Disturbance Observer-Based Terminal Sliding Mode Control Approach for Virtual Coupling Train Set

Abstract

To enhance line capacity in high-speed railways without new infrastructure, virtual coupling train sets (VCTSs) enable reduced inter-train distances via real-time communication and cooperative control. However, unknown disturbances and model uncertainties challenge VCTS performance, often causing chattering, slow convergence, and poor disturbance rejection. This paper proposes a novel finite-time extended state observer-based nonsingular terminal sliding mode (FTESO-NTSM) control strategy. The method integrates a nonsingular terminal sliding mode surface with a hyperbolic tangent-based reaching law to ensure fast convergence and chattering suppression, while a finite-time extended state observer estimates and compensates for lumped disturbances in real time. Lyapunov analysis rigorously proves finite-time stability. Numerical simulations under different initial statuses are conducted to validate the effectiveness of the proposed method. The results show that the maximum observation error achieves 0.0087 kN. The speed chattering magnitudes reach 0.00087 km/h, 0.0017 km/h, 0.0026 km/h, and 0.0034 km/h for the leading train and three followers, respectively. Furthermore, the convergence time of the followers is 56 s, 130 s, and 76 s, respectively. The results highlight that the proposed method can significantly improve line capacity and transportation efficiency.

1. Introduction

In the past few decades, high-speed railway (HSR) has experienced rapid development, emerging as a pivotal solution for meeting the growing demand for efficient, high-capacity, and environmentally sustainable inter-city transportation [1,2]. As of the end of 2024, the operational mileage of high-speed rail in China has reached 48,000 km, accounting for 70% of the total high-speed rail mileage worldwide. While HSR has significantly facilitated business and cultural exchanges, its rapid expansion has also introduced several operational challenges.

In China, the carrying capacity of several major corridors, such as the Beijing–Shanghai, Shanghai–Nanjing, Shanghai–Hangzhou, and Beijing–Tianjin HSR lines, is approaching saturation. At some pivotal hub stations, the headway between consecutive trains has been compressed to as low as three minutes, which is nearing the theoretical minimum under the conventional Moving Block Signaling (MBS) system. To alleviate the mounting pressure on these congested lines, authorities have resorted to the planning and construction of new parallel lines. A case in point is the Shanghai–Nanjing corridor, which currently operates four parallel lines with a fifth under construction. However, this approach of building new infrastructure is often constrained by prohibitively high costs, extended construction periods, and significant land resource consumption, rendering it an unsustainable long-term strategy. Consequently, there is an urgent need to explore innovative technologies that can drastically enhance line capacity without relying solely on new physical infrastructure. Therefore, virtual coupling (VC), first introduced by Bock [3] in 1999, is proposed as a promising alternative.

Currently, in global high-speed railway systems, the Automatic Train Protection (ATP) primarily relies on absolute position-based methods, such as Fixed Block Signaling (FBS) or MBS [4]. This conventional paradigm inherently limits line capacity. In a typical two-train following scenario, the following train must stop before the current absolute position of the leading train to ensure safety; this is a conservative principle that fails to account for the coordinated dynamics of both trains. In practice, it is highly improbable for a train to perform an emergency stop within such a short distance under normal operations.

To overcome this fundamental limitation, VCTS introduces a relative-position-based tracking mode. By utilizing real-time and continuous T2T communication, VCTS replaces the rigid safety margins of physical blocks with a flexible “virtual coupler.” This coupler dynamically synchronizes the states of adjacent trains, enabling a drastic reduction in inter-train distance while rigorously maintaining safety, thereby unlocking potential for a quantum leap in line capacity.

Research on VCTS primarily encompasses four key directions: T2T wireless communication [5,6], operating safety protection [7], cooperative control method [8,9], and transportation organization [10,11]. T2T wireless communication focuses on establishing a reliable, low-latency, and high-integrity communication link as the fundamental enabler for real-time state synchronization within the VCTS. Operating safety protection is dedicated to developing a failsafe framework and formal safety models that guarantee collision-free operation under the stringent constraints of drastically reduced inter-train distances. Cooperative control method centers on the design of distributed control algorithms that ensure precise and stable tracking of speed and relative distance, while compensating for nonlinear dynamics and external disturbances. Transportation organization involves optimizing system-level operations, including dynamic scheduling and resource management, to maximize line capacity and operational efficiency under the VC paradigm.

Focusing on the third domain, this paper investigates the design and optimization of VCTS control laws. The core objective is to dynamically regulate train speeds and spacing, ensuring operational safety under ATP supervision. Xun et al. [12] and Wu et al. [13] overviewed the latest control methods for VC in railway operation and analyzed the advantages and disadvantages of each method. Di Meo et al. [14] designed a train position and velocity tracking error feedback controller considering communication delay. Quaglietta et al. [15] developed a multi-state train-following model to evaluate the impact on capacity in the VCTS. Simulations are adopted by using part of the South West Main Line in the UK with the aim of identifying capacity performances. Cao et al. [16] introduced a generalized model predictive (GMC) and mixed artificial potential field method to perform cooperative control and prevent collision of the virtual coupling train. Liu et al. [17] proposed an optimal control method for VCTS in HSR by considering safe spacing and braking performance. The local stability and the sufficient condition for string stability have been proved. Zhang et al. [18] addressed the cooperative control problem for VC trains under fixed departure and coupling time, and presented a fixed-time tracking approach by applying distributed observers. However, the aforementioned studies have not considered the inherent unknown external disturbances which are inevitable during actual operations.

In recent years, the rapid advancement of deep learning has spurred growing interest in exploring intelligent control algorithms as a promising alternative for the cooperative control of VCTSs. Basile et al. [19] developed a Deep Deterministic Policy Gradient (DDPG)-based control strategy to coordinate and manage the nonlinear heterogeneous High-Speed Trains convoy, considering uncertain nonlinearities and unexpected external factors. Wang et al. [20] also addressed the dynamic and complex operation environment and proposed a Q-learning method for VCTS cooperative control. To reduce the calculating complexation, an artificial potential field (APF) approach is applied to optimize the reward function. Although certain uncertainties are taken into account, deep learning-based controllers for VCTS are inherently limited by their offline nature, which prevents them from adapting to unforeseen, fast-evolving disturbances in real time. Furthermore, the considerable computational cost required for training presents a notable barrier to their practical implementation.

Moreover, Model Predictive Control (MPC) has been proven to be popular for solving cooperative control problems in VCTSs. MPC operates by iteratively solving a finite-horizon optimal control problem online under multiple constraints. Felez et al. [21,22] introduced a decentralized MPC method for the leading train and following trains under nonlinear constraints. Simulation results showed that the proposed MPC-based approach can achieve shorter tracking distance compared to MBS system. Liu et al. [23] presented a distributed MPC approach aiming at minimizing the interference and maintaining constantly safe spacing between trains in the VCTS. This study designed the terminal variant set and provided mathematical proofs for its feasibility and stability. Luo et al. [24] proposed a robust MPC approach for the close following operation based on relative braking principle. They applied a semi-definite program-based controller tuning algorithm to satisfy the nonlinear safety constraint. Despite all these factors, the practical application of MPC is limited by its reliance on an accurate train model and computational cost for real-time optimization.

