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


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1 International Journal of Automotive Technology, Vol. 13, No. 3, pp (2012) DOI /s Copyright 2012 KSAE/ pissn / eissn PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM H.Z. LI 1), L. LI 1)*, L. HE 1), M.X. KANG 2), J. SONG 1), L.Y. YU 1) and C. WU 3) 1) State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing , China 2) College of Vehicles and Energy, Yanshan University, Qinhuangdao , China 3) ATOP Intelligence Lab, Clarksville, MD, 21029, USA (Received 15 June 2011; Revised 27 September 2011; Accepted 27 November 2011) ABSTRACT A Traction Control System (TCS) is used to control the driving force of an engine to prevent excessive slip when a vehicle starts suddenly or accelerates. The torque control strategy determines the driving performance of the vehicle under various driveslip conditions. This paper presents a new torque control method for various driveslip conditions involving abrupt changes in the road friction. This method is based on a PID plus fuzzy logic controller for driving torque regulation, which consists of a PID controller and a fuzzy logic controller. The PID controller is the fundamental component that calculates the elementary torque for traction control. In addition, the fuzzy logic controller is the compensating component that compensates for the abrupt change in the road friction. The simulation results and the experimental vehicle tests have validated that the proposed controller is effective and robust. Compared with conventional PID controllers, the driving performance under the proposed controller is greatly improved. KEY WORDS : Traction control system, Slip control, Torque control, Fuzzy control, PID plus fuzzy control 1. INTRODUCTION In a vehicle Traction Control System (TCS), the slip rate of the drive wheels is regulated by the driving torque and the braking pressure. The driving torque regulation is used to control the drive wheel slip, and the braking pressure regulation is used to control the difference in the slip rates between two drive wheels. In the past decades, several controllers for wheel slip control have been proposed, such as a PI controller (Ise et al., 1990), a nonlinear PID controller (Van Zanten et al., 1996), a selftuning PID controller (Li et al., 2009), a sliding mode controller (Jung et al., 2000; Chun and Sunwoo, 2004), a fuzzy logic controller (Khatun et al., 2003; Yu et al., 2002), a selftuning fuzzy logic controller (Lee and Tomizuka, 2003), a model predictive controller (Borrelli et al., 2006), and so on. The plant of the controller is the same throttle as that referenced in Ise s (Ise et al., 1990) or Li s (Li et al., 2009) papers, which is slightly different from the torque control. The conventional PID controller for automated machines is widely accepted in industry as well as in TCSs. However, in several special driving conditions, e.g., from a road with high friction to a road with low friction, the conventional PID controller results in a long settling time. However, to improve the performance of a vehicle, the settling time *Corresponding author. should be short. The selftuning PID and the nonlinear PID can reduce the settling time as the parameters of the plant of the controller changes, and much effort has been devoted to this field (Guo et al., 2007; Sharkawy et al., 2010). The fuzzy PI and PD control algorithms were used in previous studies (Woo et al., 2000; Kukolj et al., 2001), which are frequently called fuzzy PIDs. The control signal is the output of the fuzzy logic module. Because the fuzzy logic control is able to closely emulate the movement of human beings, it has the advantage of dealing with uncertain or noisy signals arising from human motions. Fuzzy logic controllers are widely used in vehicle controller design (Li and Gatland, 1996; Eker and Torun, 2006). Shim and Margolis (2001) focused on the yaw rate control of a vehicle, where the control algorithm includes feedforward control µ plus a conventional PID control, and the simulation results showed that the performance of a vehicle using this algorithm is much better than that using just the conventional PID controller. Gao et al. (2008) focused on the steering control for offroad vehicles using a feedforward plus a conventional PID control method. Compared with the conventional PID controller, this selftuning PID has a better performance because it is able to adapt to different parameters at different conditions of the plant. However, to ensure system stability, the parameters of the selftuning PID are always limited to a certain range; thus, the settling time of the system is still not short enough, especially when the parameters of the plant change 441
2 442 H.Z. LI et al. quickly. In addition, selftuning algorithms are usually complicated. To obtain a shorter settling time and wider operating conditions, we developed a new torque control method based on a PID controller plus a fuzzy logic controller. On a road with a constant friction, only the PID controller is activated. When the road s friction changes significantly, the fuzzy logic control will be activated. This paper assumes that the road friction and the road slope are calculated by other components of the TCS. For the estimation of road frictions, the method proposed by Li et al. (2009) is used. A frontdrive vehicle is considered here, and the pressure control will not be discussed in this paper as the length limit. The paper is organized as follows. In Section 2, the vehicle and tire models are briefly presented. The adopted vehicle model is a sevendegreesoffreedom (7DOF) model. The nonlinear tire model used here is the Magic Formula (MF) tire model. In Section 