J. Cent. South Univ. (2017) 24: 1909-1921

DOI: https://doi.org/10.1007/s11771-017-3598-2

Pressure-tracking control of a novel electro-hydraulic braking system considering friction compensation

YONG Jia-wang(雍加望)^{1, 2}, GAO Feng(高峰)¹, DING Neng-gen(丁能根)¹, HE Yu-ping³

1. School of Transportation Science and Engineering, Beihang University, Beijing 100191, China;

2. Beijing Key Laboratory for High-efficient Power Transmission and System Control of New EnergyResource Vehicle, Beijing 100191, China;

3. Department of Automotive, Mechanical and Manufacturing Engineering, University of Ontario Institute of Technology, Oshawa L1H 7K4, Canada

Central South University Press and Springer-Verlag GmbH Germany 2017

Abstract: This work presents an integrated pressure-tracking controller for a novel electro-hydraulic brake (EHB) system considering friction and hydraulic disturbances. To this end, a mathematical model of an EHB system, consisting of actuator and hydraulic sub-systems, is derived for describing the fundamental dynamics of the system and designing the controller. Due to sensor inaccuracy and measurement noise, a Kalman filter is constructed to estimate push rod stroke for generating desired master cylinder pressure. To improve pressure-tracking accuracy, a linear friction model is generated by linearizing the nonlinear Tustin friction model, and the unmodeled friction disturbances are assumed unknown but bounded. A sliding mode controller is designed for compensating friction disturbances, and the stability of the controller is investigated using the Lyapunov method. The performance of the proposed integrated controller is evaluated with a hardware-in-the-loop (HIL) test platform equipped with the EHB prototype. The test results demonstrate that the EHB system with the proposed integrated controller not only achieves good pressure-tracking performance, but also maintains robustness to friction disturbances.

Key words: electro-hydraulic brake; brake-by-wire; Kalman filter; sliding mode control; pressure-tracking; friction compensation

1 Introduction

To date, most road vehicles still use conventional brake systems with vacuum boosters. There exist three distinct disadvantages of the conventional brake systems: 1) a combustion engine is required to drive the vacuum booster, and for electric vehicles, motors and vacuum pumps are needed to provide vacuum source, thereby causing the brake system bulky and expensive; 2) active brake assist, a new safety technology for providing braking support without depressing the brake pedal, can’t be conveniently utilized during emergency situations; 3) friction brakes will produce braking forces as long as the driver depresses the brake pedal. Conventional brake systems are not suitable for new energy vehicles because of the first disadvantage [1]. The second disadvantage makes conventional brake system inapplicable for advanced chassis control systems, such as anti-slip regulation (ASR), electronic stability control (ESC), and adaptive cruise control (ACC), etc. Vehicles with ASR and differential braking based ESC can realize active brake function using a hydraulic control unit (HCU) of these active control system. However, the active brake functionality is not available for some situations, such as ACC, long slope condition etc., due to the fact that the solenoid valves are not suitable for long time continuous work [2]. In electric vehicle (EV) and hybrid electric vehicle (HEV), braking forces are coordinated between regenerative braking and frictional braking [3], but a conventional brake system cannot be implemented to coordinate with the regenerative braking and frictional braking because of the third disadvantage [4-6].

Compared with traditional braking systems, a brake-by-wire (BBW) system has a faster dynamic response and is more suitable for applications that facilitate regenerative braking [7, 8]. The BBW system also has a pedal simulator, and the brake pedal feeling can be designed using different pedal simulator parameters. In addition, the brake pedal stroke and force will not fluctuate even with the intervention of an anti-lock braking system (ABS). Usually, electro- hydraulic braking (EHB) system and electro-mechanical braking (EMB) system are the two main categories of BBW systems. Majority of the traditional braking components can be adapted to develop a EHB system, including disk or drum brakes, hydraulic pipes, solenoid valves, etc.. As a consequence, the EHB system is a good compromise between the requirement for BBW and the research and development cost. Compared with the EHB system, an EMB system has a faster response dynamic response, and is more compact. However, the major redesign of traditional braking components for the EMB system will be expensive, and their fail-safe function has always been a concern.

