Data-Driven Bus Trajectory Tracking Based on Feedforward–Feedback Model-Free Adaptive Iterative Learning Control

Abstract

This paper presents a scheme for the feedforward–feedback longitudinal trajectory tracking control of buses. The scheme is specifically designed to address the periodic and repetitive nature of bus operations. First, the vehicle’s longitudinal dynamics are linearized along the iterative axis via full-form dynamic linearization (FFDL), and parameters such as the pseudo-gradient are estimated with data and a projection algorithm to grasp the dynamic characteristics of the system. To better handle complex real-world traffic conditions, we then propose the forward and backward structure. At the same time, the iterative axis design performance index is verified, and the forward partial control law, namely, model-free adaptive iterative learning control (MFAILC), is derived. In order to further enhance the robustness to disturbance and other factors, the control law of the feedback part is designed with active disturbance rejection control (ADRC). A key advantage of this control approach is its sole reliance on the data generated during vehicle operation, without the need for specific information about the controlled vehicle. This feature enables the method to be adaptable to different vehicle types and resilient to various disturbances. Finally, MATLAB simulations are used to verify the practicality of the proposed method.

1. Introduction

Over the past few years, there has been a continuous rise in vehicle numbers, leading to more severe problems like traffic safety risks, congestion, and environmental degradation, which in turn affect people’s daily commuting and travel. Finding safer, more efficient, and environmentally friendly transportation solutions has become crucial. The development of intelligent urban public transport systems represents a key direction in modern urban traffic planning, offering an effective solution to these challenges. Smart urban public transit systems enhance both the safety and efficiency of transportation networks, helping to lower fuel usage and ease traffic congestion. Therefore, it is not only a vital area of practical concern but also a significant field of research.

Controlling vehicle motion is a fundamental challenge in the study of intelligent vehicles. In this area, managing both lateral and longitudinal dynamics [1] is crucial for enabling autonomous driving. Vehicle lateral control research has mainly focused on studying the path-tracking ability of intelligent vehicles, such as controlling the steering to control the vehicle to follow a preplanned path [2]. Lateral control involves automatic steering to ensure the vehicle follows the desired path while optimizing comfort. Longitudinal control, on the other hand, centers on tracking the vehicle’s speed [3,4] and includes managing displacement (the distance to the target), speed, and acceleration. This area of longitudinal motion control plays a key role in autonomous bus studies, with notable progress being made in recent years.

The longitudinal motion control of a vehicle [5] regulates the vehicle’s longitudinal dynamics through control strategies, ensuring a safe distance between vehicles or enabling automatic adjustments in speed, such as acceleration and deceleration. The longitudinal motion control of intelligent vehicles requires high-speed control accuracy and minimal impact to achieve stable control, thus meeting ride comfort requirements.

Numerous studies on vehicle longitudinal motion control have been carried out both domestically and internationally. The control of longitudinal vehicle motion can be categorized into direct structure control and hierarchical structure control, depending on the implementation approach [6,7]. In the direct control structure [8,9], the control inputs for all subsystems are calculated by the longitudinal controller. In hierarchical structure control systems [10,11], the control layers are usually divided into upper and lower levels. The upper control layer is responsible for generating the target acceleration for the next moment based on the driving state, such as the vehicle distance and speed difference calculated by the vehicle’s sensors. The lower control layer adjusts the vehicle’s internal power system based on the target acceleration from the upper control layer, optimizing the vehicle’s acceleration.

A sliding mode control strategy using fuzzy logic has been suggested in the literature [8] to handle longitudinal vehicle speed, considering vehicle traits like parameter uncertainty and significant nonlinearity. In this approach, the control gain of the sliding mode controller is adjusted online using fuzzy logic to eliminate oscillations, handle system uncertainty and external disturbances, and ensure smooth switching between throttle and brake control laws. A longitudinal linear model for the vehicle, incorporating aggregate parameter characteristics, was established in [10]. To account for the time-varying nature of these parameters, a vehicle parameter time-varying adaptive speed control strategy was proposed, and its stability was assessed using Lyapunov’s direct method. The proposed technique’s effectiveness and robustness were verified through both simulation and experimental results. In [9], a system identification method was used to establish a longitudinal dynamics model, and an automatic longitudinal speed control strategy was proposed. This strategy combines a PD controller with an inverse dynamics model to compensate for the nonlinear vehicle characteristics. To overcome the longitudinal control issue during braking conditions, ref. [11] presented a lower-level control strategy based on variable parameter sliding mode control, which estimates the sliding mode control parameters in real time using the vehicle’s operational state information. Finally, ref. [12] introduced a control strategy based on active disturbance rejection control (ADRC) to improve the braking performance of the automobile’s anti-lock braking system (ABS). This strategy involved an ADRC control algorithm that was designed by analyzing and constructing models for the single-wheel dynamics, tire behavior, and pressure braking systems.

The longitudinal vehicle control requirements in the study of autonomous bus driving primarily arise from the need for reliable passenger travel times, along with stringent demands for punctuality and intelligent bus stations. These factors are essential in improving the operational efficiency of a bus system [13].

Nonetheless, model-based approaches face challenges in achieving effective control and accuracy in self-driving bus control. On the one hand, various factors affect the dynamics of the system during self-driving bus operation, making it challenging to establish an accurate dynamics model. On the other hand, the operation of a bus generates a substantial amount of I/O data, which provide an accurate reflection of the vehicle’s dynamic behavior. Consequently, investigating the timely parking control issue of autonomous buses using data-driven control approaches is highly significant when precise vehicle dynamic models are unavailable. This approach presents an innovative solution to address self-driving bus control problems.

Model-free adaptive control (MFAC) is a widely used data-driven approach [14]. For nonlinear discrete-time systems, it replaces the system with a dynamically linearized model at the current operating point. Furthermore, MFAC performs online estimation of the pseudo-bias derivatives in the dynamically linearized model, relying solely on the input and output data of the system. This enables model-free adaptive control [15]. MFAC has been utilized in several domains, including rehabilitation robotics [16], autonomous vehicles [17,18,19,20], urban transportation systems [21], motor drives [22], and robotic control [23]. Some of the latest MFAC applications can be found in [24,25,26,27]. Theoretical studies and real-world applications have shown that MFAC is both straightforward and resilient. Notably, it functions without requiring a system model, depending solely on the system’s input and output data, which increases its flexibility. For repetitive systems, branch, model-free adaptive iterative learning control (MFAILC) was proposed [21,24].

