Adaptive Event-Triggered Path Tracking Control for Self-Driving Vehicles Against Denial-of-Service Attacks

This paper focuses on the secure path tracking control of wireless self-driving vehicles (SDVs). To address the challenges of network security and limited network bandwidth in the design of wireless SDVs, a co-modeling approach considering denial-of-service (DoS) attack and event trigger mechanism (ETM) is introduced for path tracking control of SDVs. To achieve a high level of tracking control performance, the threshold of the proposed adaptive ETM is designed to adjust dynamically based on the state information of the SDVs, resulting in better tracking performance. Additionally, a novel distributed control strategy is developed for SDVs using a leader-following approach, and the parameters of the distributed controllers and adaptive ETMs are obtained to meet the path tracking performance requirements and ensure effective communication through the use of Lyapunov stability theory and linear matrix inequality (LMI) techniques. The proposed method is verified through a numerical simulation of SDVs subjected to DoS attacks, demonstrating its effectiveness.


I. INTRODUCTION
With the rapid advancement of autonomous driving technology and the proliferation of intelligent vehicles, the study of path tracking control has gained widespread attention and the use of self-driving vehicles (SDVs) in various applications is growing.Path tracking control is commonly utilized in the self-driving of mobile devices such as autonomous vehicles, unmanned aerial vehicles (UAVs), and mobile robots (MRs) [1], [2], [3], [4].
The main objective of path tracking control is to make the mobile device track a given reference path, which is a significant important issue in the field of autonomous driving.Recently, the design of communication mechanisms and control strategies for wireless SDVs has attracted much attention.In [5] and [6], future error of SDVs was estimated to prevent rollover while tracking desired trajectory by using the method of model predictive control (

MPC). A framework
The associate editor coordinating the review of this manuscript and approving it for publication was Heng Wang .for trajectory planning and control that employs artificial potential fields, as described in [7], is proposed to generate the target trajectory and mitigate the steady-state error.In [8], a preview steering control design that takes into account both communication and steering delays was introduced to achieve precise, smooth, and computationally efficient trajectory tracking of SDVs.The authors in [9] proposed a path tracking method that combines the characteristics of agricultural vehicles with nonlinear MPC to adapt to realworld scenarios.Based on these prior studies, it presents a significant challenge to design control systems for networked SDVs subject to cyber-attacks.Recent research on network-based SDVs considering cyber-attacks, such as [11], [12], and [13] has been extensively discussed.In [14], the distributed observer is constructed to acheive cooperative output regulation in multi-agent systems (MASs).Furthermore, distributed platooning control under the replay attacks based on proportional integral observers is studied in [15].However, there is still much room for improvement in terms of control efficacy, which is the main motivation of this study.
For networked path tracking control systems, the transmission of data over a network is prone to various challenges, such as time delays [16], packet dropouts [17], and limited communication bandwidth [18], which can compromise the stability and tracking performance of the system.To tackle these issues, two different mechanisms, Timetriggered mechanism (TTM) and Event-triggered mechanism (ETM) [21], have been proposed.TTM ensures that data sampling and transmission are conducted in a fixed time interval [19], [20], making it easy to implement in a network control system.However, this approach can result in a significant waste of network bandwidth and resources, as a large number of redundant data samples are transmitted.On the other hand, the data packet under the ETM is transmitted only when the event trigger condition is violated, reducing the communication pressure [22], [23], [24].Several ETMs have been discussed in the literature, and their applicability has been demonstrated in different systems, such as microgrid power sharing and multi-robot system queue control [25], [26].Studies have also shown the use of event-triggered impulsive control in achieving quasi-synchronization in heterogeneous dynamic networks, as well as reducing channel occupation [27].In [28], an ETM based on segment-weighted information is presented to improve the network scheduling.In [29], event-triggered output feedback control was studied for uncertain nonlinear systems.Recently, many improved ETMs, such as the design of an adaptive triggering threshold that depends on the state information of the system have been proposed [30].Futhermore, an adaptive ETM based on discrete-time is proposed in [31].In [32], authors develop a predictive control system with a machine-learning-based ETM that considers a cost function that covers the past, current, and future information.
The control signal of wireless SDVs is transmitted over the network, making it vulnerable to network attacks.To address this issue, numerous studies have been conducted over the years to enhance the security of control systems against cyber-attacks.For example, the issue of DoS attacks on networked filtering was discussed in [33] and [34] and the formation control of nonlinear multi-agent systems was studied in the presence of DoS attacks in [35].A model that considers both ETM and DoS attacks was established for networked 5-DoF suspension system in [36].In [37], a switched sampleddata approach is proposed to resist the malicious influence of DoS attack on vehicle platoon control.However, few research on the secure trajectory control for SDVs in a networked environment, which motivated this investigation.
In the previous research on the internet of vehicles, the conventional V2V approach is widely applied, which means that all vehicles are undifferentiated.To ensure that vehicles can communicate normally when the data of one of the vehicles can not be received successfully, inspired by multi-agent systems, a leader-following approach is presented.This paper aims to address the issue of trajectory consensus control of wireless SDVs under DoS attacks.The main contributions of the paper are: 1) An improved adaptive ETM is proposed to address the problem of limited communication bandwidth.The triggering thresholds of the proposed adaptive ETM is adjusted based on the state information of SDVs.This approach reduces the number of triggering events, relieves the network channel pressure, and improves control efficiency.2) A novel leader-following approach is presented to achieve trajectory consensus control and ensure the stability of wireless SDVs.This approach simplifies the network structure and improves control performance with an appropriate communication topology.Unlike previous research efforts, this study considers external disturbance and DoS attacks, providing a more realistic representation of the actual situation.
Notations are as follows: ⊗ represents Kronecker product.I n is a identity matrix with n × n dimension.diag l 1 , l 2 , . . ., l n is a diagonal matrix with elements l i .λ max (P) and λ min (P) denote the maximum eigenvalue and the minimum eigenvalue of the matrix P, respectively.

