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Abstract

This article proposes a pressure control algorithm without a pressure sensor for a four-wheel-drive unit. To develop the control algorithm, dynamic models of the four-wheel-drive unit were obtained, including the motor, pump, and clutch. For feedback control, a first-order adaptive transfer function of the modified motor speed was proposed to fulfill the transient response characteristics, and the motor input voltage generated the demanded pressure at steady state in the feedforward control. The coefficient $α$ of the adaptive transfer function was obtained with weight factors $W_{1}$ and $W_{2}$ considering the pressure difference and temperature. The performance of the proposed pressure control algorithm without a pressure sensor was evaluated by experiments at various operating temperatures. It was found that the actual pressure closely followed the demanded pressure with an acceptable error at steady state and within an acceptable rising time in a transient state.

Keywords

Pressure control four-wheel drive modified motor speed adaptive transfer function temperature

Introduction

Recently, interest in four-wheel drive (4WD) has increased as the demand for sports utility vehicles has increased. The number of vehicles equipped with 4WD has been increasing, even in the passenger car market, due to the need to improve tractive performance and safety. When two-wheel drive (2WD) vehicles are driven on icy roads, the maximum traction force of the vehicle is limited due to the small friction coefficient of the road. However, if 2WD is changed to 4WD, the maximum traction force can be increased via torque control in the 4WD unit.

In an electronically controlled 4WD unit, active torque control can be implemented via clamping force control of the clutch plates. To control the clutch clamping force, a DC motor actuator has been used to push the clutch plates directly,^1,2 or hydraulic pressure control was implemented using a solenoid valve.^3,4 In addition, mechanical pressure feedback was proposed using a piston pump and a centrifugal valve.⁵ Hydraulic pressure sensors have generally been used to control the clutch clamping force in electro-hydraulic 4WD units.⁶

If the clutch pressure is controlled without a pressure sensor, the additional cost of the pressure sensor can be saved, and the reliability of the 4WD unit can be improved by avoiding potential failure of the pressure sensor. Many studies have focused on pressure prediction without a pressure sensor. These studies employed pressure estimation of the brake piston using the on–off signal of the hydraulic valve^7–9 as well as estimation of the pump flow rate and pressure using the pump performance curve.^8,10 Pressure was calculated using structured recurrent neural network for a hydraulic actuator system¹¹ and was estimated using state variables of a frequency converter and pump performance curves for a variable speed drive.¹² However, the pressure offset between the actual pressure and the estimated pressure is a significant drawback. In addition, pressure estimation in the clutch piston in a dual clutch transmission was proposed using a proportional–integral (PI) observer.¹³ This observer was also used to estimate the clutch piston pressure in an automatic transmission.^14–16 However, a pressure sensor and a pressure control valve were used in these works.

In hydraulic control system, when the temperature increases, the oil viscosity decreases and leakage increases. When the pressure sensor is used, the effect of the oil leakage on the pressure can be compensated by pump speed control. For pressure prediction without a pressure sensor, the effect of temperature should be considered. However, few works have been reported on the pressure estimation by considering the temperature.

In this study, a pressure control algorithm without a pressure sensor is proposed for a 4WD unit considering temperature. For feedforward control, a motor input voltage that can generate the demanded pressure at a steady state was identified. Meanwhile, the feedback control satisfied the pressure response of the transient state. The performance of the pressure control algorithm proposed in this study was evaluated through experiments at various operating temperatures.

Mathematical model of the 4WD unit

Figure 1 shows the hydraulic circuit of the 4WD unit investigated in this study. The 4WD unit is an electro-hydraulic type unit, consisting of a DC motor, an oil pump, and a clutch piston. A bond graph model of the 4WD unit is shown in Figure 2.

Figure 1.

Hydraulic circuit of the 4WD unit.

Figure 2.

Bond graph model of the 4WD unit.

Motor

A DC motor was used to drive the oil pump. The state equation of the DC motor can be obtained from the bond graph model in Figure 2

S_{e} - L \overset{\cdot}{i} - R_{c} i - K_{emf} ω = 0

(1)

K_{t} i - J \overset{\cdot}{ω} - T_{f} - T_{m} = 0

(2)

where $S_{e}$ is the input voltage, $L$ is the inductance, $R_{c}$ is the resistance, $i$ is the current, $ω$ is the motor angular velocity, $K_{t}$ is the motor torque coefficient, $K_{emf}$ is the coil back emf constant, $J$ is the motor inertia, $T_{f}$ is the static friction torque, and $T_{m}$ is the motor torque to drive the oil pump.

