Improved Active Disturbance Rejection Speed Tracking Control for High-Speed Trains Based on SBWO Algorithm

Abstract

To address the problems of random noise interference, inadequate disturbance estimation and compensation, and the difficulty in controller parameter tuning in speed tracking control of high-speed trains, an improved Active Disturbance Rejection Control (ADRC) strategy combined with a Sobol-based Black Widow Optimization (SBWO) algorithm is proposed. An improved Tracking Differentiator (TD) is adopted by integrating a novel optimal control synthesis function with a phase compensator to suppress input noise and ensure a smooth transition process. A novel Extended State Observer (ESO) using a nonlinear saturation function is designed to improve the observation accuracy and decrease chattering. An enhanced Nonlinear State Error Feedback (NLSEF) law that incorporates an error integral and adaptive parameter update laws is developed to reduce steady-state error and achieve self-tuned proportional and derivative gains. A feedforward compensation term is added to provide real-time dynamic compensation for ESO estimation errors. Finally, an enhanced Black Widow Optimization (BWO) algorithm, which initializes its population with Sobol sequences to improve its global search capability, is employed for parameter optimization. The simulation results demonstrate that compared with the control methods based on Proportional–Integral–Derivative (PID) control and conventional ADRC, the proposed strategy achieves higher steady-state tracking accuracy, better adaptability to dynamic operating conditions, stronger anti-disturbance ability, and more precise stopping precision.

1. Introduction

Possessing notable advantages such as high efficiency, safety, and environmental friendliness, high-speed rail has become a strategic pillar in the evolution and modernization of global transportation systems. At present, the operation of high-speed trains predominantly relies on manual driving, which is increasingly revealing its limitations in enhancing operational efficiency and accommodating the complex demands of modern high-speed rail networks. Consequently, to align with the trend towards the intelligentization of modern railways, the implementation of Automatic Train Operation (ATO) technology has emerged as a pivotal research direction. Research on ATO technology primarily concentrates on two aspects: the optimization of the target speed profile and the design of the tracking controller for tracking the profile. Substantial advancements have been achieved in the domain of profile optimization [1,2,3,4]; however, the optimization of target profiles will lose its significance without the support of a highly effective tracking control strategy. Therefore, the design of an efficient and precise tracking control strategy constitutes the central challenge in the realization of ATO for high-speed trains [5].

Extensive research has been conducted on train speed tracking control strategies. Pu et al. [6] proposed a train speed control algorithm based on an adaptive Proportional–Integral–Derivative (PID), which achieved good tracking performance. However, it exhibited poor robustness and anti-disturbance capability in complex disturbance environments. Xu et al. [7] designed a model predictive control algorithm capable of accurately tracking the speed profile, but the method is highly dependent on a precise model. To address this, Wang et al. [8] and Kang et al. [9] introduced train speed tracking control methods that incorporate fuzzy logic. Although this approach does not require a precise model, the formulation of fuzzy rules relies heavily on engineering experience, resulting in poor adaptability and hindering real-time tracking. Liu et al. [10] proposed a fault-tolerant control method based on adaptive backstepping that utilizes a novel time-varying function to address unknown time-varying uncertainties, but it exhibits insufficient disturbance rejection capability. When subjected to the effects of multi-source disturbances in complex dynamic environments, the existing control strategies still suffer from issues such as strong model dependency, limited adaptability, insufficient disturbance rejection capability [11,12], and so on. Therefore, there is an urgent need to develop a train speed tracking controller that is independent of a precise model, provides high control accuracy, and exhibits strong adaptability and disturbance rejection capabilities.

To address issues such as excessive reliance on precise mathematical models, difficulty in adapting to time-varying uncertainties during train operation, and insufficient disturbance rejection capability due to a lack of real-time estimation and compensation mechanisms, Scholars have gradually shifted their focus to train speed tracking control methods that do not rely on precise models and have strong anti-disturbance capabilities. Among these, Active Disturbance Rejection Control (ADRC) has garnered significant attention for its independence from precise mathematical models, its ability to estimate and compensate for total disturbance online, and its superior dynamic performance and robustness compared to traditional methods. ADRC has been widely applied in various fields, including quadrotor UAV flight control [13], robot and manipulator control [14], electric vehicle control [15], motor drive control [16], and so on.

