A Fuzzy-PID Scheme for Low Speed Control of a Vehicle While Going on a Downhill Road

Abstract

We explored a vehicle hill descent control (HDC) system based on an electronic stability program (ESP) and applied this system to brake cars. The experimental results reveal that our system can effectively reduce the workload of a driver during a downhill journey. In the first phase of our work, the control strategy of the HDC system based on fuzzy-PID (Proportion Integral Differential) was built by MATLAB/Simulink. Then, the co-simulation based on MATLAB/Simulink, CarSim and AMESim was carried out. Finally, a real vehicle test was conducted to further verify the feasibility of the strategy. A series of simulation experiments and real vehicle tests show that the HDC system can assist the driver to control the vehicle while driving downhill at low speed, thus effectively improving the safety of the vehicle and reducing the workload of driver.

1. Introduction

According to statistics, more than 1.3 million people lose their lives and more than 50 million people are seriously injured every year as a consequence of road accidents [1]. Although scientists have used various methods to solve car safety problems and have achieved some success, there is still a long way to go to improve car safety [2]. Among these road traffic crashes, the death rate on downhill roads is much higher than the average rates [3]. Research data shows that the accident rate increases with the increase of the road slope; the higher the slope, the more obvious it is [4]. Thus, downhill safety is an extremely important topic in vehicle safety research. As we all know, when driving on a long steep downhill road, maintaining the vehicle speed in a certain range by using a reliable brake system can effectively reduce the workload of driver, which ensures that the driver puts more attention on the steering vehicle [5]. In the past decades, many new technologies have been applied to improve the braking safety of vehicles on downhill roads, such as hydraulic braking systems [6,7], auxiliary braking systems for the engine [8,9], retarder auxiliary braking systems [10,11], and regenerative braking [12]. However, an auxiliary braking system for the engine provides limited braking torque, which cannot be controlled accurately [13,14,15,16]. Retarder braking systems have slow responses and will lead to heat recession after long-term operation [17,18]. Regenerative braking not only provides limited braking torque, but also can only be applied to electric vehicles. Therefore, it is highly desirable to find an efficient dynamic control system to improve the downhill braking performance of vehicles.

Stable and reliable braking power can be provided by an ESP within the vehicle. Recently, various value-added functions based on ESP hydraulic circuit systems have made rapid development. These additional functions have a great advantage in providing more comfortable and convenient driving experience for the driver by automatically applying the brakes [19]. Among various additional functions, the HDC system has been found to be the most valuable function, because it can maintain vehicle speed in a certain range by automatically applying the brakes. Besides, the HDC system is also helpful in reducing the workload of the driver and helps the driver pay more attention to steering the vehicle.

Here, the fuzzy-PID control method is adopted to build up the HDC system. As we all know, PID controllers are widely used in the automotive field due to their simple application and suitable performance. However, in the case of strong nonlinearity of the vehicle system, the efficacy of simple PID controllers is not ideal. In addition, PID controller with fixed parameters greatly limits their application and effectiveness in a variety of real-world environments [20]. So far, it is of great urgency to find an efficient and simple control method to make up for these deficiencies. Fuzzy controllers logic has emerged as a complement to conventional strict methods, as the fuzzy logic based controllers can handle the uncertainty; however, normal fuzzy control cannot eliminate the steady state error efficiently [21]. Fuzzy-PID controllers combine the advantages of PID controllers and fuzzy controllers [22,23,24]. On the one hand, the performance of PID controllers can be enhanced via a fuzzy logic assisted controller by automatically changing the controller parameters. On the other hand, the steady state error can be controlled in a small range. Further more, the actual speed of the vehicle is fed back to fuzzy-PID controller in real time. The controller adjusts the target braking torque in real time to make the vehicle reach the target speed according to the difference of the target speed and actual speed. The change of the resistances related to the vehicle tractions and the slope of the terrain will lead to the change of the vehicle actual speed, so the complex interactions between vehicle user infrastructure can be simplified [25,26,27].

In this paper, the whole vehicle simulation model was established based on CarSim software; the HDC strategy simulation model were built up through MATALB/Simulink software by using control theories of fuzzy-PID logic control; the fuzzy logic control strategy were conducted through the development of fuzzy membership functions in MATALB’s Fuzzy Logic Toolbox and Simulink. A hydraulic brake system simulation model was established via AMESim software. The PID controller has been designed to be compared with fuzzy-PID controllers. The characteristics and effects of control strategy were further validated and analyzed according to the CarSim-AMESim-Simulink co-simulation. In order to verify the feasibility of the algorithm, a real vehicle test was conducted.