Pump and hydraulic circuit

A gerotor pump was used in the proposed 4WD hydraulic system. The pump pressure and flow rate were determined based on the mechanical and volumetric efficiency. The mechanical and volumetric efficiency of the pump were obtained from the experiments, and the pump efficiency maps used in this study are shown in Figure 3.

Figure 3.

Pump efficiency map for a temperature of 30° C: (a) mechanical efficiency and (b) volumetric efficiency.

The relationship between the motor torque, $T_{m}$ , and hydraulic pressure, that is, the clutch pressure, $P$ , is expressed as follows

η_{m} \cdot T_{m} = D \cdot P

(3)

where $η_{m}$ is the mechanical efficiency, $D$ is the pump displacement, and $P$ is the clutch pressure.

The relationship between the motor angular speed, $ω$ , and hydraulic flow rate, $Q$ , is expressed as follows

Q = η_{v} \cdot D \cdot ω

(4)

where $η_{v}$ is the volumetric efficiency.

The flow equation from the pump to the clutch can be represented as

Q - R_{L} P - Q_{c} = 0

(5)

where $R_{L}$ is the pump internal leakage coefficient and $Q_{c}$ is the control flow rate required to generate the clutch pressure.

From the bond graph model in Figure 2, the flow equation of the hydraulic circuit is represented as

Q_{c} - \overset{\cdot}{V} - Av = 0

(6)

where $A$ is the piston area, $v$ is the linear piston velocity, and $V$ is the change in volume inside the hydraulic circuit. The change in volume $(V)$ can be represented by the pressure as

V = \frac{(V_{0} + Ax)}{β} P

(7)

where $β$ is the bulk modulus, $V_{0}$ is the initial volume of hydraulic fluid, and $x$ is the piston displacement.

Substituting equation (7) into equation (6) gives

\overset{\cdot}{P} = \frac{β}{(V_{0} + Ax)} (Q_{c} - Av)

(8)

Substituting equations (4) and (5) into equation (8) gives

\overset{\cdot}{P} = \frac{β}{(V_{0} + Ax)} (η_{v} D ω - R_{L} P - Av)

(9)

Clutch

The clutch clamping force is initiated when the clutch plates come into contact, and the point where this happens is called the “kissing point.” Before the kissing point, the piston displacement is determined by the piston inertial force, return spring force, and pressure force as follows

m_{p} \overset{\cdot\cdot}{x} + k_{r} x - AP = 0

(10)

After the kissing point, the piston and clutch plate move together, and the reaction force by the clutch plate stiffness needs to be considered as

(m_{p} + m_{c}) \overset{\cdot\cdot}{x} + (k_{r} + k_{c}) x - AP = 0

(11)

where $m_{p}$ is the piston mass, $k_{r}$ is the return spring stiffness, $m_{c}$ is the clutch mass, and $k_{c}$ is the clutch plate stiffness. Figure 4 shows the experimental results between the clutch plate reaction force and plate deformation.

Figure 4.

Clutch plate displacement versus reaction force.

Development of a pressure control algorithm without a pressure sensor

The proposed pressure control algorithm without a pressure sensor is composed of feedforward and feedback controls.

Feedforward control

When the clutch pressure is maintained constant at a steady state, equation (9) holds

η_{v} D ω_{ss} - R_{L} P_{dmd} = 0

(12)

where subscripts ss and dmd denote the steady state and demanded, respectively. From equation (12), the motor speed, $ω_{ss}$ , that generates the demanded pressure at a steady state can be obtained. However, since the volumetric efficiency $(η_{v})$ changes depending on the pressure and speed, the relationship between $ω_{ss}$ and $P_{dmd}$ should be obtained via iterative calculations. In Figure 5, a steady-state relationship between $ω_{ss}$ and $P_{dmd}$ is shown for the operating range. As shown, $ω_{ss}$ and $P_{dmd}$ have an almost linear relationship except in the pressure range of 4–6 bar, which is caused by the nonlinear volumetric efficiency characteristics of the pump in this range.