However, application of ADRC in the field of train speed tracking control remains relatively limited [17,18,19]. In [17], a train speed tracking controller based on ADRC was designed, and its effectiveness was validated. A cooperative ADRC method was proposed in [18] for heavy-haul trains; it employs an Extended State Observer (ESO) for the online estimation and compensation of disturbances, thereby effectively mitigating speed tracking errors and overshoot in inter-train spacing. By integrating Model Reference Adaptive Control (MRAC) with ADRC, Yang et al. [19] developed a robust control method with low model dependency for magnetic levitation (Maglev) trains, which enhanced the controller’s disturbance rejection capabilities. Although the above speed tracking control strategies based on ADRC have shown good steady-state and dynamic performance, they are still beset by several limitations: the adopted conventional ADRC exhibits an insufficient capacity for the real-time, precise compensation of time-varying external disturbances [17]; the optimal control synthesis function employed by the Tracking Differentiator (TD) is structurally complex and is susceptible to issues including overshoot, chattering, and phase delay when designing the transition profile of the input signal [17]; the conventional nonlinear saturation function utilized within the ESO is non-differentiable at the origin and exhibits poor smoothness, which leads to excessively high observer gains and subsequently provokes frequent error switching and chattering phenomena [18]; and the Nonlinear State Error Feedback (NLSEF) law has demonstrated subpar steady-state performance and speed tracking precision in train speed control [19].

Moreover, a significant challenge inherent in ADRC-based speed tracking control strategies is the difficulty in tuning the numerous controller parameters. To address this issue, swarm intelligence algorithms, such as Particle Swarm Optimization (PSO) [20], Genetic Algorithms (GAs) [21], and Ant Colony Optimization (ACO) [22], have been employed to facilitate parameter optimization. These optimization approaches, however, remain susceptible to premature convergence to local optima and exhibit slow convergence rates. Although the conventional Black Widow Optimization (BWO) algorithm [23,24] alleviates these concerns to some degree through its distinctive population update mechanism, it still exhibits limitations concerning its population diversification and global exploration capability [25]. Accordingly, Yang et al. [26] employed an improved PSO algorithm to optimize the parameters of the ADRC controller. Although this improved PSO algorithm enhances the global search capability, it remains susceptible to premature convergence to local optima and exhibits slow convergence rates. Furthermore, the controller with parameters obtained via PSO shows limited adaptability to the complex and unknown variations in the external environment.

Based on the foregoing considerations, this paper proposes a novel speed tracking control strategy for high-speed trains, which is founded upon an improved adaptive active disturbance rejection controller augmented with feedforward compensation and optimized using an improved BWO algorithm. The key contributions are structured as follows. First, the TD is enhanced by introducing a novel optimal control synthesis function in conjunction with a phase compensator. Second, an improved ESO and an NLSEF control law are designed based on the caufal function; concurrently, the NLSEF control law is further enhanced with an adaptive parameter update law, which enables the self-tuning of the proportional and derivative coefficients. In addition, a feedforward compensation mechanism is incorporated to bolster the system’s real-time disturbance compensation capability. Finally, to circumvent the problem of local convergence arising from insufficient population coverage, an improved BWO algorithm initialized with Sobol sequences is employed for controller parameter optimization. The efficacy of the proposed controller is subsequently validated through simulations conducted in MATLAB R2022a.

The remainder of this paper is organized as follows. Section 2 establishes the operational model of the high-speed train. Section 3 presents the proposed speed tracking controller, which is based on an improved adaptive ADRC framework augmented with feedforward compensation. In Section 4, an improved BWO algorithm employing Sobol sequences is designed for the optimization of the controller parameters. Section 5 provides the simulation results to validate the efficacy of the proposed control strategy. Finally, Section 6 concludes the paper.

2. High-Speed Train Dynamic Model

In comparison with multi-particle dynamic models, the single-particle model is better suited for analyzing the train’s overall operational characteristics [27]. The corresponding force analysis for the single-particle model is depicted in Figure 1.

Based on Newton’s second law, the single-particle dynamic model of a train is formulated as

$$\left\{\begin{array}{l}\dot{x}=v\\ \dot{v}={\displaystyle \frac{u-P}{m(1+\rho )}}\end{array}\right.$$

(1)

where x is the train displacement; v is the train speed; u represents the control input, corresponding to the traction force (traction state, the same as the running direction) or braking force (braking state, opposite to the running direction), which are mutually exclusive; m is the train mass; $\rho$ is the wheel rotation mass coefficient; and P denotes the total disturbance, which is expressed as

$$P=w_0+w_{\mathrm{j}}+d\left(t\right)$$

(2)

where $d\left(t\right)$ represents the external disturbances, and $w_0$ and $w_{\mathrm{j}}$ are the basic running resistance and the additional resistance, which can be expressed, respectively, as

$$w_0=fmg/{10}^3$$

(3)

$$w_{\mathrm{j}}=w_{\mathrm{r}}+w_{\mathrm{s}}+w_{\mathrm{e}}$$

(4)

