Model-Free Adaptive Predictive Tracking Control for High-Speed Trains Considering Quantization Effects and Denial-of-Service Attacks

Abstract

In this paper, the problem of model-free adaptive predictive control (MFAPC) under denial-of-service attacks and quantization effects for high-speed trains with unknown models is investigated. Since the system model of the high-speed train is unknown, the data-relational description of a high-speed train system is obtained by using the dynamic linearization technique. Secondly, the challenge of periodic denial-of-service (DoS) attacks in the network channel is considered, and, assuming that the DoS attack obeys the Bernoulli distribution, a model-free adaptive predictive control scheme based on quantized signals is proposed. Then, through rigorous theoretical analyses, it is proven that the tracking error is bounded, and the final bound depends on the desired trajectory. Finally, the correctness of these theoretical analyses is verified through numerical simulation.

1. Introduction

As an important part of modern transport, high-speed trains have had a profound impact on the development of society and economy. Their efficient, fast, and comfortable features make high-speed trains play a key role in long-distance passenger and freight transport [1,2,3]. The wide application of high-speed trains not only shortens the time distance between cities but also promotes the integration of the regional economy and improves productivity and quality of life. In the process of high-speed train operation, accurate speed-tracking control is crucial. It is not only directly related to the safety and stability of train operation but also affects the passenger experience and transport efficiency [4,5]. In Ref. [6], an event-triggered adaptive control strategy is proposed to specifically address the problem of spoofing attacks faced by high-speed trains in bottleneck sections. The proposed approach reduces the communication burden of the system by reducing unnecessary communications and, at the same time, improves the robustness and adaptability of the train control, which effectively counteracts the impact of spoofing attacks on train operations. A distributed speed- and input constraint-tracking control method for high-speed train systems was proposed in Ref. [7]. This method handles the speed-tracking and input constraint problems through a distributed control architecture, which achieves coordinated control of multiple trains and enhances the stability and safety of the system. In Ref. [8], a robust distributed cruise-control method based on a perturbation observer is proposed for multiple high-speed trains. The proposed technique effectively compensates for external perturbations and uncertainties by introducing a perturbation observer, realizes coordinated control among trains, and enhances the robustness and stability of the system.

Traditional speed-tracking control methods for high-speed trains usually rely on accurate system modeling. However, in practical applications, it is often very difficult to obtain a precise system model due to the complexity of the system and the uncertainty of the environment. In addition, traditional methods are highly dependent on model parameters, which makes it difficult for them to cope with changes in system parameters and external perturbations. Therefore, it is of great research significance to explore novel control methods that do not require accurate modeling information. To address the limitations of traditional control methods, researchers have proposed data-driven control algorithms. These algorithms do not rely on the precise mathematical model of the system but instead construct the controller directly by analyzing the system operation data. The introduction of data-driven control algorithms provides a new way of thinking to solve the control problems of complex systems. Notably, the model free adaptive control (MFAC) algorithm, as a prominent data-driven algorithm for nonlinear systems, has gained a great deal of interest in the control field [9,10,11,12,13,14]. Firstly, it does not require an accurate system model, which reduces the modeling complexity. Secondly, through its adaptive mechanism, it can maintain good control performance when the system parameters change. In addition, this algorithm is robust and can effectively deal with external disturbances. In Ref. [15], a model-free adaptive and iterative learning composite control method is proposed to deal with the actuator fault problem of underground trains. Without relying on an accurate system model, this technique effectively copes with actuator faults. It is able to improve the robustness and reliability of a metro-train control system through adaptive and iterative learning mechanisms. In Ref. [16], a data-driven coordinated control method is proposed to specifically target multiple high-speed trains under fading measurements and denial-of-service (DoS) attacks. This method improves the robustness and resilience of the system in the face of measurement data loss and network attacks by means of a data-driven control strategy and a coordination mechanism, ensuring the safe and stable operation of the train. On the other hand, in the present paper, a model-free adaptive predictive tracking control scheme for high-speed trains is proposed by combining the ideas of model-free adaptive control and predictive control, and the proposed algorithm not only inherits the advantages of model-free adaptive control but also further improves the control accuracy and response speed through the prediction mechanism [17,18,19]. Moreover, Ref. [18] proposed a distributed model-free adaptive predictive control method for spoofing attacks in multiple-input multiple-output (MIMO) multi-intelligent body systems. The method does not require a system model and uses a data-driven strategy to adjust the control in real time, which improves the robustness of the system under dynamic changes and network attacks. Through a distributed collaboration mechanism, the intelligences can share information, which enhances high overall performance and security. Wang: It has been modified. In network control systems, communication networks are critical for the exchange of control signals and data. Still, they are also susceptible to network security threats such as denial-of-service (DoS) attacks [20]. Scholars have recently dedicated themselves to the study of malicious attacks, resulting in numerous advancements [21,22,23]. A DoS attack is a common type of network attack that affects the normal operation of the system by occupying network resources through a large number of invalid requests, resulting in normal communication and control signals not being transmitted. Ref. [24] proposes an adaptive quantization control method for output feedback control in nonlinear systems that addresses sensor failures and intermittent denial-of-service (DoS) attacks. The proposed method ensures the stability and performance of the system under harsh conditions by dynamically adjusting the quantization parameters and handling sensor failures. The paper also verifies the effectiveness and robustness of the control strategy through theoretical analysis and simulation experiments. Ref. [25] presents a model-free adaptive predictive control method designed to optimize the output tracking performance of a switching system under denial-of-service (DoS) attacks. The method improves the robustness of the system through an adaptive strategy and its effectiveness is verified in theory and through simulations. Ref. [26] presents an event-triggered stabilization method for simultaneous quantization and denial-of-service (DoS) attacks in linear systems. The authors devise an event-triggered mechanism to reduce the consumption of communication and computational resources and to ensure that the system remains stable when subjected to quantization errors and intermittent communication interruptions. Their method effectively enhances the system’s resistance to DoS attacks by combining event-triggering and robust control strategies.