Many studies have been carried out to improve the control performance of the EHB system, and the controller should have two features: 1) the response time of pressure generation should be faster than the traditional braking system; 2) it is able to robustly regulate the braking forces (pressure-tracking). A new kind of EHB system has been developed in this work. Although the brake pressure buildup is supplied by a high pressure accumulator for the typical EHB system, the developed EHB system generates brake pressure in the master cylinder directly by a motor and a transmission. For this kind of EHB system, the master cylinder piston does not move until the motor torque reaches the static friction torque, and the friction force varies with velocity when the piston moves. As a major disturbance source of the EHB system, the friction force variation in the EHB actuator may increase the steady- state tracking error. The friction force changes are usually attributed to variations in lubricant, temperature, wear, etc. [9]. Consequently, to achieve high-precision motion control, the nonlinear dynamic features of the friction cannot be ignored. Mathematical models to capture the steady-state friction characteristics and model-based friction compensation techniques have been proposed [10-12] and widely used in mechanical system control. These models are able to predict and compensate the friction forces only when the system is under steady-state operation conditions. However, these models are not applicable for a system under transient dynamic operating conditions.

Several dynamic friction models and adaptive friction compensation methods have been presented for handling the unmodeled disturbances. FEEMSTER et al [13] and CANUDAS et al [14] presented two adaptive controllers for a reduced-order friction model. The friction model considers static, Coulomb, viscous and Stribeck friction effects. The first adaptive controller was developed to compensate the linear parametric friction for position tracking control. The second one was designed to ensure input-to-state stability for parametric uncertainties based on Lyapunov-based adaptive method, and the Stribeck parameter is assumed known. MARTON et al [15] proposed an iterative algorithm for identification of friction model parameters based on a nonlinear exponential Stribeck friction model. The results show that the proposed method is able to determine the friction model parameters based on velocity and pressure values.

In addition to the friction effects, the hydraulic dynamics due to the master and wheel cylinders is another major disturbance source for high-precision pressure-tracking control. Little research effort has been paid to the hydraulic dynamics in EHB system. This work proposes an integrated controller for a novel EHB prototype considering friction compensation and hydraulic dynamics. To this end, we firstly fabricated the EHB prototype, which is able to realize different working modes, such as brake booster, active braking, backup braking, and regenerative braking. Distinguishing from conventional EHB and EMB systems, the proposed EHB generates pressure in the master cylinder using a direct current (DC) motor and a ball-screw assembly instead of accumulators and pumps.

2 Functionality and implementation of EHB prototype

As shown in Fig. 1, the EHB prototype consists of a brushless DC motor, a ball-screw assembly, a 3-chamber master cylinder, a pedal cylinder, a pedal stroke simulator, solenoid valves, an electric control unit (ECU) and a hydraulic control unit (HCU). The ball-screw assembly is directly attached to the DC motor shaft with splines to enhance the system dynamic response. The ball-screw assembly converts rotational motion into linear motion, thereby forcing the master cylinder piston into reciprocating motion to regulate hydraulic pressure. The EHB system is mainly electrically-driven, and this feature makes the system suitable for most vehicles, especially for electric vehicles. The 3-chamber master cylinder contains a low-pressure chamber and two high-pressure chambers. The low-pressure chamber is connected to the single chamber pedal cylinder through a normally open valve. The two high-pressure chambers of the master cylinder are connected to the HCU, which is used to choose braking wheel cylinders to realize normal braking or ABS control. The pedal stroke simulator is connected to the pedal cylinder via a normally closed valve.

The pedal cylinder and the master cylinder are physically decoupled to realize the following operating modes:

1) Brake booster mode. The driver depresses the pedal to generate a pressure in the low-pressure chamber, and the driver’s effort will be multiplied by the motor. The braking pressure caused by the driver and the motor in the high-pressure chambers is conducted to the wheel cylinders to generate braking forces via the HCU. The pedal feel is adjustable according to the driver's expectation. The pedal feel adjustment is implemented by controlling the motor for providing different booster ratio.

Fig. 1 Configuration of EHB prototype

2) Backup braking mode. The ball nut and the master cylinder piston are physically decoupled. Via the normally open valve, the hydraulic pressure in the pedal cylinder can be conducted to the low-pressure chamber of the master cylinder even in the event of electrical failure.

3) Active braking mode. If the active braking function is requested by other chassis active control systems of the vehicle, the motor will be controlled to push the piston to generate braking pressure in the high- pressure chambers directly. To prevent the low-pressure chamber from forming vacuum, a one-way valve is used to quickly compensate brake fluid to the low-pressure chamber from a reservoir.

