#### Abstract

To solve the problems of low loading precision, slow response speed, and poor adaptive ability of a mobile dynamometer in a tractor traction test, a PID control strategy based on a radial basis function neural network with self-learning and adaptive ability is proposed. The mathematical model of the loading system is established, the algorithm of adaptive control is described, and the loading control method is simulated with MATLAB software. The system, which uses the NN-PID (neural network PID) control strategy, is used to test a YTO-MF554 tractor. Then, the proposed control strategy is validated. Results show that when the traction increases from 0 to 10 kN, the response time of the test system is 1.5 s, the average traction force in the stability range is 10.13 kN, and the maximum relative error of traction force is 2.2%. This control strategy can improve the response speed and steady-state accuracy and enhance the adaptive ability of the mobile dynamometer vehicle loading system. This study provides a reference for designing the adaptive controller of the mobile dynamometer vehicle loading system.

#### 1. Introduction

The loading process of mobile dynamometer vehicles presents many uncertainties in the tractor traction test. Frequently tuning the parameters is necessary for traditional control methods, thus disrupting the test. Moreover, the traditional system has low efficiency and accuracy. A considerable number of theories and practices have shown that the application of self-adaptable control technology to the test system will improve the efficiency and accuracy of the test system [1–4].

The adaptive control algorithm and its application have been studied by many local and international researchers. Locally, Wang et al. proposed the fuzzy adaptive PID control for the nonlinear and time-variant mobile dynamometer of an automobile [5]. He et al. expounded the output prediction of complex nonlinear systems based on the radial basis function (RBF) neural network and obtained well-predicted results [6]. Xia and Wang proposed a new method that combined the RBF neural network and single neuron PID. This method was applied to the speed control of a switched reluctance motor with good control effect [7]. Wang et al. applied RBF-PID to improve the temperature control of a thermal power plant [8]. Internationally, Anwar optimized PID controller parameters based on a genetic algorithm and achieved accurate real-time control for retarder loading [9]. Dash et al. expounded the application of the RBF neural network PID in controlling an electrical power unit [10]. D. L. Yu and D. W. Yu investigated the adaptive adjustment algorithm of RBF neural networks [11]. However, there are few studies on the adaptive control of loading for a mobile dynamometer vehicle in the tractor traction test.

In this study, the RBF neural network PID control strategy is used for the load control of the mobile dynamometer vehicle. The system output traction has a good follow effect in comparison with input load. And the output load for power wagon random loading system can reproduce the tractive load for tested tractor reasonably. This control strategy can improve loading precision and response speed and enhance the self-adaptive ability of the control system. This method can also provide a reference for the investigation of mobile dynamometer vehicle loading control.

#### 2. Mathematical Model of the Mobile Dynamometer Vehicle Loading System

The mobile dynamometer vehicle is modified with a YTO-1304 tractor. The tractor power take-off is connected to an electric eddy current retarder, which brakes the mobile dynamometer vehicle by loading the transmission system.

##### 2.1. Model of the Eddy Current Retarder

The eddy current retarder is mainly composed of front and rear rotor disks and eight excitation coils with an iron core between the disks. The following simplifications and assumptions are made when the loading torque of the retarder is calculated: the rotor disks are simplified as annular plates; the magnetic field that is produced by the coil is only distributed in the circular region and magnetic flux leakage is ignored; the relative permeability of the rotor disk is considered constant; and hysteresis losses and magnetic saturation are ignored. A detailed deduction is presented in [12], wherein the loading torque is denoted by the following:where is loading torque, N·m; is the number of magnetic pole pairs; is the resistivity of rotor disk, Ω·m; is the permeability of the vacuum, N·A^{−2}; is the number of turns of excitation coils; is the excitation current, A; is the magnetic core diameter of the excitation coil, m; is the angular velocity of the magnetic field changes, rad/s; is the relative magnetic permeability of the rotor disk; is the distance between the center of the magnetic pole and the center of the rotor disk, m; is the width of air gap m; and is the conversion rate ( usually takes 1.5).