3, a PID controller and a fuzzy logic controller are proposed. Then, in Section 4, the simulation results and the experimental vehicle tests for different road conditions are shown and discussed. Finally, several conclusions are presented in Section VEHICLE MODEL AND TIRE MODEL As shown in Figure 1, we are considering the situation when a vehicle moves from a road of high friction to a road of lower friction. For this reason, we should carefully consider the load transfer between the front wheels and the rear wheels and the differences in the slip rates between the left wheels and the right wheels Vehicle Model A 7DOF vehicle model is shown in Figure 2. The 7DOF vehicle model includes both the lateral and longitudinal dynamics. The degrees of freedom associated with this model are the longitudinal and lateral velocities (V x, V y ), the yaw angle (φ), and the wheel rotational speeds (ω fl, ω fr, ω rl, ω rr ). The vehicle mass is represented by m, and the yaw moment of inertia is represented by I v. The longitudinal and cornering forces at each wheel are indicated by F xij and F yij, respectively, where the `ij' subscript denotes fl, fr, rl, and rr (f: front; r:rear; l:left; r:right ). Suspension is not included in this model. a is the distance from the front axle to the center of gravity (CG), b is the distance from the rear axle Figure 1. Simulation with vehicle driving on a road with varying frictions; m indicates the friction coefficient. Figure 2. Vehicle model (fl: front left. fr: front right. fl: rear left. rr: rear right). to the center of gravity (CG), L is the wheel base, and w is the track width. The radius of each wheel is represented by R, and the wheel inertia is represented I w. The vehicle model is as follows: mv ( x V y ϕ ) = ( F xfl + F xfr ) cosδ w ( F yfl + F yfr ) sinδ w + F xrl + F xrr mv ( y+ V x ϕ ) = ( F xfl + F xfr ) sinδ w + ( F yfl + F yfr ) cosδ w + F yrl + F yrr I ϕ v = ( F yfl + F yfr )a cosδ w + ( F yfl F yfr ) w  sin 2 δ w ( F yrl + F yrr )b + sin ( F xfl + F xfr )a δ w F xfl F xfr ( ) w  cos 2 δ w ( F xrl F xrr ) w The rotational motion of the wheels is described by the following: I w ω i = T i RF xi where the subscript in ω i, T i, and F xi represents i = fl, fr, rl, and rr. T i is the difference between the driving torque (T d ) and the braking torque (T b ) applied to the wheel. T i = T di T bi This model includes a quasistatic longitudinal load transfer. The lateral load transfer is neglected. The normal load equation for each wheel can be expressed as follows: mg F zfl F zfr b V (6) 2 L  ( x V y ϕ )h = = g gl mg F zri F zrr b (7) 2 L  ( V x V y ϕ )h = = g gl where ϕ is the yaw rate of the vehicle, δ w is the steer angle of the front wheels, F zi (i = fl, fr, rl, rr) is the vertical load, g is the gravity, and h g is the height of CG. The parameters of the model vehicle in the experimental test are listed in Table Tire Model In this paper, the Magic Formula (MF) tire model is adopted to simulate the lateral and longitudinal forces (1) (2) (3) (4) (5)
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 lowpass 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 515%. 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 realtime 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 Hardwareinthe 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 realtime 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 realtime controller. In the vehicle test, a laptop with a USBCAN device was used, and the USBCAN 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 µjump 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. HardwareintheLoop 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
7 PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM 447 PID plus fuzzy logic control. The simulation results showed that when entering the low µ road, with the PID plus fuzzy logic controller, the engine output torque was adjusted to the proper values in a much shorter time; thus, the slip time can be reduced by a large amount. Figure 11. µjump from high µ to low µ, PID controller Experimental vehicle tests In Figures 13 and 14, the vehicle tests were performed on the campus of Tsinghua University. The high µ road was made out of dry concrete, while the low µ road was made out of a plastic sheet with water and detergent on top. The test vehicle was a sedan from Brilliance Auto Company. As shown in Figure 13 (a), the road friction changed from a high µ to a low µ at approximately 2.5 s. In Figure 13(b), after entering the low µ road, the engine output torque T e was larger than the proper value; thus, the wheel slip speeds maintained large values from approximately 2.5 s to 3.5 s. In Figure 13(d), the pressure unit was in intervention several times. In Figure 14(a), the road friction changed from a high µ to a low µ at approximately 2.75 s. As shown in Figure 14(b), the T FLC decreased at a faster rate; thus, the engine torque also decreased at a faster rate. Thus, the wheel slip speed became a relatively smaller value at approximately 3.5 s. In Figure 14(d), the pressure was in intervention only two times. Generally, the shorter the pressure intervention period is, the more stable the vehicle will be. The vehicle test results showed that when the vehicle entered the low µ road from the high µ road, the engine output torque could be adjusted to a proper value in a shorter time in the PID Figure 12. µjump from high µ to low µ, PID plus fuzzy logic controller. T e decreased at a greater rate. Thus, the slip rates of the drive wheel decreased to the proper values in a relatively shorter time. The slip time was approximately 4.5 s with the conventional PID control and 1.5 s with the proposed Figure 13. µjump from high µ to low µ, PID controller.
