimes \breve{\Xi },{\breve{\Pi }}_{62}=-H\otimes W,\hfill \\ \hfill & {\breve{\Pi }}_{44}=M{H}^{T}H\otimes \breve{\Xi }-\Theta \otimes \breve{\Xi },{\breve{\Pi }}_{54}=-M{C}^{T}H\otimes \breve{\Xi },{\breve{\Pi }}_{64}=-H\otimes W,\hfill \\ \hfill & {\breve{\Pi }}_{55}=M{C}^{T}C\otimes \breve{\Xi }-I_N\otimes Y,{\breve{\Pi }}_{65}=C\otimes W,\hfill \\ \hfill & {\breve{\Pi }}_2=[d_M{I}_N\otimes AY-d_{M}H\otimes BW0-d_{M}H\otimes BW{d}_{M}C\otimes BW-d_M{I}_N\otimes B{d}_M{I}_{2nN}],\hfill \\ \hfill & {\breve{\Pi }}_3=[\beta I_N\otimes Y000000],M=dia{g}_N\left\{{\alpha }_i\right\},\hfill \\ \hfill & {\breve{\Pi }}_4=\left[\begin{array}{ccccccc}{I}_N\otimes EY& 0& 0& 0& 0& 0& 0\end{array}\right],\Theta =dia{g}_N\left\{{\alpha }_i{\varpi }_i\right\}.\hfill \end{array}$$

Furthermore, the desired controller gain can be computed as $K=W{Y}^{-1}$, and $\Xi =Y^{-1}\breve{\Xi }{Y}^{-1}$.

Proof.

Define $Y=P^{-1}$, $\mathcal{J}=diag\{I_N\otimes Y,I_N\otimes Y,I_N\otimes Y,I_N\otimes Y,I_{2nN},I_{2nN},I_{2nN},I_{2nN},I_{nN}\}$, and $\breve{Q}=YQY,\breve{R}=YRY$,$\breve{\Xi }=Y\Phi Y$,$W=KY$.

From [44], we utilize the following inequality to handle the nonlinear item $-R^{-1}$; it is true that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill -{\breve{R}}^{-1}& \le {\nu }^2\breve{R}-2\nu (I_N\otimes Y).\hfill \end{array}\end{array}$$

(32)

One can conclude that (31) is sufficient condition to guarantee (21) holds and pre- and post-multiplying (21) with $\mathcal{J}$ and its transpose. In addition, the leader-following consensus is achieved under deception attacks.

According to the analysis above, the following Algorithm 1 is given to present the method of designing leader-following consensus control and AETS for the UAV system subject to actuator saturation and deception attacks.   □

Algorithm 1: Adaptive event-triggered control algorithm.

REQUIRE    Given values of the triggering parameters in (12) and $\mu$. ENSURE    for  $t\in [t_0,t_{end}]$  do Step 1: Compute feasible solutions of $\Xi$ in (12) and K in (15) according to Theorem 2; Step 2: The sensors of UAV group sample data ${\delta }_i\left(t\right)$. Step 3: if triggering condition holds, go to Step 4;             otherwise, go to Step 5. Step 4: Drop the current data; Controller holds the last event-triggered instant (ETI) $x_i\left(t_k^i\right)$; Go to Step 3. Step 5: use a buffer to stack the released data and update the ZOH.            compute ${\mathit{e}}_i(t_k^{i}h+lh),{\chi }_i\left(t_k^{i}h\right),{\kappa }_i\left(t\right)$, update the ZOH.             end if Step 6: Update the state ${\delta }_i\left(t\right)$ at ETI. Step 7: Compute the consensus control $u_i\left(t\right)$ with (15). Step 8: Go to Step 2. end for

In particular, when setting ${\alpha }_i=0$, the proposed AETS reduces to the conventional ETS, as stated in Remark 2.

Corollary 1.

Consider the leader-following multi-UAV system (4); for the given positive constants $\beta ,d_M$ and triggering parameter ${\overline{\varpi }}_i$, the consensus of tracking error systems (14) can be achieved if there exist matrices $\breve{Q}>0,\breve{R}>0,\breve{\Xi }>0$, such that

$$\begin{array}{c}\hfill \overline{\Pi }=\left[\begin{array}{ccc}{\overline{\Pi }}_1& *& *\\ {\overline{\Pi }}_2& {\nu }^2\breve{R}-2\nu (I_N\otimes Y)& *\\ {\overline{\Pi }}_3& 0& -I_{2nN}& *\\ {\overline{\Pi }}_4& 0& 0& -I_N\otimes Y\end{array}\right]<0,\end{array}$$

