BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LV (LIX), Fasc. 2, 2009
SecŃia
AUTOMATICĂ şi CALCULATOARE
MODELLING OF AN ELECTROMAGNETIC VALVE
ACTUATOR
BY
CONSTANTIN FLORIN CĂRUNTU, MIHAELA HANAKO MATCOVSKI,
ANDREEA ELENA BĂLĂU, DANIEL IONUł PĂTRAŞCU,
CORNELIU LAZĂR and OCTAVIAN PĂSTRĂVANU
Abstract. During the last few years automotive actuators have become mechatronic
systems in which mechanical components coexist with electronics and computing devices
and because pressure control valves are used as actuators in many control applications for
automotive systems, a proper dynamic model is necessary. Starting from the modelling of
a single stage pressure reducing valve found in literature, in this paper, the concept of
modelling a real three land three way solenoid valve actuator for the clutch system in the
automatic transmission is presented. Two simulators for an input-output model and a
state-space model were developed and these were validated with data provided from
experiments with the real valve actuator on a test bench.
Key words: control valve, pressure reducing valve modelling.
2000 Mathematics Subject Classification: 93A30, 00A72.
1. Introduction
During the last few years the major advances in automotive applications
have been enabled by smart electronic devices that monitor and control the
mechanical components. Cars have become complex systems in which
electronic and mechanical subsystems are tightly connected and interact to
achieve optimal performance. Automotive actuators, in particular, have become
mechatronic systems in which mechanical components coexist with electronics
and computing devices [1].
In recent years, the use of control systems for automated clutch and
transmission actuation has been constantly increasing [2], in the powertrain
sector, the trend towards higher levels of comfort and driving dynamics while at
10
Constantin Florin Căruntu et al.
the same time minimizing fuel consumption representing a major challenge.
Electromagnetic actuation provides reliable means and a popular
alternative to hydraulic or pneumatic actuation for implementing control
systems. It is a natural connection between electrical circuits and mechanical
systems. All electromagnetic actuators work upon the same principle. By
inducing current in a coil of wire, the actuator gives rise to a magnetic force,
which is then used to affect the movement of a physical component. Such
systems have two distinct advantages: the applied force is non-contacting, and
often the response of the electromagnet is significantly faster then the dynamics
of the system being controlled [3]. Electronic devices have improved the
accuracy of control using closed loop control techniques in many applications
that have traditionally been served by hydro-mechanical open loop systems [4].
Electromagnetic valves vary in arrangement and complexity, depending
upon their function. Three broad functional types can be distinguished:
directional control valves, pressure control valves and flow control valves.
Pressure control valves act to regulate pressure in a circuit and may be
subdivided into pressure relief valves and pressure reducing valves. Pressure
relief valves, which are normally closed, open up to establish a maximum
pressure and bypass excess flow to maintain the set pressure. Pressure reducing
valves, which are normally open, close to maintain a minimum pressure by
restricting flow in the line. There are many practical applications where this
type of electromagnetic actuator is used: electromagnetic valve actuators of
combustion engines, artificial heart actuators, electromagnetic brakes,
electromagnetic actuator for the clutch system in automatic transmissions etc.
[5].
The increasing amount of power available to man that required control
and the stringent demands of modern control systems had focused attention on
modeling different types of valves even since 1967, when in [6], pressure
control valves and electro-hydraulic servovalves were analyzed.
Recent attention has focused on modeling and developing advanced
control methods for different valve types used as actuators in automotive
control systems: physics-based nonlinear model for an exhausting valve [7],
nonlinear state-space model description of the actuator that is derived based on
physical principles and parameter identification [8], [9], [10], [11] nonlinear
physical model for programmable valves [12], nonlinear model of an
electromagnetic actuator used in brake system, based on system identification
[5], mathematical model obtained using identification methods for a valve
actuation system of an electro-hydraulic engine [13], linear model constructed
based on gray-box approach which combines mathematical modelling and
system identification for an electro-magnetic control valve [14].
Starting from the equations in [6], where a single stage pressure
reducing valve is modeled, in this paper, the concept of modelling a three land
