Research on Distributed Dual-Wheel Electric-Drive Fuzzy PI Control for Agricultural Tractors

Abstract

In order to solve the problem that, when the vehicle speed of an agricultural distributed dual-wheel electric-drive tractor changes or the system is disturbed by off-load, the traditional PI control cannot be adjusted in time, resulting in the overshoot of steering control or control delay, meaning it then cannot travel along the target trajectory quickly and accurately, a parameter-adaptive dual-dimensional fuzzy PI speed and steering adjustment controller was proposed, which can adjust the PI parameters in real time based on the deviation between vehicle speed, steering, and reference value, as well as the rate of deviation change. Firstly, based on the operational characteristics of agricultural tractors, a dynamic model of a distributed dual-wheel tractor was established, and a hardware-in-the-loop (HIL) test bench was set up. Fuzzy PI controller algorithms for vehicle speed and steering were designed and developed. In addition, simulations and tests were carried out under no-load and off-load tractor operating conditions with MATLAB/Simulink, respectively. The results indicate that, compared with a traditional PI controller, the fuzzy PI controller exhibits a faster control response and better robustness, reducing overshoot by approximately 60% and the steady-state response time by approximately 25%. When subjected to off-load disturbances, the maximum trajectory offset is controlled within 0.08 m, and the maximum trajectory offset is reduced by 45% compared with a traditional PI controller; therefore, the fuzzy PI control algorithm proposed in this paper makes the tractor’s running trajectory more stable and has stronger anti-interference ability towards off-load disturbances.

1. Introduction

With the development of agricultural mechanization in China, tractors have become the most widely used piece of agricultural power equipment in agricultural production [1,2,3]. Electric tractors have the characteristics of powerful traction [4], energy conservation, emission reduction, and a simple structure [5,6,7]. Compared with traditional fuel-drive tractors, electric-drive tractors offer advantages such as a rapid torque response, flexible forward and reverse switching, and strong adaptability to harsh conditions; however, traditional centralized four-wheel-drive tractors have complex transmission structures, low transmission efficiency, and are prone to power shortages. They also face difficulties in achieving high-precision trajectory control, which hampers the intelligent development of tractors. In contrast, distributed dual-wheel tractors have advantages such as powerful traction, slip prevention, flexible steering, independently controllable wheels, and so on; however, complex agricultural operating environments, including factors like uneven terrain, also bring instability factors to the dual-wheel structure. Therefore, it is necessary to adopt control methods to improve the stability of distributed electric-drive tractors.

PI controllers are widely applied in industry due to their simple principles, easy parameter tuning, and robustness [8,9]. The basic principle of PI control is to generate a control signal by linearly combining the error between an input signal and a reference signal, as well as its integral and derivative values. Essentially, it is a form of linear control [10]; however, the coupled control of speed and direction of the distributed electric-drive tractor exhibits nonlinear and time-varying characteristics. When the speed and direction change during movement or the system is disturbed by load, a PI controller cannot be adjusted in a timely manner; in addition, conflicts exist between a PI control system’s fast response and overshoot, where fast responses may lead to overshooting.

Since the control theory expert Zadeh proposed fuzzy mathematics in 1965 at the University of California, its theories and methods have been continually perfected. In just a few decades, fuzzy control has been widely applied in natural sciences, social sciences, and engineering control fields [11,12]. Fuzzy control does not rely on a precise mathematical model of the controlled object, and it can overcome the influence of nonlinear factors, exhibiting strong robustness to parameter variations in the controlled object. Xu et al. [13] proposed a brushless DC motor speed control system based on a fuzzy adaptive PID model. Simulation results demonstrate that this control system enhances the robustness, control accuracy, and dynamic as well as static performance of brushless DC motor control systems. Yuan et al. [14] improved the stability of an AVG robot’s operation by controlling the voltage difference between two brushless DC motors using a fuzzy controller. Some scholars [15,16] conducted control studies on the constant-pressure control of LNG submersible pumps and underwater vehicles, which improved the stability of systems. The above research uses fuzzy control algorithms for various application systems to enhance real-time performance, rapidity, and stability, but faces issues of control overshoot. Moreover, the summarization of fuzzy rules and the adjustment of fuzzy membership functions primarily rely on empirical methods, which involve significant subjectivity [17].

