Research on Intelligent Vehicle Tracking Control and Energy Consumption Optimization Based on Dilated Convolutional Model Predictive Control

Abstract

To address the limitations of low modeling accuracy in physics-based methods—which often lead to poor vehicle-tracking performance and high energy consumption—this paper proposes an intelligent vehicle modeling and trajectory tracking control method based on a dilated convolutional neural network (DCNN). First, an effective dataset was constructed by incorporating historical state information, such as longitudinal tire forces and vehicle speed, to accurately capture vehicle dynamic characteristics and reflect energy variations during motion. Next, a dilated convolutional vehicle system model (DCVSM) was designed by combining vehicle dynamics with data-driven modeling techniques. This model was then integrated into a model predictive control (MPC) framework. By solving a nonlinear optimization problem, a dilated convolutional model predictive controller (DCMPC) was developed to enhance tracking accuracy and reduce energy consumption. Finally, a co-simulation environment based on CarSim and Simulink was used to evaluate the proposed method. Comparative analyses with a traditional MPC and a neural network-based MPC (NNMPC) demonstrated that the DCMPC consistently exhibited superior trajectory tracking performance under various test scenarios. Furthermore, by computing the tire-slip energy loss rate, the proposed method was shown to offer significant advantages in improving energy efficiency.

1. Introduction

With the rapid advancement of autonomous driving technology, achieving safe, stable, and efficient trajectory tracking control in complex road environments has become one of the key challenges in intelligent driving systems [1]. Trajectory tracking control aims to enable autonomous vehicles to accurately follow a predefined path in dynamic environments, while ensuring safety, ride comfort, tracking accuracy, and energy efficiency [2]. Currently, widely adopted control strategies can generally be categorized into physics-based methods and data-driven approaches [3,4,5].

In model-based trajectory tracking control algorithms, the performance of the controller depends not only on the algorithm itself but also on the accuracy of the system model. In the past, researchers have developed various vehicle models using physical principles, such as kinematic single-track models, dynamic single-track models, and tire models described by P. Stano et al. [6]. Although these models have been widely applied, they still suffer from certain limitations during modeling, particularly due to parameter uncertainties and disturbances caused by varying environmental conditions (e.g., wet or dry road surfaces), which can directly or indirectly affect the model’s accuracy and consequently the tracking performance of the controller. A common controller design approach is to formulate the control problem as a mathematical optimization problem that satisfies operational constraints while minimizing computational cost. The iterative linear quadratic regulator (iLQR), for instance, optimizes vehicle states and control trajectories [7], but this method is limited to quadratic cost functions. Based on physical modeling, model predictive control (MPC) has emerged as a mainstream approach in autonomous driving research due to its ability to accurately describe system dynamics and handle control input and state constraints effectively [8,9]. By employing a receding-horizon optimization strategy to generate control input sequences, MPC offers greater flexibility in managing complex costs, constraints, and dynamic characteristics [10]. It has been widely applied in various domains, including aerial vehicle navigation [11]. However, the performance of MPC is highly dependent on the accuracy of the vehicle dynamics model. In real-world scenarios, due to nonlinear tire behavior, road surface variations, lateral slip, and load disturbances, traditional models such as the planar single-track model often struggle to fully capture the vehicle’s dynamic behavior under complex driving conditions [12,13,14].

To overcome the limitations of physics-based models, data-driven modeling approaches have garnered increasing attention in recent years [3]. Deep neural networks (DNNs), owing to their powerful nonlinear fitting capabilities and end-to-end modeling advantages, have demonstrated remarkable performance in autonomous driving tasks such as trajectory prediction and control [15,16,17]. The neural network vehicle modeling method (NNVM) proposed by Spielberg et al. integrates historical state and control input information to accurately model the nonlinear dynamic behavior of vehicles under varying road friction conditions, outperforming traditional physics-based models in trajectory tracking control tasks [18]. To further enhance the adaptability and real-time performance of modeling, some researchers have adopted online learning approaches—such as introducing recurrent high-order neural networks (RHONNs) for the dynamic modeling of preview points along the path. When integrated with a feedforward–feedback control architecture, this enables higher tracking accuracy and lower control latency [19]. Sana Stihi et al. introduced a prescribed performance neural network model predictive control (PPNNMPC) method for a four-degree-of-freedom robot, which incorporates neural networks and performance functions to optimize tracking performance [20]. Although such methods enhance prediction accuracy and adaptability, they still suffer from issues such as limited generalization under highly dynamic responses or long-term dependency modeling scenarios, slow network convergence, and low inference efficiency [21]. These challenges are particularly pronounced under complex road conditions—such as continuous curves or surfaces with varying adhesion coefficients—where effectively capturing long-term dynamic patterns from vehicle historical behaviors remains an open problem [22]. To address this, Fang Peijun et al. proposed an improved neural network vehicle model that integrates long short-term memory (LSTM) techniques, which was subsequently used in a feedforward–feedback control architecture [23]. Other studies have adopted deep learning modeling approaches based on Koopman operators, which map complex nonlinear vehicle dynamics into an interpretable linear feature space, thereby enhancing multi-step prediction capability and model interpretability [24]. These methods have further advanced the practical application of deep learning in trajectory tracking control.