three way pressure reducing valve used as actuator for the clutch system in the
automatic transmission of a VW vehicle is presented. Two models were
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
11
developed: a linearized input-output model and a state-space model then
implemented in Matlab/Simulink and validated by comparing the results with
data obtained on a real test-bench provided by Continental Automotive
Romania.
The rest of the paper is organized as follows. In Sec. 2, the structure and
the functional operation of the valve are presented and in Sec. 3 the
mathematical models of the solenoid three way valve are developed. Sec. 4
presents the simulators for the electromagnetic valve and the concluding
remarks are given in Sec. 5.
2. Valve Structure and Operation
Pressure control valves employ feedback and may be properly regarded
as servo control loops. Therefore proper dynamic design is necessary to achieve
stability.
In Fig. 1a a section through a real three stage pressure reducing valve is
represented. Schematics of the three land three way pressure reducing valve are
shown in Fig. 1b. The pressure to be controlled is sensed on the spool end areas
C and D and compared with a magnetic force Fmag which actuates on the
plunger. The feedback force Ffeed = FC – FD is the difference between the force
applied on the left sensed pressure chamber FC, and the force applied on the
right sensed pressure chamber FD.
The difference in force is used to actuate the spool valve which controls
the flow to maintain the pressure at the set value. In the charging phase,
illustrated in Fig. 1c, the magnetic force is greater than the feedback force (Ffeed
< Fmag) and moves the plunger to the left (x > 0), connecting the source with the
hydraulic load. In the discharging phase, illustrated in Fig. 1d, the feedback
force becomes grater that the magnetic force (Ffeed > Fmag) and the plunger is
moved to the right (x < 0); the connection between the source and the hydraulic
load is closed, the hydraulic load being connected to the tank.
Using the magnetic force and the feedback force it results a force
balance which describes the spool motion and the output pressure. This
equation of force balance is the same for both positive and negative
displacement of the spool:
(1)
Fmag − AC PC + AD PD = M v s 2 X + K e X ,
where PC represents the pressure in the left sensed chamber (C) that acts on the
AC area, PD represents the pressure in the right sensed chamber (D) that acts on
the AD area, Mv is the spool mass, Ke = 0.43w(PS0 – PR0) represents the flow
force spring rate, PS is the supply pressure, PR is the reduced pressure, w
represents the area gradient of the main orifice, X = X(s) is the Laplace
transform of the spool displacement and s represents the Laplace operator.
12
Constantin Florin Căruntu et al.
output
pressure
line pressure PS
PR
C
Ffeed
D
+1mm
x -1mm
Fmag
T
hydraulic damper
plunger
tank pressure PT
b
a
c
d
Fig. 1 − Section through a real three stage pressure reducing valve (a); Three stage
valve schematic representation (b); Charging phase of the pressure reducing valve (c);
Discharging phase of the pressure reducing valve (d).
In Fig. 1a a hydraulic damper that acts to reduce the input pressure
spike, which has negative effects on the output pressure, is also represented.
3. Valve Modelling
The models designed in this Section are based on physical principles for
flow and fluid dynamics and parameter identification.
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
13
3.1. Input-Output Mathematical Model for the Charging Phase
The charging phase of the pressure reducing valve has been illustrated
in Fig. 1c. A positive displacement of the spool allows connection between the
source and the hydraulic load, while the channel that connects the hydraulic
load with the tank is kept closed.
The linearized continuity equation from [6] was used to describe the
dynamics from the sensed pressure chambers:
VC
(2)
QC = K1 ( PR − PC ) =
(3)
QD = K 2 ( PR − PD ) =
βe
VD
βe
sPC − AC sX ,
sPD + AD sX ,
where K1, K2 are the flow-pressure coefficients of restrictors, VC, VD are the
sensing chamber volumes and βe represents the effective bulk modulus.
Using the flow through the left and right sensed chambers, the flow
through the main orifice (from the source to the hydraulic load) and the load
flow, the linearized continuity equation at the chamber of the pressure being
controlled is [15]:
(4) K C ( PS − PR ) − QL − kl PR − K1 ( PR − PC ) − K 2 ( PR − PD ) + K q X =
Vt
βe
sPR ,
where QL is the load flow, KC is the flow-pressure coefficient of main orifice, Kq
is the flow gain of main orifice, kl is the leakage coefficient and Vt represents
the total volume of the chamber where the pressure is being controlled.
These equations define the valve dynamics and combining them into a
more useful form, solving Eq. (2) and Eq. (3) w.r.t. PC and PD and substituting
into (4) yields after some manipulation:
  1
 s