where $f=r_{\mathrm{a}}+r_{\mathrm{b}}v+r_{\mathrm{c}}{v}^2$, $r_{\mathrm{a}}$, $r_{\mathrm{b}}$, and $r_{\mathrm{c}}$ are the coefficients of rolling mechanical resistance, frictional resistance, and aerodynamic resistance, respectively; $w_{\mathrm{r}}=mg\sin \theta$, $w_{\mathrm{s}}=10.5{\alpha }_{\mathrm{s}}mg/(1000l_{\mathrm{s}})$, and $w_{\mathrm{e}}=0.000131l_{\mathrm{e}}mg/{10}^3$ denote the additional resistances due to the track gradient, curvature, and tunnels, respectively [26], in which $\theta$ is the grade angle, $l_{\mathrm{s}}$ is the curve length, ${\alpha }_{\mathrm{s}}$ is the central angle of the curve, and $l_{\mathrm{e}}$ is the tunnel length.

3. Speed Tracking Controller Design Based on Improved ADRC

Active Disturbance Rejection Control, which does not rely on a precise model and possesses strong disturbance rejection capabilities, has been employed to design train speed tracking controllers [17]. However, when confronted with the complex nonlinear system and significant disturbances inherent in train operations, conventional ADRC exhibits several deficiencies [28], such as an insufficient ability to manage the transient process of the input signal, inadequate performance in disturbance estimation and compensation, poor adaptability to changes in the external environment, and so on. To this end, this paper proposes an improved ADRC strategy for train speed tracking, the system architecture of which is depicted in the block diagram in Figure 2.

3.1. Tracking Differentiator Design

When designing the transient process for the input signal, the conventional TD exhibits significant drawbacks, including overshoot, chattering, phase delay, and so on [29]. To overcome these limitations, an improved TD constructed with a composite control architecture is proposed. The architecture is developed by first determining the boundary layer of a time-optimal discrete second-order system via the isochronous region method [30] and then combining this layer with the reachable set. It employs a mechanism integrating multi-step prediction, boundary layer adjustment, and a signum-function-based decision logic. The resulting controller incorporates a novel optimal control synthesis function and a phase compensation mechanism, with its discrete expression given by

$$\left\{\begin{array}{l}{v}_1\left(k+1\right)=v_1\left(k\right)+h{v}_2\left(k\right)\\ v_2\left(k+1\right)=v_2\left(k\right)+hq\left(k\right)\\ q\left(k\right)=\mathrm{fort}\left(v_1\left(k\right)-v_{\mathrm{d}}\left(k\right),v_2\left(k\right),r,h_0\right)\end{array}\right.$$

(5)

where $v_{\mathrm{d}}$ is the desired input, $v_1$ is the tracking signal of $v_{\mathrm{d}}$, $v_2$ is the derivative of $v_1$, $h$ is the sampling period, $r$ is the speed factor, $h_0$ is the filtering factor, and fort is a novel optimal control synthesis function, which is invoked as

$$\left\{\begin{array}{l}\Delta =r\cdot h^2\\ \chi =v_1+h\cdot v_2\\ \sigma =v_1+2h\cdot v_2\\ s_{\chi }=\left(\mathrm{sgn}(\chi +\Delta )-\mathrm{sgn}(\chi -\Delta )\right)/2\\ s_{\sigma }=\left(\mathrm{sgn}(\sigma +\Delta )-\mathrm{sgn}(\sigma -\Delta )\right)/2\\ w=s_{\chi }\cdot s_{\sigma }\\ n=v_1+{\displaystyle \frac{v_2\cdot \left|v_2\right|}{2r}}+{\displaystyle \frac{3}{2}}h\cdot v_2+{\displaystyle \frac{1}{2}}{h}^2\cdot r\cdot \mathrm{sgn}(v_2)\\ {\Delta }_0=h\cdot \left|v_2\right|+r\cdot h^2/2\\ s_n=\left(\mathrm{sgn}(n+{\Delta }_0)-\mathrm{sgn}(n-{\Delta }_0)\right)/2\\ q=-r\left({\displaystyle \frac{w\cdot \sigma }{\Delta }}+(1-w)\left(s_n\cdot {\displaystyle \frac{\gamma }{{\Delta }_0}}+(1-s_n)\cdot \mathrm{sgn}(n)\right)\right)\end{array}\right.$$

(6)

To further address the phase-delay problem that occurs during the processing of noisy signals, a phase compensator is integrated to predict the tracking and differential signals output by the TD and to correct the phase lag [31]. The algorithm is formulated as follows:

$$\left\{\begin{array}{l}{v}_1(k+1)=v_1(k)+h{v}_2(k)\\ v_2(k+1)=v_2+hq(k)\\ v_1^{\prime }(k+1)=v_1(k+1)+\vartheta h{v}_2(k+1)\\ v_2^{\prime }(k+1)=v_2(k+1)+\vartheta hq(k)\end{array}\right.$$

(7)

where $\vartheta$ is the phase compensator parameter.