To address the data scarcity issue under extreme operating conditions, some researchers have proposed a latent-feature space sampling method based on LSTM-VAE, which generates control sequences that simulate vehicle behavior near critical dynamic states, thus providing valuable data for controller training and validation [25]. In response to challenges related to safety and sample efficiency, reinforcement learning approaches that integrate model-reference control frameworks with robust control barrier functions (CBFs) have also been introduced. These methods enable safe exploration while achieving effective and constraint-satisfying control performance [26]. Additionally, Bai Shaojie et al. introduced the concept of dilated convolution in temporal convolutional networks (TCNs). The dilated convolutional neural network (DCNN), by expanding the receptive field of convolution kernels without increasing the number of parameters, provides a natural advantage in handling long-term temporal dependencies and complex dynamic sequence modeling [27]. While trajectory tracking remains a central task, energy consumption optimization is also a critical performance metric in intelligent driving control systems. During vehicle operation, energy losses can arise from tire slip, frequent acceleration/deceleration, and delays in powertrain response [28]. In recent years, several studies have incorporated energy consumption modeling into the objective function of optimization algorithms, thereby improving system energy efficiency without compromising overall control effectiveness [29].

Based on the research background, this paper proposes a control framework based on a dilated convolutional neural network model predictive control (DCMPC), which aims to enhance the trajectory tracking accuracy and energy efficiency of intelligent vehicles in complex dynamic environments. The proposed method constructs a dilated convolutional vehicle system model (DCNNVSM), which incorporates historical vehicle states and control inputs as features to capture time-varying and nonlinear dynamic characteristics. This model is then embedded within an MPC optimization framework to perform path prediction and control input optimization without explicitly relying on physical parameter estimation. Furthermore, the tire-slip energy consumption rate is introduced as an additional system performance index to evaluate the energy-saving capability of the controller under different vehicle velocities and road adhesion conditions.

The main contributions of this research are as follows:

• A vehicle dynamics modeling method based on a dilated convolutional neural network is developed to more accurately capture complex vehicle dynamic behavior;
• A novel model-predictive control strategy (DCMPC) that integrates a DCNN is proposed, enabling joint optimization of trajectory tracking accuracy and energy consumption;
• Third-bullet system-level simulations are conducted under various scenarios, including high/low friction coefficient roads, different vehicle velocity, and double lane change/lane switching maneuvers. The results demonstrate that the DCMPC outperforms both neural network-based MPC (NNMPC) and traditional MPC in terms of tracking accuracy, control smoothness, and energy efficiency.

This study provides a new perspective for the integration of data-driven modeling and intelligent vehicle trajectory tracking control and offers a feasible approach to advancing energy-efficient autonomous driving systems.

2. Vehicle System Modeling

The key parameters and symbols used in the paper are first defined and summarized, as shown in

Table 1. The main parameters required for the article.

Name/Unit	Symbolic	Unit
Travel distance	$s$	m
Vehicle heading angle	$\theta$	rad
Lateral error	$e$	M
Curvature	$k$	1/m
Yaw rate	$r$	rad/s
Longitudinal velocity	$U_x$	m/s
Lateral velocity	$U_y$	m/s
Steering angle	$\delta$	rad
Longitudinal force of the front wheel	$F_{xf}$	N
Front wheel slip angle	${\alpha }_f$	rad
Rear wheel slip angle	${\alpha }_r$	rad
Front/rear tire	$\mathrm{\Theta }$	/
The sideslip angle in the state of complete slip	${\alpha }_{sl}$	rad
Sideslip angle	${\alpha }_{\mathrm{\Theta }}$	rad
Normal load	$F_{Z\mathrm{\Theta }}$	N
Vehicle friction coefficient	$\mathrm{\mu }$	/
Vehicle cornering stiffness	$C_{\mathrm{\Theta }}$	N/rad
Distance from front axle to center of mass	$a$	M
Distance from rear axle to center of mass	$b$	M
Moment of inertia	$I_Z$	Kg·m
Relaxation factor weighting coefficient	$\rho$	/
Minimum steering angle	${\delta }_{LB}$	rad
Maximum steering angle	${\delta }_{UB}$	rad
Minimum rate of change of steering angle	${\dot{\delta }}_{LB}$	rad/s
Maximum rate of change of steering angle	${\dot{\delta }}_{UB}$	rad/s
Longitudinal tire slip energy loss	$P$	J
Longitudinal tire slip force	$F_{si}$	N
Longitudinal tire slip speed	$v_{si}$	m/s

.

2.1. Bicycle Model

As shown in Figure 1, the desired path is used as a reference benchmark for the vehicle’s position. The distance traveled along this path is represented as $s$, with the relative heading angle denoted as $\theta$, and the lateral offset between the vehicle’s center of mass and the reference path is denoted as $e$. Additionally, each point on the path has a continuously varying curvature $k$, which can be calculated by integrating the heading angle and specific position of the reference path.

