© The Author(s) 2023. Published by EDP Sciences and China Science Publishing & Media Ltd.

1. Introduction

Multiple unmanned systems are gradually replacing human work in many fields such as collaborative task execution, large-scale environmental monitoring, and high-risk industry work due to their advantages of high efficiency, intelligence, security, and strong scalability [1, 2]. In order to improve the collaboration capability and security of unmanned systems, extensive research has been carried out on the consensus control of unmanned systems, which has become a hot spot in recent years [3]. Consensus control is the basis of unmanned systems cooperative formation control. The key to consensus is the information interaction between unmanned systems. However, as the scale of unmanned systems deployment expands, the communication network between agents will become complex. Therefore, the shortcomings of the unmanned system network vulnerabilities will gradually be exposed.

In the applications of the control systems via networks, cyberattacks can disrupt network connections on a large scale, which degrades the execution efficiency of the cyber system and even causes task failure. Cybersecurity has attracted more and more attention in the field of automatic control [4, 5]. Considering the followers with the unmodeled dynamics in the communication network, a fully distributed adaptive control strategy is designed in [6] based on neural networks for achieving the followers synchronized to the leader. For guaranteeing the cyber security of the unmanned system under network attacks, using resilient control methods to ensure security has become an important research direction [7]. In the network control system, network attacks refer to attacks on the controlled object at the network level. They can illegally hinder and damage the information interaction in the unmanned system network. It includes two main types of cyberattacks, denial-of-service (DoS) attacks and deception attacks [8]. In this paper, the distributed event-triggered resilient consensus control is investigated for unmanned systems against periodic DoS attacks.

DoS is one of the common types of cyberattacks, which has been widely studied [9]. The purpose of the DoS attack is to destroy the network link between neighboring agents and disconnect the information exchange, thereby hindering the consensus of the unmanned system. DoS attacks can have devastating effects on the unmanned system, which reflect the urgent need to solve the DoS attack problem. In [10], an asynchronous DoS description model is proposed, which is close to the network attacks that occur in reality. In [11], a periodic DoS attack model is introduced into the output feedback control of cyber-physical systems and can be easily combined with asynchronous attacks on different channels. In [3] the transmission nonlinearities with gain and bias in the network link are considered for the leader-following consensus problem of multi-agent systems. The consensus can be achieved by designing distributed adaptive control schemes. The research scenario is similar to cyber-attacks.

Considering with distributed and resilient control of the unmanned systems against DoS attacks, linear systems were mainly used as investigated objects in the early stage [12]. In this field, robust control methods were the main measures to resist DoS attacks. For resilient control of linear systems, the linear matrix inequality (LMI) method is a tool for solving the feedback controller parameters [13, 14]. However, practical unmanned systems are nonlinear dynamic models such that linear control methods are not applicable in most cases. In addition, the input-to-state stable (ISS) control methods are widely used to study the resilient control method against DoS attacks [15, 16]. But for many nonlinear unmanned systems, the ISS stability condition is difficult to satisfy. Recently, more and more researchers focus on the resilient control of nonlinear unmanned systems [17, 18]. Recently, type-2 fuzzy logic systems are widely used for dynamic model transformation. In [19], based on the type-2 fuzzy logic system approximating the unknown smooth function. A distributed adaptive supervisory control method is proposed in the paper. Although excellent control performance is obtained in these papers, the stability of the system cannot be guaranteed during a cyberattack due to the broken feedback loop.

Recently, event-triggered control is one of the hotspots in the field of network attack resilient control [20]. Event-triggered control aims to decrease the waste of communication resources for agents and reduce the frequency of controller updates [21, 22]. Through the action of the event-triggered mechanism (ETM), the feedback control loop will be closed when the state of the agent exceeds the designed event-triggered setting threshold [12, 23]. The controller output will not change until the conditions for triggering the next event are met. When a DoS attack occurs, the ETM works and the actuator of the agent maintains a constant output state until the network attack disappears. Although the control performance will be reduced, the stability of the system can be guaranteed. In the field of consensus control, a large number of applications combine distributed control with ETM. In [24], based on dynamic ETM, a distributed control protocol is proposed to ensure the consensus of linear agents. Due to the Zeno behavior in the ETM, it can be excluded by presetting the minimum trigger interval [25].

