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Real-Time Experimental Evaluation and Analysis of PID and MPC Controllers Using HIL Setup for Robust Steering System of Autonomous Vehicles
利用 HIL 设置对 PID 和 MPC 控制器进行实时实验评估和分析，用于自动驾驶汽车的鲁棒转向系统

S. GOKUL KRISHNAN ® ®  ^("® "){ }^{\text {® }}, P. SURESH KUMAR © ©  ^("© "){ }^{\text {© }}©, NADHEEM NASSAR MATARA © ©  ^("© "){ }^{\text {© }}©, AND YONG WANG © ©  ^("© "){ }^{\text {© }}©
S.GOKUL KRISHNAN ® ®  ^("® "){ }^{\text {® }} , P. SURESH KUMAR © ©  ^("© "){ }^{\text {© }}© , NADHEEM NASSAR MATARA © ©  ^("© "){ }^{\text {© }}© , AND YONG WANG © ©  ^("© "){ }^{\text {© }}© 1 1 ^(1){ }^{1} School of Electrical Engineering, Vellore Institute of Technology, Vellore 632014, India
1 1 ^(1){ }^{1} 韦洛尔理工学院电气工程学院，印度韦洛尔 632014 2 2 ^(2){ }^{2} Automotive Research Centre, Vellore Institute of Technology, Vellore 632014, India
2 2 ^(2){ }^{2} 印度韦洛尔 632014，韦洛尔理工学院汽车研究中心 3 3 ^(3){ }^{3} Department of Automotive Engineering, Vellore Institute of Technology, Vellore 632014, India
3 3 ^(3){ }^{3} 维洛尔理工学院汽车工程系，印度维洛尔 632014 4 4 ^(4){ }^{4} Department of Systems Science and Industrial Engineering, Binghamton University, Binghamton, NY 13902, USA
4 4 ^(4){ }^{4} 宾汉姆顿大学系统科学与工业工程系，美国纽约州宾汉姆顿市 13902 号Corresponding author: P. Suresh Kumar (suresh.kumar@vit.ac.in)
通讯作者：P：P. Suresh Kumar (suresh.kumar@vit.ac.in)

Abstract  摘要

The development of autonomous vehicles has recently received substantial impetus, fueled by researchers and industry personnel. The need for powerful steering control in autonomous vehicles is critical for assuring the vehicle’s safety and reliability. Robust steering control allows for precise and accurate maneuvering, allowing the vehicle to traverse complicated road conditions. Comparative research on the certification of a robust steering system for autonomous vehicles is presented in this paper. Traditional controllers (PD and PID) are compared with a modern Model Predictive Control (MPC) controller that uses a multi-turn potentiometer and incremental encoder for position feedback. The controllers are designed in MATLAB Simulink and deployed for real-time testing on a Speedgoat performance realtime target Hardware-in-the-Loop (HIL) machine. The study focuses on evaluating the steering system’s real-time performance in terms of accuracy and robustness. The novelty is that this work is carried out in a real experimental modified electric vehicle and presents real-time results obtained using the HIL machine and Rapid Control Prototyping (RCP) technique. The research covers a thorough examination of the experimental hardware configuration, system identification, controller design, and data-gathering technologies. A significant contribution of this research is the use of the HIL machine for real-time performance testing of different controllers with different velocities and sample times, specifically in a speed breaker scenario. To analyze each controller’s response, real-time data is logged at a high sampling rate of 0.1 milliseconds. The research contributes to the advancement of driverless vehicles by providing insights into the optimal performance of steering systems. It also emphasizes the importance of real-time testing of the robust performance of different controllers to ensure human safety in driverless cars.
最近，在研究人员和行业人员的推动下，自动驾驶汽车的发展获得了巨大的推动力。自动驾驶汽车需要强大的转向控制，这对于确保汽车的安全性和可靠性至关重要。强大的转向控制可以实现精准的操纵，使车辆能够穿越复杂的路况。本文介绍了有关自动驾驶汽车稳健转向系统认证的比较研究。传统控制器（PD 和 PID）与现代模型预测控制 (MPC) 控制器进行了比较，后者使用多圈电位计和增量编码器进行位置反馈。控制器在 MATLAB Simulink 中设计，并在 Speedgoat 性能实时目标硬件在环 (HIL) 机器上进行实时测试。研究的重点是评估转向系统在精度和鲁棒性方面的实时性能。这项工作的新颖之处在于，它是在真实的改装电动汽车实验中进行的，并展示了使用 HIL 机器和快速控制原型（RCP）技术获得的实时结果。研究涵盖了对实验硬件配置、系统识别、控制器设计和数据采集技术的全面检查。本研究的一个重要贡献是使用 HIL 机器对不同速度和采样时间的控制器进行实时性能测试，特别是在速度断路器场景中。为了分析每个控制器的响应，以 0.1 毫秒的高采样率记录实时数据。 这项研究有助于深入了解转向系统的最佳性能，从而推动无人驾驶汽车的发展。研究还强调了对不同控制器的稳健性能进行实时测试的重要性，以确保无人驾驶汽车中的人类安全。

INDEX TERMS Steering system, autonomous vehicle, model predictive control (MPC), rapid control prototyping (RCP), hardware-in-the-loop (HIL), robust control.
索引词：转向系统、自动驾驶汽车、模型预测控制（MPC）、快速控制原型（RCP）、硬件在环（HIL）、鲁棒控制。

NOMENCLATURE  名称

PARAMETERS  参数
K p K p K_(p)K_{p} - Proportional Gain.
K p K p K_(p)K_{p} - 比例增益。
K i K i K_(i)K_{i} - Integral Gain.
K i K i K_(i)K_{i} - 积分增益。
The associate editor coordinating the review of this manuscript and approving it for publication was Ton Duc Do ( 1 ) ( 1 ) ^((1)){ }^{(1)}.
负责协调本稿件审核并批准发表的副主编是 Ton Duc Do ( 1 ) ( 1 ) ^((1)){ }^{(1)} 。
K d K d K_(d)K_{d} - Derivative Gain.
K d K d K_(d)K_{d} - 衍生收益。
N N NN - Filter Coefficient.
N N NN - 滤波系数。
u ( t ) u ( t ) u(t)u(t) - Output of PID Controller.
u ( t ) u ( t ) u(t)u(t) - PID 控制器的输出。
U t ∗ ( x ( t ) ) U t ∗ ( x ( t ) ) U_(t)^(**)(x(t))U_{t}^{*}(x(t)) - Output of MPC Controller.
U t ∗ ( x ( t ) ) U t ∗ ( x ( t ) ) U_(t)^(**)(x(t))U_{t}^{*}(x(t)) - MPC 控制器的输出。
J J JJ - Cost Function.
J J JJ - 成本函数。
Nc - Control Horizon.
Nc - 控制地平线。
N p N p NpN p - Prediction Horizon.
N p N p NpN p - 预测地平线。
T s T s TsT s - Sample Time.   T s T s TsT s - 采样时间。

ABBREVIATIONS  缩略语

CNN - Convolution Neural Network.
CNN - 卷积神经网络。
EPS - Electric Power Steering.
EPS - 电动助力转向。
HIL - Hardware-in-the-Loop.
HIL - 硬件在环。
I/O - Input Output.
I/O - 输入输出。
IC - Internal Combustion.
IC - 内燃机。
LQR - Linear Quadratic Regulator.
LQR - 线性二次调节器。
MPC - Model Predictive Controller.
MPC - 模型预测控制器。
PD - Proportional-Derivative.
PD - 比例-派生。
PID - Proportional-Integral-Derivative.
PID - 比例-积分-微分。
PWM - Pulse Width Modulation.
PWM - 脉宽调制。
RCP - Rapid Control Prototyping.
RCP - 快速控制原型。
TTL - Transistor-Transistor Logic.
TTL - 晶体管-晶体管逻辑。