Due to network bandwidth or capacity limitations, sensor signals must be quantized before transmission [27,28,29]. Quantization effects in nonlinear systems have been extensively studied in the last decade. Ref. [30] presents a synchronized quantization control method for segmented affine systems, aiming to ensure the stability of the system under synchronized clocks by handling the quantization error through a unified framework. This method takes into account quantization effects, time delays, and perturbations, and provides a control strategy to improve the robustness and performance of the system. Ref. [31] presents a quantized stabilization control method, specifically for affine systems with state-dependent switching, that takes into account the quantization of control inputs and state measurements. The authors design a control strategy based on Lyapunov’s method that is able to ensure the asymptotic stability of the system with a finite number of quantization levels. Their method effectively solves the stability problem when there is simultaneous quantization of the control inputs and state measurements. Data quantization introduces quantization errors that affect the accuracy of the control system [32,33,34]. In high-speed train speed-tracking control, finding the optimal way to effectively deal with quantization errors in order to ensure the performance of the control system is an urgent matter. In Ref. [35], a model-free adaptive consensus tracking method based on an active quantizer is proposed to specifically cope with multiple high-speed trains under the influence of sensor bias. This process utilizes the active quantizer technique to achieve inter-train consensus tracking through an adaptive control strategy, which effectively reduces the impact of sensor bias on the system performance and improves the stability and accuracy of the control system. The quantization mechanism’s theoretical relevance and practical usefulness are important, but its implications for high-speed train speed-tracking control need to be further studied. There are several outstanding questions in the literature on quantization effects that contribute to our research.

By addressing these challenges through a novel model-free adaptive predictive control algorithm, this research aims to enhance the robustness and efficiency of speed-tracking control in high-speed trains, ensuring their safe and reliable operation amidst the complexities of modern communication networks and cybersecurity threats. The main contributions of this study are concluded as follows:

(1)

For high-speed train systems with complex models, the systems are transformed into equivalent data-relational descriptions using dynamic linearization techniques, and then a model-free adaptive predictive control strategy is proposed.

(2)

Periodic DoS attacks and speed error quantization effects are considered, and a rigorous theoretical analysis shows that the system remains stable with the proposed strategy even under DoS attacks and speed error quantization effects.

In this paper, some relevant mathematical symbols are used, and their definitions are shown in

Table 1. Notations and descriptions.

Notation	Description
$v\left(k\right)$	the train speed
$F\left(k\right)$	the train traction/braking force
T	the sampling step size
k	the sampling moment
$f_d\left(k\right)$	the train’s additional resistance
$f_t\left(k\right)$	tunnel resistance
$a_1\left(k\right)$, $a_2\left(k\right)$, $a_3\left(k\right)$	unknown resistance coefficients
${\mathbb{A}}^{-1}$	the inverse of a matrix ${\mathbb{A}}^{-1}$
${\mathbb{A}}^T$	transposing the matrix ${\mathbb{A}}^T$
$\mathbb{E}\left(\mathbb{A}\right)$	taking the expectations for $\mathbb{A}$

.

2. Preparatory Discussion and Problem Statement

This section considers a single-point-mass high-speed train operation model. This high-speed train single-mass dynamic model is used to view the N trains as a whole, which are analyzed in terms of forces, ignoring the forces between the trains, as shown in Figure 1.

2.1. High-Speed Train Model

For a single-mass model of a high-speed train based on Newton’s second law,

$$v(k+1)=v\left(k\right)+T\left(F\left(k\right)-f_a\left(k\right)-f_b\left(k\right)\right)$$

(1)

where k is the sampling moment, $v\left(k\right)$ is the train speed, $F\left(k\right)$ is the train traction/braking force, T is the sampling step size, $f_a\left(k\right)$ is the train’s additional resistance, and $f_b\left(k\right)$ is the train’s base resistance:

$$\begin{array}{cc}\hfill & f_b\left(k\right)=a_1\left(k\right)+a_2\left(k\right)v\left(k\right)+a_3\left(k\right)v^2\left(k\right)\hfill \\ \hfill & f_a\left(k\right)=f_r\left(k\right)+f_c\left(k\right)+f_t\left(k\right)\hfill \end{array}$$

(2)

where $f_r\left(k\right)$ denotes ramp resistance, $f_c\left(k\right)$ is the curve resistance, $f_t\left(k\right)$ is the tunnel resistance, and $a_1\left(k\right)$, $a_2\left(k\right)$, and $a_3\left(k\right)$ are unknown resistance coefficients.

It can be seen that the dynamic model contains strong nonlinear and unknown dynamics. Using the traction/braking force $F\left(k\right)$ as the input $u\left(k\right)$ to the system, the following general model is obtained:

$$v(k+1)=f\left(v\left(k\right),u\left(k\right),f_b\left(k\right),f_a\left(k\right)\right)$$

(3)

where $f(·)$ is an unknown nonlinear function of the high-speed train model. The following assumptions are made for the train system (3):

Assumption 1. 

The partial derivatives of $f(·)$ with respect to $u\left(k\right)$ are continuous.

Assumption 2. 

The nonlinear function $f(·)$ satisfies the generalized Lipschitz condition; that is, for all $k, |\Delta v(k+1\left)\right|\le b|\Delta u(k\left)\right|$ and $\Delta u\left(k\right)\ne 0$, where $\Delta v(k+1)=v(k+1)-v\left(k\right),$ $\Delta u\left(k\right)=u\left(k\right)-u(k-1)$, and b is a normal number.

Remark 1. 