Iterative learning control (ILC) is a powerful control technique [28] designed for systems that involve periodic and repetitive tasks. It leverages error data from past operations along with the control input to modify the current control input, improving tracking accuracy and system convergence. ILC typically requires a known linear model and initial conditions for controller design. For unknown nonlinear systems, a model-free adaptive iterative learning control approach, which integrates ILC with MFAC and uses system I/O data, can improve system robustness. This method also ensures complete tracking control within a finite time and guarantees both system accuracy and the monotonic convergence of tracking error. In the tracking control of several nonlinear systems [29,30,31,32,33], MFAILC has been successfully employed. Practical applications, simulation studies, and theoretical analyses have shown that the MFAILC technique is effective in addressing control problems for systems exhibiting strong nonlinearity and time-varying characteristics.

Feedforward–feedback is a control structure that uses not only historical data to provide control form but also online data to respond quickly. In this paper, ADRC is selected for the feedback part. ADRC was developed from the PID principle of “eliminating errors through error correction”. It incorporates advances in modern control theory and is a practical technology that leverages special nonlinear effects [34,35,36,37]. In its early years, ADRC was extensively applied in areas like chemical processing, power systems, precision machining equipment, and advanced weapon systems in China. More recently, some studies [38,39,40] have applied ADRC to vehicle trajectory control in traffic systems.

Buses primarily operate in environments where they execute repetitive tasks in fixed sections periodically. The time for each bus run is also fixed and repetitive, with the driving time between any two fixed positions being constant and the cycle repeating with strong repeatability. Currently, a large body of both domestic and international research on longitudinal vehicle control is available. Although some data-driven control methods have been proposed, they have struggled to achieve satisfactory control performance in the longitudinal speed control of buses. Therefore, in this paper, we propose a feedforward–eedback MFAILC method based on full-format dynamic linearization for bus longitudinal control. The key contributions of this study, in comparison to previous work, are threefold:

(1)

Based on the full-format dynamic linearization (FFDL) method, an iterative dynamic linearization method of FFDL is extended and proposed.

(2)

Based on the FFDL-MFAC algorithm, combined with the design index of the FFDL iterative dynamic linearization method, a full-form model-free adaptive iterative learning control (FFDL-MFAILC) scheme is proposed.

(3)

To enhance the applicability of the FFDL-MFAILC scheme in complex public transportation environments and bolster the robustness of its control method, the scheme is integrated with ADRC and structured into a hierarchical control system. The FFDL-MFAILC feedforward and feedback control (FFDL-MFAFILC) algorithm is designed.

The remainder of this paper is structured as follows: Section 2 outlines the establishment of the longitudinal bus dynamics model. Section 3 presents the design of the full-format dynamic linearization, FFDL-MFAILC algorithm, and FFDL-MFAFILC algorithm. In Section 4, the superiority of FFDL-MFAFILC is validated through MATLAB simulations. Finally, Section 5 summarizes the conclusions of this study.

2. Problem Formulation

2.1. Longitudinal Vehicle Model

A simplified longitudinal dynamics model of the bus was developed for generating simulation data and comparing different methods. The forces acting along the vehicle’s longitudinal axis are illustrated in Figure 1. Where, the dotted line represents the horizontal line; $\theta$ represents the angle between the road and the horizontal line, that is, the slope of the road.

The acceleration of the vehicle is calculated from Newton’s second law as follows:

$$\dot{\nu }\left(t\right)={\displaystyle \frac{1}{m}}(F\left(t\right)-H\left(t\right))$$

(1)

where m represents the vehicle’s weight, $\nu \left(t\right)$ denotes the longitudinal velocity, and F(t) refers to the longitudinal force generated by the tires. H(t) represents the time-varying aggregated external disturbances, including factors such as the aerodynamic drag force $F_{aero}$ and the gravitational force $F_g$ caused by road gradient and rolling resistance $R_x$, as expressed below:

$$H\left(t\right)=F_{aero}\left(t\right)+F_g+R_x\left(t\right)$$

(2)

where

$$F_{aero}\left(t\right)={\displaystyle \frac{1}{2}}\rho C_d{A}_F{(\nu \left(t\right)+{\nu }_{wind}\left(t\right))}^2$$

$$F_g\left(t\right)=mgsin\theta$$

$$R_x\left(t\right)=f_{R}mgcos\theta$$

where ${\nu }_{wind}\left(t\right)$ is the wind speed, and $f_R$ is the friction coefficient.

Using the vehicle travel distance, denoted as s(t), the model for the longitudinal dynamics of the vehicle can be formulated as follows:

$$\begin{array}{cc}\hfill \dot{s}\left(t\right)& =\nu \left(t\right)\hfill \\ \hfill \dot{\nu }\left(t\right)& ={\displaystyle \frac{1}{m}}(F\left(t\right)-F_{aero}\left(t\right)-F_g-R_x\left(t\right))\hfill \\ \hfill & ={\displaystyle \frac{1}{m}}(F\left(t\right)-{\displaystyle \frac{1}{2}}\rho C_d{A}_F{(\nu \left(t\right)+{\nu }_{wind}\left(t\right))}^2-mgsin\theta -f_{R}mgcos\theta )\hfill \end{array}$$

(3)

2.2. Description Along the Iteration Domain and Assumptions

The design of iterative learning controllers is carried out in the iterative domain; therefore, in this section, ℓ, representing the iteration number, is added to the original nonlinear system for augmentation [41]. Considering the repetitive operational pattern of bus vehicles, we assume that, over a defined time interval [0, T], the bus repeatedly executes a series of actions. Under this assumption, the formulation of system (3) can be written as follows:

$$\begin{array}{cc}\hfill \dot{s}(\ell ,\vartheta )& =\nu (\ell ,\vartheta )\hfill \\ \hfill \dot{\nu }(\ell ,\vartheta )& ={\displaystyle \frac{1}{m}}(F(\ell ,\vartheta )-F_{aero}(\ell ,\vartheta )-F_g-R_x(\ell ,\vartheta ))\hfill \\ \hfill & ={\displaystyle \frac{1}{m}}(F(\ell ,\vartheta )-{\displaystyle \frac{1}{2}}\rho C_d{A}_F{(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2-mgsin\theta -f_{R}mgcos\theta )\hfill \end{array}$$

(4)

where ℓ = 1, 2,…signifies the count of recurrent operations, and $\vartheta \in \{0,1,\cdots ,\mathit{T}\}$ represents the sampling time instances.

Given the complex attributes of a bus control system, characterized by nonlinearity, time variation, and complexity within a discrete-time framework, the formulation of system (4) is expressed as follows:

$$\begin{array}{c}\hfill y(\ell ,\vartheta +1)=f(y(\ell ,\vartheta ),\dots ,y(\ell ,\vartheta -n_y),u(\ell ,\vartheta ),\dots ,u(\ell ,\vartheta -n_u))\end{array}$$

(5)

where the output y corresponds to the displacement distance s; the input u corresponds to the longitudinal force F.

