Research Article
Modeling and pressure tracking control of a novel electro-hydraulic braking system Zhe Xiong1 , Xuexun Guo1 , Bo Yang1, Xiaofei Pei2
Advances in Mechanical Engineering 2018, Vol. 10(3) 1–16 Ó The Author(s) 2018 DOI: 10.1177/1687814018764153 journals.sagepub.com/home/ade
and Jie Zhang3
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
Date received: 8 November 2016; accepted: 8 February 2018 Handling Editor: Fakher Chaari
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 electromechanical brake (EMB) systems. Compared with 1
Hubei Key Laboratory of Advanced Technology of Automotive Components, Wuhan University of Technology, Wuhan, P.R. China 2 Hubei Collaborative Innovation Center for Automotive Components Technology, Wuhan University of Technology, Wuhan, P.R. China 3 Wanxiang Group Technical Center, Xiaoshan District, P.R. China Corresponding author: Bo Yang, Hubei Key Laboratory of Advanced Technology of Automotive Components, Wuhan University of Technology, 122 Luoshi Road, Wuhan 430070, Hubei, P.R. China. Email:
[email protected]
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 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 (ECBÒ) 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 (SBCÒ) 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 iBoosterÒ by Bosch, MK-C1Ò by Continental Teves, IBCÒ by ZF-TRW, and ESBÒ 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 MKC1, 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-inloop (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-
Advances in Mechanical Engineering 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
Xiong et al.
Figure 1. HCU and PFS prototype.
3 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.
RCP test environment
Figure 2. Configuration diagram of the EHB system.
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. 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
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, ControldeskÒ 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 openloop and closed-loop configurations.
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:
4
Advances in Mechanical Engineering
Figure 3. Operation block diagram of the EHB system.
Figure 4. RCP test environment.
Motor
Wheel cylinder 12V um = Rm Im + Cemf v Cemf Im = kfric v + Jm v_ + Tload
ð1Þ y_ (t) =
b qHo (t t gap ) Vw
ð5Þ
Pump
4qTi = Dv kleak pTi Tload = hDpTi
ð2Þ
mech
Brake tube
pTi = pTo + RT qTo qTo (t) = qTi (t tT )
ð3Þ
pTo = pHo + RH qHo qTo = qHo + Hh p_ Ho
ð4Þ
Brake hose
where, in equation (1), um is the controller output PWM of the motor, Rm is the equivalent resistance of the motor, Im is the current, v is the angular speed of the motor, kfric is the composite viscous friction coefficient of the rotor and the eccentric sleeve, Jm is the inertia moment of the rotor and the connected plungers, Tload is the equivalent torque of the load, and Cemf is the back electromotive force (EMF) constant. In equation (2), D is the pump displacement, pTi is the pressure at the pump outlets and the tube inlets, hmech is the mechanical efficiency of the pump, and kleak is the leakage coefficient, which is determined by the Hagen– Poiseuille coefficient of the system. In equation (3), pTo
Xiong et al.
5
is the pressure at the tube outlet, qTo is the flow rate of the fluid at the tube outlet, RT is the flow resistance of the tube, and tT is the flow delay in the tube. In equation (4), pHo is the pressure at the hose outlet, qHo is the flow rate of the fluid at the hose outlet, RH is the flow resistance of the hose, and Hh is the volume loss in the hose. In equation (5), y is the pressure in the wheel cylinder, b is the bulk modulus of the brake fluid, Vw is the maximum volume of the wheel cylinder, and tgap 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 out in carrier frequency. The average flow rate of fluid Q one period (Tv = 20 ms) can be expressed as follows12,13 out = Cv av Tv yuv uv Q
ð6Þ
where Cv is the throttle coefficient determined by the shape of the orifice, the flow conditions, and the fluid properties; av is the cross-sectional area of the valve; uv is the exponential coefficient determined by the shape of the valve orifice; and uv 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 qleak at the pressure holding stage can be approximately expressed as follows qleak = kleak y = y_
Vw b
ð7Þ
Equation can be log-linearized to d ln y = kleak bVw1 dt
Figure 5. Pressure responses of different drive inputs: (a) pressurizing and (b) depressurizing.
ð8Þ
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
Figure 6. Linearization of the inputs: (a) motor and (b) valve.