Figure 1 illustrates the schematic diagram, where each wheel on the axle is simplified to a single equivalent tire. The vehicle’s motion state is described by the yaw rate $r$, lateral velocity $U_y$, and longitudinal velocity $U_x$. In terms of lateral control, the system’s steering input is represented by $\delta$, which is the steering angle of the wheels. Additionally, the longitudinal force $F_{xf}$ on the front wheel serves as an additional input for the lateral dynamics, simulating the drive and braking forces on the front axle. Based on the vehicle’s speed state and steering input, each equivalent tire generates corresponding slip angles ${\alpha }_f$ and ${\alpha }_r$, as defined in Equations (1) and (2).

$${\mathrm{\alpha }}_f=\arctan \left({\displaystyle \frac{U_y+ar}{U_x}}\right)-\mathrm{\delta }$$

(1)

$${\mathrm{\alpha }}_r=\arctan \left({\displaystyle \frac{U_y-br}{U_x}}\right)$$

(2)

The vehicle’s slip angles generate corresponding lateral forces on the front tires $F_{yf}$ and rear tires $F_{yr}$. These tire forces are modeled using the single friction coefficient Fiala tire model. This model is applicable to both the front and rear tires, $\mathrm{\Theta }$ representing the respective forces on the front and rear tires, with specific parameters assigned to each tire. The slip angle of each axle generates a lateral force, which is calculated using Equation (3) as follows [30]; this formulation captures the nonlinear relationship between the lateral tire force and the slip angle, accounting for the transition from linear response to partial sliding and ultimately full saturation. Specifically, it reflects how the lateral force increases with slip angle under partial adhesion, then plateaus once the tire reaches its frictional limit. Such modeling is essential for accurately simulating vehicle behavior in high-demand maneuvers, such as sharp cornering or emergency lane changes, where tire dynamics significantly affect the overall stability and control performance.

$$F_y=\left\{\begin{array}{l}-C_{\mathrm{\Theta }}\tan {\mathrm{\alpha }}_{\mathrm{\Theta }}+{\displaystyle \frac{C_{\mathrm{\Theta }}^2}{3{\mathrm{\mu }}^2{F}_{Z\mathrm{\Theta }}^2}}\left|\tan {\mathrm{\alpha }}_{\mathrm{\Theta }}\right|\tan {\mathrm{\alpha }}_{\mathrm{\Theta }}\\ -{\displaystyle \frac{C_{\mathrm{\Theta }}^3}{27{\mathrm{\mu }}^2{F}_{Z\mathrm{\Theta }}^2}}{\tan }^3{\mathrm{\alpha }}_{\mathrm{\Theta }},if\left|{\mathrm{\alpha }}_{\mathrm{\Theta }}\right|<{\mathrm{\alpha }}_{sl}\\ -\mathrm{\mu }{F}_{Z\mathrm{\Theta }}\mathrm{sgn}{\mathrm{\alpha }}_{\mathrm{\Theta }}\hspace{1em}\hspace{1em}otherwise\end{array}\right.$$

(3)

where ${\alpha }_{sl}$ represents the slip angle at which the tire enters a fully sliding state, determining the maximum lateral force limit that each axle can provide. The tire’s lateral force characteristics are parameterized by the normal load $F_{Z\mathrm{\Theta }}$, which reflects the tire’s mechanical properties under different operating conditions. The ${\alpha }_{sl}$ calculation formula is given by Equation (4) as follows:

$${\mathrm{\alpha }}_{sl}=\arctan \left({\displaystyle \frac{3\mathrm{\mu }{F}_{Z\mathrm{\Theta }}}{C_{\mathrm{\Theta }}}}\right).$$

(4)

On each axle of the vehicle, $C_{\mathrm{\Theta }}$ represents the tire’s cornering stiffness, which determines the tire’s response characteristics when subjected to lateral forces. Additionally, $\mu$ represents the coefficient of friction between the tire and the road surface, a parameter that directly affects the vehicle’s traction and handling performance. The key parameters of the Fiala tire model are derived by fitting data from experimental vehicles tested in high- and low-friction environments [18]. These tire parameters are significantly influenced by road surface types and environmental conditions, so different testing scenarios lead to notable variations in the parameters.

Ultimately, these mechanical parameters are used to compute the vehicle’s motion using Equations (5) and (6), thereby constructing a more accurate dynamic model that better reflects real-world operating conditions, enhancing the simulation accuracy of the vehicle’s trajectory tracking and handling stability.

$${\dot{U}}_y={\displaystyle \frac{F_{yr}+F_{yf}\cos \delta +F_{xf}\sin \delta }{m}}-r{U}_x$$

(5)

$$\dot{r}={\displaystyle \frac{a{F}_{yf}\cos \delta +a{F}_{xf}\sin \delta -b{F}_{yr}}{I_Z}}$$

(6)

2.2. Data-Driven Vehicle Model

2.2.1. Dilated Convolution Vehicle System Model

A dilated convolution vehicle system model (DCNNVSM) replaces traditional physical models with neural networks to enhance adaptability to complex environments. Its inputs include the vehicle’s historical states and control inputs, while the output is the predicted state change.