In the research on distributed dual-wheel control, Zhang et al. [18] proposed a distributed-drive control strategy for automobiles, which improved the efficiency of the motor, while Lampl et al. [19] optimized the performance of the transportation bureau through a distributed drive. Liu Zhijun et al. [19] studied the torque distribution in a distributed drive; Qiu et al. [20] selected the virtual steering center position by calculating the front- and rear-wheel yaw angle based on a model, and generated the linear velocity of each driving wheel according to the extended Ackerman steering principle (ASP) to achieve vehicle control; Li et al. [21] adopted a trajectory-tracking control method, combining optimal control theory and fuzzy control to achieve vehicle control through the wheel torque distribution; and Yu et al. [22] generated an optimal motion trajectory through path planning, and used the heading angle compensation algorithm to correct the driving trajectory of a bus. The abovementioned research on distributed-drive systems is currently primarily applied to scenarios, such as automotive stability control, that require high-precision and complex systems. Torque as well as speed distribution and path planning are also tested with power mechanism under ideal conditions. There is currently limited research on the control performance of agricultural distributed dual-wheel electric-drive tractors and their anti-interference in complex agricultural environments. Compared to the above control strategies, fuzzy PI control has lower implementation complexity and lower requirements for system models, making it more suitable for systems such as tractors with large changes in operating conditions. It can effectively handle the nonlinear characteristics and uncertainties of workloads in complex agricultural work environments, providing a more accurate operating experience.

In this paper, for agricultural distributed dual-wheel electric-drive tractors, a parameter-adaptive two-dimensional fuzzy PI speed and steering adjustment controller was proposed. The PI parameters can be adjusted dynamically based on deviations between vehicle speed, yaw angle and reference values, and deviation rates, which combine the strengths of both PI and fuzzy controllers. On this basis, a mathematical model of a distributed dual-wheel tractor was established, and a hardware-in-the-loop (HIL) test bench was built. PI parameters were adjusted by the fuzzy controller in real time based on the response timeliness and stability requirements of a distributed-drive control system, as well as the real-time errors and error change rate of the distributed vehicle speed and yaw angle. In addition, simulation and experimental verification were verified by MATLAB_R2023a/Simulink and the hardware-in-the-loop (HIL) test bench, which demonstrate that the fuzzy PI controller can improve system steady-state response speed, reduce overshoot, and enhance the driving stability of the system when it is subjected to off-load disturbances.

2. Materials and Methods

The distributed dual-wheel structure studied in this paper is shown in Figure 1. The triangular track is directly driven by two permanent magnet synchronous motors, which are directly controlled by the driver, so as to realize the functions of stepless transmission and electronic differential. Compared with the existing dual-wheel-drive tractors, it not only significantly reduces the size of the tractor and improves its flexibility but also makes electrical braking, electromechanical composite braking, and regenerative braking easier to implement.

This section mainly constructs a mathematical model of the distributed dual-wheel tractor and analyzes the disturbance factors during the tractor’s driving process.

2.1. Distributed Tractor Mathematical Model

Based on the structural characteristics of the distributed dual-wheel tractor, a model in an XOY coordinate system is established, as shown in Figure 2 [23].

In Figure 2, $L$ and $R$ represent the left- and right-drive wheels of the distributed dual-wheel tractor, respectively. The positive semi-axis of the X-axis is defined as the positive direction of the tractor’s straight-line motion, and the Y-axis positive semi-axis is defined as the positive direction. The center of motion of the tractor is denoted by point $C$, with coordinates (x, y). The straight-line distance between the dual wheel is the wheelbase, $l$. Let the angular velocity of the tractor be $\omega$, the radius of rotation of the tractor around the instantaneous center, $O_c$, is $r$, and the angle between the tractor and the positive semi-axis of the X-axis is the yaw angle, $\theta$. The speed at point $C$ is the tractor’s instantaneous linear velocity, $V_c$, while the instantaneous linear velocities of the left and right driving wheels are denoted by $V_l$ and $V_r$, respectively.

The counterclockwise direction is taken as the positive direction for the yaw angle. The tractor motion can be represented by vector $C=[x,y,\theta ]$. The matrix expressions are shown in Equations (1) and (2):

$$\left[\begin{array}{c}x\\ y\\ \theta \end{array}\right]=\left[\begin{array}{cc}\frac{\cos \theta }{2}& \frac{\cos \theta }{2}\\ \frac{\sin \theta }{2}& \frac{\sin \theta }{2}\\ -\frac{1}{l}& \frac{1}{l}\end{array}\right]·\left[\begin{array}{c}{V}_l\\ V_r\end{array}\right]$$

(1)

$$\theta ={\displaystyle \int \omega dt}$$

(2)

From the above analysis, it can be seen that the dual-wheel-drive model described in this paper is a globally controllable system. By controlling the linear velocities $V_l$ and $V_r$, it is theoretically possible to achieve the movement of the tractor in any given pose. At the same time, due to the constraints of the system, it is assumed in the kinematic model analysis that the vehicle and the ground are pure rolling without lateral sliding.

Equations (3) and (4) are obtained by the forward kinematics solution of a dual-wheel tractor:

$$V_l=V_c-\frac{\omega l}{2}$$

(3)

$$V_r=V_c+\frac{\omega l}{2}$$

(4)

The relationship between the drive wheel speed and drive wheel rotational speed is shown in Equation (5):

$$V_l=\frac{\pi d{n}_l}{60}$$

(5)

where $n_l$ is the left-drive-wheel rotational speed, with the unit of r/min.

