Abstract
For electric power steering system (EPS), road interference, noise of the sensor, and the uncertainty of the steering system may make EPS control effect and the driver's road sense worse. EPS system which takes advantage of good current tracking ability, good anti-interference ability, and good operation stability is becoming more and more important in automotive research. The traditional H∞ control algorithm can solve the system uncertainty theoretically, but it cannot solve the contradiction between robustness and performance without considering the performance of the system. Therefore, this paper proposes a EPS current tracking method based on the hybrid sensitivity H∞ control algorithm, which takes the current tracking performance as one of the control objectives, so that the system can maximize the robustness and performance. Firstly, the dynamic model of EPS is established. Then, the two-degree-of-freedom vehicle model and tire model are introduced. The state space equation of the system is constructed on the basis of the system state space with random disturbance signals, the hybrid sensitivity H∞ controller is designed in the sensitivity index design, and the proposed algorithm can use weighting function to minimize the performance of the current tracking error as well as the robustness of the yaw rate error in response to robustness. Simulation analysis and experimental verification of EPS system are also carried out. The results show that the control method of the hybrid sensitivity H∞ can better achieve EPS target current tracking, effectively suppress the effect of external interference and noise, improve the system performance and robustness, ensure the driver get good road sense, and improve the system of steering stability.
1. Introduction
Electric power steering (EPS) system is the core technology of the automobile steering system. It is the key component of intelligent driving. It has the advantages of portability, agility, energy saving, environmental protection, and convenient installation. The EPS system is based on the mechanical steering system, and the auxiliary motor is added to control the motor to provide the auxiliary torque to achieve the purpose of power steering. Great progress has been made in EPS system and it has been widely used in automobile industry. The EPS control based on the classical control theory can realize the power steering function, but the EPS feel is not as well as hydraulic power system (HPS) [1]. It cannot restrain the disturbance caused by nonlinear factors due to the uncertainties of the steering system.
Classical control theory, modern control theory, and advanced control theory have been applied to the study of EPS system one after another. Classical control theory was first applied to EPS. Tan Guangxing et al. [2] proposed a kind of EPS control based on immune fuzzy PID. This kind of control based on fuzzy PID can greatly improve EPS power steering performance and portability. Then, PID control of EPS is further developed by using modern control theory. Zhang jianwei et al. [3] proposed a PID control strategy based on genetic algorithm, which is used to optimize the PID controller parameters by using genetic algorithm so as to achieve better control effect.
The above research does not fundamentally solve the current tracking problem, especially anti-interference and robustness problem. Then, the robustness of EPS system has become the focus of research. In order to improve the anti-interference ability of EPS system, advanced control theory is applied to EPS research. Various anti-interference algorithms have been studied experimentally. Qiu Ming et al. [4] studied the EPS based on the control principle and discussed the uncertainty of the system model and the method of minimizing interference to achieve robust control of the EPS. Wang Qirui et al. [5] studied the standard control problem of EPS and designed the robust controller for the road sense. Zhao Wanzhong et al. [6] studied EPS based on hybrid control using to minimize the impact of interference on the output and simultaneously carry out the optimization. Weng Jingliang et al. [7] studied the robust control strategy of EPS based on handling stability. The stability and tracking problem of the system were transformed into a hybrid sensitivity problem. She Guoqin et al. [8] studied the robust control of EPS and its influence on handling stability by using hybrid sensitivity method. Chen long et al. [9] designed a new μ controller by selecting the related performance indexes based on μ synthesis theory and robust control method. The stability and safety of vehicle are improved effectively. Frédéric Wilhelm et al. [10] considered the effects of friction in EPS and proposed an active compensation control strategy which could estimate the internal friction of the system and compensate for it through the motor input. It is composed of two feedback loops: internal loop for system friction estimation and external loop for minimizing the tracking error. Ying-Chih Hung et al. [11] proposed a wavelet fuzzy neural network using asymmetric membership function (WFNN-AMF) with improved differential evolution (IDE) algorithm to control the EPS. It improved the stability and comfort of the vehicle to great extent. But the performance of EPS is given less attention. Wonhee Kim et al. [12] proposed a lane-keeping system of automated vehicle based on EPS which takes unknown parameters and external disturbance, along with their derivatives into the designment of the augmented observer and nonlinear damping controller. Alaa et al. [13] studied a kind of new EPS control strategy. They, respectively, test the reference target and determine the reference model and finally realize the EPS control by using the two-level synovial control strategy to track the motor corner. Dongwook Lee et al. [14] and Cassio T. Faria et al. [15] all identified the external disturbance, nonlinearity, and uncertainty of EPS system, and the controller design is carried out based on them. Some results have been achieved.
