Observer-based adaptive memory event-triggered consensus tracking control for high-speed train under DoS attacks

Abstract

This paper focuses on the event-triggered consensus tracking of high-speed trains under denial-of-service (DoS) attacks. A novel anti-disturbance control strategy is proposed, based on an adaptive memory state observer and a disturbance observer. A memory is introduced to buffer system output information which enhancing the precision of the state observer, and the gust disturbance during the high-speed train operation is estimated by the disturbance observer. The Linear Matrix inequality technique is used to obtain the observer feedback coefficient and the controller gain, and the Lyapunov theory is employed to demonstrate the controller’s consistency tracking performance. Applying the memory to the event-triggered mechanism, a new adaptive memory event-triggered scheme is designed to better ensure system performance and effectively prevent the occurrence of Zeno behavior. To achieve precise identification of DoS attacks in the event-triggered environment, a new detection algorithm is proposed. Finally, simulation examples are used to validate the effectiveness and practicality of the proposed scheme.

1 Introduction

In recent years, with the continuous growth of high-speed railway mileage in the world, high-speed railway has become the main body of short-range and medium-range rapid transport [1]. However, the passenger and freight capacity of many interregional lines cannot meet people’s demand. To solve this problem, we should not only build new lines but improve the control strategy of high-speed trains [2]. Now train control systems are rapidly developing towards intelligent autonomous driving to improve the operational efficiency and reliability [3]. Benefit from the application of Global System for Mobile Railway Communication (GSM-R) and Radio Block Center (RBC), high-speed trains can realize fast information exchange, which makes it possible to integrate more algorithms into high-speed train control. For example, adaptive iterative control, model predictive control, and sliding mode control have been successfully applied to high-speed train control [4,5,6].

However, as a wireless communication network system that is exposed to the open environment along the railway for a long time, GSM-R is vulnerable to cyber attacks in actual operation. In recent years, there have been different degrees of cyber attacks on the railway systems in the United States, Germany, Denmark and other places [7,8,9]. With the development of network technology, traditional network attacks (such as DoS attacks, spoofing attacks, replay attacks, etc.) are becoming diversified and complicated [10]. In particular, DoS attacks can prevent the transmission of signals and even cause the paralysis of communication networks, leading to degradation or loss of group consensus performance [10, 11]. Many scholars have studied group consensus under DoS attacks in different fields [12,13,14]. For example, Ngo [12] designed elastic controllers to ensure the consensus performance of robot systems under DoS attacks. Wen [13] added an event-triggered mechanism into the security protocol of autonomous underwater vehicle group to reduce the impact of DoS attacks on system consistency, and Ye [14] introduced the elastic algorithm into the controller of autonomous underwater vehicle. Currently, consensus control has only been employed by some scholars to enhance the operational efficiency of high-speed railway systems [15,16,17], but has not been applied to solve the control problem of high-speed train under DoS attacks. Existing research on high-speed trains under DoS attacks predominantly focuses on cooperative control and elastic control. For instance, Zhao [18] devised an elastic controller for achieving cooperative control among multi-high-speed trains during DoS attacks, and Gao [19] enhanced the communication strategy and protocol at the communication layer and physical layer of CBTC, proposing a state observer-based elastic control approach for trains under DoS attacks. Cooperative control and elastic control are transformed on the basis of consensus control [20], thus the research on high-speed train tracking consensus under DoS attacks is significant.

In the previous research, scholars also required the system state of multi-high-speed trains were measurable, such as [18, 21]. However, it is difficult to directly obtain system states through the actual measurement values [22, 23]. To address this issue, many research-ers have employed state observers to refactor the input and output information of the system in order to obtain the state variables and provide feedback to the controller [19, 24]. Nevertheless, during DoS attacks, system information may be delayed or unavailable. The-refore, ensuring accurate observation of the system by the state observer under such circumstances becomes a valuable research direction. Furthermore, in practical operations, high-speed trains are often influenced by external environmental disturbances, especially gusts. Henceforth, one of the main objectives of this paper is to design an observer-based anti-disturbance controller capable of achieving high-speed train consensus tracking control under DoS attacks.

In addition, there is a problem of limited communication resources in GSM-R [25]. In the multi-agent systems, event-triggered mechanisms have been widely employed to solve this problem [26,27,28]. Yang [26] and Liu [27] utilized a fixed threshold parameter in their designed event-triggered schemes, while Liu [28] designed a distributed adaptive event-triggered strategy that can be dynamically adjusted according to system states and consumes few network resources. Ji [29] also applied this method to the cooperative control of virtual coupled trains. However, using the event-triggered mechanism may lead to degrade system performance. Thus, it is necessary to design an adaptive adjustment trigger threshold that can ensure the train safety and system performance. Furthermore, due to information transmission suspension caused by event triggering, DoS attacks may be falsely detected. At present, Gao [19] designed a distributed detection method for the physical layer equipment of the train communication network system, Zhao [18] applied the network security detection method in the smart grid to the detection of DoS attacks suffered by the train, Yue [30] proposed a DoS attack detection method based on the train communication network protocol. However, none of these methods can solve the problem of misjudgment of DoS attacks caused by event-triggered strategies. Therefore, it is imperative to design a new detection algorithm for GSM-R that is suitable for event-triggered mechanisms.

Based on the above discussions, for the problem of consensus tracking and limited communication bandwidth of multi-high-speed train group consisting of one leader and multiple followers under DoS attacks, this paper proposes an observer-based disturbance rejection control scheme in an event-triggered environment. At the same time, this article achieves accurate detection of DoS attacks in an event-triggered environment. The main contributions of this paper can be summarized as follows:

• (1) A new anti-disturbance control scheme is designed, based on state and disturbance observers. Different from [19, 24], this paper introduces a memory to store the state of the previous moment, enhancing the accuracy of the state observer under DoS attacks. The disturbance observer can overcome the challenges caused by the unknown state and realize the observation of the external disturbance. This control scheme ensures safer and smoother operation of high-speed trai-ns under DoS attacks and external disturbances.

• (2) A novel AMETS is designed. Unlike references [28, 29], it utilizes the state information of recent moments in the memory, and the trigger threshold can be adjusted adaptively. The scheme not only ensures the system performance but also saves network resources. Moreover, the update mechanism proposed in this paper is able to eliminate Zeno behavior.

• (3) A new DoS attack detection algorithm is proposed. Different from the algorithms in [18, 19, 30], this paper considers the practical application characteristics of GSM-R and the impact of event-triggered mechanism, and designs an attack detection algorithm based on identification signal. It solves the problem of misjudgment of DoS attacks caused by the non-triggered state.

The remaining sections of this paper are organized as follows. Section 2 introduces the dynamic model of a high-speed train and the communication network under DoS attack, along with relevant details. Section 3 presents the designed controller, AMETS and attack detection algorithm. In Sect. 4, the effectiveness and feasibility of the proposed method are proved by simulation examples. Finally, Sect. 5 provides the conclusions.

Notations ${\mathbb {R}} ^ {N \times N}$ represents $N \times N$ real matrices; $\textit{I}_{N}$ denotes the n-dimensional identity matrix; $1_{N}$ is the $N \times 1$ matrix with all ones; The superscript $T$ represents the transpose of the matrix. The diagonal matrices are denoted as $diag\{ \cdot \}$ and $col\{x_{1},\dots ,x_{N} \}=\{x_{1}^{T},\dots ,x_{N}^{T}\}^{T}$. Let the symbol $\parallel \cdot \parallel$ represent the Euclidean norm and the symbol $\otimes$ denote the Kronecker product.

2 Problem formulation and preliminary

2.1 Graph theory

This article represents the directed topological structure ${\mathcal {G}}=({\mathcal {V}}, {\mathcal {E}})$ composed of one leader high-speed train and N tracking high-speed trains in the form of a graph, where each high-speed train is a node. ${\mathcal {V}}=\{1,\dots ,N\}$ denotes the set of N nodes and ${\mathcal {E}} \subseteq {\mathcal {V}} \times {\mathcal {V}}$ is the set of edges. For any directed edge set ${\mathcal {E}}_{ij}$ in graph ${\mathcal {G}}$, it means that there exists an information flow from node i to node j with $(i,j) \in {\mathcal {E}}$. Let $N_{i}=\{j \in {\mathcal {V}}\mid (j,i)\in {\mathcal {E}}, j\ne i\}$ denote the set of neighbor nodes of node i. The adjacency matrix of graph ${\mathcal {G}}$ is represented as ${\mathcal {A}}=[a_{ij} ]\in {\mathbb {R}} ^ {N \times N}$, where if $a_{ij}=1$ with $(j,i)\in {\mathcal {E}}$ means there is a communication between node i and node j, otherwise $a_{ij}=0$. The Laplacian matrix of graph ${\mathcal {G}}$ is defined as ${\mathcal {L}}=[l_{ij} ]\in {\mathbb {R}} ^ {N \times N}$ where $l_{ii}=\sum _{j\in N_{i}}^{N}a_{ij}$ and $l_{ij}=-a_{ij}$, $i\ne j$. Let $\bar{{\mathcal {G}}}=(\bar{{\mathcal {V}}},\bar{{\mathcal {E}} })$ be a graph containing spanning trees with $\bar{{\mathcal {E}}} \in (\bar{{\mathcal {V}}},\bar{{\mathcal {V}} })$. Define $\bar{{\mathcal {L}}}={\mathcal {L}}+{\mathcal {A}}_{i0}$, where ${\mathcal {A}}_{i0}=diag\{a_{10},\dots ,a_{N0} \}$, and if node i can obtain information from leader node 0, then $a_{i0}=1$. Conversely, $a_{i0}=0$. The minimum and maximum eigenvalues of $\bar{{\mathcal {L}}}$ are respectively denoted as $\lambda _{min}$ and $\lambda _{max}$.

