1. Introduction
As autonomous driving technology advances, the increasing importance of accurate speed tracking in real driving scenarios such as adaptive cruise control and lane merging has led to a growing demand for improvements in longitudinal control systems. Furthermore, with the advent of Electric Vehicles (EVs) and the shift from internal combustion engines to motor-driven powertrains in recently produced vehicles, significant research and development has focused on longitudinal control methods suited to these new characteristics [
1,
2,
3]. As shown in
Figure 1, the longitudinal control systems covered in this paper output throttle or brake pedal commands to track a given speed profile created by the high-level planning system. A key requirement for these control systems is to reduce speed tracking errors without resorting to aggressive brake or throttle operations.
Traditionally, Proportional-Integral-Derivative (PID) control has been widely utilized for longitudinal speed tracking due to its simplicity and ease of implementation without relying on a system model [
4,
5,
6]. However, PID control does not account for system dynamics and external disturbances, which can lead to significant overshoot or persistent oscillations when subjected to noise or delays [
7,
8]. To address these limitations, advanced control techniques such as Sliding Mode Control (SMC) [
9,
10,
11], Linear Quadratic Regulator (LQR) [
12,
13], and Model Predictive Control (MPC) [
14,
15,
16] that directly consider vehicle dynamics have been developed. SMC, for instance, drives the system states toward a predefined sliding surface, ensuring rapid convergence to the desired state based on the designed dynamics. However, these methods are not inherently designed to minimize a given cost function or performance metric and thereby lack the ability to consider multiple objectives such as tracking performance and steering stability in the determination of control actions. In contrast, techniques like LQR and MPC solve optimization problems based on a multivariable objective function, enabling the derivation of optimal control actions that balance tracking accuracy and stability. Notably, MPC has emerged as one of the most widely studied model-based approaches in the field of autonomous driving control [
17]. MPC allows for the establishment of goals considering a variety of variables, the definition of constraints on system inputs and states during the initial design phase, and the facilitation of real-time control through a receding horizon technique that continuously updates the prediction horizon [
18].
Given the nature of MPC, which predicts and controls the vehicle’s state based on a model, using a model that accurately represents the longitudinal characteristics of actual vehicles can significantly improve tracking performance. Existing MPC studies [
14,
19,
20] have focused primarily on improving vehicle dynamics models. However, to fully replicate the longitudinal behavior of real vehicles, it is crucial to consider factors such as powertrain delays in addition to dynamics. These additional factors are vital because control signals take time to be executed in the actual vehicle and actuator lag further delays the response, regardless of the vehicle model. This delay means that the calculated longitudinal commands will not be executed instantly, leading to a model mismatch and deteriorated performance. As a result, control oscillation and instability can occur.
In longitudinal driving, any vehicle will experience delays in response to control inputs. As shown in
Figure 2, the overall signal delay observed in our experimental electric vehicle was approximately 0.2 s. Additionally, the powertrain gradually responds after a certain period when a step wheel torque input is applied. Therefore, powertrain delays can be decomposed into pure delay and actuation lag. Pure delay arises from CAN communication and computation processes between the control module and the drive/brake motor system, and driveline freeplay, where gaps in mechanical components cause delays in torque transmission, can also contribute to pure delay. Furthermore, actuation lag occurs because the drive motor or brake actuator requires time to respond to the vehicle, and this lag is influenced by the hardware capability of the actuator and the design of the low-level control system. Moreover, mechanical filtering in the driveline, caused by the elasticity of components such as drive shafts and tires, can exacerbate this lag.
To address these challenges, this paper presents a novel MPC approach for longitudinal control in autonomous electric vehicles that explicitly incorporates powertrain delays into the control model using a First Order Plus Dead Time (FOPDT) model [
21]. Unlike existing MPC methods that primarily focus on improving vehicle dynamics models without accounting for actuator delays, our approach models both the pure time delay and actuation lag inherent in electric vehicle powertrains. This integration allows the controller to anticipate and compensate for delays, significantly enhancing speed tracking accuracy and control stability.
Additionally, we develop detailed torque maps for both the motor and brake systems, capturing the non-linear torque characteristics of the electric vehicle’s powertrain. By integrating these torque maps into the control framework, the controller can accurately translate the desired drive force into appropriate throttle and brake inputs, further improving control precision.
The key contributions of this work are as follows:
By incorporating a FOPDT model into the MPC, the controller effectively handles the inherent delays in electric vehicle powertrains, which has been largely overlooked in previous studies.
The creation of detailed torque maps for the motor and brake systems, enabling precise control over throttle and brake inputs in response to the target wheel torque.