2. Methodology

When the HDC is motivated, the target vehicle speed is calculated, which is used by the control logic of fuzzy-PID theories as a control variable with actual vehicle speed that is from Vehicle model built via CarSim. The result of the control strategy is target braking pressure. The different valves will be motivated to achieve the target pressure in AMESim circumstance. The specific process is shown in Figure 1a,b (The meaning of the symbols are shown in

Table 1. List of main symbols in this article.

Symbol	Description	Unit
Ax/g	The measured/Gravitational acceleration	m/s2
gGear	The gear of vehicle	1
CForWard/CReverse	Forward/Reverse gears	1
α	The slope of ramp (The value is positive when the car’s nose is uphill and it is negative when downhill.)	100%
θ	The angle of ramp	rad
αThreHdc	The threshold of slope of ramp	100%
fHdcAct	The activation flag of HDC (The value is 1 when the HDC is motivated and 0 when HDC is shut down)	1
fEscOff	The close flag of ESC (The value is 1 when the ESC is open and 0 when ESC is shut down)	1
fHdcReq	The requirement flag of HDC (The value is 1 when the HDC is requested and 0 when it is not)	1
fxHdc	The function to calculate the target brake torque	1
hg	The height of vehicle center of mass	m
L	Wheel base	m
Lf/Lr	The distance between the front/rear axle and the center of mass	m
m	The mass of vehicle	Kg
VAct/VSo	Actual/Target vehicle speed	km/h
CVThreHdcMax	The maximum vehicle speed that HDC can be motivated	km/h
CVSoHdcInt	The initial target speed when HDC is motivated	km/h
MbHdcSo	Target brake torque	Nm
Pw	Actual brake pressure of wheel cylinder (From the pressure sensor)	bar
PSoAVe	Average target brake pressure	bar
PSof/PSor	The target pressure of the front rear axle	bar
PMaxf/PMaxr	The maximum pressure that the front/rear axle can withstand	bar
Pm	Master cylinder pressure	bar
Rf/Rr	The front/rear wheel radius	m
u	The road adhesion coefficient	1
CPThrInc	The threshold to increase the pressure of wheel cylinder	bar
CtMaxThreHdcAct	Maximum duration of HDC operation	s
fMotor	The motor start flag	1
fGasPedal/fBrakePedal	The flag of acceleration/brake pedal (The value is 1 when the pedal is pushed and 0 when the pedal is released)	1
fGasPedalK1/fBrakePedalK1	The flag of acceleration/brake in the Last Cycle	1
CVSoHdcInc/CVSoHdcDec	Increment/Decrement value of target speed	km/h
CVSoHdcMin/CVSoHdcMax	Minimum/Maximum target vehicle speed	km/h
Kp/Ki/Kd	The proportional/integral/differential parameter	1
Cpf/Cpr	The front /rear axle ratio of brake torque to brake pressure	Nm/bar
DutyUSV/DutyHSV	The duty of USV/HSV	%
HDC	Hill descent control	1
PID	Proportion Integral Differential	1
ESP	Electronic stability program	1
E	The difference between the target speed and the actual speed	1
EC	The differential of the difference between the target speed and the actual speed	1
HCU	Hydraulic control unit	1
ECU	Electronic control unit	1
CAN	Controller area network	1

).

2.1. The HDC Strategy

2.1.1. Activation Logic of HDC

Figure 2 (The meaning of the symbols are shown in Table 1) reveals the activation logic of the HDC. In this system, if HDC is requested and the ESP opened during driving downhill, the HDC will be motivated when the vehicle speed goes below the maximum threshold. HDC will be closed in the following four cases: 1. the HDC turned off; 2. the ESC turned off; 3. vehicle downhill ended; 4. the activation time reaches the maximum value. These designs effectively prevent the HDC system from being motivated on a flat road or running for too long time. Besides, the driver can turn it off at any time. In this logic the initial target speed is C_VSoHdcInt, it will be updated when the driver is detected pushing the brake pedal or the accelerator pedal. It is worth noting that the target speed limited between C_VSoHdcMin and C_VSoHdcMmx.