II. PROBLEM FORMULATION A. MODELING OF AGENT-BASED SDVs CONSENSUS CONTROL
Suppose the SDVs have one leader and N followers, and denote G = (P, S, Â) as a directed graph, where P = p 1 , p 2 , . . ., p N and S ⊆ P × P are the sets of nodes and edges, respectively.A directed edge from p j to p i of G is represented by is a weighted adjacency matrix with a ij ≥ 0, and it is assumed that a ii = 0 and a ij > 0 if s ji ∈ S. A directed path from p j to p i is a sequence of distinct nodes p σ 1 , p σ 2 , . . ., p σ n with p σ 1 = p j and p σ n = p i , such that (p σ i , p σ i+1 ) ∈ E for i = 1, 2, . . ., N − 1.The Laplacian matrix of G is defined as Ā = Ã− Â where Ã = diag Ã1 , Ã2 , . . ., ÃN and Ãi = a ij .

B. DYNAMICS MODEL
In the section, we firstly give a diagrammatic sketch of path tracking control as shown in Fig. 1.From Fig. 2, the dynamic steering equations of the 2-DOF SDV are described as follows [39]: where the symbols are given in Table 1.
Similar to [40], F yf and F yr are depend on slip angle α f and α r , which are given respectively by:   where c f and c r represent the generalized cornering stiffness of the front and rear wheels, the α f and α r are defined as: Combining ( 1)-( 3), we can deduce the precise equations of the 2-DOF SDV as: By undergoing a straightforward transformation, the following formulas can be derived: For the i-th SDV, we define the state vector T and the control input signal u i (t) = δ fi (t), one can obtain the dynamic of the i-th SDV as follows: where Considering the disturbance of each SDV, one can obtain the equetions of i-th SDV and the leader SDV: where z i (t) ∈ R n and z 0 (t) ∈ R n are the state vector of the i-th following SDV and the leader SDV, respectively.
Assumption 1: In this study, the communication graph between the following SDVs and the leader SDV is assumed to be a directed graph that contains a spanning tree, and the leader is the root node.