Figure 5.

Demanded pressure versus motor speed at steady state.

Feedforward voltage can be calculated from the steady-state equation of the 4WD unit as follows. Neglecting the derivative terms, equation (2) becomes

K_{t} i_{ss} - T_{f} - T_{m} = 0

(13)

where $i_{ss}$ is the motor current at steady state.

From equation (3), the demanded motor torque to supply the demanded clutch pressure is obtained as

T_{dmd} = T_{m} + T_{f} = \frac{D P_{dmd}}{η_{m} (P_{dmd}, ω_{ss})} + T_{f}

(14)

where $T_{dmd}$ is the demanded torque to supply the demanded pressure, $P_{dmd}$ is the demanded pressure, and $ω_{ss}$ is the motor speed used to generate the demanded pressure at steady state. For a given demanded pressure, the motor speed is obtained from the pressure–motor speed map (Figure 5), and $i_{ss}$ is expressed in terms of equations (13) and (14) as follows

i_{ss} = \frac{T_{dmd}}{K_{emf}}

(15)

From equations (1) and (15), the feedforward voltage can be obtained as follows

S_{e_FF} = R_{c} i_{ss} + K_{emf} ω_{ss}

(16)

where $S_{e_FF}$ is the feedforward input voltage.

Performance of the feedforward control was investigated by simulation. For the simulation, a co-simulator that consists of a 4WD unit model was developed using AMESim and a control algorithm based on MATLAB/Simulink. Figure 6 shows the simulation results for a stepwise input of the demanded pressure. The simulation results showed that the actual clutch pressure follows the demanded pressure at a steady state. The clutch pressure begins to increase when the clutch piston reaches the kissing point. In addition, the time to reach the kissing point increases as the demanded pressure decreases, and the rising time to the target pressure becomes faster as the demanded pressure increases. This is because the motor driving voltage (i.e. the motor speed) decreases as the demanded pressure decreases, which requires more time to build up the pressure. From the simulation results in Figure 6, it is apparent that the rising time to the demanded pressure is about 500–600 ms, which is much longer than the design specification of 200 ms.

Figure 6.

Feedforward co-simulation results for step input.

Feedback control

Using the feedforward input voltage, $S_{e_FF}$ , we can reach the target demanded pressure at steady state. However, we cannot guarantee the transient response of the pressure, as shown in Figure 6. In the clamping pressure control of the 4WD unit, it is also important to satisfy the transient characteristics such as the pressure rising time. Therefore, feedback control is required.

When the clutch plates come into contact, deformation occurs due to the clutch plate stiffness. Since this deformation is very small, the corresponding piston acceleration can be neglected after the clutch plates come into contact. Neglecting the acceleration term in equation (11) gives

x = A \frac{1}{(k_{r} + k_{c})} P

(17)

Differentiating equation (17) gives the following relationship

v = A \frac{1}{(k_{r} + k_{c})} \overset{\cdot}{P}

(18)

From equations (9) and (18), the clutch pressure P can be represented as

(\frac{(V_{0} + Ax)}{β} + A^{2} \frac{1}{(k_{r} + k_{c})}) \overset{\cdot}{P} = η_{v} D ω - R_{L} P

(19)

A Laplace transformation of equation (19) gives the transfer function for $P (s) / ω (s)$ as

P (s) = \frac{\frac{η_{v} D (k_{r} + k_{c}) β}{((V_{0} + Ax) (k_{r} + k_{c}) + A^{2} β)}}{s + \frac{R_{L} (k_{r} + k_{c}) β}{((V_{0} + Ax) (k_{r} + k_{c}) + A^{2} β)}} ω (s) = \frac{a_{2}}{s + a_{1}} ω (s)

(20)

It is seen from equation (20) that the system pressure is represented as a first-order transfer function of the motor speed. To obtain the pressure, the coefficients $a_{1}$ and $a_{2}$ of the transfer function need to be determined. However, $a_{1}$ varies depending on the present piston displacement $x$ , and $a_{2}$ varies depending on the present piston displacement $x$ and the volumetric efficiency, which itself is a function of the pressure and pump speed. Hence, it is very difficult to determine the coefficients $a_{1}$ and $a_{2}$ for a given operating condition.

