Decentralized Adaptive Event-Triggered Fault-Tolerant Cooperative Control of Multiple Unmanned Aerial Vehicles and Unmanned Ground Vehicles with Prescribed Performance under Denial-of-Service Attacks

Abstract

This paper proposes a decentralized adaptive event-triggered fault-tolerant cooperative control (ET-FTCC) scheme for multiple unmanned aerial vehicles (UAVs) and unmanned ground vehicles (UGVs) with actuator faults and external disturbances under denial-of-service (DoS) attacks. The multiple UAVs and UGVs have a larger search radius, which is important in both the civilian and military domains. The different dynamics between UAVs and UGVs result in unbalanced interactions in the communication topologies, which increases the complexity of cooperative control. DoS attacks are conducted in both sensor and control channels. The dynamic models of UAVs and UGVs are introduced firstly, and the unified heterogeneous multiagent system model with actuator faults is established. The composite observer is designed to obtain the information of state and lumped disturbance, which is used to design the controller. In order to save the limited communication network resources, the event-triggered mechanism is introduced. The transformed error is presented by using the prescribed performance function (PPF). Then, the sliding-mode manifold is presented by combining the event-triggered control scheme to achieve the tracking purpose with actuator faults, external disturbances, and DoS attacks. Based on the Lyapunov function approach, the tracking errors are bounded within the prescribed boundary. Finally, the effectiveness of the proposed method is verified by qualitative analysis and quantitative analysis of the simulation results. This study can enhance the security and reliability of heterogeneous multiagent systems, providing technical support for the safe operation of unmanned systems. This paper mainly solves the FTCC problem of second-order nonlinear heterogeneous multiagent systems, and further research is needed for the FTCC problem of higher-order nonlinear heterogeneous multi-agent systems. In addition, the system may encounter multiple cyber attacks. As one of the future research works, we can extend the results of this paper to high-order nonlinear systems under multiple cyber attacks, which contain DoS attacks and deception attacks, and achieve fault-tolerant cooperative control of heterogeneous multiagent systems.

1. Introduction

Heterogeneous multiagent systems (HMASs) are widely employed in many fields, such as area searching, spacecraft formation flying, and so on [1,2,3]. An HMAS consists of different types of agents that offer more advantages than the homogeneous MASs. For example, multiple UAVs and UGVs have a larger search radius, which is important in both the civilian and military domains. However, due to the heterogeneous characteristics such as structure, and state dimensions between UAVs and UGVs, the execution of air ground formation tasks is hindered. Due to the heterogeneous characteristics of UAVs and UGVs, as well as the influence of parameter uncertainties, air–ground heterogeneous tracking control faces several challenges: UAVs and UGVs differ notably in structure, state dimensions, and model parameters; faults in the physical layer and DoS attacks in the cyber layer of multiple UAVs and UGVs may affect tracking performance. Recently, various cooperative control issues have been investigated for MASs to achieve consensus, formation, or containment control [4,5,6]. But, the existing cooperative control strategies illustrated may not be directly scalable to the cooperative control of HMASs, since there are unexpected actuator faults and DoS attacks.

In practical applications, the actuator in the physical layer may encounter faults. Because the agent is connected through the communication topology, faults will spread through the network, and may result in critical destruction of the system. To ensure the systems’ safe operation, fault-tolerant control (FTC) is an efficient method, and has drawn much attention in safety-critical systems [7,8,9]. In [10], the fault-tolerant tracking control based on distributed observer for MASs is investigated under directed topology. In [11], the FTC protocol utilizing local state information of neighbor agents is presented for MASs subjected to actuator faults. In [12], the FTC scheme by using distributed observer and auxiliary controller gain is investigated for MASs subjected to actuator faults. In [13], the distributed FTC strategy is presented for nonlinear MASs with external disturbances under switched topologies. Nevertheless, in addition to faults in physical layer, attacks in cyber layer can also block the transmission of information, which can be very destructive. In [14], the distributed FTC scheme is investigated by utilizing fuzzy logic systems method and switching mechanism for MASs with node fault and DoS attacks. In [15], the distributed FTC protocol based on resilient observer for HMASs is illustrated under DoS attacks. In [16], the distributed output-feedback control strategy is presented for linear HMASs with sampling and random DoS attacks. In [17], the distributed consensus problem is studied for MASs subjected to DoS attacks under directed topology. In [18], the distributed formation control strategy is designed for MASs against DoS attacks. However, there are few results on FTCC of HMASs with actuator faults subjected to DoS attacks.

Transient performance is another important problem for tracking control of MASs; it is significant to take transient and steady-state performance into account in the post-fault phase. In [19], the adaptive FTC scheme which can ensure that the consensus errors satisfy the prescribed performance requirements is investigated for nonlinear MASs. In [20], the prescribed performance control strategy is presented for a vehicular platoon by utilizing a Gaussian error function and finite-time PPF. In [21], the finite-time consensus issue is illustrated for MASs within prescribed performance. In [22], the tracking control issue is presented for underwater vehicle with model uncertainties by using PPF and disturbance observer. In [23], the formation-containment control scheme is presented for HMASs with prescribed performance. Also considering the limited communication resource, the event-triggered mechanism is proposed to obtain better performance with limited communication resources. In [24], the distributed ET containment control scheme by using a disturbance observer, which does not require the continuous sharing of state information, is studied for linear MASs with DoS attack and external disturbances. In [25], the distributed ET consensus problem is presented for linear MASs under DoS jamming attacks. In [26], the distributed ET formation control scheme is investigated for nonlinear MASs against DoS attacks. In [27], the ET consensus issue is studied for linear MASs with frequency cyber attacks. Although several FTC strategies are investigated for MASs under DoS attacks, the adaptive ET-FTCC strategy for HMASs with actuator faults and prescribed performance against DoS attacks is a challenging problem.

Motivated by the above-mentioned analyses, this paper investigates the ET-FTCC scheme for HMASs with actuator faults subjected to DoS attacks. The observer is designed to obtain the unmeasurable states. The PPF is utilized to convert the tracking error. Utilizing the transformed error, the ET-FTCC with prescribed performance can be achieved.

In comparison with other relevant works, the main contributions are concluded as follows:

• In order to save the limited communication network resources, the novel ET mechanism is illustrated, in comparison with some existing results [28,29], in which only the ET mechanism in the control channel is considered. The event-triggered mechanism is also presented by using the triggered state directly, which may cause chattering of the controller once actuator faults or DoS attacks occur. In this work, the ET mechanism in the sensor and controller channel are both considered. The presented ET mechanism can balance the limited communication network resources and information utilization. Furthermore, the Zeno behavior is excluded.
• The composite observer is presented, in which the state is estimated and the lumped disturbances are reconstructed utilizing the estimated state and the sensor channel triggered outputs without using the disturbance upper bound. In this way, the unknown variables are estimated with less information. In contrast to some existing results [30,31], in which the upper bound information of disturbances should be known when designing the disturbance observer, the results of this work provide another method to estimate the disturbances.
• An observer-based ET-FTCC scheme, including FTC compensation, disturbances degradation, and DoS prevention components, is studied. To obtain good system performance, a modified PPF is presented to convert the tracking errors, thereby relaxing the requirement for the exact initial conditions. In contrast to the work in [32], in which the control input is blocked when the DoS attack is active, the data at the instant when the attacker is switched between asleep and active in this work is triggered and utilized as a compensation component.

The remainder of this paper is presented as follows. Some preliminaries and problem formulation are introduced in Section 2 and Section 3. Section 4 gives the main theoretical results. The simulation studies are provided in Section 5. Section 6 concludes the paper.

2. Preliminaries

2.1. Graph Theory

A graph is given to describe interaction topology of agents. The undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ consists of a node set $\mathcal{V}=\{1,2,\dots ,N\}$ and an edge set $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$. Let $\mathcal{A}=\left[a_{ij}\right]\in R^{N\times N}$ be adjacency matrix of $\mathcal{G}$, where $a_{ij}=1$ if $(j,i)\in \mathcal{E}$ and $a_{ij}=0$ otherwise. Let $\mathcal{D}=\mathrm{diag}\{d_1,d_2,\dots ,d_N\}$ be degree matrix with $d_i={\sum }_{j=1}^N{a}_{ij}$. The Laplacian matrix is defined as $\mathcal{L}=\left[l_{ij}\right]=\mathcal{D}-\mathcal{A}\in R^{N\times N}$.

