Distributed Adaptive Fault-Tolerant Control for High-Speed Trains Based on a Multi-Body Dynamics Model

Abstract

The safe and efficient operation of high-speed trains is highly dependent on the reliability of their actuation systems, where actuator faults and input saturation pose significant challenges to control performance. Existing centralized control strategies often lack the flexibility to handle asymmetric actuator degradation and saturation across different carriages. To overcome this limitation, this paper leverages the inherent distributed structure of a train consist and proposes an distributed adaptive fault-tolerant control (DAFTC) scheme based on a multi-body dynamics model. The controller is designed at the carriage level to explicitly handle unknown actuator faults, input saturation, and parametric uncertainties. It incorporates an adaptive law for online parameter estimation and a second-order auxiliary system—a dynamic compensator—to mitigate saturation effects. Simulation results demonstrate the controller’s effectiveness in achieving accurate dual-loop tracking of both speed and position. Quantitative comparisons show that the proposed method reduces the average speed and position-tracking errors to 0.021 km/h and 0.426 m, respectively, outperforming conventional centralized approaches.

1. Introduction

With the advancement of rail transit towards higher speeds, greater density, and enhanced intelligence, the complexity of train operation control systems has significantly increased [1]. Against this backdrop, train coordinated operation control technology has emerged as a core research field for improving overall system performance. This technology is generally categorized into two key spatial dimensions: single-train internal coordination [2] and multi-train system coordination [3]. The former treats each carriage as an independent subsystem that collaboratively accomplishes operational tasks through coordinated control. The latter focuses on achieving global optimization objectives such as safety, punctuality, and energy efficiency at the train formation level, for instance, through distributed algorithms for speed tracking and cooperative scheduling. Addressing the control needs within this first category naturally necessitates a distributed framework, where each carriage’s controller can autonomously respond to local faults and saturation constraints while maintaining cooperative objectives.

However, ideal coordinated control strategies often encounter challenges in practical engineering due to physical limitations, among which actuator saturation is a prevalent constraint. Saturation nonlinearity can lead to performance degradation or even system instability, prompting the development of numerous robust control methods. To address issues such as input saturation, parameter uncertainties, and resource constraints, researchers have developed approaches including adaptive control [4], self-triggered model predictive control [5], and multi-agent reinforcement learning-based resource allocation algorithms [6]. Furthermore, studies such as [7,8,9,10,11] introduced mechanisms such as disturbance observation, iterative learning, and event triggering, collectively enhancing the control system’s adaptability under non-ideal conditions. These efforts demonstrate that integrating coordinated control theory with robustness design is an effective approach to addressing practical engineering constraints.

In ensuring system reliability, fault-tolerant control is crucial for maintaining safe operation under component failures. If key components such as actuators malfunction or degrade without effective fault-tolerant mechanisms, operational safety can be directly compromised. Current research in this area has branched into several technical routes. One category focuses on passive fault tolerance, enhancing the controller’s inherent robustness through methods such as sliding mode control [12,13,14], stochastic system modeling [15], and composite anti-disturbance designs [16,17]. Another stream emphasizes active fault tolerance, which maintains performance via fault estimation and system reconfiguration, exemplified by neural network-based adaptive iterative learning fault-tolerant control [18] and active fault-tolerant schemes for partial actuator failures [19,20]. In recent years, model-free adaptive fault-tolerant control [21,22] has also shown promise by adopting data-driven approaches that reduce dependence on precise mathematical models and improve applicability in complex failure scenarios.

A synthesis of existing research reveals that train coordinated control, robust anti-saturation design, and high-performance fault-tolerant mechanisms have formed an interconnected technological framework. Nevertheless, faced with increasingly complex operational environments and higher intelligence requirements, the challenge remains to deeply integrate these technologies and develop comprehensive control solutions that combine high precision, strong adaptability, high resource efficiency, and inherent operational resilience. Inspired by the aforementioned research, this paper investigates the problem of precise tracking control for high-speed trains with simultaneous actuator faults, parametric uncertainties, and input saturation. The main contributions are as follows:

• The development of a distributed adaptive fault-tolerant control scheme for a heterogeneous multi-agent train model, explicitly considering both multiplicative and additive actuator faults.
• The design of a novel second-order auxiliary system to dynamically compensate for input saturation constraints, ensuring closed-loop stability.
• Rigorous stability analysis via Lyapunov theory and comprehensive simulations demonstrating superior tracking performance and resilience compared to a centralized control approach.

The remainder of this paper is organized as follows. Section 2 details the controller design and presents the stability analysis. Section 3 validates the performance of the proposed method through numerical simulations. Section 4 provides a discussion of the results, outlining the limitations of this work and potential directions for future research. Finally, Section 5 concludes the paper.

2. Controller Design and Analysis

Controller design is a key component in ensuring the safe, efficient, and stable operation of a high-speed train. This chapter will begin by establishing the longitudinal dynamic model of the train. Based on this model, the characteristics and constraints of the system will be analyzed, and corresponding control strategies will then be designed.

2.1. Dynamic Model

A high-speed train typically comprises both motor cars, equipped with independent power systems, and trailer cars, which provide emergency braking capability only. This hybrid composition is modeled as a heterogeneous multi-agent system. As illustrated in Figure 1, an eight-carriage train configuration is presented, where motor cars and trailer cars are distinguished by yellow and gray tires, respectively. Based on Newton’s second law, the longitudinal dynamics of the k-th carriage can be expressed as follows:

$$m_k{\ddot{x}}_k\left(t\right)=F_k\left(t\right)+f_k\left(t\right)+I_k\left(t\right)+d_k\left(t\right),$$

(1)

where t denotes time, the subscript k represents the carriage number, $m_k$ represents the total weight of the k-th carriage, and represents the acceleration of the k-th carriage. $F_k$ represents the traction/braking force of the k-th carriage, $f_k$ represents the basic resistance of the k-th carriage, I represents the internal force between adjacent carriages, and $d_k$ represents the unknown interference of the external environment on the k-th carriage.