2.1.2. Fuzzy-PID Logic Control of HDC

The fxHdc is the function to calculate the target brake torque, which is designed by feedback control of fuzzy-PID logic. The block diagram of the proposed self tuning fuzzy-PID type controller is given in Figure 3 (The meaning of the symbols are shown in Table 1). According to the need of fuzzy logic controller design, the quantificational field of fuzzy variables E and EC is defined as follows: input of Fuzzy system are E (the difference between the target speed and the actual speed) and EC (differential of the difference between the target speed and the actual speed). Output of Fuzzy system are Kp (the proportional parameter), Ki (the integral parameter) and Kd (the differential parameter).

The fuzzy sets of E and EC are defined as {NB, NM, ZO, PM, PB}, and their universes are defined as [–20, 10] and [–10, 10]. The membership functions of E and EC is brought in Figure 4a,b, and their division is shown in

Table 2. Division of universe E.

Membership Functions	NB	NM	ZO	PM	PB
Range	[–30 –20 –10 –2]	[–8 –3.5 0]	[–2 0 2]	[0 3.5 8]	[2 10 20 30]

and

Table 3. Division of universe EC.

Membership Functions	NB	NM	ZO	PM	PB
Range	[–20 –10 –5 –1]	[–3 –2 0]	[–1 0 1]	[0 2 3]	[1 5 10 20]

. The fuzzy sets of Kp, Ki and Kd are defined as {O, S, M, B}, and their universes are defined as [0, 500] and [0, 20] and [0, 10]. The membership functions of Kp, Ki and Kd are revealed in Figure 4c–e, and their division is shown in

Table 4. Division of universe Kp.

Membership Functions	O	S	M	B
Range	[–200 0 70]	[0 100 200]	[70 220 350]	[200 350 610 775]

,

Table 5. Division of universe Ki.

Membership Functions	O	S	M	B
Range	[–8 0 5]	[0 5 12]	[5 10 16]	[12 16 24 30]

and

Table 6. Division of universe Kd.

Membership Functions	O	S	M	B
Range	[–4 0 2]	[0 2 4]	[2 4 6]	[4 6 10 20]

. The rules of fuzzy logic control can be expressed by conditional statement of “if … then …”, which represent decision result derived from many change premises.

In general, fuzzy logic control form is used to express those rules, as shown in

Table 7. Kp rules of fuzzy control.

EC Kp E	NB	NM	ZO	PM	PB
NB	S	S	M	S	O
NM	M	M	B	M	M
ZO	B	M	B	M	B
PM	M	M	B	M	M
PB	S	S	M	S	S

,

Table 8. Ki rules of fuzzy control.

EC Kp E	NB	NM	ZO	PM	PB
NB	O	O	S	O	O
NM	O	S	M	S	O
ZO	S	M	B	M	S
PM	O	S	M	S	O
PB	O	O	S	O	O

and

Table 9. Kd rules of fuzzy control.

EC Kp E	NB	NM	ZO	PM	PB
NB	O	M	B	M	O
NM	S	S	M	S	S
ZO	O	O	S	O	O
PM	S	S	M	S	S
PB	O	M	B	M	O

. Wherein, NB represent Negative Big, NM is Negative Medium, NS is Negative Small, ZE is Almost Zero, PS is Positive Small, PM is Positive Medium, PB is Positive Big, O is Almost Zero, S is Small, M is Medium, B is Big. In this system, the output of the fuzzy system will be used as a parameter of the PID controller, and the final output of the fuzzy-PID controller is the target brake torque.

2.1.3. PID Logic Control of HDC

To compare the proposed fuzzy-PID controller, we set up the PID controller. The corresponding scheme is given in Figure 5 (The meaning of the symbols are shown in Table 1). In this system, inputs of the controller are E and EC, and its output is target brake torque (similar to fuzzy-PID controller). By contrast with the fuzzy-PID controller, the PID controller is carried out with fixed parameters (Kp, Ki, Kd are constant).