This paper employs a sliding mode control (SMC) strategy. By designing a sliding surface, SMC ensures the system trajectories converge to the desired dynamics and exhibit inherent robustness against model inaccuracies and disturbances, thereby circumventing the need for a precise model. Wang et al. [25] dealt with uncertain resistance parameters and unknown disturbances by proposing an adaptive cruising controller based on SMC and APF mechanism. Park et al. [26] designed a gap controller based on SMC to ensure the separation is completed before a given location. They adopted a position error correction scheme based on balises to reduce controller perturbations. Wang et al. [27] investigated the formation control problem of virtual coupling trains under unknown parameters and external disturbance. A robust formation control law based on SMC is developed to coordinate the movement of the VCTS. Theoretically, SMC is not entirely model-independent. Its control output synthesizes a model-based equivalent control and a discontinuous switching control. This switching action, while essential for robustness, inevitably causes chattering. The severity of this chattering is inversely related to the accuracy of the nominal train model used, as model inaccuracies must be compensated for by a larger switching gain, thereby presenting a fundamental trade-off between robustness and control smoothness.

Motivated by the studies mentioned above, this paper proposes a finite-time nonsingular terminal sliding mode (NTSM) control method with error observer scheme for VCTS to deal with the parametric certainty, multiple constraints, and time-varying external disturbances. The main contributions of this paper can be highlighted as follows:

• A novel reaching law incorporating a constant velocity term and a tanh function is proposed for the NTSM in the VCTS. This formulation enables the system to swiftly converge towards the sliding surface from large initial tracking errors and seamlessly transitions to a chattering suppression mode with sustained convergence speed when the error is small, ensuring both rapid response and robust performance.
• To counteract the performance degradation caused by model-plant mismatch and unknown disturbances, we augment the NTSM controller with an extended state observer (ESO) designed to estimate the total lumped disturbance. The finite-time stability of the close-loop system under the proposed composite controller is then rigorously proven using Lyapunov theory.
• Simulation tests are established and validated to evaluate the proposed FTESO-NTSM controller under high-speed railway scenarios. Through comparative experiments with several SM-based controllers, the simulation results demonstrate quantitatively that the proposed method achieves improvement in terms of convergence speed, tracking accuracy, and chattering attenuation, thereby providing a practical control solution for VCTS under complex operational conditions.

The remainder of this paper is organized as follows. Section 2 describes the dynamic train model formulation and lists some useful lemmas. Section 3 illustrates the proposed FTESO-NTSM controller, and proves the controller convergence and stability. Section 4 conducts several simulation experiments and analyzes the corresponding results. Finally, Section 5 concludes the contribution of this paper.

2. Problem Description

2.1. VCTS Model Establishment

As shown in Figure 1, the VCTS adopts leader–follower communication topology as the fundamental model, which consists of 1 leading train and n following trains. Each train is considered as an individual agent which can decide its own optimal strategy independently. The leading train only tracks the optimized profile and omnidirectionally transmits state information (e.g., position, speed, acceleration) to its follower. While the rest of trains in the VCTS share state information with adjacent trains bidirectionally.

Assumption 1:

There is no delay in T2T wireless communication and the connection is reliable.

In the VCTS, we set the index of leading train as 0 and the followers are set as 1, 2, …, n. According to Newton’s Law, the longitude dynamic model of each train in the convoy can be formulated as follows:

$$\left\{\begin{array}{l}{\dot{s}}_i\left(t\right)=v_i\left(t\right)\hfill \\ m_i{\dot{v}}_i\left(t\right)=u_i\left(t\right)-f_{b,i}\left(v_i\left(t\right)\right)-f_{a,i}\left(s_i\left(t\right)\right)\hfill \end{array}\right.$$

(1)

where $s_i\left(t\right)$, $v_i\left(t\right)$ denote the position and speed of train i at time t. $m_i$ represents the mass of train i. $u_i$ is the traction/braking force. $f_{b,i}$ denotes the basic resistance of train i which is related to mechanical, rolling, and air resistance. $f_{a,i}$ represents the additional resistance which mainly depends on the actual train operating environment.

The basic resistance can be calculated by adopting the classical Davis Equation as follows:

$$f_{b,i}\left(v_i\left(t\right)\right)=m_i·\left[a+b·v_i\left(t\right)+c·v_i^2\left(t\right)\right]$$

(2)

where a, b, c denote the Davis coefficients, respectively.

The additional resistance originates from real railway line conditions subject to slope, curve, and tunnel. The formulation can be described as follows:

$$f_{a,i}\left(s_i\left(t\right)\right)=m_i·\left[w_{slope,i}\left(s_i\left(t\right)\right)+w_{curve,i}\left(s_i\left(t\right)\right)+w_{tunnel,i}\left(s_i\left(t\right)\right)\right]$$

(3)

where $w_{slope,i}\left(s_i\left(t\right)\right)$, $w_{curve,i}\left(s_i\left(t\right)\right)$, $w_{tunnel,i}\left(s_i\left(t\right)\right)$ signify the unit slope resistance, curve resistance, and tunnel resistance, respectively.

The aforementioned equations express the theoretical longitude dynamic model of trains in the VCTS. However, due to the complex and variable running environment of trains, their actual operation is frequently subjected to dynamic, time-varying, and unknown disturbances from both internal and external sources. For instance, the control input is susceptible to deviations caused by fluctuations in the catenary voltage or uncertainties in the traction/braking systems. Meanwhile, the aerodynamic resistance is prone to discrepancies from Equation (2) due to structural factors of the train and external weather conditions (e.g., strong winds or rainfall).

Assumption 2:

The additional resistance is unaffected by external disturbances.

The additional resistance is primarily determined by the inherent line conditions, which remain relatively stable during train operation.