I. INTRODUCTION  I.引言

The development of autonomous ground vehicles, also referred to as self-driving cars or autonomous vehicles, has significant implications and potential applications in future smart and sustainable transportation. It is a very crucial turning point in transportation, with implications for safety, efficiency, accessibility, and environmental sustainability. The ability to improve road safety is one of the key benefits of autonomous ground vehicles. Human distractions are the major source of accidents, and autonomous cars can reduce or eliminate them by utilizing modern sensors, artificial intelligence, and real-time data processing. Autonomous vehicles, by reducing the need for human drivers, have the potential to reduce accidents and save lives. Furthermore, self-driving ground vehicles have the potential to improve transportation efficiency. Because they can communicate with other vehicles and with the infrastructure, these cars can optimize routes, minimize traffic congestion, and reduce fuel use.
自主地面车辆（也称为自动驾驶汽车或自主车辆）的发展对未来智能和可持续交通具有重大影响和潜在应用。这是交通领域一个非常关键的转折点，对安全、效率、交通便利性和环境可持续性都有影响。能够改善道路安全是自动驾驶地面车辆的主要优势之一。人类分心是事故的主要根源，而自动驾驶汽车通过利用现代传感器、人工智能和实时数据处理，可以减少或消除人类分心。自动驾驶汽车通过减少对人类驾驶员的需求，有可能减少事故和挽救生命。此外，自动驾驶地面车辆还有可能提高运输效率。由于可以与其他车辆和基础设施进行通信，这些车辆可以优化路线，最大限度地减少交通拥堵，并降低燃料消耗。
The autonomous vehicle steering system can improve safety, by eliminating driver errors. Its performance majorly depends on the sensor’s resolution and its range. The autonomous vehicle system receives feedback information from LiDAR, radar, camera, steering angular position, and velocity measurement sensors. Robust steering control in autonomous vehicles is critical to guarantee the safe and reliable functioning of self-driving cars. The capacity of an autonomous vehicle to regulate its steering inputs reliably and precisely in a variety of complex environmental conditions is referred to as robust steering control. Improved safety is one of the key advantages of precise steering control. Autonomous vehicles must be capable of dealing with unforeseen scenarios such as unexpected barriers, unpredictable weather, or complex road terrains. Robust steering control enables the vehicle to respond rapidly and efficiently to these situations, assisting in collision avoidance and ensuring the safety of passengers, pedestrians, and other vehicles on
自动驾驶汽车转向系统可以消除驾驶员的失误，从而提高安全性。其性能主要取决于传感器的分辨率和范围。自动驾驶汽车系统接收来自激光雷达、雷达、摄像头、转向角位置和速度测量传感器的反馈信息。自动驾驶汽车的稳健转向控制对于保证自动驾驶汽车的安全可靠运行至关重要。自动驾驶汽车在各种复杂环境条件下可靠、精确地调节转向输入的能力被称为鲁棒转向控制。提高安全性是精确转向控制的主要优势之一。自动驾驶汽车必须能够应对不可预见的情况，如意外障碍、不可预测的天气或复杂的道路地形。稳健的转向控制使车辆能够快速、高效地应对这些情况，协助避免碰撞，确保乘客、行人和其他车辆的安全。
the road. A robust steering control system provides better performance for driver-less cars, in the presence of difficult road conditions like uneven terrain, slippery roads, speed breakers, potholes, heavy rain and snow.
路况。强大的转向控制系统能为无人驾驶汽车在不平坦的地形、湿滑的路面、减速带、坑洼路面、暴雨和大雪等恶劣路况下提供更好的性能。
In recent literature, different types of Model Predictive Control (MPC) controllers are used for autonomous vehicle robust steering control systems. The design of the MPC controller that utilized lateral and steering angle deviation, along with relative yaw angle to control steering angle for collision avoidance based on the LiDAR data is presented in [1]. MPC controller is used to compensate the side slip for improving the tracking performance for higher speeds [2], and its performance is evaluated based on the actuator’s bandwidth [3]. A linear model-based path tracker using an MPC controller by linearizing the sequence of future steering angles has been discussed [4]. Three different types of algorithms like Ziegler-Nichol’s, WAF-tune, and twiddle are used to tune the PID controller gains to improve the steering performance [5].
在最近的文献中，不同类型的模型预测控制（MPC）控制器被用于自动驾驶汽车的稳健转向控制系统。文献[1]介绍了基于激光雷达数据，利用横向偏差、转向角偏差和相对偏航角控制转向角以避免碰撞的 MPC 控制器的设计。MPC 控制器用于补偿侧滑，以提高更高速度下的跟踪性能[2]，并根据致动器的带宽对其性能进行了评估[3]。有学者讨论了一种基于线性模型的路径跟踪器，它使用 MPC 控制器对未来转向角序列进行线性化处理 [4]。有三种不同类型的算法，如 Ziegler-Nichol's、WAF-tune 和 twiddle，用于调整 PID 控制器增益以提高转向性能 [5]。
To ensure safe cooperation between humans and machines, several studies have proposed various methods to enhance autonomous vehicle steering performance. A model reference adaptive control approach is proposed by [6] to ensure cooperation between humans and machines, which will give robust performance for disturbances. A methodology for ensuring a smooth transition between autonomous steering control and driver input for low and high-speed maneuvers is given in [7]. Small-gain theory and an iterative scheme to learn and adapt to the driver’s steering torque, to jointly operate the vehicle is analyzed in [8].
为了确保人机之间的安全合作，一些研究提出了各种方法来提高自动驾驶汽车的转向性能。文献[6]提出了一种模型参考自适应控制方法，以确保人机之间的合作，从而在干扰情况下提供稳健的性能。文献[7]给出了一种确保低速和高速机动时自主转向控制与驾驶员输入之间平稳过渡的方法。文献[8]分析了学习和适应驾驶员转向扭矩的小增益理论和迭代方案，以共同操作车辆。
The design and development of an MPC controller for high-speed accident avoidance is proposed in [9]. A hierarchical control architecture that consists of a decisionmaking layer and a motion control layer, is validated with hardware-in-the-loop (HIL) testing. Reference [10] presented a new strategy that combines differential braking with autonomous steering for collision avoidance using an MPC controller to track the center line of the road and end the swerving maneuver is discussed in detail. In case of steer-by-wire system failure, [11] proposed a torque vectoring system in conjunction with the VSC system as a redundant measure to ensure the safety of the autonomous collision avoidance system. Takagi-Sugeno (T-S) fuzzy model based on a fuzzy Lyapunov control framework for automatic lane keeping under varying system constraints such as unknown crosswinds and varying road curvatures is presented in [12]. Artificial neural networks such as Convolution Neural Network (CNN) and Long-Short-Term-Memory Network are used to predict the steering angle for lane keeping even during poor visibility conditions [13]. A novel algorithm is proposed for automated lane changing using yaw rate, steering angular position, and steering wheel feedback torque from the electric power steering (EPS) system [14].
文献[9]提出了设计和开发用于高速事故规避的 MPC 控制器。通过硬件在环（HIL）测试，验证了由决策层和运动控制层组成的分层控制架构。参考文献[10]提出了一种新策略，将差速制动与自主转向相结合，使用 MPC 控制器跟踪道路中心线并结束转向动作，以避免碰撞。在线控转向系统失效的情况下，[11] 提出了一种与 VSC 系统相结合的扭矩矢量系统，作为确保自主防撞系统安全的冗余措施。文献[12]提出了基于模糊 Lyapunov 控制框架的 Takagi-Sugeno (T-S) 模糊模型，用于在未知横风和不同路面曲率等不同系统约束条件下的自动车道保持。卷积神经网络（CNN）和长短期记忆网络等人工神经网络被用于预测转向角，即使在能见度较差的条件下也能保持车道[13]。利用偏航率、转向角位置和来自电动助力转向（EPS）系统的方向盘反馈扭矩，提出了一种自动变道的新算法[14]。
A precise path-tracking control algorithm is proposed to accommodate varying vehicle weight, weight distribution, and velocity [4]. An optimal feedback controller is used
提出了一种精确的路径跟踪控制算法，以适应车辆重量、重量分布和速度的变化 [4]。优化反馈控制器用于
to handle the tracking errors and a feedforward controller is used to anticipate upcoming road turns to improve the autonomous vehicle steering tracking performance [5]. A trajectory tracking algorithm is analyzed in [15], using an MPC controller by considering a nonlinear multi-input and multi-output system with independent wheel steering capability. The adaptive steering controller [16] performance of MPC for autonomous vehicle steering systems is presented in [17].
来处理跟踪误差，并使用前馈控制器来预测即将出现的道路转弯，以提高自动驾驶汽车的转向跟踪性能 [5]。文献[15]通过考虑具有独立车轮转向能力的非线性多输入和多输出系统，使用 MPC 控制器分析了轨迹跟踪算法。文献[17]介绍了用于自主车辆转向系统的 MPC 自适应转向控制器[16]的性能。
Addressing trajectory planning for autonomous vehicle docking, [18] introduced an algorithm to determine the optimal path for seamless docking. Reference [19] compared the performance of Linear Quadratic Regulator (LQR) and MPC controllers in various scenarios, such as lane changing, perpendicular parking, and parallel parking, using a HIL system. Reference [20] presented an autonomous vehicle steering controller that utilizes an MPC controller to track the desired path, considering disturbances and time-varying parameter uncertainties. Reference [21] proposed a precise trajectory tracking control algorithm suitable for network delay in autonomous vehicles. To address vehicle stability regarding longitudinal velocity and mass, [22] designed an H -infinite robust controller that applies direct yaw moment control and active front steering control. Reference [6] developed a model reference adaptive controller capable of achieving robust performance despite disturbances and parameter uncertainties. [23] provided an insightful analysis of reference path-tracking control strategies and the advantages and limitations of robust and observer-based control strategies. Reference [24] contributed to this field by presenting robust control policies. In the context of lateral dynamics stability during maneuvers like single lane changes and J-turns, [25] proposed an adaptive backstepping control technique that effectively handles external disturbances and parameter uncertainties. Furthermore, [26] developed an adaptive two-layer control framework for a two-axle autonomous bus to prevent sideslips and rollovers, prioritizing safety, [27] designed a digital twin for safety system of an electric vehicle where several test scenarios can be simulated by tweaking many system parameters to observe the system state.
针对自主车辆对接的轨迹规划，[18] 提出了一种算法，用于确定无缝对接的最优路径。参考文献[19]使用 HIL 系统比较了线性二次调节器 (LQR) 和 MPC 控制器在变道、垂直停车和平行停车等不同场景中的性能。参考文献[20]介绍了一种自主车辆转向控制器，该控制器利用 MPC 控制器来跟踪所需的路径，同时考虑到干扰和时变参数的不确定性。参考文献 [21] 提出了一种适用于自动驾驶车辆网络延迟的精确轨迹跟踪控制算法。为了解决纵向速度和质量方面的车辆稳定性问题，文献[22]设计了一种 H-无限鲁棒控制器，应用直接偏航力矩控制和主动前转向控制。参考文献 [6] 开发了一种模型参考自适应控制器，能够在干扰和参数不确定的情况下实现稳健性能。参考文献[23]对参考路径跟踪控制策略以及鲁棒控制策略和基于观测器的控制策略的优势和局限性进行了深入分析。参考文献[24]提出了鲁棒控制策略，为这一领域做出了贡献。在单线变道和 J 型转弯等机动过程中的横向动力学稳定性方面，[25] 提出了一种自适应反步态控制技术，可有效处理外部干扰和参数不确定性。 此外，[26] 还为双轴自主公共汽车开发了一种自适应双层控制框架，以防止侧滑和翻车，并将安全放在首位；[27] 为电动汽车的安全系统设计了一个数字孪生系统，通过调整许多系统参数来模拟多个测试场景，从而观察系统状态。
These recent studies have significantly advanced the field of autonomous vehicle control, incorporating deep reinforcement learning, trajectory planning, robust control algorithms, adaptive control techniques, and various control strategies to enhance vehicle performance, safety, and stability. Most of the above works are carried out in a simulation platform or a constrained experimental setup.
这些最新研究极大地推动了自主车辆控制领域的发展，将深度强化学习、轨迹规划、鲁棒控制算法、自适应控制技术和各种控制策略结合在一起，以提高车辆性能、安全性和稳定性。上述研究大多在仿真平台或受限实验装置中进行。
The novel highlights of this paper are,
本文的新颖亮点在于
  • The physical vehicle is converted to a testbed and the experiment is performed in the actual situational cases for the evaluation of developed modern controllers using the HIL machine for real-time control and data acquisition which overcomes the limitations of other works enclosed.
    物理车辆被转换为试验台，并在实际情况下进行实验，以评估使用实时控制和数据采集 HIL 机器开发的现代控制器。
  • This paper is also focused on the work of precise and accurate maneuvering in complex and challenging road conditions for autonomous vehicle safety and reliability with the evaluation of the real-time performance of a robust controller using HIL testing.
    本文还侧重于在复杂和具有挑战性的道路条件下进行精确和准确的操纵，以提高自动驾驶汽车的安全性和可靠性，并利用 HIL 测试评估鲁棒控制器的实时性能。
  • A comparative study is also carried out in this paper to validate a robust steering system for autonomous vehicles by comparing traditional controllers (PD and PID) with a modern Model Predictive Control (MPC) controller.
    本文还进行了一项比较研究，通过比较传统控制器（PD 和 PID）与现代模型预测控制 (MPC) 控制器，验证了用于自动驾驶汽车的稳健转向系统。
    The study utilizes a multi-turn potentiometer and incremental encoder for position feedback. Performance tests at different velocities and sampling times, particularly in a speed breaker scenario, have been conducted.
    研究利用多圈电位计和增量编码器进行位置反馈。在不同速度和采样时间下进行了性能测试，特别是在速度断路器情况下。
The rest of this paper is organized into four sections. The following Section II describes the hardware experimental setup with the data acquisition system. Section III discusses the system identification and design of various controllers for steering system. Section IV provides the real-time experimental results that are obtained for the various test scenarios. Finally, the conclusion of the paper is presented in Section V.
本文其余部分分为四节。第二节介绍了带有数据采集系统的硬件实验装置。第三节讨论系统识别和转向系统各种控制器的设计。第四部分提供了各种测试场景下获得的实时实验结果。最后，第五节是本文的结论。