Assumption 1. 

The partial derivative of $f(\dots )$ concerning the variable is continuous.

Assumption 2. 

The system (5) satisfies the generalized Lipschitz condition along the iteration axis, that is,

$$\begin{array}{c}\hfill \Delta y(\ell ,\vartheta +1)\le b\left|\right|\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )\left|\right|\end{array}$$

(6)

where $\Delta y(\ell ,\vartheta +1)=y(\ell ,\vartheta +1)-y(\ell -1,\vartheta +1),b>0$ is a constant,

$$\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )={\left[\Delta y(\ell ,\vartheta ),\dots ,\Delta y(\ell ,\vartheta -L_y+1),\Delta u(\ell ,\vartheta ),\dots ,\Delta u(\ell ,\vartheta -L_u+1)\right]}^T.$$

Remark 1. 

In this context, the assumptions imposed on the system are both reasonable and acceptable. In many standard control methods, a nonlinear system is required to meet the conditions of Assumption 1. Assumption 2 imposes an upper bound on the rate of change of the system output in response to variations in the control input along the iteration axis. From an energy standpoint, the input energy change is bounded, which in turn limits the corresponding change in output energy. This assumption holds for many practical systems, such as traffic control and temperature regulation systems.

3. Controller Design

3.1. Full-Format Dynamic Linearization Method for Iterative Domains

The impact of previous inputs, outputs, and latent time-varying parameters on the system is considered. In the context of trajectory tracking for buses performing repetitive tasks, historical control inputs and tracking error data from previous cycles can be used to adjust and optimize current control inputs, thereby improving the accuracy of the ongoing trajectory tracking process.

In this paper, a data-driven model-free adaptive iterative learning control (MFAILC) scheme is proposed. This control method is based on a novel dynamic linearization technique along the iteration axis. The partial-form dynamic linearization (PFDL) method is proposed for a class of repetitive nonlinear discrete-time SISO systems [41]. And, it is proved by theoretical analysis that MFAILC can guarantee monotonic convergence of the iterative maximum error. The full-format dynamic linearization (FFDL) method, an extension of PFDL, provides additional PG components and is better suited to the complex dynamics of a controlled system. The FFDL theorem applicable to the nonlinear motion system of buses throughout the iterative domain is presented below.

Theorem 1. 

For a nonlinear system satisfying Assumptions 1 and 2, given $L_y$ and $L_u$, when $\left|\right|\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )\left|\right|\ne 0$, there must exist an iteratively related vector of ${\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )$ called the pseudo-gradient (PG) such that the nonlinear system (5) is transformed into a full-format dynamic linearized data model on the iterative axis of the form

$$\begin{array}{c}\hfill \Delta y(\ell ,\vartheta +1)={\mathbf{\Phi }}_{L_y,L_u}^T(\ell ,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )\end{array}$$

(7)

and, for any moment ϑ, any number of iterations ℓ, ${\mathbf{\Phi }}_{L_y,L_u}$ are bounded. Where ${\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )={\left[{\mathbf{\Phi }}_1(\ell ,\vartheta ),\dots ,{\mathbf{\Phi }}_{L_y}(\ell ,\vartheta ),{\mathbf{\Phi }}_{L_{y+1}}(\ell ,\vartheta ),\dots ,{\mathbf{\Phi }}_{L_y+L_u}(\ell ,\vartheta )\right]}^T$

Proof of Theorem 1. 

$$\begin{array}{cc}\hfill & \Delta y(\ell ,\vartheta +1)=y(\ell ,\vartheta +1)-y(\ell -1,\vartheta +1)\hfill \\ \hfill & =f(y(\ell ,\vartheta ),y(\ell ,\vartheta -1),\cdots ,y(\ell ,\vartheta -n_y),u(\ell ,\vartheta ),u(\ell ,\vartheta -1),\cdots ,u(\ell ,\vartheta -n_u))-\hfill \\ \hfill & f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -n_y),u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),\cdots ,\hfill \\ \hfill & u(\ell -1,\vartheta -n_u))\hfill \\ \hfill & =f(y(\ell ,\vartheta ),y(\ell ,\vartheta -1),\cdots ,y(\ell ,\vartheta -n_y),u(\ell ,\vartheta ),u(\ell ,\vartheta -1),\cdots ,u(\ell ,\vartheta -n_u))-\hfill \\ \hfill & f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -L_y+1),y(\ell ,\vartheta -L_y),\cdots ,y(\ell ,\vartheta -n_y),\hfill \\ \hfill & u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),\cdots ,u(\ell -1,\vartheta -L_u+1),u(\ell ,\vartheta -L_u),u(\ell ,\vartheta -n_u))+\hfill \\ \hfill & f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -L_y+1),y(\ell ,\vartheta -L_y),\cdots ,y(\ell ,\vartheta -n_y),\hfill \\ \hfill & u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),\cdots ,u(\ell -1,\vartheta -L_u+1),u(\ell ,\vartheta -L_u),u(\ell ,\vartheta -n_u))-\hfill \\ \hfill & f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -n_y),u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),u(\ell -1,\vartheta -n_u))\hfill \end{array}$$

Denote

$$\begin{array}{cc}\hfill & \mathbf{\Psi }(\ell ,\vartheta )=f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -L_y+1),y(\ell ,\vartheta -L_y),\cdots ,y(\ell ,\vartheta -n_y),\hfill \\ \hfill & u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),\cdots ,u(\ell -1,\vartheta -L_u+1),u(\ell ,\vartheta -L_u),u(\ell ,\vartheta -n_u))-\hfill \\ \hfill & f(y(\ell -1,\vartheta ),y(\ell -1,\vartheta -1),\cdots ,y(\ell -1,\vartheta -n_y),u(\ell -1,\vartheta ),u(\ell -1,\vartheta -1),u(\ell -1,\vartheta -n_u))\hfill \end{array}$$