Therefore, in this study, instead of using the exact transfer function of equation (20), we propose a new concept of modified motor speed which can imitate the pressure response characteristics using adaptive transfer function as

P (s) ≅ ω_{\mod} (s) = H (s) ω (s), H (s) = \frac{α}{s + α}

(21)

where $ω_{\mod}$ is the modified motor speed. The adaptive transfer function $H (s)$ was obtained from experiments.

Experimental results are shown in Figure 7 for the stepwise input of the motor voltage from 1.5 to 5.2 V when the demanded pressure changes from 1 to 14 bar. The motor speed, $ω$ , and modified motor speed, $ω_{\mod}$ , are the same (680 r/min) at steady state. This is the required motor speed for the demanded pressure (14 bar) at steady state from equation (12). Figure 7(c) shows the experimental results of $ω_{\mod}$ for various $α$ values in $H (s)$ . As shown, the response speed of $ω_{\mod}$ becomes slower as $α$ becomes smaller. It is noted from Figure 7(c) that the modified motor speed, $ω_{\mod}$ , shows a similar response to the actual pressure when $α$ is 3.

Figure 7.

Experimental results of the 4WD unit for various $α$ values when the pressure difference is 14 bar.

Experimental results are shown in Figure 8 for the stepwise input of the motor voltage from 2.5 to 5.2 V. The target pressure is the same as the previous experiment in Figure 7, but the pressure difference, $Δ P$ , between the initial and final values was reduced from $Δ P = 13 bar$ to $Δ P = 8 bar$ . Figure 8(c) shows that the modified motor speed with $α = 3.5$ closely describes the pressure response. This means that different $α$ values are required for the same target pressure when the pressure difference $Δ P$ varies.

Figure 8.

Experimental results of the 4WD unit for various $α$ values when the pressure difference is 8 bar.

Since $Δ P$ can be represented as $Δ ω$ , which is the speed difference between the target motor speed and initial motor speed at steady state, $α$ was obtained from the experiment by varying $P_{dmd}$ and $Δ ω$ .

An $α - map$ constructed based on the experiments is shown in Figure 9, which also illustrates that $α$ becomes smaller for the same demanded pressure as $Δ ω$ becomes larger. This is because the response speed of the pressure (i.e. $ω_{\mod}$ ) becomes slower as $Δ P (Δ ω)$ becomes larger. In addition, $α$ becomes larger for the same $Δ ω$ as the demanded pressure becomes higher since the response speed of the pressure becomes faster for the same $Δ ω$ as the pressure becomes higher.

Figure 9.

3D map of $α$ under pressure increase.

Similar experiments were performed to construct an $α - map$ when the demanded pressure decreased since the response characteristic of the decreasing pressure was different from that of the increasing pressure.

α-map considering the temperature

When the temperature increases, the oil viscosity decreases and leakage increases, which causes a decrease in the volumetric efficiency. In this case, the pump (motor) speed should increase to supply the demanded pressure, and the speed should decrease when the temperature decreases. Since the α map was constructed for the temperature 30° C, the effect of the temperature change on the modified motor speed needs to be considered both in steady state and in a transient state.

Steady state

The $α$ of the variable transfer function $H (s)$ is determined based on $Δ ω$ and $P_{dmd}$ . From equation (4), the required motor speed varies depending on the volumetric efficiency, $η_{v}$ , which changes depending on the oil temperature. When the volumetric efficiency changes, the motor speed needs to be changed to supply the demanded pressure. This implies that $Δ ω$ varies with temperature.

In this study, a motor speed/demanded pressure map was constructed from the experiments using equations (4) and (12) for the temperature range −20° C to 90° C (Figure 10). For instance, when the pressure increased from 1 to 10 bar $(Δ P = 9 bar)$ at T = 30° C, the motor speed must be increased from 60 to 460 r/min $(Δ ω = 400 r / \min)$ . However, the motor speed should be increased from 125 to 875 r/min $(Δ ω = 750 r / min)$ for the same demanded pressure at T = 60° C. Therefore, to use the $α - map$ in Figure 10, the motor speed difference $Δ ω$ needs to be corrected by considering the effect of the temperature.