4) Regenerative braking mode. Under the regenerative braking or brake-by-wire mode, the normally open valve will be closed, and the normally closed valve will be opened. The pedal simulator receives the brake fluid from the pedal cylinder, and generates pedal feel. If the required deceleration exceeds the regenerative braking limit, the motor will be controlled to push the piston to generate and adjust pressure in the master cylinder to realize additional frictional braking force, and the system is working under coordination between regenerative braking and frictional braking.

The EHB prototype composition is shown in Fig. 2.

3 EHB system modeling

This section describes the mathematical models used for the development of the pressure-tracking controller of the EHB prototype. This work is mainly focused on the pressure-tracking control, and the pressure is generated by the EHB actuator. Thus, we concentrate on modeling the actuator (motor and transmission mechanism) and the hydraulic dynamics of the master cylinder and wheel cylinders. Shown in Fig. 3 is the schematic diagram for modeling the EHB system, where q_m represents the rotor angle, q_p stands for the pedal link rotating angle, T_m is the motor control torque to be discussed in Section 4, T_f is the torque due to friction losses to be introduced in Section 3, T_p is the load torque caused by master cylinder pressure. The dynamic model of the EHB system is developed, and the control-oriented model is generated for the controller design. The linear friction model is then derived for online estimation of the friction force, which will be compensated for improving the pressure-tracking precision.

3.1 EHB system dynamic model

For the purpose of modeling, the EHB system has been divided into the actuator and hydraulic sub-systems. Figure 4 shows the block diagram of the EHB system,consisting of the actuator and hydraulic sub-systems. The actuator sub-system, in turn, consists of the DC motor and ball-screw assembly, and the hydraulic sub-system involves the master cylinder, pipeline, and wheel cylinders.

Fig. 2 Composition of EHB prototype

Fig. 3 Schematic diagram for modeling EHB system

Fig. 4 Block diagram of EHB

3.1.1 Actuator sub-system model

The governing equation of motion for the actuator can be expressed as

                    (1)

where J_e stands for the equivalent total moment of inertia of the motor and the transmission mechanism, and B_m denotes the motor damping coefficient.

Considering the motion of the 3-chamber master cylinder, we have

                          (2)

                     (3)

                               (4)

where m_m, x_m and A_m represent the mass, the stroke, and the cross section area of the 3-chamber master cylinder piston, respectively, F_m denotes the applied force on the piston, p_m is the hydraulic pressure in the high-pressure chamber of the cylinder, η_h and P_h are the transmission efficiency and lead of the ball-screw assembly, respectively.

3.1.2 Hydraulic sub-system model

The dynamics of the HCU valves is ignored due to the valves’ much faster dynamic compared against that of the mast and wheel cylinders. The hydraulic pressure in the high-pressure chambers of the master cylinder can be derived following the principle of mass conservation as [16]

                       (5)

where V_m=(L_m-x_m)A_m represents the volume of the 3-chamber master cylinder, L_m defines the maximum stroke of the cylinder, β denotes the bulk modulus of the brake fluid, q_m stands for the volumetric flow rate running out of the 3-chamber master cylinder.

The governing equation of motion of the wheel cylinder pad can be cast as

                        (6)

where m_w, A_w and x_w stand for the mass, cross section area and stroke of the wheel cylinder piston, respectively; p_w represents the wheel cylinder pressure, and k_w is the equivalent stiffness of the pad and the clamp.

If the inertial forces of the wheel cylinder pads can be ignored, Eq. (6) can be reduced to

                               (7)

With the above assumption, the dynamics of the wheel cylinder can be modeled as

                      (8)

where V_w=A_wx_w represents the volume of the wheel cylinder; q_w is the volumetric flow rate entering the wheel cylinder. Usually, and Eq. (8) can be rewritten as

                                (9)

For circular pipeline, the hydraulic drop in the pipeline can be expressed as [16]

                       (10)

                            (11)

                                (12)

where K_mw represents the laminar flow coefficient [16]; μ_p defines the brake fluid viscosity; L_p and d_p denote the length and diameter of the pipeline, respectively. The EHB system modeled in Eqs. (1)-(12) is used for dynamic simulation, while the following control-oriented EHB system model is applied to derive the pressure- tracking controller.