Therefore, for the entire control system, inputting vehicle speed and yaw angle commands can control the left and right wheel speeds of the tractor, thereby controlling the motion state of the tractor.

2.2. The Overall Configuration

The target power of the agricultural distributed dual-wheel electric-drive tractor studied in this paper is 50 kW, and its basic components are shown in Figure 1. The MCU serves as the upper computer, sends action commands to the motor driver, which drives the motor and the triangular tracks, to complete the driving action of the dual-wheel tractor. The parameters of its main components are shown in

Table 1. Main component parameters of the tractor.

Parameter	Value
Diameter of the triangular track drive wheel/mm	1000
Wheelbase/m	2
Rated power per motor/kW	25.1
Rated torque per motor/N$·$m	120
Rated speed per motor/(r$·$min−1)	2000
Gear ratio	20

, and the total weight of the tractor is approximately 1816 kg.

2.2.1. Scaling of Tractor Maximum Acceleration Analysis and Test

In analyzing the distributed dual-wheel tractor, its motion equations are as shown in Equations (6)–(9):

$$F_{\mathrm{t}}=\frac{T_{\mathrm{tq}}·{\displaystyle \sum i·{\eta }_{\mathrm{t}}}}{R_0}$$

(6)

$$F_f=\omega (f_1+f_2)$$

(7)

$$F_j=\delta m\frac{dv}{dt}$$

(8)

$$a=\frac{dv}{dt}$$

(9)

where $F_t$ is the tractor traction force, with the unit of N; $T_{tq}$ is the rated torque of the motor, with the unit of N$·$m; ${\displaystyle \sum i}$ is the total gear ratio of the transmission system; ${\eta }_t$ is the mechanical efficiency of the transmission system; $R_0$ is the drive wheel radius, with the unit of m; $F_f$ is the tractor rolling resistance, with the unit of N; $F_j$ is the acceleration resistance, with the unit of N; $\omega$ is the triangular track load wheel method load, with the unit of N; $f_1$ is the rolling friction coefficient; $f_2$ is the internal friction resistance coefficient; $\delta$ is the vehicle rotational mass conversion factor; and $a$ is the tractor acceleration, with the unit of m/s².

It can be seen from Table 1 that the rated torque of the motor is 120 N$·$m, and the radius of the triangular track drive wheel is 0.5 m. As direct drive is employed without any transmission mechanism, the total gear ratio of the transmission system is taken as 1. The triangular track is used as the walking component, and there exists friction between the rubber and the drive wheels, as well as the load wheels, which may lead to slippage. Therefore, the mechanical efficiency of the transmission system is taken as 0.9, and the coefficient of the internal friction resistance is taken as 0.07. By substituting the above parameters into Equation (6), the rated traction force of the tractor is approximately 8640 N.

By connecting Equations (6)–(9), the acceleration of the tractor can be obtained as Equation (10):

$$a=\frac{F_t-F_f}{\delta m}$$

(10)

Tractors are often used in agricultural environments, and the rolling resistance coefficient of the tracks is highest in muddy or sandy terrain, ranging from 0.10 to 0.15 [24], with a maximum value of 0.15 taken here. The mass of the dual-wheel tractor is approximately 1816 kg. The conversion coefficient for the rotational mass of triangular tracked vehicles ranges from 0.7 to 1, with a chosen value of 1. Taking the gravitational acceleration as 9.8 m/s² and substituting it into Equation (10), the maximum acceleration of the tractor is calculated to be approximately 2.6 m/s². Considering operational traction requirements, the tractor reserves 20% torque, resulting in the maximum acceleration of the tractor being 2.08 m/s².

Taking into account factors such as operational environments, mechanical components, and the stability of the dual-wheel structure, the maximum acceleration of the tractor is set to 2 m/s². When the test bench is built, the speed command of the upper computer is set as the slope curve of slope 2 to simulate the acceleration response state of the tractor in a real agricultural environment. At a vehicle speed of 1 m/s, the motor speed is approximately 17.37 r/min.

2.2.2. Analysis of Tractor Uneven Load Disturbance

In scenarios where agricultural distributed dual-wheel electric-drive tractors are easily disturbed when the materials on both sides of the road surface are inconsistent, the uneven rolling friction caused by inconsistent road surface materials results in inconsistent loads acting on the two drive wheels.

Analyzing the impact load disturbance, assuming that the vehicle speed of the tractor is 3 m/s, after being affected by the impact off-load, the speed of one side of the track decreases to 2.7 m/s and then returns to the target speed. With an effective load action time of approximately 0.5 s, the load acting on the drive wheel during the entire process is shown in Equation (11):

$$F_{ls}=m\frac{\Delta v}{\Delta t}$$

(11)

where $F_{ls}$ is the impact load, with the unit of N.

By substituting the above parameters into Equation (11), the impact off-load is calculated to be 1089.6 N. During the impact off-load test, approximately 27.24 N$·$m of the shaft impact torque load should be applied to one side of the motor.