To validate the tracking performance of the novel TD, a comparative simulation was conducted for two cases with noisy square wave and sinusoidal wave inputs, and the simulation results for the tracking performance are shown in Figure 3. The frequency of both input signals is $1/(2\pi )$ Hz, and the noise is Gaussian white noise with a signal-to-noise ratio of 40 dB. As can be observed, compared with the conventional TD based on the fhan function, the new TD using the improved fort function adapts more rapidly to the dynamic transients of the signal, effectively mitigating the lag effect. It compensates for the phase delay while effectively filtering out the noise, thereby exhibiting a superior dynamic response.

3.2. Extended State Observer Design

An ESO can estimate the unknown disturbances and unmodeled dynamics of a system in real time [32]. However, the conventional ESO, employing the fal function, is characterized by a slow convergence speed and limited anti-disturbance performance, rendering it unable to satisfy the high-precision and real-time control requirements of high-speed trains [17]. To effectively suppress chattering near the origin, improve the global convergence speed, and ensure stability under dynamic disturbances, an ESO based on an improved caufal function is introduced in this paper.

The expression for the fal function is

$$\mathrm{fal}(e,\alpha ,\tau )=\left\{\begin{array}{c}{\displaystyle \frac{e}{{\tau }^{1-\alpha }}}|e|\le \tau \hfill \\ |e{|}^{\alpha }\mathrm{sgn}(e)|e|>\tau \end{array}\right.$$

(8)

where τ determines the width of the nonlinear interval in the fal function. If τ is excessively large, the fal function degenerates into a linear function, defeating the original purpose of its non-smooth feedback design. Conversely, if τ is too small, the nonlinearity becomes overly strong, leading to chattering that causes the system to oscillate around the origin. Therefore, the selection of τ requires a trade-off between the effective utilization of nonlinearity and the avoidance of adverse effects, such as chattering or linear degradation.

Given the continuous and non-smooth characteristic of the fal function, measurement noise in the feedback loop may induce chattering due to the abrupt change in the function’s derivative [33]. To address this, a Dirac function is introduced to smooth the fal function, which satisfies the following conditions:

$$\forall x\ne 0,\delta (x)=0$$

(9)

$$\delta (0)\to \infty$$

(10)

$${\displaystyle {\int }_{-\infty }^{\infty }\delta (x)\mathrm{d}x}=1$$

(11)

$\delta (x)$ can be considered a continuous probability density function.

Assume that there exists a smooth approximation function $\phi \left(x\right)$ such that $\phi \left(x\right)\approx \delta \left(x\right)$. Then

$$g(x)={\displaystyle {\int }_{-\infty }^{+\infty }f(y)\phi (x-y)\mathrm{d}y}\approx f(x)$$

(12)

The function $\delta (x)$ is chosen as

$$\delta (x)={\displaystyle \frac{1}{\pi }}\underset{l\to 0}{\lim }{\displaystyle \frac{l}{x^2+l^2}}$$

(13)

Based on the above analysis, the smooth approximation function $\mathrm{caufal}(x,\alpha ,\tau )$ can be expressed as

$$\mathrm{caufal}(x,\alpha ,\tau )={\displaystyle {\int }_{-\infty }^{+\infty }\frac{1}{\pi }\frac{l}{{(x-y)}^2+l^2}\mathrm{fal}(y,\alpha ,\tau )\mathrm{d}y}\approx \mathrm{fal}(x,\alpha ,\tau )$$

(14)

To validate the control performance of the caufal function, a simulation is conducted for the case where parameter settings are $\tau =0.05$, $\alpha =0.25$, $l=0.02$. Figure 4 and Figure 5 show the characteristic curves and sensitivity curves for the fal and caufal functions, respectively. It can be observed that the caufal function exhibits superior smoothness, continuity, and convergence when the error approaches zero. In the small-error region, its sensitivity is lower than that of the fal function, which mitigates the chattering problem caused by excessive sensitivity while also preventing dynamic overshoot.

The total disturbance of system P is expanded into a new state variable, and the following third-order nonlinear ESO is constructed:

$$\left\{\begin{array}{l}{e}_{01}=z_1-y\hfill \\ {\dot{z}}_1=z_2-{\beta }_1{e}_{01}\hfill \\ {\dot{z}}_2=z_3-{\beta }_2\mathrm{caufal}(e_{01},{\alpha }_1,\tau )+b_{0}u\hfill \\ {\dot{z}}_3=-{\beta }_3\mathrm{caufal}(e_{01},{\alpha }_2,\tau )\hfill \end{array}\right.$$

(15)

where y is the system output; z₁, z₂, and z₃ are the estimations of the speed, acceleration, and the total disturbance of the system, respectively; b₀ is the compensation factor, given by $b_0={\displaystyle \frac{1}{m(1+\rho )}}$; and β₁, β₂, and β₃ are the observer gains.