Unlike traditional physics-based bicycle models, the DCNN dynamic model can adaptively learn and capture the complex nonlinear dynamic effects caused by changes in the road–tire friction coefficient. Compared to fixed-parameter physics models, this method can more accurately describe the dynamic characteristics that are difficult to model directly while ensuring computational efficiency. As shown in Figure 2, the vehicle states and control information are used as inputs to the data input layer of the network. To capture the temporal dynamic characteristics of the vehicle system, the input not only includes the current state information but also incorporates historical state and control data from the previous time steps. Specifically, the input features consist of the current yaw rate ${\gamma }^t$, lateral velocity $U_y^t$, longitudinal velocity $U_x^t$, front wheel steering angle ${\delta }^t$, and longitudinal force $F_{xf}^t$, along with the corresponding state and control information from the past three time steps. At each time step, the corresponding state variables are integrated into a unified state vector $x_t$. These data are arranged in a sequential manner to form the input to the vehicle system $H_t$. The structure of this input formulation is expressed in Equations (7) and (8):

$$x_t={\left[r,U_y,U_x,\delta ,F_{xf}\right]}_t$$

(7)

$$H_t=\left[x_t,\dots ,x_{t-T}\right]$$

(8)

To strengthen the intrinsic connections between different time steps and reduce the model complexity, the neural network employs a dilated convolution layer with a kernel size of 4 and a dilation rate of 5. This is followed by two fully connected layers to form the dilated convolution vehicle system model. To optimize the network architecture, combinations of different neurons were tested. Ultimately, the number of neurons was set to 128 to ensure the best balance between efficiency and accuracy.

As shown in Equations (9) and (12), where $Z_{DC}\left[j\right]$ represents the j-th element of the dilated convolution output at the current time step, $H_t\left[j+i{D}_r\right]$ represents the j-th input to the system at the current time step, $W_{DC}\left[i\right]$ is the corresponding i-th dilated convolution kernel parameter, $i\in \left[0,W_{\mathrm{cov}}-1\right]$, $j\in \left[0,W_{\mathrm{cov}}\right]$, and $b_{DC}$, $b_1$, $b_2$, $b_3$ are the bias terms used to enhance the model’s expressive power and improve training results. $h_1$, $h_2$ represents the hidden layer, while $Z_1$, $Z_2$ represents the activation layer. The network’s weight parameters are denoted as $W$. To calculate the final output of the network, the historical input state of vehicle control $H_t$ is propagated forward, and the output is obtained through a series of calculations, as shown in Equations (9)–(14). The network’s output includes the state derivatives of yaw rate $\dot{r}$ and lateral velocity ${\dot{U}}_y$ (see Equation (14)). By selecting the rate of change of the predicted velocity states instead of directly predicting the state values, the network can interpret the learned vehicle dynamics into forces and accelerations, thus more intuitively reflecting the vehicle’s dynamic behavior. This approach not only improves the physical consistency of the model but also enhances its adaptability to different driving environments.

$$Z_{DC}\left[j\right]={\displaystyle \sum _i^{w_{\mathrm{cov}-1}}{H}_t\left[j+i{D}_r\right]W_{DC}\left[i\right]+b_{DC}}$$

(9)

$$h_1=W_1^T{Z}_{DC}+b_1$$

(10)

$$Z_1=\log \left(1+e^{h1}\right)$$

(11)

$$h_2=W_2^T{Z}_1+b_2$$

(12)

$$Z_2=\log \left(1+e^{h2}\right)$$

(13)

$$\left[\begin{array}{c}{\dot{U}}_y\\ \dot{r}\end{array}\right]=W_3^T{Z}_2+b_3$$

(14)

2.2.2. Tracking Error Model

As shown in Figure 1, the vehicle’s position state describes its lateral error ($e$) and heading angle deviation relative to the reference path. Among these, the $e$ is the core component of the MPC optimization’s objective function. To make a more intuitive comparison between the NNMPC and traditional MPC, this study uses the $e$ as the primary trajectory tracking target. Because the vehicle’s kinematic characteristics primarily depend on geometric relationships and velocity states, no additional training is required, and it can be directly calculated using the mathematical relationships in Equations (15) and (16). This setup ensures a fair comparison of the three control strategies under the same kinematic constraints, allowing for a balanced evaluation of the DCMPC’s performance.

$$\dot{e}=U_x\sin \mathrm{\Delta }\psi +U_y\cos \mathrm{\Delta }\psi$$

(15)

$$\mathrm{\Delta }\dot{\psi }=r-\dot{s}k$$

(16)

2.2.3. Model Discretization

To implement the continuous-time dynamics of the bicycle model or the DCNNVSM in the MPC optimization problem, the simultaneous transcription method is used to discretize the continuous dynamics system, allowing each discrete state variable and control variable to participate directly in the optimization process. In the NNMPC, the Euler integration scheme is used for state updates. During the calculation, the control inputs remain piecewise-constant within each discrete step, mapping the continuous-time dynamics to discrete-time expressions, as shown in Equation (17). Specifically, the system state $x\left(t\right)$ at the initial time $t$ is determined by the control input $u\left(t\right)$ and the continuous dynamics equation $f_c$ and evolves to a new state $x\left(t+\mathrm{\Delta }t\right)$ after a time step $\Delta t$. This discretization strategy not only simplifies the computational burden of the nonlinear optimization solver but also ensures the real-time performance and computational efficiency of the DCNNMPC in trajectory tracking and vehicle control tasks.

$$x\left(t+\Delta t\right)=x\left(t\right)+\Delta t{f}_c\left(x\left(t\right),u\left(t\right)\right)$$

(17)

