# PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM

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3 PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM 443 Table 1. Parameters of the test vehicle. parameter value parameter value m 1500 kg w 1.56 m a 1.20 m h g 0.52 m b 1.60 m R 0.31 m L 2.80 m g 9.8 m/s 2 I v 2300 kgm 2 I w 1 kgm 2 generated by the tires. The MF model can be expressed as follows (Pacejka, 2002): F x = D x sin{ C x arctan[ B x λ( 1 E x ) + E x arctan( B x λ) ]} F y = D y sin{ C y arctan[ B y α( 1 E y ) + E y arctan( B y α) ]} where α is the slip angle of tire, λ is the slip rate of the tire, D i is the peak factor, C i is the shape factor, B i is the stiffness factor, and E i is the curvature factor, with i=x, y. These tire parameters are assumed to be constant over the vehicle s experimental period and were measured in advance in our simulation and vehicle test. 3. CONTROLLER DESIGN 3.1. Structure of the PID Plus Fuzzy Logic Controller The structure of the controller is shown in Figure 3, which consists of three parts: the torque base module, the PID controller, and the fuzzy logic controller. The output of the controller can be expressed as follows: T tar = T B + T PID + T FLC (8) (9) (10) where T tar is the target torque, which is the desired engine output torque requested by the drive slip controller in the Figure 3. Block diagram of the proposed controller. intervention process. T B is the torque base, which is the initial value of the PID controller. T PID is the PID torque, which is the torque component derived from the PID controller. T FLC is called the FLC torque, which is the torque component derived from the fuzzy logic controller. T FLC is the compensatory torque when the road friction changes quickly, which is always zero on a road with constant friction. The torque base T B is calculated in the torque base module. The PID controller contributes a torque component, which is the focal torque component computed from the wheel slip. The fuzzy logic controller contributes a compensation torque when the friction of the road changes rapidly. Each of the three modules in Figure 3 will be described in detail in the following sections Torque Base Module The engine output torque is obtained from the TCS, which is a common output in vehicles equipped with a TCS. The basic idea is to regard the engine output torque when the drive slip of the drive wheels exceeds the threshold of the slip rate as the torque base. Considering the translation delay between the engine output torque and the drive slip of the drive wheels, a time delay should be taken into account. In addition, because of the noise in the engine output torque, a low-pass filter should be implemented. The method can be expressed as follows: T ed = T e ( t τ) T = 1 edf T + Ts ed T B = T edf V s = V st (11) (12) (13) where T e is the engine output torque, τ is a constant time delay determined by vehicle tests, T ed is the engine torque after the time delay, T edf is the T ed after filtering, and T is a time constant. T B is the torque base, V s is the total slip speed of all of the drive wheels, and V st is the threshold. The gear ratio as V s exceeds V st should be recorded. When T B is used in Equation (10), the change in the gear ratio should be considered PID Controller Module Slip speed control of the drive wheel In the wheel slip control, the target of the slip rate should be approximately 5-15%. However, when the vehicle speed is low, a small fluctuation in the wheel speed will result in a large fluctuation in the slip rate. Thus, in a real-time controller, the slip speed (difference between the wheel speed and the vehicle speed) is used as a control variable. In addition, we can define the target of the total slip speed as the sum of the target slip speeds of the two drive wheels. If the vehicle speed and the target speed of the drive wheels are very low, the engine may be unstable because of improper control; thus, the target of the total slip speed