Assuming the system disturbance is zero, the ESO error state equation can be written as

$$\left\{\begin{array}{l}{\dot{e}}_{01}=e_{02}-{\beta }_1{e}_{01}\hfill \\ {\dot{e}}_{02}=e_{03}-{\beta }_2\mathrm{caufal}(e_{01},{\alpha }_1,\tau )\hfill \\ {\dot{e}}_{03}=-{\beta }_3\mathrm{caufal}(e_{01},{\alpha }_2,\tau )\hfill \end{array}\right.$$

(16)

To prove the asymptotic stability of the ESO error system at the equilibrium point, the system can be transformed into

$$\dot{\mathit{e}}=-\mathit{A}\mathsf{(}\mathit{e}\mathsf{)}\mathit{e}$$

(17)

where $\mathit{e}={[e_{01},e_{02},e_{03}]}^{\mathrm{T}}$, and $\mathit{A}\mathsf{(}\mathit{e}\mathsf{)}=\left[\begin{array}{ccc}{\beta }_1& -1& 0\\ {\beta }_2{F}_1& 0& -1\\ {\beta }_3{F}_2& 0& 0\end{array}\right]$, in which $F_1={\displaystyle \frac{\mathrm{caufal}(e_{01},{\alpha }_1,\tau )}{e_{01}}}$, $F_2={\displaystyle \frac{\mathrm{caufal}(e_{01},{\alpha }_2,\tau )}{e_{01}}}$.

Let $B={\beta }_1{\beta }_2{F}_1-{\beta }_3{F}_2$. Obviously $F_1>0$, $F_2>0$. When $B>0$, the resulting matrix $\mathit{D}$ has all positive diagonal elements, making $\mathit{D}\mathit{A}\mathsf{(}\mathit{e}\mathsf{)}$ a symmetric positive-definite matrix.

$$\mathit{D}=\left[\begin{array}{ccc}1& {\displaystyle \frac{{\beta }_2{F}_1}{B}}+{\lambda }_1& -\eta \\ -{\displaystyle \frac{{\beta }_2{F}_1}{B}}-{\lambda }_1& \eta & {\displaystyle \frac{1}{B}}+{\lambda }_2\\ \eta & -{\displaystyle \frac{1}{B}}-{\lambda }_2& \eta \end{array}\right]$$

(18)

where ${\lambda }_1,{\lambda }_2$, and $\eta$ are infinitesimal positive constants.

To prove the stability of the improved ESO, a Lyapunov function is constructed in the following form:

$$V(t)={\displaystyle {\int }_0^{\mathrm{t}}\left(\mathit{D}\mathit{A}\mathsf{(}\mathit{e}\mathsf{)}\mathit{e},\dot{\mathit{e}}\right)\mathrm{d}t}+C_0$$

(19)

Expanding Equation (19) gives

$$V(t)={\displaystyle {\int }_0^{\mathrm{t}}\left[-{({\beta }_1{e}_{01}-e_{02})}^2-\eta {({\beta }_2{F}_1-e_{03})}^2-\eta {({\beta }_3{F}_2)}^2\right]}\mathrm{d}t+C_0$$

(20)

When $\eta \to 0$, it follows that

$$V(t)={\displaystyle {\int }_0^{\mathrm{t}}-{({\beta }_1{e}_{01}-e_{02})}^2\mathrm{d}t}+C_0$$

(21)

The integral ${\displaystyle {\int }_0^t{({\beta }_1{e}_{01}-e_{02})}^2}\mathrm{d}t$ is bounded at the equilibrium point. Therefore, if $C_0$ is sufficiently large, $V(t)>0$.

Differentiating $V(t)$ with respect to t yields

$$\dot{V}(t)=-{({\beta }_1{e}_{01}-e_{02})}^2-\eta {({\beta }_2{F}_1-e_{03})}^2-\eta {({\beta }_3{F}_2)}^2\le 0$$

(22)

From Equations (21) and (22), when B > 0, the system is asymptotically stable at the equilibrium point, and the observer gains are positive and satisfy ${\beta }_1{\beta }_2{F}_1-{\beta }_3{F}_2>0$, thereby guaranteeing asymptotic stability of the ESO at the equilibrium point.