The studies mentioned above about the EPS control achieved considerable effect, but there should be more in-depth study on current tracking performance which is the main indicators of the controller. These studies are based on the simulation level, so they can not verify the real effect of the algorithm due to the inevitable gap between the designed algorithm model and the actual steering system. Therefore, a controller based on hybrid sensitivity is designed and the controller is used in the actual steering system.
This paper is arranged as follows. The second section presents the two-degree-of-freedom vehicle model, EPS steering system model, and others. The third section describes the design of the new hybrid sensitivity H∞ controller. In the fourth section, the simulation results of the PID and the new hybrid sensitivity H∞ control strategy are shown. Then, the fifth section is the comparison of experimental results between two control strategies applied on the real experimental bench. Finally, the conclusions are shown in the sixth section.
2. Mathematical Model of EPS System
2.1. Two-Degree-of-Freedom Vehicle Model
In this paper, the steering performance of automobile is studied, so a simplified model of vehicle with 2 degrees of freedom is presented [16] in Figure 1.
The model’s motion differential equation is
and are, respectively, the stiffness of front and rear wheels. β means sideslip angle of vehicle center of mass. u is vehicle centroid velocity. a and are, respectively, the distance between the car and the center of mass. means yaw speed for car. δ means front wheel corner. m means mass of vehicle. is the moment of inertia of the car around the -axis.
2.2. EPS Steering System Model
For the convenience of study, this paper simplifies the front wheel and steering gear to steering shaft [17]. We focus on the situation that the relevent forces are equivalent to the steering axis. On the basis of the reliability of the model, the relation between the input and output variables is established [18]. EPS simplified model is shown in Figure 2.
In Figure 2, means torque input for steering wheel. means measuring torque of torque sensor. means electromagnetic torque of motor. is the steering resistance moment of the road equivalent to the torque on the pinion (gear ratio is N). is the motor torque acting on the steering shaft torque. is the drive ratio which is the steering shaft to the motor. is steering angle. is the angle of motor. is steering shaft angle. is the moment of inertia of steering wheel. is the moment of inertia converted to steering shaft. is the equivalent damping coefficient of system friction.
2.2.1. The Steering Shaft Model
The dynamic analysis of the wheel part above the torque sensor can be
Dynamic analysis of the steering shaft below the torque sensor is available by
is torsion bar stiffness of sensor.
2.2.2. Electrical Machinery Model
The system uses a brushed DC motor. The differential equation can be obtained by Holzer’s law of voltage:
is the motor terminal voltage. is armature resistance. is torque coefficient of motor. is coefficient of back electromotive force of motor. is inductance coefficient of motor. is motor current.
2.2.3. Steering Resistance Moment Calculation
At the small angle, the tire deformation is approximately linear. The steering resistance moment of the pinion acting on the road through the tire [19] is
d is pneumatic trail.
2.2.4. The State Equation of the System
The state equation of the system can be obtained by formulas (1) (8):
Take state variables as . The control input is . The road signal input is . is coefficient of interference intensity. δ(t) is pavement interference noise.
Among them, the coefficient matrices A, B are
Take the output variable of the system as .
The output equation of the system is
where
3. Design of Controller Based on EPS
3.1. Hybrid Sensitivity Design Problem
The hybrid sensitivity design structure of EPS system is shown in Figure 3. P(s) is the nominal space model of the system: the state space model of EPS system model. is the reference input system, including target current, ideal torque, sensor measurement, and ideal yaw rate. is system control input: the input voltage of the motor. e(t) is the error of the signal, including the deviation of the target current from the actual current, the sensor measurement deviation, and yaw rate deviation. u(t) is the control input, and is the system output, including the actual current, the sensor torque, and the yaw rate. z is the system output evaluation, mainly referring to the impact of interference on control error (the current tracking performance of sensor, interference suppression, and handling stability), energy control input, and robust stability. is sensitivity weighting factor. is input sensitivity weighting factor, is sensitivity weighted factor, and K(s) is the controller of the system [20].