The above two assumptions are reasonable and acceptable from a practical point of view, with Assumption 1 being a constraint on the rate of change in output. From the energy point of view, a finite input quantity obtains a finite output, and the high-speed train traction system satisfies this assumption. Assumption 2 is a typical constraint on the controlled object in the process of control system design.

Theorem 1. 

For a high-speed train system (3) that fulfils the Assumptions 1 and 2 above, when $\Delta u\left(k\right)\ne 0$, there must exist a time-varying parameter known as the pseudo-partial derivative (PPD) $\Xi \left(k\right)$ that allows the system (3) to be transformed into a tight-format dynamic linearized data model, as follows:

$$\Delta v(k+1)=\Xi \left(k\right)\Delta u\left(k\right)$$

(4)

where the pseudo-partial derivative $\Xi \left(k\right)$ is bounded at any moment k.

Proof of Theorem 1. 

The proof of Theorem 1 can be obtained from Assumption 2:

$$\begin{array}{cc}\hfill \Delta v(k+1)& =v(k+1)-v\left(k\right)\hfill \\ \hfill & =f\left(v\right(k),u(k\left)\right)-f\left(v\right(k-1),u(k-1\left)\right)\hfill \\ \hfill & =f\left(v\right(k),u(k\left)\right)-f\left(v\right(k),u(k-1\left)\right)+f\left(v\right(k),u(k-1\left)\right)-f\left(v\right(k-1),u(k-1\left)\right)\hfill \end{array}$$

(5)

Let $\Theta \left(k\right)=f\left(v\right(k),u(k-1\left)\right)-f\left(v\right(k-1),u(k-1\left)\right)$. According to Cauchy’s median theorem, Assumption 1 and Equation (5) can then be obtained:

$$\begin{array}{cc}\hfill \Delta v(k+1)& =f\left(v\right(k),u(k\left)\right)-f\left(v\right(k),u(k-1\left)\right)+\Theta \left(k\right)\hfill \\ & ={\displaystyle \frac{\partial f^{*}}{\partial u\left(k\right)}}\Delta u\left(k\right)+\Theta \left(k\right)\hfill \end{array}$$

where ${\displaystyle \frac{\partial f^{*}}{\partial u\left(k\right)}}$ is the partial derivative of the function $f(·)$ with respect to $u\left(k\right)$ between points ${[v\left(k\right),u(k-1)]}^{\mathrm{T}}$, and ${[v\left(k\right),u\left(k\right)]}^{\mathrm{T}}$. Consider the equation for $p\left(k\right)$:

$$\Theta \left(k\right)=p\left(k\right)\Delta u\left(k\right)$$

Since $\Delta u\left(k\right)\ne 0$, there exists a unique solution $p^{*}\left(k\right)$ to this equation, making Equation $\Theta \left(k\right)=p^{*}\left(k\right)\Delta u\left(k\right)$ hold. Furthermore, one has

$$\begin{array}{cc}\hfill \Delta v(k+1)& ={\displaystyle \frac{\partial f^{*}}{\partial u\left(k\right)}}\Delta u\left(k\right)+\Theta \left(k\right)\hfill \\ & =\Xi \left(k\right)\Delta u\left(k\right)\hfill \end{array}$$

where $\Xi \left(k\right)={\displaystyle \frac{\partial f^{*}}{\partial u\left(k\right)}}+p^{*}\left(k\right)$. According to Assumption 2, we obtain the boundedness of $\Xi \left(k\right)$. This concludes the proof. □

2.2. Controller Design

From (4), a one-step output prediction equation of the following form is given:

$$v(k+1)=v\left(k\right)+\Xi \left(k\right)\Delta u\left(k\right)$$

(6)

According to Equation (6), the N-step forward prediction equation can be further obtained as follows:

$$\left\{\begin{array}{c}v(k+1)=v\left(k\right)+\Xi \left(k\right)\Delta u\left(k\right)\hfill \\ v(k+2)=v(k+1)+\Xi (k+1)\Delta u(k+1)\hfill \\ =v\left(k\right)+\Xi \left(k\right)\Delta u\left(k\right)+\Xi (k+1)\Delta u(k+1),\hfill \\ ⋮\hfill \\ v(k+N)=v(k+N-1)+\Xi (k+N-1)\Delta u(k+N-1)\hfill \\ =v(k+N-2)+\Xi (k+N-2)\Delta u(k+N-2)\hfill \\ +\Xi (k+N-1)\Delta u(k+N-1),\hfill \\ ⋮\hfill \\ =v\left(k\right)+\Xi \left(k\right)\Delta u\left(k\right)+\cdots +\Xi (k+N-1)\Delta u(k+N-1)\hfill \end{array}\right.$$

(7)

Clearly, the following vector form can be obtained:

$$\mathbf{V}(k+1)=\mathbf{D}v\left(k\right)+\mathbf{C}\left(k\right)\Delta {\mathbf{U}}_M\left(k\right)$$

(8)

where $\mathbf{V}(k+1)={[v(k+1)\cdots v(k+1)]}^{\mathrm{T}}$; $\mathbf{D}={[1\cdots 1]}^{\mathrm{T}}$; $\Delta {\mathbf{U}}_M\left(k\right)=[\Delta u\left(k\right)\cdots \Delta (k+M-1)]$; M is the control time-domain constant; and $\mathbf{C}\left(k\right)={\left[\begin{array}{cccc}\Xi \left(k\right)& 0& 0& 0\\ \Xi \left(k\right)& \Xi (k+1)& 0& 0\\ ⋮& ⋮& \ddots & ⋮\\ \Xi \left(k\right)& \Xi (k+1)& \dots & \Xi (k+M-1)\\ ⋮& ⋮& \dots & ⋮\\ \Xi \left(k\right)& \Xi (k+1)& \dots & \Xi (k+M-1)\end{array}\right]}_{N\times M}$.

