Modeling and pressure tracking control of a novel electro-hydraulic braking system

Abstract

A brake-by-wire system is considered the best solution to realize advanced functions in electric vehicles/hybrid electric vehicles and intelligent vehicles. In this article, a novel electro-hydraulic brake system is proposed. The main issues in the development of the proposed electro-hydraulic brake system focus on hydraulic loop design and pressure tracking control. Instead of an accumulator, a large displacement piston pump coupled to a high-power motor is applied to meet the demand of pressure boost. A passive pedal feel simulator is designed to decouple the brake pedal and the wheel cylinders. The high pressure in the wheel cylinders is controlled by de/activating the motor and valves to track the reference pressure. To minimize the pressure instability observed in previous proportional–integral–derivative controller tests, a self-tuning generalized predictive controller is applied. Models of each system’s dynamic process are derived, and the parameters are identified through test data. A sliding mode differentiator is used to filter the sensor signals and improve the pressure boost performance in hard brake maneuvers. A rapid control prototype test environment based on dSPACE is built to verify and adjust the controller. The simulation results using MATLAB/Simulink and bench tests are presented and analyzed. Tracking performances show that compared to a proportional–integral–derivative controller, the proposed controller has a better effect in lowering pressure overshoots and start–stop frequency.

Keywords

Electro-hydraulic braking system system identification Levant differentiator generalized predictive control rapid control prototype

Introduction

Brake-by-wire (BBW) systems have the inherent advantages of being small in size and flexible for brake force control. When the pedal feel simulator (PFS) decouples the drivers’ feet and the wheels, BBW systems precisely manage the brake force of each wheel without disturbing the pedal feel. With the development of hybrid electric vehicles (HEVs), electric vehicles (EVs), and intelligent vehicles, brake systems are required to offer advanced functions, such as regenerative braking (RB), active safety, and autonomous drive; BBW systems are widely considered the best option to meet these demands.

Generally, BBW systems are classified into two groups: electro-hydraulic brake (EHB) and electro-mechanical brake (EMB) systems. Compared with EMB, EHB systems have a simpler structure and smoother pressure transmission. In support of current motor and valve technologies, EHB could have a comparatively low development period and provide profits more quickly.¹

The electronically controlled brake (ECB^®) by Advics is the earliest EHB production case; it was initially developed for the Toyota Prius and Lexus hybrid models.² Around the same time, Daimler–Benz implemented another EHB, the sensotronic brake control (SBC^®) by Bosch, into its W211, R230, and W219 models.³ However, this type of EHB did not appear in the next generation of Benz models because of reliability considerations. In recent years, auto part suppliers have invented ball-screw (or worm-gear) EHBs. Typical prototypes include iBooster^® by Bosch, MK-C1^® by Continental Teves, IBC^® by ZF-TRW, and ESB^® by Honda. Among these prototypes, ESB is already in small-scale production and has been used to coordinate with Honda’s HYBRID SH-AWD technique in some high-performance models, such as the NSX and MK-C1, and it has been implemented in Alfa Romeo models.

However, the market share of EHB has not obviously increased in the past decade. Instead, the electric vacuum pump (EVP) solution has been widely applied in current EV/HEVs. Although EHB has a huge marketing potential, currently, the cost, reliability, and brake force control difficulty impede its development.

In Reuter et al.,⁴ the authors compared Delphi’s EHB with conventional vacuum-booster brake systems and highlighted various valuable design options. In Velardocchia,⁵ a methodology based on hardware-in-loop (HIL) was illustrated to evaluate the dynamic characteristics of EHB components. Beyond these examples, the works of many other researchers have inspired our team to design a low-cost EHB. Different from the cylinder volume-change concept of ball-screw (or worm-gear) EHBs, the flow-change concept is adopted in our proposed EHB configuration. An analogous concept was used in an autonomous vehicle application.⁶ In our proposed EHB, no volume recovery time (turn back screw) and infinite fluid flow in the loop could provide continuous and stable pressurization.

Non-model proportional–integral–derivative (PID) controllers have been adopted to address pressure tracking issues in many similar studies, whereby simulations and HIL tests have verified the effectiveness of the controllers.^7,8 Those results, although attractive, had deficiencies in that the simulations and bench tests did not consider humidity, pad wearing and temperature, and other uncertain factors. In fact, variation of parameters is the main cause of delay and poor control accuracy. Because of the feedback structure of normal PIDs, the effects of controller outputs always fall behind the delays, including hydraulic delay, electro-mechanical delay, signal transmitting, and processing delay. Larger overshoots lead to increases in both adjustment times and energy consumption. There is still a demand for improving the pressure control precision. In Todeschini et al.,⁹ a hybrid control based on a position-pressure map was described to address the huge overshoot caused by dead-zone delay. In Line et al.,¹⁰ predictive control was used to improve the utilization of motor torque in EMB force control. In particular, in the proposed EHB, when the motor began to work, the rotor would remain statically until the increasing current overcame the friction in the piston. This effect leads to a slightly longer time delay compared to that of other EHBs. To achieve a balance in rapidity and accuracy, in this article, a generalized predictive control (GPC) method is adopted to optimize the pressure tracking performance. GPC is suitable for this EHB system because (1) it has complete theoretical support for linear single input single output systems and effective means to solve the objective min J with variable constraints; (2) it is able to estimate future system responses and compute the corresponding future controller outputs to eliminate the overshoots caused by the delay; (3) in this study, the GPC controller uses the extended auto-regressive (ARX) model, which contains delay and disturbance, and can be feasibly identified and self-tuned online.¹¹

The rest of paper is organized as follows. In section “System descriptions,” we outline the design concept of the proposed EHB system and the rapid control prototype (RCP) test environment. In section “System modeling and identification,” the dynamic models of the pressurizing and depressurizing stages are derived and identified. In section “Pedal movement,” we propose a methodology based on the Levant differentiator (LD) to filter signals and realize the brake assist (BA) function. In section “Controller design,” self-tuning GPC laws are designed to improve the pressure tracking performance under simulation. The proposed controller is experimentally validated in section “Test verification of the pressure tracking,” and our final conclusions are given in section “Conclusion.”