1 AC AD 
+ 1
+ 1 + K q X 1 +  +
+
−
 s +

 ω1  ω2

  ω1 ω2 K q K q 
 s
( KC PS − QL ) 
(5)
 1
V ω  s

A
A  
+
+ C − D  s 2  = PR K ce  C 1 
+ 1 +
ω ω


 Vt ω3  ω2
 1 2 K qω2 K qω1  
+
where ω1 =
 
 s

VD ω2  s
s VC ω1 VD ω2   s
+
+
+ 1 ,
  + 1 
 + 1 +  1 +
Vt ω3  ω1   ω3 Vt ω3 Vt ω3   ω1  ω2

β e K1
VC
, ω2 =
βe K 2
VD
are the break frequency of the left and right
14
Constantin Florin Căruntu et al.
sensed chambers, ω3 =
β e K ce
is the break frequency of the main volume and
Vt
Kce = KC + kl represents the equivalent flow-pressure coefficient.
Considering that VC « Vt and VD « Vt, the right side can be factored to
give the final form for the reducing valve model in the charging phase:
  1
 s

1 AC AD 
+ 1
+ 1  + K q X 1 +  +
+
−
s +
 ω1  ω2

  ω1 ω2 K q K q 
 s
( KC PS − QL ) 
(6)
 1

 s
 s
 s
A
A  
+
+ C − D  s 2  = PR Kce  + 1
+ 1 
+ 1 .


 ω1  ω2
 ω3

 ω1ω2 K qω2 K qω1  
3.2. Input-Output Mathematical Model for the Discharging Phase
A negative displacement of the pressure reducing valve spool allows
connection between the hydraulic load and the tank, while the channel that
connects the source with the hydraulic load is kept closed.
The linearized continuity equations at the sensed pressure chambers for
the discharging phase of the valve, illustrated in Fig. 1d, are:
VC
(7)
−QC = K1 ( PC − PR ) = −
(8)
−QD = K 2 ( PD − PR ) = −
βe
VD
βe
sPC + AC sX ,
sPD − AD sX .
Using the flow through the left and right sensed chambers, the flow
through the main orifice (from the hydraulic load to the tank) and the load flow,
the linearized continuity equation obtained for the chamber of the pressure
being controlled is [15]:
(9)
QL + K1 ( PC − PR ) + K 2 ( PD − PR ) − K D ( PR − PT ) − kl PR + K q X =
Vt
βe
sPR ,
where KD is the flow-pressure coefficient of main orifice and PT represents the
tank pressure.
Combining these equations into a more useful form, solving Eq. (7) and
Eq. (8) for PC and PD and substituting into (9) yields after some manipulation:
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
15
  1
 s

A
A 
1
+ 1 
+ 1  + K q X 1 + 
+
+ C − D s +


 ω1
  ω2

  ω1 ω 2 K q K q 
 s
( K D PT + Q L ) 
(10)
 1
AC
AD
+
+
−
ω ω
 1 2 K q ω 2 K q ω1
+
 2
s = PR K ce
 