Consider the following input criterion function:

$$\mathbf{J}={\left[{\mathbf{V}}_d(k+1)-\mathbf{V}(k+1)\right]}^{\mathrm{T}}\left[{\mathbf{V}}_d(k+1)-\mathbf{V}(k+1)\right]+\lambda \Delta {\mathbf{U}}_M^{\mathrm{T}}\left(k\right)\Delta {\mathbf{U}}_M\left(k\right)$$

(9)

where ${\mathbf{V}}_d(k+1)={\left[v_d(k+1),\cdots ,v_d(k+1)\right]}^{\mathrm{T}}$, and $\lambda >0$ is a weighting factor.

Bringing Equation (8) into Equation (9) and letting ${\displaystyle \frac{\partial \mathbf{J}}{\partial {\mathbf{U}}_M\left(k\right)}}=0$, we have the following:

$$\Delta {\mathbf{U}}_M\left(k\right)={\left[{\mathbf{C}}^{\mathrm{T}}\left(k\right)\mathbf{C}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\mathbf{C}}^{\mathrm{T}}\left(k\right)\mathbf{D}(v_d(k+1)-v\left(k\right))$$

(10)

The model-free adaptive predictive controller is as follows:

$$u\left(k\right)=u(k-1)+{\mathbf{h}}^{\mathrm{T}}\Delta {\mathbf{U}}_M\left(k\right)$$

(11)

where $\mathbf{h}={[1,0,0\cdots 0]}^{\mathrm{T}}$.

Remark 2. 

The criterion function (9) not only takes into account the speed-tracking error of the train, but also the introduction of λ to limit the variation in the input signal F in order to ensure the smoothness of the input signal $\Delta {\mathbf{U}}_M\left(k\right)$, i.e., the comfort of passengers travelling on the train.

2.3. PPD Estimation Algorithm and Prediction Algorithm

The PPD matrix $\mathbf{C}\left(k\right)$ in Equation (10) is unknown and requires further computation. For the current time instant, the PPD matrix is estimated using an improved projection algorithm.

Consider the following PPD matrix criterion function:

$$J(\Xi \left(k\right))={|\Delta v\left(k\right)-\Xi \left(k\right)\Delta u(k-1)|}^2+\mu {|\Xi \left(k\right)-\hat{\Xi }(k-1)|}^2$$

(12)

where $\mu >0$ is a weighting factor, and $\hat{\Xi }\left(k\right)$ is an estimate of $\Xi \left(k\right)$.

Using the same method as Equation (9), the following PPD estimation algorithm is obtained:

$$\hat{\Xi }\left(k\right)=\hat{\Xi }(k-1)+{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +\Delta u{(k-1)}^2}}[\Delta v\left(k\right)-\hat{\Xi }(k-1)\Delta u(k-1)]$$

(13)

where $0<\kappa \le 1$ is the step factor.

The reset algorithm is set as follows:

$$\hat{\Xi }\left(k\right)=\hat{\Xi }\left(1\right),\mathrm{if}\left|{\hat{\Xi }}_i\left(k\right)\right|\le \epsilon ,\mathrm{or}\left|\Delta u_i(k-1)\right|\le \epsilon ,$$

(14)

Note that the PPD prediction, $\hat{\Xi }(k+1),\cdots \hat{\Xi }(k+M-1)$, cannot be computed directly from the $\mathrm{I}/\mathrm{O}$ data at time $\mathrm{k}$, so it needs to be predicted based on the existing sequence of estimated values $\hat{\Xi }\left(1\right),\cdots \hat{\Xi }\left(k\right)$.

The algorithm of Equation (13) yields a series of estimates of the PPD $\hat{\Xi }\left(1\right),\cdots \hat{\Xi }\left(k\right)$. Using these estimates, an autoregressive model is built that is satisfied by the estimated series.

$$\hat{\Xi }(k+i)={\eta }_1\left(k\right)\hat{\Xi }(k+i-1)+{\eta }_2\left(k\right)\hat{\Xi }(k+i-2)+\dots +{\eta }_{np}\left(k\right)\hat{\Xi }\left(k+i-n_p\right)$$

(15)

Define $\mathit{\eta }\left(k\right)={\left[{\eta }_1\left(k\right),\cdots ,{\eta }_{np}\left(k\right)\right]}^{\mathrm{T}}$; this can then be determined using the following equation:

$$\mathit{\eta }\left(k\right)=\mathit{\eta }(k-1)+{\displaystyle \frac{\hat{\Xi }(k-1)}{\delta +\parallel \hat{\Xi }{(k-1)\parallel }^2}}\left[\hat{\Xi }\left(k\right)-{\hat{\Xi }}^{\mathrm{T}}(k-1)\mathit{\eta }(k-1)\right]$$

(16)

where $0<\delta \le 1$, and $\Xi (k-1)$ is defined as

$$\Xi (k-1)=\left[\hat{\Xi }(k-1),\cdots \hat{\Xi }\left(k-n_p\right)\right]$$

(17)

Then, combining the control algorithm, parameter estimation algorithm, and parameter prediction algorithm, the following high-speed train speed-tracking control algorithm can be given:

$$\hat{\Xi }\left(k\right)=\hat{\Xi }(k-1)+{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +\Delta u{(k-1)}^2}}[\Delta v\left(k\right)-\hat{\Xi }(k-1)\Delta u(k-1)]$$

(18)

$$\hat{\Xi }\left(k\right)=\hat{\Xi }\left(1\right),\mathrm{if}\left|{\hat{\Xi }}_i\left(k\right)\right|\le \epsilon ,\mathrm{or}\left|\Delta u_i(k-1)\right|\le \epsilon ,$$

(19)

$$\hat{\Xi }(k+i)={\eta }_1\left(k\right)\hat{\Xi }(k+i-1)+{\eta }_2\left(k\right)\hat{\Xi }(k+i-2)+\dots +{\eta }_{np}\left(k\right)\hat{\Xi }\left(k+i-n_p\right)$$

(20)

$$\mathit{\eta }\left(k\right)=\mathit{\eta }(k-1)+{\displaystyle \frac{\hat{\Xi }(k-1)}{\delta +\parallel \hat{\Xi }{(k-1)\parallel }^2}}\left[\hat{\Xi }\left(k\right)-{\hat{\Xi }}^{\mathrm{T}}(k-1)\mathit{\eta }(k-1)\right]$$

(21)

$$u\left(k\right)=u(k-1)+{\mathbf{h}}^{\mathrm{T}}\Delta {\mathbf{U}}_M\left(k\right)$$

(22)

$$\Delta {\mathbf{U}}_M\left(k\right)={\left[{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{D}(v_d(k+1)-v\left(k\right))$$

(23)

Remark 3. 