System descriptions

Hydraulic loop design

Figure 1 shows the self-designed prototype. The components, such as the motor pump, valves, and main cylinder, were integrated into one block. A schematic portrait of the overall system is shown in Figure 2. Under normal brake situations, the system is pressurized by a high-power direct current (DC) motor coupling with a large displacement pump and depressurized by the high-speed on/off solenoid reducing valve 6. When the electrical supply is operating, the normally closed valves 1, 3, and 5 are opened, whereas the normally open valves 2 and 4 are closed; as a result, the PFS decouples the low-pressure main cylinder and the high-pressure wheel cylinders. An angle sensor detects the pedal movements and provides a reference wheel pressure for the controller. Pressure sensors located at the wheel cylinders provide feedback signals to the controller. When the electrical supply fails, an alternative hydraulic loop ensures the emergency brake is engaged. By depressing the brake pedal, brake fluid is still able to flow into the wheel cylinders through the main cylinder and valves 2 and 4.

Figure 1.

HCU and PFS prototype.

Figure 2.

Configuration diagram of the EHB system.

An apparent difference between the proposed EHB system and other EHB systems is that the proposed system has no accumulator. In accumulator-type EHBs, the life of the accumulator is a main factor limiting reliability. Instead, in the proposed EHB system, a high-power DC motor (500 W with a 12-V supply) and a four-piston pump are self-designed to meet the pressure boost requirements, that is, a controller is required to determine the motor speed and start/stop timing. Similar to anti-lock brake system (ABS), there are two valves for each wheel to regulate the pressure individually. As shown in Figure 3, the vector of the target reference pressure is determined by many factors. The vector contains a comprehensive set of logical operation functions, such as brake pedal movements, electronic brakeforce distribution/Electronic stability control (EBD/ESC), adaptive cruise control (ACC), and RB. In production vehicles, the reference pressure is provided by a vehicle control unit (VCU). The complete algorithm will not be analyzed here. In this article, we focus on the precise pressure control under normal braking situations, that is, the reference pressure is calculated based only on the brake pedal angle sensor signals.

Figure 3.

Operation block diagram of the EHB system.

RCP test environment

An RCP test environment was designed to achieve system identification and control algorithm verification. Figure 4 shows the structural diagram of the test environment. For the software, Controldesk^® serves as a bridge connecting MATLAB and dSPACE Autobox and provides configuration functions, including display, recording, signal transmission, and interface design. Control algorithms are designed in Simulink and then programmed into dSPACE. One angle sensor and four pressure sensors are responsible for signal acquisition. The pulse width modulation (PWM) generator integrated in dSPACE and a circuit board drive the actuators of the EHB. Depending on different experimental purposes, the test modes can be switched between open-loop and closed-loop configurations.

Figure 4.

RCP test environment.

System modeling and identification

GPC controllers are based on reference models. In this section, the model of each dynamic stage is discussed and identified.

Subsystem models

When there is no brake pressure distribution activation or proportional valve, the pump pushes the fluid into the four wheel cylinders. Because the motor and the valve have different control intervals (see the later discussion), for convenience, we use a continuous model to describe the dynamic processes in each subsystem and then separately study each controller of the pressurization and depressurization stages through discrete models. The dynamic equations of each subsystem during the pressurization process are as follows:

Motor

{\begin{matrix} 12 V \cdot u_{m} = R_{m} I_{m} + C_{emf} ω \\ C_{emf} I_{m} = k_{fric} ω + J_{m} \overset{\cdot}{ω} + T_{load} \end{matrix}

(1)

Pump

{\begin{matrix} 4 q_{Ti} = D ω - k_{leak} p_{Ti} \\ T_{load} = \frac{D p_{Ti}}{η_{mech}} \end{matrix}

(2)

Brake tube

{\begin{matrix} p_{Ti} = p_{To} + R_{T} q_{To} \\ q_{To} (t) = q_{Ti} (t - τ_{T}) \end{matrix}

(3)

Brake hose

{\begin{matrix} p_{To} = p_{Ho} + R_{H} q_{Ho} \\ q_{To} = q_{Ho} + H_{h} {\overset{\cdot}{p}}_{Ho} \end{matrix}

(4)

Wheel cylinder

\overset{\cdot}{y} (t) = \frac{β}{V_{w}} \cdot q_{Ho} (t - τ_{gap})

(5)

where, in equation (1), $u_{m}$ is the controller output PWM of the motor, $R_{m}$ is the equivalent resistance of the motor, $I_{m}$ is the current, $ω$ is the angular speed of the motor, $k_{fric}$ is the composite viscous friction coefficient of the rotor and the eccentric sleeve, $J_{m}$ is the inertia moment of the rotor and the connected plungers, $T_{load}$ is the equivalent torque of the load, and $C_{emf}$ is the back electromotive force (EMF) constant. In equation (2), $D$ is the pump displacement, $p_{Ti}$ is the pressure at the pump outlets and the tube inlets, $η_{mech}$ is the mechanical efficiency of the pump, and $k_{leak}$ is the leakage coefficient, which is determined by the Hagen–Poiseuille coefficient of the system. In equation (3), $p_{To}$ is the pressure at the tube outlet, $q_{To}$ is the flow rate of the fluid at the tube outlet, $R_{T}$ is the flow resistance of the tube, and $τ_{T}$ is the flow delay in the tube. In equation (4), $p_{Ho}$ is the pressure at the hose outlet, $q_{Ho}$ is the flow rate of the fluid at the hose outlet, $R_{H}$ is the flow resistance of the hose, and $H_{h}$ is the volume loss in the hose. In equation (5), y is the pressure in the wheel cylinder, $β$ is the bulk modulus of the brake fluid, $V_{w}$ is the maximum volume of the wheel cylinder, and $τ_{gap}$ is the time delay caused by the gap of the actuator.

A high-speed on/off valve, labeled valve 6, is used for depressurizing at 50 Hz, which is an appropriate carrier frequency. The average flow rate of fluid ${\bar{Q}}_{out}$ in one period ( $T_{v} = 20 ms$ ) can be expressed as follows^12,13

{\bar{Q}}_{out} = - C_{v} a_{v} T_{v} \cdot y^{φ_{v}} \cdot u_{v}

(6)

where $C_{v}$ is the throttle coefficient determined by the shape of the orifice, the flow conditions, and the fluid properties; $a_{v}$ is the cross-sectional area of the valve; $φ_{v}$ is the exponential coefficient determined by the shape of the valve orifice; and $u_{v}$ is the linearized controller output of the valve.