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. Here, we use the Hammerstein model to separate the nonlinear input part14 8 < um, v (k) = fm, v (vm, v (k)), fm, v (0) = 0 na nb P P ai y(k i) + bj um, v (k j) + j ð9Þ : y(k) = i=1
j=d +1
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.
6
Advances in Mechanical Engineering dy u y = Cv av bVw1 uv dt
Figure 7. System delay estimation in a PID test.
The results reveal the problem caused using a nonmodel 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 zeroinput response of the leakage in the system should also be considered. The above factors should be fully considered in controller design studies. Identification. From section ‘‘Subsystem models,’’ the structures of the transfer functions (without time delay) at different stages can be synthesized: Pressurizing stage
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(z1 )y(k) = B(z1 )u(k 1) + j(k)
ctn
s2
b0 ctn + a1 ctn s + a2
ð10Þ ctn
with b0
ctn
= K1 Dhmech b=12V
a0
ctn
= Tm hmech (4Vw + 4bHh + ckleak )
a1
ctn
= hmech (4Vw + 4bHh + bkleak Tm )
A(z1 ) = 1 + a1 z1 + + ana zna B(z1 ) = b0 + b1 z1 + + bnb znb
In this application, during the pressurizing stage, na = 2, nb = d + 2, and d is the integer solution of the time delay divided by the sampling time. The former d terms of B(z1 ) are 0. j(k) is the unpredictable white noise. Thus, equation (13) can be expressed as a form of least squares
ctn
= b(kleak hmech + D2 K2 )
Leakage
Depressurizing stage
where u(k) is the data vector and u is the parameter vector to be estimated. ^ can be recursively The estimated parameter vector u computed as follows
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 ^ are initialized as follows lf = 1; P(k) and u(k)
c = Vw RH + Vw RT + bHh RT
d ln y = kleak bVw1 dt
ð14Þ
8 ^ = u(k ^ 1) + K(k)½y(k) uT (k) u ^ (k 1) u(k) > > > < P(k 1) u (k) K(k) = ð15Þ T (k)P(k 1) u (k) > 1 + u > > : T P(k) = l1 f ½I K(k)u (k)P(k 1)
+ c(D2 K2 + kleak hmech ) a2
ð13Þ
with
y(k) = uT (k)u + j(k) Y (s) = a0 Um (s)
ð12Þ
ð11Þ
P(0) = 10 I ^ =0 u(0)
ð16Þ
Xiong et al.
7
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 ^ are variation are small. We set lf = 0.999, and u(k) 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(z1 )y(k) = B(z1 )um (k 1)
Figure 8. Result of EHB system identification.
ð17Þ
with
A(z1 ) = 11:842z1 + 0:8452z2 B(z1 ) = 0:0466 + 0:0441z1
For the pressure decreasing stage, based on the later discussed equation (39), at sample time of 0.02 s, we have Figure 9. Sectional view of the PFS. 1
1
(1 z ) ln y(k) = 1:1014 3 10 uv (k 1)
ð18Þ
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.
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-PD 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 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
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 Fp Lp As = dmc ki lp Amc
ð19Þ
8
Advances in Mechanical Engineering
Figure 10. F-D curve of one VBB and the designed PFS.
Figure 11. P-D curve in one VBB and the EHB.
with
mm idle travel into consideration, we have the fitted expression yr = fref (dmc )
8 < k a k b = ðk a + k b Þ k i = kb : kb + kc
dmc 2 ½0:5, 4) dmc 2 ½4, 12) dmc 2 ½12, 18
yr =
where Fp is the pedal force applied by the driver; Lp is the distance between the pedal and the top fixed point; lp is the distance between two fixed points of the pedal and the master cylinder piston rush rod; dmc (mm) is the main cylinder piston displacement; As and Amc are the internal sectional areas of the PFS and the main cylinder, respectively; ki is the stiffness of each stage; and ka, b, c represents the stiffness of the three springs. 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 ka, 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.
Reference pressure For normal brake situations, the reference pressure is calculated from the pedal angle. The relationship between the main cylinder piston displacement dmc raw (mm) and the unfiltered pedal angle signal up (°) is expressed as follows dmc
raw ’ plp up
180
(
ð20Þ
ð21Þ
Figure 11 shows the test data of two typical pedal depressing maneuvers in the VBB system. Taking a 2-
dmc \2
0, k1 e
dmc k2 2 k3 )
(
k4 ,
dmc 2
ð22Þ
Here, dmc is the filtered signal of dmc raw , and k1,2,3,4 = 8.35, 18.73, 8.98, and 0.24 are fitting coefficients.
