4.3. CONTROL ALGORITHM DESIGN FOR CLOSED-LOOP MODULATION 49
It can be seen from Equation (4.4) that when the PWM command value of the inlet valve
is constant, and the dynamics of the pressure is directly decided by the rotational velocity of
the pump-motor. Based on the property of PWM control, a suitable PWM command value for
the inlet valve should be chosen based on the following criteria. Regarding the first criterion,
the valve generated noise becomes most perceivable and intolerable when the PWM command
value is approximately 50, at which the impact between valve coil and seat is most dramatic.
us, less noise will be generated when the inlet valve PWM command chosen is closer to
0 or 100. Regarding the second criterion, because of the discontinuous property of the hydraulic
pump, pressure fluctuations will be generated. ese fluctuations can be reduced by opening the
inlet valve up to a small level. In other words, the PWM command is supposed to be close to
100 for a normally open inlet valve.
Figure 4.4a shows the relationship between the variation rate of the brake pressure in
wheel cylinder and the value of the PWM signal of the pump-motor. In both experiments, the
inlet valve PWM value is set to 80, at which the noise is acceptable, and the pressure fluctuation
generated by the hydraulic pump is adequately suppressed. A linear relationship between the
variation rate of the wheel cylinder pressure and the value of the PWM signal for the pump-
motor is observed, as shown in Fig. 4.4b. is result validates the correctness of Equation (4.4).
To improve the tracking performance and robustness to external disturbances, a closed-
loop control algorithm for the hydraulic pump-motor is developed, as illustrated in Fig. 4.5. e
feed-forward lookup map is derived by experiments, as shown in Fig. 4.4. Because the wheel
cylinder pressure can be obtained by the pressure sensor, feedback control is utilized to offset the
wheel cylinder pressure tracking error. Note that the relief valve, which is controlled inactively,
plays a role only of safety outlet valves in the HPBPM control method. When the upper stream
pressure of the inlet valve exceeds 15 MPa, the relief valve will be pushed open to release the
excess pressure. At other times, the upper stream pressure of the inlet valve is equal to the outlet
pressure of the hydraulic pump.
4.3 CONTROL ALGORITHM DESIGN FOR
CLOSED-LOOP PRESSURE-DIFFERENCE-LIMITING
MODULATION
e coil current-based valve control method, which is different from the control method dis-
cussed in the last section, is explored in this section. A closed-loop modulation method is de-
veloped based on the previously discussed linear property for on/off valves.
4.3.1 LINEAR MODULATION OF ON/OFF VALVE
e relative position of a normally open valve in closed-state is illustrated in Fig. 4.6. Taking the
lowest point which valve core can reach as the origin, the local coordinate system of the valves
can be set up [45].
50 4. CONTROLLER DESIGN OF CYBER-PHYSICAL VEHICLE SYSTEMS
7
6
5
4
3
2
1
0
12
10
8
6
4
2
0
Start
Stop
Wheel Cylinder Pressure (MPa)Change Rate of Wheel
Cylinder Pressure (MPa/s)
0 1 2 3 4 5
t (s)
6 7 8 9 10
100959085807570656055504540
PWM55
PWM60
PWM65
PWM70
Fitted Curve
Experiment Data
PWM75
PWM80
PWM85
(a)
PWM
(b)
Figure 4.4: (a) Relationship between the wheel cylinder pressure and the value of the PWM
signal of the pump-motor (with the PWM value of the inlet valve at 80); and (b) relationship
between the wheel cylinder pressure and different PWM command values of the pump-motor
(with the PWM value of the inlet valve at 80).
4.3. CONTROL ALGORITHM DESIGN FOR CLOSED-LOOP MODULATION 51
PWM
Map
PI
Controller
Motor
and Pump
Wheel
Cylinder
s
dP
w_cmd
/dt
P
w_act
PWM
ff
PWM
fb
PWM Q
P
w_cmd
P
error
+
–
+
+
Figure 4.5: Structural diagram of the PWM control algorithm for wheel cylinder pressure.
X
O
α
F
s
F
e
F
h
R
v
P
in
P
out
F
N
Valve Core
Valve Seat
Figure 4.6: Diagram of the coordinate system of the inlet valves.
Suppose the valve core comes in contact with the valve seat after being energized, the axial
forces applied on the valve can be represented as
F
e
C F
s
C F
h
C F
N
sin ˛ D 0; (4.5)
where F
e
, F
s
, F
h
, F
N
, and ˛ are the electromagnetic force, spring, hydraulic force, supportive
force, and valve seat cone angle, respectively [45].
e valve can be defined to be in a critical balanced position when the following two
conditions are satisfied: first, the valve core is in contact with the valve seat; second, the contact
force between valve core and valve seat is zero, i.e., F
N
D 0. In this state, the valve core is just
about to leave the valve seat and the force balance equation in (4.5) is simplified as
F
e
C F
s
C F
h
D 0: (4.6)
In Equation (4.5), the only term that can be actively controlled is the electromagnetic force
F
e
. us, to realize the critical balanced state, the electromagnetic force should be regulated in
52 4. CONTROLLER DESIGN OF CYBER-PHYSICAL VEHICLE SYSTEMS
a way that makes F
N
D 0. e electromagnetic force can be expressed as
F
e
D
.NI /
2
2R
g
l
; (4.7)
where I and N are the applied current and turns number of the coil, respectively. l and R
g
are
the length and magnetic reluctance of air gap [45]. en, we linearize Equation (4.7) at a fixed
operation point and reform it as:
F
e
D
@F
e
@I
ˇ
ˇ
ˇ
ˇ
I DI.t/
I.t/ C
@F
e
@x
v
ˇ
ˇ
ˇ
ˇ
x
v
Dx
v
.t/
x
v
.t/
D K
i
I.t/ C K
x
v
x
v
.t/;
(4.8)
where x
v
is the displacement of the valve core, K
i
and K
x
v
are the first-order current-force and
displacement-force coefficient.
Note that the displacement of valve core is x
v
D 0 when the critical balanced position is
reached. en, Equation (4.8) can be simplified as
F
e
D K
i
I.t/: (4.9)
Based on the definition of the OX coordinate in Fig. 4.6, the spring force of a normally
open valve is expressed as
F
s
D k
s
.
x
0
C x
m
x
v
/
; (4.10)
where x
0
, x
m
, and k
s
present the pre-tension displacement, maximum displacement and the
stiffness of the return spring, respectively. Besides, the spring force can be simplified in this
state as
F
s
D k
s
.
x
0
C x
m
/
: (4.11)
When the valve core reaches the fully closed position, the F
h
is decided by the pressure
drop p between valve inlet and outlet and A
s
, which is the surface area exposed to the fluid in
axial direction. F
h
, A
s
, and p are calculated as follows [45]:
F
h
D p A
s
(4.12)
A
s
D R
2
v
.cos ˛/
2
(4.13)
p D p
in
p
out
; (4.14)
where R
v
is the spherical radius in the valve core.
Substituting Equations (4.9), (4.11), and (4.12) into Equation (4.6), we can derive a lin-
ear correlation between coil current and the pressure difference at valve’s critical balanced posi-
tion [45]:
p D
K
i
R
2
v
.cos ˛/
2
I
k
s
.
x
0
C x
m
/
R
2
v
.cos ˛/
2
: (4.15)
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