Research Article
Application of fuzzy logic in constant speed control of hydraulic retarder
Advances in Mechanical Engineering 2017, Vol. 9(2) 1–11 Ó The Author(s) 2017 DOI: 10.1177/1687814017690956 journals.sagepub.com/home/ade
Yulong Lei1, Pengxiang Song1, Hongpeng Zheng1, Yao Fu1, Xingzhong Li1 and Bin Song2
Abstract Hydraulic retarders are extensively used in commercial vehicles because of their advantages, such as their large braking torque and long continuous operating hours. In this article, the structure and working principles of hydraulic retarders are introduced, and their dynamic characteristics are analyzed. The theoretical model of a hydraulic retarder is then established based on the dynamic analysis of a vehicle driving downhill. The braking process that involves the hydraulic retarder is divided into three stages. Moreover, the filling ratio controller of the hydraulic retarder is designed by adopting fuzzy control theory to control the braking torque of the vehicle while driving downhill. The vehicle dynamic model and constant-speed control model were then established in the MATLAB/Simulink environment. The simulation results showed that the fuzzy logic controller designed in this study has good constant-torque control and anti-inference performances, which can accurately and immediately produce braking torque to satisfy the braking requirement, thereby enabling the vehicle to drive downhill at a constant speed. As a result, the control strategy designed in this article can lead to significant improvements toward a safe road transport. Keywords Hydraulic retarder, filling ratio, constant speed, controller, fuzzy logic
Date received: 13 October 2016; accepted: 5 January 2017 Academic Editor: Rahmi Guclu
Introduction During heavy-duty driving on long and steep downgrade sections of roads, a vehicle may continuously and frequently use its service brake, which may lead to an increase in thermal load and serious wear of the brake; consequently, braking failure may occur because of heat recession of the mechanical brake and may cause security risks and traffic accidents.1–3 Being widely used in commercial vehicles, hydraulic retarders are auxiliary devices that can reduce the vehicle speed by converting the mechanical energy of a driving vehicle into the thermal energy of the working oil.4 Compared with other auxiliary devices, hydraulic retarders have many advantages such as light mass, large braking torque, long working hours, good thermal diffusivity, and zero pollution;5 the continuous braking performance can
effectively solve the security issues caused by the excessive use of the service brake, thereby increasing the driving safety of the vehicle. One of the main functions of the hydraulic retarder is that it enables the vehicle to drive downhill at a constant speed. This function can be achieved by changing the braking torque of the hydraulic retarder. The constant-speed function determines whether the vehicle
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State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, China 2 Hangzhou Advance Gearbox Group Co., Ltd, Hangzhou, China Corresponding author: Yao Fu, State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130000, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 can drive stably and safely on long and steep downgrade sections. Therefore, studying the constant-speed control strategy during downhill driving is particularly important.6 Numerous studies have investigated the control strategy of the hydraulic retarder. Liu et al.7 employed a mathematical model to analyze the effects of rotor speed and filling ratio on the output torque. Li et al.8 proposed a filling ratio control strategy of four shifts, where the filling ratios are 25%, 50%, 75%, and 100%; the simulation was conducted on a long downhill road under various weather conditions. Lu and Cheng9,10 proposed a constant-speed control strategy by changing the control cycle, initial liquid charge, and variation in liquid charge in each control cycle. Song11 designed a proportional–integral–derivative (PID) controller for the hydraulic retarder to control the filling ratio, and a comprehensive mechanical electrohydraulic simulation was conducted. Zhang et al.12 proposed a type of intelligent control method based on fuzzy logic and conducted an experiment on actual vehicles. Yan et al.13 developed a hydraulic retarder model through test data and artificial neural networks; the control strategy was verified through Simulink– AMESim co-simulation. However, in simplifying the braking process of downhill driving, previous studies ignored the influence of the thermal design power of the radiator on the braking performance of the hydraulic retarder; the constant-speed control strategy of the hydraulic retarder cannot accurately calculate the target braking torque required, and the vehicle speed error is relatively large. Considering the thermal design power of the radiator, this study analyzes in detail the braking process of the vehicle using a hydraulic retarder. The vehicle speed error, vehicle acceleration, and target filling ratio of the hydraulic retarder based on fuzzy control theory are analyzed, a filling ratio controller is designed, and the dynamic model of the vehicle driving downhill and the constant-speed control model of the hydraulic retarder are established in the MATLAB/ Simulink environment. The simulation is carried out, and the simulation results as well as the comparison among different control strategies are also analyzed.