Leakage generally exists in high-pressure hydraulic systems. A leakage model should be considered when the system model must accurately simulate real hardware and when a controller is required for better accuracy and robustness. Assuming that there are no leakages from the interior of the hydraulic circuit to the exterior, the total quantity of brake fluid in the system remains unchanged. The leakages occur at the orifices, where a large pressure difference exists between the inlet and the outlet. From equation (2), the flow rate of system leakage $q_{leak}$ at the pressure holding stage can be approximately expressed as follows

q_{leak} = - k_{leak} \cdot y = \overset{\cdot}{y} \cdot \frac{V_{w}}{β}

(7)

Equation can be log-linearized to

\frac{d \ln y}{dt} = - k_{leak} β V_{w}^{- 1}

(8)

System identification

Nonlinear input separation

For a motor pump, there is a memoryless static nonlinear relationship between the actual drive voltage and the output speed corresponding to the inherent load force of friction and the spring in the pump. When the rotor is driven, it does not rotate immediately until the rising current overcomes the load force; as a result, the proposed EHB has a larger delay than accumulator-type EHBs. Similarly, for a valve, both the dead-zone and saturation-zone nonlinear input exist. At small and large PWM inputs, there are nearly no changes in the fluid flow rates. Several tests were performed to estimate the static nonlinear properties; some typical data are shown in Figure 5. As seen in Figure 5(a), when the actual driven PWM input was lower than approximately 0.3, the motor pump did not operate. Moreover, in Figure 5(b), the effective operation range of the valve was narrower, and the flows for large or small driven inputs were almost identical. If we take the initial slope values of the curves as fit points, then we have the approximate static nonlinear properties of two inputs, as shown in Figure 6.

Figure 5.

Pressure responses of different drive inputs: (a) pressurizing and (b) depressurizing.

Figure 6.

Linearization of the inputs: (a) motor and (b) valve.

Here, we use the Hammerstein model to separate the nonlinear input part¹⁴

{\begin{matrix} u_{m, v} (k) = f_{m, v} (v_{m, v} (k)), f_{m, v} (0) = 0 \\ y (k) = \sum_{i = 1}^{n_{a}} a_{i} y (k - i) + \sum_{j = d + 1}^{n_{b}} b_{j} u_{m, v} (k - j) + ξ \end{matrix}

(9)

where $u (k)$ is introduced as a controller output for the linear dynamic part and the actual drive PWM $v (k)$ is the result of $f^{- 1} (u (k))$ .

Delay estimation

Pressure delay results from mechanical inertia, inherent forces of the pump, fluid delay, filtering processes, and other factors. A closed-loop test with a normal PID controller is used to measure time delay. The results reveal the problem caused using a non-model PID in non-minimum phase systems. As shown in Figure 7, the time delay was approximately 0.05 s for pressurization and 0.03 s for depressurization. Before 0.7 s, the pressure in one wheel cylinder was stable and tracked the reference well. A disturbance of the angle sensor signal, that is, a disturbance of the reference pressure, triggered motor activation. After the first increasing overshoot, valve 6 opened, and then the second decreasing overshoot arrived soon afterward. Moreover, for the pressure storage system, the zero-input response of the leakage in the system should also be considered. The above factors should be fully considered in controller design studies.

Figure 7.

System delay estimation in a PID test.

Identification

From section “Subsystem models,” the structures of the transfer functions (without time delay) at different stages can be synthesized:

Pressurizing stage

\frac{Y (s)}{U_{m} (s)} = \frac{b_{0_ctn}}{a_{0_ctn} s^{2} + a_{1_ctn} s + a_{2_ctn}}

(10)

with

b_{0_ctn} = K_{1} D η_{mech} β / 12 V

a_{0_ctn} = T_{m} η_{mech} (4 V_{w} + 4 β H_{h} + c k_{leak})

a_{1_c t n} = η_{m e c h} (4 V_{w} + 4 β H_{h} + β k_{l e a k} T_{m}) + c (D^{2} K_{2} + k_{l e a k} η_{m e c h})

a_{2_ctn} = β (k_{leak} η_{mech} + D^{2} K_{2})

c = V_{w} R_{H} + V_{w} R_{T} + β H_{h} R_{T}

Leakage

\frac{d \ln y}{dt} = - k_{leak} β V_{w}^{- 1}

(11)

Depressurizing stage

\frac{dy}{dt} \cdot y^{- φ} = - C_{v} a_{v} β V_{w}^{- 1} \cdot u_{v}

(12)

In predictive controller design, an accurate structure and the parameters of the model are required. In subsystem modeling, many parameters are difficult to measure. It is impossible to accurately describe the model with subjectively estimated parameter values. Therefore, because the model structures of the system are derived, system identification is aimed at estimating the parameters. There are two types of algorithms: in simulations, the reference models are the result of offline identification, which used the recursive least square (RLS) algorithm; in tests, we use the forgetting factor recursive least square (FFRLS) algorithm to perform online identification. For system identification and GPC controller design, the above continuous models must be discretized. The zero-order-hold (ZOH) algorithm is used to transform the continuous models into discrete ARX models

A (z^{- 1}) y (k) = B (z^{- 1}) u (k - 1) + ξ (k)

(13)

with

{\begin{matrix} A (z^{- 1}) = 1 + a_{1} z^{- 1} + \dots + a_{n_{a}} z^{- n_{a}} \\ B (z^{- 1}) = b_{0} + b_{1} z^{- 1} + \dots + b_{n_{b}} z^{- n_{b}} \end{matrix}

In this application, during the pressurizing stage, $n_{a} = 2$ , $n_{b} = d + 2$ , and $d$ is the integer solution of the time delay divided by the sampling time. The former d terms of $B (z^{- 1})$ are 0. $ξ (k)$ is the unpredictable white noise. Thus, equation (13) can be expressed as a form of least squares

y (k) = φ^{T} (k) θ + ξ (k)

(14)

where $φ (k)$ is the data vector and $θ$ is the parameter vector to be estimated.