Given that , the loading torque is simplified as follows:where , , and are parameters that are related to the structure and material of the eddy current retarder.

A simplified derivation of the retarder transfer function is as follows: the excitation coil of the eddy current retarder can be simplified as the resistance and inductance in series [13]. Therefore, the transfer function of the excitation voltage and current is as follows:where is the coil inductance of the eddy current retarder, H; is the coil resistance of the retarder, Ω; and is the excitation voltage, V.

A simplified method for establishing the transfer function of the DC dynamometer is based on [14]. The eddy current retarder with nonlinear properties can be simplified as a linear element with the assumption that the change rate of the excitation current is approximately constant when the excitation voltage changes. Combining (2) and (3) yields the transfer function of the retarder:In the formula, is a variable, which is related to the structure, material, and current change rate of the eddy current retarder.

##### 2.2. Model of Loading Transmission System

###### 2.2.1. Eddy Current Retarder

The moment of inertia of the rotor disk is considered in establishing the transfer function of the retarder. The retarder is a brake component, which is an external drive that provides power. The dynamic variables of the retarder are shown in Figure 1.where is the moment of inertia of the retarder rotor disk; kg·m^{2}; is the rotation angle of the retarder rotor disk rad; is the input torque of the retarder N·m; and is the loading torque of the retarder N·m.

###### 2.2.2. Gear Pairs

The system variables of the gear pairs are shown in Figure 2.where is the moment of inertia of gear pairs, kg·m^{2}; is the damping coefficient of the friction torque, N·m·s·rad^{−1}; is the rotation angle of the transmission output shaft, rad; is the transmission ratio of the gear pair; and is the torque of the transmission output shaft, N·m.

###### 2.2.3. Final Drive

The system variables of the final drive are shown in Figure 3.where is the moment of inertia of the final drive, kg·m^{2}; is the damping coefficient of the final drive friction torque, N·m·s·rad^{−1}; is the final drive ratio; is the torque of wheels N·m; and is the rotation angle of the final drive output shaft rad.

###### 2.2.4. Wheels

The slip ratio of mobile dynamometer vehicle is ignored and the wheels are purely rolling. The transient dynamic analysis is performed on the entire system when the test tractor is loaded. Then, the following conclusions can be obtained:where is the moment of inertia of the wheels, kg·m^{2}; is the damping coefficient of the friction torque of the wheels, N·m·s·rad^{−1}; is the traction of the system, N; is the rolling resistance of the mobile dynamometer, N; is the rolling radius of the driving wheels of the mobile dynamometer vehicle, m; is the driving force of the testing tractor, N; is the rolling resistance of the tested tractor, N; is the rotation angle of the driving wheels of the mobile dynamometer vehicle, rad; is the quality of the mobile dynamometer vehicle, kg; is the quality of the tested tractor, kg; is the conversion factor of tested tractor’s revolving mass, ; and is the conversion factor of mobile dynamometer vehicle’s revolving mass, .

Based on the preceding summary, the following conclusions can be obtained: If we assume that , , thenwhere is the equivalent moment of inertia, kg·m^{2}, and is the equivalent damping coefficient of friction torque, N·m·s·rad^{−1}.

Assuming that the initial value of the system is zero, the rolling resistance is very small and can be ignored. Laplace transform is conducted for formulas (9) and (11), can be removed, and the result is as follows:

Based on the preceding summary, the transfer function of the entire system is shown as follows:

#### 3. Design of the Neural Network PID Controller

Frequently tuning the parameters is necessary if the nonlinearity, hysteresis, time-variation, and model uncertainty of the system are controlled by a traditional PID controller. The neural network control technology does not need the accurate mathematical model of the object and has fine self-learning and self-adaptation abilities. The parameters of the RBF-PID controller can be automatically corrected based on the changing conditions; therefore, control precision and response speed are improved [15, 16].

##### 3.1. The RBF Neural Network Model

The RBF neural network function solves nonlinear problems, which change into linear ones, by mapping the low-dimension original space to the high-dimension feature space and approximating any continuous function with the weighted sum of multiple basis functions [17, 18]. The RBF neural network structure is shown in Figure 4.