The rolling resistance coefficient of the track on muddy or sandy terrain is approximately 0.10 to 0.15, with a median value of 0.12 within the range and a rolling resistance coefficient of 0.07 on solid soil, respectively. After substituting them into Equation (7), it can be seen that the rolling resistance loads of the driving wheels on both sides are 1690.7 N and 1245.78 N, respectively, with a difference of 444.92 N. During the test, except for the part affected by the load on the mold, a continuous axial torque load of about 11.12 N$·$m should be applied to one side of the motor.

2.3. HIL Test Bench Design

Based on the power system composition of agricultural distributed tractors, the hardware-in-the-loop simulation test bench setup and MCU interface are shown in Figure 3 and Figure 4, respectively.

The real-time semi-physical in-loop simulation test bench is composed of dSPACE, a power supply, a motor driver, two permanent magnet synchronous motors with a rated power of 25.1 kW, and an electromagnetic powder brake. dSPACE is used as an MCU to collect data on the vehicle speed and yaw angle. After speed calculation, the speed commands of the two motors are transmitted to the motor driver through CAN bus communication to control the motor to perform actions. Meanwhile, the encoder on the motors and the sensor on the driver provide feedback on the motor speed and torque on the motor shaft to dSPACE through CAN bus communication, forming a closed-loop feedback control and calculating the trajectory of tractor motion. To simulate the biased load and relative load between the two driving wheels during the motion of the power platform, a magnetic powder brake is used to load one side of the motor.

2.4. Fuzzy Controller Design

The fuzzy PI control algorithm studied in this paper combines a fuzzy controller with a PI controller in a dual-input and dual-output configuration. The fuzzy controller utilizes fuzzy reasoning based on the real-time error, $e$, and the error change rate of the system, $e_c$, to derive the correction amount, $\Delta K_p$ and $\Delta K_i$, for PI control parameters, so as to adjust the PI parameters in real time. This section presents the PI controller and the fuzzy controller.

2.4.1. PI Controller Design

Closed-loop control is carried out for the vehicle speed and yaw angle during the control of the tractor’s driving process. The control block diagram is shown in Figure 5.

In Figure 5, $K_{vp}$, $K_{v\mathrm{i}}$ and $K_{wp}$, $K_{wi}$ are the control parameters of the PI controller for the vehicle speed loop and the yaw angle loop, respectively; $V_c^{*}$ and ${\omega }^{*}$ are the vehicle speed and yaw angle input for the command, respectively; $V$ and $\omega$ denote the actual vehicle speed and yaw angle calculated based on the inverse kinematics of the motor feedback speed, respectively; $n_l^{*}$ and $n_r^{*}$ are the motor speed of the left- and right-drive wheels calculated by the control system, respectively; and $n_l$ and $n_r$ denote the actual motor speeds as feedback.

The motor control in Figure 5 adopts a closed-loop vector control scheme for permanent magnet synchronous motors, as shown in Figure 6.

The transfer function of motor speed closed-loop control is shown in Equations (12) and (13):

$$G_1=K_T·\frac{k_{sp}{k}_p{S}^2+\left(k_{sp}{k}_i+k_{si}{k}_p\right)S+k_{si}{k}_i}{1.5J{L}_{\mathrm{q}}{T}_{pwm}{S}^5+J\left(L_{\mathrm{q}}+1.5R{T}_{pwm}\right)S^4+J\left(R+k_p\right)S^3+\left(J{k}_i+K_T{k}_{sp}{k}_p\right)S^2+K_T\left(k_{sp}{k}_i+k_{si}{k}_p\right)S+K_T{k}_{si}{k}_i}$$

(12)

$$K_T=\frac{3}{2}{n}_p{\phi }_f$$

(13)

where $k_{sp}$ is the speed loop proportional coefficient; $k_{si}$ is the speed loop integral coefficient; $k_p$ is the current loop proportional coefficient; $k_i$ is the current loop integral coefficient; $L_q$ is the equivalent q-axis inductance, with the unit of H; $R$ is the resistance of the permanent magnet synchronous motor, with the unit of Ω; ${\phi }_f$ is the rotor permanent magnet flux, with the unit of Wb; $n_p$ is the number of motor poles; $T_{pwm}$ is the number of motor poles; $K_T$ is the torque coefficient; and $G_1$ is the speed closed-loop transfer function.

The parameters of the motor and the closed-loop control parameters are shown in

Table 2. Control parameters and motor-related parameters.

Parameter	Value
$k_{sp}$	0.63
$k_{si}$	0.01
$k_p$	0.1
$k_i$	2
$L_q$/H	0.003483
$R$/Ω	0.04785
${\phi }_f$/Wb	0.21088
$n_p$	4
$T_{pwm}$	10−4

.