II. EXPERIMENTAL SETUP  II.实验装置

The complete test setup utilized a 7 -seater Maruti Suzuki Versa car, equipped with a Speedgoat HIL machine and a high-performance PC. This setup incorporated the highresolution 3D HDL-32E and VLP-16 Velodyne LiDARs, along with the Intel D435 depth camera. These sensors were employed for the perception, planning, and control of autonomous vehicles. Fig. 1 illustrates the autonomous vehicle with the speed breaker being examined and Fig. 2 shows the line diagram of the experimental setup. The test bench setup in the vehicle is shown in Fig. 3. Furthermore, this section covers the traction setup, steering setup, specifications of the HIL machine, data acquisition, noise removal, and filtering methods.
整个测试装置使用了一辆 7 座 Maruti Suzuki Versa 汽车，配备了一台 Speedgoat HIL 机器和一台高性能 PC。该装置集成了高分辨率 3D HDL-32E 和 VLP-16 Velodyne 激光雷达以及英特尔 D435 深度摄像头。这些传感器用于自动驾驶车辆的感知、规划和控制。图 1 展示了正在检测的带有速度断路器的自动驾驶车辆，图 2 展示了实验装置的线图。图 3 显示了车辆中的测试台设置。此外，本节还包括牵引装置、转向装置、HIL 机器规格、数据采集、噪声消除和滤波方法。

A. TRACTION SETUP  A.跟踪设置

A separately excited DC motor is used as a traction motor with the specifications given in Table 1. The Kelly KDH12801E motor driver serves to manage the armature supply of the motor. The field supply is maintained at a steady 12 V , facilitated by a 72 V to 12 V DC-DC converter. A Li-Ion battery pack, specified in Table 2, is utilized to fuel the vehicle. The schematic diagram of the car is depicted in Fig. 4. For precautionary measures, kill switches are affixed on each side of the car. Moreover, a 72 V DC to 220 V AC inverter caters to the power needs of the onboard desktop units, and a 220 V AC to 12 V DC converter provides for the vehicle’s low-voltage systems.
牵引电机采用单独励磁的直流电机，其规格如表 1 所示。Kelly KDH12801E 电机驱动器用于管理电机的电枢电源。通过一个 72 V 至 12 V 的 DC-DC 转换器，电场电源保持在稳定的 12 V。表 2 所示的锂离子电池组用于为汽车提供燃料。汽车原理图如图 4 所示。为了采取预防措施，汽车两侧都安装了断电开关。此外，72 伏直流电转换为 220 伏交流电的逆变器满足了车载台式设备的电力需求，而 220 伏交流电转换为 12 伏直流电的转换器则为车辆的低压系统提供电力。

B. STEERING SETUP  B.转向设置

The vehicle features a standalone EPS system, employing a 12 V DC brushed clutch motor. This motor is rerouted and
该车采用了独立的 EPS 系统，使用 12 V 直流有刷离合器电机。该电机经过改装，可
FIGURE 1. Autonomous car in front of the test speed breaker.
图 1.测试速度断路器前的自动驾驶汽车。
FIGURE 2. Line diagram of the experimental setup.
图 2.实验装置线图。

TABLE 1. Motor specifications.
表 1.电机规格。

Description  说明 Value  价值
Motor Power Rating  电机额定功率 10 kW  10 千瓦
Motor Armature Voltage  电机电枢电压 72 V
Motor Field Voltage  电机场电压 36 V
Motor Torque  电机扭矩 23 Nm  23 牛米
Description Value Motor Power Rating 10 kW Motor Armature Voltage 72 V Motor Field Voltage 36 V Motor Torque 23 Nm| Description | Value | | :--- | :--- | | Motor Power Rating | 10 kW | | Motor Armature Voltage | 72 V | | Motor Field Voltage | 36 V | | Motor Torque | 23 Nm |

TABLE 2. Battery specifications.
表 2.电池规格。

Description  说明 Value  价值
Battery Power Rating  电池额定功率 3.3 kW  3.3 千瓦
Battery Capacity  电池容量 46 Ah
Battery Configuration  电池配置 20 s 20 p
Battery Nominal Voltage  电池标称电压 72 V
Battery Maximum Voltage  电池最大电压 84 V
C Rating  C 级 1.4 C
Description Value Battery Power Rating 3.3 kW Battery Capacity 46 Ah Battery Configuration 20 s 20 p Battery Nominal Voltage 72 V Battery Maximum Voltage 84 V C Rating 1.4 C| Description | Value | | :--- | :--- | | Battery Power Rating | 3.3 kW | | Battery Capacity | 46 Ah | | Battery Configuration | 20 s 20 p | | Battery Nominal Voltage | 72 V | | Battery Maximum Voltage | 84 V | | C Rating | 1.4 C |
linked to a Cytron MDDS30 motor driver. It is supplemented with an incremental encoder, boasting a resolution of
与 Cytron MDDS30 电机驱动器相连。它还配有一个增量式编码器，分辨率为
TABLE 3. Target hardware specifications.
表 3.目标硬件规格。
Description  说明 Value  价值
CPU Intel Core i7 4.2 GHz , 4 4.2 GHz , 4 4.2GHz,44.2 \mathrm{GHz}, 4  英特尔酷睿 i7 4.2 GHz , 4 4.2 GHz , 4 4.2GHz,44.2 \mathrm{GHz}, 4
cores  内核
Memory (RAM)  内存（RAM） 4 GB DDR4
Main Drive  主驱动器 1 TB SSD  1 TB 固态硬盘
PCIe IO133
Description Value CPU Intel Core i7 4.2GHz,4 cores Memory (RAM) 4 GB DDR4 Main Drive 1 TB SSD PCIe IO133| Description | Value | | :--- | :--- | | CPU | Intel Core i7 $4.2 \mathrm{GHz}, 4$ | | | cores | | Memory (RAM) | 4 GB DDR4 | | Main Drive | 1 TB SSD | | PCIe | IO133 |
1024 pulses per revolution, mounted on the steering column. Moreover, a multiturn 10 k Ω 10 k Ω 10kOmega10 \mathrm{k} \Omega potentiometer, geared to the steering, facilitates the measurement of the steering position via voltage conversion. The installation of the encoder and the potentiometer are depicted in Fig. 5 (a) and (b), respectively.
每转 1024 个脉冲，安装在转向柱上。此外，一个与转向装置相连的多圈 10 k Ω 10 k Ω 10kOmega10 \mathrm{k} \Omega 电位器通过电压转换来测量转向位置。编码器和电位器的安装情况分别如图 5 (a) 和 (b) 所示。

C. HARDWARE-IN-THE-LOOP (HIL) MACHINE SETUP
C.硬件在环（HIL）机器设置

All designed control algorithms are dispatched to a hardware system for real-time testing. For rapid control prototyping, we have utilized the Speedgoat performance real-time target machine, with its specifications detailed in Table 3. This robust machine carries out computations, actuation, and data acquisition in real-time. It interfaces directly with MATLAB Simulink version 2022b using the Simulink Real-Time Target Packages.
所有设计的控制算法都会被发送到硬件系统中进行实时测试。为了快速制作控制原型，我们使用了 Speedgoat 高性能实时目标机，其规格详见表 3。这台坚固耐用的机器可实时进行计算、执行和数据采集。它使用 Simulink 实时目标包直接与 MATLAB Simulink 2022b 版本连接。
IO133 is a sophisticated analog input and output module (I/O), featuring 16 simultaneous-sampling analog input channels with 16 -bit resolution, and 8 analog output channels with concurrent update and 16-bit resolution. Additionally, it possesses 14 configurable digital I/O pins for TransistorTransistor Logic (TTL). It is predominantly utilized for rapid control prototyping and HIL testing, as displayed in Fig. 6(a). The module manages a traction motor through an analog output ranging from 0 − 5 V 0 − 5 V 0-5V0-5 \mathrm{~V}; with 0 V indicating no power flow and 5 V representing the maximum power supply to the traction motor. The steering motor operates through a driver configured for 0 to 100 % 100 % 100%100 \% Pulse Width Modulation (PWM). A 50 % 50 % 50%50 \% duty cycle leaves the steering motor stationary, while alterations to 0 % 0 % 0%0 \% and 100 % 100 % 100%100 \% trigger anticlockwise and clockwise rotations respectively. The PWM signal to the driver is dispatched via the HIL target machine utilizing the IO133 module. Feedback from the steering position is collected through a potentiometer, translating to an analog value of 0 − 5 V 0 − 5 V 0-5V0-5 \mathrm{~V}, wherein 2.427 V corresponds to the steering wheel’s central position. This data is fed into the target machine using the IO133 module’s analog input lines. Furthermore, the system interfaces with an incremental encoder, linking its A and B pulses to the digital I/O lines of the IO133 module within the target machine. The arrangement of the HIL machine setup can be seen in Fig. 6(b).
IO133 是一款先进的模拟输入和输出模块（I/O），具有 16 个同步采样、16 位分辨率的模拟输入通道，以及 8 个同步更新、16 位分辨率的模拟输出通道。此外，它还有 14 个可配置的数字输入/输出引脚，用于晶体管晶体管逻辑（TTL）。如图 6（a）所示，它主要用于快速控制原型开发和 HIL 测试。该模块通过范围为 0 − 5 V 0 − 5 V 0-5V0-5 \mathrm{~V} 的模拟输出来管理牵引电机，0 V 表示无功率流，5 V 表示向牵引电机提供的最大功率。转向电机通过配置为 0 至 100 % 100 % 100%100 \% 脉冲宽度调制 (PWM) 的驱动器运行。 50 % 50 % 50%50 \% 占空比使转向电机静止，而 0 % 0 % 0%0 \% 和 100 % 100 % 100%100 \% 的改变则分别触发逆时针和顺时针旋转。驱动器的 PWM 信号通过使用 IO133 模块的 HIL 目标机发出。转向位置的反馈通过电位计收集，并转换为 0 − 5 V 0 − 5 V 0-5V0-5 \mathrm{~V} 的模拟值，其中 2.427 V 相当于方向盘的中心位置。这些数据通过 IO133 模块的模拟输入线输入目标机器。此外，系统还与增量式编码器连接，将其 A 和 B 脉冲与目标机器内 IO133 模块的数字 I/O 线路相连接。HIL 机器设置的安排见图 6(b)。