Structure and working principles of hydraulic retarder The hydraulic retarder in this study is parallel to the transmission system through an increasing gear;11 the mounting position is shown in Figure 1. The hydraulic retarder is composed of the retarder body, control device, and electric control unit. In the retarder body, the rotor and stator constitute the working chamber. When the retarder is functioning, the working oil flows into the working chamber and the rotor moves with the
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Figure 1. Mounting position of the hydraulic retarder.
Figure 2. Working principle of the hydraulic retarder. 1: oil tank; 2: oil filter; 3: oil pump; 4: two-position two-way pilot control valve; 5: pressure regulating valve; 6: stator; 7: rotor; 8: drive shaft; 9: hydraulic retarder; 10: electromagnetic relief valve; 11: radiator.
shaft to drive the working oil to rotate. Meanwhile, the stator is fixed, which generates the reaction force for the working oil, the reaction force is transmitted to the rotor to hinder its rotation, and then the braking torque is generated. The braking torque decelerates the vehicle, and the rotational mechanical energy is converted into the internal energy of the working oil; the heat generated is removed by the radiator.14 A schematic diagram of the working principle of the hydraulic system of the retarder is shown in Figure 2. The effect of pressure regulating valve 5 is to regulate the pressure of the hydraulic system. When the retarder is activated, two-position two-way pilot control valve 4 electrifies, and then the working oil is pumped into the working chamber of stator 6 by oil pump 3 through oil filter 2. Subsequently, rotor 7 rotates with drive shaft 8 to drive the working oil to rotate. Afterward, the working oil flows between hydraulic retarder 9 and radiator 11, and the braking torque is generated to decelerate the vehicle. The filling ratio of the retarder is regulated by electromagnetic relief valve 10, which produces back pressure to limit the flow rate of the outlet as well as to
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Model of hydraulic retarder The braking torque of the hydraulic retarder is determined by the rotor speed and the filling ratio of the working chamber. The braking torque has a linear relation with the filling ratio and has a nonlinear relation with the rotor speed which is calculated as equation (2)16 Mr = a Mrmax = a lrnr 2 D5 Figure 3. Force analysis of vehicle driving downhill.
where
control the filling ratio of the working chamber of the hydraulic retarder.
nr = nig ir The braking power of the hydraulic retarder is calculated as equation (3)
Dynamic model of vehicle driving downhill Pr =
Dynamic model of vehicle The force analysis of the heavy-duty vehicle using hydraulic retarder to brake when driving downhill is shown in Figure 3. To establish a precise mathematical model, this study considers the effect of rolling resistance and aerodynamic resistance on the vehicle. Based on Newton’s second law, the dynamic equation of the vehicle driving downhill is established as equation (1)15 m
dua = Gx F f F w F r dt
ð1Þ
where m is the vehicle mass (kg); ua is the current vehicle speed (km/h); Ff is the rolling resistance (N); Fw is the aerodynamic resistance (N); Fr is the braking force of the hydraulic retarder (N); Gx is the component force of gravity along the slope (N), which is mg sin u; g is the acceleration of gravity, which is 9.8 m/s2; and u is the slope of the road (rad). The retarder braking force is expressed as Fr =
Mr i 0 h r
where Mr is the braking torque of the hydraulic retarder (N m), i0 is the gear ratio of the main reducer, h is the efficiency between the retarder and the wheel, and r is the radius of the wheel (m). The rolling resistance is expressed as Ff = Gy f = mg cos u f where Gy is the normal load (N) and f is the rolling resistance coefficient, which is f = 0:00076 + 0:000056ua . The aerodynamic resistance is expressed as Fw =
ð2Þ
1 CD Arua 2 2
where CD is the aerodynamic resistance coefficient, A is the windward area (m2), and r is the air density (kg/m3).