The core issue of defending against cyberattacks is to maintain the stability of the system under attacks and to quickly recover consensus after the attacks. Although some switching control methods can be introduced into resilient control to mitigate the effect of DoS attacks [26, 27], they must rely on system model features and reference commands. Furthermore, the feedback loop is still broken under DoS attacks. In order to maintain the feedback loop, some scholars use the method of state holding or safety observer to reconstruct virtual reference information [28, 29].

However, the observer can only obtain the last information of the neighbor agents before the attack in practical application situations. Model-based observers will not be able to obtain a satisfactory estimate if the state of the neighbor agents is variable. Many existing methods of state predictors have been developed for dealing with communication transmission issues. In [30], for the power system, an optimal wide-area controller is designed with the state predictor for compensating the error caused by transport lags. In [31], a predictor-based extended-state-observer is designed for estimating the state of neighbor agents. A leader-follower consensus protocol is proposed against communication delays and disturbances efficiently. Considering data-based prediction methods, an active communication delay compensation mechanism with a data-driven state predictor is proposed in [32] to estimate the current states of neighbor agents using the delayed state information. Combined with the prescribed performance method and neural networks, a kind of back-stepping control method is developed for autonomous underwater vehicle formation control. Many existing methods have been developed related to the state predictor for compensating communication delays. However, many existing states predicted methods rely on the model of agents and do not involve the resilient control method against cyber-attacks. The autoregressive integrated moving average (ARIMA) model [33] is widely used for trend and data prediction. Many predicting methods related to ARIMA have been proposed for applications [34–36]. Compared with other data-based intelligent prediction methods, ARIMA has the advantage that it does not require pre-training. Therefore, real-time predictions can be made using historical data.

Motivated by the above existing works, combined with the ETM and the resilient control method, a distributed control method is proposed for the unmanned systems against the periodic asynchronous multi-channel DoS attacks. ARIMA is introduced into the control scheme to predict the state information of the neighbor unmanned system in real-time during DoS attacks. The prediction mechanism will play a key role when the communication network suffers from DoS attacks. The main contributions of this paper are summarized as follows.

1. Compared with the existing work of ARIMA to classify network attacks [37], this is different from the research field of this paper. Aiming at the issue that the feedback loop is broken by the DoS attacks, to the best of the authors’ knowledge, the proposed data-based state predictor is introduced for the first time into the model-based resilient control against the DoS attacks in this paper.

2. Since ARIMA is a data-based state prediction method, it is difficult to prove the convergence of differences between the real state and predicting state for unmanned systems. In this paper, a disturbance observer is proposed to estimate the uncertain disturbance of the system caused by state estimation.

3. Comparing with the existing works related to ETM and resilient control against DoS attacks, a control scheme is proposed for a nonlinear unmanned system to guarantee the system consensus under the periodic DoS attack. Besides, the dynamic ETM is designed to reduce the actuator triggering frequency.

The paper structure is arranged as follows. The preliminaries about the unmanned systems model, graph theory, the DoS attack model, and the proposed ARIMA prediction method are shown in Section 2. The main results related to the proposed ETM, disturbance observer, and resilient controller are proposed in Section 3. Further, the simulation has been carried out to verify the resilient capacity against DoS attacks of the proposed method in Section 4. Finally, the conclusion of the paper is given in Section 5.

2. Preliminaries and problem statement

2.1. Unmanned system model description

Consider using the following typical second-order dynamic system to describe a single agent to constitute the unmanned system

(1)

where x ₁(t) ∈ ℝ ⁿ and x ₂(t) ∈ ℝ ⁿ represent the dynamic state of an individual in the unmanned system. In this paper, the individual in unmanned systems is named an agent. u(t) is the control signal of the agent. F ₁, G ₁, F ₂, and G ₂ reflect the dynamics characteristics of the agent. F ₁ and F ₂ are the nonlinear differentiable terms correlating with the dynamic of the agent. G ₁ and G ₂ are the nonlinear control input matrices for the agent, defined by the physical parameters of the system. G ₁ and G ₂ are bounded and nonsingular, rendering the agent dynamics controllable.