 
 VC ω1  s

+ 1 +



 Vt ω 3  ω 2
.
V ω
 
 s

VD ω 2  s
V ω  s
s
+ 1 + 1 +
+ C 1 + D 2 
+ 1 
+ 1

ω 3 Vt ω 3 Vt ω 3   ω1
Vt ω 3  ω1
 
  ω2

In an entire analogue manner, again making the assumption that VC « Vt
and VD « Vt like for the charging phase model and considering KD = KC the final
form for the reducing valve in the discharging phase was obtained:
  1
 s

1 AC AD 
s+
+ 1
+ 1  + K q X 1 +  +
+
−


 ω1  ω2

  ω1 ω2 K q K q 
 s
( K D PT + QL ) 
(11)
 1

 s
 s
 s
A
A  
+
+ C − D  s 2  = PR K ce  + 1
+ 1 
+ 1
ω ω

 ω1  ω2
 ω3

 1 2 K qω2 K qω1  
.
3.3. Transfer Function Block Diagram
Equations (1), (2), (3), (6), for the charging phase of the valve, and Eqs
(1), (7), (8), (11) for the discharging phase of the valve, define the pressure
reducing valve dynamics and can be used to construct the transfer function
block diagram represented in Fig. 2.
Fig. 2 − Transfer function block diagram of valve.
16
Constantin Florin Căruntu et al.
Considering the resulting force between the magnetic and the feedback
force:
F1 = Fmag − AC PC + AD PD ,
(12)
solving PC and PD from the linearized continuity Eqs. (2) and (3) and
substituting in the Eq. (1) of force balance, the following equation was obtained
[15]:
(13)
Fmag − AC
K1PR + AC sX
+ AD
 s

 + 1 K1
 ω1 
K 2 PR − AD sX
= M v s2 X + Ke X .
 s

+ 1 K 2

 ω2

Ke
,
Mv
representing the mechanical natural frequency, and substituting (12) into (1)
yields:
After some manipulations, where it was considered that ωm =
(14)
 AC2

AD2


 s2

K1
K2 
+
F1 − 
sX = K e  2 + 1 X ,
ω

 s
  s

 m

+ 1 
  + 1 

  ω1   ω2
where
(15)
AD 
 A
F1 = Fmag −  C +
 PR ,
s
s
+1
+1

ω2
 ω1

illustrating the closed loop model from Fig. 2 for the displacement x. A switch
is used in order to commutate between the three phases of the pressure reducing
valve. Like seen in Fig. 2, switching between the charging and the discharging
phase can be realized by selecting different disturbances for positive and
negative displacement of the spool.
3.4. State-Space Model
Starting from Eqs. (1), (2), (3) and (4), respectively Eqs. (1), (7), (8)
and (9) a state-space model is designed [16]:
(16)
xɺ ( t ) = Ax ( t ) + Bu ( t )
,

y ( t ) = Cx ( t ) + Du ( t )
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
where:
x (t ) = [ v (t )
u (t ) =  PS ( t )
x (t )
PT ( t )
PC (t )
QL (t )
PD (t )
Fmag (t ) 
PR (t ) ]
T
T
is
17
is
the
the
state vector,
input
vector
and
y (t ) = [ x(t ) PR (t ) ] is the output vector. The A, C and D matrices are:
T
(17)

 0

 1
 β
 e
 AC
A = βe 
 VC
 A
− D
 VD

 0


−
Ke
M v βe
−
AC
M v βe
AD
M v βe
0
0
K1
VC
0
0
−
0
0
0
Kq
K1
Vt
Vt
−
K2
VD
K2
Vt




0



K1
,
VC


K2


VD

( Kce + K 1+ K 2 ) 
−

VD

0
0 1 0 0 0 
C=
 , D = O2×4 ,
0 0 0 0 1 
and the matrix B has the Bc expression in the charging phase (for x > 0) and
the B d expression in the discharging phase (for x < 0), where:
(18)

 0


 0


Bc = βe  0


 0

 KC
V
 t
0
0
0
0
0
0
0
0
0 −
1
Vt


0
0

M v βe




0
0 
0




0
0  , B d = β e 0




0
0 
0


0 − KC
0 


Vt


1
0
0
0
0
1
Vt

M v βe 


0 


0 .