D. DATA ACQUISITION FROM STEERING MOUNTED POTENTIOMETER AND ENCODER
D.从安装在转向装置上的电位计和编码器获取数据

The wiper pin of the steering’s potentiometer and the signal pins A and B of the steering’s encoder are connected to the Speedgoat HIL machine, which serves as a data
转向器电位计的刮片引脚和转向器编码器的信号引脚 A 和 B 与 Speedgoat HIL 机器相连，后者充当数据中心。
FIGURE 3. Experimental test bench setup in a real modified electric car.
图 3.真实改装电动汽车的实验测试台设置。
FIGURE 4. Circuit diagram of HV side of the vehicle.
图 4.车辆高压侧电路图。
FIGURE 5. Steering angular position acquisition sensors.
图 5：转向角位置采集传感器。
acquisition device. A Simulink model is created, and the data is acquired using MATLAB Simulink’s data inspector toolbox. To decode the encoder’s data, a pre-built block called the quadrature shaft decoder is utilized, which converts the
采集设备。我们创建了一个 Simulink 模型，并使用 MATLAB Simulink 的数据检查器工具箱采集数据。为了解码编码器的数据，使用了一个名为正交轴解码器的预建块，它将编码器的数据转换为正交轴解码器的数据。

(a)

(b)
FIGURE 6. Connection from the HIL machine setup.
图 6.HIL 机器设置的连接。
A and B encoder pulses into radians. The linear property of the potentiometer is employed to map the analog voltage of the potentiometer to radians. For manual measurement, the analog voltage corresponding to the zero radians position is recorded as 2.427 V , and the analog voltage for a full right turn of the steering is measured as 0.299 V , which corresponds to 16.638 radians. By establishing a straight-line equation between these two points, we determine the relationship between the analog voltage and the steering’s position.
将 A 和 B 编码器脉冲转换成弧度。电位计的线性特性用于将电位计的模拟电压映射为弧度。手动测量时，零弧度位置对应的模拟电压记录为 2.427 V，而转向器完全右转时的模拟电压测量值为 0.299 V，对应 16.638 弧度。通过建立这两点之间的直线方程，我们可以确定模拟电压与转向器位置之间的关系。

E. DATA FILTERING AND NOISE REMOVAL
E.数据过滤和噪音消除

The raw data that is directly obtained from the sensor consists of a lot of noises due to the surrounding electrical interference
从传感器直接获取的原始数据会因周围的电气干扰而产生大量噪音。
FIGURE 7. Connection from the HIL machine setup.
图 7.HIL 机器设置的连接。
as unprocessed, unfiltered data that is shown in Fig. 7 The noise component is the data will spoil the signal quality and will have a great impact in the accuracy of the signal. The raw data should be filtered before using it for processing in the control system application. A generalised low pass filter with a cut-off frequency of 10 rad / sec 10 rad / sec 10rad//sec10 \mathrm{rad} / \mathrm{sec} has been designed as a transfer function model which is given by Eq. (1). The raw data from the steering block is fed to it as an input and the filtered noise-less data has been obtained as an output of the transfer function.
如图 7 所示，原始数据是未经处理和过滤的数据，数据中的噪声成分会破坏信号质量，并对信号的准确性产生重大影响。在控制系统应用中处理原始数据之前，应对其进行过滤。我们设计了一个截止频率为 10 rad / sec 10 rad / sec 10rad//sec10 \mathrm{rad} / \mathrm{sec} 的通用低通滤波器，其传递函数模型如公式 (1) 所示。来自转向块的原始数据作为输入输入到滤波器中，经过滤波的无噪声数据作为传递函数的输出输出。
1 0.1 s + 1 1 0.1 s + 1 (1)/(0.1 s+1)\frac{1}{0.1 s+1}

III. METHODOLOGY  III.方法论

A. SYSTEM IDENTIFICATION
A.系统识别

System identification involves the mathematical modeling of a dynamic system by establishing a relationship between its input and output. This process includes acquiring data for the input and output signals, selecting an appropriate model structure, estimating the system model using various techniques, and evaluating the accuracy of the obtained model. Obtaining the system model through first-principles modeling can be challenging as the mathematical parameters of a pre-designed system are difficult to determine. The strategy outlined in this work is regarded as a black box technique since it does not require knowledge of the system’s physical dynamics. The system identification toolbox in MATLAB Simulink has a basic feature that allows researchers to tune the model’s parameters until the model’s output closely matches the observed output. The flowchart in Fig. 8 depicts the processes taken to obtain the system model.
系统识别包括通过建立输入和输出之间的关系来建立动态系统的数学模型。这一过程包括获取输入和输出信号的数据、选择适当的模型结构、使用各种技术估算系统模型以及评估所获模型的准确性。通过第一原理建模获取系统模型可能具有挑战性，因为预先设计的系统的数学参数很难确定。本工作中概述的策略被视为一种黑盒技术，因为它不需要了解系统的物理动态。MATLAB Simulink 中的系统识别工具箱有一个基本功能，允许研究人员调整模型参数，直到模型输出与观测输出密切吻合。图 8 中的流程图描述了获取系统模型的过程。

1) STEERING SYSTEM DYNAMICS MODELING
1) 转向系统动力学建模

The transfer function of a system represents the mathematical relationship between the input and output. In this case, the input is a voltage supply ranging from -12 V to 12 V , which is applied to the electric steering motor connected to the steering column. A voltage of -12 V causes the steering to rotate counterclockwise, while 12 V rotates it clockwise,
系统的传递函数表示输入和输出之间的数学关系。在本例中，输入是-12 V 至 12 V 的电压，施加到连接转向柱的电动转向电机上。-12 V 的电压使转向器逆时针旋转，而 12 V 则使其顺时针旋转、
FIGURE 8. Flowchart of the system model design.
图 8.系统模型设计流程图。
with 0 V representing the rest position. The output of the system is the position of the steering in radians. To obtain the transfer function, MATLAB’s system identification toolbox is used to analyze the input and output data. The frequency response modeling using the chirp signal [28] method is chosen for its ability to provide rapid stability and transient response information. Sinusoidal voltages with frequencies ranging from 0.75 rad / sec 0.75 rad / sec 0.75rad//sec0.75 \mathrm{rad} / \mathrm{sec} to 1.5 rad / sec 1.5 rad / sec 1.5rad//sec1.5 \mathrm{rad} / \mathrm{sec} and a peak-to-peak voltage of -12 V to 12 V are used as inputs to the system. Both distinct frequency modeling and multifrequency modeling are performed using the same data set in the system identification toolbox. Fig. 9 shows the graph depicting the relationship between the input voltage and angular position of the steering system for frequencies ranging from 0.75 rad / sec 0.75 rad / sec 0.75rad//sec0.75 \mathrm{rad} / \mathrm{sec} to II rad / sec / sec //sec/ \mathrm{sec}. The transfer function parameters for the multi-frequency modeling, obtained from MATLAB’s system identification toolbox are the gain of the system k = 4.730 k = 4.730 k=4.730k=4.730, damping ratio ζ = 3.648 ζ = 3.648 zeta=3.648\zeta=3.648 and the undamped
0 V 代表静止位置。系统的输出是以弧度为单位的转向位置。为获得传递函数，使用 MATLAB 的系统识别工具箱分析输入和输出数据。选择使用啁啾信号 [28] 方法进行频率响应建模，是因为该方法能够快速提供稳定性和瞬态响应信息。频率为 0.75 rad / sec 0.75 rad / sec 0.75rad//sec0.75 \mathrm{rad} / \mathrm{sec} 至 1.5 rad / sec 1.5 rad / sec 1.5rad//sec1.5 \mathrm{rad} / \mathrm{sec} 的正弦电压和峰峰值为 -12 V 至 12 V 的电压被用作系统的输入。在系统识别工具箱中，使用相同的数据集进行不同频率建模和多频率建模。图 9 显示了频率范围从 0.75 rad / sec 0.75 rad / sec 0.75rad//sec0.75 \mathrm{rad} / \mathrm{sec} 到 II rad / sec / sec //sec/ \mathrm{sec} 的输入电压与转向系统角度位置之间关系的曲线图。从 MATLAB 的系统识别工具箱中获得的多频率建模的传递函数参数为系统增益 k = 4.730 k = 4.730 k=4.730k=4.730 、阻尼比 ζ = 3.648 ζ = 3.648 zeta=3.648\zeta=3.648 和无阻尼系数 k = 4.730 k = 4.730 k=4.730k=4.730 。
natural frequency ω n = 1.119 ω n = 1.119 omega_(n)=1.119\omega_{n}=1.119. These parameters are plugged into the Eq. (2) to get the final transfer function of the system as given in Eq. (3).
固有频率 ω n = 1.119 ω n = 1.119 omega_(n)=1.119\omega_{n}=1.119 。将这些参数代入公式 (2)，即可得到公式 (3) 所给出的系统最终传递函数。
θ ( s ) V ( s ) = ω n 2 k s 2 + 2 ζ ω n s + ω n 2 θ ( s ) V ( s ) = 5.922 s 2 + 8.164 s + 1.252 θ ( s ) V ( s ) = ω n 2 k s 2 + 2 ζ ω n s + ω n 2 θ ( s ) V ( s ) = 5.922 s 2 + 8.164 s + 1.252 {:[(theta(s))/(V(s))=(omega_(n)^(2)k)/(s^(2)+2zetaomega_(n)s+omega_(n)^(2))],[(theta(s))/(V(s))=(5.922)/(s^(2)+8.164 s+1.252)]:}\begin{aligned} \frac{\theta(s)}{V(s)} & =\frac{\omega_{n}^{2} k}{s^{2}+2 \zeta \omega_{n} s+\omega_{n}^{2}} \\ \frac{\theta(s)}{V(s)} & =\frac{5.922}{s^{2}+8.164 s+1.252} \end{aligned}
where θ ( s ) θ ( s ) theta(s)\theta(s) represents the angular position of the steering system, and V ( s ) V ( s ) V(s)V(s) represents the steering motor’s input voltage.
其中， θ ( s ) θ ( s ) theta(s)\theta(s) 表示转向系统的角度位置， V ( s ) V ( s ) V(s)V(s) 表示转向电机的输入电压。