2.2. Preliminaries for graph theory

The set of agents can be defined as the virtual nodes v _N . Furthermore, the set can be described as 𝒱 = (v ₁,v ₂,…,v _N ). The edge set can be denoted as ℰ ⊆ 𝒱 × 𝒱. 𝒜 = [a _{i j} ] ∈ R ^N × N is the adjacency matrix. If e _{i j}  = (v _i ,v _j ) ∈ ℰ, there is a _{i j}  = 1. It means that agent i can acquire the information from the agent j, otherwise, a _{i j}  = 0. It is noticed that a _{i j} (t ₀) = 0 will exist constrainedly when the edge e _{i j} suffers from a DoS attack at t ₀. The interaction communication among agents can be described as the directed graph 𝒢 = (𝒱, ℰ, 𝒜). Define the degree matrix 𝒟 = diag(d _i ) ∈ R ^N × N , where . There is the Laplacian matrix ℒ = 𝒟 − 𝒜.

Consider introducing the reference command into the unmanned system, which can be defined as a virtual node v _N . The adjacency matrix is defined as for N − 1 agents. The augmented adjacency matrix including the virtual command can then be described as

(2)

where β _i  ∈ R ^{1 × (N − 1)} represents that the information transfers from the virtual node to the following agents. There is no information transmitted from followers to the virtual node. Therefore, the last row of the matrix is the zero vector. In addition, the element in including β _i is defined as . The initial directed graph denotes the communication relationship of the unmanned systems, where denotes the nodes set including followers and the virtual node, and . For satisfying resilient controller design requirements, assumed that the original communication topology is the strong connection, which will suffer from the DoS attack to affect the information interaction. DoS attacks would be discussed and dealt with in the following sections. Moreover, it is necessary for each agent to have access to at least one neighboring agent to receive consensus reference information related to the virtual node.

2.3. DoS attacks model

For the periodic DoS attacks, a period can be described as T _n , with T _n  ≥ 0, including a duration of DoS-on and duration of DoS-off. By defining the attack period as T _p  = T _i + 1 − T _i , the attack duration can be denoted as T _d , with the initial time h _n . The attack rate is F _a  = T _d /T _p .

For the periodic DoS attacks, the attack sequence can be described as follows:

(3)

The sets of DoS-on and DoS-off on each channel can be described as 𝒯(T _i ) and ,

(4)

According to the description of the DoS attacks model by (4), a kind of periodic DoS attack sequence on a channel is shown as follows.

Figure 1. A classical periodic DoS attack

As shown in Figure 1, a new period of DoS begins with T _n at a DoS-off moment. The network attack DoS-on lasts for T _d , which is represented by the gray region. The period of the DoS attack is T _p .

2.4. Agent states predictor under DoS attacks

In this paper, for predicting the states of agents under DoS attacks, the ARIMA model is used to design the state predictor. The model can be described as

(5)

where ϵ is the estimation constant term. ε _t is the estimation error sequence. p and q represent the autoregressive number and moving average number in ARIMA, respectively. a and b are the autocorrelation and partial autocorrelation coefficients, respectively.

The states of unmanned systems can be estimated recursively by historical data. In addition, the prerequisite for the effectiveness of ARIMA is that the estimated agent should not be an “anti-predictive behavior”. Considering the consensus control of the unmanned system with the limitation of kinematic characteristics, it is assumed that there is a cooperative relationship between unmanned systems. Assumed that under the control of reference commands, large state changes, such as highly dynamic behaviors, would not be undergone by the cooperative unmanned systems during DoS attacks. The prerequisite is satisfied.

As a numerical simulation, define p = 2 and q = 2. The historical state is used to predict the future state during DoS attacks. Compared with state holding, the prediction ability of ARIMA is shown in the following figures.

Figure 2. Comparison of the two methods for state prediction

Figure 3. State prediction error of two methods

Figures 2 and 3 show the prediction ability of ARIMA. By using a sine signal as the simulated state, the error is much less than that of state holding. Therefore, this paper uses the ARIMA as the predictor for the agent state against the DoS attack. The numerical simulation and comparison are carried out in Section 4.