Thus, based on the motor transfer function and the Figure 5 control diagram, the transfer function of the closed-loop control system for the tractor is established, as shown in Equations (14)–(19):

$$a\left(s\right)=\frac{G_1{R}_0}{1-0.25{\left(\frac{k_{wp}s+k_{wi}}{s}\right)}^2{R}^2{G}_1^2}$$

(14)

$$G_v{\left(s\right)}_{open}=\frac{1}{2}·\frac{k_{vp}s+k_{vi}}{s}·\left[a\left(s\right)+G_1{R}_0\right]$$

(15)

$$G_v\left(s\right)=\frac{G_v{\left(s\right)}_{open}}{1+G_v{\left(s\right)}_{open}}$$

(16)

$$b\left(s\right)=\frac{G_{1}R+\frac{k_{vp}s+k_{vi}}{s}{R_0}^2{G_1}^2}{1+0.25{\left(\frac{k_{vp}s+k_{vi}}{s}\right)}^2{R_0}^2{G_1}^2}$$

(17)

$$G_{\omega }{\left(s\right)}_{open}=\frac{1}{2}·\frac{k_{wp}s+k_{wi}}{s}·\left[b\left(s\right)+G_{m2}{R}_0\right]$$

(18)

$$G_{\omega }\left(s\right)=\frac{G_{\omega }{\left(s\right)}_{open}}{1+G_{\omega }{\left(s\right)}_{open}}$$

(19)

where $k_{vp}$ is the vehicle speed loop proportional coefficient; $k_{vi}$ is the vehicle speed loop integral coefficient; $k_{wp}$ is the yaw angle loop proportional coefficient; $k_{wi}$ is the yaw angle loop integral coefficient; $a\left(s\right)$ and $b\left(s\right)$ are the intermediate variables; $G_v{\left(s\right)}_{open}$ is the open-loop transfer function of the vehicle speed loop; $G_v\left(s\right)$ is the closed-loop transfer function of the vehicle speed loop; $G_w{\left(s\right)}_{open}$ is the open-loop transfer function of the yaw angle loop; and $G_w\left(s\right)$ is the closed-loop transfer function of the yaw angle loop.

It can be seen from the transfer functions of the vehicle speed loop and the yaw angle loop that the system is a nonlinear time-varying cross-coupled dual-input dual-output control system. Setting one of the loops to zero during parameter tuning has a minimal impact on the tuning loop; the two interact with each other during the actual operation of the tractor. After multiple simulations, the control parameters of the vehicle speed loop are determined to be $k_{vp}$ = 0.2 and $k_{vi}$ = 35, while for the yaw angle loop the control parameters are $k_{wp}$ = 0.2 and $k_{wi}$ = 35.

2.4.2. Fuzzy Controller Design

On the basis of the PI controller, the fuzzy controller is employed to perform fuzzy inference on the error and the error change rate of the system, and fuzzy decisions are made according to fuzzy rules, ultimately obtaining the output of the PI control parameter increment [25]. Taking the vehicle speed loop as an example, the PI controller in Figure 4 is transformed into the fuzzy controller, and the fuzzy control block diagram is shown in Figure 7:

In Figure 7, $e$ represents the system error; $e_c$ represents the rate of change of the error; and $\Delta K_p$ and $\Delta K_i$ represent the PI parameter increments output by the fuzzy controller.

During the control process of the fuzzy controller, the observed system data need to be fuzzified. After multiple simulations and experimental validations of the tractor model established in this paper, the actual ranges of the error, $e$, and the rate of change of the error, $e_c$, for the vehicle speed loop are determined to be [−1, 1] and [−150, 150], respectively, which are taken as the domains of the error and the rate of change of the error. After several simulation trials, the ranges of $k_{vp}$ and $k_{vi}$ are determined to be [0, 0.7] and [0, 60], respectively, which are subtracted from the initial domain adjusted in the previous section. The above domains are discretized into seven levels, namely {NB, NM, NS, ZO, PS, PM, PB}, where N represents negative, P represents positive, ZO represents zero, and S, M, and B represent small, medium, and large, respectively [26,27,28]. The membership functions of the error, $e$, and the rate of change of the error, $e_c$, use Gaussian functions, while the membership functions of $\Delta K_p$ and $\Delta K_i$ use triangular functions. The membership functions of the error, $e$ and $\Delta K_p$, are shown in Figure 8.

The control strategy of fuzzy rules summarizes the adjustment strategy of the system according to the response of experienced personnel to the system; therefore, suitable fuzzy rules need to be established based on the response of the distributed tractor to adjust the PI controller parameters.

The following are the principles of the fuzzy controller to be established in this paper [29,30]:

(1) When the error, $e$, is large, larger $k_{vp}$ and smaller $k_{vi}$ should be output to accelerate the response speed of the system.

(2) When the error, $e$, is small, $k_{vp}$ and $k_{vi}$ should be reduced to decrease the system overshoot.

(3) When the rate of change of the error, $e_c$, is large, $k_{vp}$ and $k_{vi}$ should be appropriately reduced to decrease the oscillation caused by the integral component.