3.2 Control-oriented EHB system model

For easily analyzing the dynamics of the EHB system, Eqs. (1)-(12) can be rewritten in the state-space form with the state variable vector defined as

                        (13)

where

          (14)

Let

                       (15)

Combining Eqs. (14) and (15), we have

                          (16)

                                  (17)

Combining Eq. (16) and Eq. (17) yields

                  (18)

With Eq. (16) and Eq. (18), we can determine the pressure in the high-pressure chambers of the master cylinder as

        (19)

Let

         (20)

Equation (19) can be rewritten as

                       (21)

3.3 Friction model for actuator

The pressure-tracking performance is negatively influenced by friction force arising from the contact surfaces of the actuator. The friction is a complicated and nonlinear phenomenon. In this work, we mainly focus on the static, Coulomb, viscous and Stribeck friction effects of the EHB actuator [17-19].

A simplified friction model results from the Tustin model, which can be expressed as [20]

(22)

where F_f represents the total friction force; f_c denotes the Coulomb friction force; f_s stands for the maximum static friction force; f_v is the viscous friction coefficient; is the piston (ball nut) velocity; and v_s is the Stribeck speed.

The Tustin friction curve is shown in Fig. 5, which may be simplified with four lines. The lines L_p1 and L_p2 intersect at the point (v_pm, F_f(v_pm)), the lines L_n1 and L_n2 intersect at the point (v_nm, F_f(v_nm)). The slopes of the fourlines can be determined by , respectively. Together with (0₊, F_f (0₊)), (v_pmax, F_f (v_pmax)), (0_-, F_f(0_-)), (v_nmax, F_f (v_nmax)), we can describe the four lines as follows

                 (23)

Fig. 5 Tustin friction curve

       (24)

                 (25)

         (26)

Let

(27)

Combining Eqs. (3), (13), (23)-(27), yields

    (28)

From Eq. (3), the required motor torque for compensating friction force can be calculated as

                  (29)

where T_q(x₁, x₂) denotes the friction modeling errors, which relates to the position and velocity of the motor rotor. It is bounded as

                              (30)

4 Integrated pressure-tracking controller

Shown in Fig. 6 is the diagram of the integrated controller, the design objective of the integrated controller is to find a control scheme for generating a motor control torque, which drives the piston to make the hydraulic pressure x₃ converge to the desired one x_3d as fast as possible. To this end, we shall identify a desired smooth pressure trajectory. The push rod stroke directly relates to the master cylinder pressure. The push road stroke is proportional to the pedal stroke, since the push rod and the pedal are fastened by means of a clevis. Note that the ratio of the brake pedal stroke to the push rod stroke for the EHB prototype examined the HIL tests is 3.4. The pedal angle measured by a low-cost angular sensor is not smooth due to sensor inaccuracy and measurement noise. To address this problem, a Kalman filter (K_F module shown in Fig. 6) is designed for real-time estimation of the pedal angle. Thus, the desired pressure may be generated according to the relation between the push rod stroke and master cylinder pressure (D_P module). To overcome the unmodeled friction disturbances, a sliding mode controller is proposed to guarantee a well pressure-tracking performance.

Fig. 6 Block diagram of integrated controller

4.1 Kalman filter design

For a linear system, the state and measurement equations of the Kalman filter model can be cast as [21]

                      (31)

where is system state vector for its discrete form; u_k is a known target system input; y_k is the measurement output, w_k~N(0, S_w) and z_k~N(0, S_z) are the process and measurement noise, respectively, w_k and z_k are assumed to be mutually independent. If we define u_k as target pedal acceleration, y_k as measured pedal stroke, T_s=0.001 s as sample time, the stroke and velocity of the pedal can be expressed as

                  (32)

                          (33)

Thus, the state transition matrix for its discrete form can be expressed as

                              (34)

With Eq. (31), the optimal estimation for the system states is determined as [22]

               (35)

where and are the predicted and corrected system state estimates, respectively; and are the predicted and corrected system state error covariance; accordingly, K_k+1 is the Kalman gain. Pedal stroke data sets are measured when the pedal is depressed at constant speed (u_k=0), and then the variances of w_k and z_k have been calculated. After sufficient iterative processing using real experimental data, the values of P_k and K_k converge to the constants as follows

                     (36)