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 dmc 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 difference17,18 8 < x_ = dmc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dmc = d1 ldif jx dmc raw (t)j sign(x dmc :_ d 1 = adif sign(x dmc raw (t))
raw (t))
ð23Þ
Xiong et al.
9
Figure 12. Fast pedal (a) depressing and (b) releasing.
Figure 14. Differential outputs of the three methods: (a) differential outputs of raw signals, (b) differential outputs of KFfiltered signals, and (c) differential outputs of LD.
Figure 13. Filter performances of the KF and LD methods.
8 < Cdif .0 s:t: adif .Cdif : l2 4C (a + C )=(a C ) dif dif dif dif dif dif where x is the auxiliary variable, dmc raw (t) is the raw signal calculated from the angle sensor outputs (in this application, it refers to the pedal displacement), dmc is the filtered output, d_ 1 is the differential output, adif and ldif are the design parameters of the differentiator, and Cdif 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, Cdif certainly exists. Regarding the parameter property, a larger ldif leads to an inferior filtering performance, and a smaller ldif corresponds to a smaller frequency band. Although a smaller ldif 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 Cdif , 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, adif = 100 and ldif = 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 dmc raw . For the KF, although it is smoother, as shown in the third enlarged map, it has more delay, and detail is missing. 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.
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,
10
Advances in Mechanical Engineering
Figure 15. Differential output during a fast pedal depressing maneuver.
the motor will be driven by the maximum power to achieve a greater boost speed, which is expressed as follows um =
1, pressure tracking,
d_ 1 .10 and dmc .8 otherwise
ð24Þ
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 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.
Figure 16. Experimental verification of BA control.
This control method can be considered both feasible and effective.
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.
Xiong et al.
11
Figure 17. Control flow scheme of the simulation and the test.
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.
8 Ej (z1 ) = 1 + ej, 1 z1 + + ej, nej znej > > > > > < Gj (z1 ) = gj, 0 + gj, 1 z1 + + gj, n zngj gj > Fj (z1 ) = fj, 0 + fj, z z1 + + fj, nfj znfj > > > > : deg Ej = j 1, deg Gj = na , deg Fj = nb + j 1 The prediction error at (k + j) is ~e(k + jjk) = y(k + j) ^y(k + jjk)
GPC algorithm
ð29Þ
Its variance can be expressed as follows
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 Tmotor .Tload . Similar to equation (13), the ARX model is expressed as follows A(z1 )y(k) = zd B(z1 )u(k) + j(k)
ð25Þ
By transforming equation (25) to an incremental form, we have 1 )y(k) = B(z1 )Du(k d) + j(k) A(z
J = E ~e2 (k + jjk)
ð30Þ
The optimal prediction error ~e is ~e (k + jjk) = Ej (z1 )j(k + j)
ð31Þ
We define a quadratic cost function as follows n o min J = E ½Y Yr T ½Y Yr + DUT GDU
ð32Þ
with ð26Þ Y = ½y(k + 1), y(k + 2), . . . , y(k + Np )T
with
Yr = ½yr (k + 1), yr (k + 2), . . . , yr (k + Np )T
1 ) = A(z1 ) (1 z1 ) = 1 + a1 z1 + a2 z2 + a3 z3 A(z
DU = ½Du(k), Du(k + 1), . . . , Du(k + Nu 1)T 1
where Du(k) = u(k) (1 z ) is the incremental output of the controller. The prediction model at the sample time (k + j) is y(k + j) = Gj y(k) + Fj Du(k + j 1) + Ej j(k + j) ð27Þ where Gj and Fj satisfy the Diophantine equations
with
1 )Ej (z1 ) + zj Gj (z1 ) 1 = A(z 1 Fj (z ) = B(z1 )Ej (z1 )
ð28Þ
G = diag(g 1 , g2 , . . . , g Nu ) where Y is the vector of the future predictive pressure, Yr is the vector of the reference pressure, DU is the vector of the predictive incremental output of the controller, and G is the control weight matrix. Np is the prediction horizon and Nu is the control horizon. Based on equation (27), Y can be expressed as Y = F1 DU + FDU(k j) + GY(k) + Ej with
ð33Þ
12
Advances in Mechanical Engineering 2
6 6 6 6 6 F1 = 6 6 6 6 4
f1, 0 f2, 1 .. .