2.2. Convert Target Torque of HDC to Pressure

The brake torque is proportional to the brake pressure and the proportional relation is as follows:

$$P_{SoAve}=M{b}_{HdcSo}/(C{p}_f+C{p}_r)$$

(1)

If the required pressure is evenly distributed between the front and rear axles, the target pressure of the front and rear axle will be the same as P_SoAve. However, the maximum pressure that can be applied to the front and rear axles are usually different when the vehicle goes downhill, because the load of the rear axle will be transferred to the front axle. The maximum pressure that the front axle can withstand is shown as Formula (2), The maximum pressure of the rear axle is shown as Formula (3).

$$P_{Maxf}=(m·g\frac{l_r}{L}-m·\frac{h_g}{L}·A_x)·u·R_f/C{p}_f$$

(2)

$$P_{Maxr}=(m·g\frac{l_f}{L}+m·\frac{h_g}{L}·A_x)·u·R_r/C{p}_r$$

(3)

We suppose that the road surface is high adhesion, and the road adhesion coefficient u is “1”. There are only three types of the relationships between the pressure and the target pressure of the front and rear axle are shown in the following formula:

• If P_SoAve ≤ P_Maxf and P_SoAve ≤ P_Maxr,

  $$P_{Sof}=P_{SoAve}$$

  (4)

  $$P_{Sor}=P_{SoAve}$$

  (5)
• If P_SoAve ≥ P_Maxf and P_SoAve < P_Maxr,

  $$P_{Sof}=P_{Maxf}$$

  (6)

  $$P_{Sor}=(P_{SoAve}-P_{Maxf})·C{p}_f/C{p}_r+P_{SoAve}$$

  (7)
• If P_SoAve < P_Maxf and P_SoAve ≥ P_Maxr,

  $$P_{Sof}=(P_{SoAve}-P_{Maxr})·C{p}_r/C{p}_f+P_{SoAve}$$

  (8)

  $$P_{Sor}=P_{Maxr}$$

  (9)

It will indicates the slope is too large and the vehicle cannot descend normally if P_SoAve > P_Maxf and P_SoAve > P_Maxr. The meaning of the symbols above are shown in Table 1.

2.3. Convert Target Pressure of HDC to the Duty of Valves

The schematic diagram of the HCU (hydraulic control unit) and the model of the HCU built in AMESim are shown in Figure 6 and Figure 7, respectively. HCU consists of two normally open and two closed type solenoid valves. USV (USV_I, USV_II) and EV (EV_I_L, EV_I_R, EV_II_L, EV_II_R) are normally the open linear solenoid valve, which will be closed when energized. The pressure required to open USV and EV is proportional to the electromagnetic force from the electromagnetic coil. HSV (HSV_I, HSV_II) and AV (AV_I_L, AV_I_R, AV_II_L, AV_II_R) are the normally closed switching solenoid valve, and their states (closed or opened) are controlled electronically. The working principle of the wheel cylinder to establish pressure is as follows. The brake fluid flows through the valve USV and EV from the master cylinder to the wheel cylinders when the driver pushes the brake pedal. While the vehicle is braked without the intervention of the driver, the electromagnetic coil USV will be energized and the valve of HSV will be opened; at the same time, the motor will be motivated, the brake fluid flows through the valve HSV and EV from the master cylinder to the wheel cylinders.

Pressure corresponding to the different duty of USV is given in Figure 8. The pressure that wheel cylinders can reach is proportional to the duty of the USV, and 100% corresponds to the current of 2 amperes of the electromagnetic coil. As we all know, the greater the current, the greater the electromagnetic force. Therefore, we can convert the target pressure to the load of the USV to brake the vehicle at a certain brake pressure.

The control logic of USV, HSV and Motor are shown in Figure 9 (The meaning of the symbols are shown in Table 1). In this logic, it is worth noting that the USV, HSV and motor will be uncontrolled if Pm is bigger than PSo. If the difference between PSo and Pw is greater than CPThreInc, the pressure of the wheel cylinder must be increased by starting the motor, opening the HSV and closing the USV (the duty of USV can be found from Figure 8). If the difference between PSo and Pw is lower than CPThreInc, the pressure of the wheel cylinder can be adjusted by simply adjusting the duty of USV.

3. Simulations and Analysis

In order to further confirm the rationality and feasibility of our model algorithm, a series of simulation experiments were carried out. All simulation tests were performed in MATLAB/Simulink, CarSim and AMESim co-simulation environment.

CarSim is a simulation software specifically for vehicle dynamics. The CarSim user interface is shown in Figure 10, it mainly includes Test Specifications, Run control with Simulink and Results (Post Processing). The vehicles parameters and simulation conditions can be defined in Test Specifications; Simulation solution configurations is set in Run control with Simulink; Results (Post Processing) Can display 3D animation and draw simulation curves.

Vehicle parameters, road information and simulation conditions are set in CarSim in advance, the CarSim can simulate real car movement, the actual speed and acceleration of the vehicle will be output when the brake pressure of the wheel cylinders are input to the CarSim.

