Secure platooning control of connected vehicles against data injection attacks

Juan Liu; Shuyi Tang; Sheng Gao; Zhuping Wang; Hao Zhang; Gerhard Rigoll

doi:10.1051/sands/2025002

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Volume 4 (2025)

Security and Safety, 4 (2025) 2025002

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Security and Safety in Artificial Intelligence

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Issue		Security and Safety Volume 4, 2025 Security and Safety in Artificial Intelligence


Article Number		2025002
Number of page(s)		16
Section		Industrial Control
DOI		https://doi.org/10.1051/sands/2025002
Published online		24 July 2025

Security and Safety, Vol. 4, 2025002 (2025)

Research Article

Secure platooning control of connected vehicles against data injection attacks

Juan Liu¹, Shuyi Tang¹, Sheng Gao¹, Zhuping Wang¹^,2^*, Hao Zhang¹^,2^* and Gerhard Rigoll³

¹ School of Electronics and Information Engineering, Tongji University, Shanghai, 201804, China
² Shanghai Research Institute for Intelligent Autonomous Systems, Shanghai, 201210, China
³ Institute for Human-Machine Communication, Technical University of Munich, Munich 80333, Germany

^* Corresponding authors (email: elewzp@tongji.edu.cn (Zhuping Wang); zhanghao@tongji.edu.cn (Hao Zhang))

Received: 25 October 2024
Revised: 14 December 2024
Accepted: 20 January 2025

Abstract

This paper presents an attack-defense formation control framework for connected vehicle platoons. This framework effectively addresses the challenges posed by malicious attacks on multi-intelligent vehicle systems in open network environments. First, from the attacker’s perspective, an optimal data injection attack strategy is designed, considering energy constraints and the complexities of the communication environment. This strategy aims to disrupt the multi-intelligent vehicle system with minimal energy expenditure. Next, from the defender’s perspective, a formation compensation control strategy is proposed to enhance the system’s resilience against such attacks. This defense control strategy includes real-time monitoring and adjustments of communication data between vehicles, ensuring that vehicle formation remains stable and coordinated even in the case of data injection attacks. Finally, simulation experiments are conducted to validate the effectiveness and robustness of the proposed methods.

Key words: connected vehicle platoon / data injection attack / defense control / formation

Citation: Liu J, Tang S and Gao S et al. Secure platooning control of connected vehicles against data injection attacks. Security and Safety 2025; 4: 2025002. https://doi.org/10.1051/sands/2025002

© The Author(s) 2025. Published by EDP Sciences and China Science Publishing & Media Ltd.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

With the rapid development of intelligent transportation systems, multi-agent vehicle platooning systems, as an important application of cyber-physical systems (CPS), are gradually showing great potential in fields such as autonomous driving and vehicle cooperative control. These systems achieve efficient coordination between vehicles through wireless communication and cooperative control mechanisms, thereby improving road safety, driving efficiency, and energy utilization [1–3]. However, as platoon systems become increasingly complex and information-driven, network security issues have gradually become a major bottleneck restricting their widespread application. In particular in open network environments, attackers can initiate network attacks on multi-agent systems in various ways, leading to a degradation in system performance or even system collapse [4, 5].

Among them, false data injection (FDI) attacks are a common and serious form of network attack, widely applied in power grids [6], control systems [7], and connected vehicles [8]. FDI attacks work by falsifying control signals or sensor data, causing the system to make incorrect control decisions, which in turn affects the stability and coordination of the entire platoon. In multi-agent vehicle platoon systems, attackers can not only tamper with the control information of individual vehicles but also introduce network transmission disturbances, causing the entire platoon system to collapse [9, 10].

Currently, research on defending against FDI attacks focuses mainly on two types of strategies: attack detection [11] and attack defense [12]. Attack detection methods, such as chi-square detection [13] and entropy detection [14], are widely applied in multi-agent systems and can detect potential attacks by monitoring abnormal signals in the network. However, these traditional methods have certain limitations. First, attackers can exploit the shortcomings of existing detection mechanisms to design more covert and complex attack signals, avoiding detection using traditional methods [15]. Second, most existing detection methods focus on the identification of attack signals, and lack sufficient dynamic response and stability analysis of the system under complex attack scenarios [16]. Therefore, despite some progress in attack detection technology, it remains difficult to achieve effective defense when facing highly complex attacks. In contrast to attack detection, attack defense strategies focus more on taking appropriate control actions when an attack occurs to ensure system stability and security [17, 18]. In recent years, defense strategies based on control theory have become a research hotspot. Taking into account both attackers and defenders, optimal defense strategies have been proposed [19, 20]. These methods can effectively enhance the system’s ability to resist known attacks.

In multi-agent vehicle systems, dedicated short range communication (DSRC) enables each vehicle to acquire real-time state information from other vehicles in the platoon, allowing it to adjust its own driving strategy, optimize energy management, and coordinate scheduling to complete tasks with minimal energy consumption. However, with the continuous evolution of attack strategies, DSRC communication channels face challenges in defending against various complex attacks. For instance, existing literature mentions that sliding mode observers can be used to monitor the expected distance differences in vehicle communication networks, responding to both attack and non-attack conditions [21]. However, such sliding mode observers have high requirements for external attack signals and can only handle relatively simple attacks. Furthermore, real-time diagnostic strategies based on sliding mode control and adaptive estimation theory, while effective in dealing with specific threats like denial of service (DOS) attacks [22], face significant difficulties in detecting more covert attacks, such as data injection attacks. Currently, there is limited research on defending multi-agent vehicles against data injection attacks. Therefore, designing secure control algorithms to safeguard against network attacks and enhance the security of multi-agent vehicle systems has become a key focus in current research.

To address the challenges mentioned above, this paper proposes a novel defense control framework against false data injection attacks. The research primarily solves the following problems: In response to the specific nature of false data injection attacks in multi-agent vehicle platoon systems, a defense control framework based on the attack is proposed. This framework comprehensively considers multiple factors, including network topology and the attacker’s energy constraints, and designs optimal attack and defense control strategies. The main contributions of this chapter can be summarized as follows:

(1)
A novel attack-defense formation control framework for connected vehicle platoons is proposed, considering the perspectives of both attackers and defenders to develop their respective optimal strategies.
(2)
The optimal strategy for data injection attacks is introduced, in which attackers inject data that combine false information and its derivatives into a healthy multi-agent vehicle system, utilizing minimal energy to cause significant damage and disrupt the system’s equilibrium position.
(3)
Unlike existing defense methods [15, 16], our approach emphasizes the system’s dynamic response and the design of control strategies to combat 0-day attacks. A 0-day attack refers to an attack strategy that exploits previously unknown vulnerabilities. By analyzing these stealthy attack strategies, we provide a secure defense strategy that guarantees the stability of the vehicle platoon system under optimal data injection attacks.