2.2 Dynamic model of high-speed train

When analyzing the dynamics of high-speed trains, it is often treated as a single mass point [21]. According to the Newton’s laws of motion, the dynamic model of a high-speed train in the direction of motion is defined

$$\begin{aligned} m_{i}{\ddot{S}}_{i}(t)=F_{i}(t)-F_{i_{R}}(t) \end{aligned}$$

(1)

where ${\ddot{S}}_{i}(t)$ is the second derivative of $S_{i}(t)$ which denotes the position of the ith train. $F_{i}(t)$ is the force provided by the Traction Control Unit (TCU) / Braking Control Unit (BCU) of the high-speed train. $m_{i}$ denotes the mass of the high-speed train i. $F_{i_{R}}(t)=f_{i}(t)+f_{a}(t)$ represents the resistance encountered during actual operation of the high-speed train. In this formula, $f_{a}(t)=f_{r}(t)+f_{c}(t)+f_{t}(t)$ represents the additional resistance, where $f_{r}(t)$, $f_{c}(t)$ and $f_{t}(t)$ represent the slope resistance, curve resistance and tunnel resistance respectively and they are known quantities determined by the railway line; $f_{i}(t)$ represents the basic resistance, which is usually related to the speed of the train. It can be expressed by Davis equation as $f_{i}(t)=c_{0}+c_{1} v_{i}(t)+c_{2}v_{i}^{2}(t)$, where $c_{0}$, $c_{1}$ and $c_{2}$ are known parameters. The force supplied by the TCU/BCU satisfies the equation below

$$\begin{aligned} F_{i}(t)=m_{i}u_{i}(t)+F_{i_{R}}(t) \end{aligned}$$

(2)

where $u_{i}(t)$ represents the new control input to be designed. According to Eqs. (1) and (2), the high-speed train dynamics model can be set as a single mass point model

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{S}}_{i}(t)=v_{i}(t)\\ {\dot{v}}_{i}(t)=\frac{F_{i}(t)-F_{i_{R}}(t)}{m_{i}}=u_{i}(t) \end{array}\right. } \end{aligned}$$

(3)

By rewriting (3) according to (2), we obtain a linearized model for a high-speed train. The state of the high-speed train is defined in matrix form as follows

$$\begin{aligned} x_{i}(t)=[S_{i}(t)+i\times s,v_{i}(t)]^{T} \end{aligned}$$

(4)

where s is the distance between two trains, and $v_{i}(t)$ denotes the speed of the ith train.

During the operation of high-speed trains, it is inevitable to be disturbed by the external environment, which significantly impact their stability. With the continuous increase of train speed, gusts become the primary external disturbance. According to existing results [31], the gusts experienced by high-speed trains during operation can be expressed as

$$\begin{aligned} G_{i}(t)=\frac{1}{2}\varpi \left( 1-\cos \left( 2\pi \frac{t-t_{0}}{t_{f}-t_{0}}\right) \right. \end{aligned}$$

(5)

where $\varpi$ denotes the amplitude of the gust; $t_{f}$ and $t_{0}$ represent the end and start time of the gust. The gusty disturbance model can be transformed into a harmonic disturbance model [32], resulting in the following disturbance model

$$\begin{aligned} {\dot{d}}_{i}(t)={\check{\chi }}_{i}d_{i} \end{aligned}$$

(6)

where ${\check{\chi }}_{i}= \begin{bmatrix} 0 &{} \chi _{i}\\ -\chi _{i} &{} 0\\ \end{bmatrix} , \chi _{i}$ denotes the frequency of gusts and $d_{i}$ signifies the amplitude.

According to Eqs. (3) and (6), Eq. (4) can be rewritten as

$$\begin{aligned} {\dot{x}}_{i}(t)=Ax_{i}(t)+Bu_{i}(t)+Dd_{i}(t) \end{aligned}$$

(7)

where $i=1,\dots ,N$; A, B, and D are known matrices with appropriate dimensions. The state equation of the leader train being given by

$$\begin{aligned} {\dot{x}}_{0}(t)=Ax_{0}(t) \end{aligned}$$

(8)

where $x_{0}(t)$ denotes the state of the leader train.

Assumption 1

The matrix (A, B) is stable, and the matrix $({\check{\chi }}_{i},D)$ is observable.

Assumption 2

The eigenvalues of the matrix ${\check{\chi }}_{i}$ are distinct and distributed along the imaginary axis.

Assumption 3

The disturbance of each tracking train satisfies the condition that there exists a matrix F of appropriate dimension satisfying $D=BF$.

Remark 1

Assumption 1 is frequently found in existing conclusions [17, 33]. Assumption  2 indicates that the disturbance $d_{i}(t)$ is a non-vanishing harmonic disturbance, suitable for modeling gusts during the high-speed train operation. Assumption  3 is a widely used disturbance matching condition in most control systems [34].

2.3 High-speed train control system based on GSM-R

China Train Control System-3 (CTCS-3) is a typical train control system that transmits train control information based on GSM-R platform and realizes automatic driving of high-speed trains [35]. As depicted in Fig. 1, the CTCS-3 system comprises trackside devices, network transmission equipment, and onboard equipment. Its communication principle is that the train sends the information to the RBC through GSM-R, and then the RBC relays it to both the Computer Based Interlocking system (CBI) and the Centralized Traffic Control system (CTC). Subsequently, the RBC sends the command information generated by CBI and CTC to each train. High-speed trains accomplish indirect train-train communication through this train-RBC and RBC-train communication method. In addition, CBI can obtain the train information through the Track Circuit Reader (TCR) and the Lineside Electronic Unit (LEU).

Fig. 1

CTCS-3 train control system

Remark 2

CTC, RBC, CBI, TCR and LEU employ wir-ed communication methods, so the communication between them is reliable. LEU obtains information through balise, and TCR obtains train information through track circuit occupancy status [36].

2.4 Train communication network under DoS attacks

GSM-R is responsible for communication between trains and ground equipment, and it achieves indirect communication between multiple trains through a two-way transmission mode. Therefore, it plays a crucial role in train control. Due to the long-term exposure of GSM-R, it is more valuable and easier to launch a DoS attack on it. Once the DoS attack succeeds, it will lead to the GSM-R network be paralyzed and unable to transmit information normally, thus affecting the safety of train operation. For example, when two adjacent trains communicate under DoS attacks, the information cannot be exchanged and they may not be able to receive the information sent by the leader train. Once the leader train or the adjacent train suddenly decelerates, the following train would be unable to maintain its corresponding speed trajectory, and even makes the wrong judgment to accelerate. This will shorten the distance between two trains, and even lead to a collision and other major safety accidents. Therefore, this article focuses on the control of the train when the GSM-R network is subject to DoS attacks, as shown in Fig. 2.

Fig. 2

Communication framework between trains under DoS Attack

Remark 3

DoS attacks commonly include techniques such as Ping flood, Smurf-Attack, and SYN flood. The objective of the DoS attack discussed in this article is to disrupt communication between high-speed trains, leading to the collapse of the communication topology.

3 Main results

3.1 DoS attack detection algorithm

There are some redundant lines in the design of GSM-R. If these redundancies are subjected to DoS attacks, the tracking train can still obtain key information to maintain normal operation. This paper argues that the communication connection between multiple trains is remained, and this attack is called communication maintenance attack. Another case, if the attack is launched against critical paths within GSM-R, the tracking train cannot obtain the information of the leader or adjacent trains. This article argues that the communication connection between multiple trains is completely interrupted, and this attack is called communication interruption attack. At this time, a reliable control method must be implemented to ensure the safe operation of the train. Therefore, in order to ensure train safety and save resources, it is necessary to design a DoS attack detection method to identify communication interruption attacks. Inspired by [20], this paper proposes a DoS attack detection method based on signal confirmation.