4.2 Desired master cylinder pressure

The relationship between the master cylinder pressure and the push rod stroke (P-S characteristic) can be determined when a braking system is designed and installed. For measuring the P-S characteristic, a refitted experimental vehicle is used, and the relationship between pedal force and pedal stoke is also measured for designing the pedal stroke simulator. The measured P-S characteristic is shown in Fig. 7. The length of the brake idle stroke, which is caused by the master cylinder idle stroke and the air gap between clamp and disc, is 2.8 mm. After the idle stroke is compensated, the pressure and stroke data can be fitted quite well with a quadratic curve. The desired master cylinder pressure can be expressed as a function of the push rod stroke as

Fig. 7 Master cylinder pressure vs push rod stroke

    (37)

4.3 Robust controller

The sliding mode control (SMC) method selected is a nonlinear control technique to drive the system states onto a sliding surface. The resulting control system is robust and insensitive to parameter variations and external disturbances. The SMC technique has been proved to be applicable to a wide range of problems in electric drives, vehicle dynamic control, and robot motion control [23]. Therefore, a sliding mode controller is proposed to improve the pressure-tracking performance. Note that the system is of second order according to Eq. (14), and the number of the relative degrees is required to be one for attenuating the impact of the chattering phenomenon. Therefore, the sliding surface s is defined as

                    (38)

where x_3d is the desired master cylinder pressure, which is determined by the push rod stroke; l is a positive constant. We define as the pressure tracking error, it is obviously that if s is maintained small or is adjusted to converge to zero, then the pressure tracking error e will be small or converge to zero sincethe transfer function is stable,which also means that x₃ will be converged to the desired master cylinder pressure. The Lyapunov function is defined as

                               (39)

Differentiating Eq. (39), with Eq. (21), we have

 (40)

Substituting Eqs. (14) and (29) into Eq. (40) yields

     (41)

Let

          (42)

and then Eq. (41) becomes

           (43)

The following inequality is assumed

              (44)

The control law is designed as follows:

         (45)

where K>Γ₀ is assumed, and Eq. (43) can be rewritten as

         (46)

From Eq. (39) and (46), we have

                    (47)

Integrating Eq. (47), yields

                (48)

Substituting Eq. (39) into Eq. (48), we have

                    (49)

It is assumed that the initial tracking error s(0) is finite. Eq. (49) indicates that states will reach the sliding surface s=0 in a finite time, and the convergence rate is ensured. The controller is stable according to Eq. (46).

The sign function in Eq. (45) is discontinuous, and it may cause chattering, thereby resulting in noise, vibration, and energy dissipation. To address the chattering problem as well as to enhance the control scheme, a boundary layer is created, and Eq. (45) is rewritten as

 (50)

where is the boundary layer thickness, and

              (51)

5 HIL test results and discussion

A hardware-in-the-loop (HIL) test platform is developed to identify the model parameters and evaluate pressure-tracking controller performance. The HIL test is conducted on the UOIT vehicle simulator, which was designed to investigate the vehicle-driver-road interactions in a safe and effective way [24, 25]. As shown in Fig. 8, the HIL platform consists of a host PC, EHB prototype, data acquisition system (DAQ), pedal and motor angle sensors, hydraulic pressure sensors, four disc brakes, and pipelines. The EHB prototype is controlled by a low-cost digital signal processor (DSP), Freescale KEA128RM. The DSP samples the pedal and motor angle signals at a rate of 1000 Hz, which is also the execution rate of the proposed controller. Since the sample signals are discrete, an approach of forward difference approximation is used for discretizing the state-space equations, and the motor velocity can be rewritten in difference equation as

                 (52)

where Δ(k) is the approximation error. The proposed controller is discretized and coded using C language in CodeWarrior 10.1 software, and the compiled file is downloaded to the DSP to perform the HIL tests. Table 1 shows the EHB prototype parameters.

Fig. 8 System architecture of HIL platform

Table 1 Parameters of EHB prototype

5.1 Validation of hydraulic sub-system model

HIL tests are conducted to validate the fundamental model introduced and described in Section 3. As shown in Eq. (21), a limited random sequence motor current, which may cause varied motor angular velocity x₂, is used as a control input, and the master cylinder pressure x₃ is the output. Note that the master cylinder pressure is related to the motor velocity,

Figure 9 shows the motor angular velocity caused by the input motor control current. Figure 10 presents the measured master cylinder pressure of the EHB prototype and the estimated pressure derived from the hydraulic sub-system model. It is shown that the estimated result matches the measured one quite well. The test result implies that the proposed control-oriented model well simulates the hydraulic dynamics of the EHB prototype.