Figure 10.

Pressure versus motor speed at steady state for temperatures of −20° C, 30° C, 60° C, and 90° C.

To capture the effect of temperature on the motor speed difference, $Δ ω$ , a weight factor, $W_{1}$ , was introduced as follows

Δ ω_{T} = W_{1} (T) \times Δ ω

(22)

The weight factor, $W_{1}$ , is shown in Figure 11 as a function of temperature. As shown, the corrected motor speed difference, $Δ ω$ , decreases with increasing temperature according to $W_{1}$ .

Figure 11.

Weight factor, $W_{1}$ , to correct $Δ ω$ as a function of temperature.

Transient state

Since the oil viscosity varies depending on the temperature, it affects the transient response of the pressure. To capture this effect, the value of $α$ in $H (s)$ needs to be corrected by considering the temperature as follows

α_{T} = W_{2} (T) \times α

(23)

where $W_{2}$ is the weight factor in the transient state.

The weight factor $W_{2}$ is shown in Figure 12 as a function of temperature. When the temperature increases, the corrected $α$ increases according to $W_{2}$ to ensure the same response speed to reach the demanded pressure.

Figure 12.

Weight factor, $W_{2}$ , to correct $α$ as a function of temperature.

Block diagram of pressure control without a pressure sensor

Figure 13 shows a block diagram of the pressure control developed in this study. In the feedforward loop, the motor speed required to supply the demanded pressure at steady state is calculated for the given demanded pressure and temperature. In addition, the feedforward voltage $(S_{e_FF})$ is determined by considering the mechanical efficiency, $η_{m}$ .

Figure 13.

Block diagram of the pressure control algorithm.

In the feedback loop, $α$ is determined by $W_{1}$ and $W_{2}$ using the actual motor speed and the temperature, which reflect the steady-state and transient characteristics. The feedback voltage $S_{e_FB}$ was obtained through PI control.

Evaluation of the pressure control algorithm using a test bench

Test bench

To evaluate the performance of the pressure control algorithm proposed in this study, experiments were performed for hydraulic oil temperatures of 30° C, 90° C, and −20° C. A test bench is shown in Figure 14. The 4WD unit was installed inside the temperature chamber. The temperature inside the chamber was maintained at the demanded value by the temperature control panel. The motor speed was measured by the encoder.

Figure 14.

Test bench.

Evaluation of the pressure control algorithm at 30° C

Test results for the stepwise input are shown at a temperature of 30° C in Figure 15. In response to a stepwise change in the demanded pressure, the input voltage (c) showed an overshoot and undershoot before reaching the demanded pressure. The actual motor speed (b) also showed an overshoot and undershoot to supply the hydraulic oil flow required to generate the stepwise pressure. The modified motor speed (b) showed a stepwise shape following the demanded pressure. The actual pressure (a) followed the demanded pressure closely and showed a much smaller overshoot and undershoot compared with the actual motor speed. The average pressure error at a steady state and the rising time were 3.63% and 174 ms for the test in Figure 15, respectively, which are within the design specifications of 5% and 200 ms. The small oscillation of the actual pressure at steady state is due to the fluctuating motor speed. Equation (12) indicates that the pressure response is determined by the oil flow rate, which varies depending on the motor speed and leakage. Since the oil leakage at 30° C is relatively small, the relatively large variation in the motor speed causes a pressure fluctuation.

Figure 15.

Performance of the pressure control algorithm for a stepwise input at 30° C.

The vehicle launching test results are shown in Figure 16 at a temperature of 30° C. The pressure measured from the vehicle launching test was applied as the demanded pressure. As shown, the actual pressure closely followed the demanded pressure. When the demanded pressure changed in a stepwise manner at the starting point, the actual motor speed and input voltage showed an overshoot before reaching the demanded pressure. It is also noted that the modified motor speed showed a shape similar to the demanded pressure.

Figure 16.

Performance of the pressure control algorithm for launching at 30° C.