The estimated parameter vector $\hat{θ}$ can be recursively computed as follows

{\begin{matrix} \hat{θ} (k) = \hat{θ} (k - 1) + K (k) [y (k) - φ^{T} (k) \hat{θ} (k - 1)] \\ K (k) = \frac{P (k - 1) φ (k)}{1 + φ^{T} (k) P (k - 1) φ (k)} \\ P (k) = λ_{f}^{- 1} [I - K (k) φ^{T} (k)] P (k - 1) \end{matrix}

(15)

The offline identification is a kind of preliminary work: several short-time tests were done; the data of input and pressure were collected, saved, and identified. Because the lengths of data were short, we can set $λ_{f} = 1$ ; $P (k)$ and $\hat{θ} (k)$ are initialized as follows

{\begin{matrix} P (0) = 10 \cdot I \\ \hat{θ} (0) = 0 \end{matrix}

(16)

In actual applications, certain parameters, including bulk modulus, dynamic viscosity, fluid density, and delay, vary slowly in time because they are affected by factors such as environmental temperature, air content of brake fluid, and brake pad wear. GPC controllers are always accompanied by online identifications. Although there are no laws available to predict the parameter variation, the amplitudes of the parameter variation are small. We set λ_f = 0.999, and $\hat{θ} (k)$ are initialized by the previously identified models in memory. For short durations, under the same operating conditions (such as temperature and fluid bulk modulus), although any online identification results have different parameter values, their step response curves below 10 MPa are almost identical. If the system was affected by the change in temperature or other factors, then the FFRLS algorithm would have a strong ability of identifying the changing parameters.

For pressure tracking control, we use 0.01 s control interval in pressurizing GPC controller and 0.02 s in depressurizing controller. It can be sure that the delay time is larger than 0.02 s, if take a larger frequency, the delay compensation length (prediction vector length) would be longer, and the computation would be much more expensive. A set of typical identified models are as follows. For the pressure increasing stage, at a sample time of 0.01 s, we have

A (z^{- 1}) y (k) = B (z^{- 1}) u_{m} (k - 1)

(17)

with

{\begin{matrix} A (z^{- 1}) = 1 - 1.842 z^{- 1} + 0.8452 z^{- 2} \\ B (z^{- 1}) = 0.0466 + 0.0441 z^{- 1} \end{matrix}

For the pressure decreasing stage, based on the later discussed equation (39), at sample time of 0.02 s, we have

(1 - z^{- 1}) \ln y (k) = - 1.1014 \times 10^{- 1} u_{v} (k - 1)

(18)

Figure 8 illustrates a group of response curves of the experimental data and the identified models under the same inputs. The fitting results of the identified models to the unfiltered experimental data were on average 94.8%. We considered the proposed modeling and identification methods to be effective. The identified models accurately represented the actual system and were used as a reference models in later simulations.

Figure 8.

Result of EHB system identification.

Pedal movement

Theoretically, because the pedal and wheel cylinders are decoupled by the PFS, the pressure–displacement (P-D) curve can be customized as needed, and the pedal force–displacement (F-D) relation is the objective of the PFS design process. For comparison to conventional brake systems, we used sets of force–pressure–displacement (F-P-D) experimental data of one vacuum-booster brake (VBB) system as the reference. Although the F-P-D curve during the pedal depressing process differs from that of the pedal releasing process,¹⁵ we use the lookup curve of the depressing stage in PFS design and reference pressure calculation.

PFS

As shown in Figure 9, a passive PFS uses a combination of springs to provide a similar pedal feel to that of VBB systems. The forces acting on the PFS’s piston satisfy the following equation

F_{p} L_{p} A_{s} = d_{mc} k_{i} l_{p} A_{mc}

(19)

with

k_{i} = {\begin{matrix} k_{a} k_{b} / (k_{a} + k_{b}) \\ k_{b} \\ k_{b} + k_{c} \end{matrix} \begin{matrix} d_{mc} \in [0.5, 4) \\ d_{mc} \in [4, 12) \\ d_{mc} \in [12, 18] \end{matrix}

(20)

where $F_{p}$ is the pedal force applied by the driver; $L_{p}$ is the distance between the pedal and the top fixed point; $l_{p}$ is the distance between two fixed points of the pedal and the master cylinder piston rush rod; $d_{mc}$ (mm) is the main cylinder piston displacement; $A_{s}$ and $A_{mc}$ are the internal sectional areas of the PFS and the main cylinder, respectively; $k_{i}$ is the stiffness of each stage; and $k_{a, b, c}$ represents the stiffness of the three springs.

Figure 9.

Sectional view of the PFS.

Figure 10 shows two groups of pedal feel test data of the VBB and the PFS and a designed curve of the PFS. The F-D data for the depressing pedal process of the VBB were used to calculate the stiffness of the springs. As shown by the red curve, the F-D curve could be divided into four pieces; except for the preload force of approximately 50 N, the remaining pieces were used to solve for $k_{a, b, c}$ via equation (20). By observing the test data of the PFS, although friction forces made the test pedal forces slightly larger than the designed curve, the actual performance of the restoring pedal feel is still satisfactory.

Figure 10.

F-D curve of one VBB and the designed PFS.

Reference pressure

For normal brake situations, the reference pressure is calculated from the pedal angle. The relationship between the main cylinder piston displacement $d_{mc_raw}$ (mm) and the unfiltered pedal angle signal $θ_{p}$ (°) is expressed as follows

d_{mc_raw} \approx π l_{p} θ_{p} / 180

(21)

Figure 11 shows the test data of two typical pedal depressing maneuvers in the VBB system. Taking a 2-mm idle travel into consideration, we have the fitted expression $y_{r} = f_{ref} (d_{mc})$

y_{r} = {\begin{matrix} 0, d_{mc} < 2 \\ κ_{1} \cdot e^{- {(\frac{d_{mc} - κ_{2}}{κ_{3}})}^{2}} - κ_{4}, d_{mc} \geq 2 \end{matrix}

(22)

Here, $d_{mc}$ is the filtered signal of $d_{mc_raw}$ , and κ_1,2,3,4 = 8.35, 18.73, 8.98, and 0.24 are fitting coefficients.

Figure 11.

P-D curve in one VBB and the EHB.

Differentiator

Based on Figure 11, due to the restriction of the vacuum structure, the P-D curves of slow and fast pedal depression are almost identical. A conventional VBB system is unable to realize the BA function. When equipped with an active pressure source, for example, the ABS module, common solutions use the combination of the pedal speed and the pedal force as a BA triggering condition for redundancy consideration.¹⁶ However, in a BBW system with a PFS, the relationship between the pedal force and the pedal displacement can be approximately described by a fixed curve. Removing the pedal force sensor has a huge benefit in lowering the cost and minimizing hardware design difficulties. This possibility inspired us to study the method of realizing the BA function using only one angle sensor.

