Yaw Stability Control for in-Wheel-Motored Electric Vehicle with a ...

Yaw Stability Control for in-Wheel-Motored Electric Vehicle with a Fuzzy PID Method Hao Zhou2 , Hong Chen1,2 , Bingtao Ren2 , Haiyan Zhao1,2 1. The National Automobile Dynamic Simulation Laboratory, Changchun, 130025, P. R. China E-mail: [email protected] 2. Jilin University Department of Control Science and Engineering,Changchun, 130025, P. R. China E-mail: [email protected]

Abstract: In order to improve in-wheel-motored electric vehicle(EV) yaw stability, the yaw stability control system based on fuzzy PID controller is carried out in this paper. Firstly, we set up the original models for controller: a 2DOF EV dynamics model and wheels with in-wheel motors model. Secondly, a control system which includes three parts: reference model, upper yaw stability fuzzy PID controller and lower moment distribution controller is designed. Besides, under the help of lookup table, the problem that fuzzy toolbox can’t be downloaded into dSPACE is solved. Finally, we run the off-line and real-time simulation on a 8 degrees of freedom(DOF) in-wheel-motored EV model with unitire tire model to verify the effectiveness of the fuzzy PID algorithm. Key Words: EV, Yaw Stability, Fuzzy PID Algorithm, Off-line and Real-time Simulation

1 INTRODUCTION Four wheel drive(4WD) in-wheel-motored EVs eliminate the traditional system of conventional vehicle, and energy’s utilization efficiency is up to 80 percent, and the motor torque generated is fast and accurate [1]. Audi company attempted to find the reasons for the loss of vehicle’s control through a series of investigation [2], and they discovered that the yaw stability problem was the root cause of most accidents. Therefore, EV with its yaw stability problem is a current hot topic. Reference [3] designed a hierarchical yaw stability controller. The vehicle model was divided into body dynamics and wheel dynamics, and the virtual control yaw moment was introduced to integrate high-level and low-level controllers. A control system based on the theory of optimal control was proposed in [4]. This system improved vehicle handling performance and stability by the yaw moment produced by drive force or brake power. Direct yaw moment control, regarded as one of promising chassis technologies, has been studied by controlling brakes independently and adjusting yaw moment on vehicles directly in [5]. Reference [6] investigated the application of generalized predictive algorithm on the yaw stabilThe study is supported by the 973 Program (No. 2012CB821202), the National High Technology Research and Development Program of China (863 Program) (2012AA110701).

c 978-1-4799-7016-2/15/$31.00 2015 IEEE

ity control with the control of yaw moment by braking. Drakunov et al [7] investigated the application of sliding mode control on the yaw stability control for an automobile. The control law was based on optimum search for minimum yaw rate via sliding mode control. Paper [8] proposed a integrated controller of the yaw and rollover stability controls based on the prediction model. Robust yaw stability control based on active steering control for road vehicles was designed, and its performance was verified through a Hardware-in-the-Loop simulation [9]. In this paper, we use fuzzy PID method. Fuzzy algorithm can improve control system accuracy, and reduce the overshoot, and improve certain anti-interference ability. In PID algorithm, three parameters are adjusted to get a satisfied control effect. Fuzzy PID algorithm can lead to adjustment of the three parameters of PID algorithm as the changement of system state so as to get a better environmental adaptability. We solve the problem that fuzzy toolbox can’t be downloaded in dSPACE in real-time simulation.

2 ORIGINAL MODEL FOR CONTROLLER The 2DOF vehicle model which includes the freedoms of yaw rate and sideslip angle has a perfect representation of the yaw rate, so we design a linear 2DOF vehicle model as the desired model. To design the yaw stability controller, we build a 8DOF EV dynamics model. Based

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on the thought of modular model, the 8DOF EV model is divided into three parts: vehicle, tires, wheels with inwheel motors. 2.1

2DOF EV Dynamics Model

The 2DOF linear vehicle model doesn’t consider the nonlinearity and cornering characteristics of the tire [10]. It only considers the lateral movement and the yaw movement around the axis. 2DOF vehicle dynamics model [11] is shown in Fig. 1. Linear differential equa-

3 CONTROL SYSTEM DESIGN In order to improve EV yaw stability, we design a control system which includes a upper yaw stability controller and a lower moment distribution controller. The block diagram of the control system is given in Fig. 2.