A comprehensive evaluation of the proposed controller through both simulation and real-world experiments, demonstrating significant improvements in speed tracking accuracy and control smoothness compared to conventional MPC and PID controllers.
The remainder of this paper is organized as follows.
Section 2 introduces the proposed system architecture for longitudinal speed tracking control.
Section 3 delves into the MPC considering powertrain delays, while
Section 4 discusses the torque map for the throttle and brake pedals.
Section 5 presents the experimental results and analysis from both simulations and real vehicles. Finally,
Section 6 concludes the paper.
2. Overall System Architecture
As illustrated in
Figure 3, the proposed longitudinal control system for Autonomous Electric Vehicles (AEVs) tracks the reference speed profile planned by the higher-level module and outputs the throttle and brake pedal values for the vehicle. This control system is composed of two key modules: (1) a longitudinal MPC module considering powertrain delays and (2) a wheel torque table module for drive/brake torque and throttle/brake pedal values.
The MPC module first receives a reference speed profile, , composed of target speeds over a fixed horizon as input, and outputs the target drive force, F. The currently measured longitudinal speed of the vehicle is used as the initial state of the MPC. The MPC solves an optimization problem using a model that considers powertrain delays, aiming to minimize the difference between the given reference speed profile and the predicted vehicle longitudinal speed.
The wheel torque table module then receives the F output from the MPC module and converts it to the wheel torque . Based on the input, the current vehicle speed, and the throttle torque table during acceleration, or the brake torque table during deceleration, it determines the throttle and brake values. These throttle and brake values are then communicated to the AEV.
This wheel torque table module is designed by acquiring driving data from the vehicle CAN, which consists of pedal positions, motor torque, acceleration, and speed, and then interpolating and smoothing the data to accurately reflect the torque characteristics of the drive motor and brakes.
4. Wheel Torque Table Module
The output of the MPC module is the driving force,
F, while the actual control input of the AEV is the values of the throttle and brake pedal values. Therefore, the wheel torque table module is designed to effectively convert
F into throttle and brake pedal values. As illustrated in
Figure 4, the
F is multiplied by the wheel radius,
, to convert it into the driving wheel torque,
, in the initial stage of the module.
This torque is then used as the input for the wheel torque tables for throttle and brake.
In the wheel torque table module, the amount of brake or throttle pedal input is determined based on the target wheel torque input, . Therefore, modeling is required for the drive motor used during acceleration and the brake system used during deceleration. To prevent simultaneous throttle and brake inputs, we use the regenerative braking torque value when both the throttle and brake pedals are at zero as a neutral value. If exceeds this neutral value, the throttle torque table is used; if it is below, the brake torque table is employed to determine the control inputs.
4.1. For the Throttle Pedal
In conventional internal combustion engine vehicles, converting torque values to throttle pedal position required torque tables for each gear, taking into account engine RPM and the current gear ratio [
4,
25]. However, electric vehicle powertrains mostly operate with a single gear without a transmission, eliminating the need for gear shifting and thereby requiring only one torque table for the drive motor. Given that motor torque decreases as motor RPM increases, achieving a specific torque requires different throttle positions depending on the speed of the vehicle. Therefore, in this study, motor torque data relative to vehicle speed and throttle position were collected from both on-road natural driving and experimental driving on the proving ground.
As shown in the data acquired section of
Figure 5, the orange data obtained from the proving ground tests with constant throttle inputs and the blue data distribution obtained from normal on-road driving are presented. The necessary vehicle speed, throttle position, and motor torque data for the table generation were collected through the vehicle CAN. Then, as illustrated in the wheel torque table of
Figure 5, these data were interpolated into continuous values using the cubic spline approximation method [
26] to create a lookup table. Through this process, when the current speed and target wheel torque values are input, the appropriate amount of throttle pedal is determined.
4.2. For the Brake Pedal
The vehicle’s brake system is designed based on hydraulic brakes, which typically provide braking force proportional to the brake pedal input. Therefore, when determining the required brake pedal input for a target drive wheel torque, speed information is not necessary, unlike in the case of the throttle pedal, and the modeling can be conducted based on the relationship between the brake pedal input and the drive wheel torque. However, unlike the motor torque used for the throttle pedal, the torque values during braking are not provided by the vehicle’s CAN bus. Thus, to convert the longitudinal acceleration data obtained from the CAN into wheel torque, we inversely applied Equations (
1) and (
8) inversely to convert the longitudinal acceleration value obtained from the CAN into wheel torque. Subsequently, as illustrated in
Figure 6, this relationship was modeled using a cubic polynomial, such that when a target wheel torque below the neutral value is input, the appropriate brake pedal input is determined.