Combining DoS attacks with state estimation, the states of neighbor unmanned systems for an agent can be defined as a switching function

(6)

where x _i (t) represents the states of the ith neighbor agent in the case of DoS-off. is the estimation states of neighbor agent i in the case of DoS-on according to the state prediction method in (5) or state holding method. Noted that is a switching function caused by the DoS attack. The derivative term is hard to obtain and introduced into the controller design. In the actual operation process of the unmanned system, the state time series of the neighbor agent can be obtained. The can be calculated using the following equation.

(7)

where t _k and t _k − 1 represent the sampling times, respectively. and denote the states of the ith neighbor agent at t _k and t _k − 1. Considering the error between the real state and the estimated state, the states of neighbor agent j can be rewritten as

(8)

where ε _j represents an uncertain term for describing the error, which will be addressed in the following steps. The term ε _j emerges as a result of DoS attacks, which represents the discrepancy between the real state x _j (t) and estimated state under the DoS attacks. The boundedness of ε _j should be discussed. Therefore, an existing assumption is introduced in the paper.

The periodic DoS attack is investigated in the paper. Therefore, the attack duration is finite time. According to Assumption ², ε _j is bounded under existing conditions. By the proposed estimation mechanism in this paper, the feedback control loop can be held during the DoS attacks.

In addition, some existing lemmas that are necessary for designing the resilient controller in this paper are presented as follows:

3. Main results

3.1. Consensus control protocol

According to the definition of communication topology nodes, the reference command information can be regarded as a virtual node in the augmented adjacency matrix (2). The consensus protocol for MAS can be achieved if is satisfied. To develop the distributed controller for agents, the consensus control measurement error function can be defined as

(9)

(10)

where is the states of the ith agent. represents the states of the neighbor agent j. N represents the number of neighbor agents. is the communication topology.

The consensus of MAS can be achieved if is bounded. According to the definition of x _j in (8), it can be obtained that

(11)

Taking the derivative of (11) and by virtue of (1), it can be obtained that

(12)

where is obtained by the augmented degree matrix.

3.2. Disturbance observer for DoS attacks

According to (11), the uncertain term ε _j caused by the state predictor is treated as the disturbances that the degrade system performance under the DoS attack. Noted that is hard to obtain since the state of the neighbor agents is defined as the switching function. Compared with the intrinsic disturbances in agents, ε _j is only discovered under the DoS attacks and caused by the state predictor leading to a signal vibration. Developing a disturbance observer to compensate for the error term ε _j and degrade the signal vibration is crucial for improving the system’s resiliency. An estimation term can be immediately integrated into the resilient controller to compensate for the disturbance and protect the system security.

According to (12), the system uncertain disturbance term can be denoted as

(13)

According to the above analysis, it is noticed that the disturbance does not always exist since the DoS attack cannot always occur. For dealing with the issue, define the observer of Δ _i as

(14)

where represents the estimation term of 𝒟 _i which is defined as

(15)

In the proposed disturbance observer, is an indirect estimator related to Δ _i for dealing with the issue that the disturbance does not always exist. is a given term by neighbor agents information via communication topological network. can be acquired by (14). Substituting 𝒟 _i into (12) one yields

(16)

The observer error can be calculated as

(17)

Furthermore, define the error term between 𝒟 _i and as . By virtue of (16), it can be obtained that

(18)

Assume , where δ _i represents the unknown upper bound of . The update law of 𝒟 _i and estimation adaptive law of δ _i are designed as

(19)

(20)

(21)

(22)

where k _d represents a diagonal matrix. k _δ , k _ε , and ε ^* are the positive parameters.

3.3. Event-triggered control protocol

Considering the agent model (1) and the consensus control protocol (10), design the virtual control signal φ(ξ _i ) and the control signal u _i (t) for the ith agent to realize the consensus of unmanned systems.

By substituting agent model (1), the consensus control differences (12) can be rewritten as

(23)

For the stability of agent, the virtual control signal φ(ξ ⁱ ) for is designed as

(24)

in which

(25)

Noted that G ₁ is nonsingular, and is avoided as every following agent has a neighbor agent or the virtual node at least to receive the information according to Assumption ¹ established about communication topology with a spanning tree in the paper. Therefore, can be calculated. Furthermore, defined the adaptive gain error term

(26)

where k ^*, k _η , and k _c are positive definite matrices. k _φ is an adaptive control matrix related to ξ _i for improving the ability of resilience. It is noted that sign(ξ _i )k _η ξ _i  ≥ 0 is expected. There is . It can be obtained . If , and k _φ (0) >  k ^*, there is k _φ  >  k ^* >  0.