Figure 2: Block Diagram of the Control System

Figure 1: 2DOF Vehicle Dynamics Model tions obtained from 2DOF vehicle model based on Newton’s laws are: 𝑏𝐾𝑓 − 𝑎𝐾𝑟 𝑚𝑉𝑥 (𝛽˙ 𝑑 + 𝛾𝑑 ) = −(𝐾𝑓 + 𝐾𝑟 )𝛽 + 𝛾 + 𝐾𝑓 𝛿 𝑉𝑥 𝑏𝐾𝑟 − 𝑎𝐾𝑓 𝐼𝑧 𝛾˙ 𝑑 = (𝑏2 𝐾𝑟 − 𝑎2 𝐾𝑓 )𝛽𝑑 − 𝛾𝑑 + 𝑎𝐾𝑓 𝛿 𝑉𝑥 (1) where 𝐾𝑓 and 𝐾𝑟 denote the equivalent cornering stiffness of the front and rear axle. 𝑉𝑥 denotes the vehicle longitudinal speed. 𝑎 denotes the distance between centroid and the front axle. 𝑏 denotes the distance between centroid and the rear axle. 𝐼𝑧 denotes the moment of inertia about 𝑧 axis of the vehicle. 𝛿 denotes the steering wheel angle. 2.2 Wheels With in-Wheel Motor Model Relationship between motor torque and the input voltage can be expressed as a first-order lag system, that is: 𝑇𝑚 (𝑠) =

𝐾𝑡 /𝑅𝑚 𝑢𝑚 1 + (𝐿𝑚 /𝑅𝑚 )𝑠

(2)

where 𝑇𝑚 denotes the motor’s torque. 𝐾𝑡 denotes a motor’s constant. 𝑅𝑚 denotes the motor’s resistance. 𝐿𝑚 denotes the motor’s inductance. 𝑢𝑚 denotes the control voltage. The relationship between motor’s torque and longitudinal force is: 𝐹𝑥 =

𝑇𝑚 (𝑠) 𝑅𝑒

(3)

where 𝑅𝑒 denotes the rolling radius. In this way, the relationship between vehicle’s yaw control input and motor’s motion response could be established.

It is assumed that the yaw rate 𝛾 is measured and the sideslip angle 𝛽 has been well estimated. Depending on 𝛿 which can be gained by the driver action and 𝑉𝑥 , the expected vehicle’s sideslip angle 𝛽𝑑 and expected yaw rate 𝛾𝑑 can be calculated through the 2DOF model. The sideslip angle’s deviation (𝛽𝑑 − 𝛽) and yaw rate’s deviation (𝛾𝑑 − 𝛾) are the input variables for the fuzzy PID controller. The control input is the yaw moment 𝑀𝑧 which controls the vehicle’s stability. In order to ensure the yaw stability, it is converted to each wheel’s driving or braking torque using in-wheel motors. 3.1 Reference Model By formula (1) and Laplace transformation, according to article [12], 𝛽𝑑 and 𝛾𝑑 can be calculated as, 𝛾𝑑 =

𝑉𝑥 /(𝑎 + 𝑚𝑎𝑏𝑉𝑥2 /(2𝐶𝑓 𝑎(𝑎 + 𝑏))) 𝛿 1 + 𝐼𝑧 𝑉𝑥 /(2𝐶𝑓 𝑎(𝑎 + 𝑏) + 𝑚𝑏𝑉𝑥2 )𝑠

(4)

where 𝑚 denotes the vehicle mass, and 𝐶𝑓 denotes the front cornering stiffness. Subject to the restrictions attached to the condition of the tires, 𝛾𝑑 should meet the following condition: ∣ 𝛾𝑑 ∣≤

𝜇𝑔 𝑉𝑥

(5)

where 𝜇 denotes the road surface friction coefficient. So 𝛾𝑑 must be the minimum value of formula (4) and formula (5). The sideslip angle is the vehicle’s stability index, which affects the driver’s driving comfort. The theoretical 𝛽𝑑 is set to 0. 𝛽𝑑 = 0 (6) Therefore, the considered control problem here is converted into a tracking problem by the approach of setting the reference model.