So, the expression of actual longitude dynamic model of trains can be rewritten as follows:

$$\left\{\begin{array}{l}{\widehat{u}}_i\left(t\right)=u_i\left(t\right)+\Delta u_i\left(t\right)\hfill \\ {\widehat{f}}_{b,i}\left(v_i\left(t\right)\right)=m_i\left[\left(a+\Delta a_i\left(t\right)\right)+\left(b+\Delta b_i\left(t\right)\right)·v_i\left(t\right)+\left(c+\Delta c_i\left(t\right)\right)·v_i^2\left(t\right)\right]\hfill \\ {\widehat{f}}_{a,i}\left(s_i\left(t\right)\right)=m_i·\left[w_{slope,i}\left(s_i\left(t\right)\right)+w_{curve,i}\left(s_i\left(t\right)\right)+w_{tunnel,i}\left(s_i\left(t\right)\right)\right]\hfill \\ m_i{\dot{v}}_i\left(t\right)={\widehat{u}}_i\left(t\right)-{\widehat{f}}_{b,i}\left(v_i\left(t\right)\right)-{\widehat{f}}_{a,i}\left(s_i\left(t\right)\right)-d_i\left(t\right)\hfill \end{array}\right.$$

(4)

where ${\widehat{u}}_i\left(t\right)$, ${\widehat{f}}_{b,i}\left(v_i\left(t\right)\right)$, ${\widehat{f}}_{a,i}\left(s_i\left(t\right)\right)$ represent the actual traction/braking force, aerodynamic drag, and additional resistance, respectively. $\Delta u_i\left(t\right)$ denotes uncertain impact on the output force. $\Delta a_i\left(t\right)$, $\Delta b_i\left(t\right)$, $\Delta c_i\left(t\right)$ represent the bias of the Davis coefficients. $d_i\left(t\right)$ represents the uncertainty during the train operation.

For the convenience of modeling and controller design, the train longitude dynamic model can be divided into two parts: equivalent model and bias model. Equation (1) is designed for the equivalent model, while all the extra external disturbances can be combined as the bias model, which is given by the following:

$$\left\{\begin{array}{l}{m}_i{\dot{v}}_i\left(t\right)=\underset{\mathrm{equivalent} \mathrm{model}}{\underbrace{u_i\left(t\right)-f_{b,i}\left(v_i\left(t\right)\right)-f_{a,i}\left(s_i\left(t\right)\right)}}-\underset{\mathrm{bias} \mathrm{model}}{\underbrace{D_i\left(t\right)}}\hfill \\ D_i\left(t\right)=-\Delta u_i\left(t\right)-m_i·\left[\Delta a_i\left(t\right)+\Delta b_i\left(t\right)v_i\left(t\right)+\Delta c_i\left(t\right)v_i^2\left(t\right)\right]+d_i\left(t\right)\hfill \end{array}\right.$$

(5)

where $D_i\left(t\right)$ signifies the combination of uncertainty disturbances, such as catenary fluctuations, traction/braking system uncertainties, aerodynamic variations due to weather, unmodeled dynamics, and parameter drifts.

Assumption 3:

The external unknown disturbances $D_i\left(t\right)$ is bounded by $D_i\left(t\right)<D_{\max }$.

2.2. Constraints

In the VCTS, the design of train controllers is subject to multiple complex constraints, including fluctuating operational conditions, inherent nonlinear and time-varying dynamic properties of trains, and strict safety mandates for maintaining inter-train distances. These multifaceted limitations impose rigorous bounds on state variables, such as position, velocity, acceleration, and control inputs. To achieve efficient and safe operation of the VCTS while ensuring reliability and passenger comfort, it is imperative to systematically integrate these constraints into both the train dynamic modeling and the controller synthesis process.

Constraint 1:

The inter-station travel time and distance of trains are constrained by the predetermined timetable. The limitations can be expressed as 

$$0\le t\le T$$

(6)

$$0\le s_i\left(t\right)\le S$$

(7)

 where T denotes the total travel time. S represents the distance between stations.

Constraint 2:

The control input is derived from traction and braking system, and is bounded by traction/braking characteristic profiles. Such constraints can be given by 

$$-u_{b,\max }\le u_i\left(t\right)\le u_{t,\max }$$

(8)

 where $u_{b,\max }$ denotes the maximum braking force. $u_{t,\max }$ represents the maximum traction force.

Constraint 3:

For running safety, the speed of trains must be controlled under the permitted restriction from Automatic Train Protection (ATP). Hence, the train operating speed satisfies 

$$0\le v_i\left(t\right)<v_{\lim }$$

(9)

 where $v_{\lim }$ denotes the speed limit which is set according to the line condition and so on.

Constraint 4:

In the VCTS, adjacent trains should remain at a safe distance based on the hit-soft-wall theory, as shown in Figure 2.

In this paper, the target safe distance between adjacent trains is dynamically computed based on the relative braking principle, ensuring that even in emergency braking, trains maintain a non-collision gap.

Thus, the tracking gap between neighboring trains should satisfy

$$s_i\left(t\right)-s_{i+1}\left(t\right)-L_i\ge G_{i,i+1}$$

(10)

where $L_i$ denotes the length of train i. $G_{i,i+1}$ represents the expected tracking gap between train i and train i + 1. And $G_{i,i+1}$ is defined based on the relative braking principle as

$$G_{i,i+1}=\left\{\begin{array}{ll}{G}_{\min }\hfill & , v_i\left(t\right)\ge v_{i+1}\left(t\right)\hfill \\ G_{\min }+\left({\displaystyle \frac{v_{i+1}^2\left(t\right)}{2a_{eb,i+1}}}-{\displaystyle \frac{v_i^2\left(t\right)}{2a_{eb,i}}}\right)\hfill & , v_i\left(t\right)<v_{i+1}\left(t\right)\hfill \end{array}\right.$$

(11)

where $G_{\min }$ denotes the preset minimum tracking distance between adjacent trains, which is equal to zero for the leading train. $a_{eb,i}$ signifies the emergency braking acceleration of train i.

2.3. Some Useful Lemmas

Lemma 1 

[27]: Consider the nonlinear system below,

$$\dot{x}=f\left(x\right), f\left(0\right)=0, x\in {ℝ}^n$$

(12)

Suppose there exists a continuous positive definite function V(x), and real numbers c > 0 and $\alpha \in \left(0,1\right)$, such that the following inequality holds:

$$\dot{V}\left(x\right)+cV{\left(x\right)}^{\alpha }\le 0, \forall x\in \mathrm{\Phi }$$

(13)

Then the origin is finite-time stable. The convergence time satisfies

$$T_s\le {\displaystyle \frac{V{\left(x\right)}^{1-\alpha }}{c\left(1-\alpha \right)}}$$

(14)

Lemma 2 

[28]: For a second-order system considering the external disturbances as

$$\left\{\begin{array}{l}{\dot{x}}_1=x_2\hfill \\ {\dot{x}}_2=f\left(t,x_1,x_2\right)+bu+w\hfill \\ y=x\hfill \end{array}\right.$$

(15)

By defining ${\epsilon }_i={\widehat{x}}_i=x_i$ as the system state error, we can formulate an error system as

$$\left\{\begin{array}{l}{\dot{\epsilon }}_1={\epsilon }_2-k_{1}si{g}^{\left(\lambda +1\right)/2}\left({\epsilon }_1\right)\hfill \\ {\dot{\epsilon }}_2={\epsilon }_3-k_{2}si{g}^{\left(\lambda +1\right)/2}\left({\epsilon }_1\right)\hfill \\ {\dot{\epsilon }}_3=-k_{3}si{g}^{\lambda }\left({\epsilon }_1\right)-\dot{L}\hfill \end{array}\right.$$

(16)

With the lumped disturbance satisfies the condition that $\dot{L}$ is bounded by L_M, with $0<\lambda <1$, $\eta ={\left[si{g}^{\left(\lambda +1\right)/2}\left({\epsilon }_1\right),{\epsilon }_2,{\epsilon }_3\right]}^T$, there exist constants $k_1,k_2,k_3>0$, and the observation error η becomes sufficiently small in a finite amount of time.