Figure 1. Schematic diagram of the multi-body dynamics model of a high-speed train.

As shown in Figure 1, the traction/braking efficiencies of different carriages of high-speed trains vary, and the actuators of motor cars and trailer cars also differ. The forward direction of the train is regarded as the reference direction. The expressions of $I_k$ and $f_k$ are as follows. Among them, $I_k=i_k+i_d$, where $i_k$ and $i_d$ are the internal forces of the train acting on the k-th carriage from adjacent vehicles, respectively.

$$i_k\left(t\right)=\left\{\begin{array}{c}-\mathrm{{\rm Y}}\left(t\right)[x_k\left(t\right)-x_{k+1}\left(t\right)-l],k=1\hfill \\ -\mathrm{{\rm Y}}\left(t\right)[x_k\left(t\right)-x_{k+1}\left(t\right)-l]\hfill \\ +\mathrm{{\rm Y}}\left(t\right)[x_{k-1}\left(t\right)-x_k\left(t\right)-l],k=2,\cdots ,n-1\hfill \\ \mathrm{{\rm Y}}\left(t\right)[x_{k-1}\left(t\right)-x_k\left(t\right)-l],k=n\hfill \end{array}\right.$$

(2)

$$i_d\left(t\right)=\left\{\begin{array}{c}-\lambda \left(t\right)[{\dot{x}}_k\left(t\right)-{\dot{x}}_{k+1}\left(t\right)],k=1\hfill \\ -\lambda \left(t\right)[{\dot{x}}_k\left(t\right)-{\dot{x}}_{k+1}\left(t\right)]\hfill \\ +\lambda \left(t\right)[{\dot{x}}_{k-1}\left(t\right)-{\dot{x}}_k\left(t\right)],k=2,\cdots ,n-1\hfill \\ \lambda \left(t\right)[{\dot{x}}_{k-1}\left(t\right)-{\dot{x}}_k\left(t\right)],k=n\hfill \end{array}\right.$$

(3)

$$f_k\left(t\right)=\left\{\begin{array}{c}-m_k\left(t\right)[a_0\left(t\right)+a_1\left(t\right){\dot{x}}_k\left(t\right)]\hfill \\ -M\left(t\right)a_2\left(t\right){\dot{x}}_k^2,k=1\hfill \\ -m_k\left(t\right)[a_0\left(t\right)+a_1\left(t\right){\dot{x}}_k\left(t\right)],k=2,\cdots ,n\hfill \end{array}\right.$$

(4)

Among them, n represents the total number of carriages in the train, and l represents the length of a carriage. The elastic coefficient and damping coefficient of the coupler are denoted as $\mathrm{{\rm Y}}$ and $\lambda$, respectively. $M\left(t\right)$ represents the total weight of the entire train, which can be calculated by summing the weights of each carriage, where $m_k\left(t\right)$ represents the weight of the k-th carriage. In addition, $a_0$ and $a_1$ are mechanical resistance coefficients, and $c_3$ represents the air resistance coefficient. Usually, the air resistance only affects the first carriage of the train. It should be noted that the uncertain parameters handled by the controller are $\mathrm{{\rm Y}},\lambda ,$ $a_0,a_1$ and $a_2$.

2.2. Actuator Failure Model

Actuator faults, inevitably arising from long-term cyclic operation in high-speed train traction systems, can lead to issues such as overheating and severe vibration. These anomalies may result in partial or complete failure of traction/braking actuators, manifesting as over-voltage in the traction transformer, over-current in the traction converter, or over-heating of the asynchronous motor. When a fault occurs in the actuator during the operation of a high-speed train, the actual force $F_k\left(t\right)$ applied to the train can be equivalently expressed as:

$$F_k\left(t\right)={\rho }_k\left(t\right)u_k\left(t\right)+{\delta }_k\left(t\right).$$

(5)

Among them, $u_k$ represents the traction/braking force expected to be input by the controller, ${\rho }_k\in \left(0,1\right)$ is the health factor of the actuator, that is, the multiplicative fault coefficient of the k-th carriage, and $\left|{\delta }_k\right|\le {\delta }_{max}<\infty$ is the additive fault coefficient, where ${\delta }_{max}$ is an unknown upper bound.

The maximum allowable operating speed of a high-speed train directly determines the maximum traction capacity of the traction system and its redundant design. However, in the input saturation state, the high-speed train will not be able to maintain normal operating performance. Therefore, based on the input limitation conditions of the traction motor, the traction/braking force of the high-speed train needs to meet the following constraint requirements:

$${\overline{u}}_k\left(t\right)=sa{t}_{u_k}\left(u_k\left(t\right)\right)=\left\{\begin{array}{c}{u}_{max}\left(t\right),u_k\left(t\right)\ge u_{max}\left(t\right)\hfill \\ u_{min}\left(t\right),u_k\left(t\right)\le u_{min}\left(t\right)\hfill \\ u_k\left(t\right),u_{min}\left(t\right)<u_k\left(t\right)<u_{max}\left(t\right)\hfill \end{array}\right.$$

(6)

Among them, $u_{max}$ represents the maximum traction/braking force that the traction motor can provide, and $u_{min}$ represents the minimum traction/braking force that the traction motor can provide. Since the traction/braking force provided by the motor is converted from the input power, $u_{max}$ and $u_{min}$ are functions related to the train speed. The established traction force model (6) captures the uncertainties inherent in the train’s actuators. Define the input error as:

$$\Delta u_k\left(t\right)={\overline{u}}_k\left(t\right)-u_k\left(t\right).$$

(7)