2015 27th Chinese Control and Decision Conference (CCDC)

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3.2

Design of Upper Yaw Stability Controller

PID control method occupies 90 percent of the industrial control field. The control effect can be satisfied only by adjusting the proportional coefficient 𝐾𝑝 , integral coefficient 𝐾𝑖 and differential coefficient 𝐾𝑑 . Deviation equation is: 𝑒(𝑡) = 𝑟(𝑡) − 𝑦(𝑡)

(7)

where 𝑟(𝑡) denotes the target value and 𝑦(𝑡) denotes the actual output value. The control input 𝑢(𝑡) equation in the time domain is: ∫ 𝑑𝑒(𝑡) (8) 𝑢(𝑡) = 𝐾𝑝 𝑒(𝑡) + 𝐾𝑖 𝑒(𝑡) + 𝐾𝑑 𝑑𝑡 Fuzzy algorithm can improve the accuracy of control system, and reduce the overshoot, and improve certain anti-interference ability. Due to the effect of quantization factor, the input variables vary within the fuzzy domain range. Then by fuzzy processing, according to the preset fuzzy inference rules, defuzzification dealing, and the effect of scale factor, the fuzzy control results can be obtained. The upper yaw stability controller is a combination of fuzzy controller and PID controller. The former is used to adjust the parameters of the latter. The error 𝑒 and the error rate of changement 𝑒𝑐 are the input variables of the conventional two-dimensional fuzzy controller. The input variable 𝑒 can improve the control system’s accuracy. The input variable 𝑒𝑐 can reduce the overshoot, and increase system’s stability, and improve certain antiinterference ability. Corresponding schematic is shown in Fig. 3.

Figure 3: The Schematic of Fuzzy PID Controller ”Mamdani” fuzzy inference rules are adopted here. We choose the Gaussmf form as membership functions of 𝑒 and 𝑒𝑐 which are the input variables, and choose the trimf form as membership functions of the 𝐾𝑝 , 𝐾𝑖 and 𝐾𝑑 which are the output variables. The fuzzy subset distribution of 𝑒 is shown in Fig. 4, and the fuzzy subset distribution of 𝑒𝑐 is the same as it. The fuzzy subset distribution of the output 𝐾𝑝 is shown in Fig. 5, and the fuzzy subset distributions of 𝐾𝑖 and 𝐾𝑑 are the same as it. 1878

Figure 4: The Fuzzy Subset Distribution of the Input 𝑒

Figure 5: The Fuzzy Subset Distribution of Output 𝐾𝑝

3.3 Design of Lower Moment Distribution Controller To ensure the vehicle’s yaw stability, the traditional internal combustion engine vehicles(ICVs) primarily use the braking force at the expense of reducing the velocity. But each 4WD EV’s wheel can be controlled independently by the driving torque or braking torque to change the yaw moment of the vehicle so as to improve handling and stability. Article [13] proposes a appropriate allocation strategy for four wheels as follows: 𝑀𝑧 𝑅 𝑒 𝑇𝑡 − 2(1 + 𝑎/𝑏) (𝑑1 + 𝑑2 )(1 + 𝑎/𝑏) 𝑀𝑧 𝑅 𝑒 𝑇𝑡 𝑇 𝑓 𝑟 = 𝑅 𝑒 𝐹𝑓 𝑟 = + 2(1 + 𝑎/𝑏) (𝑑1 + 𝑑2 )(1 + 𝑎/𝑏) 𝑇𝑡 𝑀𝑧 𝑅 𝑒 − 𝑇𝑟𝑙 = 𝑅𝑒 𝐹𝑟𝑙 = 2(1 + 𝑏/𝑎) (𝑑1 + 𝑑2 )(1 + 𝑏/𝑎) 𝑇𝑡 𝑀𝑧 𝑅 𝑒 + 𝑇𝑟𝑟 = 𝑅𝑒 𝐹𝑟𝑟 = 2(1 + 𝑏/𝑎) (𝑑1 + 𝑑2 )(1 + 𝑏/𝑎) (9) where 𝑇𝑡 denotes the total driving torque. 𝐹𝑥 (𝑥 = 𝑓 𝑙, 𝑓 𝑟, 𝑟𝑙, 𝑟𝑟) denotes the driving force. 𝑑1 denotes the front-wheel tread, and 𝑑2 denotes the rear-wheel tread. 𝑇 𝑓 𝑙 = 𝑅 𝑒 𝐹𝑓 𝑙 =