The process of updating the control signal is designed to be efficient and practical, as it does not require the use of neural networks or complex matrix operations. This simplicity ensures ease of implementation in real-world scenarios.

Remark 4. 

The optimal method for parameter selection remains an open challenge for control strategies. However, for the control algorithm proposed here, the following recommendations are provided:

1. 

Select the controller parameter $\left(\lambda \right)$ to satisfy the inequality in Equation (11);

2. 

Constrain the parameter η within the interval $(0,1]$;

3. 

Ensure that the parameters μ and κ are both positive ($\mu ,\kappa >0$);

4. 

Typically set the constant ϵ to ${10}^{-5}$.

These guidelines aim to facilitate effective parameter selection for the proposed control algorithm.

2.4. Quantifier Model

In this paper, it is assumed that the output signal is quantized before it is transmitted to the controller. The quantizers are assumed to be of the logarithmic kind and are defined through the following equation:

$$U=\left\{\pm l_i:l_i={\theta }^i{i}_0,\pm 1,\pm 2,\dots \right\}\cup \left\{0\right\},0<\theta <1,l_0>0$$

(24)

where $l_i$ is the quantization interval, the parameter $\theta$ is linked with the quantization density, i signifies the quantization level, and $l_0$ is an initial value. The associated quantizer $q(·)$ is defined as

$$q\left(x\right)=\left\{\begin{array}{ccc}{l}_i\hfill & \mathrm{if}\hfill & {\displaystyle \frac{1}{1+\zeta }}x<x\le {\displaystyle \frac{1}{1-\zeta }}x\hfill \\ 0\hfill & \mathrm{if}\hfill & x=0\hfill \\ -q(-x)\hfill & \mathrm{if}\hfill & x<0\hfill \end{array}\right.$$

(25)

where $\zeta ={\displaystyle \frac{1-\theta }{1+\theta }}\in [0,1]$ reflects the quantization accuracy. We remark that the quantizer $q(·)$ in (25) is both symmetrical—that is, $q(-v)=-q\left(v\right)$—and time-invariant.

The sector-boundary approach is utilized in this paper to deal with quantization errors, as follows:

$$q(\Lambda )=(1+\Delta (\Lambda \left)\right)\Lambda$$

where $|\Delta (\Lambda \left)\right|<\rho$. The quantization effect can then be translated into fan-boundary uncertainty.

By utilizing the sector-boundary method, we obtain

$$e_q^{\Delta }\left(k\right)=q(\Delta v\left(k\right))-\Delta v\left(k\right)={\sum }_{\Delta }\Delta v\left(k\right),$$

(26)

$$e_q^e\left(k\right)=q\left(e\left(k\right)\right)-e\left(k\right)={\sum }_{e}e\left(k\right),$$

(27)

where $e_q^{\Delta }\left(k\right)$ and $e_q^e\left(k\right)$ are the quantization errors of quantizers $q_{\Delta }(·)$ and $q_e(·)$, respectively, and ${\Sigma }_{\Delta }$ and ${\Sigma }_{\mathrm{e}}$ indicate two constants with $\left|{\Sigma }_{\Delta }\right|\le {\zeta }_{\Delta }$ and $\left|{\Sigma }_{\mathrm{e}}\right|\le {\zeta }_e$, where ${\zeta }_{\Delta }={\displaystyle \frac{1-{\theta }_{\Delta }}{1+{\theta }_{\Delta }}}\in [0,1]$ and ${\zeta }_e={\displaystyle \frac{1-{\theta }_e}{1+{\theta }_e}}\in [0,1]$.

2.5. DoS Attack Model

This study examines potential network attacks on traction/braking force and speed data transmitted in high-speed train operations. We focus on the MFAPC scheme’s response to periodic DoS attacks. The vulnerability of the sensor–controller channel to such attacks is acknowledged.

The attacker’s capabilities are constrained, preventing sustained attacks. A periodic DoS attack model, detailed in Figure 2 with parameters $n, t_{\mathrm{off}}$, and R representing the period number, the sleeping interval length, and the total period duration, respectively, is considered. The attacker accumulates energy during dormant intervals and executes DoS attacks during designated periods. To evade detection, $\alpha \left(k\right)$ represents the attack probability variable, following a Bernoulli distribution. A successful attack results in blocked transmission of the latest information $\alpha \left(k\right)=0$; otherwise, data transmission proceeds $\alpha \left(k\right)=1$. With the expected value $\mathbb{E}\left\{\alpha \left(k\right)\right\}=\overline{\alpha }\in [0,1]$, the specifics of the attack strategy are described below.