Mr n r alrnr 3 D5 = 9549 9549
ð3Þ
where a is the filling ratio of the working chamber (%), Mrmax is the braking characteristic when the filling ratio of the working chamber is 100% (original characteristics), r is the density of the working oil (kg/m3), l is the performance factor, D is the profile diameter (m), nr is the rotor speed (r/min), n is the engine speed (r/min), ir is the gear ratio of the increasing gear, and ig is the gear ratio of the transmission. According to equation (2), the hydraulic retarder can produce large braking torque at high speed and high filling ratio to achieve a good braking effect. However, large braking torque indicates large braking power, and the braking power of the hydraulic retarder determines the heat generated by the retarder. Due to the limitation of the thermal design power (Pc) of the radiator in the hydraulic retarder system, the maximum braking torque of the retarder is restricted by the thermal design power of the radiator, as shown in Figure 4. To increase the braking torque of the hydraulic retarder and improve its continuous braking performance, on the basis of sufficient arrangement space of the transmission system, the radiator that has a large radiating area and high heat transfer coefficient should be used to increase the thermal design power.17,18 Given the restriction of the thermal design power of the radiator, the braking process when the vehicle drives downhill using the hydraulic retarder is divided into three stages, as shown in Figure 5: Stage 1 (AB) is the braking process under high rotor speed based on the thermal design power of the radiator. To ensure the continuous braking performance of the retarder, its braking power is equal to the radiating power of the radiator; the braking torque increases with the deceleration of the rotor speed, and the filling ratio increases gradually
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Figure 4. Braking torque changing over rotor speed and thermal design power of the radiator.
Stage 1 :
8 P = Pc > < r Mr = Pc 9549 nr > : Mr a = Mrmax =
Figure 5. Braking characteristics of the retarder under different stages.
Pc 9549 =Mrmax nr
=
Pc 9549 lrnr 3 D5
Stage 2 (BC) is the braking process under medium or low rotor speed based on the original characteristics of the retarder. To produce the maximum braking torque, the filling ratio of the working chamber at this stage is 100%; the braking power and braking torque of the retarder decrease with the deceleration of the rotor speed
Stage 2 :
8 lrnr 3 D5 Mrmax nr > < Pr = 9549 = 9549 Mr = Mrmax = lrnr 2 D5 > : a = 100%
Stage 3 (CD) is the braking process in which the vehicle reaches the target speed with constant torque based on the target vehicle speed. When the vehicle reaches the target speed, the constant-speed control strategy is activated, the controller calculates the target filling ratio that the vehicle requires under current road conditions, and then the vehicle drives downhill at this target speed. The filling ratio of the working chamber at this stage is atarget; the braking torque and braking power both have a constant value
Stage 3 :
8 atarget Mrmax nr a lrn 3 D5 > = target9549 r < Pr = 9549 Mr = atarget Mrmax = atarget lrnr 2 D5 > : a = atarget
Filling ratio controller While driving downhill, when the hydraulic retarder is needed to decelerate the vehicle, the driver gets the
retarder started and the working oil starts to flow into the working chamber to produce braking torque. After Stage 1 (constant-power brake) and Stage 2 (original characteristics’ brake) of the braking process, when the vehicle decelerates to the target speed that the driver expected, the driver turns on the constant-speed switch, then the braking process enters into Stage 3 (constant-speed brake), and the electronic control unit (ECU) records the current vehicle speed as the target speed. The error signal between the current and target speed is inputted to the constant-speed fuzzy controller. The controller then changes the braking torque of the retarder by changing the filling ratio of the working chamber to reach the target of driving downhill at a constant speed. The control logic is shown in Figure 6. The filling ratio controller designed in this study is intended to adjust the braking torque of the hydraulic retarder. Based on the analysis of the braking process of the retarder in section ‘‘Model of hydraulic retarder,’’ the controller is divided into two parts, namely, decelerating controller and constant-speed fuzzy controller. Moreover, the control logic of the main controller is shown in Figure 7.
Decelerating controller At the start of the brake, the vehicle speed is relatively high, and the decelerating controller starts to function. The decelerating controller consists of two parts, which includes the constant-power brake (Stage 1) and original characteristics’ brake (Stage 2). At the stage of constant-power brake (Stage 1), the braking power of the hydraulic retarder is equal to the thermal design power of the radiator (Pr = Pc); thus, the filling ratio of the working chamber of retarder in this stage is expressed as
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Figure 8. Control logic of the constant-speed fuzzy logic controller. Figure 6. Control logic of the constant-speed control strategy.
Figure 9. Fuzzy logic controller with its inputs and output.