2.3. Multi-Body Dynamics Model of the Train

Based on Formulas (1)–(4), the multi-agent system model of the high-speed train is constructed as follows:

$${\Xi }_m\ddot{x}=U+{\Xi }_E\Pi {\phi }_x+{\Xi }_C\Pi \dot{x}+{\Xi }_0+{\Xi }_1\dot{x}+{\Xi }_2{\varphi }_x+{\Xi }_D.$$

(8)

Among them:

$$\Pi ={\left[\begin{array}{ccccc}-1& 1& 0& 0& \cdots \\ 1& -2& 1& 0& \cdots \\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& \cdots & 1& -2& 1\\ 0& \cdots & 0& 1& -1\end{array}\right]}_{n\times n},$$

$${\phi }_x={\left[x_1,x_2+l,\dots ,x_n+(n-1)l\right]}^T\in {\mathbb{R}}^{n\times 1},$$

$${\varphi }_x={\left[{\dot{x}}_1^2,0,\cdots ,0\right]}^T\in {\mathbb{R}}^{n\times 1},$$

$${\Xi }_m={\left({\Xi }_A^{\prime }\right)}^{-1}diag\left[m_1,m_2,\dots ,m_n\right]\in {\mathbb{R}}^{n\times n},$$

$${\Xi }_E={\left({\Xi }_A^{\prime }\right)}^{-1}\mathrm{{\rm Y}}\in \mathbb{R},{\Xi }_C={\left({\Xi }_M^{\prime }\right)}^{-1}\lambda \in \mathbb{R},$$

$${\Xi }_0={\left({\Xi }_A^{\prime }\right)}^{-1}{\left[-m_1{c}_1,-m_2{c}_1,\dots ,-m_n{c}_1\right]}^T\in {\mathbb{R}}^{n\times 1},$$

$${\Xi }_1={\left({\Xi }_A^{\prime }\right)}^{-1}diag\left[-m_1{c}_2,-m_2{c}_2,\dots ,-m_n{c}_2\right]\in {\mathbb{R}}^{n\times n},$$

$${\Xi }_2=-{\left({\Xi }_A^{\prime }\right)}^{-1}M{c}_3\in {\mathbb{R}}^{n\times 1},$$

$${\Xi }_D={\left({\Xi }_A^{\prime }\right)}^{-1}{\left[{\theta }_{f,1}^{\prime },{\theta }_{f,2}^{\prime },\cdots ,{\theta }_{f,n}^{\prime }\right]}^T\in {\mathbb{R}}^{n\times 1}.$$

Among them, the known parameter vector of the train is denoted as $\ddot{x}$, $\dot{x}$, ${\phi }_x$ and ${\varphi }_x$. There are slowly-varying unknown parameters in the train dynamics model. ${\Xi }_m$, ${\Xi }_E$, ${\Xi }_C$, ${\Xi }_0$, ${\Xi }_1$, ${\Xi }_2$ represent the matrix or vector composed of unknown slow-time-varying parameters. ${\Xi }_D^{\prime }=d_k+{\delta }_k$, $k=1,2,\cdots ,n$ represents the sum of the unknown external disturbances of the carriages and the unknown disturbances of the internal actuators, and this part of the parameters is fast-time-varying. U is the control input calculated by the controller. The superscript T is the transpose of a vector or a matrix, and $\mathrm{diag}$() is the diagonal matrix operator.

Assumption 1.

The desired states of each carriage of the train are unknown, the speed trajectory is continuously differentiable, and the desired acceleration of the carriages is known.

The calculation process of the desired operation trajectory of the high-speed train’s Automatic Train Operation (ATO) system is to obtain the speed and position by integrating the desired acceleration. In this way, the problem of calculation expansion caused by multiple differentials of $x_d$ will not occur.

Assumption 2.

The external disturbance vector is bounded, i.e., $\left|{\Xi }_D\right|\le \Lambda ,\Lambda \in {\mathbb{R}}^{n\times 1}$ is a known vector.

Assumption 3.

${\Xi }_m$, ${\Xi }_E$, ${\Xi }_C$, ${\Xi }_0$, ${\Xi }_1$, ${\Xi }_2$ is an unknown slow-time-varying parameter. For example, its derivative is 0, i.e., ${\dot{\Xi }}_m\approx 0$.

${\Xi }_m$, ${\Xi }_E$, ${\Xi }_C$, ${\Xi }_0$, ${\Xi }_1$, ${\Xi }_2$ are related to the carriage structure, coupler coefficient, characteristics and materials of the traction/braking system, and its rate of change is extremely small.

2.4. Design of Distributed Fault-Tolerant Controller for Trains

An auxiliary system is constructed to address stability issues arising from actuator saturation. To design a controller in the presence of actuator faults and saturation, a second-order secondary system is introduced as follows:

$$\begin{array}{c}{\dot{\lambda }}_1=-k_1{\lambda }_1^{p/q}+{\lambda }_2,\hfill \\ {\dot{\lambda }}_2=-k_2{\lambda }_2^{p/q}+k_3\Delta u.\hfill \end{array}$$

(9)

Among them, $k_1$, $k_2$, $k_3$, p, q are design parameters. ${\lambda }_1\in {\mathbb{R}}^{n\times 1}$ and ${\lambda }_2\in {\mathbb{R}}^{n\times 1}$ are the states of the auxiliary system. $\Delta u\in {\mathbb{R}}^{n\times 1}$ is the input signal of the system. The positive parameters $k_1,k_2,k_3$ determine the convergence rate of the auxiliary system and tracking errors, typically tuned within $\left[0.01,10\right]$ through simulation to balance response speed and noise sensitivity. The ratio $p/q$ (with $p,q$ being positive odd integers and $p<q$) governs the finite-time convergence property, where values like $p/q=3/5$ ensure satisfactory transient performance. Specific values used in simulations are provided in Section 3.1.