By Assumption 2 and Cauchy’s mean value Assumption 2, $\Delta y(\ell ,\vartheta +1)$ can be rewritten as

$$\begin{array}{cc}\hfill & \Delta y(\ell ,\vartheta +1)={\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta )}}\left[y(\ell ,\vartheta )-y(\ell -1,\vartheta )\right]+{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -1)}}\left[y(\ell ,\vartheta -1)-y(\ell -1,\vartheta -1)\right]\hfill \\ \hfill & +\cdots +{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -L_y+1)}}\left[y(\ell ,\vartheta -L_y+1)-y(\vartheta -L_y+1,j-1)\right]\hfill \\ \hfill & +{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta )}}\left[u(\ell ,\vartheta )-u(\ell -1,\vartheta )\right]+{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta -1)}}\left[u(\ell ,\vartheta -1)-u(\ell -1,\vartheta -1)\right]+\cdots \hfill \\ \hfill & +{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta -L_u+1)}}\left[u(\ell ,\vartheta -L_u+1)-u(\ell -1,\vartheta -L_u+1)\right]+\mathbf{\Psi }(\ell ,\vartheta )\hfill \\ \hfill & ={\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta )}}\Delta y(\ell ,\vartheta )+{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -1)}}\Delta y(\ell ,\vartheta -1)+\cdots +{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -L_y+1)}}\Delta y(\ell ,\vartheta -L_y+1)\hfill \\ \hfill & +{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta )}}\Delta u(\ell ,\vartheta )+{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta -1)}}\Delta u(\ell ,\vartheta -1)+\cdots +{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta -L_u+1)}}\Delta u(\ell ,\vartheta -L_u+1)\hfill \\ \hfill & +\mathbf{\Psi }(\ell ,\vartheta )\hfill \end{array}$$

where ${\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -p)}}$, $0\le p\le L_y-1$, ${\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -q)}}$, $0\le q\le L_u-1$ denote the partial derivatives of $f(\dots )$ with respect to the $(p+1)$th variable and the $(n_y+2+q)$th variable at a certain point between

$$\begin{array}{cc}\hfill & [y(\ell ,\vartheta )-y(\ell ,\vartheta -1),\dots ,y(\ell ,\vartheta -L_y+1),y(\ell ,\vartheta -L_y),\dots ,y(\ell ,\vartheta -n_y),u(\ell ,\vartheta ),u(\ell ,\vartheta -1),\dots ,\hfill \\ \hfill & u(\ell ,\vartheta -L_u+1),u(\ell ,\vartheta -L_u),u(\ell ,\vartheta -n_u){]}^T\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & [y(\ell -1,\vartheta )-y(\ell -1,\vartheta -1),\dots ,y(\ell -1,\vartheta -L_y+1),y(\ell ,\vartheta -L_y),\dots ,y(\ell ,\vartheta -n_y),u(\ell -1,\vartheta ),\hfill \\ \hfill & u(\ell -1,\vartheta -1),\dots ,u(\ell -1,\vartheta -L_u+1),u(\ell ,\vartheta -L_u),u(\ell ,\vartheta -n_u){]}^T\hfill \end{array}$$

For each fixed iteration $\vartheta$ and each fixed time $\vartheta$, consider the following equation with the variable $\eta (\ell ,\vartheta )$:

$$\begin{array}{c}\hfill \mathbf{\Psi }(\ell ,\vartheta )={\eta }^T(\ell ,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )\end{array}$$

(8)

Since $\left|\right|\Delta H_{L_y,L_u}(\ell ,\vartheta )\left|\right|\ne 0$, there must exist a unique solution ${\eta }^{*}(\ell ,\vartheta )$ to Equation (8). Let

$$\begin{array}{cc}\hfill & {\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )={\eta }^{*}(\ell ,\vartheta )+[{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta )}},{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -1)}},\dots ,{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -L_y+1)}},{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta )}},{\displaystyle \frac{\partial f^{*}}{\partial u(\ell ,\vartheta -1)}},\hfill \\ \hfill & \cdots ,{\displaystyle \frac{\partial f^{*}}{\partial y(\ell ,\vartheta -L_u+1)}}]\hfill \end{array}$$

Then, $\Delta y(\ell ,\vartheta +1)$ is rewritten as $\Delta y(\ell ,\vartheta +1)={\mathbf{\Phi }}_{L_y,L_u}^T(\ell ,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}(\ell ,\vartheta )$.

It follows that if the components of ${\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )$ are unbounded, then Assumption 2 cannot hold, and, therefore, ${\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )$ is bounded. □

3.2. Iterative Controller Design

Given a desired trajectory $y_d\left(\vartheta \right)$, $\vartheta \in \left\{0,1,\dots ,T\right\}$, the control objective is to find a sequence of appropriate control inputs $u(\ell ,\vartheta )$ such that the tracking error $e(\ell ,\vartheta +1)=y_d(\ell ,\vartheta +1)-y(\ell ,\vartheta +1)$ converges to zero as the iteration number $\vartheta$ approaches infinity.

Consider the cost function of the control input as follows:

$$\begin{array}{c}\hfill C\left(u(\ell ,\vartheta )\right)={\left|e(\ell ,\vartheta +1)\right|}^2+\lambda {|u(\ell ,\vartheta )-u(\ell -1,\vartheta )|}^2\end{array}$$

(9)

where $\lambda$ > 0 is a weighting factor.

By substituting Equation (7) and the definition of $e(\ell ,\vartheta +1)$ into Equation (9) as well as setting $u(\ell ,\vartheta +1)$ to zero, we derive the iterative learning control algorithm with full-format dynamic linearization as follows:

$$\begin{array}{cc}\hfill u(\ell ,\vartheta )& =u(\ell -1,\vartheta )+{\displaystyle \frac{{\varphi }_{Ly+1}(\ell ,\vartheta )}{\lambda +|{\varphi }_{L_y+1}{(\ell ,\vartheta )|}^2}}[{\rho }_{L_y+1}e(\ell -1,\vartheta +1)-\sum _{c=1}^{L_y}{\rho }_c{\varphi }_c(\ell ,\vartheta )\Delta y(\ell ,\vartheta -c+1)\hfill \\ \hfill & -\sum _{c=L_y+2}^{L_y+L_u}{\rho }_c{\varphi }_c(\ell ,\vartheta )\Delta u(\ell ,\vartheta +L_y-c+1)],(L_u\ge 2)\hfill \\ \hfill u(\ell ,\vartheta )& =u(\ell -1,\vartheta )+{\displaystyle \frac{{\varphi }_{Ly+1}(\ell ,\vartheta )}{\lambda +|{\varphi }_{L_y+1}{(\ell ,\vartheta )|}^2}}[{\rho }_{L_y+1}e(\ell -1,\vartheta +1)\hfill \\ \hfill & -\sum _{c=1}^{L_y}{\rho }_c{\varphi }_c(\ell ,\vartheta )\Delta y(\ell ,\vartheta -c+1)],(L_u=1)\hfill \end{array}$$

(10)

where the step factor $\rho \in (0,1],c=1,2,\dots ,L_y+L_u$.