The sensitivity control function S, the input sensitivity function , and the complement sensitivity function are included in the design of the hybrid sensitivity H∞ control system [21]. Its Laplasse transform [22], respectively, is
is the unit matrix.
is the infinite norm of the sensitivity function and it represents the tracking performance of the closed-loop control system to the target. In order to improve the tracking performance of the system, should be as small as possible. is the infinite norm of the complement sensitivity function and it represents the measure of the perturbation allowed by the closed-loop system. In order to ensure the robustness of the system, should be as small as possible. However, there is a limitation of at the same system frequency, so and cannot reach the minimum at the same time [21]. By introducing the sensitivity weighting matrix, , and the complement sensitivity weighting matrix, , the relative performance and stability of the system can be achieved. In addition, in order to avoid the actuator (motor) saturation or overload, the input sensitivity function weighting matrix, , is introduced to limit the overoutput [23].
The hybrid sensitivity design problem is transformed into a controller K(s) about system stability, disturbance rejection, and current tracking problems. The closed-loop control system from ω to z, including the generalized controlled object G(s), is stable and minimal. It can be described as
The hybrid sensitivity control problem can be reduced to a H∞ control problem. For generalized space G,
The solution of the control problem is to find controller K, stabilize , and make .
3.2. The Design of Evaluation Index
3.2.1. Tracking Performance
Due to the influence of sensor noise and other system uncertainties, there is a certain deviation between the actual auxiliary current and the target current value . It can be shown as
In order to reduce the effect of error and improve the current tracking performance, this paper makes a weighted control of the current deviation between the actual current and the target current obtained through the help characteristic curve to obtain better tracking performance.
It can be expressed as
is the current tracking weighting function.
3.2.2. Road Feeling
Road feeling is the reaction of pavement information on the steering wheel. The pavement is measured by the reaction of torque sensor by fixing the steering wheel. We use to represent the measurement of torque sensor. The road surface information is reflected by the steering resistance moment . To get good road sense, we need to control the error between the two.
It can be expressed as
φ is the road sense coefficient and is the road sense weighting function [24].
3.2.3. Handling Stability
The handling stability is mainly reflected in the changes of yaw rate. The target value of yaw rate is determined by the steady-state yaw rate gain function [20].
where
In order to obtain better handling and stability, the deviation between the actual yaw rate and the target yaw rate is controlled as
is the stability weighting function.
3.2.4. Avoiding Current Overload
In order to avoid the excessive input current of the motor, the function of input is introduced. It makes as small as possible to reduce energy consumption and device loss.
3.2.5. The Robust Stability of the System
In order to avoid the system instability due to the performance of the system, the output of the system is weighted by .
is the complementary sensitivity weighting factor.
3.3. Selection of Weighting Function
The selection of weighting function reflects the requirement of the performance index of the control system and the choice should be based on the specific control target and the control requirement.
In general, the order of a weighting function is not too high. The controller is simple in order to solve it. So we choose the first-order weighting function as follows:
The choice of weighting function needs iterative computation. In this paper, , , and are obtained by simulation as
According to the controlled object and the generalized weighting function selected, the controller K(s) is solved by using the MATLAB robust toolbox [25].
4. Simulation Results
In order to verify the control strategy to improve the EPS road sense and effect on vehicle handling performance, this paper builds a EPS simulation framework according to the control model of the system and the hybrid sensitivity H∞ control model. It is shown in Figure 4.
In the figure, and are, respectively, sensor noise and road interference. is ideal yaw rate. is steady-state yaw angle velocity function. is stability factor. The ideal yaw rate can be calculated in this way: . is ideal torque sensor measurement. φ is coefficient of road inductance. The ideal torque sensor has a φ times relationship with the measured resistance torque [24]. It is shown as . is ideal current. The steering wheel torque is manipulated by hand, and the ideal target current is calculated by the boost characteristic curve.
The selection of simulation experiment parameters is shown in Table 1.
4.1. Current Tracking Performance
The tracking speed of the actual current of the controlled motor to the target current and the tracking performance determine the response performance of EPS system. The tracking condition of the target current is observed by a step signal of the steering system. Figure 5 is the tracking response of the actual current of the controlled motor to the target current under the PID control strategy and the hybrid sensitivity H∞ control strategy, respectively.