3. Controller Design

In this section, we will investigate the FTESO-NTSM control algorithm for each train in the VCTS by considering nonlinear external disturbances. First, we formulate the control objective cooperative operation model which accounts for tracking distance and velocity. Then, we adopt the NTSM control method as the baseline to avoid singularity caused by the classical terminal sliding mode (TSM) algorithm. Finally, an extended state observer based on tracking error is designed to estimate the external unknown disturbances. The goal of the proposed controller is to converge the tracking distance error and speed error to a balanced surface in finite time.

3.1. Cooperative Control Model

The primary control objective for VCTS is to maintain a safe and stable formation while ensuring all trains track the desired operational profile. Unlike traditional moving block systems that rely on fixed absolute braking distances, VCTSs require a cooperative control strategy where trains dynamically adjust their states based on the real-time information of adjacent trains. This necessitates precise tracking of both relative distance and velocity.

In this paper, relative distance and velocity are taken as the optimization target of controller design. According to Equation (10), the tracking error can be described as

$$\left\{\begin{array}{l}{e}_{s,i}\left(t\right)=s_i\left(t\right)-s_{i+1}\left(t\right)-G_{i,i+1}\left(t\right)\hfill \\ e_{v,i}\left(t\right)=v_i\left(t\right)-v_{i+1}\left(t\right)\hfill \end{array}\right., i=0,1,2,\dots ,n$$

(17)

where $e_{s,i}\left(t\right)$, $e_{v,i}\left(t\right)$ denote the relative distance and velocity error of neighboring trains.

For the nonlinear train dynamics subject to unknown disturbances $D_i\left(t\right)$, the objective of the proposed controller is to design a distributed control law $u_i\left(t\right)$ such that the state tracking errors $e_{s,i}\left(t\right)$ and $e_{v,i}\left(t\right)$ are driven to converge to a sufficiently small neighborhood of zero within a finite time $\tilde{T}$, i.e.,:

$$\underset{t\to \tilde{T}}{{\displaystyle \lim }}\left(\left|e_{s,i}\left(t\right)\right|+\left|e_{v,i}\left(t\right)\right|\right)\to 0, \forall t\le T$$

(18)

3.2. Nonsingular Terminal Sliding Mode Controller Design

Applying the tracking error expressions defined in Equation (17), the NTSM controller is designed to achieve finite-time convergence of the tracking errors. The classical TSM control, while offering finite-time stability, suffers from the singularity problem that occurs when the system state is far from the equilibrium point and the derivative of the sliding variable becomes unbounded. The NTSM surface is introduced to circumvent this issue while preserving the finite-time convergence property.

The NTSM surface $h_i$ for the ith train is defined as

$$h_i=e_{s,i}+{\displaystyle \frac{1}{\beta }}{e_{v,i}}^{p/q}$$

(19)

where β denotes a gain parameter such that β > 0. p and q represent the positive odd integers satisfying the condition $1<p/q<2$.

The time derivative of the sliding surface $h_i$ is given by the following:

$${\dot{h}}_i=e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e_{v,i}}^{\left(p/q-1\right)}{\dot{e}}_{v,i}$$

(20)

Substituting the error dynamics from Equation (17) into Equation (20), combining train longitude dynamic model Equation (5), we achieve the following:

$${\dot{h}}_i=e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e_{v,i}}^{\left(p/q-1\right)}\left({\dot{v}}_i-{\displaystyle \frac{u_{i+1}-f_{b,i+1}-f_{a,i+1}-D_{i+1}}{m_{i+1}}}\right)$$

(21)

Assuming that Equation (21) is equal to zero, we achieve the following:

$$\begin{array}{ll}0& =e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e_{v,i}}^{\left(p/q-1\right)}\left({\dot{v}}_i-{\displaystyle \frac{u_{i+1}-f_{b,i+1}-f_{a,i+1}-D_{i+1}}{m_{i+1}}}\right)\hfill \\ & =m_{i+1}{\displaystyle \frac{\beta q}{p}}{e_{v,i}}^{\left(2-p/q\right)}+m_{i+1}{\dot{v}}_i-\left(u_{i+1}-f_{b,i+1}-f_{a,i+1}-D_{i+1}\right)\hfill \end{array}$$

(22)

Thus, the theoretically calculated output of train i + 1 can be expressed as

$$u_{i+1}=m_{i+1}{\displaystyle \frac{\beta q}{p}}{e_{v,i}}^{\left(2-p/q\right)}+M_{i+1}{\dot{v}}_i+f_{b,i+1}+f_{a,i+1}+D_{i+1}$$

(23)

According to Equation (5), Equation (23) contains the unknown external disturbances $D_{i+1}$ which is nonlinear, time-varying, and hard to estimate. To ensure the tracking errors reach and remain on the sliding surface in finite time, the control law in Equation (24) should be split into two parts, i.e., an equivalent control term $u_{eq,i}\left(t\right)$ and a switching control term $u_{sw,i}\left(t\right)$ as follows:

$$u_{i+1}\left(t\right)=u_{eq,i+1}\left(t\right)+u_{sw,i+1}\left(t\right)$$

(24)

From Equation (24), the equivalent control is derived under the assumption of no disturbances, as follows:

$$u_{eq,i}\left(t\right)=m_i{\displaystyle \frac{\beta q}{p}}{e_{v,i-1}}^{\left(2-p/q\right)}+M_i{\dot{v}}_{i-1}+f_{b,i}+f_{a,i}$$

(25)

To guarantee robustness against disturbances and model uncertainties while mitigating the chattering phenomenon inherent in conventional sliding mode control, the switching control law is designed based on an improved exponential reaching law. In this paper, a hyperbolic tangent function is introduced to replace the traditional method. The switching control law is formulated as

$$\left\{\begin{array}{l}{u}_{sw,i}\left(t\right)=k_1\tanh \left(\mu d_i\right)+k_2{d}_i\hfill \\ \tanh \left(\mu d_i\right)={\displaystyle \frac{e^{\mu d_i}-e^{-\mu d_i}}{e^{\mu d_i}+e^{-\mu d_i}}}\hfill \end{array}\right.$$

(26)

where k₁, k₂ represent the controller gains which satisfy k₁ > 0 and k₂ > 0. μ denotes the scaling parameter which satisfies μ > 0.