Define virtual control error

(27)

By virtue of (1), the derivative of (27) is derived as

(28)

where the actuator control signal u _i (t _k ) of the event-triggering time sequence is defined as t _k  = {t ₀,t ₁,…,t _n }. The purpose of introducing the ETM into the proposed resilient controller is to reach an equilibrium between control performance and service life of actuators under DoS attack in practical design and to make the controller resilient under DoS attacks. In the proposed resilient controller, the ETM can decrease the actuator triggering frequency under the DoS attacks. Under normal conditions, actuators would be triggered at a fixed maximum frequency for the optimal control performance. However, the controller is threatened by the reference signal vibration due to the DoS attacks. By using the ETM, the controller acting on the actuator will decrease the triggering frequency, making the controller insensitive to the reference signal involving the cyber-attack. The resiliency can be reflected in the process of consensus control under the DoS attacks. By reducing the actuator trigger frequency, the service life of agents can be prolonged. It is a way to ensure the security of the systems.

The measurement error for ETM is defined as

(29)

The event time instants for actuator controller u(t _k ) are determined by the following approach

(30)

where f(t) satisfies the following definition

(31)

In the paper, a dynamic ETM method is designed to improve system performance. Compared with some existing work such as [24], the proposed dynamic ETM term μ _i is updated by the virtual control error (27) without measurement error for ETM. The coupling between event trigger error and condition is decreased. The dynamic triggering term μ _i is designed as

(32)

The control signal u(t _k ) can be designed as the function related to ζ _i (t _k ), which is represented as u(ζ _i (t _k )). According to ETM (29), u(t _k ) can be rewritten as u(e _i (t)+ζ _i (t)). The controller input can be carried out by

(33)

where φ(ξ _i ) is designed by (24) and (28). However, it is noticed that is hard to acquire. For solving the difficulty, the high-order-differentiator (HOD) is introduced.

The differentiator can be designed with a compact form

(34)

where , φ = φ(ξ _i ), and , .

3.4. Stability analysis

Theorem. Consider the unmanned systems described by (1) under the directed communication topology (2). The state predictor is designed as (5). The consensus error is constructed as (10). The dynamic ETM is defined as (30). A disturbance observer focusing on DoS attacks is developed as (14) following the update laid by (19)–(22). Furthermore, the virtual control signal is designed as (24) with the adaptive laws (25) and (26), and the control signal is designed as (33). The consensus of unmanned systems can be guaranteed, and errors are uniformly ultimately bounded (UUB).

For the disturbance observer, define the observer estimation error . Since δ _i is a constant, . For the agent i in the MAS, the Lyapunov function candidate is defined as follows:

(35)

According to (22), it is noticed that and ε ^* >  0. It is easy to verify that ε _δ  >  0. The derivative of V _d is derived as

(36)

Define , it is noticed that κ _i  >  0. Furthermore, there is ε _δ  >  0. According to Lemma ¹, one yields

(37)

Substituting (37) into (36) has

(38)

where K _d  = k _d k _δ . Therefore, . If there is k _ε  ≥ 0.2785, such that .

According to ε _δ  >  0, one yields

(39)

where α _d  = 2λ _min(K _d ). It can be obtained that

(40)

According to (35), it means that and are UUB. The error convergence of the state predictor can be proved.▫

For the proof of the event-triggered consensus control protocol, choose the following candidate Lyapunov function for the ith agent as

(41)

According to (31) and (32), one has

(42)

It is noticed that , it has

(43)

Hence, the condition V _i  ≥ 0 is always satisfied. Taking the derivative of (41), one has

(44)

Define , where and . Substituting (23), (27) into and according to Lemma ², one has

(45)

According to Lemma and (17), one has

(46)

where ∥⋅∥ _F represents Frobenius norm.

Furthermore, substituting (28) and (33) into , and according to Lemma ², one has

(47)

By virtue of ETM (30)–(32), the matrix Π and Λ can be selected as k _ζ and , respectively, such that

(48)

Substituting (46) and (48) into (44), then one can obtain

(49)

where ς₁, ς₂ are given as

(50)

where . Further, is the control matrix of MAS, which is bounded on the basis of the system characteristics. According to (40), ς₂ is bounded. Therefore, ξ _i , , and ζ _i are UUB.