Assumption 1.

The topology $\mathcal{G}$ is connected.

Lemma 1

([33]). If Assumption 1 is satisfied and $\mathcal{U}\in R^N$ is a nonzero non-negative vector, then $\mathcal{L}$ and $\mathcal{L}+\mathrm{diag}\left(\mathcal{U}\right)$ are both symmetric positive definite matrices.

2.2. Actuator Fault Model

The actuator fault model is presented as

$$u_{ih}^F\left(t\right)={\rho }_{ih}{u}_{ih}\left(t\right)+u_{ihb}\left(t\right),$$

(1)

where ${\rho }_{ih}$ denotes the unknown effectiveness factor, $u_{ihb}\left(t\right)$ denotes the bias, $u_{ih}\left(t\right)$ and $u_{ih}^F\left(t\right)$ denote the desired and actual control input signal.

From (1), one has

$$\begin{array}{ccc}\hfill u_i^F\left(t\right)& = & {\rho }_i{u}_i\left(t\right)+u_{ib}\left(t\right),\hfill \end{array}$$

(2)

with ${\rho }_i=\mathrm{diag}\{{\rho }_{i1},{\rho }_{i2},\dots ,{\rho }_{in}\}$, $u_{ib}\left(t\right)={[u_{ib1}\left(t\right),u_{ib2}\left(t\right),\dots ,u_{ibn}\left(t\right)]}^T$.

Assumption 2.

There exist a lower bound ${\underline{\rho }}_i$ and upper bound ${\overline{\rho }}_i$ satisfying $0<{\underline{\rho }}_i\le {\rho }_i\le {\overline{\rho }}_i\le 1$. $\parallel {\dot{\rho }}_i\parallel \le {\rho }_{ic}$ with ${\rho }_{ic}$ being a positive constant.

2.3. DoS Attack Model

DoS attacks can interfere with the communication channel and damage the communication equipment, causing data transmission to be interrupted. It can attack the single communication channel or attack measurement, communication, and control channels all at once. The DoS attack has great destructive power, but it needs to stop to recover energy, due to energy limitations. The attack’s purpose is to obliterate the transmitted data before reaching the destination, which is expressed as

$$\begin{array}{c}\hfill {\mathcal{S}}_{DoS}=\left\{\begin{array}{c}0,t\in {\Xi }_{n,1}:=[p_{dn},p_{dn}+l_{dn}),\hfill \\ 1,t\in {\Xi }_{n,2}:=[p_{dn}+l_{dn},p_{dn+1}),\hfill \end{array}\right.\end{array}$$

(3)

where $p_{dn}$ and $l_{dn}$ represent the instant and the length of the n-th attack, respectively. ${\Xi }_{n,1}$ and ${\Xi }_{n,2}$ represent the sleeping period and active period of the DoS attack, respectively.

Assumption 3.

Let $n_a({\tau }_d,t)$ be the number of attacks on $[{\tau }_d,t)$; there exist ${\xi }_d\ge 0$ and ${\tau }_S>0$ satisfying

$$n_a({\tau }_d,t)\le {\xi }_d+{\displaystyle \frac{t-{\tau }_d}{{\tau }_S}}.$$

Assumption 4.

Let $|\Xi (\tau ,t\left)\right|$ be the length of $\Xi ({\tau }_d,t)$; there exist $o_d\ge 0$ and $T>1$ satisfying

$$|\Xi (\tau ,t)|\le o_d+{\displaystyle \frac{t-{\tau }_d}{T}}.$$

3. Problem Formulation

In this section, $M_1$ quadrotor UAVs and $M_2$ two-wheeled mobile robot UGVs are considered. For convenience, let ${\Pi }_1=\{1,2,\dots ,M_1\}$, ${\Pi }_2=\{M_1+1,M_1+2,\dots ,M_1+M_2\}$, and $\Pi ={\Pi }_1\bigcup {\Pi }_2$.

3.1. UAV Dynamics

The structure of the quadrotor UAV is shown in Figure 1. In order to describe the dynamic characteristic of the quadrotor UAV, the inertia frame and the body-fixed frame are employed. The axes of the inertia frame are denoted as $(x_e,y_e,z_e)$, and the axes of the body-fixed frame are denoted as $(x_b,y_b,z_b)$. The translational dynamics of the i-th ($i\in {\Pi }_1$) UAV are described as [34]

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_{ai}{\ddot{p}}_{aix}=(cos{\varphi }_{ai}sin{\theta }_{ai}cos{\psi }_{ai}+sin{\varphi }_{ai}sin{\psi }_{ai})u_{pi}-d_{ix}{\dot{p}}_{aix},\hfill \\ m_{ai}{\ddot{p}}_{aiy}=(cos{\varphi }_{ai}sin{\theta }_{ai}sin{\psi }_{ai}+sin{\varphi }_{ai}cos{\psi }_{ai})u_{pi}-d_{iy}{\dot{p}}_{aiy},\hfill \\ m_{ai}{\ddot{p}}_{aiz}=(cos{\theta }_{ai}cos{\varphi }_{ai})u_{pi}-m_{ai}g-d_{iz}{\dot{p}}_{aiz},\hfill \end{array}\right.\end{array}$$

(4)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{J}_{ax}{\ddot{\varphi }}_{ai}={\dot{\theta }}_{ai}{\dot{\psi }}_{ai}(J_{ay}-J_{az})-J_{ar}{\dot{\theta }}_{ai}{\overline{d}}_{ai}+{\tau }_{\varphi i}-d_{i\varphi }{\dot{\varphi }}_{ai},\hfill \\ J_{ay}{\ddot{\theta }}_{ai}={\dot{\varphi }}_{ai}{\dot{\psi }}_{ai}(J_{az}-J_{ax})-J_{ar}{\dot{\varphi }}_{ai}{\overline{d}}_{ai}+{\tau }_{\theta i}-d_{i\theta }{\dot{\theta }}_{ai},\hfill \\ J_{az}{\ddot{\psi }}_{ai}={\dot{\varphi }}_{ai}{\dot{\theta }}_{ai}(J_{ax}-J_{ay})+{\tau }_{\psi i}-d_{i\psi }{\dot{\psi }}_{ai},\hfill \end{array}\right.\end{array}$$

(5)

where ${[p_{aix},p_{aiy},p_{aiz}]}^T$ and ${[{\varphi }_{ai},{\theta }_{ai},{\psi }_{ai}]}^T$ denote the position and attitude states, $d_{ix}$, $d_{iy}$, and $d_{iz}$ denote the aerodynamic damping coefficients, $m_{ai}$, $u_{pi}$, and g denote the mass, control thrust, and gravitational acceleration, ${\tau }_{\varphi i}$, ${\tau }_{\theta i}$, and ${\tau }_{\psi i}$ denote the three control torques, $J_{ax}$, $J_{ay}$, and $J_{az}$ denote the moments of the inertia, $J_{ar}$ denotes the moment of rotor’s inertia, and ${\overline{d}}_{ai}$ is the overall residual rotor angle.

Consider a new control signal $u_{ai}$; (4) can be rewritten as

$${\ddot{p}}_{ai}=u_{ai}+f_{ai},$$

(6)

where $f_{ai}={[-d_{ix}{\dot{p}}_{aix}/m_{ai},-d_{iy}{\dot{p}}_{aiy}/m_{ai},-d_{iz}{\dot{p}}_{aiz}/m_{ai}-g]}^T$, $p_{ai}={[p_{aix},p_{aiy},p_{aiz}]}^T$, $u_{ai}={[u_{aix},u_{aiy},u_{aiz}]}^T$ with $u_{aix}=(cos{\varphi }_{ai}sin{\theta }_{ai}cos{\psi }_{ai}+sin{\varphi }_{ai}sin{\psi }_{ai})u_{pi}/m_{ai}$, $u_{aiy}=(cos{\varphi }_{ai}sin{\theta }_{ai}sin{\psi }_{ai}-sin{\varphi }_{ai}cos{\psi }_{ai})u_{pi}/m_{ai}$, $u_{aiz}=(cos{\varphi }_{ai}cos{\theta }_{ai})u_{pi}/m_{ai}-g$.

Remark 1.