2. Problem formulation

2.1. Control objective

As shown in Figure 1, the platoon consists of a leader vehicle 0 and N following vehicles, i = 1, 2, …, N. Each follower vehicle i transmits its position q_i and velocity p_i information to the data processing center through DSRC. The data processing center sends the computed control signals to each vehicle via DSRC. When the network is subjected to an FDI attack, the previous control signal u_i(t) for each following vehicle i will be replaced by ${\tilde{u}}_{i} (t)$ $\tilde{u}_i(t)$ , and the stability of the platoon will be compromised.

Figure 1.

The vehicular platoon under FDI attack

To address this issue, the following assumptions are made.

Assumption 1.

Assume that the original multi-vehicle control system and the attacker’s communication channel are ideal, without considering the effects of communication noise, packet loss, or delay. Additionly, the attacker’s energy is sufficient to implement an optimal attack strategy.

The following control objectives are proposed to achieve the stable formation of the multi-agent system under FDI attacks.

Definition 1.

Consider a multi-agent vehicle platoon subjected to FDI attacks, the objective is to ensure that all vehicles maintain the same velocity and desired driving distance. That is

1) lim_{t → ∞}∥p_i(t)−p₀(t)∥ = 0,

2) lim_{t → ∞}∥q_i(t)−q_i − 1(t)∥ = d_des,

where p_i is the velocity of the following vehicle i, p₀ is the velocity of leader vehicle 0. q_i is the position of the following vehicle i, q_i − 1 is the position of the following vehicle i − 1, d_des is the desired distance between the following vehicle i and i − 1.

2.2. System Dynamic of Connected Platoon

In studying the platoon control of multi-vehicle intelligent systems, the initial step is to model intelligent vehicles within the system. This can be achieved by representing each vehicle as a node and adopting a second-order model. The lead vehicle can then be represented as follows:

$\begin{matrix} \begin{matrix} {\dot{q}}_{0} (t) & = p_{0} (t), \\ {\dot{p}}_{0} (t) & = u_{0} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{q}_0(t)&=p_0(t),\\ \dot{p}_0(t)&=u_0(t), \end{aligned} \end{aligned}$ (1)

where q₀(t) and p₀(t) denote the position and velocity of the lead vehicle, respectively. u₀(t) is the control input of the lead vehicle. The following vehicles can be represented as

$\begin{matrix} \begin{matrix} {\dot{q}}_{i} (t) & = p_{i} (t), \\ {\dot{p}}_{i} (t) & = u_{i} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{q}_i(t)&=p_i(t),\\ \dot{p}_i(t)&=u_i(t), \end{aligned} \end{aligned}$ (2)

where q_i(t), p_i(t) and u_i(t) denote the position, velocity and control input of the following vehicle i, respectively, i = 1, …N. Denote $x_{i} (t) = [\begin{matrix} q_{i} (t) \\ p_{i} (t) \end{matrix}]$ $x_{i}(t){=}{\left[\begin{matrix}{q_{i}(t)}\\{p_{i}(t)}\\\end{matrix}\right]}$ , taking the derivative of x_i(t), we have

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = [\begin{matrix} {\dot{q}}_{i} (t) \\ {\dot{p}}_{i} (t) \end{matrix}] = [\begin{matrix} p_{i} (t) \\ b r e v e_{i} (t) \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] [\begin{matrix} q_{i} (t) \\ p_{i} (t) \end{matrix}] + [\begin{matrix} 0 \\ 1 \end{matrix}] u_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_{i}\left(t\right)=\begin{bmatrix}\dot{q}_i(t)\\\dot{p}_i(t)\end{bmatrix}=\begin{bmatrix}p_i(t)\\u_i(t)\end{bmatrix} =\begin{bmatrix}0&1\\0&0\end{bmatrix}\begin{bmatrix}q_i(t)\\p_i(t)\end{bmatrix}+\begin{bmatrix}0\\1\end{bmatrix}u_i(t). \end{aligned} \end{aligned}$ (3)

Defining $A = [\begin{matrix} 01 \\ 00 \end{matrix}]$ $A=\left[\begin{array}{ll}{0}{1}\\{0}{0}\end{array}\right]$ , $B = [[\begin{matrix} 0 \\ 1 \end{matrix}]]$ $B=\left[\begin{bmatrix}{0}\\{1}\\\end{bmatrix}\right]$ , we have

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = A x_{i} (t) + B u_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_i\left(t\right)=Ax_i\left(t\right)+Bu_i\left(t\right). \end{aligned} \end{aligned}$ (4)

The control input u_i(t) can be rewritten as

$\begin{matrix} \begin{matrix} u_{i} (t) = {\dot{p}}_{i} (t) = [\begin{matrix} 0 & 1 \end{matrix}] [\begin{matrix} {\dot{q}}_{i} (t) \\ {\dot{p}}_{i} (t) \end{matrix}] . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} u_i(t)=\dot{p}_i(t)=\begin{bmatrix}0&1\end{bmatrix}\!\begin{bmatrix}\dot{q}_i(t)\\\dot{p}_i(t)\end{bmatrix}. \end{aligned} \end{aligned}$ (5)

Defining $C = [\begin{matrix} 01 \end{matrix}]$ $C{=}\left[\begin{array}{ll}0 1\end{array}\right]$ , then

$\begin{matrix} \begin{matrix} u_{i} (t) = C {\dot{x}}_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} u_i(t)=C\dot{x}_i(t). \end{aligned} \end{aligned}$ (6)

Substituting Equation (6) into Equation (5) results into,

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = A x_{i} (t) + B C {\dot{x}}_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_i\left(t\right)=Ax_i\left(t\right)+BC\dot{x}_i\left(t\right). \end{aligned} \end{aligned}$ (7)

The equivalent transformation of the above formula is obtained

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = {(I - B C)}^{- 1} A x_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_{i}(t)=\left(I-BC\right)^{-1}Ax_{i}(t). \end{aligned} \end{aligned}$ (8)

One of the primary objectives of multi-vehicle intelligent control is to ensure that each following vehicle maintains the same speed as the lead vehicle. To simplify the analysis, the following vehicle’s velocity is considered the output.