In the DCNNVSM, the continuous-time dynamics of the system are determined by the historical states and control inputs. Specifically, the control input at the current time step and the historical state records form the input vector $H_t$, which is fed into the dilated convolution network dynamics model $f_{DCNN,c}$, as shown in Equation (18). This method can effectively capture the complex nonlinear characteristics of vehicle dynamics and adapt to the dynamic changes in different road conditions. In contrast, the continuous-time dynamics $f_{bike,c}$ of the bicycle model only depend on the current state and control input without utilizing historical information, as shown in Equation (19). While this physics-based modeling approach is computationally efficient, it may struggle to accurately describe the true motion characteristics of the vehicle under complex road conditions.

$$x\left(t+\Delta t\right)=x\left(t\right)+\Delta t{f}_{DCNN,c}\left(h_t\right)$$

(18)

$$x\left(t+\Delta t\right)=x\left(t\right)+\Delta t{f}_{bike,c}\left(x\left(t\right),u\left(t\right)\right)$$

(19)

In the DCMPC’s framework, the introduction of higher-order integration methods can significantly increase the computational complexity due to the historical state dependency of the neural network. Compared to more complex higher-order integration methods, Euler’s Method offers higher computational efficiency. During the discretization of dynamics, this method requires a minimal number of neural network evaluations to obtain the next state within the optimization time window, thus effectively reducing computational overhead and improving the real-time performance of the overall optimization solver.

2.2.4. Dataset Composition

The dataset constructed in this study includes data from multiple sources to ensure that the neural network model can comprehensively learn the vehicle dynamics under different friction conditions. Specifically, we collected 200,000 driving data points from high-friction environments, 200,000 hybrid data points combining different modeling methods, and 119,966 real vehicle operation data points to systematically analyze the impacts of different data types on vehicle modeling accuracy. Additionally, we further investigated the interaction between the physical vehicle model’s described modeling methods and real data to optimize the model’s dynamic prediction capabilities.

The real dataset primarily comes from the high-friction oval track test data released by Nathan A. Spielberg et al. [18], which provided reliable real-world reference data for the neural network. To further enhance the model’s generalization ability under different friction environments, we also collected 203,200 additional data points. These data cover the lateral acceleration range of the vehicle driving along the oval track under two different road surface friction coefficients, allowing the model to more accurately learn tire-road interaction characteristics. Ultimately, by integrating the data from these various sources, we constructed a comprehensive and diverse dataset, which provided a solid foundation for training the dilated convolutional neural network vehicle model.

2.2.5. Data Processing

Before training, both the real-world data and the collected dataset were preprocessed. To enable the network to better learn the vehicle’s dynamic characteristics, five representative motion states of the vehicle, denoted as $x_t$, were extracted from the full dataset and used as the network input. The dataset was then divided into training, validation, and test sets in a ratio of 70%, 15%, and 15%, respectively, ensuring consistent data distribution across the subsets. Each data sample consists of the vehicle’s state information at the current time step and the three preceding time steps. However, there is no temporal continuity between the adjacent samples, which ensures that the model learns generalizable motion patterns and improves its performance under varying driving conditions. To provide a clearer view of the dataset’s characteristics, Figure 3 presents feature histograms illustrating the distributions across the training, validation, and test sets.

Figure 3 presents stacked histograms illustrating the distribution of different features across the training, validation, and test datasets. Overall, the feature distributions among the datasets exhibit a high degree of consistency, indicating that the training data effectively represents the characteristics of the development and test sets. The distributions vary across different features: yaw rate and lateral velocity show relatively concentrated distributions, while longitudinal velocity, steering angle, and front-axle longitudinal tire force exhibit wider ranges, reflecting more complex dynamic behavior. As time increases, slight shifts in the distribution of some features are observed; however, the overall trends remain stable.

2.2.6. Model Training

As previously described, the dataset was divided into training, development, and test sets in a ratio of 70%, 15%, and 15%, respectively. To ensure temporal correlation within each individual input data segment and independence between different input segments, Gaussian-based randomization was applied during the dataset construction process. Model training was conducted on a computer equipped with an Intel i7-10700 processor and 16 GB of RAM. During training, the mean squared error (MSE) was used as the loss function, which is defined as follows:

$$\underset{Z_{DC}}{\mathrm{minimize}}{\displaystyle \frac{1}{N}}{{\displaystyle \sum _{i=1}^N‖{\left[\begin{array}{c}{\dot{U}}_y\\ \dot{r}\end{array}\right]}_{\mathrm{t}+1}-{\left[\begin{array}{c}{\widehat{\dot{U}}}_y\\ \widehat{\dot{r}}\end{array}\right]}_{t+1}‖}}_2,$$

(20)

where N denotes the total number of trainings samples, and ${\left[\begin{array}{c}{\widehat{\dot{U}}}_y\\ \widehat{\dot{r}}\end{array}\right]}_{t+1}$ represents the predicted value of the target vehicle state.

Based on the loss function defined in the formula, the Adam optimization algorithm is employed for first-order optimization of the network parameters; the parameters of the optimizer are set to their default values (β₁ = 0.9, β₂ = 0.999, epsilon = 1 × 10⁻⁸), with the learning rate set to 0.0001. In each training epoch, only 1000 mini-batches are updated, and the training is continued for 1500 epochs to enhance model convergence and prediction accuracy.