2) MODEL VALIDATION  2) 模型验证

Model validation plays a critical role in ensuring the accuracy, completeness, and cleanliness of the obtained system model. It is crucial to validate the system model to design the controller with the desired specifications. The obtained system model should be tested using multiple different data sets to assess its validity, and it should demonstrate decent and acceptable accuracy. If the model performs with good accuracy and the model fit is satisfactory with the test and validation sets, then it can be considered final for further system processing. However, if the obtained model fails to perform well or lacks a good model fit, the estimation of the mathematical model equation, specifically the transfer function, should be repeated while increasing the order of the system. The obtained transfer function model as shown in the Eq. (3) is verified with 2 different test data known as the validation data set. The results show 78.65 % 78.65 % 78.65%78.65 \% as the best fit for the system.
模型验证在确保所获系统模型的准确性、完整性和简洁性方面起着至关重要的作用。验证系统模型对于设计出符合预期规格的控制器至关重要。应使用多个不同的数据集对所获得的系统模型进行测试，以评估其有效性，并应证明其准确性和可接受性。如果模型精度良好，且测试和验证集的模型拟合度令人满意，则可将其视为最终模型，用于进一步的系统处理。但是，如果获得的模型性能不佳或缺乏良好的模型拟合，则应在增加系统阶数的同时重复数学模型方程的估算，特别是传递函数的估算。公式 (3) 所示的传递函数模型由两个不同的测试数据（即验证数据集）进行验证。结果表明 78.65 % 78.65 % 78.65%78.65 \% 是系统的最佳拟合。

B. CONTROLLER DESIGN  B.控制器设计

1) DESIGN OF PD AND PID CONTROLLER
1) 设计 PD 和 PID 控制器

PID controllers are widely used in the field of control engineering for regulating dynamics of the systems. They are renowned for their simplicity, effectiveness, and versatility in a wide range of applications. The developed system model utilized the transfer function toolbox in Simulink. The output from the PID controller block set serves as the input to the transfer function. The output from the system is then fed back to the sum block with a negative gain, forming a closedloop negative feedback system. A step input of 10 radians is introduced into the system, and PID parameters, specifically K p K p K_(p)K_{p} (proportional gain), K d K d K_(d)K_{d} (derivative gain), and K i K i K_(i)K_{i} (integral gain), were fine-tuned using the MATLAB’s real-time PID auto-tuner, which relies on a system model-based approach, focusing on system response tuning and transient response tuning. After introducing the plant to the tuner, it linearizes the system in preparation for tuning. Finally, the optimal gains and the filter coefficient obtained from the toolbox are K p = 28.446 , K i = 2.11 , K d = 4.699 K p = 28.446 , K i = 2.11 , K d = 4.699 K_(p)=28.446,K_(i)=2.11,K_(d)=4.699K_{p}=28.446, K_{i}=2.11, K_{d}=4.699 and N = 118.794 N = 118.794 N=118.794N=118.794. These gain values are plugged into the Eq. (4) to form a PID controller.
PID 控制器广泛应用于控制工程领域，用于调节系统的动态。PID 控制器以其简单、有效和应用广泛而闻名。所开发的系统模型使用了 Simulink 中的传递函数工具箱。PID 控制器模块组的输出作为传递函数的输入。然后，系统的输出以负增益反馈到总和模块，形成一个闭环负反馈系统。在系统中引入 10 弧度的阶跃输入，并使用 MATLAB 的实时 PID 自动调节器微调 PID 参数，特别是 K p K p K_(p)K_{p} （比例增益）、 K d K d K_(d)K_{d} （导数增益）和 K i K i K_(i)K_{i} （积分增益），该调节器依赖于基于系统模型的方法，侧重于系统响应调节和瞬态响应调节。将工厂引入调节器后，调节器将系统线性化，为调节做好准备。最后，从工具箱中获得的最佳增益和滤波器系数分别为 K p = 28.446 , K i = 2.11 , K d = 4.699 K p = 28.446 , K i = 2.11 , K d = 4.699 K_(p)=28.446,K_(i)=2.11,K_(d)=4.699K_{p}=28.446, K_{i}=2.11, K_{d}=4.699 和 N = 118.794 N = 118.794 N=118.794N=118.794 。将这些增益值插入公式 (4) 中，即可形成 PID 控制器。
In system response tuning, the system’s rapid or gradual response to the input is considered a design parameter.
在系统响应调整中，系统对输入的快速或渐进响应被视为一个设计参数。
In transient method tuning, the parameters of interest are the system’s robustness and aggressiveness. Given the system’s performance and response, the robustness and aggressiveness of the controller were adjusted to the anticipated level, and the relevant settings were applied. After achieving the desired system response, the tuning parameters and Filter Coefficient were extracted from the tuner. Saturation parameters were established to confine the output of the PID controllers within the maximum acceptable rating of the hardware, with an Upper Limit = + 12 = + 12 =+12=+12 and Lower Limit = − 12 [ 9 ] = − 12 [ 9 ] =-12[9]=-12[9].
在瞬态方法调整中，关注的参数是系统的鲁棒性和攻击性。根据系统的性能和响应，将控制器的鲁棒性和攻击性调整到预期水平，并应用相关设置。达到预期的系统响应后，从调节器中提取调节参数和滤波系数。建立饱和参数是为了将 PID 控制器的输出限制在硬件可接受的最大额定值范围内，上限为 = + 12 = + 12 =+12=+12 ，下限为 = − 12 [ 9 ] = − 12 [ 9 ] =-12[9]=-12[9] 。
u ( t ) = K p e ( t ) + K i ∫ 0 t e ( t ) d t + K d d e ( t ) d t u ( t ) = K p e ( t ) + K i ∫ 0 t   e ( t ) d t + K d d e ( t ) d t u(t)=K_(p)e(t)+K_(i)int_(0)^(t)e(t)dt+K_(d)(de(t))/(dt)u(t)=K_{p} e(t)+K_{i} \int_{0}^{t} e(t) d t+K_{d} \frac{d e(t)}{d t}
where u ( t ) u ( t ) u(t)u(t) represents the output of the controller and e ( t ) e ( t ) e(t)e(t) represents the error signal from the reference point.
其中， u ( t ) u ( t ) u(t)u(t) 表示控制器的输出， e ( t ) e ( t ) e(t)e(t) 表示来自参考点的误差信号。
Closed-loop transfer function of the system is calculated by obtaining the Laplace transform for the Eq. (4) as shown in Eq. (5) and by plugging in Eq. (5) to Eq. (6). The stability analysis of the PID controller is performed by obtaining the ZPK representation of the closed-loop transfer function of the Eq. (6) as shown in the Eq. (7). From Eq. (7) it is observed the locations of closed-loop poles are lying on the left of the S-plane. So, the designed closed-loop system is stable.
如公式 (5) 所示，通过对公式 (4) 进行拉普拉斯变换，并将公式 (5) 插入公式 (6) 来计算系统的闭环传递函数。如式 (7) 所示，通过获得式 (6) 的闭环传递函数的 ZPK 表示，可以对 PID 控制器进行稳定性分析。从式（7）可以看出，闭环极点的位置位于 S 平面的左侧。因此，设计的闭环系统是稳定的。
C ( s ) = K p + K i s + K d N 1 + N 1 s Y ( s ) R ( s ) = G ( s ) C ( s ) 1 + G ( s ) C ( s ) 3474 ( s + 5.689 ) ( s + 0.07511 ) ( s + 5.221 ) ( s + 0.07481 ) ( s 2 + 121.7 s + 3800 ) C ( s ) = K p + K i s + K d N 1 + N 1 s Y ( s ) R ( s ) = G ( s ) C ( s ) 1 + G ( s ) C ( s ) 3474 ( s + 5.689 ) ( s + 0.07511 ) ( s + 5.221 ) ( s + 0.07481 ) s 2 + 121.7 s + 3800 {:[C(s)=K_(p)+(K_(i))/(s)+K_(d)(N)/(1+N(1)/(s))],[(Y(s))/(R(s))=(G(s)C(s))/(1+G(s)C(s))],[(3474(s+5.689)(s+0.07511))/((s+5.221)(s+0.07481)(s^(2)+121.7 s+3800))]:}\begin{aligned} C(s) & =K_{p}+\frac{K_{i}}{s}+K_{d} \frac{N}{1+N \frac{1}{s}} \\ \frac{Y(s)}{R(s)} & =\frac{G(s) C(s)}{1+G(s) C(s)} \\ & \frac{3474(s+5.689)(s+0.07511)}{(s+5.221)(s+0.07481)\left(s^{2}+121.7 s+3800\right)} \end{aligned}
where, C ( s ) C ( s ) C(s)\mathrm{C}(\mathrm{s}) is the controller transfer function, R ( s ) R ( s ) R(s)\mathrm{R}(\mathrm{s}) is the closed loop transfer function and G ( s ) G ( s ) G(s)G(s) is the plant transfer function given by Eq. (3).
其中， C ( s ) C ( s ) C(s)\mathrm{C}(\mathrm{s}) 是控制器传递函数， R ( s ) R ( s ) R(s)\mathrm{R}(\mathrm{s}) 是闭环传递函数， G ( s ) G ( s ) G(s)G(s) 是公式 (3) 给出的植物传递函数。