Firstly, a definition is provided. If the tracking train has no communication link with the lead train, such train is referred to as a “Communication Interruption Train” , otherwise it is designated as a “Normal Train”. To identify the “Communication Interruption Train”, an identification signal $I_{0}(t)=\ln \alpha (t+1)+\epsilon$ is established, where $\alpha >1$ and $\epsilon >0$ are constant values. Notably, $I_{0}(t)$ is observed to increase monotonically over time. The lead train broadcasts this identification signal to its adjacent trains, and the signal is transmitted between the tracking trains. Define the adjacent train of the tracking train i as j. The normal train is defined by $\beta _{i}(t)=0$, and the communication interruption train is $\beta _{i}(t)=1$. In the initial state, all tracking trains are considered to be normal, i.e. $\beta _{\nu }=0,\nu =0,1,2\ldots ,N$, and the identification signal of each tracking train is set to 0, i.e. $I_{\nu }=0,\nu =0,1,2\ldots ,N$. If train i does not satisfy the set event-triggered condition, it will periodically transmit the maximum identification signal what it has received to its adjacent trains. Assuming the signal period of $\tau _{I}$, it represents the maximum of the identification signal periods sent by all following trains. The tracking train i updates its identification signal $I_{i}(t)$ based on the maximum identification signal it receives from its neighboring train j, that is, $I_{i}(t)=max\{I_{j}(t),j\in N_{i}\}$. If at time t, the maximum identification signal received from its adjacent train j is larger than the previous one, its identification signal will be updated, and $\beta _{i}(t)$ will be set to 0, indicating that the tracking train i is a normal train and can communicate with the leader. If ith train cannot receive an identification signal from its neighbors or if the identification signal it receives from its neighbors does not change, then $I_{i}(t)$ will not be updated. The following three scenarios may lead to the failure of updating the identification signal

• (1) At least one of its adjacent trains satisfies the triggering conditions, but that train is experiencing communication interruption attacks.

• (2) All its neighboring trains fail to meet the triggering conditions, and all these trains are experiencing communication interruption.

• (3) None of its adjacent trains satisfy the trigger condition, but there exists a normal train among them.

If the maximum identification signal received by tracking train i from its adjacent trains does not change within either time t or time interval $(t,t+ \tau _{I}]$, tracking train i confirms that its communication with all neighboring trains is interrupted and $\beta _{i}(t)$ sets to 1. If the maximum identification signal does not change at time t but becomes larger within $(t,t+\tau _{I}]$, it indicates that there is at least one normal train among its neighbors and this adjacent train is in a non-triggered state, thus $\beta _{i}(t)$ will be set to 0. In this way, the algorithm prevents misjudgment of DoS attacks in the third case.

Remark 4

DoS attacks consume a huge amount of energy, yet energy of the attacker is limited. Therefore, the attacker cannot simultaneously target all train-to-train communication lines. Moreover, a period of sleep time is required between two consecutive DoS attacks to accumulate energy [20]. Different from the detection of DoS attacks in [18, 19], this paper considers multiple possibilities of the impact of DoS attacks on GSM-R. The algorithm designed in this article effectively solves the problem of detecting communication interruption attacks in event-triggered environment. Theoretically, the false positive rate of the DoS attack detection algorithm designed in this paper can reach 0%. In the actual operation of high-speed trains, the false alarm rate will also be affected by various factors such as communication delay and packet loss. In addition, the maximum transmission period of identification signals in the non-triggered state of the train should be less than the duration of the DoS attack. The selection of the maximum transmission period should also consider the detection speed and resource consumption, ensuring both detection speed and avoiding excessive resource consumption.

3.2 Design of AMETS and observers-based controllers

Under the communication interruption attack, the GSM-R network communication system falls into a paralyzed state, which will delay the data transmission or lead to the failure of data transmission. As a result, the tracking train cannot obtain information quickly and accurately, and the observer may produce wrong estimation or infinite estimation. In this paper, a memory is added to the state observer. The memory can store the train state at the previous transmission time. When a communication interruption attack occurs, the state observer can call the stored data to complete the accurate estimation of the leader train state. In addition, in order to avoid unnecessary communication resource occupation, this article designs a new adaptive event-triggered control protocol based on memory.

First, the tracking error of the ith tracking train is defined as

$$\begin{aligned} \xi _{i}(t)=x_{i}(t)-x_{0}(t) \end{aligned}$$

(9)

To facilitate the subsequent discussion, let $\{t_{k}^{i}\}$ denote the event triggering time of the ith tracking train. As shown in Fig. 2, when the ith train satisfies the event-triggered condition, $x_{i}(t_{k}^{i})$ will be written to the memory and transmitted to the memory of the adjacent train j through GSM-R. $x_{j}(t_{k}^{j})$ triggered by adjacent train j has the same transmission path. Based on the received information, one can obtain

$$\begin{aligned}{} & {} \theta _{i}(t)=\sum _{j\in N_{i}}a_{ij}(\xi _{i}(t)-\xi _{j}(t))+a_{i0}\xi _{i}(t) \end{aligned}$$

(10)

$$\begin{aligned}{} & {} e_{i,t_{k-q+1}}(t)=\theta _{i}(t_{k-q+1}^{i})-\theta _{i}(t) \end{aligned}$$

(11)

where $q\in N$ denotes the quantity of data stored in memory; $e_{i,t_{k-q+1}}(t)$ is the difference between the data error stored in the memory and the error at the current time. Combining formulas (9)–(11), the new AMETS is designed as

$$\begin{aligned} h_{i}(t)= & {} \Bigl (\sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigr )^{T}\varTheta \Bigl (\sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigr )\nonumber \\{} & {} -\varrho _{i}(t)\theta _{i}^{T}(t)\varTheta \theta _{i}(t) \end{aligned}$$

(12)

where $\varTheta =C^{T}C$ represents the weight matrix triggered by events and it is positive definite, with C being a matrix of appropriate dimensions; $\omega _{q}$ is a given positive constant and $\sum _{q=1}^{m}\omega _{q}=1$, indicating the weight of historical data stored in the memory. $\varrho _{i}(t)$ is the adaptive trigger parameter, which satisfies the following adaptive update law

$$\begin{aligned} \varrho _{i}(t){} & {} =\varrho _{i}\tanh (\gamma \Vert \theta _{i}(t)\Vert ^{2})\nonumber \\ {}{} & {} \quad +\varrho _{M}(1-\tanh (\gamma \Vert \theta _{i}(t)\Vert ^{2})) \nonumber \\ \end{aligned}$$

(13)

where $\varrho _{i}(t)\in [\varrho _{i},\varrho _{M}]\subset (0,1]$; $\gamma$ is a nonzero positive constant, which is used to adjust the sensitivity to changes with error.

Remark 5

It is worth noting that the latest published packets have higher weights, so the $\omega _{1}$ should be larger than the other $\omega _{q}$. At the same time, the $\omega _{q}$ also affects the number of trigger points. A smaller $\omega _{1}$ and a larger m will increase the number of trigger points. When $\omega _{1}=1$ and $m=1$, the above event-triggered scheme will degenerate to a normal adaptive triggering mechanism. Additionally, the triggering speed is determined by $\varrho _{i}(t)$. The smaller $\varrho _{i}(t)$, the more triggering points, the better system performance. Conversely, the fewer triggering points, the worse system performance. Clearly, the more triggering points, the heavier the communication burden. When $\varrho _{i}(t)=0$, the AMETS loses its effectiveness. $\varrho _{i}$ and $\gamma$ can adjust the change rate of $\varrho _{i}(t)$, thus ensuring the effectiveness of the event-triggered mechanism when selecting parameters, taking into account both system performance and communication burden.

Similar to [39], the next triggering time is defined as

$$\begin{aligned} t_{k+1}^{i}= {\left\{ \begin{array}{ll} \inf \{t>t_{k}^{i}|h_{i}(t)>0\} &{}\beta _{i}(t)=0\\ t_{k}^{i}+\vartheta &{}\beta _{i}(t)=1 \end{array}\right. } \end{aligned}$$

(14)

where $\vartheta <\tau _{I}$ , is a given positive constant.

Remark 6

Different from the traditional distributed ev-ent triggered condition [29], the recently triggered data packets are stored in the memory of size m added in (12). These newly released data packets avoid the mutation of measurement data caused by the communication interruption attack and ensure the integrity of state information. Therefore, the AMETS designed in this paper can better resist DoS attacks. Compared with the existing adaptive event-triggered strategy, the AMETS in this paper makes use of the previous state information of the train, which makes the event trigger point more accurate and effectively reduces unnecessary communication. In addition, the event-triggered strategy will inevitably affect the performance of the system [37]. In this article, the AMETS can quickly adjust the number of trigger points by changing the values of $\omega _{q}$ and m to ensure the best performance of the system. Compared with the traditional event-triggered strategy, the AMETS greatly reduces the amount of calculation. At the same time, this paper also considers the tracking error of the trains to improve the trigger mechanism and ensure the tracking performance of the multi-high-speed train system.

The control objective of this article is to design a control strategy, based on the observation data of the state and disturbance observers, to ensure the consistent tracking of the follower trains to the leader, especially when communication between multi-high-speed trains is subjected to DoS attacks and external disturbances, that is $\lim _{t\rightarrow \infty }\Vert x_{i}(t)-x_{0}(t)\Vert =0$. The following lemmas are used to achieve the control objectives of this paper.