Evaluation of the pressure control algorithm at 90° C

Test results are shown for a temperature of 90° C in Figure 17. As shown, the actual pressure (a) closely followed the demanded pressure to ensure a stepwise change in the demanded pressure. The input voltage at 90° C is higher than that at 30° C. Accordingly, the actual motor speed was higher than that at 30° C. This is because the leakage in the hydraulic system increased as the volumetric efficiency decreased at high temperature. We also noted that the amplitude of the pressure fluctuation at steady state was reduced compared with that at 30° C since the motor speed variation was reduced at high temperature. The modified motor speed (b) showed a stepwise pressure shape following the demanded pressure. The average pressure error at steady state and the rising time were 2.46% and 159 ms for the test in Figure 17, respectively, which are within the design specifications.

Figure 17.

Performance of the pressure control algorithm for a stepwise input at 90° C.

Test results for vehicle launching are shown at a temperature of 90° C in Figure 18. As shown, the actual pressure closely followed the demanded pressure. When the demanded pressure changed in a stepwise manner at the starting point, the actual motor speed and input voltage showed an overshoot before reaching the demanded pressure. The modified motor speed showed a shape similar to the demanded pressure.

Figure 18.

Performance of the pressure control algorithm for launching at 90° C.

Evaluation of the pressure control algorithm at −20° C

Test results at a temperature of −20° C are shown in Figure 19. The actual pressure (a) followed the demanded pressure with some differences. As shown, the pressure difference increased as the demanded pressure decreased. We also noted that the motor speed was very small (at about 23 r/min) for the demanded pressure of 6 bar. This is because a small amount of oil flow is enough to generate the demanded pressure since the leakage is very small at extremely low temperatures (equation (12)). The average pressure error at steady state and the rising time were 9.17% and 130 ms, respectively, for the test shown in Figure 19. When the temperature decreased, the oil viscosity increased and the leakage decreased. Therefore, the demanded pressure was generated by a relatively low motor speed, and it is difficult to obtain an accurate pressure since the pressure changes significantly based on small variations in the motor speed (oil flow rate).

Figure 19.

Performance of the pressure control algorithm for a stepwise input at −20° C.

The test results for vehicle launching are shown in Figure 20 at a temperature of −20° C. The actual pressure followed the demanded pressure, but there is a relatively large pressure error due to difficulties in pressure control at low speed.

Figure 20.

Performance of the pressure control algorithm for launching at −20° C.

From the test results in Figures 15 –20, it was found that the pressure control algorithm proposed in this study showed a good tracking performance at normal operating temperatures with an acceptable steady-state error and transient rising time. Compared to the pressure prediction method in the previous works, the pressure control algorithm can predict the actual pressure for a wide range of the operating temperature with an average error of 3.63% at 30° C and 2.46% at 90° C even if the pressure error increased to 9.17% at extremely low temperature. In contrast, few works have been done with detail description on how to consider the effect of temperature.

Conclusion

A pressure control algorithm without a pressure sensor was proposed for a 4WD unit. To develop the control algorithm, dynamic models of the 4WD unit were obtained, including the motor, pump, and clutch. In the pump model, the mechanical and volumetric efficiency were considered.

For feedforward control, the motor input voltage generated the demanded pressure at steady state. To fulfill the transient response characteristics such as pressure rising time, a first-order adaptive transfer function of the modified motor speed was proposed in order to match the response of the motor speed with the response of the pressure, which was implemented for feedback control. Experiments showed that the modified motor speed can closely simulate the transient pressure response. The coefficient $α$ of the adaptive transfer function was obtained from the experiments. In addition, since the transient response of the pressure varied depending on the pressure difference and temperature, weight factors $W_{1}$ and $W_{2}$ were introduced. The performance of the proposed pressure control algorithm without a pressure sensor was evaluated through experiments at various operating temperatures. We found that the actual pressure followed the demanded pressure with an acceptable error at steady state and within an acceptable rising time in a transient state. The relatively large pressure error at extremely low temperatures (−20° C) was due to difficulty with pressure control since the pressure varied significantly based on small oil flows.

We expect that the pressure control algorithm developed in this study can be used for a 4WD unit without a pressure sensor and can assure a robust system in the case of pressure sensor failure in existing pressure feedback systems.

Footnotes

Handling Editor: Elsa de Sa Caetano

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hyunsoo Kim

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