2) DESIGN OF MPC CONTROLLER
2) MPC 控制器的设计

The MPC is a modern, rapidly evolving, high-performance system known for its constraint satisfaction. However, because there is no defined way for tuning the controller parameters, creating an MPC controller is difficult. The prediction horizon ( Np ) ( Np ) (Np)(\mathrm{Np}), control horizon ( Nc ) ( Nc ) (Nc)(\mathrm{Nc}), input constraints, weights, and sample time are among the parameters. We can change the system’s performance parameters, such as the controller’s robustness and aggressiveness, similarly to PID tuning. Furthermore, the MPC controller’s state estimation response, which determines the system’s weight, can be modified [17].
MPC 是一种现代的、快速发展的高性能系统，以满足约束条件而闻名。然而，由于没有确定的控制器参数调整方法，创建 MPC 控制器非常困难。参数包括预测范围 ( Np ) ( Np ) (Np)(\mathrm{Np}) 、控制范围 ( Nc ) ( Nc ) (Nc)(\mathrm{Nc}) 、输入约束、权重和采样时间。我们可以改变系统的性能参数，如控制器的鲁棒性和攻击性，与 PID 调整类似。此外，还可以修改 MPC 控制器的状态估计响应，它决定了系统的权重 [17]。
Initially, the MPC structure is defined with a single manipulated variable and a single input system. The physical system constraints are outlined in Table 4. The MPC parameters are tuned based on the closed-loop system behavior. First, a longer N p N p NpN p is selected and a nominal N c N c NcN c is selected along with the input and output weights of the system. The behavior of the system is monitored and the parameters are finetuned manually. The rate weight on the input side is set at 0.01353 , while the output weight is fixed at 7.3890 . The system response was evaluated using various values for
最初，MPC 结构是以单一操纵变量和单一输入系统来定义的。物理系统约束条件如表 4 所示。MPC 参数根据闭环系统行为进行调整。首先，选择较长的 N p N p NpN p 和标称的 N c N c NcN c 以及系统的输入和输出权重。对系统行为进行监控，并手动对参数进行微调。输入端的速率权重设置为 0.01353，而输出权重则固定为 7.3890。使用不同的参数值对系统响应进行了评估。
FIGURE 9. Frequency response modeling.
图 9.频率响应建模。
the prediction and control horizons. It is important to note that the system’s robustness and stability increase with an increase in N p N p NpN p. However, beyond a certain N p N p NpN p value, further increases in this variable do not significantly impact the system. Conversely, when the control horizon ( N c ) ( N c ) (Nc)(N c) increases, the system’s aggressiveness also increases, but at the expense of stability. In our case, the goal is to design a robust steering controller. As a compromise, the value of Np is set higher, Np = 20 Np = 20 Np=20\mathrm{Np}=20 than the value of N c , N c = 2 N c , N c = 2 Nc,Nc=2N c, N c=2. This value of Nc is chosen according to the general rule, 0.1 N p ≤ N c ≤ 0.1 N p ≤ N c ≤ 0.1 Np <= Nc <=0.1 N p \leq N c \leq 0.2 N p 0.2 N p 0.2 Np0.2 N p. The MPC’s sampling time ( T s T s TsT s ) is kept constant at T s = 0.001 T s = 0.001 Ts=0.001T s=0.001 seconds, playing a crucial role in the computational performance of the controller. These parameters are used to design the MPC controller in MATLAB and the overall representation and cost function of the MPC system in given in Eq. (8) and (9) respectively.
的预测和控制范围。值得注意的是，随着 N p N p NpN p 的增加，系统的鲁棒性和稳定性也随之增加。 然而，超过一定的 N p N p NpN p 值后，该变量的进一步增加不会对系统产生显著影响。相反，当控制范围 ( N c ) ( N c ) (Nc)(N c) 增加时，系统的攻击性也会增加，但这是以牺牲稳定性为代价的。在我们的案例中，我们的目标是设计一个稳健的转向控制器。作为折中方案，Np 的值 Np = 20 Np = 20 Np=20\mathrm{Np}=20 高于 N c , N c = 2 N c , N c = 2 Nc,Nc=2N c, N c=2 。Nc 的值根据一般规则 0.1 N p ≤ N c ≤ 0.1 N p ≤ N c ≤ 0.1 Np <= Nc <=0.1 N p \leq N c \leq 0.2 N p 0.2 N p 0.2 Np0.2 N p 选择。MPC 的采样时间 ( T s T s TsT s ) 保持不变，为 T s = 0.001 T s = 0.001 Ts=0.001T s=0.001 秒，这对控制器的计算性能起着至关重要的作用。这些参数用于在 MATLAB 中设计 MPC 控制器，MPC 系统的总体表示和成本函数分别见式 (8) 和 (9)。
U t ∗ ( x ( t ) ) = arg ⁡ min U t ∑ k = 0 N − 1 q ( x t + k , u i ( t + k ) ) U t ∗ ( x ( t ) ) = arg ⁡ min U t   ∑ k = 0 N − 1   q x t + k , u i ( t + k ) U_(t)^(**)(x(t))=arg min_(U_(t))sum_(k=0)^(N-1)q(x_(t+k),u_(i(t+k)))U_{t}^{*}(x(t))=\arg \min _{U_{t}} \sum_{k=0}^{N-1} q\left(x_{t+k}, u_{i(t+k)}\right)
where U t U t U_(t)U_{t} represents the optimization variable, x ( t ) x ( t ) x(t)x(t) is measurement, x t + k x t + k x_(t+k)x_{t+k} is state constraints and u i ( t + k ) u i ( t + k ) u_(i(t+k))u_{i(t+k)} is input constraints of the systems.
其中， U t U t U_(t)U_{t} 表示优化变量， x ( t ) x ( t ) x(t)x(t) 为测量值， x t + k x t + k x_(t+k)x_{t+k} 为状态约束条件， u i ( t + k ) u i ( t + k ) u_(i(t+k))u_{i(t+k)} 为系统输入约束条件。
J = ∑ i = 1 p w e e k + i 2 + ∑ i = 0 p − 1 w Δ u Δ u k + i 2 J = ∑ i = 1 p   w e e k + i 2 + ∑ i = 0 p − 1   w Δ u Δ u k + i 2 J=sum_(i=1)^(p)w_(e)e_(k+i^(2))+sum_(i=0)^(p-1)w_(Delta u)Deltau_(k+i^(2))J=\sum_{i=1}^{p} w_{e} e_{k+i^{2}}+\sum_{i=0}^{p-1} w_{\Delta u} \Delta u_{k+i^{2}}
where J J JJ represents the Cost function, w w ww is weights of the system, e e ee is error and Δ u Δ u Delta u\Delta u is controller action.
其中， J J JJ 表示成本函数， w w ww 表示系统权重， e e ee 表示误差， Δ u Δ u Delta u\Delta u 表示控制器动作。
Closed-loop stability analysis of the MPC controller is determined by the root locus method by obtaining the total system’s transfer function including the controller as discussed in Section III. From the obtained root locus it is observed the locations of closed-loop poles are lying on the left of the S-plane. So, the closed-loop system is said to be stable.
MPC 控制器的闭环稳定性分析是通过根位置法确定的，即获得包括控制器在内的整个系统的传递函数，如第三节所述。从获得的根位置可以看出，闭环极点的位置位于 S 平面的左侧。因此，闭环系统可以说是稳定的。
  1. PIN CONFIGURATION  引脚配置
All electrical connections between the Speedgoat target machine and the actual hardware are made through the
Speedgoat 目标机与实际硬件之间的所有电气连接均通过
TABLE 4. MPC constraints.
表 4.MPC 约束条件。
Channel  频道 Type  类型 Min  最小 Max  最大 RateMin  最低费率 RateMax  最大费率
Input u ( t ) u ( t ) u(t)\mathrm{u}(\mathrm{t})  输入 u ( t ) u ( t ) u(t)\mathrm{u}(\mathrm{t}) MV 2 -12 12 + Inf
Output y ( t ) y ( t ) y(t)\mathrm{y}(\mathrm{t})  输出 y ( t ) y ( t ) y(t)\mathrm{y}(\mathrm{t}) MO 13 16 - -
Channel Type Min Max RateMin RateMax Input u(t) MV 2 -12 12 + Inf Output y(t) MO 13 16 - -| Channel | Type | Min | Max | RateMin | RateMax | | :--- | :--- | :--- | :--- | :--- | :--- | | Input $\mathrm{u}(\mathrm{t})$ | MV | 2 | -12 | 12 | + Inf | | Output $\mathrm{y}(\mathrm{t})$ | MO | 13 | 16 | - | - |
Speedgoat’s terminal board. The encoder section of the steering wheel provides high-frequency digital pulses as input to the system, while the steering’s multi-turn potentiometer serves as the analog input. The steering wheel actuation is achieved through the Cytron motor, which requires highfrequency PWM digital signals as input. In the Simulink environment, these pins are configured as shown in Fig. 10. The Pin setup block is used to declare the analog and digital inputs and outputs. A sampling time of 0.0001 seconds is set for each block. The analog input to the system has a maximum sample rate of 200 kSPS and is equipped with an anti-aliasing filter with a -3 dB cutoff frequency. The digital I/O lines operate at a Low Voltage-TTL level of 3.3 V with a 5 V tolerance [29]
Speedgoat 的接线板。方向盘的编码器部分为系统提供高频数字脉冲输入，而转向器的多圈电位计则作为模拟输入。方向盘驱动通过 Cytron 电机实现，该电机需要高频 PWM 数字信号作为输入。在 Simulink 环境中，这些引脚的配置如图 10 所示。引脚设置块用于声明模拟和数字输入输出。每个块的采样时间设置为 0.0001 秒。系统的模拟输入最大采样率为 200 kSPS，并配有一个截止频率为 -3 dB 的抗混叠滤波器。数字输入/输出线的工作电压为 3.3 V，容差为 5 V [29] 。
2) IMPLEMENTATION OF PD, PID, AND MPC CONTROLLERS
2) 实施 PD、PID 和 MPC 控制器
The PID block in Simulink simplifies the system by directly handling the steering’s position error in radians. The PID output is limited to a range of -12 V to +12 V , as the input voltage to the steering motor is proportional to the position. Therefore, the PID controller’s output serves as the motor voltage. To convert the voltage output from the PID controller to PWM values, a mapping is applied. A voltage of -12 V corresponds to a PWM value of 0 , while +12 V corresponds to a PWM value of 1 . For example, a PWM value of 0.5 results in 0 V being applied to the steering motor. This mapping is based on the steering motor driver’s datasheet and straightline equation that is represented by Eqs. (10)-(12). The PD and the PID controller are implemented similarly, with the inclusion of the K i K i K_(i)K_{i} term in the controller’s gain. The control structure of PID implemented in the MATLAB Simulink is shown in Fig. 12. Fig. 11 shows the root locus for the designed PID closed loop system with the desired specifications. The
Simulink 中的 PID 模块通过直接处理以弧度为单位的转向位置误差来简化系统。由于转向电机的输入电压与位置成正比，因此 PID 输出的范围仅限于 -12 V 至 +12 V。因此，PID 控制器的输出可用作电机电压。为了将 PID 控制器输出的电压转换为 PWM 值，需要应用映射。电压为 -12 V 时，PWM 值为 0；电压为 +12 V 时，PWM 值为 1。例如，PWM 值为 0.5 时，转向电机的电压为 0 V。这一映射是基于转向电机驱动器的数据表和直线方程（如公式 (10)-(12) 所示）。PD 和 PID 控制器的实现方式类似，只是在控制器的增益中加入了 K i K i K_(i)K_{i} 项。在 MATLAB Simulink 中实现的 PID 控制结构如图 12 所示。图 11 显示了所设计的 PID 闭环系统的根位置，该系统具有所需的规格。图 11
FIGURE 10. I/O configuration with HIL and Simulink.
图 10.使用 HIL 和 Simulink 进行输入/输出配置。
FIGURE 11. Root locus of the closed-loop PID system.
图 11.闭环 PID 系统的根位置。
poles location of the root locus matches with the ZPK form of the closed-loop transfer function shown in Eq. (7).
根点的极点位置与公式 (7) 所示的闭环传递函数的 ZPK 形式相吻合。
To implement the MPC controller in Simulink, the actual steering position angle is directly fed into the measured output port of the MPC controller block. The desired steering position reference is provided to the controller’s ref port. The manipulated variable, which ranges from -12 V to 12 V , is the output from the controller. This value is then converted to PWM values, as the motor driver requires PWM to actuate the motor [30]. The control structure of MPC implemented in the MATLAB Simulink is shown in Fig. 13.
为了在 Simulink 中实现 MPC 控制器，实际转向位置角被直接输入 MPC 控制器模块的测量输出端口。控制器的 ref 端口提供所需的转向位置参考。控制器的输出为受控变量，其范围为 -12 V 至 12 V。然后将该值转换为 PWM 值，因为电机驱动器需要 PWM 来驱动电机 [30]。在 MATLAB Simulink 中实现的 MPC 控制结构如图 13 所示。
y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 y − 1 0.5 − 1 = x − 12 0 − 12 y = 0.0416666 ( x ) + 0.5 y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 y − 1 0.5 − 1 = x − 12 0 − 12 y = 0.0416666 ( x ) + 0.5 {:[(y-y_(1))/(y_(2)-y_(1))=(x-x_(1))/(x_(2)-x_(1))],[(y-1)/(0.5-1)=(x-12)/(0-12)],[y=0.0416666(x)+0.5]:}\begin{aligned} \frac{y-y_{1}}{y_{2}-y_{1}} & =\frac{x-x_{1}}{x_{2}-x_{1}} \\ \frac{y-1}{0.5-1} & =\frac{x-12}{0-12} \\ y & =0.0416666(x)+0.5 \end{aligned}
where x x xx represents the output voltage from the designed controller, and y y yy represents the steering motor driver’s input PWM signal.
其中， x x xx 表示设计控制器的输出电压， y y yy 表示转向电机驱动器的输入 PWM 信号。