Fig. 9 Time history of motor angular velocity

5.2 Friction model parameters identification

To identify the friction model parameters, a PI controller shown in Fig. 11 is proposed for stabilizing the master cylinder piston (ball nut) velocity, and the velocity is permitted to vary from -100 to 100 mm/s with the resolution of 5 mm/s. The friction torque at each velocity is measured for five times, and the average of the measured values is calculated as one measurement point.

Fig. 10 Measured and estimated master cylinder pressure

Fig. 11 PI controller for master cylinder piston velocity

The measured friction torque data for negative and positive piston velocity are shown in Figs. 12 and 13. Four lines are fitted using the method of least squares, and the identified friction model parameters are shown in Table 2. It is shown that simulation result for the proposed friction model with the identified parameters listed in Table 2 attaches the measured data quite well.

Fig. 12 Measured friction torque versus negative master cylinder piston velocity

5.3 Kalman filter validation

Figures 14 and 15 show the time history of the push rod stroke estimated with the proposed Kalman filter and measured in the HIL test at a high and a low braking pedal operating frequency, respectively. The figures also illustrate the time history of the push rod velocity estimated with the Kalman filter. As shown in Figs. 14 and 15, the measurement error may be attributed to noise and sensor resolution. The benchmark discloses that the estimated push rod stroke based on the Kalman filter and the stroke measured in the HIL test achieve an excellent agreement.

Fig. 13 Measured friction torque versus positive master cylinder piston velocity

Table 2 Identified friction model parameters

Fig. 14 Time history of push rod stroke at a high braking pedal operating frequency

Fig. 15 Time history of push rod stroke at a slow braking pedal operating frequency

5.4 Transient dynamic responses of EHB prototype

The time duration required to generate the master cylinder pressure from 0 MPa to 10 MPa is crucial for road vehicle braking systems, and this process takes about 300 ms for traditional vacuum brake systems [26]. To examine the dynamic features of the fabricated EHB prototype and to validate the proposed pressure-tracking controller, we investigate the transient dynamic responses to a rectangular and a step inputs with the HIL platform.

Figure 16 shows the transient dynamic response of the EHB prototype in terms of the time history of the master cylinder pressure under the rectangular impulse input. The time history of the master cylinder pressure indicates how the fabricated EHB prototype would behave when an actual emergency braking occurs. As shown in Fig. 16, only 200 ms is required to raise the master cylinder pressure to 10 MPa from 0 MPa, and the steady-state pressure tracking error is within a reasonable range (less than 0.1 MPa). A small pressure overshoot occurs due to the fast master cylinder piston movement in the dead zone. The result also indicates that theproposed controller is able to control the master cylinder piston to cross the dead zone fast enough, and then track the desired pressure without exhibiting a delay.

Fig. 16 Time history of master cylinder pressure in response to a rectangular impulse input

We investigate the step variation response of the controller to ensure the consistency of step tracking performance. Shown in Fig. 17 is the time history of the master cylinder pressure of the hydraulic sub-system, which exhibits the performance of the proposed controller in response to 1 MPa step variation. It takes about 100ms to reach the reference pressure. The steady-state pressure tracking error is approximate 0, and the root mean square error (RMSE) of the proposed controller is 0.05 MPa, where RMSE is determined as

               (53)

where p_des(t) and p_act(t) stand for the desired pressure and actual measured pressure at the time instant t, accordingly. N is the total number of samples collected.

Fig. 17 Time history of master cylinder pressure in response to 1 MPa step variation

5.5 Performance of pressure-tracking controller considering friction compensation

A skilled driver participated in testing the fabricated EHB prototype with the proposed pressure-tracking controller on the HIL test platform. Similar to the result shown in Fig. 14, the time history of the push rod stroke estimated with the Kalman filter looks like a distorted sinusoidal pattern. With the estimated time history of the push rod stroke, the desired time history of the master cylinder pressure can be derived using the governing equation expressed in Eq. (28). The proposed pressure- tracking controller makes the actual pressure to track the desired one considering the friction variations.