Based on the above analysis, an n × n matrix is constructed with an n-dimensional vector as the diagonal elements of the matrix. The following tracking error is defined:

$$\begin{array}{c}{e}_1=x-x_d-{\lambda }_1,\hfill \\ e_2=\dot{x}-{\dot{x}}_d-{\lambda }_2.\hfill \end{array}$$

(10)

Among them, $x={\left[x_1,x_2,\cdots ,x_n\right]}^T\in {\mathbb{R}}^{n\times 1}$ and $\dot{x}={\left[{\dot{x}}_1,{\dot{x}}_2,\cdots ,{\dot{x}}_n\right]}^T\in {\mathbb{R}}^{n\times 1}$ are the column vectors of the actual position and speed of the train, and $x_d={\left[x_{d.1},x_{d.2},\cdots ,x_{d,n}\right]}^T\in {\mathbb{R}}^{n\times 1}$ and ${\dot{x}}_d={\left[{\dot{x}}_{d.1},{\dot{x}}_{d.2},\cdots ,{\dot{x}}_{d,n}\right]}^T\in {\mathbb{R}}^{n\times 1}$ are the column vectors of the desired position and desired speed. It is worth noting that if $\overline{u}=u$, $\Delta u=0$, then ${\lambda }_1$ and ${\lambda }_2$ are still zero. By constructing the auxiliary system, the influence of input saturation on the stability of the closed-loop control system can be solved. From the perspective of train control, the speed change of the auxiliary system can be adjusted through the parameters $k_1$, $k_2$, $k_3$, p, q. By introducing the auxiliary system states ${\lambda }_1$ and ${\lambda }_2$ into the tracking errors $e_1$ and $e_2$, the problem of the rapid increase of the tracking error caused by actuator faults can be corrected. The introduction of the auxiliary states ${\lambda }_1$ and ${\lambda }_2$ serves to reshape the error dynamics by providing a buffer against abrupt changes caused by actuator faults and saturation. When an actuator fault occurs or saturation is encountered, the term $\Delta u$ becomes non-zero, activating the auxiliary system. This causes ${\lambda }_1$ and ${\lambda }_2$ to deviate from zero, thereby modifying the tracking errors $e_1$ and $e_2$ through Equations (9) and (10). This modification effectively redistributes the tracking burden over time and across carriages, preventing the rapid accumulation of tracking errors that would otherwise lead to performance degradation or instability.

A sliding surface is introduced as follows:

$$s=c{e}_1+e_2,$$

(11)

where $c>0$. Aiming at the faults, saturation and uncertainties of the actuators of high-speed trains, the distributed adaptive fault-tolerant controller is designed as follows:

$$\begin{array}{c}u=-k_3^{-1}(e_2+k_1{\lambda }_1^{p/q}+\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q})\\ -(-{\widehat{\Xi }}_M\ddot{x}+{\widehat{\Xi }}_E\Pi {\phi }_x+{\widehat{\Xi }}_C\Pi \dot{x}\\ +{\widehat{\Xi }}_0+{\widehat{\Xi }}_1\dot{x}+{\widehat{\Xi }}_2{\varphi }_x)-{\overline{\Xi }}_D·\mathrm{sgn}\left(s\right)\end{array}.$$

(12)

The adaptive rate of the design parameters is as follows:

$${\dot{\widehat{\Xi }}}_M=-{\gamma }_M^{-1}{k}_3\mathrm{diag}\left(s^T\right)\ddot{x}$$

(13)

$${\dot{\widehat{\Xi }}}_E={\gamma }_E^{-1}{k}_3{s}^T\Pi {\phi }_x$$

(14)

$${\dot{\widehat{\Xi }}}_C={\gamma }_C^{-1}{k}_3\mathrm{diag}\left(s^T\right)\mathrm{diag}\left(\Pi \dot{x}\right)$$

(15)

$${\dot{\widehat{\Xi }}}_0={\gamma }_0^{-1}{k}_3{s}^T$$

(16)

$${\dot{\widehat{\Xi }}}_1={\gamma }_1^{-1}{k}_3\mathrm{diag}\left(s^T\right)\mathrm{diag}\left(\dot{x}\right)$$

(17)

$${\dot{\widehat{\Xi }}}_2={\gamma }_2^{-1}{k}_3\mathrm{diag}\left(s^T\right)\mathrm{diag}\left({\varphi }_x\right)$$

(18)

Remark  1.

The diagonal structure of the parameter estimation matrices stems from the independence of uncertain parameters across different carriages. The trace operator is employed in the adaptive laws as it provides a compact notation for updating the diagonal elements of parameter matrices while maintaining dimensional consistency. To prevent parameter drift—a common issue in adaptive control—the update laws incorporate inherent robustness through the sliding variable s, which ensures boundedness of estimates under persistent excitation conditions. Specifically, the coupling between parameter updates and tracking errors guarantees that estimates remain bounded as long as the system operates within the defined stability bounds.

Proof. 

To prove the convergence of the proposed controller, the Lyapunov function is constructed as follows:

$$\begin{array}{cc}\hfill V& ={\displaystyle \frac{1}{2}}{s}^{T}s+{\displaystyle \frac{1}{2}}{\gamma }_M\mathrm{tr}({\tilde{\Xi }}_M^2)+{\displaystyle \frac{1}{2}}{\gamma }_E\mathrm{tr}({\tilde{\Xi }}_E^2)+{\displaystyle \frac{1}{2}}{\gamma }_C\mathrm{tr}({\tilde{\Xi }}_C^2)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\gamma }_0\mathrm{tr}({\tilde{\Xi }}_0^2)+{\displaystyle \frac{1}{2}}{\gamma }_1\mathrm{tr}({\tilde{\Xi }}_1^2)+{\displaystyle \frac{1}{2}}{\gamma }_2\mathrm{tr}({\tilde{\Xi }}_2^2).\hfill \end{array}$$