The six-degree-of-freedom quadrotor dynamic systems can be controlled by utilizing attitude subsystems and trajectory subsystems control framework. The trajectory subsystems, which are capable of guiding the quadrotor UAV to an expected position, are more significant in the UAV tracking control design than the attitude subsystems [35]. Therefore, the adaptive event-triggered fault-tolerant cooperative control scheme studied in this paper considers the trajectory systems.

3.2. UGV Dynamics

The two-wheeled mobile robot UGV is driven by two wheels at the rear, with a supporting universal wheel at the front to maintain stability. The structure of the UGV is shown in Figure 2. The dynamic model of the i-th ($i\in {\Pi }_2$) two-wheeled mobile robot UGV is described as [36]

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{x}}_{gi}\hfill & =v_{gi}cos{\theta }_{gi},\hfill \\ {\dot{y}}_{gi}\hfill & =v_{gi}sin{\theta }_{gi},\hfill \\ {\dot{\theta }}_{gi}\hfill & ={\omega }_{gi},\hfill \end{array}\right.\end{array}$$

(7)

where $x_i={[x_{gi},y_{gi}]}^T$ denotes the position states, ${\theta }_{gi}$ denotes the orientation, $v_{gi}$ and ${\omega }_{gi}$ are the linear and angular velocity, respectively, ${\dot{v}}_{gi}=F_{gi}/M_{gi}$, ${\dot{\omega }}_{gi}={\tau }_{gi}/J_{gi}$, $M_{gi}$ and $J_{gi}$ are the mass and moment of the inertia, and $F_{gi}$ and ${\tau }_{gi}$ are the force and torque.

Due to the non-holonomic constraint, a hand position approach is utilized. The front point $(p_{gix},p_{giy})$ of the UGVs is defined as the hand point. The coordinate transformation is given as $p_{gix}=x_{gi}+L_{gi}cos{\theta }_{gi}$ and $p_{giy}=y_{gi}+L_{gi}sin{\theta }_{gi}$, where $L_{gi}$ is the distance between the position $(x_{gi},y_{gi})$ and the new defined hand point $(p_{gix},p_{giy})$. Then, the system (7) is rewritten

$${\ddot{p}}_{gi}=u_{gi}+f_{gi},$$

(8)

where

$$\begin{array}{ccc}\hfill u_{gi}& = & \left[\begin{array}{cc}cos{\theta }_{gi}& -L_{gi}sin{\theta }_{gi}\\ sin{\theta }_{gi}& L_{gi}cos{\theta }_{gi}\end{array}\right]\left[\begin{array}{c}{\dot{v}}_{gi}\\ {\dot{\omega }}_{gi}\end{array}\right],\hfill \\ \hfill f_{gi}& = & \left[\begin{array}{c}-v_i{\omega }_{gi}sin{\theta }_{gi}-L_{gi}{\omega }_{gi}^{2}cos{\theta }_{gi}\\ v_i{\omega }_{gi}cos{\theta }_{gi}-L_{gi}{\omega }_{gi}^{2}sin{\theta }_{gi}\end{array}\right].\hfill \end{array}$$

3.3. Model Transformation

In comparison with MASs, the difficulty of cooperative control of HMASs is that the dynamic characteristics and even the dimensions differ between the agents. To solve this control problem, the unified control model with actuator faults model (2) and external disturbances $d_i$ is written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_{i1}=x_{i2},\hfill \\ {\dot{x}}_{i2}={\rho }_i{u}_i+u_{ib}+d_i+f_i,\hfill \end{array}\right.\end{array}$$

(9)

where $x_{i1}={[p_{aix},p_{aiy},p_{aiz}]}^T$ denotes the position of the i-th UAV, ${\rho }_i=\mathrm{diag}\{{\rho }_{aix},{\rho }_{aiy},{\rho }_{aiz}\}$ denotes the effectiveness factor, $u_i={[u_{aix},u_{aiy},u_{aiz}]}^T$ denotes the virtual control input signal, $f_i=f_{ai}$ denotes the nonlinear term, $u_{ib}={[u_{aibx},u_{aiby},u_{aibz}]}^T$ denotes the bias, $d_i={[d_{aix},d_{aiy},d_{aiz}]}^T$ denotes the external disturbance ($i\in {\Pi }_1$), $x_{i1}={[p_{gix},p_{giy},0]}^T$ denotes the position of the i-th UGV, ${\rho }_i=\mathrm{diag}\{{\rho }_{gix},{\rho }_{giy},0\}$ denotes the effectiveness factor, $u_i={[u_{gix},u_{giy},0]}^T$ denotes the virtual control input signals, $f_i=f_{gi}$ denotes the nonlinear term, $u_{ib}={[u_{gibx},u_{giby},0]}^T$ denotes the bias, and $d_i={[d_{gix},d_{giy},0]}^T$ denotes the external disturbance, $i\in {\Pi }_2$.

Hence, (9) can be rewritten as

$$\begin{array}{c}\hfill {\dot{x}}_i=A_i{x}_i\left(t\right)+B_i{\rho }_i{u}_i\left(t\right)+R_i\left(t\right),\end{array}$$

(10)

where $A_i=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $B_i={[0,1]}^T$, $R_i\left(t\right)=B_i{d}_i+B_i{u}_{ib}+B_i{f}_i$.

Remark 2.

Since the dynamics of UAVs and UGVs are three dimensional and two dimensional, respectively, it is hard to design collaborative control scheme straightforwardly. Hence, the dimension of the UGV is upgraded, i.e., the position, velocity, and control input on Z-axis are zero. In addition, the unified control model (9) is transformed from the actual physical models of quadrotor UAVs and two-wheeled mobile robot UGVs; therefore, the complexity of UAVs and UGVs can be reflected. Consequently, the unified control model (9) accommodates both UAVs and UGVs.

3.4. Control Objective

The control objective is to present ET-FTCC scheme for HMASs (9) subjected to actuator faults and disturbances against DoS attacks, such that the tracking errors can be restrained to the prescribed bounds.

4. Main Results

In this section, the ET-FTCC scheme is designed for HMASs subjected actuator faults and disturbances against DoS attacks. Figure 3 shows the diagram.

4.1. Event-Triggered Mechanism Design

To save the limited communication resource, the ET mechanism in the sensor and controller channel are both considered. For the sensor channel, the ET time instants are designed as

$$t_{k+1}^{i}h=\inf \{qh>t_k^{i}h:{\mathcal{F}}_i\left(qh\right)>0\},$$

(11)

where ${\mathcal{F}}_i\left(qh\right)=e_{zi}{\left(t_k^{i}h\right)}^T{\mathcal{Q}}_i{e}_{zi}\left(t_k^{i}h\right)-\sigma {ϵ}_i{(t_k^{i}h+qh)}^T{\mathcal{Q}}_i{ϵ}_i(t_k^{i}h+qh)$, with $\sigma \in [0,1)$, ${\mathcal{Q}}_i>0$, $q\in N$, $e_{zi}\left(t_k^{i}h\right)=y_i\left(t_k^{i}h\right)-y_i(t_k^{i}h+qh)$, ${ϵ}_i(t_k^{i}h+qh)={\sum }_{j\in N_i}{a}_{ij}[y_i(t_k^{i}h+qh)-y_j(t_k^{j}h+qh)]$.

The DoS attack can interfere with communication channel, which indicates that the k-th triggered data may not be delivered to the target when the DoS attack is active. Figure 4 shows an example of the ET mechanism and sequence of DoS attacks. In Figure 4, the n-th DoS attack period ${\Xi }_n$ includes the sleeping period ${\Xi }_{n,1}$ and the active period ${\Xi }_{n,2}$. To ensure the performance, the reasonable assumption is made that the packets at the end of each attack period $p_{dn}$ should be released regardless of whether it satisfies the condition (11) in the ET mechanism. The specific meaning of each kind of packet is shown in Figure 4. For example, at instant $10h$, the packet does not satisfy the condition (11), but it is necessary to improve the performance. Hence, the resilient ET mechanism is given as

$$t_{k,n+1}^{i}h\in \left\{t_s^{i}h\mathrm{satisfying}\left(11\right)\right|t_s^{i}h\in {\Xi }_{n,1}\}\cup \left\{p_{dn}\right\},$$

(12)

where $t_{k,n+1}^{i}h$ denotes the instant when the data are transmitted to sensor. $k\in \{0,1,\dots ,k(n\left)\right\}$ with $k\left(n\right)=sup\{k\in N|t_{k,n+1}^{i}h<p_{dn}+l_{dn}\}$.