(4) When the error, $e$, and the rate of change of the error, $e_c$, are of medium size, the values of $k_{vp}$ and $k_{vi}$ remain basically unchanged to ensure the responsiveness and stability of the system.

According to the above principles, fuzzy control rule tables for $\Delta K_p$ and $\Delta K_i$ are established as shown in

Table 3. Control rules of $\Delta K_p$.

$\mathbf{\Delta }{\mathit{K}}_{\mathit{p}}$		${\mathit{e}}_{\mathit{c}}$
		NB	NM	NS	ZO	PS	PM	PB
$e$	NB	NB	NB	NM	NM	NS	ZO	ZO
	NM	NB	NB	NM	NS	NS	ZO	PS
	NS	NM	NM	NM	NS	ZO	PS	PS
	ZO	NM	NM	NS	ZO	PS	PM	PM
	PS	NS	NS	ZO	PS	PS	PM	PM
	PM	NS	ZO	PS	PM	PM	PM	PB
	PB	ZO	ZO	PM	PM	PM	PB	PB

and

Table 4. Control rules of $\Delta K_i$.

$\mathbf{\Delta }{\mathit{K}}_{\mathit{i}}$		${\mathit{e}}_{\mathit{c}}$
		NB	NM	NS	ZO	PS	PM	PB
$e$	NB	PB	PB	PM	PM	PS	ZO	ZO
	NM	PB	PB	PM	PS	PS	ZO	ZO
	NS	PB	PM	PS	PS	ZO	NS	NS
	ZO	PM	PM	PS	ZO	NS	NM	NM
	PS	PM	PS	ZO	NS	NS	NM	NB
	PM	ZO	ZO	NS	NS	NM	NB	NB
	PB	ZO	ZO	NS	NM	NM	NB	NB

, respectively.

The logical definition represented in Table 3 and Table 4 are as follows: if the error, $e$, is PB and the error change rate of the system, $e_c$, is PM, then $\Delta K_p$ and $\Delta K_i$ are PB and NB, respectively.

Since the control quantity output by the fuzzy controller is the combination of different membership function curves, which cannot be directly applied to the controlled variable, it is therefore necessary to determine a value from the output variables of the fuzzy controller, which is then specified as a clear variable. The weighted average method is used in this paper to defuzzify the arrays between membership function curves. The fuzzy arrays between membership function curves, $G$, are divided into n different discrete pieces of data. Each piece of data, ${\lambda }_i$, is multiplied by its corresponding membership degree, ${\mu }_G({\lambda }_i)$, and then the sum obtained by adding them is divided by the sum of the membership degrees to obtain the defuzzified output value, $U_G$, as shown in Equation (20):

$$U_G=\frac{{\displaystyle \sum _{i=1}^n{\mu }_G({\lambda }_i){\lambda }_i}}{{\displaystyle \sum _{i=1}^n{\mu }_G({\lambda }_i)}}$$

(20)

Under the domain of the error, $e$, and the error change rate of the system, $e_c$, the output surface is illustrated in Figure 9, and it can be seen that its output is a nonlinear and continuously changing surface.

PI control parameters acting on the speed loop of the distributed dual-wheel tractor are obtained by adding its initial values, $k_{vp}\prime$ and $k_{vi}\prime$, to the increment values, $\Delta K_p$ and $\Delta K_i$, from the fuzzy control output, as shown in Equation (21):

$$\left\{\begin{array}{c}{k}_{vp}=k_{vp}\prime +\Delta K_p\\ k_{vi}=k_{vi}\prime +\Delta K_i\end{array}\right.$$

(21)

3. Results and Discussion

3.1. Simulation Results and Analysis of Vehicle Speed Response

Taking the vehicle speed control of distributed dual-wheel electric-drive tractors as an example, combined with Figure 3 and Figure 4, a simulation model of distributed dual-wheel tractors was established in MATLAB/Simulink to test the vehicle speed response and stability under no-load and off-load conditions.

3.1.1. Simulation and Analysis of Vehicle Speed Response under No-Load Conditions

Setting the target vehicle speed to 1 m/s and the yaw angle to 0.1 rad/s, the simulation results are shown in Figure 10.

As shown in Figure 10, the overshoot of traditional PI is 0.08%, while the fuzzy PI control cannot generate overshoot under the condition of a fast response speed. Compared with traditional PI control, the steady-state response time of the fuzzy PI controller is reduced by 0.3 s.

3.1.2. Simulation and Analysis of Vehicle Speed Response under Off-Load Conditions

Setting the target vehicle speed to 1 m/s and the yaw angle to 0.1 rad/s, an impact off-load of 27.24 N$·$m and a continuous off-load of 11.12 N$·$m were applied to the right-drive motor after 4 s. The simulation results are shown in Figure 11.