C. PERIODIC DoS ATTACK MODEL
The network security should be considered since the signal of SDVs is transmitted over the wireless network.Here, we consider a periodic DoS attack that aim to block the network channels intermittently, and the stability of wireless-based SDVs will be seriously threatened.The periodic DoS attack is modeled as where and T is the attack period; T s is length of the sleep duration satisfying 0 < T s < T .In ( 8), (t) = 1 denotes the attack in sleep duration; otherwise, (t) = 2 indicates network channel is under DoS attacks.Evidence shows that if adversaries launch DoS attacks on SDVs, a large amount of useful control information is blocked, resulting poor tracking performance.

D. DISTRIBUTED ADAPTIVE EVENT-TRIGGERED MECHANISM
In order to alleviate the load of network, an ETM is introduced to select some necessary data as the control input.As is shown in Fig. 3, the signal of SDVs is periodically sampled.The transmission instant is not the same as sampling instant but determined by where and δ 0 ,δ m and ρ are given positive constants; i > 0 is a weight matrix to be designed; h is the sampling period; lh = t i n,k h + η i h with η i ∈ 0, 1, 2, . . ., η imax and η imax h < t i n,k+1 h − t i n,k h.In addition, k max satisfies t i n,k max h < nT + T s .Remark 1: In (9), the set of releasing instants during [nT + T s , (n+1)T ) is ∅.It means that the data packets are discarded during the active period of DoS attacks.Now, we construct the following tracking error controller for the i-th SDV: where K ∈ R 1×n is the local controller gain to be designed, a ij is the coupling weight between followers.That is a ij > 0 if the j-th SDV can deliver information to the i-th SDV; otherwise a ij = 0. b i > 0 is the coupling weight between leader and each followers.
Remark 2: The leader-following approach is introduced in event-triggered condition and controller design as an effective method for simplifying the network structure and improving tracking performance in MASs, as previously demonstrated in [11].

E. CLOSED-LOOP OF SDVs
From the definition in (9), one knows that there exist k max triggering intervals during the time interval I 1,n .As is shown in Fig. 3, define the triggering interval Defining the sample intervals within the k-th triggering interval as From the definition in (9), one knows that Define the consensus error as x i (t) = z i (t) − z 0 (t), then we can obtain from ( 9) and ( 10), where Combining ( 7), ( 9) and ( 10), the closed-loop tracking error system of SDVs can be modeled as the following switched system: where The aim of this paper is to design the distributed tracking error controller in (13) and the adaptive ETM in (9) using the leader-following approach for SDVs DoS attacks to ensure the SDVs (14) achieve exponential tracking consensus.

III. MAIN RESULTS
In this section, our objective is to achieve exponential stability of wireless SDVs in (14).To do so, we will first obtain the conditions for stability and then provide a design method for both the tracking error controller and the novel adaptive ETM.Theorem 1: Given scalars δ, h, τ M , r, α > 0 and the gain matrix K , if there exist positive definite matrices P , Q , R > 0, i ( = 1, 2; i = 1, 2, 3, . . ., N ) and the matrix S with appropriate dimensions such that < 0, (15) where Then, the wireless-based closed-loop tracking error system in ( 14) is exponentially stable with the decay rate ϑ = ϖ 2T .
136594 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
In the following, we will discuss the case for = 1 and = 2, respectively.For = 1, deriving (22) along the trajectory of ( 14), one can obtain Applying the Jensen's inequality [43] leads to where From the definition of δ i (t), one knows that δ 0 ≤ δ i (t) ≤ δ 0 + δ m = δ.Then one can obtain e T (t) e T (t) < H T (t) ˆ H(t). ( from the adaptive ETM in (9), where H(t) = x(t −τ (t))−e(t) and = diag 1 , 2 , . . ., N .Applying Schur complement in (15), we can get and combining ( 23)-( 25), one can obtain where , and it yields that From Theorem 1, we know that where nT = t 1,n ≤ t 2,n = nT + T s .Similar to the method in [44], taking the derivation along the trajectory of the closed-loop tracking error system (14), one can obtain from (30), we can easily get Thus, for t ∈ I 1,n , it follows from ( 29) and ( 31) that Recalling the definition of Combining ( 32) and ( 33), we have Similarly, for t ∈ I 2,n , one has Let M = max e ϖ Ts T , 1 ϵ 2 , and we have From ( 22), we have where 1 = min λ min (P i ) , 2 = max λ max (P i ) + Combining (36) and (37), we can conclude that From Definition 1, one can conclude that the closed-loop tracking error system of SDVs in ( 14) is exponentially stable under periodic DoS attacks.This completes the proof.
Next we will design the parameters of the tracking error controller in (13) and the adaptive ETM in (9) based on Theorem 1.
Theorem 2: Given scalars δ, h, τ M , r and α > 0 ( = 1, 2), the tracking error system of SDVs in ( 14) is exponentially stable if there exist positive definite matrices P , Q , R , ˜ i ( = 1, 2; i = 1, 2, 3, . . ., N ), and matrices S and K such that where Then, the distributed controller gains and the weight matrix of the proposed adaptive ETM can be obtained by K = K P−1 15) we can obtain