Remark 3.

According to Equation (11), the next transferred data will be triggered if the condition is met. Since the DoS attacks can block the communication, the triggered data should not be within the active period of the DoS attacks. Hence, the resilient ET mechanism (12) is proposed.

4.2. Observer Design

The state of the HMASs is treated as not fully measurable; the observer is presented for (10) at $t\in [t_k^{i}h,t_{k+1}^{i}h)$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{\widehat{x}}}_i\left(t\right)\hfill & =A_i{\widehat{x}}_i\left(t\right)+B_i{\widehat{\rho }}_i{u}_i\left(t\right)+L_i(y_i\left(t_{k,n+1}^{i}h\right)-{\widehat{y}}_i\left(t\right))+{\Delta }_i\left(t\right),\hfill \\ {\widehat{y}}_i\left(t\right)\hfill & =C_i{\widehat{x}}_i\left(t\right),\hfill \end{array}\right.\end{array}$$

(13)

where ${\widehat{x}}_i\left(t\right)$, ${\widehat{\rho }}_i$, ${\widehat{\theta }}_i$, and ${\widehat{y}}_i\left(t\right)$ denote the estimations of $x_i\left(t\right)$, ${\rho }_i$, ${\theta }_i^{*}$, and $y_i\left(t\right)$, respectively. $L_i$ denotes observer gain. ${\Delta }_i\left(t\right)$ is an estimation error compensator, which is given as

$$\begin{array}{ccc}\hfill {\Delta }_i\left(t\right)& = & {\displaystyle \frac{P_i{e}_{xi}\left(t\right)}{{∥e_{xi}{\left(t\right)}^T{P}_i∥}^2}}\left({\widehat{\iota }}_i∥e_{xi}{\left(t\right)}^T{P}_i∥{\Psi }_i\left(t\right)-{\kappa }_1{\parallel {\Psi }_i\left(t\right)\parallel }^2\right.\hfill \\ & & \left.-∥e_{xi}{\left(t\right)}^T{P}_i{L}_{ci}{e}_{xi}\left(t\right)∥-∥e_{xi}{\left(t\right)}^T{P}_i{L}_i{e}_{yi}\left(t\right)∥+{\eta }_{ci}\right),\hfill \end{array}$$

(14)

where $e_{xi}\left(t\right)=x_i\left(t\right)-{\widehat{x}}_i\left(t\right)$, $e_{yi}\left(t\right)=y_i\left(t_{k,n+1}^{i}h\right)-y_i\left(t\right)$, ${\eta }_{ci}=({\overline{\rho }}_i-{\underline{\rho }}_i)({\overline{\rho }}_i-{\underline{\rho }}_i+{\rho }_{ic})$, $P_i$ is a positive symmetry matrix, $L_{ci}$ is a designed matrix, ${\widehat{\iota }}_i$ is the estimations of ${\iota }_i$, and ${\kappa }_1$ is a positive constant.

The adaptive law of ${\widehat{\rho }}_i$ is designed as

$${\dot{\widehat{\rho }}}_i={\mathrm{Proj}}_{[{\underline{\rho }}_i,{\overline{\rho }}_i]}\left\{\mathcal{P}\right\}=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{\widehat{\rho }}_i={\overline{\rho }}_i\mathrm{and}\mathcal{P}\ge 0\hfill \\ & \mathrm{or}{\widehat{\rho }}_i={\underline{\rho }}_i\mathrm{and}\mathcal{P}\le 0,\hfill \\ \mathcal{P},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(15)

where $\mathcal{P}=-B_i{u}_i{\left(t\right)}^T{P}_i{e}_{xi}\left(t\right)$.

Hence, the error system is obtained as

$$\begin{array}{ccc}\hfill {\dot{e}}_{xi}\left(t\right)& = & B_i{\tilde{\rho }}_i{u}_i\left(t\right)+(A_i-L_i{C}_i)e_{xi}\left(t\right)+R_i\left(t\right)-L_i{e}_{yi}\left(t\right)-{\Delta }_i\left(t\right)\hfill \end{array}$$

(16)

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

In this work, the lumped disturbance is assumed to meet $R_i\left(t\right)\le {\iota }_i\parallel {\Psi }_i\left(t\right)\parallel$, with ${\iota }_i$ being an unknown constant. The auxiliary vector is given as

$$\begin{array}{ccc}\hfill {\Psi }_i\left(t\right)& = & {\int }_0^t{\kappa }_2{e}_{xi}\left(t\right)dt+e_{xi}\left(t\right)-{\xi }_i\left(t\right),\hfill \end{array}$$

(17)

with ${\kappa }_2$ being a positive constant. ${\xi }_i\left(t\right)$ is an internal variable, which is given as

$$\begin{array}{ccc}\hfill {\dot{\xi }}_i\left(t\right)& = & B_i{\tilde{\rho }}_i{u}_i\left(t\right)+(A_i-L_i{C}_i)e_{xi}\left(t\right)+({\kappa }_1+{\widehat{\iota }}_i){\Psi }_i\left(t\right)\hfill \\ & & +{\kappa }_2{e}_{xi}\left(t\right)-L_i{e}_{yi}\left(t\right)-{\Delta }_i\left(t\right).\hfill \end{array}$$

(18)

The adaptive law ${\widehat{\iota }}_i$ is designed as

$$\begin{array}{ccc}\hfill {\dot{\widehat{\iota }}}_i& = & {\kappa }_1(\parallel {\Psi }_i{\left(t\right)\parallel }^2+\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel ).\hfill \end{array}$$

(19)

In addition, the lumped disturbances can be estimated as

$$\begin{array}{ccc}\hfill {\dot{\widehat{R}}}_i\left(t\right)& = & ({\kappa }_1+{\widehat{\iota }}_i){\Psi }_i\left(t\right).\hfill \end{array}$$

(20)

Theorem 1.

With the estimators (13) and (20), and the adaptive law (15), the estimation errors of the close-loop systems is guaranteed if the following condition holds:

$$\begin{array}{c}\hfill {(P\Gamma +P{L}_c)}^T+(P\Gamma +P{L}_c)<0,\end{array}$$

(21)

with $P=\mathrm{diag}\{P_1,P_2,\dots ,P_N\}$, $L_c=\mathrm{diag}\{L_{c1},L_{c2},\dots ,L_{cN}\}$, $\Gamma =[A_1-L_1{C}_1,A_2-L_2{C}_2,\dots ,A_N-L_N{C}_N]$.

Proof. 

Consider the Lyapunov function

$$\begin{array}{ccc}\hfill V_1\left(t\right)& = & \sum _{i=1}^N(e_{xi}^T\left(t\right)P_i{e}_{xi}\left(t\right)+{\Psi }_i\left(t\right){\Psi }_i\left(t\right)+{\displaystyle \frac{{\tilde{\iota }}_i^T{\tilde{\iota }}_i}{{\kappa }_1}}+{\tilde{\rho }}_i^T{\rho }_i),\hfill \end{array}$$

(22)

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

The derivative of (22) is presented as

$$\begin{array}{ccc}\hfill {\dot{V}}_1\left(t\right)& = & 2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i(B_i{\tilde{\rho }}_i{u}_i\left(t\right)+(A_i-L_i{C}_i)e_{xi}\left(t\right)-L_i{e}_{yi}\left(t\right)\hfill \\ & & +R_i\left(t\right)-{\Delta }_i\left(t\right))+2\sum _{i=1}^N({\Psi }_i\left(t\right){\dot{\Psi }}_i\left(t\right)+{\displaystyle \frac{{\tilde{\iota }}_i^T{\dot{\tilde{\iota }}}_i}{{\kappa }_1}}+{\tilde{\rho }}_i^T{\dot{\tilde{\rho }}}_i).\hfill \end{array}$$

(23)