It can be seen from Figure 5, due to the influence of system stiffness and damping, the step signal will generate about 0.5s oscillation of the target current of the system. When the motor current of PID control is tracking the target, it will generate a certain amount of oscillation in 0-0.5 s and the response time is relatively long. There is a certain overshoot in the time of 0.8-2 s. In addition, the hybrid sensitivity control can eliminate jitter; meanwhile, it can respond more quickly and it has a good effect on tracking current.
4.2. Road Feeling
Road sense is the steering wheel torque that the driver feels during the manipulation of the vehicle. It is represented by the measurement value of the torque sensor to a certain extent. This paper mainly tests the effect of the controller on filtering interference by inputting the step response of the road and adding the noise of the road and the sensor. Set the road step as to simulate. Simulation results of step response under PID control and hybrid sensitivity H∞ control are obtained. They are shown in Figure 6.
It can be seen from Figure 6 that the measurement value of the torque sensor of PID control can track the target current very well, but it cannot eliminate the influence of jamming noise which makes the driver's road feeling worse. Compared with PID control, torque sensor measurement value controlled by hybrid sensitivity H∞ control can effectively eliminate the influence of jamming noise, lower overshoot, and shorten the stability time, so that the driver can have a better sense of road in all kinds of interferences and noises.
4.3. Handling Stability
Yaw velocity of the vehicle is the main performance index of the vehicle steering stability. The stability of the vehicle under the control strategy can be verified by the change of the angular velocity of the pavement when road step is added. As shown in Figure 7, when the speed is set to 30 m/s, the yaw velocity under the hybrid sensitivity control and the PID control is changed as shown.
Figure 7 shows that the yaw velocity of the PID control is about 7% overshoot in the 0.5-1 s period and a longer stability time is needed, which makes the vehicle's handling stability worse. The design of hybrid sensitivity H∞ control of yaw velocity response can be very stable and response very quickly. This ensure the vehicle yaw velocity response to the driver's control command be fast and accurate. At the same time, it can improve the handling and stability of vehicle.
5. A Hybrid Sensitivity Robust Controller Is Added to the EPS System Experiment
In the simulation test section, hybrid sensitivity H∞ controller is verified to a certain extent. In this section, the designed hybrid sensitivity H∞ controller is embedded into the built EPS test bench [26, 27], and the designed controller mainly investigates the tracking situation of the actual current to the target current to verify the performance of the controller [28]. EPS test bench is shown in Figure 8. In situ simulation conditions, the torque is applied to the steering wheel. The tracking data of the actual current to the target current is recorded in real time by DSpace software control desk [29]. The power performance of EPS system under hybrid sensitivity controller and PID controller is analyzed and compared. The test results are shown in Figures 9 and 10.
When turning in the same place, it can be seen from Figure 9 that the curve tracking of actual current to target current is almost coincident with the hybrid sensitivity H∞ controller. It has achieved good tracking performance. As can be seen from Figure 10, the tracking curve of actual current to target current fluctuates under PID control, especially in 13 and 20 seconds, and is not ideal. The experimental results show that the proposed hybrid sensitivity H∞ control strategy is better than the PID power assisted control strategy. It can ensure both robustness and good performance. Compared with the EPS system under the PID control, the EPS system under the hybrid sensitivity H∞ control can have faster response speed, better robustness, and better performance.
6. Conclusion
(A)This paper establishes the mathematics model of EPS system and it takes better tracking performance of current, better driver's sense of road, and better handling stability as the control targets. The state space equation of the system is constructed by using modern cybernetics, and a hybrid sensitivity H∞ controller is designed.(B)To verify the design of hybrid sensitivity H∞ controller, the simulation model is built. The simulation results show that the designed controller has good current tracking, which can guarantee that the driver has a good sense of road and can make the vehicle have good maneuvering stability. It can effectively suppress the noise and can improve the system performance while ensuring the robustness and robustness stability of the system compared with the PID controller.(C)To verify the control effect of the controller, it is embedded into experimental bench experiments. The experimental results show that the design controller can realize the current tracking in the actual system and improve the performance, robustness, and robustness stability of the system compared with PID control.(D)In this paper, the main factors affecting EPS system are mainly considered. The change of the system stiffness, the perturbation of the motion, and the parameters of the system are not considered. The next step will be to take more overall consideration of more indicators to establish a more eligible control system.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors acknowledge the support from Natural Science Foundation of Shandong Province (ZR2016EEQ06).