According to the simulation results shown in Figure 11, under impact off-load conditions, the overshoot of traditional PI is 0.03% and the steady-state response time is 1.5 s., while the overshoot of fuzzy PI control is 0.02% and the steady-state response time is 1.1 s. Compared with the traditional PI control strategy, the overshoot of the fuzzy PI control strategy is reduced by 30%, and the steady-state response time is reduced by 0.4 s. Under continuous off-load conditions, the speed variation in distributed dual-wheel electric-drive tractors using fuzzy PI control is nearly identical to that with traditional PI control; however, when the fuzzy PI controller is used, the vehicle speed changes smoothly during the process of recovering to the target vehicle speed.

Through the simulation of vehicle speed response, it has been confirmed that the fuzzy PI controller improves the response characteristics and anti-interference stability of the tractor control system.

Therefore, the fuzzy PI controller was deployed on a real-time in-the-loop simulation test platform for a hardware-in-the-loop test, and the tractor was subjected to a vehicle speed loop response test and simulated for tractor driving tests under no-load, impact off-load, and continuous off-load operating conditions. By comparing it with a traditional PI controller, the advantages of a fuzzy PI controller in complex agricultural environments are further confirmed.

3.2. Vehicle Speed Response Test

The vehicle speed response tests were conducted on control systems employing a fuzzy PI controller and a traditional PI controller. The acceleration and deceleration processes were set at a speed of 0–2 m/s to obtain tractor speed response data. The results are shown in Figure 12.

As shown in Figure 12, the traditional PI controller exhibits a fast response during continuous acceleration and deceleration processes, but has an overshoot of approximately 9%. The fuzzy PI controller responds quickly during continuous acceleration and deceleration processes, with an overshoot of only 3%, which is 60% less than that of the traditional PI controller. Considering the characteristics of distributed power system control, the fuzzy PI controller’s response is more ideal and can improve the stability of the control system.

3.3. Tractor Straight-Line Driving Test under Various Operating Conditions

Tractor motion trajectories were collected and analyzed using the real-time hardware-in-the-loop simulation platform, as shown in Figure 13. Test trajectories, the yaw angle, real-time motor speeds, and torque information were output to an oscilloscope, a YOKGAWA-DLM3034, through dSPACE’s DAC output function. The variations in tractor motion trajectories before and after applying fuzzy control strategies and under loading conditions were compared. In the obtained tractor driving trajectory data, the maximum deviation of the trajectory compared to the ideal trajectory (y = 0) when the tractor is subjected to off-load disturbance is measured, and the stability of the anti-interference control strategy of the distributed dual-wheel power system is analyzed.

3.3.1. No-Load Straight-Line Driving Condition Test

The no-load straight-line driving test was carried out on the tractor to analyze the quality of the trajectory. The tractor speed was set to 3 m/s; the DAC output function of dSPACE was used to collect the tractor trajectory data, and they were compared with those of the traditional PI controller trajectory. The test lasted approximately 100 s, and the results are shown in Figure 14.

During the approximately 100 s driving test, the maximum deviation of the tractor trajectory under the traditional PI control strategy was about 0.1 m, while under the fuzzy PI control strategy it was about 0.04 m. Throughout the test, the deviation angle between the PI control trajectory direction and the expected trajectory direction continued to increase, while the fuzzy control trajectory showed higher overlap with the expected trajectory and smaller deviation angles.

3.3.2. Impact Off-Load Straight-Line Driving Condition Test

The linear driving test of the tractor with impact off-load was carried out, and the driving speed of the tractor was set at 3 m/s. When the driving stabilized, an impulse off-load of 27.24 N$·$m was applied on the shaft. The test lasted approximately 100 s, and tractor driving trajectory data were collected, as shown in Figure 15.

As shown in Figure 15, it can be seen that after applying the impulse off-load to the tractor, the maximum deviation of the fuzzy control trajectory was approximately 0.08 m, while the maximum deviation of the traditional PI control trajectory was about 0.14 m. Compared with traditional PI control, the fuzzy control resulted in a smaller deviation angle between the tractor and the X-axis, indicating better control performance and potential for further enhancing system stability.

3.3.3. Continuous Off-Load Straight-Line Driving Condition Test

The continuous off-load straight-line driving test was carried out on the tractor, and the driving speed of the tractor was set at 3 m/s. When the driving stabilized, a continuous off-load of 11.12 N$·$m was applied. The test lasted approximately 100 s, and tractor driving trajectory data were collected, as shown in Figure 16.

As shown in Figure 16, it can be observed that under continuous off-load conditions, the maximum deviation of the tractor trajectory with the traditional PI controller was 0.12 m, while with the fuzzy PI controller it was 0.08 m. Compared to traditional PI control, fuzzy PI control resulted in a smaller deviation angle between the tractor and the X-axis, indicating more stable tractor operation and a more ideal trajectory.