$\begin{matrix} \begin{matrix} y_{i} (t) = p_{i} (t) = [\begin{matrix} 0 & 1 \end{matrix}] [\begin{matrix} q_{i} (t) \\ p_{i} (t) \end{matrix}] = C x_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} y_{i}(t)=p_i(t)=\begin{bmatrix}0&1\end{bmatrix}\biggl [\begin{matrix}q_i(t)\\ p_i(t)\end{matrix}\biggr ] =Cx_{i}(t). \end{aligned} \end{aligned}$ (9)

Currently, many multi-agent vehicle formation control strategies employ a second-order modeling approach. However, such a method treats the multi-agent vehicle platoon as a mass-spring-damping system, which is overly idealistic and fails to account for factors such as mass, aerodynamic drag, and mechanical efficiency in real-world environments. Consequently, a third-order model for the multi-agent vehicles is proposed [23], with the lead vehicle denoted as follows:

$\begin{matrix} \begin{matrix} {\dot{q}}_{0} (t) = p_{0} (t), \\ {\dot{p}}_{0} (t) = a_{0} (t), \\ {\dot{a}}_{0} (t) = u_{0} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{gathered} \dot{q}_{0}(t)=p_{0}(t), \\ \dot{p}_{0}(t)=a_{0}(t), \\ \dot{a}_{0}(t)=u_{0}(t), \end{gathered} \end{aligned}$ (10)

where q₀(t),p₀(t),a₀(t),u₀(t) represent the position, velocity, acceleration and control inputs of the lead vehicle, respectively. The following vehicle is denoted as

$\begin{matrix} \begin{matrix} {\dot{q}}_{i} (t) = p_{i} (t), \\ \frac{τ_{T, i}}{r_{w, i}} T_{i} (t) = m_{i} {\dot{p}}_{i} (t) + C_{A, i} p_{i}^{2} (t) + m_{i} g f_{i}, \\ η_{i} {\dot{T}}_{i} (t) + T_{i} (t) = T_{d e s, i} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\dot{q}_{i}(t)=p_{i}(t), \\&\frac{\tau _{T,i}}{r_{w,i}}T_{i}(t)=m_{i}\dot{p}_{i}(t)+C_{A,i}p_{i}^{2}(t)+m_{i}gf_{i}, \\&\eta _{i}\dot{T}_{i}(t)+T_{i}(t)=T_{des,i}(t), \\ \end{aligned} \end{aligned}$ (11)

where q_i(t) and p_i(t) represent the position and velocity of the following vehicle i for i = 1, 2, 3, …, N. The variable η_i denotes the time delay constant of the longitudinal dynamic system for following vehicle i, m_i is the mass of following vehicle i, τ_T, i represents the mechanical efficiency of the transmission system of following vehicle i, and r_w, i is the wheel radius of following vehicle i. The term T_i(t)> 0 signifies the actual driving or braking torque of following vehicle i. Additionally, C_A, i is the lumped air resistance coefficient for following vehicle i, g is the gravitational acceleration constant, f_i is the rolling resistance coefficient of following vehicle i, and T_des, i(t) is the torque corresponding to the desired driving or braking force of vehicle i.

The position q_i(t) of the following vehicle i is denoted as

$\begin{matrix} \begin{matrix} {\dot{q}}_{i} (t) = p_{i} (t), \\ {\dot{p}}_{i} (t) = a_{i} (t), \\ {\dot{a}}_{i} (t) = \frac{1}{m_{i}} (τ_{_{T, i}} \frac{\dot{T_{i}} (t)}{r_{_{w, i}}} - 2 C_{_{A, i}} p_{_{i}} (t) {\dot{p}}_{_{i}} (t)) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\dot{q}_{i}(t) =p_{i}(t), \\&\dot{p}_{i}(t) =a_i(t), \\&\dot{a}_{i}\left(t\right) =\frac{1}{m_{i}}(\tau _{_{T,i}}\frac{\dot{T_{i}}(t)}{r_{_{w,i}}}-2C_{_{A,i}}p_{_i}(t)\dot{p}_{_i}(t)). \end{aligned} \end{aligned}$ (12)

Continuing with further derivation using ${\dot{a}}_{i} (t)$ $\dot{a}_i(t)$ , we obtain

$\begin{matrix} \begin{matrix} {\dot{a}}_{i} (t) & \begin{matrix} = & \frac{1}{m_{i}} (τ_{_{T, i}} \frac{T_{i} (t)}{r_{_{w, i}}} - 2 C_{_{A, i}} p_{i} (t) {\dot{p}}_{i} (t)) \end{matrix} \\ = \frac{1}{m_{i}} (τ_{T, i} \frac{T_{d e s, i} (t) - T_{i} (t)}{η_{i} r_{w, i}} - 2 C_{A, i} p_{i} (t) {\dot{p}}_{i} (t)) \\ = - \frac{1}{η_{i}} α_{i} (t) + \frac{1}{η_{i}} \frac{1}{m_{i}} (τ_{T, i} \frac{T_{d e s, i} (t)}{r_{w, i}} - C_{A, i} p_{i} (t) (2 η_{i} {\dot{p}}_{i} (t) + p_{i} (t)) - m_{i} g f_{i}) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{a}_i(t)&\begin{aligned}=&\frac{1}{m_i}(\tau _{_{T,i}}\frac{T_i(t)}{r_{_{w,i}}}-2C_{_{A,i}}p_i(t)\dot{p}_i(t))\end{aligned} \\&=\frac{1}{m_i}(\tau _{T,i}\frac{T_{des,i}(t)-T_i(t)}{\eta _ir_{w,i}}-2C_{A,i}p_i(t)\dot{p}_i(t)) \\&=-\frac{1}{\eta _i}\alpha _i(t)+\frac{1}{\eta _i}\frac{1}{m_i}(\tau _{T,i}\frac{T_{des,i}(t)}{r_{w,i}}-C_{A,i}p_i(t)(2\eta _i\dot{p}_i(t)+p_i(t))-m_igf_i). \end{aligned} \end{aligned}$ (13)

The nonlinear model accurately reflects the node dynamics of the actual vehicle in a real-world environment. However, analyzing performance changes under platoon control using this model can be challenging. Therefore, it is essential to linearize the model and adopt a feedback linearization strategy.