0
0 0 .. .
f2, 0 .. . .. .
fNp , Np 1 2 f1, 1 6 f2, 2 6 F2 = 6 .. 4 .
.. .
f 1, 2 f2, 3 .. .
3
0 0 .. .
7 7 7 7 7 7 7 7 7 5
fNu , 0 .. .
.. .
fNp , Np 2
fNp , Np fNp , Np + 1 2 g1, 0 g1, 1 6 g2, 0 g2, 1 6 G = 6 .. .. 4 . . gNp , 0 gNp , 1
.. .
fNp , Np Nu 3
f1, nb f2, nb + 1 .. .
Np 3 Nu
7 7 7 5
fNp , Np + nb 1 Np 3 nb 3 g1, na g2, na 7 7 .. 7 .. . 5 . gNp , na Np 3 (na + 1)
Figure 18. Pressurizing response at different Np.
Substituting equation (33) into equation (32), for min J (∂J =∂DU = 0), we have the following DU(k) = (FT1 F1 + G)1 FT1 ½Yr F2 DU(k j) GY(k) ð34Þ Note that the driving voltages of the motor and valves are limited. When constraints exist, DU(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)
0 um, v (k + i jk ) 1 y0
ð35Þ
Transforming the constraint into incremental form, we have CDu(k) L with C=
Dumin C0 ,L= C0 Dumax
ð36Þ 2
1 61 , C0 = 6 4 ...
3 1
0 ..
7 7 5
.
1
1
Dumin = ½u(k 1), . . . , u(k 1)T(Nu 3 1) Dumax = ½1 u(k 1), . . . , 1 u(k 1)T(Nu 3 1) Substituting equation (27) into equation (32), the general quadratic programming form is obtained min f (DU) =
1 DUT HDU + dT DU 2
s:t: CDu L ð37Þ
with H = 2(FT1 F1 + G), dT = 2½Yr Y1 T F1 . 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 Np 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 Np and the same Nu = 1. The results are shown in Figures 18 and 19. The overshoot is approximately 2 MPa at Np = 1. For linear systems, it is better to choose a value of Np that is larger than the rise time of the step response. Thus, in the situation of Np = 20 and 30, negligible overshoots occur. However, the defects of these curves are more time-consuming. When Np 10, the rising curves have the same preferable rapidity but different overshoots. The motor operation during this period is almost driven under the maximum um (k), that is, the motor operates at its highest efficiency and power range. Different values of Np have different stop times. In this simulation, a larger Np leads to less overshoot. 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
Xiong et al.
Figure 19. Controller outputs at different Np.
13
Figure 21. Depressurizing response at different Np.
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 z1 ) ln y(k) = Tc Cv av uv
Figure 20. Pressurizing response at different Nu.
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 um comes into the dynamic condition Tmotor \Tload , the motor stops immediately and um 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 um should be close to Tmotor = Tload . Because leakage in the pressure holding is inevitable, a small overshoot is allowed. The control effects of different Nu on a specified Np are shown in Figure 20. Theoretically, a small Nu could have a restraining effect on DU(k), thereby enhancing the system stability. A larger Nu corresponds to greater flexibility but a larger calculation amount. If Nu was set to a relatively large value, then the control actions of future predictive U of um and uv would interact to reduce the stability. In this application, Np = 8 and Nu = 2 are chosen.
b (y(k))uv 1 Vw
ð38Þ
We have a uv 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))uv 1 to 0.25. Thus, the nonlinear equation of equation (38) can be approximately linearized to equation (39) in terms of the input uv (k) and output ln y(k) ln y(k + 1) ln y(k) = 0:005Cv av
b d z uv (k) Vw ð39Þ
Figure 21 shows the simulation results of different Np and Nu = 1 with decreasing pressure. Note that we set the motor to remain stationary in these simulations. When the predictive length Np is close to the delay time, pressure tracking is the most rapid. The larger Np indicates decreasing rapidity and more on-off operation time. To lower the calculation amounts and avoid the interference effect of um and uv , a value of Np close to the delay time and Nu = 1 is appropriate.