Considering the consensus error of MAS for all agents, according to (9), (10) and (49), the system consensus error satisfies

(51)

It can be concluded that is uniformly convergent. Furthermore, the stability of the system consensus can be demonstrated.

3.5. Excluding Zeno behavior

In this subsection, it has been proved that there will be no Zeno behavior in the proposed dynamic ETM (30). For verifying that there is no Zeno behavior, the inequality t _k + 1 − t _k  >  0 must be satisfied.

By defining e _i  = ζ _i (t _k ) − ζ _i (t) during one trigger interval, it is noticed that ζ _i (t _k ) is a constant in [t _k , t _k + 1].

(52)

where 𝒦 is a positive constant.

According to (52), the integral of satisfies

(53)

Furthermore, the following constraint exists

(54)

According to dynamic ETM (30), (31), and (43), there are and μ >  0. It can be obtained that e _i  >  0. One yields

(55)

According to (55), there is a minimum time interval between the trigger of two events. The condition that inequality t _k + 1 − t _k  >  0 is satisfied. Therefore, the Zeno behavior is excluded.

Figure 4. Directed communication topology of MAS

4. Simulation result

4.1. Experimental simulation model

In this section, the effectiveness of the proposed ETM consensus control method with predictor is verified by simulation scenarios as follows. A multiple unmanned aerial vehicles (multi-UAV) system is used to represent the MAS in this paper. The multi-UAV system consists of 4 UAVs as 4 nodes labeled as 1 to 4 and a virtual reference node labeled as 5. The directed communication topology is established as (2) in Figure 4. The derivation process and specific system parameters of the UAV model used can be found in [41]. The dynamic model of the ith UAV is shown as follows:

(56)

(57)

(58)

where V _i , χ _i , and γ _i represent the velocity, flight path angle, and heading angle of the ith UAV, respectively. μ _i , α _i , and β _i represent its angle of attack, sideslip angle, and bank angle, respectively. p _i , q _i , and r _i denote its angular rate, respectively. For fitting in the proposed second-order MAS consensus control model (1) to design the resilient consensus controller in the paper, the multi-UAV attitude consensus control is considered to verify the effectiveness of the controller. By defining and , the dynamic model of UAV can be rewritten as the form of (1).

In Figure 4, chi, i = 1, 2, …, 6, represents the communication channel, in which the state information transforms from the UAV that sends out commands to the object UAV. The dotted circle represents the virtual reference node signal. The arrows of channels represent the direction of transmission. The communication between nodes is carried out by directed routing. Parameters of the DoS attack model for every channel are T _n  = 10 s and F _a  = 9%. Further, T _d  = 0.9 s. The delay times for every channel are set as T _delay = 1.3 s to realize the asynchronous DoS attacks. The beginning time of the DoS attacks T _begin = 5 s. The timing sequences of DoS attacks on six channels are shown in Figure 5.

Figure 5. DoS attack on six channels

For researching consensus control based on dynamic ETM under the DoS attacks, the time-varying reference command must be considered. Because the consensus control and ETM will not work if the command is a constant when MAS has been stable. In addition, the system stability will not suffer from the DoS attacks, since the desired state is constant whether it exists or not a DoS attack.

In the proposed event-triggered resilient controller with the DoS attack disturbance observer scheme, the actuator control signal for each agent is adopted as (33). The virtual control signal is adopted as (24), which follows the updated laws as (25) and (26). The disturbance observer is designed as (14) and (19). Observer parameters are updated by (20)–(22). The actuator triggers follow the dynamic ETM as (30) and (31), where the dynamic parameters are satisfied (32).

The initial states of agents are taken as , , , , , , , , , , , , , , , , , , , , , , , . For the disturbance observer, k _d  = diag(1,1,1), k _δ  = diag(1,1,1), ε ^* = 0.2, and k _ε  = 0.3. For the virtual control signal, k ^* = diag(10,10,12), k _c  = diag(5,5,5), k _η  = diag(1,1,1), and h _η  = 0.5. For the HOD, a ₀ = 2, a ₁ = 10. For the actuator control signal, k _ζ  = diag(10,20,10). For the ETM, h ₁ = 0.5, k _μ  = 5, and k _{ζ μ}  = diag(0.5,0.5,0.5).