$\begin{matrix} T_{d e s, i} (t) = \frac{1}{τ_{_{T, i}}} (C_{_{A, i}} p_{_{i}} (t) (2 η_{_{i}} {\dot{p}}_{_{i}} (t) + p_{_{i}} (t)) + m_{_{i}} g f_{_{i}} + m_{_{i}} u_{_{i}} (t)) r_{_{w, i}}, \end{matrix}$ $\begin{aligned} T_{des,i}(t)=\frac{1}{\tau _{_{T,i}}}(C_{_{A,i}}p_{_i}(t)(2\eta _{_i}\dot{p}_{_i}(t)+p_{_i}(t))+m_{_i}gf_{_i}+m_{_i}u_{_i}(t))r_{_{w,i}}, \end{aligned}$ (14)

where u_{_i}(t) is the control input of the following vehicle i after feedback linearization. Consequently, the general formula of dynamic third-order modeling is obtained.

$\begin{matrix} \begin{matrix} {\dot{q}}_{i} (t) = p_{i} (t), \\ {\dot{p}}_{i} (t) = a_{i} (t), \\ {\dot{a}}_{i} (t) = - \frac{1}{η} a_{i} (t) + \frac{1}{η} u_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\dot{q}_i(t) =p_i(t), \\&\dot{p}_{i}(t) =a_i(t), \\&\dot{a}_i(t) =-\frac{1}{\eta }a_i(t)+\frac{1}{\eta }u_i(t). \end{aligned} \end{aligned}$ (15)

Define $x_{i} (t) = [\begin{matrix} q_{i} (t) \\ p_{i} (t) \\ a_{i} (t) \end{matrix}]$ $x_i(t){=}\begin{bmatrix}q_i(t)\\p_i(t)\\a_i(t)\end{bmatrix}$ , by deriving x_i(t), we have

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) & = [\begin{matrix} {\dot{q}}_{i} (t) \\ {\dot{p}}_{i} (t) \\ {\dot{a}}_{i} (t) \end{matrix}] = [\begin{matrix} q_{i} (t) \\ a_{i} (t) \\ {\dot{a}}_{i} (t) \end{matrix}] \\ = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & - \frac{1}{η} \end{matrix}] [\begin{matrix} q_{i} (t) \\ p_{i} (t) \\ a_{i} (t) \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ \frac{1}{η} \end{matrix}] u_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_i(t)&= \begin{bmatrix} \dot{q}_i(t) \\ \dot{p}_i(t) \\ \dot{a}_i(t) \end{bmatrix} = \begin{bmatrix} q_i(t) \\ a_i(t) \\ \dot{a}_i(t) \end{bmatrix} \\&= \begin{bmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&-\frac{1}{\eta } \end{bmatrix} \begin{bmatrix} q_i(t) \\ p_i(t) \\ a_i(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ \frac{1}{\eta } \end{bmatrix} u_i(t). \end{aligned} \end{aligned}$ (16)

Define $M = [\begin{matrix} 010 \\ 001 \\ 00 - \frac{1}{η} \end{matrix}], N = [\begin{matrix} 0 \\ 0 \\ \frac{1}{η} \end{matrix}]$ $M=\left[\begin{array}{lll}0 1 0\\0 0 1\\00-\dfrac{1}{\eta}\end{array}\right], N=\left[\begin{array}{l}0\\0\\\dfrac{1}{\eta}\end{array}\right]$ , we have

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = M x_{i} (t) + N u_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_i(t)=Mx_i(t)+Nu_i(t). \end{aligned} \end{aligned}$ (17)

The control input u_i(t) is as follows:

$\begin{matrix} \begin{matrix} u_{i} (t) = η {\dot{a}}_{i} (t) + a_{i} (t) = [\begin{matrix} 0 & 1 & η \end{matrix}] [\begin{matrix} {\dot{q}}_{i} (t) \\ {\dot{p}}_{i} (t) \\ {\dot{a}}_{i} (t) \end{matrix}] . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} u_i(t) =\eta \dot{a}_i(t)+a_i(t) =\begin{bmatrix}0&1&\eta \end{bmatrix}{\begin{bmatrix}\dot{q}_i(t)\\\dot{p}_i(t)\\\dot{a}_i(t)\end{bmatrix}}. \end{aligned} \end{aligned}$ (18)

Define $D = [\begin{matrix} 01 η \end{matrix}]$ $D{=}\left[\begin{array}{ll}01\eta\end{array}\right]$ , we have

$\begin{matrix} \begin{matrix} u_{i} (t) = D {\dot{x}}_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} u_i(t)=D\dot{x}_i(t). \end{aligned} \end{aligned}$ (19)

The equivalent transformation of the above equation is obtained as follows:

$\begin{matrix} \begin{matrix} {\dot{x}}_{i} (t) = {(I - N D)}^{- 1} M x_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{x}_{i}(t)=\left(I-ND\right)^{-1}Mx_{i}(t). \end{aligned} \end{aligned}$ (20)

Further, the model of the following vehicle can be rewritten as:

$\begin{matrix} \begin{matrix} y_{i} (t) = p_{i} (t) = [\begin{matrix} 0 & 1 & 0 \end{matrix}] [\begin{matrix} q_{i} (t) \\ p_{i} (t) \\ a_{i} (t) \end{matrix}] = S x_{i} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} y_{i}(t) =p_{i}(t) =\begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}q_i(t)\\p_i(t)\\a_i(t)\end{bmatrix} =Sx_{i}(t), \end{aligned} \end{aligned}$ (21)

where matrix $S = [\begin{matrix} 010 \end{matrix}]$ $S=\left[\begin{array}{lll}010\end{array}\right]$ . Based on the above analysis, the formulas obtained from both second-order and third-order modeling are similar. Therefore, the following analysis of attack strategies and formation control will be conducted using the third-order system as an example.

3. Control strategy design against data injection attacks

3.1. Optimal data injection attack strategy

Consider a multi-agent vehicle platoon under attack, as shown in Figure 1. The attacker compromises the control signal from the system controller by attacking the network communication channel. The control signal after the attack is denoted as ${\tilde{u}}_{i} (t)$ $\tilde{u}_i(t)$ , replacing the original control signal u_i(t), which is

$\begin{matrix} \begin{matrix} {\tilde{u}}_{i} (t) = u_{i} (t) + u_{ai} (t) + {\dot{u}}_{ai} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \tilde{u}_i(t)=u_i(t)+u_{ai}(t)+\dot{u}_{ai}(t), \end{aligned} \end{aligned}$ (22)

where u_ai(t) represents the injected signal for the data injection attack, and the derived signal is intended to enhance the effectiveness of the attack. u_i(t)=K_i x_i(t) represents the healthy control input of the following vehicle i, with K_i being the control gain of the following vehicle i.