The cost function for the estimated value of the pseudo-gradient (PG) vector is

$$\begin{array}{cc}\hfill C\left({\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )\right)& =|\Delta y(\ell -1,\vartheta +1)-{\mathbf{\Phi }}_{L_y,L_u}^T(\ell ,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}{(\ell -1,\vartheta )|}^2\hfill \\ \hfill & +\mu \left|\right|{\mathbf{\Phi }}_{L_y,L_u}(\ell ,\vartheta )-{\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell -1,\vartheta ){\left|\right|}^2\hfill \end{array}$$

(11)

where $\mu$ > 0 is a weighting factor; ${\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell -1,\vartheta )$ is the estimated value of ${\mathbf{\Phi }}_{L_y,L_u}(\ell -1,\vartheta )$.

According to the optimality condition ${\displaystyle \frac{1}{2}\frac{\partial J}{\partial {\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )}=0}$, the algorithm for estimating the value of the PG vector can be obtained as

$$\begin{array}{cc}\hfill & {\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )={\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell -1,\vartheta )\hfill \\ \hfill & +{\displaystyle \frac{\eta \Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )}{\mu +\parallel \Delta {\mathit{H}}_{L_y,L_u}{(\ell -1,\vartheta )\parallel }^2}}\times \left[\Delta y(\ell -1,\vartheta +1)-{\widehat{\mathbf{\Phi }}}_{L_y+L_u}^T(\ell -1,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )\right]\hfill \end{array}$$

(12)

where the step factor $\eta \in (0,2]$ is added to make the estimation algorithm more generic.

To ensure that the parameter estimation algorithm (12) exhibits strong tracking performance, we introduce the following resetting algorithm:

$$\begin{array}{c}\hfill {\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )={\widehat{\mathbf{\Phi }}}_{L_y,L_u}(1,\vartheta ),if\left|\right|{\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )\left|\right|\le \epsilon ,or\left|\right|\Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )\left|\right|\le \epsilon \end{array}$$

(13)

where $\epsilon$ is a small positive constant, and ${\widehat{\mathbf{\Phi }}}_{L_y,L_u}(1,\vartheta )$ is the initial value of ${\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )$.

3.3. Modification for Feedforward–Feedback Structure

FFDL-MFAILC is a feedforward control method that leverages the repetitive operational characteristics of a bus system to improve control performance. However, due to the numerous disturbances and emergencies encountered during actual bus operations, simple feedforward control alone cannot fully meet the control requirements. Therefore, building on the proposed FFDL-MFAILC algorithm, this section introduces a feedback control component and the design of the full-format dynamic linearization model-free adaptive feedforward–feedback iterative learning control (FFDL-MFAFILC) algorithm. In the feedback component, the extended state observer (ESO) from ADRC is employed to adjust the input, thereby reducing errors and mitigating the impact of disturbances.

For system (4), let $y_s(\ell ,\vartheta )=s(\ell ,\vartheta ),x_1(\ell ,\vartheta )=s(\ell ,\vartheta ),x_2(\ell ,\vartheta )={\dot{x}}_1(\ell ,\vartheta )$, then

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{x}}_1(\ell ,\vartheta )& =x_2(\ell ,\vartheta )\hfill \\ \hfill {\dot{x}}_2(\ell ,\vartheta )& ={\displaystyle \frac{1}{m}}(F(\ell ,\vartheta )-{\displaystyle \frac{1}{2}}\rho C_d{A}_F{(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2\hfill \\ \hfill & -mgsin\theta -f_{R}mgcos\theta )\hfill \\ \hfill y_s(\ell ,\vartheta )& =x_1(\ell ,\vartheta )\hfill \end{array}\right.\hfill \end{array}$$

(14)

Let ${\displaystyle b_0=\frac{1}{m}}$, $u_s(\ell ,\vartheta )=F(\ell ,\vartheta )$, ${\displaystyle k_v=-\frac{\rho C_d{A}_F}{2m}}$, $k_f=-g(sin\theta +f_{R}cos\theta )$, which, when substitutes into (14), produces

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{x}}_1(\ell ,\vartheta )& =x_2(\ell ,\vartheta )\hfill \\ \hfill {\dot{x}}_2(\ell ,\vartheta )& =b_0{u}_s(\ell ,\vartheta )+k_v{(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2+k_f\hfill \\ \hfill y_s(\ell ,\vartheta )& =x_1(\ell ,\vartheta )\hfill \end{array}\right.\hfill \end{array}$$

(15)

Since both $\nu (\ell ,\vartheta )$ and ${(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2$ are non-negative numbers, $y_1(\ell ,\vartheta )=\nu (\ell ,\vartheta )$ and $y_2(\ell ,\vartheta )={(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2$ are increasing functions with respect to the variable $\nu (\ell ,\vartheta )$. In addition, ${\nu }_{wind}(\ell ,\vartheta )$ is a relatively small time-varying function that can be used as a perturbation. Therefore, $k_v{(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2$ can be approximated. Let $k_1\nu (\ell ,\vartheta )\approx k_v{(\nu (\ell ,\vartheta )+{\nu }_{wind}(\ell ,\vartheta ))}^2$, then (15) can be transformed into the following form:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{x}}_1(\ell ,\vartheta )& =x_2(\ell ,\vartheta )\hfill \\ \hfill {\dot{x}}_2(\ell ,\vartheta )& =b_0{u}_s(\ell ,\vartheta )+k_1\nu (\ell ,\vartheta )+k_f+w(\ell ,\vartheta )\hfill \\ \hfill y_s(\ell ,\vartheta )& =x_1(\ell ,\vartheta )\hfill \end{array}\right.\hfill \end{array}$$

(16)

where $w(\ell ,\vartheta )$ is a disturbance, which includes braking disturbance, wind resistance disturbance, and controller disturbance.

Let $x_3(\ell ,\vartheta )=k_{1}v(\ell ,\vartheta )+k_f+w(\ell ,\vartheta )$, ${\dot{x}}_3(\ell ,\vartheta )=h$, then (16) can be transformed into the following form:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{x}}_1(\ell ,\vartheta )& =x_2(\ell ,\vartheta )\hfill \\ \hfill {\dot{x}}_2(\ell ,\vartheta )& =b_0{u}_s(\ell ,\vartheta )+x_3(\ell ,\vartheta )\hfill \\ \hfill {\dot{x}}_3(\ell ,\vartheta )& =h\hfill \\ \hfill y_s(\ell ,\vartheta )& =x_1(\ell ,\vartheta )\hfill \end{array}\right.\hfill \end{array}$$

(17)

According to (17), the extended state observer (ESO) is designed:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{z}}_1(\ell ,\vartheta )& =z_2(\ell ,\vartheta -1)+{\beta }_1(y_s(\ell ,\vartheta -1)-z_1(\ell ,\vartheta -1))\hfill \\ \hfill {\dot{z}}_2(\ell ,\vartheta )& =b_0{u}_s(\ell ,\vartheta -1)+z_3(\ell ,\vartheta -1)+{\beta }_2(y_s(\ell ,\vartheta -1)-z_1(\ell ,\vartheta -1))\hfill \\ \hfill {\dot{z}}_3(\ell ,\vartheta )& ={\beta }_3(y_s(\ell ,\vartheta -1)-z_1(\ell ,\vartheta -1))\hfill \end{array}\right.\hfill \end{array}$$

(18)

where $z_a(\ell ,\vartheta )$ is the observed value of $x_a(\ell ,\vartheta )$ (a = 1, 2, 3).