2.2.7. Training Results Analysis

This section provides a comparative analysis of DCVSM against the bicycle model (BM) and the neural network-based vehicle system model (NNVSM). Additionally, a conventional convolutional neural network-based vehicle system model (CNNVSM) is included for comparison. The objective is to validate the superiority and applicability of the proposed DCVSM in vehicle modeling.

Figure 4a illustrates the variation trends of mean squared error (MSE) loss on the training and validation sets for BM, NNVSM, DCVSM, and CNNVSM over 1500 training epochs. As shown in the figure, BM, NNVSM, and CNNVSM stabilize at a very early stage. Although they converge quickly, their learning capacities are limited. In contrast, DCVSM continues to exhibit a significant decline in loss beyond 400 epochs, indicating higher learning efficiency and a sustained convergence process. CNNVSM performs relatively poorly during both training and validation, which can be attributed to the convolutional neural network’s need for pooling and padding operations. These steps increase the data volume and consequently reduce the learning efficiency while substantially increasing the computational cost. This observation further supports the effectiveness of the proposed method. Figure 4b presents a further quantitative comparison of the prediction errors on the test set for the four models. The corresponding MSE values for BM, NNVSM, DCVSM, and CNNVSM are 5.4672 × 10⁻⁴, 4.7194 × 10⁻⁴, 4.5278 × 10⁻⁵, and 1.0395 × 10⁻⁶, respectively. Comparing these results, it is evident that DCVSM achieves the best convergence in terms of prediction error and delivers the highest prediction accuracy, demonstrating its robust predictive capability under various driving conditions.

In summary, compared with BM, NNVSM, and CNNVSM, the proposed DCVSM integrates physical model information to some extent, which enhances prediction stability. Moreover, it maintains high modeling accuracy even with fewer trainable parameters, further validating the effectiveness of the proposed approach.

3. Design of Model Predictive Control Algorithm

3.1. Application of Dilated Convolution in Model Predictive Control

To verify the trajectory tracking ability of the DCMPC under the influence of complex vehicle dynamics, this paper constructs an optimization control problem based on the steering system. In this control framework, because the vehicle’s longitudinal velocity $U_x$ and longitudinal forces $F_{xf}$ are external inputs to the lateral dynamics, their computation is not directly included in the optimization problem. During the optimization process, the steering angle velocity $\dot{\delta }$ is used as the control input, with steering angle $\delta$ being treated as a state variable of the system. This modeling approach extends the vehicle’s state vector representation, allowing it to simultaneously include longitudinal inputs ($U_x$ and $F_{xf}$). Ultimately, a complete system state equation is derived, as shown in Equation (21).

$$x_t={\left[r,U_y,U_x,\delta ,F_{xf}\right]}_t$$

(21)

In the DCMPC, to ensure that the vehicle can accurately track the reference trajectory, the system adopts the cost function defined by Equation (22). To enable a direct and fair comparison of the performance of the DCMPC, traditional MPC, and NNMPC within the bicycle model framework, this study uses a commonly employed trajectory tracking cost function in the autonomous driving field, simplifying it to its basic form, which only constrains $e$ and steering angle velocity $\dot{\delta }$. It is important to note that, because the steering angle $\dot{\delta }$ is directly used as the control input in the optimization problem, its rate of change can be directly reflected in the cost function and constrained using a quadratic weight $Q_{\delta }$ to ensure the smoothness of the steering angle variation. Additionally, to ensure trajectory tracking accuracy, the lateral error $e^k$ is also included in the cost calculation and weighted by a quadratic weight $Q_e$. Furthermore, this study introduces a relaxation factor $\epsilon$ to the traditional cost function [31]. During the actual operation of the controller, strict constraints may prevent the optimal solution from being found within the limited computation time, thus affecting the system’s real-time performance and stability. Therefore, the relaxation factor is introduced as an optimization variable, allowing some constraints to be relaxed while maintaining high solution efficiency and control performance, thereby improving the robustness and real-time computational capability of the controller.

$$J={\displaystyle \sum _{k=1}^N\left(Q_e{\left(e^k\right)}^2+Q_{\delta }{\left({\dot{\delta }}^k\right)}^2+\rho {\epsilon }^2\right)}$$

(22)

Incorporating the neural network model into the optimization results forms the following nonlinear problem, as shown in Equation (23):

$$\begin{array}{cc}\hfill \min & J\hfill \\ \hfill s.t.& h^1=h^{meas}\hfill \\ & x^{k+1}=f_{NN}\left(h^k\right),k=1,\dots ,N-1\hfill \\ & {\dot{\delta }}_{LB}\le {\dot{\delta }}^k\le {\dot{\delta }}_{UB},k=1,\dots ,N\hfill \\ & {\delta }_{LB}\le {\delta }^k\le {\delta }_{UB}\hfill \end{array}$$

(23)

During the optimization process, the discretized neural network model is constructed as a function of historical state information and represented by DCNN. The simultaneous transcription method is used to unfold the DCNNVSM at each optimization stage within the prediction time domain, as shown in Figure 5. In the initial optimization phase, the neural network’s input contains only historical state information from real measurements, ensuring that the optimization problem is solved starting from the real physical state. In the subsequent optimization phases, the input data gradually shift from historical states to optimization variables (predictive states), allowing the DCNN to leverage its learned system dynamics.