4.2. Simulation scenario 1

Considering periodic DoS attacks, we use the time-varying reference command in the control scheme to verify the resilient capability of the proposed controller without introducing the state predictor in the first step. By using the state holding of the agent during the DoS attacks proposed in Section 2, the resilient control effect is as Figure 6.

Figure 6. Attitude response of multi-UAV with state holding under DoS attacks

In order to show the MAS tracking effect by the proposed resilient controller, UAVs attitude tracking effectiveness simulation results on three attitude angles [μ,α,β] from to reference signal are shown in Figure 7. It shows that the tracking performance on three attitudes can be guaranteed by the proposed resilient controller. The tracking error will exist during the DoS attack as the reference command is unavailable, after which the tracking error will be eliminated.

Figure 7. State response on three attitudes under DoS attacks

Figure 8. Attitude tracking error of multi-UAV with state holding

The tracking error between agent state signal and reference state signal x ⁰(t) is defined as . The attitude tracking error [μ _i ,α _i ,β _i ] of UAVs are shown in Figure 8. It can be shown that the maximum of the tracking error is close to 0.01 in the simulation due to the periodicDoS attacks. Besides, the error will approach zero during the duration of DoS-off. The resilient ability can be verified.

4.3. Simulation scenario 2

For comparison, using the same reference signal, the resilient controller is utilized with the state predictor in the second step. The control performance of the three attitude angles [μ,α,β] from is shown as Figure 9. The tracking performance on three attitudes is shown in Figure 10.

Figure 9. Attitude response of multi-UAV with ARIMA under DoS attacks

Figure 10. State response on three attitudes under DoS attacks

Compared with the resilient control scheme without a state predictor, state tracking trajectories of followers are smoother during the duration of the DoS-on attack. This means that the system has better security. The error of attitude tracking is given in Figure 11.

Figure 11. Attitude tracking error of multi-UAV with ARIMA

The attitude tracking error is less than 1 × 10⁻³ by utilizing the ARIMA under the periodic DoS attacks, which is much lower than that without the ARIMA state predictor. The merit of introducing ARIMA into the resilient control scheme can be verified.For demonstrating the estimation effectiveness of the proposed indirect disturbance observer, the reconstruction result of the disturbance from the observer in the situation of the proposed controller by introducing the predictor is shown in Figure 12.

Figure 12. Disturbance observation results for attitudes of multi-UAV

In order to verify the consensus control effectiveness of the proposed control scheme, define the consensus evaluating indicator as

(59)

where respects the average states information of all follower UAVs. According to the consensus control measurement error (10), the evaluating indicator (59) can be used to describe the consensus measurement error of an agent.

Figure 13. Attitude tracking consensus error of multi-UAV

Figure 13 shows the simulation results about consensus error by using the evaluation indicator (59). It can be seen from Figure 13 that there is an obvious consensus deviation at the initial moment of simulation due to the different initial states of the multi-UAV. After a short adjustment by the proposed controller, MAS can achieve convergence of consensus. Further, the consensus errors of multi-UAV in three attitudes [μ _i , α _i , β _i ] are less than 5 × 10⁻⁴ even by utilizing a time-varying reference command following ETM under the periodic DoS attacks. The consensus performance is never compromised by network attacks and time-varying reference commands. The attitude tracking performance can be ensured simultaneously. It can be illustrated that the proposed resilient controller has the ability to maintain the tracking performance while ensuring the consensus under the periodic DoS attacks.

The sequence of events triggered by four agents during 1 s is described in Figure 14. In the simulation scenarios, the initial controller refresh frequency is set as 1 × 10³ Hz. As shown in Figure 14, the controller trigger frequency is far less than the control system refresh frequency. As the time-varying attitude reference command is deployed, the controller trigger must exist for guaranteeing the system consensus and tracking capability.