We assume that the signals $\Delta v\left(k\right)$ and $e\left(k\right)$ are quantized by two quantizers, $q_{\Delta }(·)$ and $q_e(·)$, before entering; considering the effect of a DoS attack, the following model-free adaptive predictive control algorithm for high-speed trains considering quantization effects and DoS attacks can be obtained:

$$\hat{\Xi }\left(k\right)=\hat{\Xi }(k-1)+{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +\Delta u{(k-1)}^2}}[q(\Delta v\left(k\right))-\hat{\Xi }(k-1)\Delta u(k-1)]$$

(28)

$$u\left(k\right)=u(k-1)+\alpha \left(k\right){\mathbf{h}}^{\mathrm{T}}\Delta {\mathbf{U}}_M\left(k\right)$$

(29)

$$\Delta {\mathbf{U}}_M\left(k\right)={\left[{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{D}q\left(e\left(k\right)\right)$$

(30)

The PPD prediction algorithm and the reset algorithm are the same as those described in Equations (18)–(21) and are thus omitted here to reduce repetition. The model-free adaptive predictive control algorithm (28)–(30) can be summarized as the pseudo-code shown in Algorithm 1, and its structure is shown in Figure 3.

Remark 5. 

Regarding the prediction step N, for a simple system, N can be set to 1. For a complex system, in order to obtain a satisfactory transition process and tracking performance, N should be set to a larger value, but the amount of computation will increase. Generally, N can be set in the range of 4–10. In this paper, we choose $N=5$.

Algorithm 1 MFAPC Algorithm

Input:  $v\left(k\right)$   1: for $k=1$ to end, do   2:       step 1: Get $\mathbf{V}\left(k\right)$;   3:       step 2: Update $\hat{\Xi }\left(k\right)$ by (13);   4:       step 3: Reset the algorithm by (18);   5:       step 4: Calculate $\Xi (k-1)$ by (17);   6:       step 5: Update $\theta \left(k\right)$ by (16);   7:       step 6: Initialize $\hat{\Xi }(k+1)\cdots \hat{\Xi }(k+N)$ with zero matrix;   8:       step 7:   9:       for $i:=1\mathrm{to}{N}_{\mathrm{u}}$ do 10:         $\mathrm{Calculate}\hat{\Xi }(k+i)\mathrm{by}\left(15\right)$ 11:       end for 12:       Step 8: Construct the matrix C according to (7) 13:       Step 9: Predict $\mathbf{V}(k+i),i=1,\cdots ,N_p\mathrm{by}\left(8\right)$ 14:       Step 10: Calculate the control variables increment by (29) 15:       Step 11: 16:       if $\alpha \left(k\right)=1$ then 17:            $u\left(k\right)=u(k-1)+{\mathbf{h}}^{\mathrm{T}}\Delta {\mathbf{U}}_M\left(k\right)$ 18:       else 19:            $u\left(k\right)=u(k-1)$; 20:       end if 21:       Step 12: Update the control variables by (28) 22: end for Output:  $u\left(k\right)$

3. Stability Analysis

Theorem 2. 

If high-speed train systems (4) under the presence of quantization effects and DoS attacks satisfy Assumptions 1 and 2, and the model-free adaptive predictive control scheme (28)–(30) is used to select appropriate controller parameters, there exists a tracking error ${lim}_{k\to \infty }\mathbb{E}\left\{\right|e(k+1)\left|\right\}\le {\displaystyle \frac{c_1}{1-c_1}}d$ when $\lambda >{\lambda }_{min}$, where $c_1$ is a positive constant.

Proof of Theorem 2. 

The boundedness of the estimation value of $\Xi \left(k\right)$ is firstly proven. Define $\tilde{\Xi }\left(k\right)=\hat{\Xi }\left(k\right)-\Xi \left(k\right)$ as the estimation error of PPD $\Xi \left(k\right)$. Subtracting $\Xi \left(k\right)$ from the estimation algorithm (28), it becomes

$$\begin{array}{cc}\hfill \tilde{\Xi }\left(k\right)& =\tilde{\Xi }(k-1)-\Delta \Xi \left(k\right)+{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +{|\Delta u(k-1)|}^2}}\left(\Delta v\left(k\right)-\hat{\Xi }(k-1)\Delta u(k-1)+{\Sigma }_{\Delta }\Delta v\left(k\right)\right)\hfill \\ \hfill & =\left(1-{\displaystyle \frac{\kappa \Delta u^2(k-1)}{\mu +\Delta u^2{(k-1)}^2}}\right)\tilde{\Xi }(k-1)+{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +\Delta u^2(k-1)}}{\Sigma }_{\Delta }\Xi (k-1)\Delta u(k-1)-\Delta \Xi \left(k\right)\hfill \end{array}$$

(31)

From (31) and (26), we can obtain

$$\begin{array}{cc}\hfill \tilde{\Xi }\left(k\right)& =\left|1-{\displaystyle \frac{\kappa \Delta u^2(k-1)}{\mu +\Delta u^2{(k-1)}^2}}\right||\tilde{\Xi }(k-1)|+\left|{\displaystyle \frac{\kappa \Delta u(k-1)}{\mu +\Delta u^2(k-1)}}\right|\left|{\Sigma }_{\Delta }\right||\Xi (k-1)||\Delta u(k-1)|-\Delta |\Xi \left(k\right)|\hfill \\ \hfill & \le \left|1-{\displaystyle \frac{\kappa \Delta u^2(k-1)}{\mu +\Delta u^2{(k-1)}^2}}\right||\tilde{\Xi }(k-1)|+{\delta }_{\Delta }\left|{\displaystyle \frac{\kappa \Delta u^2(k-1)}{\mu +\Delta u^2(k-1)}}\right||\Xi (k-1)|+2b\hfill \end{array}$$

(32)

Choosing $\mu >0,0<\kappa <1$, then $\kappa \Delta u^2(k-1)<\Delta u^2(k-1)<\mu +\Delta u^2(k-1)$. Thus, it follows that

$$0<\left|1-{\displaystyle \frac{\kappa \Delta u^2(k-1)}{\mu +\Delta u^2{(k-1)}^2}}\right|\triangleq h<1$$

(33)