IV. SIMULATION
In this section, the effectiveness of the proposed adaptive ETM-based control method is demonstrated through a numerical example.The simulation involves five SDVs with parameters in Table 2.The communication network topology is shown in Fig. 4.
Then, we can get the following coefficient matrices: Select the following initial parameters: h = 0.01, δ = 0.07, τ M = 0.03 and r = 20.Similar to the method of Algorithm 1 in [44], we can get α 1 = 0.01, α 2 = 0.005, µ = 2, ϵ 1 = 0.95, ϵ 2 = 0.95, T = 0.5 and T s = 0.4.From Fig. 4, we can obtain the Laplacian matrix and adjacency matrix as follows: From Theorem 2, the gain of path tracking controller and the parameters of the proposed adaptive ETM can be solved as follows: Suppose the initial states and the disturbance of road conditions are z 18t sin(t), 0], respectively.The responses of sideslip angle ς and yaw rate γ of SDVs are depicted in Fig. 5.It can be concluded that the consensus performance of path tracking control for SDVs under DoS attacks can be ensured using the proposed control and communication strategy.
To examine the effectiveness of the communication strategy, we present the sequences of releasing instants and the releasing intervals for 4 SDVs in Fig. 6.It can be observed that the intervals of data releasing instant are fixed (with h = 0.01s) but vary over time determined by distributed adaptive ETMs.The threshold of the proposed adaptive ETM is presented in Fig. 7, which indicates that when the tracking error tends to be stable, the threshold approaches to an equilibrium point with a larger value compared to the oscillating region   of the SDV states.As result, during these periods, the data release rate (DRR) is higher than the one during other periods, which can be observed in Fig. 6.It can also be calculated that the number of data releases (NDRs) of 4 SDVs within 0-3s are all 300 when using TTM, whereas under the proposed distributed adaptive ETM the number of data releases is much lower than that under the TTM.The details are given in Table 3, from which one can see that the DDRs are not more than 23.67%.It means that the proposed communication strategy can effectively reduces network burden.

V. CONCLUSION
In this study, the problem of the path tracking control for SDVs subject to DoS attacks has been investigated.The tracking error model of SDVs is converted into a switched system model due to the DoS attack, and the distributed adaptive ETM is proposed to mitigate the communication network burden, with a threshold designed to adapt the variation of SDVs.Sufficient conditions are derived to obtain both the SDV controller gains and the weight matrices of distributed adaptive ETMs.The effectiveness of path tracking control consensus is manifested though numerical SDVs subject to DoS attacks.Future work will focus on improving the dynamic performance of attitude control by investigating the physical model of wireless SDVs.

FIGURE 1 .
FIGURE 1. Diagrammatic sketch of path tracking control.

FIGURE 3 .
FIGURE 3. The valid releasing sequence under the AETM and DoS attacks.

FIGURE 5 .
FIGURE 5.The sideslip angle and yaw rate of SDV i under DoS attacks.

FIGURE 6 .
FIGURE 6.The release instants and intervals of the i -th SDV.

FIGURE 7 .
FIGURE 7. The threshold of distributed adaptive ETMs.

TABLE 1 .
Physical meaning of the states.

TABLE 3 .
The i -th NDR and DDR.