Lemma 1

[38] Suppose a and b are positive real numbers satisfying $1/a+1/b=1$, then

$$\begin{aligned} {\mathcal {P}}{\mathcal {Q}}\le \frac{{\mathcal {P}}^{a}}{a}+\frac{{\mathcal {Q}}^{b}}{b} \end{aligned}$$

where ${\mathcal {P}}$,${\mathcal {Q}}$ are nonnegative real numbers.

Lemma 2

[39] For a positive definite function V(t), if it satisfies

$$\begin{aligned} (1)\;\text {if } \beta _{i}(t)= & {} 0,{\dot{V}}(t)\le -\varepsilon V(t) \\ (2)\;\text {if } \beta _{i}(t)= & {} 1,{\dot{V}}(t)\le \varLambda \max \{V(t_{s}),V(t_{s+1})\} \end{aligned}$$

where $\varepsilon$ and $\varLambda$ are unknown positive constants, $t_{s}$ represents the instant of the last successful trigger and $t_{s+1}$ is the next instant of $t_{s}$. Then, the following inequality holds

$$\begin{aligned} V(t)\le e^{[-\varepsilon (t-t_{0}-n(t_{0},t)\vartheta )+\varLambda n(t_0,t)\vartheta ]}V(t_{0}) \end{aligned}$$

where $n(t_{0},t)$ denotes the count of attempts made by the tracking train to initiate communication connections amidst the communication interruption attack.

Lemma 3

[40] For the known matrix equation $W=R$, it can be transformed into the LMI form as

$$\begin{aligned} \begin{bmatrix} -\varphi I &{} (W-R)^{T} \\ W-R &{} -\varphi I \\ \end{bmatrix} <0 \end{aligned}$$

where $\varphi$ is a positive constant, and I is an identity matrix of the appropriate dimension.

The state observer of the ith train is constructed as below

$$\begin{aligned} \dot{{\hat{x}}}_{i}(t){} & {} =A{\hat{x}}_{i}(t)+Bu_{i}(t)+LC \nonumber \\{} & {} \times \biggl [\beta _{i}(t)\biggr (\sum _{q=1}^{m}\omega _{q}\theta _{i}(t_{s-q+1}^{i})-{\hat{\theta }}_{i}(t)\biggl )\nonumber \\{} & {} \quad +(1-\beta _{i}(t))\biggr (\sum _{q=1}^{m}\omega _{q}\theta _{i}(t_{k-q+1}^{i})-{\hat{\theta }}_{i}(t)\biggl )\biggr ]\nonumber \\ \end{aligned}$$

(15)

where ${\hat{x}}_{i}(t)\in R^{n}$ represents the estimated state of the ith train; ${\hat{\theta }}_{i}(t)=\sum _{j\in N_{i}}a_{ij}({\hat{x}}_{i}(t)-{\hat{x}}_{j}(t))+a_{i0}{\hat{x}}_{i}(t)$; and L is the observer gain to be designed. In addition, the disturbance observer is designed as

$$\begin{aligned} {\hat{d}}_{i}(t)= & {} \eta _{i}(t)+E\psi _{i}(t) \end{aligned}$$

(16)

$$\begin{aligned} {\dot{\eta }}_{i}(t)= & {} {\check{\chi }}_{i}\eta _{i}(t)+({\check{\chi }}_{i}E-EA)\psi _{i}(t)+EBK{\hat{\theta }}_{i}(t)\nonumber \\ \end{aligned}$$

(17)

where E denotes the feedback matrix of the disturbance observer, and $\psi _{i}(t)$ denotes

$$\begin{aligned} \psi _{i}(t)= {\left\{ \begin{array}{ll} \theta _{i}(t_{k}^{i}) &{}\beta _{i}(t)=0\\ \theta _{i}(t_{s}^{i}) &{}\beta _{i}(t)=1 \end{array}\right. } \end{aligned}$$

(18)

The controller, based on the state observer (15) and disturbance observer (16), is designed as follows

$$\begin{aligned} u_{i}(t)=-K{\hat{x}}_i(t)-F{\hat{d}}_{i}(t) \end{aligned}$$

(19)

where K is the controller gain to be designed.

Remark 7

It is worth mentioning that different from the observer proposed in [19, 22, 24], this paper adds a memory to the observer (15), which makes it possible to accurately estimate the state even through some recently released packets in the reservoir under the communication interruption attack. Furthermore, the disturbance observer is added to the control strategy to reduce the influence of external disturbance on the train. In this way, it is ensured that the control strategy (19) can accurately control the high-speed train using only the observation information.

3.3 Consistency tracking analysis

To facilitate subsequent analysis, the notations are simplified

$$\begin{aligned} e_{x_{i}}(t)= & {} x_{i}(t)-{\hat{x}}_{i}(t) \end{aligned}$$

(20)

$$\begin{aligned} e_{d_{i}}(t)= & {} d_{i}(t)-{\hat{d}}_{i}(t) \end{aligned}$$

(21)

where $e_{x_{i}}(t)$ represents the state estimation error and $e_{d_{i}}(t)$ denotes the disturbance estimation error. Through Eqs. (7) and (19), one can deduce

$$\begin{aligned} {\dot{x}}(t)= & {} (I_{N}\otimes A)x(t)-(I_{N}\otimes BK){\hat{x}}(t)\nonumber \\{} & {} +(I_{N}\otimes D)e_{d}(t) \end{aligned}$$

(22)

where $x(t)=col\{x_{1}(t),\dots ,x_{N}(t)\}$, ${\hat{x}}(t)=col\{ {\hat{x}}_{1}(t), \dots ,{\hat{x}}_{N}(t)\}$ and $e_{d}(t)=col\{e_{d_{1}}(t),\dots ,e_{d_{N}}(t)\}$.

The derivative of the tracking error (9) with respect to time can be deduced from (8) and (22)

$$\begin{aligned} {\dot{\xi }}(t)= & {} (I_{N}\otimes A)\xi (t)-(I_{N}\otimes BK){\hat{x}}(t)\nonumber \\{} & {} +(I_{N}\otimes D)e_{d}(t) \end{aligned}$$

(23)

where $\xi (t)=col\{\xi _{1}(t),\dots ,\xi _{N}(t)\}$. By utilizing (6) and (16), we can obtain the derivative of (21)

$$\begin{aligned} {\dot{e}}_{d_{i}}(t)= & {} {\check{\chi }}_{i}e_{d_{i}}(t)+EA\psi _{i}(t)\nonumber \\{} & {} -EBK{\hat{\theta }}_{i}(t)-E{\dot{\psi }}_{i}(t) \end{aligned}$$

(24)

Let $\psi (t)=col\{\psi _{1}(t),\dots ,\psi _{N}(t)\}$, it is easy to obtain $\psi (t)=(\bar{{\mathcal {L}}}\otimes I_{N})\xi (t)$. According to (23), it can be inferred that

$$\begin{aligned} {\dot{\psi }}(t)= & {} (I_{N}\otimes A)\psi (t)-(I_{N}\otimes BK) {\hat{\theta }}(t)\nonumber \\{} & {} +(\bar{{\mathcal {L}}}\otimes D)e_{d}(t) \end{aligned}$$

(25)

where ${\hat{\theta }}(t)=col\{{\hat{\theta }}_{1}(t),\dots ,{\hat{\theta }}_{N}(t)\}$. Based on (24) and (25), the derivative of $e_{d}(t)$ can be simplified as

$$\begin{aligned} {\dot{e}}_{d}(t)=(I_{N}\otimes {\check{\chi }}_{i}-\bar{{\mathcal {L}}}\otimes ED)e_{d}(t) \end{aligned}$$

(26)

Next, the main results of this article will be expressed in the form of the following theorems.

Theorem 1

Assume that Assumptions 1,2,3 hold. Under the control of controller (19) and event-triggered function (12) with trigger sequence (14), if there are positive definite matrices $P_{1}$, $P_{2}$ and matrices L, K, G satisfying the following conditions, then the high-speed train can achieve consensus tracking, and the designed observer (15) and (16) can accurately estimate the state of the high-speed train.

$$\begin{aligned} G\bar{{\mathcal {L}}}+\bar{{\mathcal {L}}}^{T}G\ge \lambda _{min}I_{N}>0 \end{aligned}$$

(27)

where $G=(\bar{{\mathcal {L}}})^{-1} 1_{N}=diag\{g_{1},g_{2},\dots ,g_{N} \}$, and $g_{i}>0 (i=1,2,\ldots ,N)$.

$$\begin{aligned} \zeta -1\ge \frac{{\mathcal {H}}_{2} \lambda _{max} (F^{T} F))}{\lambda _{min} (\mu )} \end{aligned}$$

(28)

where $\mu$ is defined below.

$$\begin{aligned} \begin{bmatrix} \varGamma _{1} &{} K^{T}B^{T}P_{2} \\ *&{} \varGamma _{2} \\ \end{bmatrix} <0 \end{aligned}$$

(29)

$$\begin{aligned} \begin{bmatrix} \varGamma _{3} &{} K^{T}B^{T}P_{2} \\ *&{} \varGamma _{4} \\ \end{bmatrix} \le 0 \nonumber \\ (\rho _{1}\varrho _{M}+2\rho _{2}){\lambda _{max}}^{2}C^{T}C-\varLambda P_{1}\le 0 \end{aligned}$$