3.4. Discussion

In this section, the fuzzy control algorithm is deployed on the simulation test bench to test and verify the stability of the control system. The maximum overshoot, steady-state response time, and maximum trajectory offset under traditional PI control and fuzzy PI control are analyzed and compared, as shown in

Table 5. Comparison of maximum overshoot and steady-state response time.

Controller	Acceleration Process		Deceleration Process
	Maximum Overshoot (%)	Steady-State Response Time (s)	Maximum Overshoot (%)	Steady-State Response Time (s)
Traditional PI controller	0.09	1.15	0.08	1.02
Fuzzy PI controller	0.03	0.67	0.03	0.94

and

Table 6. Comparison of Maximum Trajectory Deviation.

Controller	No Load	Impulse Off-Load	Continuous Off-Load
	Maximum Trajectory Offset (m)
Traditional PI controller	0.10	0.14	0.12
Fuzzy PI controller	0.04	0.08	0.08

.

The results indicate the following:

(1) In the whole-vehicle speed loop tests, during continuous acceleration the traditional PI controller exhibits maximum overshoot and steady-state response times of 0.09% and 1.15 s, respectively, while under fuzzy PI control the maximum overshoot and steady-state response time are 0.03% and 0.67 s, respectively. After fuzzy parameter adjustment of the PI control, the maximum overshoot and steady-state response time decrease by 66.7% and 41.7%, respectively. Similarly, during continuous deceleration, the maximum overshoot and steady-state response time of the traditional PI controller are 0.08% and 1.02 s, respectively, while under fuzzy PI control the maximum overshoot and steady-state response time are 0.03% and 0.94 s, respectively. After fuzzy parameter adjustment, the maximum overshoot and steady-state response time decrease by 62.5% and 7.8%, respectively. Considering the characteristics of distributed power systems, the fuzzy PI controller performs more effectively, significantly enhancing the stability of the control system.

(2) In the straight-line driving test of the tractor, under the traditional PI controller strategy the maximum deviation of tractor trajectories under no-load, impact off-load, and continuous off-load conditions are 0.10 m, 0.14 m, and 0.12 m, respectively, while the maximum driving offset of the tractor under the fuzzy PI controller is 0.04 m, 0.08 m, and 0.08 m, respectively. After fuzzy parameter adjustment of the PI control, the maximum trajectory deviations decrease by 60.0%, 42.9%, and 33.3% for the three conditions. Although the maximum offset under both control strategies is within the ideal offset error, based on the analysis of the angle between the tractor’s driving direction and the X-axis at the end of the experiment, compared with traditional PI control, fuzzy PI control has a smaller angle between the tractor’s driving trajectory and the X-axis, and can control the maximum trajectory offset within 0.08 m, significantly improving driving accuracy and driver maneuverability. This improvement in control accuracy not only enables drivers to operate tractors more accurately, but also ensures stable driving on uneven terrain, reducing operational difficulty and potential risks caused by trajectory deviation.

In summary, the study of a distributed dual-wheel tractor with a fuzzy PI control strategy has enhanced the tractor’s operational precision and stability in real agricultural environments, making it significant for activities like field cultivation and planting. It also extends the application of fuzzy control theory to nonlinear and time-varying systems. Despite the tractor’s compact size, strong power, and high flexibility, practical applications may face challenges such as extreme weather conditions that could impact controller performance. Additionally, current battery technology limits endurance in heavy-load scenarios, making range-extending power sources a feasible solution for the future.

4. Conclusions

This paper mainly investigates the control optimization of distributed dual-wheel tractors in complex agricultural environments. In order to solve the nonlinear and time-varying coupled control issues of distributed dual-wheel electric-drive tractor motion systems, a distributed dual-wheel electric-drive system controller based on vehicle speed and a steering fuzzy PI controller was designed for the continuous off-load condition affecting the stable operation of the distributed tractor. Simulink simulation and a hardware-in-the-loop (HIL) test bench were used to analyze the response speed, quality, and stability of the tractor system under off-load disturbance.

The results indicate the following: (1) Compared to traditional PI control, fuzzy PI control can significantly enhance system response accuracy with a reduction in overshoot by approximately 60% and the steady-state response time by approximately 30%, thereby improving system stability. When the system is disturbed by off-load, the fuzzy PI controller has stronger anti-interference and a smoother recovery path to the target state. (2) The fuzzy PI controller proposed in this paper has been tested in a straight line, and the results show that the maximum offset of the tractor trajectory is controlled within 0.04 m under the condition of no load; under conditions of tractor disturbance due to loading, the maximum deviation is controlled within 0.08 m. Compared to the traditional PI controller, the maximum trajectory offset has been reduced by 45%, the tractor’s driving trajectory is more stable, and the overall control performance is better.

The application of distributed dual-wheel electric-drive tractors in complex agricultural environments is a future research hotspot. The research on a distributed dual-wheel tractor based on a fuzzy PI control strategy in this paper provides theoretical and data-based support for the subsequent development and improvement of distributed agricultural equipment electric-drive systems.