In the network structure, , is the input, is the vector of radial basis, and is usually the Gauss function:where is the center vector of the th node and is the width of the basis vector.

is the weight vector of the network, and the output of the network is shown as follows:The optimization objective function of the identifier is shown as follows:According to the gradient descent method, the iteration algorithms of the network parameters are as follows:where is the learning rate and is the momentum factor.

The Jacobin matrix indicates that, with regard to the sensitivity of the output to input changes, the algorithm is as follows:where .

##### 3.2. Control Principle of RBF-PID

The control principle of the loading system is shown in Figure 5. According to the experimental system proposed in this paper, the input is the desired traction; the output is the actual loading force, which is measured by drawing the force transducer; and the control object is the loading system. The RBF neural network realizes online identification. The parameters of the PID controller are dynamically tuned in accordance with the identification information and optimization index.

The control algorithm uses the increasing type of PID. The error is shown as follows:The algorithm is described as The optimization index of the neural network is, , and are tuned according to the gradient descent method.where is the Jacobin information of the controlled object, which can be calculated by the neural network; is the learning rate of the proportional unit; is the learning rate of the integral unit; and is the learning rate of the differential unit.

#### 4. Simulation Analysis of Loading Control System

The structure of the RBF neural network is 3-6-1, wherein , , and are inputs. After simulation and tests, the initialization parameters are determined as , , = 15, = 0.1, = 0.1, = 0.68, = 0.68, and = 0.35. The initial value of the weight vector, the node center, and the width value of the basis function are shown as follows:

A detailed simulation was conducted with MATLAB software. According to the characteristics of progressive loading, the traction is loaded from 0 to 10 k N. The response curves of two control strategies are shown in Figure 6. The RBF neural network PID, compared with traditional PID, effectively inhibits severe oscillation and realizes steady-state, error-free tracking.

#### 5. Verification of the Control Strategy by Vehicle Test

All the selected instruments in the testing system are shown in Table 1.

The loading test of the YTO-MF554 tractor was conducted based on the designed testing system. The test picture is shown in Figure 7. The test pavement is concrete, the temperature is 20°C, the pressure is 100.1 kPa, and the relative humidity is 65%. The height of the traction point is 580 mm and the front and rear tire pressures are both 100 kPa.

The output traction data is acquired when the traction is loaded from 0 to 10 kN, drawing the target value and the actual output value curve shown in Figure 8. The traction error in the stability interval (1.5–10 s) is shown in Figure 9. As shown in Figures 8 and 9, the response time of the complete loading process is 1.5 s, no obvious overshoot exists, the maximum traction error is 0.22 kN, and the traction mean value is 10.13 kN in the stability interval. The parameters of the RBF-PID controller are self-adjusted; therefore, the response speed is improved and the loading process is more stable. Moreover, the severe oscillation of the test system is avoided, which helps improve the test accuracy of each testing parameter.

#### 6. Conclusion

A RBF neural network for the PID control strategy of the mobile dynamometer vehicle loading system is designed. The structure and initial parameters of the RBF neural network are determined by a method that combined simulations and tests. RBF-PID can optimize PID parameters in real time. The simulation result shows that RBF-PID prevents the severe oscillation of the test system by improving the speed of dynamic response and steady-state precision of the system.

Based on the RBF-PID controller design, the loading test of the YTO-MF554 tractor was conducted according to the regulations of the tractor traction performance test. In the test, the mobile dynamometer vehicle is loaded from 0 to 10 kN and the loading time is 10 s. The response time is 1.5 s and no obvious overshoot occurred during the loading process. The maximum traction error is 0.22 kN and the mean traction value is 10.13 kN in the stability interval. Test efficiency and precision are obviously improved. Frequently tuning the parameters of the controller is no longer necessary.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research is supported by the 2016 National Key Research and Development Plan (no. 2016YFD0701002-2) and Key Technology Research Project of Henan Province (no. 102102210165).