Figure 7. Control logic of the filling ratio controller.
a=
Pc 9549 lrnr 3 D5
in which the amount of the working oil in the working chamber of the retarder increases with the deceleration of the vehicle speed until the working chamber is filled with working oil. At the stage of original characteristics’ brake (Stage 2), the filling ratio of the working chamber of the hydraulic retarder is a constant value which is a = 100%. The braking torque completely changes according to the original characteristics of the retarder; thus, the braking torque gradually decreases with the deceleration of the vehicle speed until the vehicle speed reaches the target speed.
Constant-speed fuzzy logic controller Fuzzy control is a computer control technology based on natural language control rules and fuzzy logic inferences. Fuzzy control is extensively used in engineering because of its simplicity, flexibility, and good performance. The core of the filling ratio controller is the constant-speed fuzzy logic controller which corresponds to the constant-speed brake (Stage 3) of the retarder braking process. This controller uses a method in which the control target is based on the target vehicle speed. This method is stable and reliable and can control the vehicle speed at a rate that is near the target vehicle speed during the braking process of driving
downhill. The diagram of the fuzzy control logic is shown in Figure 8. As shown in equation (4), the constant-speed fuzzy logic controller includes two inputs, namely, vehicle speed error e and the rate of error ec (
e = ua utarget ec =
d(ua utarget ) dt
=a
ð4Þ
where utarget is the target vehicle speed (km/h), and a is the longitudinal acceleration of the vehicle (m/s2). The output of the fuzzy logic controller is the variation in the expectant target filling ratio Datarget, the target filling ratio required atarget is acurrent + Datarget, and the braking torque of the hydraulic retarder required Mreq can be obtained as Mreq = atarget lrnr 2 D5
ð5Þ
The schematic diagram for the fuzzy logic controller with its inputs and output is shown in Figure 9. The basic domain of vehicle speed error e is in the range of [20.5, 0.5] m/s, and the domain of error rate ec is in the range of [21.5, 1.5] m/s2. The quantitative factor is a kind of domain conversion coefficient from basic domain to fuzzy domain. The fuzzy domain can be achieved by multiplying the quantitative factor and the basic domain. The quantitative factors ke and kec are set as 6 and 2, respectively. The basic domain of the filling ratio variation Datarget is in the range of [0%, 100%], and the quantitative factor ku is implemented using coordinate transformation u = (Datarget 0:5) 3 6. Then all the variables of the fuzzy logic
6 controller are in the fuzzy set (fuzzy domain) of {23, 22, 21, 0, 1, 2, 3}, and the corresponding linguistic variable set is {NB, NM, NS, ZO, PS, PM, PB}. Selecting the membership function depends on different control objects; generally, the membership function of the fuzzy subset should be a continuous function and a symmetrical convex fuzzy set in addition to the boundary of the universe of discourse.19 In this study, all the variables of input and output languages of the constant-speed fuzzy controller use the triangle membership function, which has the advantages of easy implementation, simple operation preferable control performance, and widespread used in fuzzy control,20 as shown in Figure 10. PB means positive big, PM means positive medium, PS means positive small, ZO means zero, NS means negative small, NM means negative medium, and NB means negative big. When the current vehicle speed is significantly higher than the target vehicle speed, which means that the error e is PB, and if the rate of error ec is also PB, then to eliminate the error immediately and prevent the vehicle speed from increasing rapidly during downhill driving, the fuzzy logic controller controls the hydraulic retarder to increase the filling ratio rapidly and thereby to increase the braking torque, indicating that the Datarget is PB. However, if the rate of error ec is NS, which means that the speed error tends to decrease, then the current acceleration of the vehicle decreases; thus, the fuzzy logic controller controls the hydraulic retarder to increase the filling ratio appropriately, indicating that the Datarget is PS. The overshoot of the system is avoided and the vehicle speed tends to reach the target value immediately.
Advances in Mechanical Engineering If the current vehicle speed is significantly lower than the target vehicle speed, then the error e is NB; if the rate of error ec is also NB, then the slope of the road decreases. To prevent the vehicle speed from becoming extremely low, the fuzzy logic controller controls the hydraulic retarder to decrease the filling ratio rapidly and braking torque, indicating that the Datarget is NB. However, if the rate of error ec is PS, which means that the error decreases, then the fuzzy logic controller controls the hydraulic retarder to decrease the filling ratio appropriately, indicating that the Datarget is NS. The other fuzzy control rules of the constant-speed fuzzy logic controller are presented in Table 1. The relationship between e, ec, and Datarget is shown in Figure 11.