The matrix form of Formula (18) is as follows:

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{\mathbf{Z}}}_{\vartheta }=\left[\begin{array}{ccc}-{\beta }_1& 1& 0\\ -{\beta }_2& 0& 1\\ -{\beta }_3& 0& 0\end{array}\right]{\mathbf{Z}}_{\vartheta -1}+\left[\begin{array}{cc}0& {\beta }_1\\ b_0& {\beta }_2\\ 0& {\beta }_3\end{array}\right]\left[\begin{array}{c}{u}_s(\ell ,\vartheta -1)\\ y_s(\ell ,\vartheta -1)\end{array}\right]\hfill \\ y_s(\ell ,\vartheta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]{\mathbf{Z}}_{\vartheta -1}\hfill \end{array}\right.\hfill \end{array}$$

(19)

where ${\dot{\mathbf{Z}}}_{\vartheta }={\left[z_1(\ell ,\vartheta ),z_2(\ell ,\vartheta ),z_3(\ell ,\vartheta )\right]}^T$, ${\dot{\mathbf{Z}}}_{\vartheta -1}={\left[z_1(\ell ,\vartheta -1),z_2(\ell ,\vartheta -1),z_3(\ell ,\vartheta -1)\right]}^T$

The characteristic equation of system (19) is

$$s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3=0$$

let $s^3+{\beta }_1{s}^2+{\beta }_{2}s+{\beta }_3={(s+w_0)}^3$, then ${\beta }_1=3w_0,{\beta }_2=3w_0^2,{\beta }_3=w_0^3$. Substituting it into (19), we have

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{\mathbf{Z}}}_{\vartheta }=\left[\begin{array}{ccc}-3w_0& 1& 0\\ -3w_0^2& 0& 1\\ -w_0^3& 0& 0\end{array}\right]{\mathbf{Z}}_{\vartheta -1}+\left[\begin{array}{cc}0& 3w_0\\ b_0& 3w_0^2\\ 0& w_0^3\end{array}\right]\left[\begin{array}{c}{u}_s(\ell ,\vartheta -1)\\ y_s(\ell ,\vartheta -1)\end{array}\right]\hfill \\ y_s(\ell ,\vartheta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]{\mathbf{Z}}_{\vartheta -1}\hfill \end{array}\right.\hfill \end{array}$$

(20)

According to Formula (17), there are

$$\ddot{s}(\ell ,\vartheta )=b_0{u}_s(\ell ,\vartheta )+x_3(\ell ,\vartheta )$$

(21)

In order to eliminate the disturbance, there is

$$u_s(\ell ,\vartheta )={\displaystyle \frac{u_0(\ell ,\vartheta )-z_3(\ell ,\vartheta )}{b_0}}$$

(22)

where $u_0(\ell ,\vartheta )$ is the input of the system.

Since ADRC is only used as the feedback part, $u_0(\ell ,\vartheta )=0$, the feedback input is

$$u_s(\ell ,\vartheta )=-{\displaystyle \frac{z_3(\ell ,\vartheta )}{b_0}}$$

(23)

The FFDL-MFAFILC algorithm is as follows:

$$\begin{array}{c}\left\{\begin{array}{c}{u}_f(\ell ,\vartheta )=u_f(\ell -1,\vartheta )+{\displaystyle \frac{{\varphi }_{Ly+1}(\ell ,\vartheta )}{\lambda +|{\varphi }_{L_y+1}{(\ell ,\vartheta )|}^2}}[{\rho }_{L_y+1}e(\ell -1,\vartheta +1)\hfill \\ -\sum _{c=1}^{L_y}{\rho }_c{\varphi }_c(\ell ,\vartheta )\Delta y(\ell ,\vartheta -c+1)-\sum _{c=L_y+2}^{L_y+L_u}{\rho }_c{\varphi }_c(\ell ,\vartheta )\Delta u(\ell ,\vartheta +L_y-c+1)]\hfill \\ u_b(\ell ,\vartheta )=-{\displaystyle \frac{z_3(\ell ,\vartheta )}{b_0}}\hfill \\ u(\ell ,\vartheta )={\beta }_f{u}_f(\ell ,\vartheta )+{\beta }_b{u}_b(\ell ,\vartheta )\hfill \\ {\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )={\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell -1,\vartheta )+{\displaystyle \frac{\eta \Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )}{\mu +\parallel \Delta {\mathit{H}}_{L_y,L_u}{(\ell -1,\vartheta )\parallel }^2}}\hfill \\ \times \left[\Delta y(\ell -1,\vartheta +1)-{\widehat{\mathbf{\Phi }}}_{L_y+L_u}^T(\ell -1,\vartheta )\Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )\right]\hfill \\ {\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )={\widehat{\mathbf{\Phi }}}_{L_y,L_u}(1,\vartheta ),if\left|\right|{\widehat{\mathbf{\Phi }}}_{L_y,L_u}(\ell ,\vartheta )\left|\right|\le \epsilon ,or\left|\right|\Delta {\mathit{H}}_{L_y,L_u}(\ell -1,\vartheta )\left|\right|\le \epsilon \hfill \\ {\dot{\mathbf{Z}}}_{\vartheta }=\left[\begin{array}{ccc}-3w_0& 1& 0\\ -3w_0^2& 0& 1\\ -w_0^3& 0& 0\end{array}\right]{\mathbf{Z}}_{\vartheta -1}+\left[\begin{array}{cc}0& 3w_0\\ b_0& 3w_0^2\\ 0& w_0^3\end{array}\right]\left[\begin{array}{c}u(\ell ,\vartheta -1)\\ y(\ell ,\vartheta -1)\end{array}\right]\hfill \end{array}\right.\hfill \end{array}$$

(24)

where $u_f(\ell ,\vartheta )$ is the input of the feedforward part, $u_b(\ell ,\vartheta )$ is the input of the feedback part, and $0<{\beta }_f\le 1$ and $0<{\beta }_b\le 1$ are the weight factors.