Thus, the proposed NTSM control law is formulated as

$$u_i\left(t\right)=m_i{\displaystyle \frac{\beta q}{p}}{e_{v,i-1}}^{\left(2-p/q\right)}+m_i{\dot{v}}_{i-1}+f_{b,i}+f_{a,i}-k_1\tanh \left(\mu h_i\right)-k_2{h}_i$$

(27)

Theorem 1:

Considering the tracking error dynamics (12) and the proposed NTSM control law (27), the sliding surface  $h_i$ will converge to zero in finite time. Meanwhile, the tracking distance error and speed error will also converge to zero in finite time.

Proof: 

Consider the Lyapunov function candidate $V_i={\displaystyle \frac{1}{2}}{h}_i^2$. Differentiating $V_i$ with respect to time and substituting the control law and tracking error yields

$$\begin{array}{ll}{\dot{V}}_i& =h_i{\dot{h}}_i\hfill \\ & =h_i\left[e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e_{v,i}}^{\left(p/q-1\right)}\left({\dot{v}}_i-{\displaystyle \frac{u_{i+1}-f_{b,i+1}-f_{a,i+1}-D_{i+1}}{m_{i+1}}}\right)\right]\hfill \\ & \le h_i\left(-k_1\tanh \left(\mu h_i\right)-k_2{h}_i+D_{i+1}\right)\hfill \\ & =-k_1{h}_i\tanh \left(\mu h_i\right)-k_2{h}_i^2+h_i{D}_{i+1}\hfill \end{array}$$

(28)

The property $h_i\tanh \left(\mu h_i\right)\ge 0$ is equal to zero when $h_i=0$. Applying the function $\left|h_i\right|\ge \left|h_i\tanh \left(\mu h_i\right)\right|$, Equation (28) can be reformulated as follows:

$${\dot{V}}_i\le -\left|h_i\right|\left(k_1\left|\tanh \left(\mu h_i\right)\right|-\left|D_{i+1}\right|\right)-k_2{h}_i^2$$

(29)

Given that $\left|\tanh \left(\mu h_i\right)\right|\le 1$, the gain conditions $k_1>D_{i+1}$ and $D_{i+1}$ are bounded, and we achieve the following:

$${\dot{V}}_i\le -{\eta }_i\left|h_i\right|-k_2{h}_i^2\le 0$$

(30)

where ${\eta }_i=k_1-\left|D_{i+1}\right|>0$. This inequality confirms that ${\dot{V}}_i$ is negative definite, which means the designed controller is robust and the tracking error will converge to a neighborhood of zero, ultimately.

Since $-k_2{h}_i^2\le 0$ stands, Equation (30) can be rewritten as

$${\dot{V}}_i\le -{\eta }_i\left|h_i\right|=-{\eta }_i\sqrt{2}{V}_i^{1/2}$$

(31)

Once the system approaches the balanced sliding surface, according to Lemma 1, the tracking distance and tracking speed converge to a finite time $T_{s,i}\le {\displaystyle \frac{2V_i{\left(0\right)}^{1/2}}{\sqrt{2}{\eta }_i}}$. □

3.3. Finite-Time Extended State Observer Design

Although the NTSM controller developed in the previous subsection can achieve finite-time convergence and robustness to a certain extent, its performance is constrained by the need for a priori knowledge of the disturbance bound, often leading to conservative control gains and chattering. To address this limitation, an FTESO is proposed in this section.

The lumped disturbance $D_{i+1}$, as defined in Equation (5), is treated as an extended state. The system tracking error can be subsequently expressed as the following extended state-space model:

$$\left\{\begin{array}{l}{\dot{e}}_{s,i}=e_{v,i}\hfill \\ {\dot{e}}_{v,i}={\mathrm{\Theta }}_i\left(u_i,t\right)+D_{i+1}/m_{i+1}\hfill \\ {\dot{D}}_{i+1}={\zeta }_{i+1}\hfill \end{array}\right.$$

(32)

where ${\mathrm{\Theta }}_i\left(u_i,t\right)={\dot{v}}_i-{\dot{\tilde{v}}}_{i+1}$ denotes the known nominal dynamics. ${\zeta }_{i+1}$ represents the derivative of the lumped disturbance, which is assumed to be bounded.

Assumption 4:

The derivative of the lumped disturbance is assumed to be bounded, i.e., $\left|{\zeta }_{i+1}\right|\le L_M$.

Aiming for finite-time convergence of the estimation errors, the FTESO is constructed as follows:

$$\left\{\begin{array}{l}{\dot{z}}_{1,i}=z_{2,i}-{\gamma }_{1}si{g}^{{\sigma }_1}\left(z_{1,i}-e_{s,i}\right)\hfill \\ {\dot{z}}_{2,i}={\mathrm{\Theta }}_i\left(u_i,t\right)+z_{3,i}-{\gamma }_{2}si{g}^{{\sigma }_2}\left(z_{1,i}-e_{s,i}\right)\hfill \\ {\dot{z}}_{3,i}=-{\gamma }_{3}si{g}^{{\sigma }_3}\left(z_{1,i}-e_{s,i}\right)\hfill \end{array}\right.$$

(33)

where $z_{1,i}$, $z_{2,i}$, $z_{3,i}$ denote the estimates of $e_{s,i}$, $e_{v,i}$ and ${\dot{e}}_{v,i}$, respectively. ${\gamma }_1$, ${\gamma }_2$, ${\gamma }_3$ are observer gains. ${\sigma }_1=\left(\lambda +1\right)/2$, ${\sigma }_2=\left(\lambda +1\right)/2$, ${\sigma }_1=\lambda$ are preset constant values. The function $si{g}^{{\sigma }_k}\left(z_{1,i}-e_{s,i}\right)$ is formulated as

$$si{g}^{{\sigma }_k}\left(z_{1,i}-e_{s,i}\right)={\left(z_{1,i}-e_{s,i}\right)}^{{\sigma }_k}\mathrm{sgn}\left(z_{1,i}-e_{s,i}\right), k=1,2,3$$

(34)

Furthermore, we define the estimation error of ESO as ${\tilde{z}}_{1,i}=z_{1,i}-e_{s,i}$, ${\tilde{z}}_{2,i}=z_{2,i}-e_{v,i}$, ${\tilde{z}}_{3,i}=z_{3,i}-D_{i+1}$. Equation (33) can be rewritten as

$$\left\{\begin{array}{l}{\dot{\tilde{z}}}_{1,i}={\tilde{z}}_{2,i}-{\gamma }_{1}si{g}^{{\sigma }_1}\left({\tilde{z}}_{1,i}\right)\hfill \\ {\dot{\tilde{z}}}_{2,i}={\tilde{z}}_{3,i}-{\gamma }_{2}si{g}^{{\sigma }_2}\left({\tilde{z}}_{1,i}\right)\hfill \\ {\dot{\tilde{z}}}_{3,i}=-{\zeta }_{i+1}-{\gamma }_{3}si{g}^{{\sigma }_3}\left({\tilde{z}}_{1,i}\right)\hfill \end{array}\right.$$

(35)

Theorem 2:

Considering the longitude dynamic model for each train in the VCTS when Assumption 4 is satisfied, there exist gain parameters ${\gamma }_1,{\gamma }_2,{\gamma }_3>0$ and $0<{\sigma }_k<1$ that make the observation error converge to a neighborhood of zero in finite time.