(30)

$$\begin{aligned} \frac{n(t_{0},t)}{t-t_{0}}\le \frac{\varepsilon -\sigma }{(\varepsilon +\varLambda )\vartheta } \end{aligned}$$

(31)

where

$$\begin{aligned} \varGamma _{1}= & {} A^{T}P_{1}+P_{1}A-\lambda _{min}(P_{1}LC+C^{T}L^{T}P_{1}) \\{} & {} +\frac{1}{{\mathcal {X}}_{1}} P_{1}P_{1}+\frac{1}{{\mathcal {X}}_{2}} P_{1} LL^{T}P_{1}+\varepsilon P_{1}\\ \varGamma _{2}= & {} A^{T}P_{2}+P_{2}A-P_{2}BK-K^{T}B^{T}P_{2}+\frac{1}{{\mathcal {H}}_{1}}P_{2}P_{2} \\{} & {} +\frac{1}{{\mathcal {H}}_{2}} P_{2}BB^{T}P_{2}+{\mathcal {X}}_{2}\varrho _{M}\lambda _{max}^{2}C^{T}C+\varepsilon P_{2}\\ \varGamma _{3}= & {} A^{T}P_{1}+P_{1}A-\lambda _{min}(P_{1}LC+C^{T}L^{T}P_{1}) \\{} & {} +\frac{1}{{\mathcal {X}}_{1}} P_{1}P_{1}+\rho P_{1} LL^{T}P_{1}-\varLambda P_{1}\\ \varGamma _{4}= & {} A^{T}P_{2}+P_{2}A-P_{2}BK-K^{T}B^{T}P_{2}+\frac{1}{{\mathcal {H}}_{1}}P_{2}P_{2}\\{} & {} +\frac{1}{{\mathcal {H}}_{2}} P_{2}BB^{T}P_{2}+2\rho _{2}{\lambda _{max}}^{2}C^{T}C-\varLambda P_{2} \end{aligned}$$

Here, ${\mathcal {X}}_{1}$, ${\mathcal {X}}_{2}$, ${\mathcal {H}}_{1}$, ${\mathcal {H}}_{2}$, $\rho _{1}$, $\rho _{2}$, $\varepsilon$ and $\varLambda$ are all positive constants. And $\rho =\frac{\rho _{1}+\rho _{2}}{\rho _{1}\rho _{2}}$, $\sigma$ satisfies $0<\sigma <\varepsilon$.

Next, we will give the proof of Theorem 1.

Proof

The appropriate Lyapunov function is chosen as

$$\begin{aligned} V(t)=V_{1}(t)+\zeta V_{2}(t) \end{aligned}$$

(32)

where

$$\begin{aligned} V_{1}(t)= & {} e_{x}^{T}(t)P_{1}e_{x}(t)+\xi ^{T}(t)P_{2}\xi (t) \end{aligned}$$

(33)

$$\begin{aligned} V_{2}(t)= & {} e_{d}^{T}(t)(G\otimes I_{n}) e_{d}(t) \end{aligned}$$

(34)

Due to the existence of DoS attacks, it is necessary to consider two cases where $\beta _{i}(t)$ is 0 or 1. Consequently, the proof of Theorem 1 in this article will be divided into the following two cases.

Case 1

If $\beta _{i}(t)=0$, it means that the tracking train i keeps communication with its neighbors, and the latest sampling information can be acquired.

The derivative of $e_{x_{i}}(t)$ is obtained from (7) and (15) as follows

$$\begin{aligned} {\dot{e}}_{x}(t)= & {} (I_{N}\otimes A-\bar{{\mathcal {L}}}\otimes LC)e_{x}(t)+(I_{N}\otimes D)d(t)\nonumber \\{} & {} -(I_{N}\otimes LC)\sum _{q=1}^{m}\omega _{q} e_{t_{k-q+1}} (t) \end{aligned}$$

(35)

where

$$\begin{aligned}{} & {} d(t)=col\{d_{1}(t),\dots ,d_{N}(t)\} \\{} & {} e_{x}(t)=col\{e_{x_{1}}(t),\dots ,e_{x_{N}}(t)\}\\{} & {} \sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)=col\left\{ \sum _{q=1}^{m}\omega _{q}e_{1,t_{k-q+1}}(t),\right. \\{} & {} \qquad \qquad \qquad \qquad \qquad \left. \dots ,\sum _{q=1}^{m}\omega _{q}e_{N,t_{k-q+1}}(t)\right\} \end{aligned}$$

According to (23) and (35), the derivative of $V_{1}(t)$ with respect to time is given by

$$\begin{aligned} {\dot{V}}_{1}(t)={} & {} e_{x}^{T}(t)[I_{N}\otimes (A^{T}P_{1}+P_{1}A)-\bar{{\mathcal {L}}}\otimes (P_{1}LC\nonumber \\{} & {} +C^{T}L^{T}P_{1})]e_{x}(t)+2e_{x}^{T} (t)(I_{N}\otimes P_{1}D)d(t)\nonumber \\{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\nonumber \\{} & {} +\xi ^{T}(t)[I_{N}\otimes (A^{T}P_{2}+P_{2}A-P_{2}BK\nonumber \\{} & {} {-}K^{T}B^{T}P_{2})]\xi (t)+2\xi ^{T}(t)(I_{N}{\otimes } P_{2}BK)e_{x}(t)\nonumber \\{} & {} -2\xi ^{T}(t)(I_{N}\otimes P_{2}BK)x_{0}(t)\nonumber \\{} & {} +2\xi ^{T}(t)(I_{N}\otimes P_{2}D)e_{d}(t) \end{aligned}$$

(36)

It follows from Lemma 1 that

$$\begin{aligned}{} & {} 2e_{x}^{T}(t)(I_{N}{\otimes } P_{1}D)d(t)\nonumber \\ {}{} & {} \le \frac{1}{{\mathcal {X}}}_{1} e_{x}^{T}(t)(I_{N}\otimes P_{1}P_{1})e_{x}(t)\nonumber \\{} & {} \quad +{\mathcal {X}}_{1}d^{T}(t)(I_{N}\otimes D^{T}D)d(t) \end{aligned}$$

(37)

$$\begin{aligned}{} & {} \quad -2\xi ^{T}(t)(I_{N}\otimes P_{2}BK) x_{0}(t)\nonumber \\ {}{} & {} \le \frac{1}{{\mathcal {H}}_{1}} \xi ^{T}(t)(I_{N}\otimes P_{2} P_{2} )\xi (t)\nonumber \\{} & {} \quad +{\mathcal {H}}_{1} x_{0}^{T}(t)(I_{N}\otimes K^{T} B^{T} BK) x_{0}(t) \end{aligned}$$

(38)

$$\begin{aligned}{} & {} \quad 2\xi ^{T}(t)(I_{N}\otimes P_{2}D) e_{d}(t)\nonumber \\ {}{} & {} \le \frac{1}{{\mathcal {H}}_{2}}\xi ^{T}(t)(I_{N}\otimes P_{2}BB^{T}P_{2})\times \xi (t)\nonumber \\ {}{} & {} \quad +{\mathcal {H}}_{2}e_{d}^{T}(t)(I_{N}\otimes F^{T}F) e_{d}(t) \end{aligned}$$

(39)

$$\begin{aligned}{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\nonumber \\{} & {} \quad \le \frac{1}{{\mathcal {X}}_{2}} e_{x}^{T}(t)(I_{N}\otimes P_{1}LL^{T}P_{1}) e_{x}(t)\nonumber \\{} & {} \quad +{\mathcal {X}}_{2}\biggl (\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\biggr )^{T} (I_{N}\otimes C^{T}C)\nonumber \\{} & {} \quad \times \biggl (\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\biggr ) \end{aligned}$$

(40)

Based on the event-triggering condition (12)

$$\begin{aligned}{} & {} \biggl (\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\biggr )^{T} (I_{N}\otimes C^{T}C)\biggl (\sum _{q=1}^{m}\omega _{q}e_{t_{k-q+1}}(t)\biggr ) \nonumber \\{} & {} \quad \le \varrho (t)\theta ^{T}(t)(I_{N}\otimes C^{T}C)\theta (t) \nonumber \\{} & {} \quad =\varrho (t) \xi ^{T}(t)(\bar{{\mathcal {L}}}^{2}\otimes C^{T}C)\xi (t) \end{aligned}$$

(41)

where $\varrho (t){=}diag\{\varrho _{1}(t),\dots ,\varrho _{N}(t)\}$, $\theta (t){=}col\{\theta _{1}(t), \dots ,\theta _{N}(t)\}$. According to (26), the derivative of $V_{2}(t)$ with respect to time can be deduced

$$\begin{aligned} {\dot{V}}_{2}(t)= & {} e_{d}^{T}(t)[G\otimes ({\check{\chi }}_{i}^{T}+{\check{\chi }}_{i})\nonumber \\{} & {} -(G\bar{{\mathcal {L}}}+\bar{{\mathcal {L}}}^{T}G)\otimes ED]e_{d}(t) \end{aligned}$$