In fact, the research object is the calculation of the differential value of $d_{mc_raw}$ . However, the raw signals of the angle sensor are accompanied with noise, and their differential values are unsuitable for use as the pedal speed. Here, a nonlinear tracking differentiator via a two-sliding mode (LD) is introduced to synchronously filter the raw signal and calculate the difference^17,18

{\begin{matrix} \overset{\cdot}{x} = d_{mc} \\ d_{mc} = d_{1} - λ_{dif} \sqrt{| x - d_{mc_raw} (t) |} \cdot sign (x - d_{mc_raw} (t)) \\ {\overset{\cdot}{d}}_{1} = - α_{dif} \cdot sign (x - d_{mc_raw} (t)) \end{matrix}

(23)

s . t . {\begin{matrix} C_{dif} > 0 \\ α_{dif} > C_{dif} \\ λ_{dif}^{2} \geq 4 C_{dif} (α_{dif} + C_{dif}) / (α_{dif} - C_{dif}) \end{matrix}

where $x$ is the auxiliary variable, $d_{mc_raw} (t)$ is the raw signal calculated from the angle sensor outputs (in this application, it refers to the pedal displacement), $d_{mc}$ is the filtered output, ${\overset{\cdot}{d}}_{1}$ is the differential output, $α_{dif}$ and $λ_{dif}$ are the design parameters of the differentiator, and $C_{dif}$ is the upper bound of Lipchitz’s constant of the difference of the raw signal.

The LD is suitable for a brake system because the internal friction damping limits the maximum pedal speed;¹⁹ thus, $C_{dif}$ certainly exists. Regarding the parameter property, a larger $λ_{dif}$ leads to an inferior filtering performance, and a smaller $λ_{dif}$ corresponds to a smaller frequency band. Although a smaller $λ_{dif}$ could enhance the filtering effect, the tracking performance of high-frequency signals would be worse. For example, in fast pedal depressing or releasing maneuvers, there will be a significant delay between the raw and filtered signals.

To estimate $C_{dif}$ , several tests of fast depressing and releasing were performed. A typical dataset is shown in Figure 12. With the help of the opposite pedal force, the releasing process occurs at a higher speed. In Figure 12(b), the estimated maximum speed is approximately 80 mm/s. Based on this value, simulations are performed to determine an optimal parameter combination that can achieve both a fast response and effective filtering. Here, $α_{dif} = 100$ and $λ_{dif} = 44$ are appropriate for this pedal mechanism and sensor. Figure 13 shows the filtering performance of the LD; a well-tuned second-order Kalman filter (KF) is used for contrast. For the LD, in the stages of rising and falling shown in the enlarged maps, vibrations of the raw signal are smoothly processed and weakened. There are almost no overshoots at large change points of $d_{mc_raw}$ . For the KF, although it is smoother, as shown in the third enlarged map, it has more delay, and detail is missing.

Figure 12.

Fast pedal (a) depressing and (b) releasing.

Figure 13.

Filter performances of the KF and LD methods.

Figure 14 shows the differential values of the raw signal and the KF and LD. In Figure 14(a), the noise of the raw signal is amplified by dividing the sample time; as a result, it is unable to distinguish pedal speed by observing such output data. In Figure 14(b), the differential value of the KF-filtered signals is still too noisy to accurately identify the pedal speed. For Figure 14(c), the output curve shows a clear distinction of movements. Although the LD’s outputs are not absolutely equal to the real speed, we can still use an appropriate threshold value to distinguish fast and slow pedal movements.

Figure 14.

Differential outputs of the three methods: (a) differential outputs of raw signals, (b) differential outputs of KF-filtered signals, and (c) differential outputs of LD.

Test verification of the BA function

Here, we use the LD’s output to identify fast or slow depression of the pedal. In the fast depressing mode, the motor will be driven by the maximum power to achieve a greater boost speed, which is expressed as follows

u_{m} = {\begin{matrix} 1, {\overset{\cdot}{d}}_{1} > 10 and d_{mc} > 8 \\ pressure tracking, otherwise \end{matrix}

(24)

The pedal depression process in Figure 12(a) was adopted to verify BA control. Figure 15 shows the pedal speed during this process. In fact, for the 100 Hz raw signal, noise produces 100 times more disturbance on its differential value. For this sensor, the noise frequency is random and uncertain, and the amplitudes of the noise are approximately equal to the sensor’s minimum accuracy or its integral multiples (in this sensor, the minimum accuracy is approximately 0.1 mm, but it may vary under different working environments). From Figure 15, we can find that the amplitude of the noise differential is approximately 20 mm/s. However, as shown by the red curve, the negative oscillation of the differential value is significantly suppressed. When the pedal began to move at 0.7 s, the differentiator output exceeds +10 at 0.8 s, that is, there is an acceptable 0.1 s delay.

Figure 15.

Differential output during a fast pedal depressing maneuver.

Figure 16 shows a group of measured pressures in the BA tests. By observing the blue curve, because the pedal rod has a rigid connection with the hydraulic loop, the pressure response of VBB has a smaller delay than that of EHB. However, regardless of how fast the pedal is depressed, the relationship between pressure and displacement remains unchanged. For EHB with BA control, the pressure is less than that of the VBB in the initial 0.2 s. For BA, by triggering for a period of 0.8–1.3 s, the EHB outputs the maximum power. After a transport delay, the BA’s pressure exceeds that of no BA at 0.84 s and that of the vacuum booster at 0.88 s and then reaches a maximum pressure soon thereafter. This control method can be considered both feasible and effective.

Figure 16.

Experimental verification of BA control.

Controller design

Normally, many applications use an inner current loop to indirectly control the motor torque because the current has an approximately proportional relationship to the load torque. However, there are a few reasons why we do not use current-loop control: (1) the brake pressure can be regarded as the load on the motor pump, thus eliminating the need to estimate torque through the current when using pressure sensors; (2) when the motor’s eccentric shaft is rotating, the force acting on the piston vibrates cyclically, and, accordingly, the current oscillates, making it difficult to use such current to control torque precisely; and (3) because of the effect of operating temperatures on the motor starting current and pump internal friction variation, it is difficult to identify an accurate fixed relation between the current and the load torque via calibration tests. In this article, the control object in the pressurizing stage is the motor speed, which controls the pressurizing rates and time. The existing PWM speed control technique is mature and suitable for this application.