Proof: 

Inspired by Lemma 2, the observation error is set as $\eta ={\left[si{g}^{\left(\lambda +1\right)/2}\left({\epsilon }_1\right),{\epsilon }_2,{\epsilon }_3\right]}^T$. The positive definite matrix P is $\left[\begin{array}{ccc}2{\gamma }_1/\lambda +1& -{\gamma }_2& -{\gamma }_2\\ -{\gamma }_2& 2& 0\\ -{\gamma }_2& 0& 2\end{array}\right]$.

The Lyapunov candidate function is proposed as

$$V\left(\eta \right)={\eta }^{T}P\eta$$

(36)

where η denotes the observation error. P is a positive definite matrix.

According to Lemma 2, the derivative of η with respect to time can be expressed as

$$\begin{array}{ll}\dot{\eta }& =\left[\begin{array}{c}{\displaystyle \frac{\lambda +1}{2}}{\left|{\tilde{z}}_{1,i}\right|}^{\left(\lambda -1\right)/2}\left({\tilde{z}}_{2,i}-{\gamma }_{1}si{g}^{\left(\lambda +1\right)/2}\left({\tilde{z}}_{1,i}\right)\right)\\ {\tilde{z}}_{3,i}-{\gamma }_{2}si{g}^{\left(\lambda +1\right)/2}\left({\tilde{z}}_{1,i}\right)\\ -{\zeta }_{i+1}-{\gamma }_{3}si{g}^{\lambda }\left({\tilde{z}}_{1,i}\right)\end{array}\right]\hfill \\ & =\left[\begin{array}{ccc}-r\rho {\gamma }_1& r\rho & 0\\ -{\gamma }_2& 0& 1\\ -\rho {\gamma }_3& 0& 0\end{array}\right]\eta +\left[\begin{array}{c}0\\ 0\\ -1\end{array}\right]h_{i+1}\hfill \\ & =A\left(r,\rho \right)\eta +B{\zeta }_{i+1}\hfill \end{array}$$

(37)

where $r=\left(\lambda +1\right)/2>0$, $\rho ={\left|{\tilde{z}}_{1,i}\right|}^{\left(\lambda -1\right)/2}>0$. Then the characteristic function of A(r, ρ) is given by

$$G\left(s\right)=s^3+r\rho {\gamma }_1{s}^2+r\rho {\gamma }_{2}s+r{\rho }^2{\gamma }_2$$

(38)

Since ${\gamma }_1,{\gamma }_2,{\gamma }_3>0$ and $\rho >0$, all coefficients in $G\left(s\right)$ are positive, which means $G\left(s\right)$ satisfies the Hurwitz conditions. Furthermore, matrix A(r, ρ) is also Hurwitz compliant. So, there exists a positive definite matrix Q that satisfies $A^{T}P+PA=-Q$.

Differentiating $V\left(\eta \right)$ with respect to time, and by applying Equation (37), we achieve the following:

$$\begin{array}{ll}\dot{V}\left(\eta \right)& ={\dot{\eta }}^{T}P\eta +{\eta }^{T}P\dot{\eta }\hfill \\ & ={\left(A\eta -B{h}_{i+1}\right)}^{T}P\eta +{\eta }^{T}P\left(A\eta -B{h}_{i+1}\right)\hfill \\ & ={\eta }^T\left(A^{T}P+PA\right)\eta +2{\zeta }_{i+1}{\tilde{B}}^T\eta \hfill \end{array}$$

(39)

where ${\tilde{B}}^T=-B^{T}P=\left[\begin{array}{ccc}{\gamma }_3& 0& -2\end{array}\right]$. If we let $l={‖\tilde{B}‖}_2=\sqrt{{\gamma }_3^2+4}$, then we achieve the following:

$$\begin{array}{ll}\dot{V}\left(\eta \right)& =-{\eta }^{T}Q\eta +2h_{i+1}{\tilde{B}}^T\eta \hfill \\ & \le -{\varphi }_{\min }\left\{Q\right\}{‖\eta ‖}_2^2+2l{L}_M{‖\eta ‖}_2\hfill \end{array}$$

(40)

where ${\varphi }_{\min }\left\{Q\right\}$ denotes the minimum characteristic value. When ${‖\eta ‖}_2\le 2l{L}_M/{\varphi }_{\min }\left\{Q\right\}$ by selecting proper value of ${\gamma }_1,{\gamma }_2,{\gamma }_3$, $\dot{V}\left(\eta \right)\le 0$ holds. □

3.4. NTSM Controller with FTESO

The preceding subsections have individually detailed the design of the NTSM controller, which guarantees finite-time convergence of tracking errors, and the FTESO, which provides accurate and timely estimation of the lumped disturbance. To significantly enhance the system’s robustness and accuracy, these two components are now integrated into a composite control strategy. The structure of the proposed controller is shown in Figure 3.

The equivalent control law in Equation (25) is modified by adding the unknown disturbance D_i(t) with its estimate ${\widehat{D}}_i\left(t\right)$. Consequently, the integrated control law for train i is formulated as

$$u_i\left(t\right)=u_{eq,i}\left(t\right)+u_{sw,i}\left(t\right)+{\widehat{D}}_i\left(t\right)$$

(41)

Theorem 3:

For system (5), if we consider the tracking error dynamics (12), the FTESO-NTSM control law is designed as

$$u_i\left(t\right)=m_i{\displaystyle \frac{\beta q}{p}}{e_{v,i-1}}^{\left(2-p/q\right)}+m_i{\dot{v}}_{i-1}+f_{b,i}+f_{a,i}+{\widehat{D}}_i\left(t\right)-k_1\tanh \left(\mu h_i\right)-k_2{h}_i$$

(42)

The sliding surface$h_i$will converge to zero in finite time. Meanwhile, the tracking distance error and speed error will also converge to zero in finite time.