(19)

Among them, ${\tilde{\Xi }}_M={\Xi }_M-{\widehat{\Xi }}_M$ represents the estimation error. ${\tilde{\Xi }}_E, {\tilde{\Xi }}_C, {\tilde{\Xi }}_0, {\tilde{\Xi }}_1, {\tilde{\Xi }}_2$ have the same definition. $\mathrm{tr}$() represents the operator of the matrix trace, which is the sum of the diagonal elements of the matrix. Taking the derivative of (19), we have:

$$\begin{array}{cc}\hfill \dot{V}& =s^T\dot{s}+{\gamma }_M\mathrm{tr}{\tilde{\Xi }}_M{\dot{\tilde{\Xi }}}_M+{\gamma }_E\mathrm{tr}{\tilde{\Xi }}_E{\dot{\tilde{\Xi }}}_E+{\gamma }_C\mathrm{tr}{\tilde{\Xi }}_C{\dot{\tilde{\Xi }}}_C\hfill \\ \hfill & +{\gamma }_0\mathrm{tr}{\tilde{\Xi }}_0{\dot{\tilde{\Xi }}}_0+{\gamma }_1\mathrm{tr}{\tilde{\Xi }}_1{\dot{\tilde{\Xi }}}_1+{\gamma }_2\mathrm{tr}{\tilde{\Xi }}_2{\dot{\tilde{\Xi }}}_2\hfill \\ \hfill & =s^T\dot{s}+{\gamma }_M\mathrm{tr}{\tilde{\Xi }}_M({\dot{\Xi }}_M-{\dot{\widehat{\Xi }}}_M)\hfill \\ \hfill & +{\gamma }_E\mathrm{tr}{\tilde{\Xi }}_E({\dot{\Xi }}_E-{\dot{\widehat{\Xi }}}_E)+{\gamma }_C\mathrm{tr}{\tilde{\Xi }}_C({\dot{\Xi }}_C-{\dot{\widehat{\Xi }}}_C)\hfill \\ \hfill & +{\gamma }_0\mathrm{tr}{\tilde{\Xi }}_0({\dot{\Xi }}_0-{\dot{\widehat{\Xi }}}_0)+{\gamma }_1\mathrm{tr}{\tilde{\Xi }}_1({\dot{\Xi }}_1-{\dot{\widehat{\Xi }}}_1)\hfill \\ \hfill & +{\gamma }_2\mathrm{tr}{\tilde{\Xi }}_2({\dot{\Xi }}_2-{\dot{\widehat{\Xi }}}_2)\hfill \\ \hfill & =s^T\dot{s}-{\gamma }_M\mathrm{tr}({\tilde{\Xi }}_M{\dot{\widehat{\Xi }}}_M)-{\gamma }_E\mathrm{tr}({\tilde{\Xi }}_E{\dot{\widehat{\Xi }}}_E)\hfill \\ \hfill & -{\gamma }_C\mathrm{tr}({\tilde{\Xi }}_C{\dot{\widehat{\Xi }}}_C)-{\gamma }_0\mathrm{tr}({\tilde{\Xi }}_0{\dot{\widehat{\Xi }}}_0)\hfill \\ \hfill & -{\gamma }_1\mathrm{tr}({\tilde{\Xi }}_1{\dot{\widehat{\Xi }}}_1)-{\gamma }_2\mathrm{tr}({\tilde{\Xi }}_2{\dot{\widehat{\Xi }}}_2).\hfill \end{array}$$

(20)

The derivative of Formula (11) yields:

$$\dot{s}=c{\dot{e}}_1+{\dot{e}}_2.$$

(21)

Substituting Formula (9)–(12) into Formula (21) yields the result:

$$\begin{array}{cc}\hfill \dot{s}& =c{\dot{e}}_1+{\dot{e}}_2\hfill \\ \hfill & =e_2+k_1{\lambda }_1^{p/q}+\left(\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}-k_3\Delta u\right)\hfill \\ \hfill & =e_2+k_1{\lambda }_1^{p/q}+k_3\left(-{\Xi }_M\ddot{x}+\overline{u}+{\Xi }_E\Pi {\phi }_x\right.\hfill \\ \hfill & \left.+{\Xi }_C\Pi \dot{x}+{\Xi }_0+{\Xi }_1\dot{x}+{\Xi }_2{\dot{\phi }}_x+{\Xi }_D\right)+\left(\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}-k_3\Delta u\right)\hfill \\ \hfill & =e_2+k_1{\lambda }_1^{p/q}+k_3\left(-{\Xi }_M\ddot{x}+{\Xi }_E\Pi {\phi }_x\right.\hfill \\ \hfill & \left.+{\Xi }_C\Pi \dot{x}+{\Xi }_0+{\Xi }_1\dot{x}+{\Xi }_2{\dot{\phi }}_x+{\Xi }_D\right)\hfill \\ \hfill & +k_3\left(\overline{u}-\Delta u\right)+\left(\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}\right)\hfill \\ \hfill & =e_2+k_1{\lambda }_1^{p/q}+k_3\left(-{\Xi }_M\ddot{x}+{\Xi }_E\Pi {\phi }_x\right.\hfill \\ \hfill & \left.+{\Xi }_C\Pi \dot{x}+{\Xi }_0+{\Xi }_1\dot{x}+{\Xi }_2{\dot{\phi }}_x+{\Xi }_D\right)\hfill \\ \hfill & +k_{3}u+\left(\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}\right)\hfill \end{array}$$

(22)