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
14
Advances in Mechanical Engineering
Figure 22. Tracking performance of four methods.
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=
½um , 0T , ½0, uv T ,
e.km yr e\ kv yr
ð40Þ
where e is the error, yr is the reference pressure, and km , kv ( 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 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.
Figure 24. Experimental tracking result of slow pedal depression (PID).
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
Xiong et al.
Figure 25. Experimental tracking result of slow pedal release (PID).
Figure 26. Experimental tracking result of slow pedal depression (GPC).
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. 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
15
Figure 27. Experimental tracking result of slow pedal release (GPC).
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 um 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 antijamming properties. 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:
16
Advances in Mechanical Engineering 1.
2.
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 accumulatortype 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.
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.
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 https://orcid.org/0000-0001-7620-6142 Xuexun Guo https://orcid.org/0000-0003-2390-3301 https://orcid.org/0000-0002-8283-9277 Xiaofei Pei
References 1. Jonner W, Winner H, Dreilich L, et al. Electrohydraulic brake system—the first approach to brake-by-wire technology. SAE paper 960991, 1996.
2. Nakamura E, Soga M, Sakai A, et al. Development of electronically controlled brake system for hybrid vehicle. SAE paper 2002-01-0300, 2002. 3. Kant B. Sensotronic brake control (SBC). In: Reif K (ed.) Automotive mechatronics. Wiesbaden: Springer Fachmedien, 2015. 4. Reuter D, Lloyd E, Zehnder J, et al. Hydraulic design considerations for EHB systems. SAE paper 2003-010324, 2003. 5. Velardocchia M. A methodology to investigate the dynamic characteristics of ESP and EHB hydraulic units. SAE paper 2006-01-1281, 2006. 6. Milane´s V, Gonza´lez C, Naranjo JE, et al. Electrohydraulic braking system for autonomous vehicles. Int J Automot Techn 2010; 11: 89–95. 7. Wu D, Ding H, Guo K, et al. experimental research on the pressure following control of electro-hydraulic braking system. SAE paper 2014-01-0848, 2014. 8. Yu Z, Jian C, Xu S, et al. Research of active power source based on electronic hydraulic braking system. SAE paper 2015-01-1211, 2015. 9. Todeschini F, Corno M, Panzani G, et al. Adaptive position–pressure control of a brake by wire actuator for sport motorcycles. Eur J Control 2014; 20: 79–86. 10. Line C, Manzie C and Good MC. Electromechanical brake modeling and control: from PI to MPC. IEEE T Contr Syst T 2008; 16: 446–457. 11. Clarke DW and Mohtadi C. Properties of generalized predictive control. Automatica 1989; 25: 859–875. 12. Wang B, Huang X, Wang J, et al. A robust wheel slip ratio control design combining hydraulic and regenerative braking systems for in-wheel-motors-driven electric vehicles. J Frankl Inst 2014; 352: 577–602. 13. Ding N, Yu G and Wang W. Estimation of brake pressure and tyre-road friction during ABS activation. Int J Vehicle Des 2012; 58: 33–4513. 14. Narendra KS and Gallman P. An iterative method for the identification of nonlinear systems using a Hammerstein model. IEEE T Automat Contr 1966; 11: 546–550. 15. Gerdes JC, Hedrick JK, Gerdes JC, et al. Brake system modeling for simulation and control. J Dyn Syst: T ASME 1999; 121: 496–503. 16. Hirose T, Taniguchi T, Hatano T, et al. A study on the effect of Brake Assist Systems (BAS). SAE Int J Passeng Cars Mech Syst 2009; 1: 729–735. 17. Levant A. Robust exact differentiation via sliding mode technique. Automatica 1998; 34: 379–384. 18. Levant A. Principles of 2-sliding mode design. Automatica 2007; 43: 576–586. 19. Park S, Bae S and Lee JM. Numerical evaluation of braking feel to design optimal brake-by-wire system. Int J Vehicle Des 2005; 37: 1–23. 20. Mayne DQ, Rawlings JB, Rao CV, et al. Constrained model predictive control: stability and optimality. Automatica 2000; 36: 789–814. 21. Camacho EF. Constrained generalized predictive control. IEEE T Automat Contr 1993; 38: 327–332.