4. Simulation Results

In this study, numerical simulations were conducted using MATLAB to verify the effectiveness, superiority, and robustness of the proposed FFDL-MFAFILC algorithm. To ensure the consistency between the simulation and actual bus operations, and to validate the effectiveness of the algorithm, this study utilized real data recorded from a specific travel interval of an operating bus. The route considered in this study was Beijing Bus No. 961, starting from Zhangyicun North Station, passing through Wuzhuang South Station, and finally arriving at Lianfang East Bridge South Station. This route included two intersections, and the route diagram for Bus No. 961 is shown in Figure 2.

The actual operation data of Bus No. 961 are shown in

Table 1. Operation data of Bus No. 961.

Time (s)	Distance from Initial Position (m)	Position	Weight (kg)	Road Slope $\mathit{\theta }{(}^{\circ })$
0	0	Zhangyicun North Station	10,000	2.3
6	0	Zhangyicun North Station	10,000	2.3
60	319	Intersection 1	10,000	1.9
106	319	Intersection 1	10,000	1.9
151	753	Wuzhuang South Station	9200	1.7
166	753	Wuzhuang South Station	9200	1.7
193	923	Intersection 2	9200	1.1
281	923	Intersection 2	9200	1.1
351	1577	Lianfang Dongqiao South Station	9700	1.4
360	1577	Wuzhuang South Station	9700	1.4

, including the displacement, position, and weight of the bus and the road slope at the corresponding time.

In the simulation in Section 4.1 and Section 4.2, the main simulation parameters of MFAFILC were $b_0=0.001,w_0=0.2$, ${\widehat{\mathbf{\Phi }}}_{L_y,L_u}(1,\vartheta )=$${[65,-60,0.04]}^T$, ${\rho }_{L_y+1}=13$, ${\rho }_j=1$. In addition, the wind speed was set to ${\nu }_{wind}(\ell ,\vartheta )=5sin\vartheta$. And, we employed two key performance metrics to evaluate the efficacy of our proposed method: maximum tracking error (MTE) and mean squared error (MSE). These metrics provided a comprehensive assessment of the tracking performance and the accuracy of our method in comparison to existing approaches.

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}\mathrm{MTE}={\max }_{\vartheta }|y_d\left(\vartheta \right)-y\left(\vartheta \right)|,\vartheta =1,2,\dots ,T\hfill \\ \mathrm{MSE}={\displaystyle \frac{1}{T}}\sum _{\vartheta =1}^T{(y_d\left(\vartheta \right)-y\left(\vartheta \right))}^2\hfill \end{array}\right.\hfill \end{array}$$

(25)

where T is the simulation period.

4.1. Trajectory Tracking Without Disturbance

In this section, we mainly simulate the effectiveness of the FFDL-MFAFILC algorithm for bus trajectory tracking control without interference.

Figure 3 presents the maximum tracking error observed at different iteration stages, with the weight factors ${\beta }_f$ and ${\beta }_b$ both set to 1. The Figure 3a depicts the maximum speed tracking error across the iterations, while the Figure 3b shows the maximum distance tracking error over the same iterations. Notably, Figure 3 highlights a clear trend: as the number of iterations increases, the tracking error steadily decreases, demonstrating the effectiveness of the iterative learning process. Specifically, the maximum path tracking error is reduced to 6.31 by the 30th iteration, indicating significant improvement in accuracy. This reduction in error underscores the robustness of the control strategy in refining the system’s performance over time. Furthermore, the gradual convergence of the tracking error suggests that the proposed method is highly effective in optimizing the system’s ability to follow the desired trajectory, with diminishing errors after each iteration. Such performance improvement is crucial for applications where precise tracking is required, such as autonomous vehicles or robotic systems.

In order to verify the optimal control effect with weight factors ${\beta }_f=1$ and ${\beta }_b=1$, different weight ratios were selected for comparison, as shown in Figure 4.

The MTE results under each weight ratio are shown in the

Table 2. MTE under different weight ratios.

Weight Ratio	MTE
${\beta }_f=1,{\beta }_b=1$	6.31
${\beta }_f=1,{\beta }_b=0.8$	6.62
${\beta }_f=0.8,{\beta }_b=0.8$	8.42
${\beta }_f=0.8,{\beta }_b=0.6$	12.61
${\beta }_f=0.6,{\beta }_b=0.6$	25.73

.

The results show that the control effect is optimal when ${\beta }_f=1$ and ${\beta }_b=1$, so this weight ratio was selected for the subsequent simulation experiments.

Figure 5 illustrates the system input, specifically the longitudinal power of the bus, at the 30th iteration. In this simulation, we took into account the realistic longitudinal force limitations of the vehicle, which were constrained by physical factors such as engine capacity and road conditions. To reflect these constraints, the upper and lower bounds of the input were set to $(\pm 5\times {10}^4)$. The results shown in Figure 5 reveal that the input magnitude remains consistently within these predefined limits throughout the iteration process. Moreover, the input exhibits only minimal variation, indicating a stable and controlled adjustment process.

This stability in the input profile is a key feature of the proposed FFDL-MFAFILC algorithm. The minimal fluctuation in the system input suggests that the algorithm effectively manages the trade-off between the feedforward and feedback components, ensuring that the system operates within the practical constraints of the vehicle while achieving optimal performance. The smoothness and bounded nature of the input adjustments further demonstrate the algorithm’s capability to adapt the control inputs in a manner that is both physically feasible and highly stable. In practical applications, such as in autonomous vehicles or automated bus systems, maintaining input values within a safe and realistic range is critical for ensuring both the safety and reliability of the control system. The FFDL-MFAFILC algorithm’s ability to maintain these conditions while also improving tracking performance underscores its potential for real-world deployment, where system stability and robustness are paramount.

Figure 6 presents the tracking performance at the 30th iteration. The Figure 6a displays the speed tracking results, while the Figure 6b illustrates the path tracking performance. The simulation outcomes clearly show that, under the control of the FFDL-MFAFILC algorithm, the bus speed remains very close to the desired speed throughout the entire iteration process. Although there are minor deviations for a few moments, these are transient, and the speed quickly converges to the desired value, demonstrating the algorithm’s robustness in regulating the speed. These brief deviations are likely due to transient dynamics or minor discrepancies in system feedback, but they do not significantly affect the overall performance.