Table 1: The Main Parameters of Model

3.4 Real-Time Implement Method Because the fuzzy toolbox can’t be downloaded into dSPACE, we use lookup table module to replace it. Application of test function of simulink system, test interface is shown in Fig. 6.

Parameter 𝑚 𝐼𝑧 𝑎 𝑏 𝐶𝑓 𝐶𝑟 𝑅𝑚 𝐿𝑚 𝑇𝑚𝑎𝑥 𝑑1 𝑑2 𝑅𝑒

Unit 𝑘𝑔 𝑘𝑔 ⋅ 𝑚2 𝑚 𝑚 𝑁/𝑟𝑎𝑑 𝑁/𝑟𝑎𝑑 Ω 𝑚𝐻 𝑁 ⋅𝑚 𝑚 𝑚 𝑚

Value 1359.8 1992.54 1.0628 1.4852 23540 23101 0.0005 0.05 500 1.414 1.422 0.29

Fig. 8(a). The magnitude of the tires’ angle 𝛿𝑚𝑎𝑥 is Figure 6: System Test Interface

(a) front wheel steering angle 0.2

front wheel steering angle

Results are saved and are converted into lookup table so as to make three input-output maps. The transformation process of Δ𝐾𝑝 is shown in Fig. 7. Δ𝐾𝑖 ’s map and Δ𝐾𝑑 ’s map are created by the same means. After veri-

angle(rad)

0.1 0 -0.1 -0.2

0

1

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5 time(s)

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the value of sideslip(rad)

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reference with control without control

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0.05 0 -0.05 -0.1 -0.15

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Figure 7: The Transformation Process of Δ𝐾𝑝 fication, when the input values of the fuzzy toolbox and the lookup table are the same, the output values of them are the same too.

the value of yaw rate(rad/s)

(c) yaw rate 0.5

0 reference with control without control -0.5

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Figure 8: The Wheel Input and the Results

4 SIMULATION AND VERIFICATION To compose a 8DOF EV dynamics model, we set up 4DOF including body freedom of vertical, horizontal, yaw and roll, as well as four wheels’ steering DOF. In order to verify the effectiveness of the EV yaw stability with fuzzy PID control system, this paper builds off-line simulation platform based on Matlab/simulink and real-time simulation platform based on dSPACE and xPC Target. The main parameters of model are shown in Table 1. 4.1

Off-Line Simulation

Do ”sine lane-change” experiment with 4WD EV’s yaw stability control system and its conditions are shown in

120∘ ; road surface friction coefficient 𝜇 is 0.8; the initial speed 𝑉0 is 80km/h; the driving torque 𝑇𝑡 is 100N⋅m; the braking torque 𝑇𝑏 is 0. The sideslip angel’s actual value 𝛽 and the expected value 𝛽𝑑 are shown in Fig. 8(b). The yaw rate’s actual value 𝛾 and the expected value 𝛾𝑑 are shown in Fig. 8(c). It can be seen that the maximum sideslip angle is 0.024 rad, which is less than the margin of safety 0.6 rad. The maximum tracking error of yaw rate is 0.0024 rad/s with controller. The results satisfy the control requirements and constitute a clear contrast with the circumstance without controller. The changements of the three parameters of PID are shown in Fig. 9. Throughout, as


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6

1.01

Value

1.005 1

x 10 2.45

60

2.4

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2.35

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2.3

45

0.995

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2.15 10

30

0.985

0

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Figure 9: Changements in the Three Parameters of PID

angle is 0.0228 rad, which is less than the margin of safety 0.6 rad. The maximum tracking error of yaw rate is 0.0029 rad/s with controller. The results satisfy the control requirements and constitute a clear contrast with the circumstance without controller. The changements of the three parameters of PID are shown in Fig. 12. Throughout, as time changes, in order to fast-track the 1.03