(33)

where

$$\begin{array}{cc}\hfill & {\overline{\Pi }}_1=\left[\begin{array}{ccccccc}{\overline{\Pi }}_{11}& *& *& *& *& *& *\\ {\overline{\Pi }}_{21}& {\overline{\Pi }}_{22}& *& *& *& *& *\\ 0& \breve{R}& -\breve{Q}-\breve{R}& *& *& *& *\\ {\overline{\Pi }}_{41}& {\overline{\Pi }}_{42}& 0& {\overline{\Pi }}_{44}& *& *& *\\ {\overline{\Pi }}_{51}& {\overline{\Pi }}_{52}& 0& {\overline{\Pi }}_{54}& {\overline{\Pi }}_{55}& *& *\\ {\overline{\Pi }}_{61}& {\overline{\Pi }}_{62}& 0& {\overline{\Pi }}_{64}& {\overline{\Pi }}_{65}& -I_{nN}& *\\ I_{2nN}& 0& 0& 0& 0& 0& -I_{2nN}\end{array}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\overline{\Pi }}_{11}=I_N\otimes Y{A}^T+I_N\otimes AY+\breve{Q}-\breve{R},{\overline{\Pi }}_{21}=-{(H\otimes BW)}^T+\breve{R},\hfill \\ \hfill & {\overline{\Pi }}_{41}=-{(H\otimes BW)}^T,{\overline{\Pi }}_{51}={(C\otimes BW)}^T,{\overline{\Pi }}_{61}=-I_N\otimes B^T,\hfill \\ \hfill & {\overline{\Pi }}_{22}=-2\breve{R}+H^T\Gamma H\otimes \breve{\Xi },{\overline{\Pi }}_{42}=H^T\Gamma H\otimes \breve{\Xi },{\overline{\Pi }}_{52}=-C^T\Gamma H\otimes \breve{\Xi },{\overline{\Pi }}_{62}=-H\otimes W,\hfill \\ \hfill & {\overline{\Pi }}_{44}=H^T\Gamma H\otimes \breve{\Xi }-I_N\otimes \breve{\Xi },{\overline{\Pi }}_{54}=-C^T\Gamma H\otimes \breve{\Xi },{\overline{\Pi }}_{64}=-H\otimes W,\hfill \\ \hfill & {\overline{\Pi }}_{55}=M{C}^T\Gamma C\otimes \breve{\Xi }-I_N\otimes Y,{\overline{\Pi }}_{65}=C\otimes W,\hfill \\ \hfill & {\overline{\Pi }}_2=[d_M{I}_N\otimes AY-d_{M}H\otimes BW0-d_{M}H\otimes BW{d}_{M}C\otimes BW-d_M{I}_N\otimes B{d}_M{I}_{2nN}],\hfill \\ \hfill & {\overline{\Pi }}_3=\left[\begin{array}{ccccccc}\beta I_N\otimes Y& 0& 0& 0& 0& 0& 0\end{array}\right],\hfill \\ \hfill & {\overline{\Pi }}_4=\left[\begin{array}{ccccccc}{I}_N\otimes EY& 0& 0& 0& 0& 0& 0\end{array}\right],\Gamma =dia{g}_N\left\{{\overline{\varpi }}_i\right\}.\hfill \end{array}$$

The designed controller gain can be obtained as $K=W{Y}^{-1}$, and $\Xi =Y^{-1}\breve{\Xi }{Y}^{-1}$.

Proof.

Choose the following Lyapunov functional:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill V\left(t\right)={\xi }^T\left(t\right)(I_N\otimes P)\xi \left(t\right)+{\int }_{t-d_M}^t{\xi }^T\left(s\right)Q\xi \left(s\right)ds+d_M{\int }_{-d_M}^0{\int }_{t+\alpha }^t{\dot{\xi }}^T\left(s\right)R\dot{\xi }\left(s\right)dsd\alpha ,\end{array}\end{array}$$

and along the proving route of Theorem 1, the results can be obtained. For brevity, it is omitted here. □

4. An Illustrative Example

A simulation example with two cases is introduced to testify the validity of the proposed consensus control method under AETS in this section.

Assuming that the UAV group flies at a fixed altitude, the tracking flight can be approximated as a two-dimensional plane motion.

The directed topology graph of the communication network is depicted in Figure 1, in which the leader UAV is indicated by 0, while 1–4 represent follower UAVs.