From (15), one can deduce that

$$\begin{array}{cc}\hfill {2e_{xi}}^T& \left(t\right)P_i{B}_i{\tilde{\rho }}_i{u}_i\left(t\right)+2{\tilde{\rho }}_i^T{\dot{\tilde{\rho }}}_i\hfill \\ \hfill & =2e_{xi}^T\left(t\right)P_i{B}_i{\tilde{\rho }}_i{u}_i\left(t\right)+2{\tilde{\rho }}_i^T({\dot{\rho }}_i-{\dot{\widehat{\rho }}}_i)\hfill \\ \hfill & =2e_{xi}^T\left(t\right)P_i{B}_i{\tilde{\rho }}_i{u}_i\left(t\right)+2{\tilde{\rho }}_i^T({\dot{\rho }}_i-e_{xi}^T\left(t\right)P_i{B}_i{u}_i\left(t\right))\hfill \\ \hfill & \le 2{\tilde{\rho }}_i^T{\dot{\rho }}_i\hfill \\ \hfill & \le 2{\eta }_{ci},\hfill \end{array}$$

(24)

In terms of (20), one has

$$\begin{array}{cc}\hfill {2e_{xi}}^T& \left(t\right)P_i{R}_i\left(t\right)+2{\Psi }_i\left(t\right){\dot{\Psi }}_i\left(t\right)+2{\displaystyle \frac{{\tilde{\iota }}_i^T{\dot{\tilde{\iota }}}_i}{{\kappa }_1}}\hfill \\ \hfill & \le 2{\iota }_i\parallel e_{xi}^T\left(t\right)P_i\parallel {\Psi }_i\left(t\right)+2{\Psi }_i{\left(t\right)}^T({\kappa }_2{e}_{xi}\left(t\right)+{\dot{e}}_{xi}\left(t\right)-{\dot{\xi }}_i\left(t\right))+2{\displaystyle \frac{{\tilde{\iota }}_i^T{\dot{\tilde{\iota }}}_i}{{\kappa }_1}}\hfill \\ \hfill & \le 2{\iota }_i\parallel e_{xi}^T\left(t\right)P_i\parallel {\Psi }_i\left(t\right)+2{\Psi }_i{\left(t\right)}^T(R_i\left(t\right)-{\widehat{R}}_i\left(t\right))+2{\displaystyle \frac{{\tilde{\iota }}_i^T{\dot{\tilde{\iota }}}_i}{{\kappa }_1}}\hfill \\ \hfill & \le 2{\iota }_i\parallel e_{xi}^T\left(t\right)P_i\parallel {\Psi }_i\left(t\right)+2{\Psi }_i{\left(t\right)}^T({\iota }_i\parallel {\Psi }_i\left(t\right)\parallel -({\kappa }_1+{\widehat{\iota }}_i){\Psi }_i\left(t\right))\hfill \\ \hfill & -2{\tilde{\iota }}_i^T(\parallel {\Psi }_i{\left(t\right)\parallel }^2+\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel )\hfill \\ \hfill & =2{\iota }_i\parallel e_{xi}^T\left(t\right)P_i\parallel {\Psi }_i\left(t\right)-2{\tilde{\iota }}_i\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel -2{\kappa }_1{\parallel {\Psi }_i\left(t\right)\parallel }^2\hfill \\ \hfill & =-2{\kappa }_1\parallel {\Psi }_i{\left(t\right)\parallel }^2+2{\widehat{\iota }}_i\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel .\hfill \end{array}$$

(25)

Combing (24) and (25), one can obtain

$$\begin{array}{ccc}\hfill {\dot{V}}_1\left(t\right)& \le & 2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i((A_i-L_i{C}_i)e_{xi}\left(t\right)-L_i{e}_{yi}\left(t\right)-{\Delta }_i\left(t\right))\hfill \\ & & -2\sum _{i=1}^N(e_{xi}^T\left(t\right)P_i{L}_{ci}{e}_{xi}\left(t\right)-e_{xi}^T\left(t\right)P_i{L}_{ci}{e}_{xi}\left(t\right))\hfill \\ & & -2\sum _{i=1}^N{\kappa }_1\parallel {\Psi }_i{\left(t\right)\parallel }^2+2\sum _{i=1}^N{\widehat{\iota }}_i\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel \hfill \\ & & +2\sum _{i=1}^N{\eta }_{ci}\hfill \end{array}$$

(26)

Substituting (14) into (26), one has

$$\begin{array}{ccc}\hfill {\dot{V}}_1\left(t\right)& \le & -2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i{\displaystyle \frac{P_i{e}_{xi}\left(t\right)}{{∥e_{xi}{\left(t\right)}^T{P}_i∥}^2}}({\eta }_{ci}-{\kappa }_1{\parallel {\Psi }_i\left(t\right)\parallel }^2\hfill \\ & & +{\widehat{\iota }}_i∥e_{xi}{\left(t\right)}^T{P}_i∥{\Psi }_i\left(t\right)-∥e_{xi}{\left(t\right)}^T{P}_i{L}_{ci}{e}_{xi}\left(t\right)∥\hfill \\ & & -∥e_{xi}{\left(t\right)}^T{P}_i{L}_i{e}_{yi}\left(t\right)∥)-2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i{L}_i{e}_{yi}\left(t\right)\hfill \\ & & +2\sum _{i=1}^N({\eta }_{ci}-{\kappa }_1\parallel {\Psi }_i{\left(t\right)\parallel }^2+{\widehat{\iota }}_i\parallel e_{xi}^T\left(t\right)P_i\parallel \parallel {\Psi }_i\left(t\right)\parallel \hfill \\ & & -e_{xi}^T\left(t\right)P_i{L}_{ci}{e}_{xi}\left(t\right))+2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i(A_i-L_i{C}_i)e_{xi}\left(t\right)\hfill \\ & & +2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i{L}_{ci}{e}_{xi}\left(t\right)\hfill \\ \hfill & \le & 2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i(A_i-L_i{C}_i)e_{xi}\left(t\right)+2\sum _{i=1}^N{e}_{xi}^T\left(t\right)P_i{L}_{ci}{e}_{xi}\left(t\right)\hfill \\ \hfill & =& e^T({(P\Gamma +P{L}_c)}^T+(P\Gamma +P{L}_c))e.\hfill \end{array}$$

(27)

where $e_x={[e_{x1},e_{x2},\dots ,e_{xN}]}^T$.

According to (21), one can conclude that ${\dot{V}}_1\left(t\right)<0$. On ${\Xi }_{n,2}$, similar results can be obtained. Hence, the convergence of the estimation errors is guaranteed. This completes the proof. □

Remark 4.

In this work, the estimation of the lumped disturbance is based on the assumption $R_i\left(t\right)\le {\iota }_i\parallel {\Psi }_i\left(t\right)\parallel$, which is reasonable. The actuator fault and the external disturbance in the practical system are both bounded, and the auxiliary vector ${\Psi }_i\left(t\right)$ has the information of the lumped disturbance. Since the lumped disturbance is bounded, by introducing the unknown scalar ${\iota }_i$, the assumption can be satisfied.

4.3. FTCC Design

The tracking error is defined as

$$\begin{array}{ccc}\hfill s_i& = & {\chi }_1{z}_i+{\chi }_2\sum _{j\in N_i}{a}_{ij}(z_i-z_j),\hfill \end{array}$$

(28)

where $z_i=y_i\left(t_{k,n+1}^{i}h\right)-y_{id}\left(t\right)$, $y_{id}\left(t\right)$ is the desired reference, ${\chi }_1$ and ${\chi }_2$ are positive parameters.

Then, (28) can be rewritten as

$$\begin{array}{ccc}\hfill S& = & [({\chi }_1{I}_N+{\chi }_2\mathcal{L})\otimes I_3]Z_1,\hfill \end{array}$$

(29)

where $S={[s_1^T,s_2^T,\dots ,s_N^T]}^T$ and $Z_1={[z_1^T,z_2^T,\dots ,z_N^T]}^T$. In terms of Lemma 1, one has $Z_1\to 0$ as $S\to 0$.

The error constraints are defined as

$$-{\underline{k}}_i{\epsilon }_i\left(t\right)\le s_i\le {\overline{k}}_i{\epsilon }_i\left(t\right),$$

(30)

where ${\underline{k}}_i$ and ${\overline{k}}_i$ are positive constants.

The PPF is chosen as

$${\epsilon }_i\left(t\right)=coth({\beta }_{1}t+{\beta }_2)-1+{\epsilon }_{i\infty },$$

(31)

where ${\beta }_1$ and ${\beta }_2$ are positive constants.