3) DATA ACQUISITION  3) 数据采集

All data from each block is logged using Simulink Data Inspector with a sampling time of 0.0001 seconds. The Speedgoat performance target machine ensures accurate and high-resolution logging of data. Analog data is recorded with a 16-bit resolution. The Simulink Data Inspector allows realtime inspection of captured data and enables comparison with time series data at multiple stages and runs simultaneously.
使用 Simulink 数据检查器记录每个区块的所有数据，采样时间为 0.0001 秒。Speedgoat 性能目标机可确保准确、高分辨率地记录数据。模拟数据以 16 位分辨率记录。Simulink 数据检查器可对捕获的数据进行实时检查，并可同时在多个阶段和运行中与时间序列数据进行比较。

IV. EXPERIMENTAL RESULTS AND DISCUSSION
IV.实验结果和讨论

The robustness of each controller is analysed by how well the controller can maintain at the given setpoint, when the vehicle travels in a speed breaker. The results were obtained for two different vehicle speeds along the power consumption of each controller.
每个控制器的鲁棒性是通过当车辆在速度断路器中行驶时，控制器在给定设定点上的稳定性来分析的。结果是在两种不同的车速下，每个控制器的功耗都不同。

1) CLOSED-LOOP CONTROLLER'S STEERING PERFORMANCE IN JACKED POSITION
1) 闭环控制仪在千斤顶位置的转向性能

The vehicle’s front wheels are lifted using two hydraulic jacks, and the closed response was logged. The experimental and simulated results for PD, PID, and MPC controllers, along with the power consumed by the controller drive system, are shown in Fig. 14 (a), (b), and ©, respectively.
使用两个液压千斤顶抬起车辆前轮，并记录闭合响应。PD、PID 和 MPC 控制器的实验和模拟结果以及控制器驱动系统消耗的功率分别如图 14 (a)、(b) 和 © 所示。
It was observed that the MPC controller outperformed the PD and PID controllers with a shorter settling time and a steady-state error of less than 1 % 1 % 1%1 \%. The power consumed by the MPC control drive was also lower compared to the other controllers. The PD controller performed better than the PID controller because the steering system had a constant error, and the integral term of the PID controller added up the error, making the controller system unstable. This constant steady-state error and high-power consumption are the reasons behind the performance of the PID controller. The simulated results matched the experimental results perfectly, validating the design of the PD, PID, and MPC controllers.
据观察，MPC 控制器的性能优于 PD 和 PID 控制器，其稳定时间更短，稳态误差小于 1 % 1 % 1%1 \% 。与其他控制器相比，MPC 控制驱动器的功耗也更低。PD 控制器的性能优于 PID 控制器，这是因为转向系统具有恒定误差，而 PID 控制器的积分项会增加误差，使控制系统不稳定。这种恒定的稳态误差和高能耗是 PID 控制器表现不佳的原因。模拟结果与实验结果完全吻合，验证了 PD、PID 和 MPC 控制器的设计。

A. OPEN-LOOP RESPONSE  A.开环响应

The behavior of the steering is logged for the open-loop steering system. The results obtained for two different velocities, 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} and 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}, are shown in Fig. 15 (a) and (b), respectively.
对开环转向系统的转向行为进行了记录。两种不同速度 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 和 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 的结果分别如图 15 (a) 和 (b) 所示。
The displacement of the steering at 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} is significantly smaller compared to the displacement at 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}. A robust controller must be developed to maintain the steering at the respective angle, even when the vehicle encounters external disturbances such as speed bumps, debris, and rough terrain.
与 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 处的位移相比， 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 处的转向位移要小得多。必须开发一种稳健的控制器，即使在车辆遇到减速带、碎石和崎岖地形等外部干扰时，也能将转向保持在相应的角度。

B. REAL-TIME TESTING OF CLOSED-LOOP CONTROLLERS PERFORMANCE WITH DIFFERENT SAMPLING TIMES
B.不同采样时间下闭环控制性能的实时测试

  1. SAMPLING TIME OF THE CONTROLLER - 0.001 SEC
    控制器采样时间 - 0.001 秒
Closed loop responses with PD, PID, and MPC controllers were logged using Simulink data inspector for two different velocities, 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} and 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}. Fig. 16 (a) and (b) depict the closed-loop response and power consumption of the PD controller at these velocities.
使用 Simulink 数据检查器记录了 PD、PID 和 MPC 控制器在 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 和 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 两种不同速度下的闭环响应。图 16 (a) 和 (b) 描述了 PD 控制器在这些速度下的闭环响应和功耗。
FIGURE 12. PID controller implementation.
图 12.PID 控制器的实现。
FIGURE 13. MPC controller implementation.
图 13.MPC 控制器的实现。
At 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h}, the PD controller exhibits decent response; however, at 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}, the PD controller loses its robustness and the response deviates. The power consumption remains similar and constant for both velocities, with a maximum requirement of approximately 40 watts and an average requirement of around 15 watts. The closed-loop response and power consumption of the PID controller are illustrated in Fig. 17 (a) and (b) respectively.
在 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 时，PD 控制器显示出良好的响应；但在 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 时，PD 控制器失去了鲁棒性，响应出现偏差。两种速度下的功耗相似且恒定，最大功耗约为 40 瓦，平均功耗约为 15 瓦。图 17 (a) 和 (b) 分别显示了 PID 控制器的闭环响应和功耗。
The response of the PID controller, at a speed of 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h}, has effectively maintained the steering position with a maximum displacement of 0.1 radians. The controller’s performance remains consistent even as the speed increases to 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}. Power consumption at these two speeds is similar and constant. The peak power requirement is approximately 25 watts, with an average requirement of around 8 watts. The closed-loop response of the MPC controller and its power consumption are shown in Fig. 18 (a) and (b), respectively, at two different velocities.
在速度为 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 时，PID 控制器的响应有效地保持了转向位置，最大位移为 0.1 弧度。即使速度增加到 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 时，控制器的性能也保持不变。这两种速度下的功耗相似且恒定。峰值功耗约为 25 瓦，平均功耗约为 8 瓦。图 18 (a) 和 (b) 分别显示了 MPC 控制器在两个不同速度下的闭环响应及其功耗。
The MPC controller exhibits significant instability, particularly at the lower speed of 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h}. Additionally, the power consumed by the MPC controller is high, with a maximum demand of around 60 watts and an average of approximately 25 watts. This instability is primarily caused by small disturbances on the steering wheels due to the irregular terrain of the road surface. Generally, the MPC controller requires an accurately modeled system with included disturbances, which necessitates estimating complex model coefficients.
MPC 控制器表现出明显的不稳定性，尤其是在 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 的较低转速下。此外，MPC 控制器的功耗很高，最大需求约为 60 瓦，平均约为 25 瓦。这种不稳定性主要是由路面不规则地形对方向盘造成的微小干扰引起的。一般来说，MPC 控制器需要一个包含干扰的精确建模系统，这就需要估计复杂的模型系数。