The corresponding Laplacian matrix L and leader adjacency matrix $\mathcal{B}$ on the basis of Figure 1 can be acquired as:

$$\begin{array}{c}\hfill L=\left[\begin{array}{cccc}0& 0& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 1& -1\\ -1& 0& 0& 1\end{array}\right],\mathcal{B}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right],E=diag\{0.02,0.01,0.02,0.01\}.\end{array}$$

Suppose the uncertain disturbance functions are $f\left({\delta }_i\left(t\right)\right)\left(t\right)=0.01sin\left({\delta }_i\left(t\right)\right),f\left({\delta }_0\left(t\right)\right)\left(t\right)$$=0.01sin\left({\delta }_0\left(t\right)\right)$.

Case 1: By using Theorem 2, we can obtain the triggering matrix $\Xi$ and controller gain K as:

$\Xi =\left[\begin{array}{cccc}-1.9234& 0& -0.6403& 0\\ 0& -1.9661& 0& -0.733\\ -0.6403& -1.4923& 0\\ 0& -0.733& 0& -1.7276\end{array}\right],$$K=\left[\begin{array}{cccc}0.3783& 0& 1.1729& 0\\ 0& 0.4095& 0& 1.2777\end{array}\right]$.

Under the initial states ${\delta }_0\left(t\right)={\left[1.22.63.22.6\right]}^T,{\delta }_1\left(t\right)={\left[1.5121.8\right]}^T,{\delta }_2\left(t\right)={\left[0.7143\right]}^T,{\delta }_3\left(t\right)={\left[2.31.51.62\right]}^T,{\delta }_4\left(t\right)={\left[4.322.23\right]}^T$, sampling period h = 0.1 s, and triggering parameters ${\alpha }_i=1,{\varpi }_1=10,{\varpi }_2=15,{\varpi }_3=12,{\varpi }_4=10$, we can obtain the following responses, which are presented in Figure 2.

Figure 2 depicts the tracking error of all follower UAVs and the leader, where one can see that the motion of the leader and follower UAVs are finally identical from 16.3 s, that is to say, the leader-following consensus of the UAV system is achieved under the proposed control law.

Case 2: If we set ${\alpha }_i=0,i=1,2,\cdots ,N$, the proposed AETS will reduce into the conventional ETS for the leader-following consensus of UAV systems.

The initial conditions in this case are chosen as the same case 1, and the triggering parameters are selected as ${\overline{\varpi }}_1=0.05,{\overline{\varpi }}_2=0.007,{\overline{\varpi }}_3=0.035,{\overline{\varpi }}_4=0.003$. The controller gain K and triggering matrix $\Xi$ according to Corollary 1 are derived as follows:

$$\begin{array}{c}\hfill \Xi =\left[\begin{array}{cccc}0.8238& -0.0067& 0.0725& 0.0098\\ -0.0067& 0.8739& 0.0893& 0.0742\\ 0.0725& 0.0893& 1.0466& 0.0012\\ 0.0098& 0.0742& 0.0012& 1.1461\end{array}\right],K=\left[\begin{array}{cccc}0.1500& 0.0078& 0.5907& 0.0103\\ 0.0065& 0.1900& -0.0038& 0.6666\end{array}\right].\end{array}$$

The tracking errors of the displacement and velocity for all UAVs under conventional ETS are shown in Figure 3. It is not difficult to see that although the system finally achieves consensus, the tracking speed is slower than that in Case 1. For the purpose of testifying the effectiveness of the designed method in data transmission, we choose $\overline{\rho }=0.03$ and sampling period $h=0.1$ s. The number of data sampling (NDS) for each UAV within 25 s is 250, and the results of number of packet-releasing (NPR) and data-releasing rate (DRR=NPR/NDS) are given in

Table 1. Comparison of triggering numbers under different schemes.

UAV	Conventional ETS		Our Method
	NPR	NDR	NPR	NDR
1	29	11.6%	20	8%
2	31	12.4%	25	10%
3	23	9.2%	21	8.4%
4	23	9.2%	20	8%

.

From Table 1, it is clear that the DRR of the proposed AETS is less than the one of conventional ETS. According to the above results, the designed scheme can not only ensure the performance of the system, but it also effectively saves communication resources. The simulation results verify the applicability of the proposed method.

5. Conclusions

In this paper, we have investigated the AETS-based leader-following consensus for multi-UAV systems subject to actuator saturation and deception attacks. By applying a novel AETS, the data-releasing rate can be greatly reduced, while the efficiency of data releasing is enhanced during the systems under deception attacks with actuator saturation. With the aid of improved Lyapunov–Krasovskii technique, some sufficient conditions have been proposed to ensure the leader-following secure consensus of the UAV systems subject to deception attacks. In addition, extending the proposed methods to a cooperative output regulation problem for more complicated dynamical systems under switching topology is the major study orientation of our future work.