0 

0 


1
This model is more precise because no approximations were used, as
for the input-output model.
18
Constantin Florin Căruntu et al.
4. Simulators for Electromagnetic Valve
In this section two simulators that were designed starting from the
models described in Sec. 2 and developed in Matlab/Simulink program are
presented.
Parameter values used for testing in Simulink are presented in Table 1.
Dimensional parameters were measured directly on the sectioned valve and the
flow coefficients were determined through experiments with the real valve on a
test bench at Continental Automotive Romania.
Table 1
Coefficients Values
Symbol
Value
Unit
Mv
2.5 e-3
kg
PS
1 e+6
N/m2
PT
1 e+5
N/m2
βe
1.6 e+9
N/m2
Ke
2 e+3
N/m
KC
7.5772 e-11
m3/s·bar
KD
7.5772 e-11
m3/s·bar
K1
1.2634 e-10
m3/s·bar
K2
1.3873 e-9
m3/s·bar
Kq
8.0127
m3/s·bar
kl
VC
2 e-9
3 e-3
7.53 e-7
m3/s·bar
m
m3
VD
1.04 e-6
m3
Vt
3 e-4
m3
AC
3.664 e-5
m2
AD
2.94 e-5
m2
ω1
2.6845 e+6
rad/s
ω2
2.1342 e+7
rad/s
ω3
1.6606 e+4
rad/s
α
2 e-5
m
w
The models were validated by comparing the results with data obtained
on a real test-bench provided by Continental Automotive Romania.
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
19
4.1. Model Testing
For testing purposes a Simulink model was created, using as input a
step signal, like shown in Fig. 3. The commutation between the charging and
the discharging phase was simulated by two switches that connect different
perturbation depending on the value of the displacement. These switches were
used in order to avoid the rapidly switching between the two flow perturbations
caused by the oscillations of the plunger, using α as the threshold.
[Ps]
-K-
Kc
Fmag , Ql
-K Fmag
x
0
Pr
In
QL
Model
Load Flow
Pt
Kd
-K -
-K -
Pr, x
Fig. 3 − Simulink model with step signal input.
In Fig. 3 two subsystems were used: one noted Model and representing
the transfer functions of the reducing valve model (Fig. 4), that were
represented as a block diagram in Fig. 2, and one noted as Load Flow
representing the load flow.
1
Fmag
MSD 1_num (s)
MSD 1_den (s)
P_num (s)
1
x
2
In
P_den (s)
MSD 2_num (s)
Flow _num (s)
MSD 2_den (s)
Flow _den (s)
FC_num (s)
FC_den (s)
FD_num (s)
FD_den (s)
Fig. 4 − Transfer functions represented in Simulink.
2
Pr
20
Constantin Florin Căruntu et al.
For a step signal as the magnetic force and a sequence of pulses as the
load flow represented in Fig. 5, the results obtained for spool displacement and
reduced pressure are presented in Fig. 6. For modeling the load flow needed to
actuate the clutch, two impulse signals, a positive one and a negative one, for 20
ms with a value of 10-4 m3/s were considered, value determined from
measurements on the test bench.
5
QL
Fmag
Fmg[N], QL [104*m3/s]
4
3
2
1
0
-1
0
0.5
1
1.5
2
2.5
time [s]
3
3.5
4
4.5
Fig. 5 − Magnetic force and load flow.
The displacement follows the step input behavior while the reducing
pressure has almost the same value like the reference signal.
7
displacement
pressure
displacement [m], pressure [bar]
6
5
4
3
2
1
0
-1
0
1
2
3
4
5
time [s]
Fig. 6 − Spool displacement and reduced pressure.
The model shows good performance, being stable for input signals
variations.
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
21
4.2. Input-Output Model Simulation
In order to validate the results obtained for the solenoid valve actuator,
a Simulink model (represented in Fig. 7) was created, using a magnetic force as
input (Magnet block). The magnetic force block implements the connection
between electric current trough solenoid and magnetic force generated by the
magnetic flux. A force sensor was utilized to measure the magnetic force and