In order to facilitate controller design, (30) can be transformed as

$$s_i={\epsilon }_i\left(t\right)\Lambda \left(e_{is}\right),$$

(32)

where $e_{is}$ is the transformed error, $\Lambda \left(e_{is}\right)={\displaystyle \frac{{\overline{k}}_i{e}^{e_{is}}-{\underline{k}}_i{e}^{-e_{is}}}{e^{e_{is}}-e^{-e_{is}}}}$.

Then, one can obtain

$$e_{is}={\displaystyle \frac{1}{2}}ln({\displaystyle \frac{{\underline{k}}_i+{\pi }_i}{{\overline{k}}_i}}-{\pi }_i),$$

(33)

where ${\pi }_i=s_i/{\epsilon }_i$.

The derivative of (33) is presented as

$$\begin{array}{ccc}\hfill {\dot{e}}_{is}& = & {\displaystyle \frac{1}{2}}({\displaystyle \frac{1}{{\pi }_i+{\underline{k}}_i}}-{\displaystyle \frac{1}{{\pi }_i-{\overline{k}}_i}})({\displaystyle \frac{{\dot{s}}_i}{{\epsilon }_i}}-{\displaystyle \frac{{\dot{\epsilon }}_i{s}_i}{{\epsilon }_i^2}})\hfill \\ \hfill & = & {\upsilon }_i({\dot{s}}_i-{\displaystyle \frac{{\dot{\epsilon }}_i{s}_i}{{\epsilon }_i}}),\hfill \end{array}$$

(34)

where ${\upsilon }_i={\displaystyle \frac{1}{2{\epsilon }_i}}({\displaystyle \frac{1}{{\pi }_i+{\underline{k}}_i}}-{\displaystyle \frac{1}{{\pi }_i-{\overline{k}}_i}})$.

The sliding mode manifold is designed as

$${\delta }_i={\int }_0^t({\varsigma }_i(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})-{\upsilon }_i\Omega C_i{B}_i{\widehat{\rho }}_i{u}_i^a\left(t\right))dt+e_{is},$$

(35)

where ${\alpha }_1$ and ${\alpha }_2$ are positive constants. ${\varsigma }_i$ is designed as

$${\varsigma }_i={\displaystyle \frac{{\alpha }_a}{{\alpha }_b+{\alpha }_{c}exp(-{\alpha }_d(\parallel {\widehat{R}}_i\parallel +\parallel e_{is}{\parallel )}^{{\alpha }_e})}},$$

where ${\alpha }_a$, ${\alpha }_b$, ${\alpha }_c$, ${\alpha }_d$, and ${\alpha }_e$ are positive constants.

The derivative of (35) is presented as

$$\begin{array}{ccc}\hfill {\dot{\delta }}_i& = & {\varsigma }_i(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})-{\upsilon }_i\Omega C_i{B}_i{\widehat{\rho }}_i{u}_i^a\left(t\right)+{\upsilon }_i({\dot{s}}_i-{\displaystyle \frac{{\dot{\epsilon }}_i{s}_i}{{\epsilon }_i}})\hfill \\ \hfill & = & {\varsigma }_i(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})-{\upsilon }_i\Omega C_i{B}_i{\widehat{\rho }}_i{u}_i^a\left(t\right)+{\upsilon }_i\Omega C_i(A_i{x}_i\left(t\right)\hfill \\ & & +B_i{\rho }_i{u}_i\left(t\right)+{\theta }_i^{*T}{\phi }_i\left({\widehat{x}}_i\right)+R_i\left(t\right))-{\upsilon }_i\Omega {\dot{y}}_{id}\hfill \\ & & +{\upsilon }_i{\chi }_2\sum _{j\in N_i}{l}_{ij}{z}_j-{\displaystyle \frac{{\upsilon }_i{\dot{\epsilon }}_i{s}_i}{{\epsilon }_i}}\hfill \\ \hfill & = & {\varsigma }_i(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})+{\upsilon }_i\Omega C_i(A_i{x}_i\left(t\right)+B_i{\widehat{\rho }}_i{u}_i^b\left(t\right)\hfill \\ & & +B_i{\tilde{\rho }}_i{u}_i\left(t\right)+{\theta }_i^{*T}{\phi }_i\left({\widehat{x}}_i\right)+R_i\left(t\right))\hfill \\ & & -{\upsilon }_i\Omega {\dot{y}}_{id}+{\upsilon }_i{\chi }_2\sum _{j\in N_i}{l}_{ij}{z}_j-{\displaystyle \frac{{\upsilon }_i{\dot{\epsilon }}_i{s}_i}{{\epsilon }_i}},\hfill \end{array}$$

(36)

where $\Omega ={\chi }_1+{\chi }_2{l}_{ii}$.

Hence, the equivalent control is obtained

$$u_i^{beq}\left(t\right)=-{\left(C_i{B}_i{\widehat{\rho }}_i\right)}^{-1}({\displaystyle \frac{{\varsigma }_i}{{\upsilon }_i\Omega }}(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})-{\dot{y}}_{id}+{\Phi }_{i1}\left(t\right)),$$

(37)

where ${\Phi }_{i1}\left(t\right)=C_i(A_i{x}_i\left(t\right)+B_i{\tilde{\rho }}_i{u}_i\left(t\right)+{\theta }_i^{*T}{\phi }_i\left({\widehat{x}}_i\right)+R_i\left(t\right))+{\displaystyle \frac{{\chi }_2}{\Omega }}{\displaystyle \sum _{j\in N_i}}{l}_{ij}{z}_j-{\displaystyle \frac{{\dot{\epsilon }}_i{s}_i}{\Omega {\epsilon }_i}}$.

The FTCC scheme is designed as

$$\begin{array}{c}\hfill u_i^a\left(t\right)=\left\{\begin{array}{c}-{\left(C_i{B}_i{\widehat{\rho }}_i\right)}^{-1}\left(({\alpha }_f+{\widehat{\varrho }}_{1i}{\varpi }_1)\mathrm{sign}\left({\delta }_i\right)\right),t\in {\Xi }_{n,1},\hfill \\ -{\left(C_i{B}_i{\widehat{\rho }}_i\right)}^{-1}\left(({\alpha }_g+{\widehat{\varrho }}_{2i}{\varpi }_2)\mathrm{sign}\left({\delta }_i\right)\right),t\in {\Xi }_{n,2},\hfill \end{array}\right.\end{array}$$

(38)

$$\begin{array}{c}\hfill u_i^b\left(t\right)=-{\left(C_i{B}_i{\widehat{\rho }}_i\right)}^{-1}({\displaystyle \frac{{\varsigma }_i}{{\upsilon }_i\Omega }}(e_{is}^{{\alpha }_1}+{\dot{e}}_{is}^{{\alpha }_2})-{\dot{y}}_{id}+{\Phi }_{i2}\left(t\right)),\end{array}$$

(39)

where ${\Phi }_{i2}\left(t\right)=C_i(A_i{\widehat{x}}_i\left(t\right)+{\widehat{\theta }}_i^T{\phi }_i\left({\widehat{x}}_i\right)+{\widehat{R}}_i\left(t\right))+{\displaystyle \frac{{\chi }_2}{\Omega }}{\displaystyle \sum _{j\in N_i}}{l}_{ij}{\widehat{z}}_j-{\displaystyle \frac{{\dot{\epsilon }}_i{\widehat{s}}_i}{\Omega {\epsilon }_i}}$, ${\widehat{z}}_j={\widehat{y}}_j-y_{jd}$, ${\widehat{s}}_i={\chi }_1{\widehat{z}}_i+{\chi }_2{\displaystyle \sum _{j\in N_i}}{a}_{ij}({\widehat{z}}_i-{\widehat{z}}_j)$. ${\alpha }_f$ and ${\alpha }_g$ are positive constants, ${\varpi }_1$ and ${\varpi }_2$ are two new vectors, which will be given later, and ${\widehat{\varrho }}_{1i}$ and ${\widehat{\varrho }}_{2i}$ are the estimations of two unknown vectors ${\varrho }_{1i}$ and ${\varrho }_{2i}$, respectively, which will also be defined later.