8 448 H.Z. LI et al. Figure 16. µjump from low µ to high µ, PID plus fuzzy logic controller. Figure 14. µjump 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 µjump 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. µjump from low µ to high µ, PID controller. Figure 17. µjump from low µ to high µ, PID controller.
9 PID PLUS FUZZY LOGIC METHOD FOR TORQUE CONTROL IN TRACTION CONTROL SYSTEM 449 varying friction conditions. 5. CONCLUSION A PID plus fuzzy logic controller is proposed to improve the slip control performance on a friction transition road. Several simulations and vehicle tests were performed to evaluate the effectiveness of the proposed PID plus fuzzy logic controller in various conditions. The simulation results and the experimental vehicle tests both show that the vehicles with the proposed controller have better driving performances in terms of shorter settling times and less state oscillations. ACKNOWLEDGEMENT The authors thank the supports from the State Key Laboratory of Automotive Safety and Energy. The authors greatly appreciate the supports from the National Natural Science Foundation of China (No ), State Key Laboratory of Automobile Safety and Energy of Tsinghua University (No. KF11011) and Disaster Research Foundation of PICC 2011 (No. 2011D05). Figure 18. µjump from low µ to high µ, PID plus fuzzy logic controller. approximately 5.1 m/s at 6 s with the PID controller, while it was approximately 8.3 m/s with the PID plus fuzzy logic controller. Thus, the simulation results show that when the vehicle enters the high µ road from the low µ road, compared with PID controller, the engine output torque can be adjusted to a proper value in a shorter amount of time with a PID plus fuzzy logic controller Experimental vehicle tests In Figures 17 and 18, the road conditions and the vehicles were the same as those in Figures 13 and 14. As shown in Figure 17(a), the road friction changed from a low µ to a high µ at approximately 3.75 s. In Figure 17(b), after the vehicle entered the high µ road, the engine output torque T e increased slowly; thus, the vehicle speed V x increased slowly. In Figure 18, the road friction changed from a low µ to a high µ at approximately 4.3 s. In Figure 18(b), the T FLC increased at a faster rate; thus, the engine torque also increased at a faster rate. Therefore, the speed of the vehicle increased at a faster rate. Thus, the vehicle test results showed that when the vehicle entered the high µ road from the low µ road, the engine output torque could be adjusted to a proper value in a shorter amount of time with the PID plus fuzzy logic controller. Compared with the conventional PID controller, the driving performance of the vehicle with the proposed PID plus fuzzy logic controller is much better on a road with REFERENCES Borrelli, F., Bemporad, A., Fodor, M. and Hrovat, D. (2006). An MPC/hybrid system approach to traction control. IEEE Trans. Control Systems Technology 14, 3, Chun, K. and Sunwoo, M. (2004). Wheel slip control with moving sliding surface for traction control system. Int. J. Automotive Technology 5, 2, Eker, I. and Torun, Y. (2006). Fuzzy logic control to be conventional method. Energy Conversion Manage 47, 4, Gao, Y., Huang, R. and Zhang, Q. (2008). A comparison of three steering controllers for offroad vehicles. Proc. IMechE Part D: J. Automobile Engineering, 222, Guo, L. Z., Zhu, Q. M. and Warwick, K. (2007). Design of a minimum variance multiple inputmultiple output neuro selftuning proportionalintegralderivative controller for nonlinear dynamic systems. Proc. IMech Part I: J. Systems and Control Engineering, 221, Ise, K., Fujita, K., Inoue, Y. and Masutomi, S. (1990). The "Lexus" traction control (TRAC) system. SAE Paper No , Jung, H., Kwak, B. and Park, Y. (2000). Slip controller design for traction control system. Int. J. Automotive Technology 1, 1, Khatun, P., Bingham, C. M., Schofield, N. and Mellor, P. H. (2003). Application of fuzzy control algorithm for electric vehicle antilock braking/traction control systems. IEEE Trans. Vehicular Technology 52, 5, Kukolj, D. D., Kuzmanovic, S. B. and Levi, E. (2001).
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