Proof: 

Consider the Lyapunov function candidate $V_i={\displaystyle \frac{1}{2}}{h}_i^2$. Differentiating $h_i$ with respect to time and substituting the control law (44), we achieve the following:

$$\begin{array}{ll}{\dot{h}}_i& =e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e}_{v,i}^{\left(p/q-1\right)}\left({\dot{v}}_i-{\displaystyle \frac{u_{i+1}-f_{b,i+1}-f_{a,i+1}-D_{i+1}}{m_{i+1}}}\right)\hfill \\ & =e_{v,i}+{\displaystyle \frac{p}{\beta q}}{e}_{v,i}^{\left(p/q-1\right)}\left(-{\displaystyle \frac{\beta q}{p}}{e}_{v,i}^{\left(2-p/q\right)}+{\displaystyle \frac{{\widehat{D}}_{i+1}-D_{i+1}-k_1\tanh \left(\mu h_i\right)-k_2{h}_i}{m_{i+1}}}\right)\hfill \\ & \le {\displaystyle \frac{{\widehat{D}}_{i+1}-D_{i+1}-k_1\tanh \left(\mu h_i\right)-k_2{h}_i}{m_{i+1}}}\hfill \end{array}$$

(43)

Then, by differentiating $V_i$ with respect to time and combining with Equation (43), we can formulate the following:

$${\dot{V}}_i=h_i{\dot{h}}_i \le h_i\left[{\widehat{D}}_{i+1}-D_{i+1}-k_1\tanh \left(\mu h_i\right)-k_2{h}_i\right]/m_{i+1}$$

(44)

Let ${\tilde{D}}_{i+1}={\widehat{D}}_{i+1}-D_{i+1}$. Then, we obtain the following:

$${\dot{V}}_i\le \left[-\left|h_i\right|\left(k_1-\left|{\tilde{D}}_{i+1}\right|\right)-k_2{h}_i^2\right]/m_{i+1}$$

(45)

The observation error of lumped disturbances $\left|{\tilde{D}}_{i+1}\right|$ will converge to a neighborhood of zero according to Section 3.3. So, Equation (45) can be given as

$${\dot{V}}_i\le \left[-{\eta }_{1i}\left|h_i\right|-k_2{h}_i^2\right]/m_{i+1}\le 0$$

(46)

where ${\eta }_{1i}=k_1-\left|{\tilde{D}}_{i+1}\right|>0$. Since the estimate error of FTESO $\left|{\tilde{D}}_{i+1}\right|$ will converge to a neighborhood of zero in finite time, by selecting an appropriate value for k₁, Equation (46) is guaranteed to be non-positive, thereby demonstrating the stability and reliability of the proposed controller.

Once the system approaches the balanced sliding surface, according to Lemma 1, the tracking distance and tracking speed converge to a finite time $T_{s,i}\le {\displaystyle \frac{2V_i{\left(0\right)}^{1/2}}{\sqrt{2}{\eta }_{1i}}}$. □

4. Numerical Simulations

In this section, several numerical experiments are conducted for tracking control in the VCTS subject to multiple external disturbances to illustrate the performance of the proposed FTESO-NTSM control algorithm. In the simulation, we take 4 CRH-3 trains as a train convoy, which consists of 1 leading train and 3 following trains. The parameters of a CRH-3 train are shown in

Table 1. Parameters of CRH-3.

Parameter	Symbol	Value
Train mass	mi	563 t
Basic resistance coefficient per mass	c	0.000115
Basic resistance coefficient per mass	b	0.0064
Basic resistance coefficient per mass	a	0.79
Maximum traction	ut,max	300 kN
Train length	Li	200 m
Emergency deceleration	aeb,max	−0.8 m/s2

. To ensure convenience during the simulating process, a simulation environment was constructed to represent a cruising scenario under temporary speed restriction. To emulate diverse tracking behaviors in the train platoon, the initial conditions of each train, including their speed and tracking distance, were configured with different values. The operation conditions of HSR in the experiment are given as follows: (1) The total length of cruising distance is set as 160 km; (2) the maximum speed limit is 300 km/h; (3) the reference cruising speed is 288 km/h; and (4) a temporary speed restriction of 270 km/h is applied between 80 km and 100 km to emulate real-world operational interventions, such as maintenance zones or adverse weather conditions. The speed limit of the line is 310 km/h, as shown in Figure 4. The experiments are operated on a PC with a 1.6 GHz processor and 16 GB memory by Matlab2023b.

The parameters of the proposed FTESO-NTSM controller are listed in

Table 2. Parameters of proposed controller.

Parameter	Symbol	Value
NTSM coefficient	β	0.1
NTSM coefficient	p	5
NTSM coefficient	q	3
Reaching law parameter	k1	10
Tanh parameter	k2	5
Tanh parameter	μ	0.2
ESO gain	γ1	40
ESO gain	γ2	1000
ESO gain	γ3	20,000

. The sampling time period is set as 0.1 s. The reference profile for the leading train to track is shown in Figure 4.

First, we evaluate the effectiveness of the proposed FTESO-NTSM control method. The initial speed of the leading train and three following trains are 288 km/h, 284.4 km/h, 280.8 km/h, and 277.2 km/h, respectively. The initial tracking distance between the leader and follower 1, follower 1 and follower 2, follower 2 and follower 3 are set as 300 m. The nominal basic resistance coefficients are adopted from Table 1, which is given as $0.79+0.0064v_i\left(t\right)+0.000115v_i^2\left(t\right)$. The external disturbance is assumed to be a preset function which is given as

$$D_i\left(t\right)=5\sin \left(2\pi t/300\right)$$

(47)

The cooperative operation performance of the trains in the VCTS is summarized in Figure 5, Figure 6 and Figure 7. Figure 5 depicts the speed profiles of the four trains under different initial conditions. The leading train accurately tracks the reference profile and cruises at 288 km/h at the beginning. When a temporary speed restriction is applied, it decelerates accordingly to 268.8 km/h and maintains this speed up to a distance of 10 km. Since the speed of all three following trains are lower than 288 km/h, they initially operate in traction phase to accelerate, reaching peak speeds of 299.8 km/h, 301.7 km/h, and 306.2 km/h for follower 1, follower 2, and follower 3, respectively. Then they start to decelerate to the target cruising speed. This coordinated acceleration–deceleration strategy enables the entire convoy to achieve the desired speed and inter-train spacing.

Figure 6 and Figure 7 illustrate the tracking speed error and tracking distance error profiles of adjacent trains in the VCTS. Since the reference profile does not account for external disturbances, the leading train exhibits fluctuation phenomenon during reference tracking. The tracking speed error varies within the range of [−0.049 km/h, 0.061 km/h] and the tracking distance is within the range of [7.7 × 10⁻⁴ m, 2.6 × 10⁻³ m] for most of the simulation duration. However, due to the temporary speed restriction, the maximum tracking distance error of the leading train increases to −0.28 m and 0.47 m. During the initial phase, followers 1, 2, and 3 operate at a speed-adjustment stage to achieve virtual coupling. Their acceleration initiation is sequentially delayed relative to their preceding trains, which can be attributed to their lower initial speeds and the dependency of their control strategies on the state of the preceding train. Ultimately, the three followers achieve coordinated operation by reaching both the target speed and the desired safe spacing at 69 s, 131 s, and 162 s, respectively.

Figure 8 illustrates the sliding mode surface profiles of the leading train and the three followers. The sliding mode surface of the leading train remains consistently near zero. In contrast, the followers exhibit initial fluctuations in their sliding surfaces due to tracking errors in both relative distance and speed. These surfaces subsequently converge to the vicinity of zero at 69 s, 131 s, and 162 s for followers 1, 2, and 3, respectively. This convergence behavior aligns well with the variation trends observed in Figure 6 and Figure 7.