Substituting the controller (12) into (22) yields:

$$\begin{array}{cc}\hfill \dot{s}& =e_2+k_1{\lambda }_1^{p/q}+k_3\left(-{\Xi }_M\ddot{x}+{\Xi }_E\Pi {\phi }_x+{\Xi }_C\Pi \dot{x}+{\Xi }_0+{\Xi }_1\dot{x}+{\Xi }_2{\varphi }_x+{\Xi }_D\right)\hfill \\ \hfill & +k_3\left[-k_3^{-1}\left(e_2+k_1{\lambda }_1^{p/q}+\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}\right)\right]\hfill \\ & -k_3\left[\left(-{\widehat{\Xi }}_M\ddot{x}+{\widehat{\Xi }}_E\Pi {\phi }_x+{\widehat{\Xi }}_C\Pi \dot{x}+{\widehat{\Xi }}_0+{\widehat{\Xi }}_1\dot{x}+{\widehat{\Xi }}_2{\varphi }_x\right)-{\Xi }_D\mathrm{sgn}\left(s\right)\right]\hfill \\ \hfill & +\left(\ddot{x}-{\ddot{x}}_d+k_2{\lambda }_2^{p/q}\right)\hfill \\ \hfill & =k_3\left[-{\tilde{\Xi }}_M\ddot{x}+{\tilde{\Xi }}_E\Pi {\phi }_x+{\tilde{\Xi }}_C\Pi \dot{x}+{\tilde{\Xi }}_0+{\tilde{\Xi }}_1\dot{x}+{\tilde{\Xi }}_2{\varphi }_x+{\Xi }_D-{\tilde{\Xi }}_D\mathrm{sgn}\left(s\right)\right].\hfill \end{array}$$

(23)

Then, substituting (23) into (20) leads to:

$$\begin{array}{cc}\hfill \dot{V}& =s^T\dot{s}-{\gamma }_M\mathrm{tr}\left({\tilde{\Xi }}_M{\widehat{\tilde{\Xi }}}_M\right)-{\gamma }_E\mathrm{tr}\left({\tilde{\Xi }}_E{\widehat{\tilde{\Xi }}}_E\right)\hfill \\ \hfill & -{\gamma }_C\mathrm{tr}\left({\tilde{\Xi }}_C{\widehat{\tilde{\Xi }}}_C\right)-{\gamma }_0\mathrm{tr}\left({\tilde{\Xi }}_0{\widehat{\tilde{\Xi }}}_0\right)-{\gamma }_1\mathrm{tr}\left({\tilde{\Xi }}_1{\widehat{\tilde{\Xi }}}_1\right)\hfill \\ \hfill & -{\gamma }_2\mathrm{tr}\left({\tilde{\Xi }}_2{\widehat{\tilde{\Xi }}}_2\right)\hfill \\ \hfill & =k_3{s}^T\left(-{\tilde{\Xi }}_M\ddot{x}+{\tilde{\Xi }}_E\Pi {\phi }_x+{\tilde{\Xi }}_C\Pi \dot{x}+{\tilde{\Xi }}_0+{\tilde{\Xi }}_1\dot{x}+{\tilde{\Xi }}_2{\varphi }_x\right)\hfill \\ \hfill & +k_3{s}^T\left[{\Xi }_D-{\tilde{\Xi }}_D\mathrm{sgn}\left(s\right)\right]\hfill \\ \hfill & -{\gamma }_M\mathrm{tr}\left({\tilde{\Xi }}_M{\widehat{\tilde{\Xi }}}_M\right)-{\gamma }_E\mathrm{tr}\left({\tilde{\Xi }}_E{\widehat{\tilde{\Xi }}}_E\right)-{\gamma }_C\mathrm{tr}\left({\tilde{\Xi }}_C{\widehat{\tilde{\Xi }}}_C\right)\hfill \\ \hfill & -{\gamma }_0\mathrm{tr}\left({\tilde{\Xi }}_0{\widehat{\tilde{\Xi }}}_0\right)-{\gamma }_1\mathrm{tr}\left({\tilde{\Xi }}_1{\widehat{\tilde{\Xi }}}_1\right)-{\gamma }_2\mathrm{tr}\left({\tilde{\Xi }}_2{\widehat{\tilde{\Xi }}}_2\right)\hfill \\ \hfill & =\mathrm{tr}\left[{\tilde{\Xi }}_M\left(-{\gamma }_M{\widehat{\tilde{\Xi }}}_M-k_3{s}^T\ddot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_E\left(-{\gamma }_E{\widehat{\tilde{\Xi }}}_E+k_3{s}^T\Pi {\phi }_x\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_C\left(-{\gamma }_C{\widehat{\tilde{\Xi }}}_C+k_3{s}^T\Pi \dot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_0\left(-{\gamma }_0{\widehat{\tilde{\Xi }}}_0+k_3{s}^T\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_1\left(-{\gamma }_1{\widehat{\tilde{\Xi }}}_1+k_3{s}^T\dot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_2\left(-{\gamma }_2{\widehat{\tilde{\Xi }}}_2+k_3{s}^T{\varphi }_x\right)\right]\hfill \\ \hfill & +k_3{s}^T\left[{\Xi }_D-{\tilde{\Xi }}_D\mathrm{sgn}\left(s\right)\right],\hfill \end{array}$$

(24)

where ${\tilde{\Xi }}_M={\Xi }_M-{\widehat{\Xi }}_M\le 0$.