Simulation analysis Simulation model The dynamic model of the vehicle driving downhill is established based on equation (1), and the simulation of the constant-speed control strategy during continuous braking process in the MATLAB/Simulink environment is analyzed,21 as shown in Figures 12 and 13. The braking performance of the hydraulic retarder is examined; thus, several assumptions propose that the service and engine brakes do not participate in this simulation model, the vehicle travels in a straight line, the road is flat and adhesion is good, and the wheels do not lock and slip. The onset time of the variation in the filling ratio is considered in the simulation model; specifically, the activation of the retarder takes 1 s, and approximately 0.3 s elapses after the vehicle speed
Figure 10. Membership functions of input and output variables for constant-speed fuzzy controller.
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Table 1. Rules in the fuzzy logic controller. Datarget
ec
e
PB PM PS ZO NS NM NB
PB
PM
PS
ZO
NS
NM
NB
PB PB PM PM PS ZO NS
PB PB PM PS PS ZO NS
PB PM PS PS ZO NS NM
PM PS PS ZO NS NS NM
PM PS ZO NS NS NM NB
PS ZO ZO NS NM NB NB
PS ZO NS NM NB NB NB
PB: positive big; PM: positive medium; PS: positive small; ZO: zero; NS: negative small; NM: negative medium; NB: negative big.
Figure 11. Relationship between e, ec, and Datarget.
reaches the target speed. The inputs of the model include the vehicle mass, the slope of the road, and other parameters which are shown in Table 2. The outputs of the model are the vehicle speed and longitudinal acceleration. The core of the simulation model is the hydraulic retarder model. The effect of filling time on the braking performance of the hydraulic retarder is considered. Based on section ‘‘Model of hydraulic retarder,’’ the restriction of the thermal design power of the radiator on the braking power of the hydraulic retarder is added to this model, where at the start of the braking process under high rotor speed, the maximum braking power of the hydraulic retarder is equal to the thermal design power of the radiator, which is 600 kW.
Simulation results for constant slope
Figure 12. Dynamic simulation model of the vehicle driving downhill.
The simulation results for the heavy-duty vehicle driving downhill at 5% constant slope (the initial vehicle speed and vehicle mass are 60 km/h and 35,000 kg, respectively) are shown in Figure 14. Specifically, Figure 14(a) presents the slope of the road, vehicle speed, and vehicle acceleration changing over time.
Figure 13. Dynamic simulation model in the of MATLAB/Simulink environment.
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Table 2. Parameters of the simulation model. Parameter
Value
Vehicle mass Initial vehicle speed Target vehicle speed Aerodynamic resistance coefficient Windward area Atmospheric pressure Air density Environment temperature Gear ratio of transmission Gear ratio of main reducer Gear ratio of increasing gear Wheel radius Maximum braking torque of retarder Maximum rotor speed of retarder Thermal design power of radiator Simulation step
35,000 kg 60 km/h 30 km/h 0.55 8.5 m2 0.1 MPa 1.205 kg/m3 20°C 1.646 4.111 1.5 0.526 m 4500 N m 2800 r/min 600 kW 0.1
Figure 14(b) depicts the characteristics of the hydraulic retarder as it changes over time. At the start of the braking process, the vehicle speed and acceleration have an increasing trend because the hydraulic retarder has a dynamic filling process at a time of about 1 s. After the initial filling process, under the restriction of the thermal design power of the radiator, the braking power of the hydraulic retarder is equal to the thermal design power of the radiator, which is at a constant value of 600 kW. The braking torque of the retarder increases with the deceleration of the vehicle speed, and the filling ratio of the working chamber increases
gradually. At 7 s, the working chamber is full of working oil, and the braking torque decreases with the deceleration of the vehicle speed based on the original characteristic of the retarder. At 13 s, the vehicle speed reaches the target speed of 30 km/h; the constant-speed fuzzy logic controller starts to function and calculates the target filling ratio required under the current working conditions based on the speed error and the rate of speed error. The vehicle continues to drive downhill at the target speed and the acceleration tends to stabilize at the constant-speed stage. The initial vehicle speed is then changed to 70 km/h, and the vehicle mass is changed to 40,000 kg; the simulation results are shown in Figure 15. Figure 15(a) shows that the required time for the vehicle speed to reach the target speed is 21 s. Meanwhile, Figure 15(b) illustrates that the corresponding filling ratio increases from 57% to 65% compared with that in Figure 14(b), and the braking torque and braking power increase accordingly. The comparison among the use of the common constant-speed control strategy (by changing the control cycle, initial liquid charge, and variation in liquid charge in each control cycle9,10), PID control strategy, and the fuzzy constant-speed strategy during downhill driving is shown in Figure 16. The simulation time is extended to 40 s. Figure 16 shows that when the filling ratio controller is used at the time of reaching constantspeed stage, speed fluctuation hardly occurs and the speed reaches a stable level of 30 km/h; furthermore, the vehicle speed error when reaching the constantspeed stage decreases from 5% to 0.3%.