To save the limited communication resources, the event condition in the control channel is considered. Hence, under the ET condition of the control channel, the triggered control $u_i^b\left(t\right)$ can be rewritten as

$$\begin{array}{c}\hfill u_i^{bet}\left(t\right)=\left\{\begin{array}{c}-{\left(C_i{B}_i{\widehat{\rho }}_i\left(t_c\right)\right)}^{-1}({\displaystyle \frac{{\varsigma }_i}{{\upsilon }_i\Omega }}(e_{is}^{{\alpha }_1}\left(t_c\right)+{\dot{e}}_{is}^{{\alpha }_2}\left(t_c\right))-{\dot{y}}_{id}\left(t_c\right)+{\Phi }_{i2}\left(t_c\right)),t\in {\Xi }_{n,1},\hfill \\ -{\left(C_i{B}_i{\widehat{\rho }}_i\left(t_{p_{dn}+l_{dn}}\right)\right)}^{-1}\left({\displaystyle \frac{{\varsigma }_i}{{\upsilon }_i\Omega }}\right(e_{is}^{{\alpha }_1}\left(t_{p_{dn}+l_{dn}}\right)\hfill \\ +{\dot{e}}_{is}^{{\alpha }_2}\left(t_{p_{dn}+ldn}\right))-{\dot{y}}_{id}\left(t_{p_{dn}+l_{dn}}\right)+{\Phi }_{i2}\left(t_{p_{dn}+l_{dn}}\right)),t\in {\Xi }_{n,2},\hfill \end{array}\right.\end{array}$$

(40)

where $t_c$ is the event-triggered instant of the control channel, which is designed as

$$t_{c+1}^i=\inf \{t>t_c^i:h_i\left(t\right)>0\}.$$

(41)

The triggered function is designed as

$$\begin{array}{cc}\hfill h_i\left(t\right)=& \parallel e_{is}^{{\alpha }_1}\left(t\right)-e_{is}^{{\alpha }_1}\left(t_c\right)\parallel -e^{{\mu }_1}-{\nu }_1\hfill \end{array}$$

(42)

$$\begin{array}{cc}& \mathrm{or}\parallel {\dot{e}}_{is}^{{\alpha }_2}\left(t\right)-{\dot{e}}_{is}^{{\alpha }_2}\left(t_c\right)\parallel -e^{{\mu }_2}-{\nu }_2,\hfill \end{array}$$

(43)

where ${\mu }_1$, ${\mu }_2$, ${\nu }_1$, and ${\nu }_2$ are positive constants.

Theorem 2.

Consider HMASs under Assumptions 1–4. The sliding mode manifold is given as (35), the triggered function is designed as (42), the control law is presented as (38) and (40), and then the sliding motion will maintain and the tracking errors of all vehicles can be guaranteed.

Proof. 

Select the Lyapunov function

$$\begin{array}{ccc}\hfill V_s\left(t\right)& = & \sum _{i=1}^N(1-\mathcal{J})V_{s1}\left(t\right)+\mathcal{J}{V}_{s2}\left(t\right),\hfill \end{array}$$

(44)

with $V_{s1}\left(t\right)={\displaystyle \frac{1}{2}}{\delta }_i{\left(t\right)}^T{\delta }_i\left(t\right)+{\displaystyle \frac{1}{2}}{({\varrho }_{1i}-{\widehat{\varrho }}_{1i})}^2$, $\mathcal{J}=0$, $t\in {\Xi }_{n,1}$; $V_{s2}\left(t\right)={\displaystyle \frac{1}{2}}{\delta }_i{\left(t\right)}^T{\delta }_i\left(t\right)+{\displaystyle \frac{1}{2}}{({\varrho }_{2i}-{\widehat{\varrho }}_{2i})}^2$, $\mathcal{J}=1$, $t\in {\Xi }_{n,2}$.

Consider that there is no DoS attack, i.e., $t\in {\Xi }_{n,1}$. Substituting (38) and (40) into (36), one can obtain

$$\begin{array}{ccc}\hfill {\dot{\delta }}_i\left(t\right)& = & {\varsigma }_i(e_{is}^{{\alpha }_1}\left(t\right)+{\dot{e}}_{is}^{{\alpha }_2}\left(t\right))-{\upsilon }_i\Omega {\dot{y}}_{id}+{\upsilon }_i\Omega {\Phi }_{i1}\left(t\right)\hfill \\ & & -{\varsigma }_i(e_{is}^{{\alpha }_1}\left(t_c\right)+{\dot{e}}_{is}^{{\alpha }_2}\left(t_c\right))+{\upsilon }_i\Omega {\dot{y}}_{id}\left(t_c\right)-{\upsilon }_i\Omega {\Phi }_{i2}\left(t_c\right)\hfill \\ \hfill & = & {\varsigma }_i(e_{is}^{{\alpha }_1}\left(t\right)-e_{is}^{{\alpha }_1}\left(t_c\right)+{\dot{e}}_{is}^{{\alpha }_2}\left(t\right)-{\dot{e}}_{is}^{{\alpha }_2}\left(t_c\right))+{\Phi }_{i3}\left(t\right)\hfill \\ & & -({\alpha }_f+{\widehat{\varrho }}_{1i}{\varpi }_1)\mathrm{sign}\left({\delta }_i\right),\hfill \end{array}$$

(45)

where ${\Phi }_{i3}\left(t\right)=-{\upsilon }_i\Omega {\dot{y}}_{id}+{\upsilon }_i\Omega {\Phi }_{i1}\left(t\right)+{\upsilon }_i\Omega {\dot{y}}_{id}\left(t_c\right)-{\upsilon }_i\Omega {\Phi }_{i2}\left(t_c\right)+({\alpha }_f+{\widehat{\varrho }}_{1i}{\varpi }_1)\mathrm{sign}\left({\delta }_i\right)$.

Let ${\Theta }_i= \parallel {\varsigma }_i(e_{is}^{{\alpha }_1}\left(t\right)-e_{is}^{{\alpha }_1}\left(t_c\right)+{\dot{e}}_{is}^{{\alpha }_2}\left(t\right)-{\dot{e}}_{is}^{{\alpha }_2}\left(t_c\right))\parallel + \parallel {\Phi }_{i3}\left(t\right)\parallel$. Then, one can obtain that ${\Theta }_i$ is unknown but bounded. The RBFNN is utilized to approximate the unknown part ${\Theta }_i$.

$${\Theta }_i={\mathcal{F}}_i^T{\psi }_i\left(t\right)+{\mathcal{M}}_i.$$

(46)

Hence, one can obtain

$$\begin{array}{ccc}\hfill {\delta }_i\left(t\right){\Theta }_i& \le & \parallel {\delta }_i\left(t\right)\parallel \parallel {\mathcal{F}}_i^T\parallel \parallel {\psi }_i\left(t\right)\parallel +\parallel {\delta }_i\left(t\right)\parallel {\mathcal{M}}_i\hfill \\ \hfill & = & \parallel {\delta }_i\left(t\right)\parallel {\varrho }_{1i}{\varpi }_{1i},\hfill \end{array}$$

(47)

where ${\varrho }_{1i}=[\parallel {\mathcal{F}}_i^T\parallel ,{\mathcal{M}}_i]$, ${\varpi }_{1i}=[\parallel {\psi }_i\left(t\right){\parallel ,1]}^T$. In addition, the adaptive law of ${\varrho }_{1i}$ is updated as ${\widehat{\varrho }}_{1i}=\parallel {\delta }_i\left(t\right)\parallel {\varpi }_{1i}$.