The multi-agent vehicle system after the attack is represented as:

$\begin{matrix} \begin{matrix} {\dot{\tilde{x}}}_{i} (t) = (A + B K_{i}) {\tilde{x}}_{i} (t) + B u_{ai} (t), \\ {\tilde{y}}_{i} (t) = C {\tilde{x}}_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\dot{\tilde{x}}_i(t)=(A+BK_i) \tilde{x}_i(t)+Bu_{ai}(t),\\ &\tilde{y}_i(t)=C\tilde{x}_i(t). \end{aligned} \end{aligned}$ (23)

The difference between the actual velocity and the desired velocity of each following vehicle is represented as ΔE_i(t), which is as follows:

$\begin{matrix} \begin{matrix} Δ E_{i} (t) = y_{i} (t) - y_{des} = C x_{i} (t) - y_{des}, \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \Delta E_i(t)=y_i(t)-y_{des}=Cx_i(t)-y_{des}, \end{aligned} \end{aligned}$ (24)

where y_des = p₀, meaning that the velocity of all following vehicles matches the velocity of the leading vehicle p₀.

This paper proposes an attack strategy aimed at compromising the vehicle network from the attacker’s perspective and discusses the impact of this attack on the control of the multi-agent vehicle system. In other words, the attacker seeks to achieve the least energy expenditure while causing the greatest state deviation in the attacked system.

Definition 2.

The attacker compromises the control signal from the system controller by attacking the DSRC network communication channel. The attacker’s objective is to minimize the energy consumption required to inject the signal u_ai(t) into the platoon system Equation (23), while simultaneously maximizing the damage to the targeted platoon system.

Remark 1.

In practical scenarios, the attacker’s energy is limited and typically sourced from electrical power. Therefore, when analyzing attack strategies and their impacts, it is crucial to consider the attacker’s energy costs.

Based on the attacker’s objectives, the network attack problem in multi-agent vehicles can be framed as the following optimization problem [24].

Problem 1.

The attack problem can be formulated as

$\begin{matrix} \begin{matrix} max J_{i} = \int_{t_{i}}^{t_{r}} (Δ E_{i}^{T} (t) P_{i} Δ E_{i} (t) - u_{ai}^{T} (t) Q_{i} u_{ai} (t)) d t, \\ s.t. {\begin{matrix} Δ E_{i} (t) = C x_{i} (t) - y_{des}, \\ P_{i} \geq 0, Q_{i} > 0, \end{matrix} \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\max J_{i}=\int _{t_{i}}^{t_{r}}\left(\Delta E_{i}^{T}(t) P_{i} \Delta E_{i}(t)-u_{a i}^{T}(t) Q_{i} u_{a i}(t)\right) dt, \\&\text{ s.t.} \left\{ \begin{array}{l} \Delta E_{i}(t)=C x_{i}(t)-y_{d e s}, \\ P_{i} \ge 0, \quad Q_{i}>0, \end{array}\right. \end{aligned} \end{aligned}$ (25)

where P and Q are the weight matrices, P_i is a positive semi-definite matrix, and Q_i is positive definite matrix.

By analyzing the objectives of the attacker, the network attack problem for multi-intelligent vehicles can be transformed into the following optimal problem. Here, $Δ E_{i}^{T} (t) P_{i} Δ E_{i} (t)$ $\Delta E_{i}^{T}(t) P_{i} \Delta E_{i}(t)$ is used to represent the offset of the multi-intelligent vehicle system, and $u_{ai}^{T} (t) Q_{i} u_{ai} (t)$ $u_{a i}^{T}(t) Q_{i} u_{a i}(t)$ denotes the energy consumed by the attacker due to the inflicted attacks.

Theorem 1.

The optimal attack strategy for multi-vehicle system Equation (8) is given by u_ai(t)=Q_i⁻¹B_i^Tλ_i(t), as shown in Equation (22), where λ_i(t) satisfies

$\begin{matrix} \begin{matrix} {\dot{λ}}_{i} (t) = - C_{i}^{T} P_{i} C_{i} {\tilde{x}}_{i} (t) + y_{des} C_{i}^{T} P - {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{\lambda }_i(t)=-C_i^TP_iC_i\tilde{x}_i(t)+y_{des}C_i^TP-(A_i+B_iK_i)^T\lambda _i(t). \end{aligned} \end{aligned}$ (26)

In this case, the attack strategy consumes less energy, u_ai^T(t)Q_iu_ai(t), and causes more damage, ΔE_i^T(t)P_iΔE_i(t), to the multi-intelligent vehicles.

Proof:

Firstly, consider the optimization problem of Equation (25). Denote η_i(t)=ΔE_i^T(t)P_iΔE_i(t)−u_ai^T(t)Q_iu_ai(t). Then Equation (25) can be written as maxJ_i = ∫_t₀^t_fη_i(t) dt. According to the definition of ΔE_i(t) and 𝒰_ai(t), it can be obtained that

$\begin{matrix} \begin{matrix} η_{i} (t) & \begin{matrix} = {(C_{i} x_{i} (t) - y_{des})}^{T} P_{i} (C_{i} x_{i} (t) - y_{des}) - u_{ai}^{T} (t) Q_{i} u_{ai} (t) \end{matrix} \\ = x_{i}^{T} (t) C_{i}^{T} P_{i} C_{i} x_{i} (t) - y_{des} x_{i}^{T} (t) C_{i}^{T} P_{i} - y_{des} C_{i} P_{i} x_{i} (t) \\ + y_{des}^{2} P_{i} - u_{ai}^{T} (t) Q_{i} u_{ai} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \eta _i(t)&\begin{aligned}=(C_ix_i(t)-y_{des})^TP_i(C_ix_i(t)-y_{des})-u_{ai}^T(t)Q_iu_{ai}(t)\end{aligned} \\&=x_i^T(t)C_i^TP_iC_ix_i(t)-y_{des}x_i^T(t)C_i^TP_i-y_{des}C_iP_ix_i(t) \\&+y_{des}^2P_i-u_{ai}^T(t)Q_iu_{ai}(t). \end{aligned} \end{aligned}$ (27)