4 444 H.-Z. LI et al. torque T ref and the balance torque T Bal. T err can be expressed as: T err = T Ref T Bal (17) where T Ref is the reference torque, which is defined as: ( 1) T Ref = T B + T I + T FLC (18) Figure 4. Target of the total slip speed design. should be a relatively large value when the vehicle speed is small. As the vehicle velocity increases, the sum of the slip speeds should be translated into a constant slip rate, as shown in Figure 4. In Figure 4, V t is the target slip speed of the drive wheels, V V1, V V2, V V3, and V t1, and V t1 is the threshold, which is determined by the vehicle tests PID controller The slip rate control using a conventional PID controller is expressed in the time domain as follows: T PID () t = T P () t + T I () t + T D () t t de() t = K p et () + K i et () dt+ K d dt 0 (14) where T P is the proportional torque, T I is the integral torque, T D is the derivative torque, e(t) is the error between the target total slip speed and the total slip speed, de(t) is the derivative of the error e(t), T PID (t) is the control torque used to control the slip speed of the drive wheels, K p is the proportional gain, K i is the integral gain, and K d is the derivative gain. When T PID is used in Equation (10), the change in the gear ratio should be considered. e(t) is expressed as: et () = V t V s (15) where V s is the sum of all of the slip speeds of the two drive wheels, which can be expressed as: R( ω V fl + ω fr ) 2V x when R( ω fl + ω fr ) > 2V x s = (16) 0 when R( ω fl + ω fr ) 2V x 3.4. Fuzzy Logic Controller Module If the road friction changes rapidly during the drive wheel slip control process, the PID controller cannot adjust the output to a proper value in a short time. Thus, a fuzzy logic controller is introduced to compensate. The inputs of the fuzzy logic module are the total slip speed of the two drive wheels V s and the torque error T err. The torque error T err is described as follows Torque error calculation The torque error T err is the difference between the reference (1) where T FLC is the FLC torque from the previous computation cycle. The balance torque T Bal is the engine output torque, which is calculated based on the vehicle dynamics. The sum of the longitudinal forces between the tires and the road of drive wheels, F x, can be expressed as: F x = F j + F i + F w + F j (19) where F f is the rolling resistance, F i is the resistance due to the gradients, F w is the aerodynamic drag, and F j is the acceleration resistance. The aerodynamic drag F w will be neglected because it is very small. Furthermore, assuming that the angle of the gradient i road is small, F x can be expressed as: F x = Gf+ Gi road + ζma x (20) where G is the gravity of the vehicle, f is the rolling resistance coefficient, a x is the longitudinal acceleration, and ζ is a mass factor. The longitudinal acceleration of the vehicle a x can be calculated from the vehicle velocity V x as: a x = ( V x V xlast ) t (21) where t is the time interval between V x and V xlast, and V xlast is the vehicle velocity from the last computation cycle. The mass factor ζ can be expressed as follows: ζ 1 I w I f i 2 g i 2 0 η = T (22) mr 2 mr 2 where I f is the inertia of the rotating parts in the power train and η T is the transmission efficiency. The vertical load F z can be expressed as: h g F z G b L i L road m h --- ΣI g w I f i g i = L LR LR ax F zw1 G Rf ---- L (23) where F zw1 is the air lifting force on the front axle, which is small and will be neglected in this paper. The maximum tire force F xp can be expressed as: F xp = µ pe F z (24) Generally, the required engine output torque T epr can be expressed as:

5 PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM 445 Figure 5. Block diagram of the fuzzy logic controller. T epr F xp R = i g i 0 η T (25) where η T is the efficiency of the power train. The balance torque T Bal is equal to the engine output torque T epr. T Bal = T epr (26) Fuzzy logic controller The fuzzy logic controller determines the gradient of T FLC based on V s and T err. The basic idea is that if T err is very big, the FLC torque will decrease; if V s is very small, the FLC torque will increase. The block diagram of the fuzzy logic controller is shown in Figure 5. The inputs of the fuzzy logic controller are the total slip speed V s and the torque error T err. The output of the fuzzy logic module ñ is the gradient of T FLC. Then, an integral module is implemented, and the output of the integral module is T FLC. (I) Fuzzification: To provide sufficient rule coverage, five fuzzy sets are used for both the inputs and the outputs of the controller. V s has a set of values (VS: very small, S: small, M: median, B:, VB: very big), defined as follows: {V s } = {VS, S, M, B, VB}. T err and ρ have a set of values (NB: negative big, NM: negative median, ZE: zero, PM: positive median, PB: positive big), defined as follows: {T err, ρ} = {NB, NM, ZE, PM, PB}. (II) The fuzzy decision process: processes a table of rules from the knowledge base using fuzzy inputs from the previous step to produce the fuzzy outputs. Table 3 shows the rules for the proposed fuzzy logic controller. These rules are introduced based on the expert knowledge and the extensive simulations performed in this study. The rules function as follows: If V s is Very Small and T err is Negative Big, this indicates that the torque target is so small at present that ρ may be Positive Big. If V s is Very Big and T err is Positive Big, this indicates that the torque target is so large at present that ρ may be Negative Big. If V s and T err satisfy the other cases, this indicates that the torque target is neither too large nor too small; thus, ρ may be Zero or a transition value to the Negative Big or Positive Big. These criteria are the rule base. Then, by performing different simulations, we can tune the rules by considering the effect of the drive wheel slip rate. The fuzzy controller uses the Mamdani Fuzzy Inference System (FIS), which is Table 3. Rule table of the fuzzy logic controller. Torque error Total slip speeds VS S M B VB NB PB PM PM ZE ZE NM PB PM ZE ZE ZE ZE PM ZE ZE ZE NM PM ZE ZE ZE NM NM PB ZE ZE NM NB NB characterized by the following fuzzy rule: IF V s is A and T err is B THEN ρ is C where A, B, and C are fuzzy sets defined in the input and output domains, respectively. (III) Defuzzification: scales and maps the fuzzy output from the fuzzy decision process to produce an output value. The defuzzification method used in this paper is the center of area method. This method determines the center of the area below the combined membership functions. Figure 6. Membership functions of V s. Figure 7. Membership functions of T err. Figure 8. Membership functions of ρ.