(42)

It can be known from (27)

$$\begin{aligned} {\dot{V}}_{2}(t)&\le e_{d}^{T}(t)[G\otimes ({\check{\chi }}_{i}^{T}+{\check{\chi }}_{i})-\lambda _{min}I_{N}\otimes ED] e_{d}(t)\nonumber \\&=-e_{d}^{T}(t)\mu e_{d}(t) \end{aligned}$$

(43)

where $\mu =-G\otimes ({\check{\chi }}_{i}^{T}+{\check{\chi }}_{i})+\lambda _{min}I_{N}\otimes ED$. By substituting (36)–(43) into ${\dot{V}}(t)={\dot{V}}_{1}(t)+ \zeta {\dot{V}}_2 (t)$, one can obtain

$$\begin{aligned}{} & {} {\dot{V}}(t)\le e_{x}^{T}(t)[I_{N}\otimes (A^{T}P_{1}+P_{1}A)-\bar{{\mathcal {L}}}\otimes (P_{1}LC\nonumber \\{} & {} \quad +C^{T}L^{T}P_{1})+\frac{1}{{\mathcal {X}}_{1}}I_{N}\otimes P_{1}P_{1}\nonumber \\{} & {} \quad +\frac{1}{{\mathcal {X}}_{2}}I_{N}\otimes P_{1}LL^{T}P_{1}] e_{x}(t)+\xi ^{T}(t)[I_{N}\otimes (A^{T}P_{2}\nonumber \\{} & {} \quad +P_{2} A-P_{2}BK-K^{T}B^{T}P_{2})+\frac{1}{{\mathcal {H}}_{1}}I_{N}\otimes P_{2}P_{2}\nonumber \\{} & {} \quad +\frac{1}{{\mathcal {H}}_{2}} I_{N}\otimes P_{2}BB^{T} P_{2}+{\mathcal {X}}_{2} \bar{{\mathcal {L}}}^{2}\otimes C^{T}C]\xi (t)\nonumber \\{} & {} \quad +2\xi ^{T}(t)(I_{N}\otimes P_{2}BK) e_{x}(t)\nonumber \\{} & {} \quad +{\mathcal {H}}_{2} e_{d}^{T}(t)(I_{N}\otimes F^{T}F) e_{d}(t)-\zeta e_{d}^{T}(t)\mu e_{d}(t)\nonumber \\ \end{aligned}$$

(44)

According to conditions (28) and (29),

$$\begin{aligned} {\dot{V}}(t){} & {} \le e_{x}^{T}(t)(-\varepsilon P_{1}) e_{x}(t)+\xi ^{T}(t)(-\varepsilon P_{2})\xi (t)\nonumber \\{} & {} \quad +\frac{{\mathcal {H}}_{2}\lambda _{max}(F^{T}F)}{\lambda _{min}(\mu )}e_{d}^{T}(t)\mu e_{d}(t)-\zeta e_{d}^{T}(t)\mu e_{d}(t) \nonumber \\{} & {} \quad {} \le -\varepsilon V_{1}(t)-e_{d}^{T}(t)\mu e_{d}(t) \end{aligned}$$

(45)

Case 2

If $\beta _{i}(t)=1$ means that the ith train is a communication interruption train and cannot receive the latest collected information from adjacent trains. In addition, this paper assumes that the communication interruption attack occurs at time $t_{s+1}^{i}$.

$$\begin{aligned} {\dot{e}}_{x}(t)= & {} (I_{N}\otimes A-\bar{{\mathcal {L}}}\otimes LC) e_{x}(t)+(I_{N}\otimes D)d(t)\nonumber \\{} & {} -(I_{N}\otimes LC)\biggl (\sum _{q=1}^{m}\omega _{q} \theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr )\nonumber \\{} & {} -(I_{N}\otimes LC)(\theta (t_{s+1})-\theta (t)) \end{aligned}$$

(46)

where $\theta (t_{s+1})-\theta (t)=col\{\theta _{1} (t_{s+1})-\theta _{1}(t),\dots , \theta _{N}(t_{s+1}) -\theta _{N}(t)\}$.

According to (46), the derivative of $V_{1}(t)$ with respect to time under communication interruption attacks can be deduced as

$$\begin{aligned} {\dot{V}}_{1}(t)= & {} e_{x}^{T}(t)[I_{N}\otimes (A^{T}P_{1}+P_{1}A)-\bar{{\mathcal {L}}}\otimes (P_{1}LC\nonumber \\{} & {} +C^{T}L^{T}P_{1})]e_{x}(t)+2e_{x}^{T}(t)(I_{N}\otimes P_{1}D)d(t)\nonumber \\{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)(\theta (t_{s+1})-\theta (t))\nonumber \\{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)\biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})\nonumber \\{} & {} -\theta (t_{s+1})\biggr )+\xi ^{T}(t)[I_{N}\otimes (A^{T}P_{2}+P_{2}A\nonumber \\{} & {} -P_{2}BK-K^{T}B^{T}P_{2})]\xi (t)\nonumber \\{} & {} +2\xi ^{T}(t)(I_{N}\otimes P_{2}BK) e_{x}(t)\nonumber \\{} & {} -2\xi ^{T}(t)(I_{N}\otimes P_{2}BK)x_{0}(t)\nonumber \\{} & {} +2\xi ^{T}(t)(I_{N}\otimes P_{2}D) e_{d}(t) \end{aligned}$$

(47)

Based on Lemma 1 and the event-triggered conditions, it can be inferred that

$$\begin{aligned}{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)\biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr )\nonumber \\{} & {} \quad \le \frac{1}{\rho _{1}}e_{x}^{T}(t)(I_{N}\otimes P_{1}LL^{T}P_{1})e_{x}(t)\nonumber \\{} & {} \quad {}+\rho _{1}\biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr )^{T}\nonumber \\{} & {} \quad {}\times (I_{N}\otimes C^{T}C)\biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr ) \nonumber \\{} & {} \biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr )^{T}(I_{N}\otimes C^{T}C)\nonumber \\{} & {} \quad \times \biggl (\sum _{q=1}^{m}\omega _{q}\theta (t_{s-q+1}^{i})-\theta (t_{s+1})\biggr )\nonumber \\{} & {} \quad \le \varrho (t_{s+1})\theta ^{T}(t_{s+1})(I_{N}\otimes C^{T}C)\theta (t_{s+1})\nonumber \\{} & {} \quad =\varrho (t_{s+1})\xi ^{T}(t_{s+1})(\bar{{\mathcal {L}}}^{2}\otimes C^{T}C)\xi (t_{s+1}) \end{aligned}$$

(48)

$$\begin{aligned}{} & {} -2e_{x}^{T}(t)(I_{N}\otimes P_{1}LC)(\theta (t_{s+1})-\theta (t))\nonumber \\{} & {} \quad \le \frac{1}{\rho _{2}}e_{x}^{T}(t)(I_{N}\otimes P_{1}LL^{T}P_{1})e_{x}(t)\nonumber \\{} & {} \quad +\rho _{2}(\theta (t_{s+1})-\theta (t))^{T}(I_{N}\otimes C^{T}C)\nonumber \\{} & {} \quad \times (\theta (t_{s+1})-\theta (t))\nonumber \\{} & {} (\theta (t_{s+1})-\theta (t))^{T}(I_{N}\otimes C^{T}C)(\theta (t_{s+1})-\theta (t))\nonumber \\{} & {} \quad \le 2\theta ^{T}(t_{s+1})(I_{N}\otimes C^{T}C)\theta (t_{s+1})\nonumber \\{} & {} \quad +2\theta ^{T}(t_{s})(I_{N}\otimes C^{T}C)\theta (t_{s}) \end{aligned}$$

(49)

The analysis of other parts refers to the proof in the Case 1. Taking (48) and (49) into (47) and combining condition (30), the derivative of V(t) with respect to time at this case can be derived.

$$\begin{aligned} \dot{V}(t)\le \varLambda max\{V_{1}(t),V_{1}(t_{s+1})\}-e_{d}^{T}(t)\mu e_{d}(t) \end{aligned}$$

(50)

By Lemma 2 and (45), (50) can be obtained

$$\begin{aligned} V_{1}(t)\le e^{[\varepsilon -(t-t_{0}-n(t_{0},t)\vartheta )+\varLambda n(t_{0},t)\vartheta ]}V(t_{0}) \end{aligned}$$

Then we have $\lim _{t\rightarrow \infty }V_{1}(t)=0$. In addition, it can be inferred from the above derivation that $\zeta {\dot{V}}_{2}(t)\le -e_{d}^{T}(t)\mu e_{d}(t)$. Then it is easy to obtain that $V_{2}(t)\ge 0$, ${\dot{V}}_2(t)\le 0$. And $\lim _{t\rightarrow \infty }V_{2}(t)=V_{2}(\infty )$, so $V_{2}(\infty )\le V_{2}(t)\le V_{2}(t_{0})$. By integrating ${\dot{V}}_{2}(t)$, we obtain

$$\begin{aligned} V_{2}(t_{0})-V_{2}(\infty )\le \int _{t_{0}}^{\infty }(-e_{d}^{T}(t)\mu e_{d}(t))dt \end{aligned}$$

According to Barbarat’s lemma, it can be concluded that $\lim _{t\rightarrow \infty }e_{d}(t)=0$. Based on the above proof, we know that when $t\rightarrow \infty$, the tracking error, the state estimation error and the disturbance estimation error all tend to 0, and the tracking consensus is achieved. $\square$

Remark 8

In this paper, based on the LMI toolbox, the observer and controller gains are solved at the same time, avoiding the adjustment of various parameters and symmetric positive definite matrix in the early stage. This makes the solution of the observer and controller gains more accurate. In addition, due to the introduction of the detection algorithm, the limits of the DoS attack duration and frequency are not given separately in this paper. By $n(t_{0},t)$, the DoS attack is indirectly limited.