Safety is the priority of brake systems. A faster response corresponds to a shorter brake distance. A consistent relationship between a vehicle’s brake force and a driver’s brake pedal input leads to a comfortable and confident brake feel. The large pressure overshoots may lead to an unnecessary triggering of the ABS and extension of the actuator working time. Previously, several PID parameter combinations were attempted. Clearly, stable PID controllers exist; however, the response time of such controllers is longer. When the actual pressure is close to the reference pressure, the stable PID controller would then take more time to reach the reference. If the derivative effect of the PID is weakened, overshoots would emerge again.

To address this dilemma, predictive control was found to be the optimum method. Figure 17 shows the schematic diagram of the control loop (in this section, the BA function is not used). The GPC controllers contain two sub-controllers: a motor and reducing valve; each controls the pressure increase or decrease process. The EHB block can switch between the simulation model and the test bench.

Figure 17.

Control flow scheme of the simulation and the test.

GPC algorithm

Taking the pressurizing stage as an example, although the mechanical model of the motor pump has dynamic nonlinearity, the pressure increasing model could still be expressed as a linear model when $T_{motor} > T_{load}$ . Similar to equation (13), the ARX model is expressed as follows

A (z^{- 1}) y (k) = z^{- d} B (z^{- 1}) u (k) + ξ (k)

(25)

By transforming equation (25) to an incremental form, we have

\bar{A} (z^{- 1}) y (k) = B (z^{- 1}) Δ u (k - d) + ξ (k)

(26)

with

\bar{A} (z^{- 1}) = A (z^{- 1}) \cdot (1 - z^{- 1}) = 1 + {\bar{a}}_{1} z^{- 1} + {\bar{a}}_{2} z^{- 2} + {\bar{a}}_{3} z^{- 3}

where $Δ u (k) = u (k) \cdot (1 - z^{- 1})$ is the incremental output of the controller.

The prediction model at the sample time $(k + j)$ is

y (k + j) = G_{j} y (k) + F_{j} Δ u (k + j - 1) + E_{j} ξ (k + j)

(27)

where $G_{j}$ and $F_{j}$ satisfy the Diophantine equations

{\begin{matrix} 1 = \bar{A} (z^{- 1}) E_{j} (z^{- 1}) + z^{- j} G_{j} (z^{- 1}) \\ F_{j} (z^{- 1}) = B (z^{- 1}) E_{j} (z^{- 1}) \end{matrix}

(28)

with

{\begin{matrix} E_{j} (z^{- 1}) = 1 + e_{j, 1} z^{- 1} + \dots + e_{j, n_{ej}} z^{- n_{ej}} \\ G_{j} (z^{- 1}) = g_{j, 0} + g_{j, 1} z^{- 1} + \dots + g_{j, n_{gj}} z^{- n_{gj}} \\ F_{j} (z^{- 1}) = f_{j, 0} + f_{j, z} z^{- 1} + \dots + f_{j, n_{fj}} z^{- n_{fj}} \\ \deg E_{j} = j - 1, \deg G_{j} = n_{a}, \deg F_{j} = n_{b} + j - 1 \end{matrix}

The prediction error at $(k + j)$ is

\tilde{e} (k + j | k) = y (k + j) - \hat{y} (k + j | k)

(29)

Its variance can be expressed as follows

J = E {{\tilde{e}}^{2} (k + j | k)}

(30)

The optimal prediction error ${\tilde{e}}^{*}$ is

{\tilde{e}}^{*} (k + j | k) = E_{j} (z^{- 1}) ξ (k + j)

(31)

We define a quadratic cost function as follows

min J = E {{[Y - Y_{r}]}^{T} [Y - Y_{r}] + Δ U^{T} Γ Δ U}

(32)

with

\begin{matrix} Y = [y (k + 1), y (k + 2), \dots, y (k + N_{p})]^{T} \\ Y_{r} = [y_{r} (k + 1), y_{r} (k + 2), \dots, y_{r} (k + N_{p})]^{T} \\ Δ U = [Δ u (k), Δ u (k + 1), \dots, Δ u (k + N_{u} - 1)]^{T} \\ Γ = diag (γ_{1}, γ_{2}, \dots, γ_{N_{u}}) \end{matrix}

where $Y$ is the vector of the future predictive pressure, $Y_{r}$ is the vector of the reference pressure, $Δ U$ is the vector of the predictive incremental output of the controller, and $Γ$ is the control weight matrix. $N_{p}$ is the prediction horizon and $N_{u}$ is the control horizon.

Based on equation (27), $Y$ can be expressed as

Y = F_{1} Δ U + F Δ U (k - j) + GY (k) + E ξ

(33)

with

F_{1} = {[\begin{matrix} f_{1, 0} & 0 & 0 & \dots & 0 \\ f_{2, 1} & f_{2, 0} & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ ⋮ & f_{N_{u}, 0} \\ ⋮ & ⋮ \\ f_{N_{p}, N_{p} - 1} & f_{N_{p}, N_{p} - 2} & \dots & \dots & f_{N_{p}, N_{p} - N_{u}} \end{matrix}]}_{N_{p} \times N_{u}}

F_{2} = {[\begin{matrix} f_{1, 1} & f_{1, 2} & \dots & f_{1, n_{b}} \\ f_{2, 2} & f_{2, 3} & \dots & f_{2, n_{b} + 1} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ f_{N_{p}, N_{p}} & f_{N_{p}, N_{p} + 1} & \dots & f_{N_{p}, N_{p} + n_{b} - 1} \end{matrix}]}_{N_{p} \times n_{b}}

G = {[\begin{matrix} g_{1, 0} & g_{1, 1} & \dots & g_{1, n_{a}} \\ g_{2, 0} & g_{2, 1} & \dots & g_{2, n_{a}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ g_{N_{p}, 0} & g_{N_{p}, 1} & \dots & g_{N_{p}, n_{a}} \end{matrix}]}_{N_{p} \times (n_{a} + 1)}

Substituting equation (33) into equation (32), for $min J (\partial J / \partial Δ U = 0)$ , we have the following