6 446 H.-Z. LI et al. (IV) Integral: the output of the fuzzy logic is a torque gradient; thus, there is an integral module as follows: T FLC = ρdt (27) Figures 6, 7, and 8 show the membership functions and the ranges of values of V s, T err, and ρ, respectively. 4. RESULTS AND DISCUSSION The simulation was carried out on the Hardware-in-the- Loop (HIL) simulation platform, as shown in Figure 9. The HIL platform has six components: the upper computer, the lower computer, the sensors, the actuators, the ESC controller, and the other peripheral components, such as the acceleration pedal and the brake pedal. The upper computer is a PC used for monitoring during the simulation and for analysis after the simulation. The lower computer runs a real-time operation system used for computing the vehicle movement and for providing signals required by the ESC controller. The sensors are pressure sensors. The actuators include a HCU (Hydraulic Control Unit), brake pipes, brake wheel cylinders, and so on. The ESC controller is the same as the one that was installed in the test vehicle. Performance comparisons between the proposed method and the conventional PID control are presented as follows. The parameters of the vehicle are shown in Table 1 in Section 2.1. In all of the tests in this section, the parameters of the PID controller are the same. The P gain K P was adjusted to 3, the I gain K I was 0.02, and the D gain K D was 5. The steering angle was zero. Additionally, in all of the figures in this section, when T tar was 250 Nm, there was no intervention. The vehicle test was performed on a Zunchi vehicle in which the engine can adjust the torque output by a CAN bus, which is a widely used bus in vehicles. To perform the vehicle test, several modifications were made to the vehicle. First, pressure sensors and the HCU were added to the vehicle. The pressure sensors were in the Figure 10. Test system in the experimental vehicle test. braking pipes, and the HCU was in the engine room. Then, an ESC controller was developed and installed in the vehicle. The ESC controller is developed based on a single chip microcomputer, which is a real-time controller. In the vehicle test, a laptop with a USB-CAN device was used, and the USB-CAN device was connected to the vehicle CAN bus to record the data, as shown in Figure 10. In the vehicle test, several signals were recorded, and different types of signals were obtained through different techniques. The torque and the throttle were obtained from the CAN bus. Pressure signals were acquired from the analog to digital channels of the pressure sensors. The wheel velocities were obtained by the capture channels of the ESC controller from the wheel speed sensors. In addition, the velocity of the vehicle was calculated from the wheel speeds by the ESC controller. All of the data to be logged were available on the CAN bus µ-jump from Low µ to High µ Simulation results The simulation settings of the throttle, the gear, and the road friction were equal in Figures 11 and 12. V fl and V fr in Figure 11(c) represent the wheel speeds, which can be expressed as: V fl V fr = Rω fl = Rω fr (28) (29) Figure 9. Hardware-in-the-Loop simulation platform. P fl and P fr in Figure 11(d) represent the wheel cylinder pressures in the front left wheel and the front right wheel, respectively. As shown in Figure 11(b), the engine output torque decreased slowly as it entered the low µ road with the conventional PID controller. The slip rates of the drive wheels remained large for a relatively long time, and the pressure intervention was activated for several instances. With PID plus fuzzy logic controller, as shown in Figure 12(b), the FLC torque T FLC decreased by approximately 120 Nm in approximately 0.6 s; thus, the engine output torque

8 448 H.-Z. LI et al. Figure 16. µ-jump from low µ to high µ, PID plus fuzzy logic controller. Figure 14. µ-jump from high µ to low µ, PID plus fuzzy logic controller. plus fuzzy logic controller. Then, the wheel slip can be adjusted to a smaller value in a shorter amount of time µ-jump from Low µ road to High µ Road Simulation results The simulation settings of the throttle, the gear, and the road friction are equal in Figures 15 and 16. The road friction changes from a low µ to a high µ at 4.06 s. Because the pressures remained zero throughout the simulation, the pressures are not given. As shown in Figure 15(b), the engine torque T e increased slowly when the vehicle entered the high µ road with the PID controller. In Figure 15(c), the speed of the vehicle V x increased slowly. With PID plus fuzzy logic controller, as shown in Figure 16(b), the T FLC increased by approximately 135 Nm within approximately 1.2 s, and the engine torque increased at a much faster rate. Consequently, the speed of the vehicle increases at a faster rate. The vehicle speed was Figure 15. µ-jump from low µ to high µ, PID controller. Figure 17. µ-jump from low µ to high µ, PID controller.

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