As shown in Fig. 18(a) in terms of the time history of the master cylinder pressures, the measured actual pressure tracks the desired one quite well without significant tracking error throughout the duration of the real-time braking testing. This implies that the proposed control scheme achieves a satisfactory pressure-tracking performance. Shown in Fig. 18(b) is the time history of the pressure-tracking error corresponding the result illustrated in Fig. 18(a). It is observed that the peak tracking error occurs when the ramp switches occur, at which the motor angular velocity changes direction, for example at the time instants of t=1.7 s, t=2 s, etc. Once the motor angular velocity changes direction, it leads to the direction change of the friction force. The friction force direction changes, in turn, may be attributed to the peak tracking errors. Figure 18(c) shows the time history of the motor control and friction compensation torques. Figure 18(d) illustrates the time history of the motor angular velocity. A close observation of Figs. 18(c) and (d) reveals that once the motor angular velocity direction changes, the friction torque (T_f) oscillates, and the proposed controller manipulates the DC motor to generate the motor control torque (T_m) in phase with T_f. Thus, the motor torque manipulated by the controller compensates the friction torque and makes the tracking error within a reasonable range.

To identify the contribution of the friction compensation to the performance improvement of the controller, we tested the controller without considering the friction compensation. The RMSE of the controller without the friction compensation term is 0.86 MPa, whereas that of the proposed controller with the friction compensation term is 0.18 MPa. Compared with the baseline controller without considering the friction compensation, the proposed controller reduces the RMSE (root of mean square steady-state master pressure tracking error) by 79.1%. In other words, the friction compensation term makes the control scheme robust to friction disturbances.

6 Conclusions

An integrated robust controller is proposed for the master cylinder pressure-tracking control of a novel electro-hydraulic braking (EHB) system. In order to capture the dynamic features and identify the system parameters of the EHB system, a sophisticated EHB system model is generated, which consists of an actuator and hydraulic sub-system model. Based on the EHB system model, a second-order control-oriented model is derived to design the proposed controller. For the purpose of increasing the reliability and improving measuring accuracy for the pressure-tracking control, a Kalman filter is designed to estimate the time-varying brake pedal stroke. Since the brake pedal and the master cylinder push rod are fastened with a clevis, we can determine a smooth time-varying push rod movement trajectory based on the estimated time-varying brake pedal stroke. Built upon the resulting time-varying push rod movement trajectory, the desired master cylinder pressure trajectory is generated for pressure tracking control. A linear friction model for the EHB actuation system is constructed for online friction torque estimation. The linear friction model is derived by linearizing the nonlinear Tustin friction model, and the parameters of the linear model are identified with real-time simulation. It is assumed that the friction model error is unknown but bounded, and a sliding mode controller is designed for coping with the friction disturbances. A hardware-in-the-loop (HIL) test platform is developed to identify the EHB model parameters and to evaluate the performance of the proposed integrated controller.

Fig. 18 Testing result of EHB prototype with pressure-tracking controller in terms of time history:

HIL testing result shows that the EHB system model is able to capture the fundamental dynamics of the EHB prototype. It is demonstrated that the designed Kalman filter is able to reliably and accurately estimate the required state variables in the presence of measurement noises. The proposed controller is able to control the master cylinder piston to cross the dead zone fast enough, and then track the desired pressure without exhibiting a delay. Finally, the effectiveness and pressure-tracking performance of the controller is demonstrated and evaluated. Considering the friction torque compensation, the proposed controller can significant reduce the pressure-tracking error, thereby exhibiting the controller’s robustness to friction disturbances. The eventual objective of the proposed pressure-tracking controller and the fabricated EHB prototype is to enhance vehicle stability. This goal can be realized by determining desired master cylinder braking pressure to reduce wheel slip under emergency conditions and by distributing the braking pressure to individual wheel to produce required yaw moment. This may lead to the application of low-cost ABS and electronic stability systems. To this end, in the near future, the fabricated EHB prototype with the proposed controller will be further explored using numerical simulation and will be further validated through in-vehicle field and road tests.

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(Edited by DENG Lü-xiang)

Cite this article as: YONG Jia-wang, GAO Feng, DING Neng-gen, HE Yu-ping. Pressure-tracking control of a novel electro-hydraulic braking system considering friction compensation [J]. Journal of Central South University, 2017, 24(8): 1909-1921. DOI: https://doi.org/10.1007/s11771-017-3598-2.

Foundation item: Projects(51405008, 51175015) supported by the National Natural Science Foundation of China; Project(2012AA110904) supported by the National High Technology Research and Development Program of China

Received date: 2016-10-17; Accepted date: 2016-12-08

Corresponding author: DING Neng-gen, Associate Professor, PhD; Tel: +86-10-82338040; E-mail: dingng@buaa.edu.cn