The test results are as follows:

3.1. The Result Carried out with Fuzzy-PID Controller

The first test circumstance of simulation was set up in CarSim, and the slope of ramp was set as 20%, 40% and 15%, respectively. The vehicle accelerated from rest, and at the same time the HDC was requested. The test results are shown in Figure 11a–c, and the driver does nothing with the brake and acceleration pedals, the actual speed can follow the target speed well when the slope of the ramp is varied.

The second test circumstance of the simulation was set up in CarSim, the slope of ramp was constant with a value of 20%. The vehicle accelerated from rest, at the same time the HDC was requested. During this process, the brake and acceleration pedal were pushed at random. The specific test results are shown in Figure 12a–c, and target speed decreases when the driver pushes the brake pedal and increases when the driver pushes the acceleration pedal. The results indicate that the actual speed has a good effect compared to the varied target speed.

3.2. The Result Carried out with PID Controller

The same circumstance of simulation was set up in the CarSim and the operation of the car was the same as above. The test results are shown in Figure 13a–c and Figure 14a–c:

We can see from the simulation result that the actual speed can follow the target speed well. However, the overshoot of PID controller and the time to reach steady state are greater than that of fuzzy-PID controller. The fuzzy-PID controller has a great advantage over PID controller.

4. Real Vehicle Tests and Analysis

The control strategy of HDC was generated to be the code by the code generation tools of MATLAB/Simulink, and was integrated with existing ESP products. Then, we validated it in a real vehicle.

The structural explosion diagram of electronic control unit (ECU), hydraulic control unit (HCU) and real picture of ESP are shown in Figure 15. The image of the proving ground and the location of the ESP on the vehicle are shown in Figure 16.

The physical effects of longitudinal acceleration are measured by the longitudinal acceleration sensor when the vehicle travels on the slope. From the operating principle, this acceleration consists of two parts, one of them is the component of gravity in the direction of the slope and the other can be expressed by the differential of the speed. It can be described as the following formula:

$$Ax=g·\sin \theta +\stackrel{•}{V_{Act}}$$

(10)

For small angles the angle, its sinus and its tangent, are the same. The real road slope is small, so the slope of the ramp can be calculated by the following formula:

$$\alpha =\frac{Ax-\stackrel{•}{V_{Act}}}{g}·100\%$$

(11)

The meaning of the symbols in Formulas (10) and (11) are shown in Table 1.

The signal of vehicle speed, brake pedal and accelerator pedal are from the vehicle controller area network (CAN). The supervisory control and data acquisition are from Vector Tools.

The first test, drive the car to the ramp and push the HDC button (the HDC is requested when this button is pushed), release the brake and accelerator pedals. The test results are as follows:

As shown in Figure 17, Figure 18 and Figure 19, the driver does nothing with the brake and acceleration pedals, and the actual speed can follow target speed well on the different slope of ramp, which are 10%, 15% and 20%, respectively. The results are consistent with the results of the Co-simulation.

The second test, we drive the car to the ramp with slope is 20% and push the HDC button, and release the brake and accelerator pedals at the same time. When the vehicle speed is steady, the brake pedal will be pushed randomly. The test results are as follows:

As shown in Figure 20, target speed of vehicle decrease when the driver pushes the brake pedal. The results show that the actual speed has a better following effect than the varied target speed, which is consistent with the results of the co-simulation.

In the third test, we drove the car to the ramp with a slope of 20% and pushed the HDC button, and released the brake and accelerator pedals. The accelerator pedal was pushed randomly after the vehicle speed was steady. The test results are as follows:

As shown in Figure 21, the target speed of vehicle increases when the driver pushes the acceleration pedal. The results reveal that, compared to the varied target speed, the actual speed has a better following effect, which is consistent with the results of the co-simulation.

5. Conclusions

In order to reduce the workload of the driver during downhill, we explored an efficient vehicle HDC system based on ESP. In this work, both simulations and experiments were performed to validate the effectiveness of the proposed strategy. The experimental results show that both the overshoot and the time to reach a steady state of fuzzy PID are shorter than that of PID. The simulation and actual vehicle test results co-confirm that the vehicle can maintain very low speed, which is the varied target speed, during downhill driving whether it is a ramp change or driver intervention. Therefore, it is very reasonable for us to believe that the HDC is a feasible and effective system to assist the driver in controlling the vehicle speed during driving downhill. In addition, our system also improves the vehicle’s safety and reduces the driver’s workload.