Then,

$$\begin{array}{cc}\hfill \tilde{\Xi }\left(k\right)& \le \left|h\right||\tilde{\Xi }(k-1)|+{\delta }_{\Delta }|\Xi (k-1)|+2b\hfill \\ \hfill & \le \left|h\right||\tilde{\Xi }(k-1)|+{\delta }_{\Delta }b+2b\hfill \\ \hfill & \le {\left|h\right|}^2|\tilde{\Xi }(k-2)|+|h|c\hfill \\ \hfill & ⋮\hfill \\ \hfill & \le {\left|h\right|}^{k-1}\left|\tilde{\Xi }\left(1\right)\right|+{\displaystyle \frac{c}{1-h}}\hfill \end{array}$$

(34)

where $c={\delta }_{\Delta }b+2b$; one can conclude that $\tilde{\Xi }\left(k\right)$ is bounded, which implies that $\hat{\Xi }\left(k\right)$ is bounded, and so $\Xi \left(k\right)$ is also bounded.

Next, we proceed to show that the tracking error is consistently bounded. Define $e(k+1)=v_d(k+1)-v(k+1)$. Subtracting $v_d\left(k\right)$ from both sides of Equation (4) simultaneously, we can obtain the following:

$$e(k+1)=e\left(k\right)-\Xi \left(k\right)\Delta u\left(k\right)+\Delta v_d(k+1)$$

(35)

Substituting (29) into (27), we have

$$\begin{array}{cc}\hfill \mathbb{E}\{\Delta u(k\left)\right\}& =\overline{\alpha }{\mathbf{h}}^{\mathrm{T}}{\left[{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{D}\mathbb{E}q\left(e\left(k\right)\right)\hfill \\ \hfill & =\overline{\alpha }\mathbf{\Gamma }\left(k\right)\mathbb{E}q\left(e\left(k\right)\right)\hfill \\ \hfill & =\overline{\alpha }\mathbf{\Gamma }\left(k\right)\mathbb{E}\left\{e\left(k\right)+{\Sigma }_{e}e\left(k\right)\right\}\hfill \\ \hfill & =\overline{\alpha }\mathbf{\Gamma }\left(k\right)\left(1+{\Sigma }_e\right)\mathbb{E}\left\{e\left(k\right)\right\}\hfill \end{array}$$

(36)

where $\mathbf{\Gamma }\left(k\right)={\mathbf{h}}^{\mathrm{T}}{\left[{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{D}$.

From (36) and (35), the following can be obtained:

$$\begin{array}{cc}\hfill \mathbb{E}\left\{e\right(k+1\left)\right\}& =\mathbb{E}\left\{e\left(k\right)\right\}-\Xi \left(k\right)\mathbb{E}\{\Delta u\left(k\right)\}+\Delta v_d(k+1)\hfill \\ \hfill & =\mathbb{E}\left\{e\left(k\right)\right\}-\Xi \left(k\right)\overline{\alpha }\mathbf{\Gamma }\left(k\right)\left(1+{\Sigma }_e\right)\mathbb{E}\left\{e\left(k\right)\right\}+\Delta v_d(k+1)\hfill \\ \hfill & =\left(1-\Xi \left(k\right)\mathbf{\Gamma }\left(k\right)\overline{\alpha }\left(1+{\Sigma }_v\right)\right)\mathbb{E}\left\{e\left(k\right)\right\}+\Delta v_d(k+1)\hfill \end{array}$$

(37)

Furthermore, $v_d(k+1)$ is bounded, so that $\left|\Delta v_d(k+1)\right|\le d$. Consider the matrix $\left(\mathbf{Q}={\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right)$. Given that $\left({\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)\right)$ is a positive semidefinite matrix and $\lambda >0$, it follows that both $\mathbf{Q}$ and ${\mathbf{Q}}^{-1}$ are positive definite matrices. Therefore, we have the following implications:

$$\begin{array}{c}{\mathbf{Q}}^{-1}={\displaystyle \frac{{\mathbf{Q}}^{*}}{det\left(\mathbf{Q}\right)}}\\ {\mathbf{Q}}^{*}=\left[\begin{array}{ccc}{Q}_{11}& \cdots & Q_{\left(M\right)1}\\ ⋮& \ddots & ⋮\\ Q_{1\left(M\right)}& \cdots & Q_{\left(M\right)\left(M\right)}\end{array}\right]\end{array}$$

Let $\mathbf{Q}$ be a matrix and ${\mathbf{Q}}^{*}$ be the adjoint matrix of $\mathbf{Q}$. The element $Q_{ij}$ is the algebraic cofactor of $\mathbf{Q}$. Hence, the following relationship holds:

$$\begin{array}{cc}\hfill & {\mathbf{h}}^{\mathrm{T}}{\left[{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\hat{\mathbf{C}}\left(k\right)+\lambda \mathbf{I}\right]}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{E}\left(k\right)\hfill \\ \hfill & ={\mathbf{h}}^{\mathrm{T}}{\mathbf{Q}}^{-1}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{E}\left(k\right)\hfill \\ \hfill & ={\mathbf{h}}^{\mathrm{T}}{\displaystyle \frac{{\mathbf{Q}}^{*}}{det\left(\mathbf{Q}\right)}}{\hat{\mathbf{C}}}^{\mathrm{T}}\left(k\right)\mathbf{E}\left(k\right)\hfill \\ \hfill & =\sum _{a=1}^M{\displaystyle \frac{(N-a+1){\mathbf{Q}}_{iai}\hat{\Xi }(N-a+1)}{det\left(\mathbf{Q}\right)}}\hfill \end{array}$$

(38)

Since $\hat{\Xi }\left(k\right)$ is bounded for any time k, Equation (38) is also bounded. Furthermore, the top limit of the value is a fixed constant that is not influenced by the variable of time k.