The Hamiltonian function is defined as

$\begin{matrix} \begin{matrix} H_{i} (η_{i}, λ_{i}, u_{ai}, t) = \frac{1}{2} η_{i} (t) + λ_{i}^{T} (t) [(A_{i} + B_{i} K_{i}) {\tilde{x}}_{i} (t) + B_{i} u_{ai} (t)] . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} H_i(\eta _i,\lambda _i,u_{ai},t)=\frac{1}{2}\eta _i(t)+\lambda _i^T(t)\Big [(A_i+B_iK_i)\tilde{x}_i(t)+B_iu_{ai}(t)\Big ]. \end{aligned} \end{aligned}$ (28)

According to the optimal control principle, the co-state equation is described as

$\begin{matrix} \begin{matrix} \frac{\partial H_{i}}{\partial u_{ai} (t)} & = - {\dot{λ}}_{i} (t) \\ = C_{i}^{T} P_{i} C_{i} {\tilde{x}}_{i} (t) - y_{des} C_{i}^{T} P + {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \frac{\partial H_i}{\partial u_{ai}(t)}&=-\dot{\lambda }_i(t)\\ &=C_i^TP_iC_i\tilde{x}_i(t)-y_{des}C_i^TP+\left(A_i+B_iK_i\right)^T\lambda _i(t). \end{aligned} \end{aligned}$ (29)

Take the partial derivative of Equation (29), one obtains

$\begin{matrix} \begin{matrix} \frac{\partial H_{i}}{\partial u_{ai} (t)} = - Q_{i} u_{ai} (t) + B_{i}^{T} λ_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \frac{\partial H_{i}}{\partial u_{ai}(t)}=-Q_{i}u_{ai}(t)+B_{i}^{T}\lambda _{i}(t). \end{aligned} \end{aligned}$ (30)

Using the maximum principle conditions. Setting $\frac{\partial H_{i}}{\partial u_{ai} (t)} = 0$ $\frac{\partial H_i}{\partial u_{ai}(t)}{=}0$ , Equation (29) results into

$\begin{matrix} \begin{matrix} \dot{λ_{i}} (t) & = - C_{i}^{T} P_{i} C_{i} \tilde{x_{i}} (t) + y_{des} C_{i}^{T} P + {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t) = 0, \\ \Rightarrow \dot{λ_{i}} (t) & = - C_{i}^{T} P_{i} C_{i} \tilde{x_{i}} (t) + y_{des} C_{i}^{T} P - {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} \dot{\lambda _i}(t)&=-C_i^TP_iC_i\tilde{x_i}(t)+y_{des}C_i^TP+\left(A_i+B_iK_i\right)^T\lambda _i(t){=}0, \\\Rightarrow \dot{\lambda _i}(t)&=-C_i^TP_iC_i\tilde{x_i}(t)+y_{des}C_i^TP-\left(A_i+B_iK_i\right)^T\lambda _i(t), \end{aligned} \end{aligned}$ (31)

and Equation (30) becomes

$\begin{matrix} \begin{matrix} - Q_{i} u_{ai} (t) + B_{i}^{T} λ_{i} (t) = 0, \\ \Rightarrow u_{ai} (t) = Q_{i}^{- 1} B_{i}^{T} λ_{i} (t) . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&-Q_iu_{ai}(t)+B_i^T\lambda _i(t){=}0, \\ &\Rightarrow u_{ai}(t)=Q_i^{-1}B_i^T\lambda _i(t). \end{aligned} \end{aligned}$ (32)

Combining Equation (31) and Equation (32), when u_ai(t)=Q_i⁻¹B_i^Tλ_i(t) and ${\dot{λ}}_{i} (t) = - C_{i}^{T} P_{i} C_{i} {\tilde{x}}_{i} (t) + y_{des} C_{i}^{T} P - {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t)$ $\dot{\lambda}_{i}(t)=-C_{i}^{T}P_{i}C_{i}\tilde{x}_{i}(t)+y_{des}C_{i}^{T}P-\left(A_{i}+B_{i}K_{i}\right)^{T}\lambda_{i}(t)$ are satisfied, the optimization problem Equation (25) is fulfilled and J_i reaches its maximum value. At this point, the deviation of the multi-intelligent vehicle system, ΔE_i^T(t)PΔE_i(t), and the difference in the attacker’s energy consumption, u_ai^T(t)Qu_ai(t), reach their maximum, which can be considered as the completion of the attack objective. ▪

3.2. Control Strategy Design Against Data Injection Attacks

In Theorem 1, the optimal attack strategy u_ai(t) is obtained. As defined in Definition 1, the objective of the defense is to ensure that the multi-intelligent vehicles can still perform relatively stable formation control under optimal injection attacks u_ai(t).

Problem 2

To compensate for the offset caused by the attack u_ai(t), it need to design the control strategy u_bi(t) against data injection attacks.

To compensate for the offset caused by the attack, the control signal after the attack is ${\tilde{u}}_{i} (t) = u_{i} (t) + u_{aj} (t) + {\dot{u}}_{a_{ij}} (t)$ $\tilde{u}_{i}(t)=u_{i}(t)+u_{a j}(t)+\dot{u}_{a_{i j}}(t)$ . The control strategy against data injection attacks is derived as

$\begin{matrix} \begin{matrix} u_{si} (t) = {\tilde{u}}_{i} (t) + u_{bi} (t), \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned} u_{si}(t) = {\tilde{u}_i}(t) + {u_{bi}}(t), \end{aligned} \end{aligned}$ (33)

where $u_{bi} (t) = Q_{i}^{- 1} B_{i}^{T} C_{i}^{T} P_{i} (C_{i} x_{i} (t) - y_{des}) - Q_{i}^{- 1} B_{i}^{T} [E + {(A_{i} + B_{i} K_{i})}^{T}] λ_{i} (t)$ $u_{b i}(t)=Q_{i}^{-1} B_{i}^{T} C_{i}^{T} P_{i}\left(C_{i}x_i(t)-y_{\text {des }}\right)-Q_{i}^{-1} B_{i}^{T}\left[E+\left(A_{i}+B_{i} K_{i}\right)^{T}\right] \lambda_{i}(t)$ , and ${\dot{λ}}_{i} (t) = - {C_{i}}^{T} P_{i} C_{i} {\tilde{x}}_{i} (t) + y_{des} {C_{i}}^{T} P - {(A_{i} + B_{i} K_{i})}^{T} λ_{i} (t)$ ${\dot \lambda _i}(t) = - {C_i}^T{P_i}{C_i}{\tilde x_i}(t) + {y_{des}}{C_i}^TP - {({A_i} + {B_i}{K_i})^T}{\lambda _i}(t)$ .