Δ U (k) = (F_{1}^{T} F_{1} + Γ)^{- 1} F_{1}^{T} [Y_{r} - F_{2} Δ U (k - j) - GY (k)]

(34)

Note that the driving voltages of the motor and valves are limited. When constraints exist, $Δ U (k)$ does not have an analytic solution, as indicated by equation (34). We define the upper bound to be 1 and the lower bound to be 0 (PWM) of the controller output $U$ , as in equation (35). In addition, the nonlinear parts can be calculated using equation (9)

{\begin{matrix} 0 \leq u_{m, v} (k + i | k) \leq 1 \\ y \geq 0 \end{matrix}

(35)

Transforming the constraint into incremental form, we have

C Δ u (k) \leq L

(36)

with

C = [\begin{matrix} - C_{0} \\ C_{0} \end{matrix}], L = [\begin{matrix} - Δ u_{min} \\ Δ u_{max} \end{matrix}], C_{0} = [\begin{matrix} 1 \\ 1 & 1 & 0 \\ ⋮ & ⋱ \\ 1 & \dots & 1 \end{matrix}]

Δ u_{min} = [- u (k - 1), \dots, - u (k - 1)]_{(N_{u} \times 1)}^{T}

Δ u_{max} = [1 - u (k - 1), \dots, 1 - u (k - 1)]_{(N_{u} \times 1)}^{T}

Substituting equation (27) into equation (32), the general quadratic programming form is obtained

min f (Δ U) = \frac{1}{2} Δ U^{T} H Δ U + d^{T} Δ U s . t . C Δ u \leq L

(37)

with $H = 2 (F_{1}^{T} F_{1} + Γ)$ , $d^{T} = - 2 [Y_{r} - Y_{1}]^{T} F_{1}$ .

To solve the quadratic cost function defined by equation (34) under the constraint equation (36), the Quadratic Programming algorithm using Active Set iterations was utilized. The specific algorithms were discussed in the previous studies.^20,21

When compared to single-step optimization controllers, the multi-step optimization of GPC has better robustness. When the $N_{p}$ value is appropriate, the dynamic characteristic of the object is fully considered. Although possible incorrect estimation or delay changing could occur, GPC with a self-tuning regulator still has adequate robustness to model inaccuracy.

Pressurizing controller

The issue of choosing the controller parameters is analyzed via simulation. Theoretically, the step response will have a maximum overshoot at the maximum slope point of the step response curve. We choose 8 MPa as a reference pressure to simulate the control effects at different $N_{p}$ and the same $N_{u} = 1$ . The results are shown in Figures 18 and 19. The overshoot is approximately 2 MPa at $N_{p} = 1$ . For linear systems, it is better to choose a value of $N_{p}$ that is larger than the rise time of the step response. Thus, in the situation of $N_{p} = 20 and 30$ , negligible overshoots occur. However, the defects of these curves are more time-consuming. When $N_{p} \leq 10$ , the rising curves have the same preferable rapidity but different overshoots. The motor operation during this period is almost driven under the maximum $u_{m} (k)$ , that is, the motor operates at its highest efficiency and power range. Different values of $N_{p}$ have different stop times. In this simulation, a larger $N_{p}$ leads to less overshoot.

Figure 18.

Pressurizing response at different N_p.

Figure 19.

Controller outputs at different N_p.

The response speed and overshoot are usually incompatible, especially in this non-minimum phase system. Nevertheless, by taking advantage of the GPC properties and the dynamic nonlinear characteristic of the motor pump, it is feasible to find a balance between rapidity and accuracy. The motor pump pushes the fluid unidirectionally, and there is almost no inertia effect, that is, when the prediction output $u_{m}^{*}$ comes into the dynamic condition $T_{motor} < T_{load}$ , the motor stops immediately and $u_{m}^{*}$ could be considered to be 0. In fact, the proposed GPC controller adopts a linear model to control a nonlinear system. Thus, the controller’s objective is as follows: when the measured pressure is close to the reference, the corresponding torque from $u_{m}^{*}$ should be close to $T_{motor} = T_{load}$ . Because leakage in the pressure holding is inevitable, a small overshoot is allowed.

The control effects of different $N_{u}$ on a specified $N_{p}$ are shown in Figure 20. Theoretically, a small $N_{u}$ could have a restraining effect on $Δ U (k)$ , thereby enhancing the system stability. A larger $N_{u}$ corresponds to greater flexibility but a larger calculation amount. If $N_{u}$ was set to a relatively large value, then the control actions of future predictive $U^{*}$ of $u_{m}$ and $u_{v}$ would interact to reduce the stability. In this application, $N_{p} = 8$ and $N_{u} = 2$ are chosen.

Figure 20.

Pressurizing response at different N_u.

Depressurizing controller

The PWM carrier frequency of the valve and its controller frequency are both 50 Hz. We use 1000 Hz as the RCP test frequency to ensure at least 5% of the duty ratio is adjustable. For the nonlinear model of the depressurizing stage (equation (12)), the pressure output $y (k)$ can be transformed to $\ln y (k)$ ; thus, we have the following discrete form

(1 - z^{- 1}) \ln y (k) = - T_{c} C_{v} a_{v} u_{v} \cdot \frac{β}{V_{w}} \cdot (y (k))^{φ_{v} - 1}

(38)

We have a $φ_{v}$ value of approximately 0.9 from the experimental data fitting. To preferentially ensure the model accuracy at high pressures, we set the value of $(y (k))^{φ_{v} - 1}$ to 0.25. Thus, the nonlinear equation of equation (38) can be approximately linearized to equation (39) in terms of the input $u_{v} (k)$ and output $\ln y (k)$

\ln y (k + 1) - \ln y (k) = - 0.005 C_{v} a_{v} \cdot \frac{β}{V_{w}} \cdot z^{- d} \cdot u_{v} (k)

(39)

Figure 21 shows the simulation results of different $N_{p}$ and $N_{u} = 1$ with decreasing pressure. Note that we set the motor to remain stationary in these simulations. When the predictive length $N_{p}$ is close to the delay time, pressure tracking is the most rapid. The larger $N_{p}$ indicates decreasing rapidity and more on-off operation time. To lower the calculation amounts and avoid the interference effect of $u_{m}$ and $u_{v}$ , a value of $N_{p}$ close to the delay time and $N_{u} = 1$ is appropriate.