At each stage of the optimization prediction, the system model is dynamically adjusted and evolves, with the final optimization formulation consisting solely of optimization state variables and control inputs. To objectively evaluate the control performance of the DCNN model, traditional MPC and NNMPC using the planar bicycle model are introduced under the same lateral control optimization framework as Equation (23) for comparative experiments. Both the NNMPC and traditional MPC use the same cost function J and weight parameters as the DCMPC, ensuring consistent optimization objectives and evaluation criteria when switching prediction models. This comparative approach effectively assesses the prediction accuracy, control stability, and tire-slip energy consumption of the DCMPC, thereby verifying its applicability and advantages under complex dynamic environments.

3.2. Tire-Slip Energy Consumption

To further verify the performance of the proposed method in terms of energy loss, we calculated the tire-slip energy loss generated during operation for the three methods. The longitudinal tire-slip energy consumption P is defined as shown in Equation (24):

$$P={\displaystyle \sum _{i=1}^4{F}_{si}·v_{si}},$$

(24)

where $F_{si}$ and $v_{si}$ represent the longitudinal tire slip force and the slip velocity, respectively.

4. Simulation Testing and Analysis

This study selects traditional MPC and NNMPC algorithms as benchmarks to validate the effectiveness of the proposed DCMPC approach in terms of trajectory tracking accuracy and stability for intelligent vehicles. Furthermore, the slip energy loss rates during the tracking process are calculated to demonstrate the energy efficiency advantage of the proposed method. A co-simulation platform is established using CarSim and Simulink, and simulation tests are conducted under double-lane change scenarios. The simulations are performed at two vehicle speeds: 36 km/h and 72 km/h. For each speed condition, two different road surface friction coefficients are used: dry road (high friction coefficient, μ = 0.85) and slippery road (low friction coefficient, μ = 0.3). The main parameters of the controlled vehicle model and its controller are listed in

Table 2. Key parameters of the vehicle dynamics and control system used in simulation studies.

Name/Unit	Symbolic	Value
Sampling time/s	T	0.05
Relaxation factor weighting coefficient	$\rho$	1000
Vehicle mass/kg	m	1732
Moment of inertia/(kg·m2)	Iz	4175
Front wheel lateral cornering stiffness/(N/rad)	Ccf	66,900
Rear wheel lateral cornering stiffness/(N/rad)	Ccr	62,700
Front wheel longitudinal cornering stiffness/(N/rad)	Clf	66,900
Rear wheel longitudinal cornering stiffness/(N/rad)	Clr	62,700
Distance from front axle to center of mass/m	a	1.232
Distance from rear axle to center of mass/m	b	1.468
Prediction horizon/step	Np	30
Control horizon/step	Nc	3

. The primary parameters of the controlled vehicle model and its corresponding controller are presented in Table 1. Within the MPC framework, the selection of the prediction horizon (NP) and control horizon (NC) significantly influences the controller’s performance. The prediction horizon, NP, determines the extent of the controller’s foresight into future system states. A shorter NP may lead to myopic control strategies and imprecise tracking, whereas a longer NP enhances predictive capability but increases computational complexity, potentially compromising real-time applicability. The control horizon, NC, defines the number of future control moves optimized at each step. A shorter NC might restrict the controller’s responsiveness to dynamic changes or disturbances, while a longer NC increases the number of decision variables, thereby escalating the computational burden. Balancing these considerations, this study adopts NP = 30 and NC = 3 for simulation experiments.

Double-Lane Change Maneuver Simulation Analysis

Figure 6, Figure 7, Figure 8 and Figure 9 illustrate the lateral displacement variations and lateral displacement errors under high- and low-friction conditions at vehicle velocities of 36 km/h and 72 km/h, respectively, for the three control algorithms. As observed from Figure 6 and Figure 8, the DCMPC demonstrates superior tracking performance during cornering, regardless of whether the vehicle is operating at low or medium-high velocity, or on high- or low-friction surfaces. According to the statistical data presented in Figure 7, at a vehicle speed of 36 km/h and under a road friction coefficient of μ = 0.85, the peak lateral errors for the DCMPC, NNMPC, and traditional MPC are 0.1024 m, 0.1090 m, and 0.3192 m, respectively, with corresponding RMSE values of 0.0335 m, 0.0342 m, and 0.0836 m, respectively. Under a lower friction coefficient of μ = 0.3, the peak lateral errors are 0.0987 m for the DCMPC, 0.1155 m for the NNMPC, and 0.3027 m for the traditional MPC, with RMSE values of 0.0330 m, 0.0349 m, and 0.0810 m, respectively. Both the DCMPC and NNMPC demonstrate better overall control performance, while the traditional MPC shows inferior results. According to the statistical data in Figure 9, at a higher vehicle speed of 72 km/h and with a friction coefficient of μ = 0.85, the peak lateral errors for the DCMPC, NNMPC, and traditional MPC increase to 0.3610 m, 0.3954 m, and 0.5945 m, with corresponding RMSE values of 0.1199 m, 0.1287 m, and 0.1831 m. Under the low-friction condition of μ = 0.3, the peak lateral errors are 0.3307 m, 0.2921 m, and 0.3757 m, and the RMSE values are 0.1028 m, 0.1040 m, and 0.1245 m for the DCMPC, NNMPC, and traditional MPC, respectively. At higher speeds, the traditional MPC clearly underperforms as compared with the other two methods. Although all three algorithms exhibit increased peak errors and RMSE values under high-speed conditions, the DCMPC consistently achieves the smallest lateral errors and the lowest fluctuation amplitude. These results further demonstrate the adaptability and effectiveness of the DCMPC under various operating conditions.