Figure 9 presents the observed disturbance profiles for all four trains in the platoon. Throughout the simulation, the maximum observation error is 0.0087 kN; these results validate the effective disturbance estimation capability of the proposed ESO module.

To benchmark the performance of the proposed algorithm, it is compared with three alternative controllers: (1) a baseline NTSM controller without ESO(NTSM-T); (2) an NTSM controller incorporating ESO with an exponential reaching law (FTESO-NTSM-E); and (3) an NTSM controller without ESO but employing an exponential reaching law (NTSM-E). The initial speeds of the four trains are set at different values (288 km/h, 291.6 km/h, 284.4 km/h, and 295.2 km/h). The initial tracking distances are also varied at 250 m, 300 m, and 350 m to assess formation stability. The configurations for basic resistance and external disturbances are identical to those in the former simulation. The comparative results, depicted in Figure 10, demonstrate the convergence and tracking performance of all four control strategies.

As illustrated in Figure 10, the proposed FTESO-NTSM algorithm generates the minimal level of chattering during the cruising phase. Such performance can be attributed to two key factors. Firstly, algorithms incorporating the ESO module exhibit significantly less chattering than those without it, owing to the ESO’s capability to estimate and compensate for external disturbances in real time. Secondly, the adoption of a tanh function-based reaching law in our proposed method offers smoother control activity compared to the conventional exponential reaching law, which is inherently prone to inducing chattering. The quantitative comparison of speed chattering magnitudes for different algorithms during the cruising phase is further summarized in

Table 3. Comparison of speed chattering magnitudes during the cruising phase.

Method	Leading Train	Follower 1	Follower 2	Follower 3
NTSM-T	0.0097 km/h	0.028 km/h	0.051 km/h	0.0659 km/h
FTESO-NTSM-E	0.0019 km/h	0.0033 km/h	0.0039 km/h	0.0036 km/h
NTSM-E	0.0192 km/h	0.0050 km/h	0.0184 km/h	0.0357 km/h
FTESO-NTSM (ours)	0.00087 km/h	0.0017 km/h	0.0026 km/h	0.0034 km/h

, which corroborates the aforementioned observations.

Figure 11 and Figure 12 depict the comparative tracking performance of speed and distance under the four control strategies in the VCTS. The proposed FTESO-NTSM algorithm demonstrates superior convergence capability, rapidly steering the system from initial state deviations to stable tracking speed and distance. As evidenced by the results, the three follower trains achieve convergence at 56 s, 130 s, and 76 s, respectively. A comparative summary of the convergence times for all strategies is quantitatively presented in

Table 4. Comparison of convergence performance.

Method	Follower 1	Follower 2	Follower 3
NTSM-T	677 s	484 s	392 s
FTESO-NTSM-E	59 s	132 s	80 s
NTSM-E	671 s	480 s	388 s
FTESO-NTSM (ours)	56 s	130 s	76 s

.

Since VCTS control must operate in real time, the computational time for each train is recorded accordingly. The average calculating time is 0.003 s, which satisfies the control period in this paper and is both practical and feasible for onboard controllers.

5. Results and Conclusions

5.1. Results

Simulation experiments demonstrated the effectiveness of the proposed method in achieving stable operation of VCTS under various initial conditions. In the first simulation test, the maximum tracking speed errors between adjacent trains are 0.061 km/h, 6.896 km/h, 10.051 km/h, and 9.433 km/h, respectively. Correspondingly, the maximum tracking distance errors are 0.47 m, 300 m, 341.53 m, and 349.36 m. The three followers converge to the target speed and the desired safe spacing at 69 s, 131 s, and 162 s, respectively. The ESO module exhibits a maximum estimation error of 0.0087 kN, demonstrating its capability to accurately estimate external disturbances and thereby enhance control precision. In the second simulation test, we compare the proposed method with three conventional sliding mode controllers. The proposed method achieves speed chattering magnitudes of 0.00087 km/h, 0.0017 km/h, 0.0026 km/h, and 0.0034 km/h for the leading train and the three followers. Compared to the NTSM-T and NTSM-E controllers, which have no ESO module, FTESO-NTSM-E and the proposed FTESO-NTSM controllers exhibit smaller speed chattering performance. This is because the ESO module can estimate and compensate the external disturbances in real time for more accurate control output. Furthermore, the adoption of the tanh function to enable continuous and smooth switching on the sliding surface endows the proposed FTESO-NTSM controller with notably reduced speed chattering in comparison with the FTESO-NTSM-E controller, such as 0.00103 km/h reduction for the leading train, and 0.0016 km/h, 0.0013 km/h, and 0.0002 km/h reduction for the followers. In terms of convergence performance, the proposed controller enabled the three following trains to achieve stable convergence within 56 s, 130 s, and 76 s, respectively. In contrast, the NTSM-T and NTSM-E controllers, which lack an ESO module, exhibited significantly slower convergence, with corresponding convergence times of approximately 670 s, 480 s, and 390 s for the three followers. The FTESO-NTSM-E controller shows similar convergence times but exhibits higher chattering due to its discontinuous switching law. These results collectively validate its effectiveness in tracking speed/distance, convergence speed, and robustness of the proposed approach.

5.2. Conclusions

This paper set out to address the critical challenge of precise and robust cooperative control for VCTS operating under unknown external disturbances and multiple operational constraints. To this end, a novel FTESO-NTSM control strategy is proposed. The core of this approach lies in the synergistic integration of a finite-time convergent disturbance observer with a nonsingular terminal sliding mode controller featuring a smooth hyperbolic tangent reaching law. Theoretical analysis, grounded in Lyapunov stability theory, rigorously proves the finite-time stability of the closed-loop system. Simulation results demonstrate that the proposed controller significantly outperforms conventional sliding mode methods in key performance metrics. It achieves rapid convergence from varied initial states, maintains precise tracking of both speed and relative distance, and effectively suppresses control chattering. The embedded extended state observer proves its effectiveness in terms of real-time estimation and compensation of lumped disturbances, which is a fundamental contributor to the enhanced robustness and accuracy of the system.

However, this work is subject to certain limitations. The simulation environment relies on idealized assumptions, including delay-free T2T communication and deterministic disturbance models, which may not fully capture the stochastic and complex nature of real-world railway operations. Furthermore, the scenarios tested, while representative, are limited in scope. Future research should therefore focus on validating the controller’s performance under more realistic conditions. This includes incorporating stochastic disturbances, communication delays, and packet losses into the model, and testing across a wider spectrum of emergency and fault scenarios.

In conclusion, this research provides a theoretically sound and simulation-validated control framework for cooperative train operation in the VCTS. It offers a promising solution for enhancing railway line capacity and operational efficiency, marking a meaningful step toward the realization of safe and reliable virtually coupled train operations.