Furthermore, the bus trajectory closely follows the desired path, indicating that the control strategy effectively handles the spatial coordination between speed and position. The Figure 6b confirms that the path tracking error is minimal, with the bus consistently staying on or very near the target trajectory. This result underscores the capability of the FFDL-MFAFILC algorithm to perform both speed and path tracking simultaneously, ensuring that the vehicle not only achieves the correct speed at each point but also adheres to the intended route with high precision. These results highlight the effectiveness of the proposed control strategy in both aspects of the tracking task. The FFDL-MFAFILC algorithm is shown to be capable of minimizing tracking errors, effectively balancing the need for speed regulation and precise path following. Such performance is crucial for applications such as autonomous driving, automated public transportation, and robotic navigation, where accurate speed control and trajectory adherence are essential for safety, efficiency, and overall system reliability. The ability of the algorithm to quickly correct deviations in both speed and position further emphasizes its potential for real-world deployment in dynamic environments.

There are several classical control algorithms for vehicle trajectory tracking, such as Model Predictive Control (MPC) [42], Proportional-Integral-Derivative Control (PID) [43], and Sliding Mode Control (SMC) [44]. To evaluate the effectiveness and superiority of the FFDL-MFAFILC algorithm, it is compared with the simulation results of FFDL-MFAILC, MPC, PID, and SMC in Figure 7. All five algorithms are capable of tracking the bus trajectory effectively under disturbance-free conditions.

By comparing the MTE and MSE of these five algorithms, it can be seen that the control effect of FFDL-MFAFILC is the best, followed by FFDL-MFAILC.

4.2. Trajectory Tracking with Disturbance

In this section, we describe simulations of the effectiveness and robustness of the FFDL-MFAFILC algorithm for bus trajectory tracking control in the presence of interference.

In actual driving environments, buses are subject to numerous positional disturbances. To evaluate the performance of the proposed control algorithm, both numerical simulations and experimental tests were conducted under significant disturbances. The disturbance considered in this study was a normally distributed white noise with a power of 0.1. The measurement noise and disturbance are illustrated in Figure 8. These disturbances affect the bus’s speed.

Figure 9 illustrates the maximum tracking errors observed at different iteration stages in the presence of disturbances. Compared to Figure 3, the maximum speed tracking error is significantly larger when interference is present, as the heavy disturbance directly affects the bus’s speed. The maximum path tracking error increases slightly under heavy interference, with the maximum error reaching 13.66 in the 30th iteration.

Figure 10 shows the system input during the 30th iteration under disturbance. Compared to Figure 5, the magnitude of and variation in the system input are slightly larger under disturbance. This increase is expected, as external disturbances often cause more fluctuations in the system’s behavior. However, despite the larger variations, the input still remains within the predefined bounds of $(\pm 5\times {10}^4)$, demonstrating that the system is still operating within a reasonable range.This indicates that the FFDL-MFAFILC algorithm continues to perform effectively under disturbance, adjusting the feedforward–feedback inputs with minimal disruption. The algorithm proves to be both stable and resilient, maintaining control over the system input even in the presence of external disturbances.

Figure 11 illustrates the tracking performance during the 30th iteration under heavy interference. The simulation results indicate a slight deviation between the bus speed and the desired speed, which is expected given the substantial disturbance impacting the system. The disturbance introduces variability that directly influences the speed, causing these minor deviations. However, despite this, the speed quickly converges back to the desired value, demonstrating the system’s ability to recover from external interference. This result highlights the robustness of the FFDL-MFAFILC algorithm, which effectively maintains tracking performance even under challenging conditions, ensuring that the bus speed remains relatively close to the desired trajectory despite the disturbances.

Although there is some fluctuation, the bus speed stays close to the desired speed. Additionally, the bus trajectory under heavy disturbance deviates slightly from the desired trajectory at certain moments but remains largely consistent for most of the time. Therefore, the FFDL-MFAFILC algorithm not only effectively tracks the bus trajectory but also demonstrates strong robustness.

To further verify the robustness of FFDL-MFAFILC, it is compared with the simulation results of FFDL-MFAILC, MPC, PID, and SMC under heavy disturbance in Figure 12.

Upon thorough examination of the data presented in Figure 7 and Figure 12 as well as

Table 3. MTE and MSE for different control strategies without disturbances.

	MTE	MSE
FFDL-MFAFILC	6.31	7.45
FFDL-MFAILC	7.91	13.93
MPC	16.83	31.15
PID	34.51	146.19
SMC	28.62	109.14

and

Table 4. MTE and MSE for different control strategies with disturbances.

	MTE	MSE
FFDL-MFAFILC	13.66	17.14
FFDL-MFAILC	17.38	28.01
MPC	19.40	47.55
PID	36.29	148.78
SMC	30.95	121.38

, it becomes evident that both PID and SMC exhibit inferior control performance and reduced resistance to interference compared to the other three algorithms. Notably, FFDL-MFAILC demonstrates commendable control performance in both interference and non-interference scenarios, ranking second only to FFDL-MFAFILC. However, it experiences a notable decline in effectiveness under interference conditions. In contrast, while the control performance of PID does not match that of FFDL-MFAILC, it exhibits less sensitivity to disturbances, indicating better robustness. The enhanced version, FFDL-MFAFILC, not only improves control performance but also shows enhanced ability to counteract disturbances. This algorithm outperforms FFDL-MFAILC, MPC, PID, and SMC in both trajectory tracking precision and overall robustness.

5. Conclusions

This paper introduces a feedforward–feedback control approach based on FFDL-MFAILC to facilitate longitudinal trajectory tracking for bus systems. The method effectively utilizes operation-generated data, eliminating the need for specific information about the controlled vehicle. This approach overcomes the challenges of precisely modeling bus dynamics and enables its applicability across diverse vehicle models. Additionally, a comparative assessment was conducted, contrasting this approach with FFDL-MFAILC, MPC, PID, and SMC strategies. The thorough analysis and evaluation of tracking errors demonstrate that the FFDL-MFAFILC algorithm offers significant advantages in terms of adaptability and precision.

Furthermore, the proposed FFDL-MFAFILC approach shows great promise for broader applications in fields requiring precise, repetitive trajectory tracking, such as robot-assisted rehabilitation and assembly line robotics. In robot-assisted rehabilitation, for example, achieving precise control over hand or limb trajectories is essential for effective therapy. Similarly, in assembly line robotics, maintaining accurate, repeatable movement patterns is critical for improving efficiency and reducing errors. By providing a robust, model-free solution that adapts to various operational conditions, FFDL-MFAFILC has the potential to significantly enhance the performance and reliability of the systems in these domains.

Moving forward, our future work will explore advanced control strategies and algorithms that are responsive to rapid environmental changes. We also intend to conduct comprehensive experimental verifications under dynamically changing conditions. Specifically, we plan collaborations with real-world partners such as the Beijing Bus Group and the Tongzhou Road Institute to test and validate our control solutions in operational urban bus systems and controlled road conditions. This empirical approach will bridge the gap between theoretical advancements and their real-world applications, ensuring that our contributions are not only innovative but also practically viable.