2) SAMPLING TIME OF THE CONTROLLER - 0.002 SEC
2) 控制器的采样时间 - 0.002 秒

The initial sampling time in the closed-loop response for all the controllers was set to 0.001 seconds. Since the sampling time depends on the microcontroller used in the actual system, it is important to vary the sampling time and validate performance. The sampling time for PD, PID, and MPC controllers was increased to 0.002 seconds, and the results were obtained at a constant speed of 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h}. The results,
所有控制器闭环响应的初始采样时间均设定为 0.001 秒。由于采样时间取决于实际系统中使用的微控制器，因此必须改变采样时间并验证性能。PD、PID 和 MPC 控制器的采样时间增加到 0.002 秒，并在 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 的恒定速度下获得结果。结果如下
FIGURE 14. Validation of experimental (Jacked) and simulation performance of different controllers for the steering system and their control input.
图 14.不同转向系统控制器及其控制输入的实验（插孔）和模拟性能验证。
FIGURE 15. Open-loop response.
图 15.开环响应。
FIGURE 16. Steering angular position and control input of PD controller with different vehicle speeds.
图 16.不同车速下的转向角位置和 PD 控制器的控制输入。
FIGURE 17. Steering angular position and control input of PID controller with different vehicle speeds.
图 17.不同车速下的转向角位置和 PID 控制器的控制输入。
controller achieved an error of less than 1 % 1 % 1%1 \%, the PID controller performed with an error below 2 % 2 % 2%2 \%, and the MPC controller struggled with stability due to constant disturbances. Power consumption was lowest for the PID controller in this scenario compared to the PD and MPC controllers.
PD 控制器的误差小于 1 % 1 % 1%1 \% ，PID 控制器的误差小于 2 % 2 % 2%2 \% ，而 MPC 控制器在持续干扰下难以保持稳定。与 PD 和 MPC 控制器相比，PID 控制器的功耗最低。
  • At 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h}, the PID controller outperformed the PD controller, which experienced instability initially but eventually recovered. The MPC controller exhibited unstable responses but improved compared to the 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} results. Power consumption remained lower for the PID controller when compared to the PD and MPC controllers.
    在 2 km / h 2 km / h 2km//h2 \mathrm{~km} / \mathrm{h} 时，PID 控制器的性能优于 PD 控制器，后者最初出现不稳定，但最终恢复了。MPC 控制器的响应不稳定，但与 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 的结果相比有所改善。与 PD 和 MPC 控制器相比，PID 控制器的功耗仍然较低。
  • At 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} with an increased sampling rate of 0.002 sec , the MPC controller excelled compared to the PD and PID controllers, which lost stability. The MPC controller maintained superior power consumption compared to
    在 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 采样率增加到 0.002 秒时，MPC 控制器比 PD 和 PID 控制器更出色，后者失去了稳定性。与 PD 和 PID 控制器相比，MPC 控制器的功耗更低。
FIGURE 18. Steering angular position and control input of MPC controller with different vehicle speeds.
图 18.不同车速下的转向角位置和 MPC 控制器的控制输入。
FIGURE 19. Steering angular position and control input of different controllers with speed as 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} and sampling time of 0.002 seconds.
图 19.速度为 1 km / h 1 km / h 1km//h1 \mathrm{~km} / \mathrm{h} 、采样时间为 0.002 秒时，不同控制器的转向角位置和控制输入。
the PD and PID controllers, with the PID controller peaking at 140 Watts due to instability.
由于不稳定，PID 控制器的峰值为 140 瓦特。

v. CONCLUSION  v.结论

The study compared PD, PID, and MPC controllers for the autonomous vehicle’s steering system using real-time data obtained from the HIL machine in a real modified electric car under various test scenarios. The results highlighted the strengths and weaknesses of each controller. The MPC
该研究利用从真实改装电动汽车的 HIL 设备中获取的实时数据，在各种测试场景下比较了自动驾驶汽车转向系统的 PD、PID 和 MPC 控制器。结果凸显了每种控制器的优缺点。MPC
controller demonstrated superior tracking performance in the absence of external disturbances. The PD controller exhibited robustness against constant small disturbances, while the PID controller performed well in the presence of impulse disturbances. Power consumption varied among the controllers, with the MPC controller consistently consuming less power. Overall, the choice of controller depends on the specific requirements and the ability to model external disturbances.
在没有外部干扰的情况下，PID 控制器表现出卓越的跟踪性能。PD 控制器对持续的小干扰表现出鲁棒性，而 PID 控制器在脉冲干扰下表现良好。不同控制器的功耗各不相同，MPC 控制器的功耗始终较低。总之，控制器的选择取决于具体要求和模拟外部干扰的能力。
Future research in autonomous vehicle development can focus on advanced control algorithms beyond PD, PID, and MPC controllers to enhance steering system performance. Sensor fusion and perception can be explored to integrate multiple sensors for improved accuracy. Safety measures and redundancy mechanisms should be developed to ensure fail-safe operation. Real-world testing and validation in various driving scenarios can provide a comprehensive evaluation. Human-machine interaction aspects, including driver behavior and interaction interfaces, should be studied for seamless collaboration. These research directions aim to advance steering system performance, improve safety, and enhance the overall development of autonomous vehicles.
未来的自动驾驶汽车开发研究可侧重于 PD、PID 和 MPC 控制器之外的先进控制算法，以提高转向系统的性能。可以探索传感器融合和感知技术，以整合多个传感器，提高精确度。应开发安全措施和冗余机制，以确保故障安全运行。在各种驾驶场景中进行真实世界测试和验证可提供全面的评估。应研究人机交互方面，包括驾驶员行为和交互界面，以实现无缝协作。这些研究方向旨在提高转向系统的性能，改善安全性，促进自动驾驶汽车的全面发展。

ACKNOWLEDGMENT  致谢

The authors would like to thank Team AutoZ and Vellore Institute of Technology for providing the required facilities and necessary support to complete the concerned work.
作者感谢 AutoZ 团队和韦洛尔理工学院为完成相关工作提供了所需的设施和必要的支持。

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S. GOKUL KRISHNAN is currently pursuing the B.Tech. degree in electrical engineering with Vellore Institute of Technology (VIT University), Vellore. He is also a Visiting Research Intern with the King Abdullah University of Science and Technology (KAUST), Saudi Arabia, working on future smart DC grid technologies. Previously, he was a Research Intern with Indian Institute of Space Science and Technology (IIST-ISRO), Thiruvananthapuram, where he was developed the electrical power systems (EPS) for small satellites. He is also an Undergraduate Researcher with the Automotive Research Center (ARC), VIT. His research interests include modern grid-connected power electronics, power systems, control systems, hardware-in-the-loop (HIL), and rapid control prototyping (RCP) systems.
S. GOKUL KRISHNAN 目前正在维罗尔的维罗尔理工大学（VIT University）攻读电气工程学士学位。他还是沙特阿拉伯阿卜杜拉国王科技大学（KAUST）的访问研究实习生，研究未来智能直流电网技术。此前，他曾在位于 Thiruvananthapuram 的印度空间科学与技术研究所（IIST-ISRO）担任实习研究员，为小型卫星开发电力系统（EPS）。他还是 VIT 汽车研究中心 (ARC) 的本科生研究员。他的研究兴趣包括现代并网电力电子技术、电力系统、控制系统、硬件在环（HIL）和快速控制原型（RCP）系统。

P. SURESH KUMAR received the master’s degree from NIT Trichy and the Ph.D. degree in design of control systems from Indian Institute of Space Science and Technology (IIST-ISRO). He is currently an Assistant Professor with the Autonomous Vehicles Research Laboratory, Automotive Research Center (ARC), Vellore Institute of Technology (VIT University), Vellore. With a strong background in modeling and controller design, he has published extensively in renowned scientific journals and presented his research at international conferences. His expertise lies mainly in autonomous aerial, ground vehicles, and electric vehicles. He actively engages in interdisciplinary collaborations, in cuttingedge research projects.
P. SURESH KUMAR 拥有印度国家理工学院（NIT Trichy）硕士学位和印度空间科学与技术研究所（IIST-ISRO）控制系统设计博士学位。他目前是维洛尔理工大学汽车研究中心（ARC）自主车辆研究实验室的助理教授。他在建模和控制器设计方面有着深厚的背景，在知名科学杂志上发表了大量论文，并在国际会议上介绍了自己的研究成果。他的专长主要是自主飞行器、地面车辆和电动汽车。他积极开展跨学科合作，参与尖端研究项目。
NADHEEM NASSAR MATARA is currently pursuing the B . T e c h B . T e c h B.TechB . T e c h. degree in mechanical engineering and in automotive engineering with Vellore Institute of Technology (VIT University), Vellore. He has played a role in Defense Research and Development Organization (DRDO) funded projects, specifically contributing to the development of a multifunctional rover for antitank landmine detection. He is also an Undergraduate Researcher with the Automotive Research Center (ARC), VIT. His profound expertise in the domains of automotive safety system, generative design, additive manufacturing, computer-aided simulation, and finite element analysis (FEA). These proficiencies underscore a multifaceted skill set, positioning him at the intersection of theoretical knowledge and hands-on application.
NADHEEM NASSAR MATARA 目前正在韦洛尔的韦洛尔理工大学攻读机械工程和汽车工程学位。他在国防研究与发展组织（DRDO）资助的项目中发挥了作用，特别是为反坦克地雷探测多功能漫游车的开发做出了贡献。他还是 VIT 汽车研究中心 (ARC) 的本科生研究员。他在汽车安全系统、生成设计、增材制造、计算机辅助模拟和有限元分析（FEA）等领域拥有深厚的专业知识。这些专长凸显了他多方面的技能，使他成为理论知识和实践应用的交叉点。
YONG WANG received the Ph.D. degree in energy and power engineering from Huazhong University of Science and Technology, China, in 2010, and the Ph.D. degree in industrial engineering and operations research from the University of Illinois at Chicago, in 2015. He was a Visiting Scholar with the Department of Mechanical Engineering, University of Michigan, Ann Arbor (2007-2009). He is currently an Associate Professor with the Department of Systems Science and Industrial Engineering, Watson College of Engineering and Applied Science, Binghamton University. His research with Binghamton University focuses on the design, modeling, and management of complex systems (energy, healthcare, manufacturing, and transportation).
王勇，2010 年获中国华中科技大学能源与动力工程博士学位，2015 年获美国伊利诺伊大学芝加哥分校工业工程与运筹学博士学位。他曾是密歇根大学安娜堡分校机械工程系的访问学者（2007-2009 年）。他目前是宾汉姆顿大学沃森工程与应用科学学院系统科学与工业工程系副教授。他在宾汉姆顿大学的研究重点是复杂系统（能源、医疗保健、制造和运输）的设计、建模和管理。