By subtracting adaptive law (13) from (18), we can obtain:

$$\begin{array}{cc}\hfill \dot{V}& =\mathrm{tr}\left[{\tilde{\Xi }}_M\left(-{\gamma }_M(-{\gamma }_M^{-1}{k}_3\mathrm{diag}\left(s^T\right)\ddot{x})-k_3{s}^T\ddot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_E\left(-{\gamma }_E({\gamma }_E^{-1}{k}_3{s}^T\Pi {\phi }_x)+k_3{s}^T\Pi {\phi }_x\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_C\left(-{\gamma }_C({\gamma }_C^{-1}{k}_3\mathrm{diag}\left(s^T\right)\Pi \dot{x})+k_3{s}^T\Pi \dot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_0\left(-{\gamma }_0\left({\gamma }_0^{-1}{k}_3\mathrm{diag}\left(s^T\right)\right)+k_3{s}^T\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_1\left(-{\gamma }_1\left({\gamma }_1^{-1}{k}_3\mathrm{diag}\left(s^T\right)\dot{x}\right)+k_3{s}^T\dot{x}\right)\right]\hfill \\ \hfill & +\mathrm{tr}\left[{\tilde{\Xi }}_2\left(-{\gamma }_2\left({\gamma }_2^{-1}{k}_3\mathrm{diag}\left(s^T\right){\varphi }_x\right)+k_3{s}^T{\varphi }_x\right)\right]\hfill \\ \hfill & +k_3{s}^T\left[{\Xi }_D-{\overline{\Xi }}_D·\mathrm{sgn}\left(s\right)\right]\hfill \\ \hfill & \le k_3{s}^T\left[{\Xi }_D-{\overline{\Xi }}_D·\mathrm{sgn}\left(s\right)\right]\hfill \\ & =k_3(s^T{\Xi }_D-|s^T\left|{\overline{\Xi }}_D\right)\le 0\hfill \end{array}$$

(25)

Among them, all the matrices involved in (25) are diagonal matrices. The operations of diagonal matrices satisfy the commutative law, that is:

$$\mathrm{diag}\left(A\right)\mathrm{diag}\left(B\right)=\mathrm{diag}\left(B\right)\mathrm{diag}\left(A\right),A\in {\mathbb{R}}^{n\times 1},B\in {\mathbb{R}}^{n\times 1}.$$

Therefore, under the control of the controller u and the adaptive law, the system will eventually reach the sliding mode surface, i.e., $s=0$, and then $e_1$ and $e_2$ will approach zero. When the actuators of the system have sufficient power, $\Delta u=0$ and the auxiliary system converges. will approach zero within a bounded time. After that, the tracking errors of the train’s speed and position will approach zero, and finally, each carriage will achieve state-consistent tracking of the desired operation trajectory.    □

The steps of DFATC method are as shown in Algorithm 1.    

Algorithm 1: Distributed Adaptive Fault—Tolerant Control

3. Simulation Verification

In this section, a numerical simulation experiment is designed to verify the control effect of the designed Distributed Adaptive Fault-Tolerant Control (DAFTC) method.

3.1. Simulation Environment Description

The computation was carried out in MATLAB R2025b on a personal computer (PC) running the Windows operating system with 8.00 GB of RAM and we set $\Delta t=0.1$. The simulation adopts a high-speed train formation of four motor cars and four trailer cars, that is, each train consists of four motor cars and four trailer cars. Among them, the motor cars can provide traction force, electric braking force, and air braking force, while the trailer cars can only provide air braking force. In distributed control, the traction/braking forces among different motor cars are different, and its expression is as follows:

$$\begin{array}{cc}\hfill u=& -k_3^{-1}\left(e_{2,1}+k_1{\lambda }_{1,1}^{p/q}+{\ddot{x}}_1-{\ddot{x}}_{d,1}+k_2{\lambda }_{2,1}^{p/q}\right)\hfill \\ \hfill & -\left(-{\widehat{\Xi }}_M{\ddot{x}}_1+{\widehat{\Xi }}_E\Pi {\phi }_x+{\widehat{\Xi }}_C\Pi {\dot{x}}_1+{\widehat{\Xi }}_0+{\widehat{\Xi }}_1{\dot{x}}_1+{\widehat{\Xi }}_2{\varphi }_x\right)\hfill \\ \hfill & -{\overline{\Xi }}_D·sgn\left(s_1\right).\hfill \end{array}$$

(26)

Among them, $e_{2,1}$ represents the first element of the column vector $e_2$. Similarly, ${\lambda }_{1,1}^{p/q}$, ${\ddot{x}}_1$, ${\ddot{x}}_{d,1}$, ${\lambda }_{2,1}^{p/q}$ have the same definition. For a fair comparative analysis, identical parameter values ($k_1=0.01,k_2=5,k_3=0.01,p=5,q=3$) are maintained consistently across both the DAFTC and CAFTC.

For a comparative baseline, a Centralized Adaptive Fault-Tolerant Control (CAFTC) strategy is designed. To ensure a fair comparison that isolates the effect of the control architecture, the CAFTC scheme utilizes the same control law as the proposed DAFTC (Equation (12)) to calculate the total control effort. However, in contrast to the distributed allocation of DAFTC, the individual control inputs $u_k$ computed for each carriage are aggregated and averaged to generate a unified force command. This command is then applied uniformly to all motor carriages. The force for each motor carriage under CAFTC is given by:

$$F_{\mathrm{CAFTC}}={\displaystyle \frac{1}{n_m}}\sum _{k=1}^n{u}_k$$

(27)

where $n_m$ is the number of motor carriages. This approach guarantees that the total traction/braking forces acting on the train is identical under both control schemes. Therefore, any performance differences observed can be unequivocally attributed to the DAFTC’s ability to perform heterogeneous force allocation in response to faults, rather than discrepancies in overall control effort or parameter tuning. This establishes CAFTC as a well-defined and fair baseline for evaluating the advantages of the distributed coordination strategy.

The carriages are assumed to be strongly coupled. Therefore, due to the coupler effects, the relative displacements and speeds between carriages remain minimal under normal operation. Therefore, the operation trajectories of each carriage are basically the same. Based on the above analysis, only the operation trajectory of the first carriage is shown in the simulation operation trajectory diagram to keep the graph concise.