Figure 14. Simulation results of constant slope (35,000 kg, 60 km/h): (a) vehicle and (b) hydraulic retarder.
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Figure 15. Simulation results of constant slope (40,000 kg, 70 km/h): (a) vehicle and (b) hydraulic retarder.
Figure 16. Comparison of different control strategies.
Simulation results for changing slope The simulation results for downhill driving of the heavy-duty vehicle at a changing slope of 4%–8% are reported in Figure 17. Figure 17(a) shows the slope of the road, vehicle speed, and acceleration change over time. Figure 17(b) depicts the characteristic of the hydraulic retarder as it changes over time. Moreover, Figure 17(a) shows that even when the slope of the road is changing and the vehicle reaches the target vehicle speed, the fuzzy logic controller designed in this study can also stabilize the vehicle speed near the target vehicle speed and the acceleration has minimal change. The reason is that during the change in the slope, the filling ratio controller monitors the change in the rate of speed error; based on the fuzzy rules, the filling ratio
of the working chamber generates real-time change to stabilize the vehicle speed to reach the target speed. Thus, the vehicle can drive downhill at a stable speed. The comparison between the current vehicle speed and target vehicle speed during the downhill drive at a changing slope is shown in Figure 18. The vehicle speed error when reaching the constant-speed stage is in the range of 6 0.3 km/h. According to the comparison, the model that uses fuzzy logic can immediately and accurately stabilize the vehicle speed in the range of the driver expectations.
Conclusion Focusing on the hydraulic retarder function of downhill driving at constant speed, this study proposed a
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Figure 17. Simulation results of changing slope: (a) vehicle and (b) hydraulic retarder.
Figure 18. Comparison of the current and target speed.
constant-speed control strategy based on fuzzy theory which could lead to significant improvements in the safety of road transport. A filling ratio controller for the hydraulic retarder was designed to immediately and accurately produce braking torque during downhill driving to satisfy the braking requirements of the vehicle. The vehicle dynamic model and hydraulic constantspeed model were established in the MATLAB/ Simulink environment. Based on the simulation results, the following conclusions are made: 1.
The braking process of the vehicle driving downhill is classified into three stages, and the computational formula of the braking torque as well as the braking power and filling ratio of
2.
3.
the hydraulic retarder is deduced. The accuracy of the established model of the hydraulic retarder was verified by comparing the simulation results with those of the theoretical analysis. During the constant-speed stage of the vehicle downhill driving, the designed filling ratio controller is responsive and can immediately and accurately calculate the target filling ratio. The hydraulic retarder then outputs the braking torque required under current conditions to enable the vehicle to drive at a constant speed, which meets the control target of the braking torque. The comparison among the common constantspeed strategy, PID control strategy, and the control strategy in this article shows that the
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controller designed in this study has better constant-torque performance. When the vehicle speed reaches the target speed, the hydraulic retarder can produce a stable braking torque, the vehicle speed error decreases from 5% to 0.3%, and the stability of the vehicle as it drives downhill improves significantly. When the vehicle drives downhill at a changing slope, the controller controls the retarder to produce real-time braking torque to enable the vehicle speed to reach the target speed as fast as possible. The error is below 1%, which shows that the designed controller has good antiinference performance.
Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research was sponsored by the Program for New Century Excellent Talents in University (NCET-08-0248), the 985 Project Automotive Engineering of Jilin University, and Graduate Innovation Fund of Jilin University (2016077).
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