3.3. NLSEF Control Law Design Based on Adaptive Parameter Update Law

To address the poor steady-state performance and insufficient speed tracking accuracy in traditional nonlinear control laws for train speed tracking control, an error integral term is introduced into the conventional NLSEF control law. This modification makes μ₃ analogous to the integral coefficient of a PID controller, thereby optimizing the nonlinear control. A novel NLSEF control law is then designed by incorporating the caufal function, yielding the control input as

$$u_0={\mu }_1\mathrm{caufal}(e_1,{\alpha }_3,\tau )+{\mu }_2\mathrm{caufal}(e_2,{\alpha }_4,\tau )+{\mu }_3\mathrm{caufal}(e_3,{\alpha }_5,\tau )$$

(23)

where $e_1=v_1-z_1$, $e_2=v_2-z_2$, and $e_3={\int }_0^t{e}_1(\tau )\mathrm{d}\tau$. ${\mu }_1$, ${\mu }_2$, and ${\mu }_3$ are the proportional, differential, and integral coefficients, respectively. ${\alpha }_3$, ${\alpha }_4$, and ${\alpha }_5$ are adjustable parameters. ${\alpha }_3$, ${\alpha }_4$, and ${\alpha }_5$ satisfy the following constraint: $0\text{<}{\alpha }_5<{\alpha }_3<1<{\alpha }_4$.

To further improve the adaptability and the disturbance rejection capability of the control strategy, an adaptive parameter update law is introduced to dynamically adjust the key parameters (${\mu }_1$ and ${\mu }_2$) in the NLSEF control law.

The displacement and speed tracking errors of the high-speed train are defined as $e_x=x-x_{\mathrm{d}}$ and ${\dot{e}}_x=v-v_{\mathrm{d}}$, respectively, where $x_{\mathrm{d}}$ and $v_{\mathrm{d}}$ are the desired displacement and speed. The adaptive parameter estimation errors are defined as ${\tilde{\mu }}_1={\mu }_1-{\widehat{\mu }}_1$ and ${\tilde{\mu }}_2={\mu }_2-{\widehat{\mu }}_2$.

The tracking error function $\kappa$ for the adaptive parameter update law is defined as

$$\kappa =\zeta e_x+{\dot{e}}_x$$

(24)

where $\zeta$ is a properly selected constant.

The adaptive parameter update laws for ${\mu }_1$ and ${\mu }_2$ are designed as

$${\widehat{\mu }}_1={\displaystyle {\int }_0^T\left\{{\phi }_{\mathrm{p}}[\kappa {\widehat{\mu }}_1\left(v_1-z_1\right)]/{\tilde{\mu }}_1\right\}\mathrm{d}t+{\mu }_{10}}$$

(25)

$${\widehat{\mu }}_2={\displaystyle {\int }_0^T\left\{{\phi }_{\mathrm{d}}[\kappa {\widehat{\mu }}_2\left(v_2-z_2\right)]/{\tilde{\mu }}_2\right\}\mathrm{d}t+{\mu }_{20}}$$

(26)

where ${\phi }_{\mathrm{p}}$ and ${\phi }_{\mathrm{d}}$ are designed constants, ${\mu }_{10}$ and ${\mu }_{20}$ are the initial parameter values, which are implemented to avoid potential singularity problems in the adaptive update law, ensuring that the variables ${\tilde{\mu }}_1$ and ${\tilde{\mu }}_2$ do not approach zero during the iteration process [34].

Thus, the control input $u_0$ can be rewritten as

$$u_0={\widehat{\mu }}_1\mathrm{caufal}(e_1,{\alpha }_3,\tau )+{\widehat{\mu }}_2\mathrm{caufal}(e_2,{\alpha }_4,\tau )+{\mu }_3\mathrm{caufal}(e_3,{\alpha }_5,\tau )$$

(27)

3.4. Feedforward Compensation Design

The estimation errors of the ESO can introduce additional disturbance into the system’s double-integrator structure. To further improve the disturbance rejection capability and tracking performance of the control system, a feedforward compensation term is introduced, which is expressed as

$$\left\{\begin{array}{l}F=k_{\mathrm{c}}(e_1+e_2+e_3)\hfill \\ e_1=v_1-z_1\hfill \\ e_2=v_2-z_2\hfill \\ e_3={\displaystyle {\int }_0^{\mathrm{t}}{e}_1(\tau )\mathrm{d}\tau }\hfill \end{array}\right.$$

(28)

where $k_{\mathrm{c}}$ is the feedforward compensation coefficient.

The second-order dynamic model of the high-speed train can be given by

$$\dot{v}=P+b_{0}u=P+b_0(u_0-{\displaystyle \frac{z_3}{b_0}}+F)=b_0{u}_0+b_{0}F+(P-z_3)$$

(29)

When an error exists between the disturbance estimated by the ESO and the actual disturbance, i.e., $P\ne z_3$, the compensation mechanism can form a dual disturbance rejection architecture combining feedforward compensation and feedback regulation. This structure can offset a portion of the disturbance estimation error, thereby achieving effective disturbance compensation.