the results were used in a form of a two dimensional look-up table, designed at
Continental Automotive Romania for this type of valve.
The blocks in the upper part of the model (Druck_A, Druck_P,
Strommesszange and Weg_Magnet) represent the real data obtained from
experiments made on a test bench at Continental Automotive Romania with this
type of valve.
The gains in the model are used to transform the values of the
parameters that are in international units in other units used for display (meter to
millimeter for the spool displacement and Pascal to bar for the reduced
pressure).
[time Druck _A]
[Pr]
Pr
[xm]
[time Druck _P]
[time Strommesszange ]
[time Weg _Magnet ]
1
[Ps]
Ps
[i]
0.003 s+1
Magnet
[i]
i
-u+2.745
x
[x]
[xm]
[x]
-Kx_s vs. x_m
[Ps]
Fmag
-K -
Kc
x
0
Pr
In
QL
Model
Load Flow
Pt
[Pr]
Kd
u-0.35
-KPr_s vs. Pr_m
Fig. 7 − Input-output Simulink model.
switching _filt
Fmag
22
Constantin Florin Căruntu et al.
In Fig. 7 the saturation block was used to allow only positive values for
the magnetic force and the filter (switching_filt) to eliminate the high
frequencies caused by the look-up table.
In Fig. 8 a real input signal is illustrated, represented either by the
magnetic force or by the current used to obtain the magnetic force through the
look-up table.
8
Fmag
i - current
7
Fmag [N], current [A]
6
5
4
3
2
1
0
-1
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 8 − Current and magnetic force used as input signals.
The results of the simulations are presented in Figs. 9 and 10, where the
spool displacement and the reduced pressure were compared with real data
obtained from experiments made on a test bench with the input signal from
Fig. 8.
1
measured
simulated
0.8
0.3
0.25
0.2
displacement [mm]
0.6
0.2
0.15
0.1
0.1
0.4
0.05
0
0.8
0
0.8
0.2
0.82
0.84
0.85
0.86
0.88
0.9
0.9
0
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 9 − Compared spool displacements for input-output model.
It can be seen that the simulated displacement of the spool has even
smaller variations than the measured displacement while the behaviour is the
same.
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
23
14
measured
simulated
12
6
10
5
8
pressure [bar]
6
5.5
5.5
5
4.5
4.5
6
4
4
3.5
3.5
0.80.8
4
0.82
0.84 0.85
0.86
0.88
0.9
0.9
2
0
-2
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 10 − Compared reducing pressures for input-output model.
Concerning the reduced pressure, the experimental results reveal that
the simulated pressure follows the measured pressure behaviour, having in the
steady state an irrelevant offset. The amplitude of the simulated pressures
variations in steady state is lower than the amplitude of the measured pressures
variations.
4.3. State-Space Model Simulation
The state-space model was represented in Simulink as shown in Fig. 11,
where a similar switch as in the input-output model was used to commutate the
reduced pressure between the charging and the discharging phases.
[time Druck _A ]
Pr
[time Druck _P ]
Ps
[Pr]
[Ps]
Load Flow
[Ps]
[time Strommesszange ]
i
[time Weg _Magnet ]
x
[Ql ]
QL
[x]
-K -
[Ps1]
[x3]
[i]
-u+2.745
[x]
-K -
[Pr]
x_s vs. x_m
u-0.35
[xm]
[Pr3]
[xm]
-K -
1
Pr_s vs. Pr_m
[Fmag ]
[i]
0.003 s+1
switching _filt
Magnet
[Ps1]
[Ps1]
Pt
[Ql]
x' = Ax+Bu
y = Cx+Du
Pt
[x1]
[Ql]
[Pr1]
x' = Ax+Bu
y = Cx+Du
[x2]
[Pr2]
State -Space _d
State -Space _c
[Fmag ]
[Fmag ]
[Pr1]
[x1]
[Pr2]
[x1]
0
[Pr3]
[x1]
0
[x2]
Fig. 11 − State-space Simulink model.
[x3]
24
Constantin Florin Căruntu et al.
The results obtained for the spool displacement using the state-space
model are similar to those obtained using the input-output model and are