3.4 LMI solves the gain matrix

In Theorem 1, the necessary conditions for the high-speed train to achieve tracking consistency and the observer to achieve accurate observation are given. However, since L and K are coupled with the unknown matrix, they cannot be solved directly by LMI. The following theorem presents a method for solving the state observer and controller gains.

Theorem 2

For the given ${\mathcal {X}}_{1}$, ${\mathcal {X}}_{2}$, ${\mathcal {H}}_{1}$, ${\mathcal {H}}_{2}$, $\varepsilon$ and $\varphi$, if there exist symmetric positive definite matrices $P_{1}$, $P_{2}$ and matrices Q, M, Z satisfying the following conditions

$$\begin{aligned}{} & {} \begin{bmatrix} \breve{\varGamma }_{1} &{} *&{} *&{}*&{}*&{} *\\ P_1&{}-{\mathcal {X}}_{1}&{}*&{}*&{}*&{}*\\ Q&{}0&{}-{\mathcal {X}}_{2}&{}*&{}*&{}*\\ BZ&{}0&{}0&{}\breve{\varGamma }_{2} &{}*&{}*\\ 0&{}0&{}0&{}P_{2}&{}-{\mathcal {H}}_{1}&{}*\\ 0&{}0&{}0&{}B^{T}P_{2}&{}0&{}-{\mathcal {H}}_{2} \end{bmatrix} <0 \end{aligned}$$

(51)

$$\begin{aligned}{} & {} \begin{bmatrix} -\varphi I &{} *\\ P_{2}B-BM &{} -\varphi I \end{bmatrix} <0 \end{aligned}$$

(52)

where

$$\begin{aligned} \breve{\varGamma }_{1} ={}&A^{T}P_{1}+P_{1}A-\lambda _{min}(Q^{T}C+C^{T}Q)+\varepsilon P_{1} \\ \breve{\varGamma }_{2} ={}&A^{T}P_{2}+P_{2}A-BZ-Z^{T}B^{T}\\ {}&+{\mathcal {X}}_{2}\varrho _{M}{\lambda _{max}}^{2}C^{T}C+\varepsilon P_{2} \end{aligned}$$

From above formulas, we can derive $L={P_{1}}^{-1} Q^{T}$ and $K=M^{-1}Z$. In this way, the observer gain L and the controller gain K can be obtained by solving LMI (51) and (52). The proof of Theorem 2 is given next.

Proof

Let $Q=L^{T}P_{1}$, $P_{2}B=BM$ and $Z=MK$. Then equation (32) can be written as

$$\begin{aligned} \begin{bmatrix} {\tilde{\varGamma }}_{1} &{} Z^{T}B^{T} \\ *&{} {\tilde{\varGamma }}_{2} \end{bmatrix} <0 \end{aligned}$$

where

$$\begin{aligned} {\tilde{\varGamma }}_{1} ={}&A^{T}P_{1}+P_{1}A-\lambda _{min}(Q^{T}C+C^{T}Q)+\frac{1}{{\mathcal {X}}_{1}}P_{1}P_{1}\\ {}&+\frac{1}{{\mathcal {X}}_{2}}Q^{T}Q+\varepsilon P_{1}\\ \breve{\varGamma }_{2} ={}&A^{T}P_{2}+P_{2}A-BZ-Z^{T}B^{T}+\frac{1}{{\mathcal {H}}_{1}}P_{2}P_{2}\\ {}&+\frac{1}{{\mathcal {H}}_{2}}BMM^{T}B^{T}+{\mathcal {X}}_{2}\varrho _{M}{\lambda _{max}}^2C^{T}C+\varepsilon P_{2} \end{aligned}$$

Applying the Schur complement lemma and Lemma 2, we can prove that (51) and (52) hold. $\square$

3.5 Zeno behavior analysis

Zeno behavior will cause the trigger to fire infinitely many times in a finite time, which is both impractical and contradictory the purpose of reducing communication burden through event triggering strategies. Therefore, Zeno behavior is a non-negligible problem in event-triggered mechanism and must be avoided. Inspired by the discussion of Zeno behavior in [41], the following theorem is given to prove that there is a positive lower bound between any two consecutive firing instants in the communication time, thereby ensuring that Zeno behavior does not occur.

Theorem 3

Under the control of the event-triggered mechanism (14), the minimum interval between any two event triggering instants of the tracking train is always positive. That is

$$\begin{aligned} t_{k+1}^{i}-t_{k}^{i}\ge \pi _{i}\ge \frac{\varPsi _{i}^{2}}{\Omega [\Vert A\Vert \varPi _{i}^{2}+\varPi _{i} \varPhi _{i} (t_{k}^{i})]}>0 \end{aligned}$$

(53)

where $\pi _{i}$ is a positive constant and the other symbols represent

$$\begin{aligned} \varPsi _{i}^{2}= & {} {\Bigg \Vert C\sum _{q=1}^{m} \omega _{q} e_{i,t_{k-q+1}}(t) \Bigg \Vert }^2 \\ \Omega= & {} 2\Vert C\Vert \\ \varPi _{i}= & {} {\Bigg \Vert \sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigg \Vert }^2 \\ \varPhi _{i} (t_{k}^{i})= & {} \Vert A\Vert \Bigg \Vert \sum _{q=1}^{m}\omega _{q}\theta _{i}(t_{k-q+1}^{i})\Bigg \Vert \\{} & {} +\Vert B\Vert \biggl ( \Bigg \Vert \sum _{j\in N_{i}}a_{ij}(u_{i}(t)-u_{j}(t))\Bigg \Vert +u_{i}(t)\biggr ) \\{} & {} +\Vert D\Vert \biggl (\Bigg \Vert \sum _{j\in N_{i}}a_{ij}(d_{i}(t)-d_{j}(t))\Bigg \Vert +d_{i}(t)\biggr ) \end{aligned}$$

Proof

For any $t\in [t_{k}^{i},t_{k+1}^{i}]$, it exists

$$\begin{aligned}{} & {} \frac{d}{dt} \biggl ( \Bigg \Vert \sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1)}}(t) \Bigg \Vert ^{2} \biggr ) \nonumber \\{} & {} \quad \le 2\Bigg \Vert \sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigg \Vert \Vert {\dot{\theta }}_{i}(t)\Vert \nonumber \\{} & {} \quad \le 2\Vert A\Vert \Bigg \Vert \sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1)}}(t)\Bigg \Vert ^{2} \nonumber \\{} & {} \qquad +2\Bigg \Vert \sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1)}}(t)\Bigg \Vert \nonumber \\{} & {} \qquad \times \biggl [ \Vert A\Vert \Bigg \Vert \sum _{q=1}^{m}\omega _{q}\theta _{i}(t_{k-q+1}^{i})\Bigg \Vert \nonumber \\{} & {} \qquad +\Vert B\Vert \biggl (\Bigg \Vert \sum _{j\in N_{i}}a_{ij}(u_{i}(t)-u_{j}(t))\Bigg \Vert +u_{i}(t) \biggr ) \nonumber \\{} & {} \qquad +\Vert D\Vert \biggl (\Bigg \Vert \sum _{j\in N_{i}}a_{ij} (d_{i}(t)-d_{j}(t))\Bigg \Vert +d_{i}(t)\biggr )\Biggr ] \nonumber \\ \end{aligned}$$

(54)

According to the above inequality, the derivative of $\varPsi _{i}^{2}$ with respect to time satisfies

$$\begin{aligned} \frac{d}{dt} (\varPsi _{i}^{2})\le \Omega [\Vert A\Vert \varPi _{i}^{2}+\varPi _{i} \varPhi _{i} (t_{k}^{i})] \end{aligned}$$

(55)

Fig. 3

Inter-train communication topology

Table 1 Values of train parameters

Fig. 4

Tracking velocity error of each train

Fig. 5

Tracking position error of each train

Fig. 6

Actual position curve in continuous operation state

In addition, it can be known from the event-triggered mechanism (12) that at any triggering moment

$$\begin{aligned}{} & {} \Bigl (\sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigr )^{T}\varTheta \Bigl (\sum _{q=1}^{m}\omega _{q}e_{i,t_{k-q+1}}(t)\Bigr )\\{} & {} \quad >\varrho _{i}(t)\theta _{i}^{T}(t)\varTheta \theta _{i}(t) \end{aligned}$$