4. Simulation verification

4.1. Optimality verification of data injection attacks

From the attacker’s perspective, the optimal attack strategy involves using minimal energy to cause significant state deviations in the targeted system. In the multi-intelligent vehicle system, the control objective is for the vehicles to maintain the same driving velocity and desired inter-vehicle distances. Thus, the deviation in the system state represents the differences in speed and position relative to the desired values.

In this section, numerical simulations are performed on a multi-intelligent vehicle platoon consisting of three follower vehicles to validate the proposed approach. Initially, these follower vehicles are positioned at 4m, 6m, and 7m from the longitudinal zero reference point, respectively, to simulate the platoon’s braking process with an expected velocity of p₀ = 0m/s. In the absence of an attack, the positions and velocities of these follower vehicles evolve as follows:

$\begin{matrix} \begin{matrix} lim_{t \to \infty} ‖ p_{i} (t) - p_{0} (t) ‖ = 0, \\ lim_{t \to \infty} ‖ q_{i - 1} (t) - q_{i} (t) ‖ = 3 . \end{matrix} \end{matrix}$ $\begin{aligned} \begin{aligned}&\lim _{t\rightarrow \infty }\parallel p_i(t)-p_0(t)\parallel =0,\\ &\lim _{t\rightarrow \infty }\parallel q_{i-1}(t)-q_i(t)\parallel =3. \end{aligned} \end{aligned}$ (34)

To verify the optimality of the proposed attack strategy, a comparative experiment is conducted between the attack energy and the attack effects on the vehicles. Using the optimal attack strategy method, the optimal attack gain is calculated as K_a = [ − 0.50.5]. Generating random numbers in MATLAB, we denote K_b = K_a + K_a * (rand(size(1)) − rand(size(1))), which can yield the random attack gains K_b.

Figures 2–4 illustrate the attack energy and the positional changes under both the optimal attack strategy K_a = [ − 0.50.5] and the random attack strategy K_b = [ − 0.1334 0.1334]. In Figure 2, the blue line represents the energy consumed by the optimal attack strategy K_a, while the orange line represents the energy consumed by the random attack strategy K_b. In Figures 3 and 4, the blue line illustrates the vehicle position and velocity changes under the optimal attack strategy K_a, and the orange line shows the changes under the random attack strategy. From Figure 2, it can be seen that under K_b = [ − 0.1334 0.1334], the random attack strategy consumes almost no energy. However, in Figures 3 and 4, it is evident that the random attack strategy causes little to no damage to the vehicle’s position and speed; the vehicle remains at the longitudinal zero reference point, and its speed stays around 0 m/s. The random attack strategy does not damage the system and is ineffective.

Figure 2.

Comparison of energy consumption under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]

Figure 3.

Vehicle position changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]

Figure 4.

Vehicle speed changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]

Figures 5–7 illustrate the attack energy, position changes, and speed changes under the optimal attack gain $K_{a} = [\begin{matrix} - 0.5 0.5 \end{matrix}]$ ${K_a}\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} { - 0.5} {0.5} \end{array}} \right]$ and the random attack gain K_b = [ − 0.6843 0.6843]. From Figure 5, it can be seen that under similar attack effects, the optimal attack gain K_a calculated using the optimal attack strategy method in this chapter consumes far less energy than the random attack gain K_b. At 10 seconds, the energy value represented by the orange line of K_b exceeds twice that of the blue line representing K_a. Figure 6 shows the position changes under different attack gains, while Figure 7 illustrates the velocity changes. Notably, during the first 0–2 seconds, the orange line K_b exhibits better destructive effects on position and velocity compared to the blue line K_a, as the energy consumption of K_a is nearly zero, indicating no effective attack yet. However, after 2 seconds, when the energy consumption of the optimal attack is less than that of the random attack gain, the destructive effects of the optimal attack gain K_a on position and velocity approach or surpass those of the random attack gain K_b.

Figure 5.

Comparison of energy consumption under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.0.6843 0.0.6843]

Figure 6.

Vehicle position changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.6843 0.6843]

Figure 7.

Vehicle velocity changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.0.6843 0.6843]

Through the comparison of simulation results between the optimal attack strategy and two different random attack strategies in Figures 2–7, it can be concluded that under the random attack strategy, when the attack energy is small, the impact on the attacked system is minimal. However, when the energy is too large, it leads waste. In contrast, the optimal attack strategy designed in this paper is able to cause greater damage to the attacked system with the minimum energy consumption.

4.2. Validation of the effectiveness of the control strategy

To validate the effectiveness of the control strategy, simulations are conducted to assess the deviations in vehicle position and velocity induced by random data input attacks. Subsequently, the impact on vehicle position and velocity is analyzed when the compensatio control strategy is applied during the attack.

Figure 8.

Deviations of the first vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations

Figure 9.

Deviations of the first vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations

Figure 10.

Deviations of the second vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations

Figure 11.

Deviations of the second vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations

Figure 12.

Deviations of the third vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations

Figure 13.

Deviations of the third vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations

Figure 14.

The initial positions and velocities of the platoon

Figure 15.

Positions and velocities of vehicle platoon at 3s

Figures 8 and 9 illustrate the position and velocity deviations of the first following vehicle during random data injection attacks and after the application of the compensation control strategy, respectively. The blue line represents the vehicle’s velocity deviation, while the red line indicates the vehicle’s position deviation. Under the attack strategy, the vehicle’s velocity oscillates continuously, making it impossible to achieve the braking objective. With the control strategy in place, the velocity and position deviations caused by the attack are partially corrected, indicating that the control strategy has a certain level of effectiveness.

Figures 10 and 11 illustrate the position and velocity deviations of the second following vehicle during random data injection attacks and after the application of the compensation control strategy, respectively. The blue line represents the vehicle’s velocity deviation, while the red line indicates the vehicle’s position deviation. Under the attack strategy, the vehicle’s velocity continuously oscillates, making it impossible to achieve the braking objective. With the control strategy in place, the velocity and position deviations caused by the attack are partially corrected, indicating that the control strategy has a certain level of effectiveness.