represented in Fig. 12.
1
measured
simulated
0.8
0.3
0.3
displacement [mm]
0.25
0.6
0.2
0.2
0.4
0.1
0.1
0.15
0.05
00
0.8
0.8
0.2
0.82
0.850.86
0.84
0.88
0.9
0.9
0
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 12 − Compared spool displacements for state-space model.
Fig. 13 illustrates the difference between the simulated and the
measured reduced pressures. It can be seen that the amplitude of the simulated
pressures variations in steady state is lower than the amplitude of the measured
pressures variations. Also, the simulated pressure has in steady state a slight
offset.
14
measured
simulated
12
66
pressure [bar]
10
5.5
5.5
55
8
4.5
4.5
6
44
3.5
3.5
0.8
0.8
4
0.82
0.85 0.86
0.84
0.88
0.9
0.9
2
0
-2
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 13 − Compared reducing pressures for state-space model.
4.4. State-Space Model Simulation Using an S-Function
After testing and validating the state-space model presented in the
previous section, the state-space model was implemented as an S-function.
Bul. Inst. Polit. Iaşi, t. LV (LIX), f. 2, 2009
25
Fig. 14 represents the Simulink model utilized for testing this block by
comparing the results obtained through simulations with the measured values on
the real test bench.
[time Druck _A]
[Pr]
Pr
[time Druck _P]
Load Flow
[Ps]
Ps
[Ps]
[time Strommesszange ]
-K-
[Ps1]
[i]
i
[time Weg _Magnet ]
[Ql ]
QL
-u+2.745
x
[x]
[xm]
[xm]
1
[Fmag ]
[i]
0.003 s+1
Saturation
Magnet
switching _filt
[Ps1]
Pt
[xs]
sfun _valve _state
[Ql]
[Prs]
S-Function
[Fmag ]
[x]
[xs]
-K -
[Pr]
u-0.35
[Prs]
-K -
x_s vs. x_m
Pr_s vs. Pr_m
Fig. 14 − State-space with S-function Simulink model.
Fig. 15 illustrates the results and the measured values for the spool
displacement and in Fig. 16 the compared reducing pressures, obtained using
the input signals illustrated in Fig. 8, are represented.
1
measured
simulated
0.8
0.3
0.25
displacement [mm]
0.6
0.2
0.2
0.15
0.4
0.1
0.1
0.05
0.2
00
0.8
0.8
0.82
0.84 0.85
0.86
0.88
0.9
0.9
0
-0.2
-0.4
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 15 − Compared spool displacements for S-function state-space model.
26
Constantin Florin Căruntu et al.
14
measured
simulated
12
66
pressure [bar]
10
5.55.5
55
8
4.54.5
6
44
3.53.5
0.8
0.8
4
0.82
0.850.86
0.84
0.88
0.9
0.9
2
0
-2
0
0.2
0.4
0.6
0.8
1
time [s]
1.2
1.4
1.6
Fig. 16 − Compared reducing pressures for S-function state-space model.
5. Conclusions
In this paper, two different models for a solenoid valve actuator used in
the automotive control systems were developed: a liniarized input-output
model, where simplifications were made in order to obtain a suitable transfer
function to be implemented in Simulink and to obtain an appropriate behavior
for the outputs, and a state-space model with no simplifications, that was
implemented as a state-space model and as an S-function in Simulink. The
results of the experiments illustrate a similar behavior of these three simulators.
The models were validated by comparing the results with data obtained on a
real test-bench provided by Continental Automotive Romania. It can be
concluded that the simulators have good results illustrated by the similar
behavior obtained for the spool displacement and the reduced pressure
compared with the measured values.
A c k n o w l e d g m e n t s. This work was partially supported by CNMP-SICONA
project and Continental Automotive Romania. The authors gratefully acknowledge the support.
Received: January 8, 2009
“Gheorghe Asachi” Technical University of Iaşi,
Department of Automatic Control and
Applied Informatics
e-mail: caruntuc@ac.tuiasi.ro