The parameters for the high-speed train, listed in Table 1, are set with reference to [23,24]. The resistance coefficients $c_0\left(t\right)$, $c_1\left(t\right)$ and $c_2\left(t\right)$ are modeled with small sinusoidal variations to simulate slow, bounded parameter uncertainties in real operations, thereby validating the robustness of the adaptive controller.

Table 1. Parameter Settings for High-Speed Trains.

Parameter	Value	Unit
$m_k$	$49\pm 5$	t
$c_0\left(t\right)$	$0.55+0.055sin\left(0.004t\right)$	N/kN
$c_1\left(t\right)$	$0.00361+0.00036sin\left(0.004t\right)$	N/kN
$c_2\left(t\right)$	$0.00011+0.000011sin\left(0.004t\right)$	N/kN
$l_h$	$25.225$	m
$l_r$	$2\times {10}^7$	N/m
$\epsilon \left(t\right)$	$5\times {10}^6$	N·s/m

3.2. Result Analysis

The situation where the actuator efficiency of each carriage during train operation decreases and is inconsistent, representative of typical actuator faults such as traction converter thermal derating, is set, and a comparative simulation is carried out between the DAFTC proposed in this paper and CAFTC. Figure 2 shows the speed-tracking trajectory diagrams of DAFTC and CAFTC. Figure 3 shows the corresponding speed-tracking errors of the two controllers. A transient speed-tracking error is observed during the initial traction phase. This is because actuator faults, saturation limitations, and the need for a certain time to adjust the parameters through adaptive control lead to an increase in the train’s error in a short time, but it can converge quickly. When the actuator efficiency is low and inconsistent, the train’s speed-tracking error increases significantly, but it will also converge to a certain range subsequently.

Figure 2. Speed-tracking curves under the control of DAFTC and CAFTC.

Figure 3. Speed-tracking errors under the control of DAFTC and CAFTC.

By comparing the speed-tracking errors of the two controllers, it can be seen that DAFTC has higher control accuracy and a faster recovery speed of the train’s tracking state when short-term execution faults occur. At the same time, from the speed-tracking error curve, it can be seen that after the speed-tracking error converges, it is always controlled within $\pm 2$ km/h. When the train’s operating conditions change, there will be a small-degree fluctuation, but it always fluctuates within a small range and can converge quickly, which has a very small impact on the riding comfort.

Figure 4 and Figure 5 show the position-tracking trajectories and position-tracking errors of DAFTC and CAFTC, respectively. By comparing the position errors of the two controllers, it can be seen that DAFTC, which adopts the distributed control strategy, can make the error converge in a relatively short time. In contrast, for CAFTC, which adopts the centralized control strategy, the position error fails to converge. The reason is that in the centralized control mode, when there are differences in the execution of each carriage, a carriage with low efficiency will always be affected by the forces from adjacent carriages to maintain the speed consistency of the whole train. Moreover, DAFTC can adaptively adjust the control parameters during the differential control of carriages. Therefore, DAFTC has higher position-tracking accuracy and stronger anti-interference ability on the line. Table 2 present the analysis results of the control performance indicators of the two controllers.

Figure 4. Position-tracking curves under the control of DAFTC and CAFTC.

Figure 5. Position-tracking errors under the control of DAFTC and CAFTC.

Table 2. Analysis of performance indicators of the two controllers.

Indicator	CAFTC	DAFTC
Average speed error	0.098 km/h	0.021 km/h
Average position error	1.512 m	0.426 m
Maximum speed error	6.882 km/h	7.002 km/h

Figure 6 and Figure 7 show the carriage control inputs of DAFTC and CAFTC, respectively. It can be seen that there are differences in the traction/braking forces among carriages in DAFTC. Due to the low efficiency of the actuators, the traction/braking forces calculated by the train controller cannot be achieved under the condition of input saturation. In summary, DAFTC can still achieve precise dual- closed-loop tracking of speed and position when the train actuators malfunction and the actuator efficiencies among different units are inconsistent.

Figure 6. Traction/braking force of the carriage under CAFTC control.

Figure 7. Traction/braking force of the carriage under DAFTC control.

4. Discussion

The simulation results demonstrate that the proposed DAFTC effectively addresses the challenges of actuator faults, input saturation, and parameter uncertainties in high-speed train systems. As shown in Table 2, compared with CAFTC, the proposed DAFTC reduces the average speed tracking error from 0.098 km/h to 0.021 km/h and reduces the average position tracking error from 1.512 m to 0.426 m. These specific numerical results fully demonstrate the advantage of DAFTC in control accuracy. This improvement can be attributed to the distributed control structure, which allows for individualized force allocation among carriages and adaptive parameter tuning in real time.

These findings align with and extend previous studies on distributed train control and fault-tolerant systems. The integration of a second-order auxiliary system successfully mitigates the adverse effects of input saturation, while the adaptive laws ensure robustness against uncertain dynamics and actuator degradation. The results also highlight the importance of considering heterogeneity among carriages—a factor often oversimplified in centralized designs. While the controller is designed to effectively manage partial and recoverable saturation, it should be acknowledged that its performance is bounded under the extreme condition of persistent, full-train saturation.

5. Conclusions

This paper has proposed a distributed adaptive fault-tolerant control (DAFTC) scheme for high-speed trains based on a carriage-level multi-body dynamics model. The scheme effectively handles system nonlinearities, parametric uncertainties, and external disturbances. Rigorous theoretical analysis using Lyapunov methods guarantees the stability of the closed-loop system under actuator faults. Simulation results demonstrate that, compared to the centralized fault-tolerant control method, the proposed distributed adaptive fault-tolerant control scheme reduces the average speed tracking error and average position tracking error by approximately 78% and 72%, respectively, enhancing the tracking accuracy and system resilience of high-speed trains under actuator faults and input saturation conditions. Future work could explore the integration of communication delays between carriages, more complex fault scenarios, and the application of data-driven methods to further enhance the adaptability and scalability of the proposed control framework.