Then, it can be obtained that

$$\begin{aligned} {\Bigg \Vert C\sum _{q=1}^{m} \omega _{q} e_{i,t_{k-q+1}}(t) \Bigg \Vert }^2>\varrho _{i}(t) \Vert C\theta _{i} (t_{k}^{i})\Vert ^{2} \end{aligned}$$

From the above deductions, it can be concluded that

$$\begin{aligned} \varPsi _{i}^{2}>\varrho _{i}(t) \Vert C\theta _{i} (t_{k}^{i})\Vert ^{2}> 0 \end{aligned}$$

(56)

Therefore, $t_{k+1}^{i}-t_{k}^{i}\ge min\{\pi _{i},\vartheta \}>0$, which indicates that the event-triggered mechanism designed in this paper can avoid Zeno behavior. $\square$

Fig. 7

Actual velocity curve in continuous operation state

Fig. 8

The position curve of each train

Remark 9

This article employs a hybrid event-triggered mechanism to determine the triggering moment. When the tracking train is subjected to the communication interruption attack, a fixed time interval $\vartheta$ is used to determine the subsequent triggering time. At this point, there is a positive lower bound between any two consecutive triggering instants, preventing Zeno behavior from occurring. $\pi _{i}$ is associated with $\varrho _{i}(t)$.

4 Simulation example

This article considers a multi-high-speed train consist of a leader train and three follower trains. The communication topology among high-speed trains is illustrated in Fig. 3, where the information of the leader train can be transmitted to each other tracking trains, and the communication matrix is

$$\begin{aligned} \bar{{\mathcal {L}}}= \begin{bmatrix} 2 &{} -1 &{} 0\\ -1 &{} 2 &{} -1\\ 0 &{} -1 &{} 1 \end{bmatrix} \end{aligned}$$

By solving, the maximum eigenvalue $\lambda _{max}=3.2407$ and the minimum eigenvalue $\lambda _{min}=0.1981$ are obtained. And system matrices are set $A= \begin{bmatrix} 0 &{} 1\\ 0 &{} 0\\ \end{bmatrix}$, $B= \begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}$, $C= \begin{bmatrix} 1&{}1 \\ \end{bmatrix}$, $D= \begin{bmatrix} 0&{}0 \\ -0.6&{}-0.6\\ \end{bmatrix}$, $E= \begin{bmatrix} 0&{}0 \\ -0.3&{}-0.3\\ \end{bmatrix}$, $F= \begin{bmatrix} -0.6&{}-0.6\\ \end{bmatrix}$. In addition, the initial state of each train is set as $x_{0}(t)=[24000\quad 60]^{T}$, $x_{1}(t)=[16000\quad 55]^T$, $x_{2}(t)=[8000\quad 65]^{T}$, $x_{3}(t)=[0\quad 70]^{T}$, and the other parameters of each train are shown in Table 1, where s(i) is the track slope of the high-speed train.

Fig. 9

Tracking velocity error under communication interruption attack

Fig. 10

Tracking position error under communication interruption attack

Fig. 11

The actual velocity of each train under communication interruption attack

According to Theorems 1 and 2, the corresponding parameters are selected ${\mathcal {X}}_{1}=1$, ${\mathcal {X}}_{2}=21$, ${\mathcal {H}}_{1}=10$, ${\mathcal {H}}_{2}=1.65$, $\varepsilon =0.25$, $\varphi =0.001$, $\varrho _{M}=0.3$, $\varLambda =5$, $\sigma =0.005$ and $\vartheta =0.04$. By solving LMI (51) and (52), the observer gain L and the controller gain K are obtained as

$$\begin{aligned} L= \begin{bmatrix} -1.2269\\ 0.2324 \end{bmatrix},K= \begin{bmatrix} -864.8314&-767.2462\nonumber \end{bmatrix} \end{aligned}$$

After comprehensively considering the system performance and communication burden, each parameter in the trigger function (12) and the adaptive rate (13) is set as $m=3$, $\omega _{1}=0.85$, $\omega _{2}=0.1$, $\omega _{3}=0.05$, $\varrho _{i}=0.05$ and $\gamma =0.1$.

Assuming the leader train is at a constant speed, the GSM-R is not under communication interruption attack, and the tracking trains are all disturbed by gusts. The frequency of gusty disturbance on each train is given as $\chi _{1}=1.2$, $\chi _{2}=2.3$, $\chi _{3}=3.1$, with amplitudes of $d_1=[1\quad 1]^{T}$, $d_{2}=[4\quad 2.5]^{T}$, $d_{3}=[3\quad 1.5]^{T}$. Figures 4 and 5 show the tracking velocity and position error of the following trains to the leader train. Through the comparison in Figs. 4 and 5, it can be seen that the disturbance observer improves the tracking consensus performance.

Under the above conditions, this paper simulated the three operating states of traction, braking and idle speed existing in the continuous operation of the leader train. Figures 6 and 7 respectively depict the actual position and velocity curves of each train during continuous operation, and they illustrate the effectiveness and practicality of the disturbance observer proposed in this study under actual continuous operating conditions.

Then, based on the above conditions, this article simulated the communication interruption attack on the 3th train. The gray lines in the figure denote the time period when the communication interruption attack occurs. Figure 8 shows the actual position of each train with and without communication interruption attack. Through comparison, it is evident that the following trains maintain good tracking performance when the 3th train is attacked, with no notable difference in its actual position compared with that without attack, and no rapid approaching or collision occurs among the trains. Figures 9 and 10 illustrate the tracking velocity and position errors of the following trains in the attack state. When the communication interruption attack occurs, due to the state information of the observer all comes from the memory, the tracking errors of each train have a small fluctuation, and then each train quickly adjusts its controller output to ensure stable operation at a safe distance.

Fig. 12

Position of each train under communication interruption attack

Fig. 13

Trigger interval of each tracking train

Similarly, this study also simulated the communication interruption attack during the continuous operation of high-speed trains. It is noteworthy that the communication interruption attacks occur during the traction state, braking state, and the transition between traction, idle speed, and braking. Therefore, the attack designed in this paper is more targeted and realistic. Figures 11 and 12 show the actual velocity and position curve of each train. When subjected to an attack during the transition between traction, idle speed, and braking, the proposed method can adjust the system output to track the leader train quickly. This proves that the proposed control strategy can guarantee the consensus tracking between trains when it is attacked during actual continuous operation.

Figure 13 shows the trigger time and communication interval of the three tracking trains within 100 s. The simulation results show that under the given maximum threshold $\varrho _{M}$, when the state of the tracking train is consistent with the leader train, the dynamic trigger threshold $\varrho _{i}(t)$ increases, and the trigger interval increases accordingly. Furthermore, it is not difficult to find from Fig. 13 that the trigger points are more intensive in the time period when the tracking state error is large. This is because more data packets are needed to reduce the error in order to achieve consistent tracking quickly, which is consistent with the theoretical analysis.

The calculation shows that each train discards more than $85\%$ of the data packets, which significantly reduces the communication burden, and $n(t_{0},t)$ satisfies condition (34). At the same time, Zeno behavior is also avoided.

Table 2 Number of trigger points for two different trigger thresholds

To further illustrate the advantages of AMETS proposed in this paper, we conducted simulations by degenerating the dynamic triggering threshold $\varrho _{i}(t)$ to a fixed triggering threshold. The fixed trigger threshold was set to $\varrho _{i}(t)=\frac{\varrho _{i}+\varrho _{M}}{2}$. Table 2 shows the number of trigger points of the two methods within 100 s. By comparison, it can be concluded that the method designed in this article can better conserve communication resources and ensure the control performance of tracking trains.

In the future, when facing more complex high-speed train network, the capacity of the memory in AMETS will be increased. At present, the on-board computer system of the high-speed train has been able to realize the accurate calculation of a large amounts of data. Therefore, the results of this paper are still feasible in the future.

5 Conclusions

This paper has delved into the tracking consensus issues of high-speed trains when the GSM-R is under DoS attacks. External disturbance, primarily composed of gusts, has been taken into account, making the train operation state in this article more realistic. To ensure the tracking consistency of high-speed trains under DoS attacks and gusty disturbances, this paper has proposed a network attack detection algorithm to detect communication interruption attacks between high-speed trains under event-triggered environment. According to the detection results, an anti-disturbance controller based on the state observer and disturbance observer has been designed, and the gains of the state observer and controller have been calculated by LMI. In addition, to conserve communication resources, an AMETS has been designed. Now, it is updated using not only the current state of the train but also the previously stored information in the memory and can eliminate Zeno behavior. Finally, the simulation results have showed the effectiveness and superiority of the proposed method. In addition, the result of this paper can also be extended to the fault-tolerant control of high-speed trains. It is worth noting that the high-speed train operation line set in this paper is the most common section without a switch, while the application of complex railway lines (section with a switch, railway station yard sections, etc.) is worth studying in the future.