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27
3. Peterson K., Stefanopoulou A., Wang Y., Haghgooie M., Nonlinear Self-Tuning
Control for Soft Landing of an Electromechanical Valve Actuator. Proc. of
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Mechanics, 74, 8, 550-562 (2005).
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Electro-Pneumatic Exhaust Valve for Internal Combustion Engines. Proc. of
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Valve Actuator. Society of Automotive Engineers (2002).
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an Electromechanical Valve Actuator Through Iterative Learning Control.
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USA, 2003.
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Electromagnetic Valves for Camless Engines. International Journal of Control,
80, 11, 1796-1813 (2007).
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Ph. D. Diss., Budapest, Hungary, 2004.
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IEEE Transactions on Control Systems Technology, 16, 34-45 (2008).
13. Liao H.- H., Roelle M.J., Gerdes J.C., Repetitive Control of an Electro-Gydralic
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of American Control Conference, Ankorage, 2002.
15. Pătraşcu D.I., Bălău A.E., Căruntu C.F., Lazăr C., Matcovschi M.H., Păstrăvanu O.,
Modelling of a Solenoid Valve Actuator for Automotive Control Systems.
accepted for the 17th International Conference on Control Systems and
Computer Science CSCS-17, Bucureşti, Mai 26-29, 2009.
16. Bălău A.E., Căruntu C.F., Pătraşcu D.I., Lazăr C., Matcovschi M.H., Păstrăvanu O.,
Modelling of a Pressure Reducing Valve Actuator for Automotive Applications.
accepted for the 3rd IEEE Multi-conference on Systems and Control MSC
2009, Saint Petersburg, Russia, July 8-10, 2009.
MODELAREA UNEI VALVE ELECTROMAGNETICE UTILIZATE CA
ELEMENT DE EXECUłIE
(Rezumat)
În ultimii ani elementele de execuŃie din autovehicule au devenit sisteme
mecatronice în care componentele mecanice coexistă cu cele electronice. Deoarece
valvele de control al presiunii sunt folosite ca elemente de execuŃie în multe aplicaŃii de
28
Constantin Florin Căruntu et al.
control în autovehicule, a fost acordată o atenŃie deosebită modelării acestor tipuri de
valve.
Valvele hidraulice utilizează mişcarea mecanică pentru a controla presiunea
unei surse de fluid, fiind folosite ca elemente de execuŃie în componenŃa structurilor de
control ale autovehiculelor. Numeroasele tipuri de valve pot fi clasificate în funcŃie de
rolul lor în: valve de control al direcŃiei fluidului, valve de control al presiunii şi valve
de control al debitului.
Obiectul studiului efectuat l-a constituit o valvă de reducere a presiunii cu 3 căi
cu comanda electromagnetică ce intră în componenŃa transmisiei automate a unui
autovehicul Volkswagen. Pornind de la ecuaŃiile generale prezentate în [6] au fost
dezvoltate două modele neliniare (de tip liniar cu comutaŃie sau liniar pe porŃiuni) ale
acestei valve: un model liniarizat intrare-ieşire şi un model de stare.
Cele două modele au fost implementate în MATLAB-Simulink, pentru
simulare fiind utilizate valorile parametrilor furnizate de firma Continental Automotive
Romania. Pentru validarea modelelor, rezultatele obŃinute prin simulare au fost
comparate cu rezultatele experimentale obŃinute de Continental Automotive Romania pe
un stand de probă utilizând o valvă electromagnetică de tipul investigat. ConcordanŃa
celor două tipuri de rezultate confirmă validitatea modelelor elaborate. Pentru modelul
de stare a fost implementat în final un bloc Simulink de tip S-function.