Figure 10 and Figure 11 illustrate the front wheel steering angle (FWSA) inputs under different vehicle velocities and road friction coefficients for the three controllers. It can be observed that DCMPC outperforms both the traditional MPC and NNMPC in terms of FWSA control across all tested conditions. As the vehicle velocity increases, both the traditional MPC and NNMPC exhibit varying degrees of oscillation in FWSA, whereas the DCMPC consistently maintains stability during the initial phase. Under the low-friction condition at 36 km/h and the high-friction condition at 72 km/h, the DCMPC keeps the FWSA within 5°. Although the FWSA peak value under the 72 km/h low-friction condition reaches −11.75°, it still meets the physical requirements for intelligent vehicles. In contrast, the peak values for the traditional MPC and NNMPC reach 21.45° and 21.41°, respectively, exceeding the vehicle’s physical limits. This demonstrates that the DCMPC offers superior control performance and generates FWSA values more in line with the physical demands of intelligent vehicles.

Figure 12 and Figure 13 also present a comparison of lateral yaw rates under different vehicle velocities and friction coefficients. The results show that the DCMPC maintains excellent stability during the initial stage under both 36 km/h and 72 km/h driving conditions. It can be observed that the amplitude of the lateral yaw rate increases with vehicle speed for all three controllers. Additionally, the DCMPC consistently achieves a lower yaw rate amplitude than the other two controllers across all conditions. Notably, under the high-speed, low-friction condition (72 km/h), the DCMPC does not exhibit significant oscillations, indicating that the proposed control method maintains robust stability even in challenging dynamic environments.

Figure 14 and Figure 15 display the centroid side-slip angle for the three controllers under different vehicle velocities and road friction conditions. Under the 36 km/h condition, all controllers exhibit relatively smooth behavior, except for slight oscillations observed during the start-up phase in the traditional MPC and NNMPC. Under the 72 km/h condition, the centroid side-slip angle for the DCMPC remains within a reasonable range with slight fluctuations, while the traditional MPC and NNMPC experience larger oscillations. In comparison, the DCMPC shows smaller fluctuations, demonstrating superior overall performance. Therefore, compared to the other two controllers, the DCMPC not only performs well at low-to-medium velocity but also provides better tracking control performance at higher velocity.

As shown by the comparisons of Figure 16 and Figure 17 with Figure 7 and Figure 9, higher tire-slip energy losses are observed during the periods of 0–3 s and 4–6 s, which correspond to the intervals in which the vehicle exhibits larger tracking errors. This phenomenon is primarily due to steering maneuvers taking place during these periods, which have a notable impact on tire slip-related energy consumption. After 6 s, the tracking error gradually decreases and stabilizes, and correspondingly, the tire slip energy loss also diminishes and stabilizes. This observation indicates a clear correlation: tire-slip energy consumption increases with larger tracking errors and decreases as tracking accuracy improves.

The simulation results demonstrate that, under the condition of smaller tracking errors, energy consumption is also reduced. Although under the 72 km/h scenario the DCMPC exhibits slight oscillations in both tracking error and tire-slip energy consumption, its overall performance surpasses that of traditional MPC and NN-MPC. Specifically, it achieves lower energy consumption while maintaining superior tracking accuracy. These findings suggest the potential of extending the DCMPC framework into a multi-objective optimization approach that simultaneously balances tracking precision and energy efficiency.

5. Conclusions

• This paper proposes an intelligent vehicle trajectory tracking and energy consumption optimization control framework that integrates dilated convolutional neural networks (DCNNs) with model predictive control (MPC). By introducing the dilated convolution structure, the method significantly improves path prediction accuracy, enhances feature extraction capabilities, and extends the perception range, providing more accurate trajectory information for subsequent model predictive control. This integration significantly improves trajectory tracking performance in complex dynamic environments.
• The proposed control strategy, combining the DCNN-based path prediction model with MPC, effectively achieves stable vehicle control without prior knowledge of the road surface friction coefficient. By coordinating control inputs with predicted trajectories, the system not only optimizes energy consumption but also maintains high trajectory tracking accuracy, demonstrating superior adaptability and robustness compared to traditional methods, especially in complex scenarios such as time-varying friction.
• In the simulation analysis, the proposed method was validated under different road surface friction coefficients and compared with traditional methods. The results show that the proposed method not only improves trajectory tracking accuracy but also demonstrates better energy consumption optimization, confirming its strong practical applicability in intelligent transportation and autonomous driving fields.
• In future research, the scope will be further expanded to focus on the robustness of the proposed method under extreme conditions and high noise environments as well as practical deployment challenges such as hardware constraints, real-time processing capabilities, and system scalability. The impact of the relaxation factor (ε) on the MPC framework will be studied in depth and compared with other stability enhancement methods. Additionally, testing scenarios will be extended to include more complex dynamic environments, such as consecutive S-curves and emergency obstacle avoidance, to comprehensively validate the robustness and adaptability of the proposed method. These efforts will contribute to the optimization of the control algorithm and enhance its practical application in intelligent vehicles.