Figures 12 and 13 illustrate the position and velocity deviations of the third following vehicle during random data injection attacks and after the application of the compensation control strategy, respectively. The blue line represents the vehicle’s velocity deviation, while the red line indicates the vehicle’s position deviation. Under the attack strategy, the vehicle’s velocity continuously oscillates, making it impossible to achieve the braking objective. With the control strategy in place, the velocity and position deviations caused by the attack are partially corrected, indicating that the control strategy has a certain level of effectiveness.

Figures 14 and 15 illustrate the state of the platoon at the initial state and the state of the platoon after running for 3 seconds, respectively. At the initial state, the speeds of the following vehicles are 10 m/s, 8 m/s, and 6 m/s. Since the simulation models the braking state of the platoon, the desired speed for all vehicles is 0 m/s. From Figure 15, it can be seen that after the platoon has traveled for 3 seconds, the speeds of all following vehicles have reached 0 m/s, indicating that the braking state has been achieved, and the vehicles maintain approximately the same distance from each other.

5. Conclusion

This paper investigates formation control of multi-agent vehicle systems against data injection attacks. As attack strategies continue to evolve, existing multi-agent vehicle control strategies struggle to cope with the impacts of complex attacks. A better understanding of attack behaviors can lead to the design of corresponding control strategies, preparing the system for secure defense. In this paper, multi-agent vehicles utilizes DSRC for inter-vehicle communication, and different scenarios are considered based on second-order and third-order modeling of multi-agent vehicles. Through the analysis of the attack objectives, the optimal data injection attack strategy is derived. The attacker injects a mix of false data and its derived data into a healthy multi-agent vehicle system, causing significant damage and shifting the balance position with minimal energy expenditure. By analyzing the attack strategy, corresponding multi-agent control strategies against data injection attacks are developed to guarantee the safety of the vehicle platoon. Future work will focus on addressing communication network issues and scalability challenges in larger systems, aiming to improve efficiency, collaboration, and stability.

Acknowledgments

We would like to thank all editors and reviewers who help us improve the paper.

Funding

This work is supported by the National Natural Science Foundation of China 62433014, in part by the Shanghai International Science and Technology Cooperation Project (22510712000), in part by National Key R&D Program of China under Grant (2020AAA0108100), in part by the Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100) and in part by the Fundamental Research Funds for the Central Universities.

Conflicts of interest

The authors declare that they have no conflict of interest.

Data availability statement

The original data are available from corresponding authors upon reasonable request.

Author contribution statement

Juan Liu wrote and constructed this paper. Shuyi Tang carried out the theoretical derivation inspection. Sheng Gao performed simulation experiment assistance. Zhuping Wang discussed the recent development, corrects typos in the paper and jointly wrote this paper. Hao Zhang mainly surveyed the related work and jointly wrote this paper. Gerhard Rigoll revised this paper.

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Juan Liu obtained her B.E. degree in automation from Hohai University, Jiangsu, China, in 2017, and later achieved her M.Sc. degree in control science and engineering from Tongji University, Shanghai, China, in 2020. Currently, she is actively engaged in her Ph.D. degree, specializing in control science and engineering at Tongji University, Shanghai, China. Her current research focuses on optimal attack strategy design and intelligent control.

Shuyi Tang received the B.Sc. degree in automation from ChangAn University, Xian, China in 2018 and M.Sc. in Control Science and Engineering with the school of Electronics and Information Engineering, TongjiUniversity, Shanghai, China, in 2022. Her research interest covers security in multi-agent vehicle formation and optimal attack strategy design problem.

Sheng Gao received his B.Sc. degree in automation from Donghua University, Shanghai, China in 2019. He is currently working toward the Ph.D. degree in control science and engineering at Tongji University, Shanghai, China. From January to March 2024, he was a Visiting Ph.D. Student with the Chair of Intelligent Control Systems, RWTH Aachen University, Aachen, Germany. His current research interests include optimal control, cyber-physical systems, robot and cyber security.

Zhuping Wang received her B.Sc. degree and the M.Sc. degrees from Northwestern Polytechnic University, China, in 1994 and 1997, respectively. She further attained her Ph.D. degree from the National University of Singapore in 2003. She is currently a Professor with the College of Electronic and Information Engineering, Tongji University, Shanghai, China. Her research interests include autonomous vehicles and intelligent control of robotic systems.

Hao Zhang obtained the B.E. degree from Wuhan University of Technology, Wuhan, China, in 2001, and the Ph.D. degree from Huazhong University of Science and Technology, Wuhan, China, in 2007. She is currently a Professor within the College of Electronic and Information Engineering at Tongji University, Shanghai, China. Her research pursuits primarily focus on intelligent control, multi-agent systems, and optimal control.

Gerhard Rigoll (Life Fellow, IEEE) received the Dr.-Ing. habil. degree from the University of Stuttgart, Germany, in 1991. He spent several years as guest researcher in USA and Japan, and was appoined in 1993 as Full Professor of computer science at Gerhard-Mercator-University Duisbug, Germany. In 2002 he joined the Technical University of Munich (TUM), Germany, where he is heading the Institute for Human-Machine Communication. His research interests include human-machine communication and multimedia information processing, covering areas, such as speech and handwriting recognition, gesture recognition, face detection and identification, action and emotion recognition, and interactive computer graphics.

All Figures

	Figure 1. The vehicular platoon under FDI attack
In the text

	Figure 2. Comparison of energy consumption under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]
In the text

	Figure 3. Vehicle position changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]
In the text

	Figure 4. Vehicle speed changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.1334 0.1334]
In the text

	Figure 5. Comparison of energy consumption under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.0.6843 0.0.6843]
In the text

	Figure 6. Vehicle position changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.6843 0.6843]
In the text

	Figure 7. Vehicle velocity changes under the optimal attack strategy K_a = [ − 0.5 0.5] and the random attack strategy K_b = [ − 0.0.6843 0.6843]
In the text

	Figure 8. Deviations of the first vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 9. Deviations of the first vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 10. Deviations of the second vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 11. Deviations of the second vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 12. Deviations of the third vehicle under random data injection attacks. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 13. Deviations of the third vehicle under the compensation control strategy. (a) Position deviations. (b) Velocity deviations
In the text

	Figure 14. The initial positions and velocities of the platoon
In the text

	Figure 15. Positions and velocities of vehicle platoon at 3s
In the text

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