Figure 21.

Depressurizing response at different N_p.

Additional error dead-zone control

Under normal braking conditions, the EHB system model contains a double input $u$ of the motor and valve, and a single output $y$ of the pressure. In consideration of lowering the start–stop frequency of the actuators, the motor and reducing valve cannot be activated concurrently. The control logic is expressed as equation (40). A dead-zone near the reference pressure is set. Therefore, the system model is decoupled into two interference-free parts

u = {\begin{matrix} {[u_{m}, 0]}^{T}, e > k_{m} \cdot y_{r} \\ {[0, u_{v}]}^{T}, e < - k_{v} \cdot y_{r} \end{matrix}

(40)

where $e$ is the error, $y_{r}$ is the reference pressure, and $k_{m}, k_{v}$ ( $\leq 5 %$ ) are the dead-zone coefficients.

The introduction of control dead-zone could not only reduce the start–stop frequency of the actuators but also restrain the overshoot caused by the delay. As shown in Figure 22, the GPC and PID controllers without the dead-zone strived to lower the error to 0 and caused the pressure fluctuate near the reference. Alternatively, by observing the pressure curve of the PID controller with the dead-zone, some overshoots are greater than the dead-zone range; thus, there are still a few adjusting cycles. The GPC controller with the dead-zone allows small overshoots to increase and has an earlier stopping time when decreasing. Figure 23 shows the outputs $u$ of two controllers during the simulations in Figure 22. In (a), the controller outputs start/stop quickly and are kept in a highly efficient operation range. The reduction of the operation duration and avoidance of inefficiency would significantly benefit both energy savings and service life. In Figure 23(b), the GPC controller without the dead-zone must drive the motor constantly to compensate for the leakage. Moreover, in the period from 5 to 6 s, predictive control outputs of the motor and reducing valve have coupled interactions that produce a pressure fluctuation near the reference pressure. In Figure 23(c) and (d), although the PID with the dead-zone control has fewer adjusting cycles than Figure 23(d), it still takes more time to address large overshoots than the GPC controllers.

Figure 22.

Tracking performance of four methods.

Figure 23.

Comparison of the outputs of four controllers:(a) GPC with dead-zone control, (b) GPC without dead-zone control, (c) PID with dead-zone control, and (d) PID without dead-zone control.

Test verification of the pressure tracking

Static closed-loop tests (wheels remain static) were performed to evaluate the tracking performance of the wheel pressure. The results of two typical brake maneuvers are shown in Figures 26 and 27. For comparison, two earlier test results using well-tuned PID controllers are plotted in Figures 24 and 25. Figure 24 shows the tracking result when the driver depresses the pedal slowly. Several positive and negative overshoots can be identified. Figure 25 shows a tracking result of slow pedal release. Similarly, in the periods of 4 to 6 s and 9 to 10 s, repeating pressure vibrations are found. During these periods, the motor and reducing valve were driven alternately to adjust the pressure and achieve the reference pressure. From these two figures, the pressure tracking of the PID controller is more likely to exhibit unsteady pressure vibration.

Figure 24.

Experimental tracking result of slow pedal depression (PID).

Figure 25.

Experimental tracking result of slow pedal release (PID).

In Figures 26 and 27, the tracking performances of the proposed controller are evaluated with slow pedal depression and release. There are nearly no overshoots. In Figure 26, at the increasing stage of the prior 4 s, compared to the reference pressure, the measured pressure does not fluctuate frequently. In the period between 4 and 7 s, the controller outputs of the motor and the reducing valve are $u = 0$ . In the prior 7 s, the reducing valve is controlled closed. These results indicate that the tracking error is maintained in an acceptable range and the operating time of the actuators is lowered significantly. In Figure 27, the measured pressure shows both high accuracy and robustness. During the releasing process, the output of the motor controller $u_{m}$ is automatically controlled to be zero, and the unnecessary motor starts can be avoided. In both plots, the curves of the measured pressure are similar to a step-shaped profile. Benefitting from the introduction of the dead-zone control, the motor and reducing valve do not switch frequently to pursue absolute accuracy. GPC controllers not only provide fine tracking performance but also satisfactory anti-jamming properties.

Figure 26.

Experimental tracking result of slow pedal depression (GPC).

Figure 27.

Experimental tracking result of slow pedal release (GPC).

From the above tests, we can conclude the following: (1) the proposed GPC controllers have fine tracking performance; (2) the design of the error dead-zone control method promotes the anti-jamming property effectively, and when controllers of the motor and reducing valve ignore the small error $e (k)$ in the dead-zone, the start-stop frequency of the actuators could be significantly decreased; (3) because of the limitation of the dead-zone nonlinear properties of the actuators, the input–output characteristics of the motor’s low speed and the valve’s low turn-on time are difficult to calibrate via testing and have a relatively wide variation range affected by load. For a model-based GPC controller, it is better to avoid actuators operating in the abovementioned low-efficiency range. Because of the introduction of the GPC and error dead-zone control, although some accuracy is sacrificed, the actuators are always operated at their most efficient ranges.

Conclusion

This article presented a novel EHB system. The characteristics and pressure control of the prototype were illustrated through simulation and RCP tests. The main results and conclusions are as follows:

Considering the brake pedal characteristics, when the pedal maximum speed can be estimated, the LD is suitable for the BA function of the EHB. This approach may provide a new method for lowering the cost and reducing the algorithm complexity. Although accumulator-type EHBs have better pressure boost ability, the proposed EHB can still have a preferable performance in practical application with the help of BA.

The proposed identification method proved to be effective; therefore, GPC controllers were introduced to optimize the pressure tracking performance. When compared with a PID controller, simulations and test results showed that a GPC controller has apparent advantages. Most of the time, GPC controllers output the maximum driven voltage in the pressure tracking process and stop in a timely manner when reaching the reference pressure. The control effect could achieve a balance between rapidity and accuracy. Moreover, the benefits of service life and energy savings are achieved.

Footnotes

Acknowledgements

The authors would like to thank the reviewers and the Associate Editor for their helpful and detailed comments, which have helped to improve the presentation of this article.

Handling Editor: Fakher Chaari

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: This work was financially supported in part by the National Natural Science Foundation of the People’s Republic of China (grant number: 51505354).

ORCID iD

Zhe Xiong

Xuexun Guo

Xiaofei Pei

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