Combining Equations (27) and (28), the control input for the train can be designed as

$$\begin{array}{l}u=u_0-{\displaystyle \frac{z_3}{b_0}}+F={\widehat{\mu }}_1\mathrm{caufal}(e_1,{\alpha }_3,\tau )+{\widehat{\mu }}_2\mathrm{caufal}(e_2,{\alpha }_4,\tau )+{\mu }_3\mathrm{caufal}(e_3,{\alpha }_5,\tau )-{\displaystyle \frac{z_3}{b_0}}\\ +k_{\mathrm{c}}(v_1-z_1+v_2-z_2+{\displaystyle {\int }_0^{\mathrm{t}}{e}_1(\tau )\mathrm{d}\tau })\end{array}$$

(30)

4. Parameter Optimization Based on SBWO

To address the issues of swarm intelligence optimization algorithms being prone to local optima and exhibiting slow convergence during parameter tuning for the ADRC speed tracking controller, a Sobol-based Black Widow Optimization (SBWO) algorithm is proposed to enhance the population update and global search capabilities.

To effectively generate uniformly distributed sample points in a high-dimensional search space, the Sobol sequence is introduced to generate the initial positions of individuals in the population [23], which can be represented as

$$P_i=S_{\mathrm{L}}+G_i\cdot (S_{\mathrm{U}}-S_{\mathrm{L}})$$

(31)

where $P_i$ denotes the generated position coordinate, $\left[S_{\mathrm{L}},S_{\mathrm{U}}\right]$ is the search range, and $G_i\in [0,1]$ represents the random number generated by the Sobol sequence.

To validate the influence of the population initialization strategy on algorithm performance, 500 individuals are initialized using a Sobol sequence and traditional pseudo-random numbers, respectively. These individuals are then mapped into a two-dimensional search space within the range of [−1, 1] to compare their position distributions [35]. The simulation results are shown in Figure 6. As can be observed, the SBWO algorithm based on the Sobol sequence, owing to its superior ergodicity and uniformity, accelerates the convergence of the algorithm and enhances its global exploration capability [36], thereby effectively avoiding entrapment in local optima.

The Integral of Time-weighted Absolute Error (ITAE) criterion is adopted as the fitness function J_ITAE, which is formulated as follows:

$$J_{\mathrm{ITAE}}={\displaystyle {\int }_0^{\infty }t|e(t)|\mathrm{d}t}$$

(32)

where e(t) is the speed tracking error.

The schematic diagram of the ADRC controller optimization process based on the SBWO algorithm is shown in Figure 7.

To demonstrate the parameter optimization performance of SBWO, a comparative simulation was conducted against the PSO algorithm, as shown in Figure 8. It is evident from the figure that the SBWO algorithm converges to a smaller optimal fitness value within approximately ten generations. Compared to the PSO algorithm, the SBWO algorithm demonstrates a faster convergence speed, superior global search capability, and greater stability.

5. Simulation Results and Analysis

5.1. Parameter Selection

To validate the performance of the designed controller, simulations are conducted. The train parameters are as follows [13]: m = 452,300 kg, f = 0.16 + 0.0053v + 0.00018v² (N/kN), $\rho =0.11$, and $d\left(t\right)=\sin (0.077t)mg/{10}^3$(N). The key controller parameters $r$, $h_0$, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, $\tau$, ${\mu }_3$, and $k_{\mathrm{c}}$ are automatically tuned using the SBWO algorithm, whereas ${\mu }_1,{\mu }_2$ are adaptively adjusted based on an adaptive parameter update law.

The additional running resistance is given as

$$w_{\mathrm{j}}=\left\{\begin{array}{l}{w}_e+w_s,t\in [0,400)\hfill \\ w_e+w_s+w_r,t\in [400,1000)\hfill \\ w_e+w_s+w_r+o(\cdot ),t\in [1000,1800)\hfill \\ \begin{array}{l}{w}_s+w_r+o(\cdot ),t\in [1800,2400)\\ w_r,t\in [2400,3000]\end{array}\hfill \end{array}\right.$$

(33)

The track parameters are set as follows: $l_{\mathrm{e}}=5000\mathrm{m}$, $\theta =\mathsf{\pi }/180\mathrm{rad}$ ${\alpha }_{\mathrm{s}}=2\mathsf{\pi }/3\mathrm{rad}$, $l_{\mathrm{s}}=400\mathrm{m}$, and $o(\cdot )=50\sin (0.02vt)$.

5.2. Simulation Results and Analysis of the Control Strategy

The simulation is configured for an operating profile that includes two acceleration phases, three cruising phases, and two deceleration phases. The maximum speed is 336 km/h, with a total running time of 3000 s and a distance of 239.148 km. The desired speed and displacement curves are shown in Figure 9.