Figure 14. Responses of triggered time for multi-UAV

5. Conclusion and future work

This paper investigates a resilient control method for MAS under periodic DoS attacks. A data-based state prediction method is introduced into a model-based controller scheme to combat DoS attacks. The prediction method adopted in this paper is to design a state predictor to estimate the state of neighbor agents under the DoS attacks. State predictors can connect the feedback loop that is disrupted by DoS attacks. For the prediction error caused by the data-based state prediction method, this paper proposes a disturbance observer to compensate for the error which is regarded as an uncertain disturbance. Furthermore, in order to guarantee the security and consensus of MAS, this paper proposes a resilient controller based on dynamic ETM. Finally, the effectiveness of the proposed resilient consensus control method is shown by numerical simulations. Meanwhile, the resilient capability of the proposed controller against the periodic DoS attacks has been demonstrated by the comparative simulations. In this paper, a resilient control method is developed against network attacks with constraint conditions without considering more general types of attacks. In our future study will be designed by using the switching function form. Then, more general DoS attacks and other kinds of network attacks will be considered, such as random DoS attacks and false data injection attacks to expand the application of the proposed resilient control method.

Conflict of Interest

The authors declare that they have no conflict of interest.

Data Availability

No data are associated with this article.

Authors’ Contributions

Haichuan Yang: Conceptualization, Methodology, Draft. Ziquan Yu: Writing – review & editing. Youmin Zhang: Writing – review & editing.

Acknowledgments

We thank the anonymous reviewers for their helpful comments.

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 61833013, 62003162, 62233009), Natural Science Foundation of Jiangsu Province of China (Nos. BK20200416, BK20222012), China Postdoctoral Science Foundation (Nos. 2020TQ0151, 2020M681590), Fundamental Research Funds for the Central Universities (No. NS2021025), Industry-University Research Innovation Foundation for the Chinese Ministry of Education (No. 2021ZYA02005), Science and Technology on Space Intelligent Control Laboratory (No. HTKJ2022KL502015), Aeronautical Science Foundation of China (No. 20200007018001), and Natural Sciences and Engineering Research Council of Canada.

References

Haichuan Yang received the M.S. degree in control theory and control engineering from Xi’an University of technology, Xi’an, China, in 2020. He is currently pursuing the Ph.D. degree in control science and engineering with Nanjing University of Aeronautics and Astronautics, Nanjing, China. His current research interests include resilient cooperative control of unmanned aerial vehicles and their applications.

Ziquan Yu received the Ph.D. degree in control science and engineering from Northwestern Polytechnical University, Xi’an, China, in 2019. From 2017 to 2019, he was a joint Ph.D. student supported by the China Scholarship Council with the Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, Montreal, QC, Canada. He is currently with the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His current research interests include fault-tolerant cooperative control of safety-critical systems, and guidance, navigation, and control of unmannedaerial vehicles.

Youmin Zhang received the B.S., M.S., and Ph.D. degrees in automatic control from Northwestern Polytechnical University, Xi’an, China, in 1983, 1986, and 1995, respectively. He is currently a Professor with the Department of Mechanical, Industrial and Aerospace Engineering and the Concordia Institute of Aerospace Design and Innovation, Concordia University, Montreal, QC, Canada. His current research interests include guidance, navigation, and control, fault detection and diagnosis, fault-tolerant control, and remote sensing with applications to unmanned aerial/space/ground/marine vehicles, smart grids, smart cities, and cyber-physical systems.

All Figures

	Figure 1. A classical periodic DoS attack
In the text

	Figure 2. Comparison of the two methods for state prediction
In the text

	Figure 3. State prediction error of two methods
In the text

	Figure 4. Directed communication topology of MAS
In the text

	Figure 5. DoS attack on six channels
In the text

	Figure 6. Attitude response of multi-UAV with state holding under DoS attacks
In the text

	Figure 7. State response on three attitudes under DoS attacks
In the text

	Figure 8. Attitude tracking error of multi-UAV with state holding
In the text

	Figure 9. Attitude response of multi-UAV with ARIMA under DoS attacks
In the text

	Figure 10. State response on three attitudes under DoS attacks
In the text

	Figure 11. Attitude tracking error of multi-UAV with ARIMA
In the text

	Figure 12. Disturbance observation results for attitudes of multi-UAV
In the text

	Figure 13. Attitude tracking consensus error of multi-UAV
In the text

	Figure 14. Responses of triggered time for multi-UAV
In the text