5

x 10 2.6 Kp Ki Kd

1.02

6

70

2.5

1.01

2.4

1

2.3

0.99

2.2

0.98

2.1

60

50

Value

time changes, in order to fast-track the changing expectations, three parameters of PID are in a dynamic process of changement. The value of 𝐾𝑖 has been in greater magnitude to make the static error a minimum value. Four wheels’ torques are shown in Fig. 10. Through analysis, trends are in line with the actual situation.

x 10

Value

Kp Ki Kd

Value

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Value

x 10

Value

1.015

40

0.97

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2 10

20

Figure 12: Changements in the Three Parameters of PID

500 400

expectations, three parameters of PID are also in a dynamic process of changement. Four wheels’ torques are shown in Fig. 13. Through analysis, trends are in line with the actual situation.

300

Torque(N m)

200 Tfl Tfr Trl Trr

100 0 -100 -200 -300

500

-400 -500

Tfl Tfr Trl Trr

400

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10 300

Torque(N m)

200

Figure 10: Four Wheel Torques

100 0 -100 -200

Do ”double lane-change” experiment and its conditions are shown in Fig. 11(a). 𝛿𝑚𝑎𝑥 is 120∘ ; 𝜇 is 0.8; 𝑉0 is

-300 -400

0

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5 time(s)

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Figure 13: Four Wheel Torques

(a) front wheel steering angle 0.2 front wheel steering angle

angle(rad)

0.1 0 -0.1 -0.2

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0.8

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9 reference with control without control

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Figure 11: The Wheel Input and the Results 80km/h; 𝑇𝑡 is 100N⋅m; 𝑇𝑏 is 0. 𝛽 and 𝛽𝑑 are shown in Fig. 11(b). 𝛾 and 𝛾𝑑 are shown in Fig. 11(c). It can be seen that the maximum sideslip 1880

Through these off-line simulation experiments, we can see, the yaw rate and the sideslip angle are closer to the expected value. Fuzzy PID algorithm leads to a perfect tracking results and makes the vehicle more stable in driving conditions. 4.2 Real-Time Simulation As a real-time simulation system, real-time simulation can consider the outside noise, the changement of the model’s internal structure, and transmission signal’s interference with parameter settings’ interference suffered by the external conditions. In addition, it could establish a more accurate system, and be more adapted to nonlinear systems, and be closer to the actual situation and the actual object. In practice, the real-time platform is showed in Fig. 14. By normalization and operation of limiting the amplitude, the reference model, PID controller with lookup table and moment distribution controller are downloaded into dSPACE, and the 4WD 8DOF EV model which is to be controlled is downloaded into xPC Target. Under the joint action of xPC Target, AD board PCL726, dSPACE


that fuzzy toolbox can’t be downloaded into dSPACE is solved. Finally, the fuzzy PID algorithm has been verified to be effective through the off-line and real-time simulation on a 8DOF in-wheel-motored EV model with unitire tire model and results are satisfactory.

REFERENCES

Figure 14: The Real-Time Platform

and DA board PCL818, we do the real-time simulation experiments. Do ”double lane-change” experiment and its initial conditions, except that 𝑇𝑡 is 200N⋅m, are shown in Fig. 11(a). 𝛽 and 𝛽𝑑 are shown in Fig. 15(a). 𝛾 and (a) sideslip angle


0.05 reference with control

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Figure 15: The Real-Time Simulation Results 𝛾𝑑 are shown in Fig. 15(b). As can be seen from the figure, the results of real-time simulation are similar to off-line simulation results. The maximum sideslip angle is 0.0444 rad, which is less than the margin of safety 0.6 rad. The maximum tracking error of yaw rate is 0.0459 rad/s with controller, which satisfied the control requirements.

5 CONCLUSION Aiming at improving in-wheel-motored EV yaw stability, fuzzy PID controller is used to ensure the yaw stability in this paper. At first, the original models are set up for controller: a 2DOF EV dynamics model and inwheel-motored wheels model. Then, reference model, upper yaw stability fuzzy PID controller and lower moment distribution controller are used to design control system. With the help of lookup table, the problem

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