The derivative of (44) yields that

$$\begin{array}{ccc}\hfill {\dot{V}}_s\left(t\right)& \le & \sum _{i=1}^N({\delta }_i{\left(t\right)}^T{\dot{\delta }}_i\left(t\right)+({\varrho }_{1i}-{\widehat{\varrho }}_{1i})({\dot{\varrho }}_{1i}-{\dot{\widehat{\varrho }}}_{1i}))\hfill \\ \hfill & \le & \sum _{i=1}^N(\parallel {\delta }_i\left(t\right)\parallel {\varrho }_{1i}{\varpi }_{1i}-{\delta }_i{\left(t\right)}^T({\alpha }_f+{\widehat{\varrho }}_{1i}{\varpi }_{1i})\mathrm{sign}\left({\delta }_i\right)\hfill \\ & & -({\varrho }_{1i}-{\widehat{\varrho }}_{1i})\parallel {\delta }_i\left(t\right)\parallel {\varpi }_{1i})\hfill \\ \hfill & = & \sum _{i=1}^N(\parallel {\delta }_i\left(t\right)\parallel ({\varrho }_{1i}{\varpi }_{1i}-{\alpha }_f-{\widehat{\varrho }}_{1i}{\varpi }_{1i})\hfill \\ & & -({\varrho }_{1i}-{\widehat{\varrho }}_{1i})\parallel {\delta }_i\left(t\right)\parallel {\varpi }_{1i})\hfill \\ \hfill & \le & -\sum _{i=1}^N{\alpha }_f\parallel {\delta }_i\left(t\right)\parallel .\hfill \end{array}$$

(48)

Consider that the DoS attack is launched, i.e., $t\in {\Xi }_{n,2}$. Similar results can be obtained through similar derivations. Hence, it can be concluded that ${\dot{V}}_s\left(t\right)\le 0$, which implies that the sliding motion will maintain and the convergence of $e_{is}$ is guaranteed. In terms of the property of the PPF, the boundedness of $e_{is}$ implies that the tracking errors $s_i$ are also bounded within the predefined bounds specified by (30). This ends the proof. □

Remark 5.

It can be easily seen from (10) that the minimum time interval will be no less than h, which implies that the time interval between two adjacent data release times will not be less than the sampling period. Hence, the Zeno behavior can be avoided in the sensor channel. A similar analysis can be applied in the controller channel. Therefore, the Zeno behavior can be also avoided in the controller channel.

Remark 6.

In some existing literature [37,38], the control input is directly set to zero. It does not consider how to utilize an observer to guarantee the safe and stable operation of the system. The existing results about event-triggered mechanism concentrates on either the sensor channel or the control channel. It is challenging to take both sensor and control channels into account, simultaneously. Hence, the novelty of this work is that an observer-based ET-FTCC scheme is proposed via dual-terminal triggered mechanisms with prescribed performance.

5. Simulation Studies

In this section, the efficiency of the presented ET-FTCC scheme is verified with actuator faults and external disturbances under DoS attacks.

5.1. Simulation Conditions

The HMASs containing two UAVs and two UGVs are considered. The topology of the HMASs is described in Figure 5.

The system parameters are given as $m_{ai}=2\mathrm{kg}$, $d_{ix}=d_{iy}=d_{iz}=0.012\mathrm{N}·\mathrm{s}·{\mathrm{rad}}^{-1}$, $M_{gi}=10\mathrm{kg}$, $J_{gi}=1\mathrm{N}·\mathrm{s}·{\mathrm{rad}}^{-1}$, $L_{gi}=0.5\mathrm{m}$. The initial values are $x_{11}\left(0\right)={[-0.9,2,2.5]}^T$, $x_{21}\left(0\right)={[-0.4,-1,0.5]}^T$, $x_{31}\left(0\right)={[2.2,-2.5,0]}^T$, $x_{41}\left(0\right)={[-1.5,1.5,0]}^T$, $x_{12}\left(0\right)={[0.1,0.2,0.5]}^T$, $x_{22}\left(0\right)={[0.3,0.6,0.4]}^T$, $x_{32}\left(0\right)={[0.1,0.3,0]}^T$, and $x_{42}\left(0\right)={[0.2,0.4,0]}^T$. The parameters are selected as ${\alpha }_1=0.7$, ${\alpha }_2=1.2$, ${\alpha }_a=0.9$, ${\alpha }_b=1.3$, ${\alpha }_c=1$, ${\alpha }_f=0.2$, ${\alpha }_g=0.5$, ${\mu }_1=0.4$, ${\mu }_2=0.4$, ${\nu }_1=0.01$, and ${\nu }_2=0.01$. The prescribed performance function parameters are given as ${\beta }_1=0.3$, ${\beta }_2=0.3$, ${\epsilon }_{i\infty }=3$. The external disturbances are given as $d_1={[0.3sin\left(t\right),0.4,0.2cos\left(t\right)]}^T$, $d_2={[0.25,0.3sin\left(t\right),0.2]}^T$, $d_3={[0.3sin\left(t\right),0.2,0]}^T$, and $d_4={[0.5cos\left(t\right),0.1,0]}^T$. The actuator faults are encountered by the followers UAV#1 and UGV#4, and they are given as ${\rho }_{11}=I_3$, $u_{11b}={[0,0,0]}^T$ for $t<9\mathrm{s}$, and ${\rho }_{11}=diag\{0.9,1,0.8\}$, $u_{11b}={[0.2,0.3,0.15]}^T$ for $t\ge 9\mathrm{s}$, ${\rho }_{41}=diag\{1,1,0\}$, $u_{41b}={[0,0,0]}^T$ for $t<15\mathrm{s}$, and ${\rho }_{41}=diag\{0.8,0.9,0\}$, $u_{41b}={[0.3,0.2,0]}^T$ for $t\ge 15\mathrm{s}$.

The performance of the presented ET-FTCC scheme is compared with the control scheme in [39]. To further quantitatively evaluate the control performance, the X-direction, Y-direction, and Z-direction tracking error metric (TEM) are defined as

$$\begin{array}{ccc}\hfill \mathrm{XTEM}& = & \sqrt{{\displaystyle \frac{1}{4}}\sum _{i=1}^4{\left|s_{ix}\right|}^2},\mathrm{YTEM}=\sqrt{{\displaystyle \frac{1}{4}}\sum _{i=1}^4{\left|s_{iy}\right|}^2},\mathrm{ZTEM}=\sqrt{{\displaystyle \frac{1}{2}}\sum _{i=1}^2{\left|s_{iz}\right|}^2}.\hfill \end{array}$$

5.2. Simulation Results

The simulation results are presented in Figure 6, Figure 7 and Figure 8. The tracking errors of UAVs and UGVs in the X-direction, Y-direction, and Z-direction are depicted in Figure 6. It can be observed from Figure 6 that the tracking errors achieve good performance, which are confined within the prescribed error bounds $({\underline{k}}_i{\epsilon }_i,{\overline{k}}_i{\epsilon }_i)$. In addition, the active period of the DoS attack is denoted by the green shadow, while the rest parts denote that there is no DoS attack. To further show the advantages of the proposed scheme, a simulation comparison of the control approach in [39] is also carried out.

From Figure 6, one can determine that the tracking performance will degraded when the DoS attacks are active. It can be observed that the proposed scheme in this work has better recovery performance under DoS attacks. The tracking performance can recover soon when the attackers are sleeping. Meanwhile, the approach in [39] cannot ensure good performance when the DoS attacks are active. As depicted in Figure 7, the proposed scheme has better recovering abilities than the approach in [39], which means that the tracking performance is recovered by utilizing the presented scheme, even subjected to actuator faults and external disturbances under DoS attacks. Figure 8 shows the release times and intervals of the event-triggered mechanism. It can be observed from Figure 8 that the communication burden is reduced Hence, the efficiency of the presented decentralized adaptive ET-FTCC scheme with prescribed performance in the presence of actuator faults, external disturbances, and DoS attacks is verified.

6. Conclusions

In this paper, the ET-FTCC is investigated for HMASs with actuator faults and external disturbances under DoS attacks. The composite observer is designed to obtain the information of state and lumped disturbance, which is used to design the controller. After that, the event-triggered sliding mode FTCC scheme is presented to compensate for actuator faults and external disturbances and defend against DoS attacks. The tracking errors are bounded within the prescribed boundary. The effectiveness of the proposed method is verified by qualitative analysis and quantitative analysis of the simulation results. This study can enhance the security and reliability of heterogeneous multi-agent systems, providing technical support for the safe operation of unmanned systems.

This article mainly solves the FTCC problem of second-order nonlinear heterogeneous multiagent systems, and further research is needed for the FTCC problem of higher-order nonlinear heterogeneous multi-agent systems. In addition, the DoS attacks are considered in this paper. However, the heterogeneous multiagent systems may encounter multiple cyber attacks, such as DoS attacks and deception attacks. In order to solve the multiple cyber attacks, a multiple cyber attack model should be modeled first. Combining the characteristics of the attack model and the unified model of the heterogeneous multiagent systems, the FTCC scheme will be investigated to compensate the influence of the faults under multiple cyber attacks, which is one of our future works.