Since $\mathbf{Q}$ is a positive definite matrix, $det\left(\mathbf{Q}\right),Q_{i1}(i=2,3,\cdots M)$ are polynomials of the orders $M, M-1$ and $M-2$ of $\lambda$, respectively, whose first-term coefficients are 1. It is noted that ${\lambda }_{min}>0$, such that when $\lambda >{\lambda }_{min}$, the sign of (38) is the same as that of ${\displaystyle \frac{{\mathbf{Q}}_{11}}{det\left(\mathbf{Q}\right)}}$. Moreover, ${\zeta }_e={\displaystyle \frac{1-{\theta }_e}{1+{\theta }_e}}$, and thus $0<{\zeta }_e<1$ and $0<{\Sigma }_e<1$. Since $0<\overline{\alpha }<1$, there is a constant satisfying the following:

$$0<1-\overline{\alpha }\Xi \left(k\right)\mathbf{\Gamma }\left(k\right)\left(1+{\Sigma }_e\right)\le c_1<1$$

(39)

From Equations (37) and (39), we have

$$\begin{array}{cc}\hfill \mathbb{E}\left\{\right|e(k+1)\left|\right\}& \le c_1\mathbb{E}\left\{\right|e\left(k\right)\left|\right\}+d\hfill \\ \hfill & \le c_1^2\mathbb{E}\left\{\right|e(k-1)\left|\right\}+c_{1}d\hfill \\ \hfill & \le ⋮\hfill \\ \hfill & \le c_1^{k-1}\mathbb{E}\left\{\right|e\left(1\right)\left|\right\}+{\displaystyle \frac{c_1}{1-c_1}}d\hfill \end{array}$$

(40)

where $0<c_1<1$. Thus, $\mathbb{E}\left\{e\right(k+1\left)\right\}$ is bounded, and this is ultimately bounded depending on the desired trajectory. □

4. Simulation

It has been determined that the duration of the operation will be 2500 s, and the sample interval will always be 1 s. The desired speed is shown in Figure 2. The following is a list of several critical parameters that are provided in support of the simulation test: $\epsilon ={10}^{-5}$; $\delta =0.1$; $M=2$; $\eta =0.8$; $\Xi \left(1\right)=0.8$; $\Xi \left(2\right)=0.6$; $\Xi \left(3\right)=0.8$; $\Xi \left(4\right)=0.4$; $\Xi \left(5\right)=1.2$; $\mathit{\eta }\left(4\right)={[0.3,0.2,0.4]}^{\mathrm{T}}$; $\mathbb{E}\left\{\alpha \right(k\left)\right\}=0.8$. Detailed information regarding the simulation coefficients that pertain to the running resistance can be found in Refs. [31,32]. It is essential to keep in mind that these resistance coefficients are utilized just to produce the fundamental and supplementary resistances encountered during the operation of the train. This particular set of coefficients is not the only one available; other values can be used in their place if required.

$$\begin{array}{cc}& a_1=0.275sin\left(3.9·{10}^{-3}k\right)+0.2977\hfill \\ & a_2=2.4·{10}^{-3}sin\left(3.9·{10}^{-3}k\right)+2.517·{10}^{-2}\hfill \\ & a_3=4.65·{10}^{-5}sin\left(3.9·{10}^{-3}k\right)+2.995·{10}^{-4}\hfill \end{array}$$

The additional resistance is as follows:

$$f_b\left(k\right)=\left\{\begin{array}{cc}0.01,\hfill & k\le 500\hfill \\ 0.04,\hfill & 500<k\le 750\hfill \\ 0.01,\hfill & 750<k\le 1500\hfill \\ 0.01,\hfill & 1500<k\le 2000\hfill \\ 0.07,\hfill & 2000<k\le 2250\hfill \\ 0.02,\hfill & 2250<k\le 2500\hfill \end{array}\right.$$

Figure 3 shows the velocity tracking of the algorithm, where the red solid line is the desired velocity profile and the blue solid line is the actual velocity profile, from which it can be seen that even in the presence of data quantization and DoS attacks, the v is still able to track the desired trajectory well under the MFAPC algorithm. Figure 4 shows the traction/braking force.

We are thus able to confirm the effectiveness of the activity quantizer. The quantization process of the tracking error is demonstrated in Figure 5. It can be observed that although the values that lie within the same quantization interval are different, the quantizer maps these data to the same values. This processing results in a significant reduction in the disorder of the original signal and the quantized signal becomes significantly more regular. This process shows that the quantizer plays an important role in reducing the frequency of controller updates, thus improving the overall efficiency and stability of the system. See Figure 6 and Figure 7. However, different quantization densities may affect the performance of the controller. In order to investigate this issue, we set different quantization densities, as shown in Figure 8, which displays the speed error curves under different quantization densities. It can be seen that a decrease in the quantization density leads to a decrease in the convergence performance of the error curves.

Finally, we compare the proposed method with the MFAC algorithm and the PID approach. Figure 9 and Figure 10 show that different data-driven control methods can quickly and reliably stabilize the MHSTs around the desired trajectories. However, by analyzing the mean tracking error curves for the velocity tracking, it is found that the fluctuation range observed with MFAPC is significantly smaller than those observed with MFAC and PID, which further proves the superiority of the MFAPC method.

5. Conclusions

In this paper, the problem of model-free adaptive predictive speed-tracking control of high-speed trains under DoS attacks and quantization effects is solved. Firstly, a complex high-speed train system is transformed into a data-relational description using the dynamic linearization technique. Considering a periodic DoS attack and quantization effects, a model-free adaptive predictive control algorithm is proposed. Simulation results show that with this algorithm, the system can be kept stable and track the desired speed well, even under DoS attacks and quantization effects. In future work, we aim to study the model-free adaptive predictive consensus tracking problem for multiple high-speed trains.