To validate the performance of the designed controller, the proposed speed tracking control strategy based on improved ADRC is compared with strategies using traditional ADRC [13] and PID control [37] through simulations. The results are presented in Figure 10. It can be seen that, during steady-state operation, the speed and displacement tracking errors of the improved ADRC are maintained within [−1.6 × 10⁻⁴, 1.2 × 10⁻⁴] km/h and [−0.14, −0.13] m, respectively, achieving the highest steady-state tracking accuracy; when operating conditions change, the improved ADRC responds more rapidly and ensures a smoother transition, demonstrating better adaptability to dynamic conditions; when the conditions of the line vary, the proposed strategy exhibits the minimum tracking error fluctuation, indicating strong anti-disturbance capability; and when stopping, the displacement error with the improved ADRC is −0.0006 m, which is the smallest, thus yielding the highest stopping precision. The simulation results confirm that the proposed speed tracking controller based on improved ADRC performs excellently in terms of steady-state tracking accuracy, dynamic condition adaptability, anti-disturbance performance, and stopping precision, thereby validating the feasibility and effectiveness of the designed controller.

To quantitatively analyze the control performance of the improved ADRC, the root mean square error (RMSE) and the integral of absolute error (IAE) are introduced as speed error performance indexes. The definitions of RMSE and IAE are as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{n=1}^n{\left(v_{\mathrm{d}}(t)-v(t)\right)}^2}$$

(34)

$$IAE=\sum _{n=1}^n\left|v_{\mathrm{d}}(t)-v(t)\right|$$

(35)

The comparison of high-speed train speed error indexes based on the proposed improved ADRC, traditional ADRC, and PID speed tracking controllers is shown in

Table 1. Comparison of speed error indexes.

Controllers	RMSE	IAE
PID	0.058728	150.531431
Traditional ADRC	0.004564	4.101577
Improved ADRC	0.000619	0.651449

. It is evident that the proposed improved ADRC yields the lowest RMSE and IAE values. The result demonstrates that the improved ADRC exhibits superior overall tracking accuracy and control performance compared to both the conventional ADRC and the PID controller.

To further validate the robustness of the proposed control strategy against the system’s total disturbance, which comprises basic resistance, additional resistance, and external disturbances in practical train operation, a simulation experiment is conducted. In the simulation, the system’s total disturbance is increased, and the basic resistance coefficients are set as follows: $r_a=0.16+0.02\ast rand$, $r_b=0.0053+0.001\ast rand$, and $r_c=0.00018+0.00002\ast rand$, where rand is a function that generates a random number uniformly from the interval [0, 1]. The additional resistance coefficient θ is adjusted to 1.2 times its original value, and the external disturbance $d\left(t\right)$ is doubled. The results are presented in Figure 11, where it can be seen that even when the total disturbance is increased, the designed controller still enables the train to accurately track the desired trajectory throughout the entire operation. The simulation results indicate that the designed controller is highly robust against the system’s total disturbance and demonstrate the strong anti-disturbance capability of the system.

The simulation results for speed and total disturbance observation based on the improved ESO are presented in Figure 12. It is evident that, during train operation, the observation errors for both speed and total disturbance are small. The designed observer achieves an accurate estimation of the train’s speed and the total disturbance, which ensures the excellent steady-state and dynamic performance of the train.

To verify the impact of the improved ESO on the speed tracking performance of the train, a quantitative study was conducted. The simulation results are presented in

Table 2. Performance comparison of control strategies with different ESOs.

Controllers	RMSE	IAE
ADRC with traditional ESO	0.004564	4.101577
ADRC with improved ESO	0.001911	1.243221

, which shows that the control strategy incorporating the improved ESO achieves lower RMSE and IAE values. The simulation results indicate that the proposed method possesses superior speed tracking performance and disturbance rejection capability.

6. Conclusions

This paper presents an improved feedforward-compensated adaptive ADRC strategy, with its parameters optimized by the SBWO algorithm. An improved TD based on the fort function is introduced, providing a smooth transition for the train’s speed signal while filtering noise and compensating for phase delay; an ESO designed with a caufal function achieves precise estimation and compensation of composite disturbances; the NLSEF control law is enhanced by incorporating an error integral term through nonlinear operations, which reduces the steady-state tracking error, while an adaptive parameter update law enables the self-adjustment of its proportional and differential coefficients; and a feedforward compensation mechanism is integrated to provide real-time dynamic compensation for ESO estimation errors. The core controller parameters are tuned using the proposed SBWO algorithm, which initializes its population with the Sobol sequence and employs a diversified search strategy, thereby enhancing global search capability, convergence speed, and optimization accuracy. This strategy effectively improves the system’s anti-disturbance performance, overcomes the challenges of parameter tuning, and achieves high-precision tracking of the train’s target curve under multi-source disturbances in complex dynamic environments.