FPST TOC
Proceedings of IMECE’04
2004 ASME International Mechanical Engineering Congress and RD&D Expo
November 13-19, 2004, Anaheim, California USA
IMECE2004-62109
SELF-CALIBRATION OF PUSH-PULL SOLENOID ACTUATORS IN
ELECTROHYDRAULIC VALVES ∗
QingHui Yuan
Dept. of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota 55455
Email: qhyuan@me.umn.edu
Perry Y. Li
Dept. of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota 55455
Email: pli@me.umn.edu
ABSTRACT
System parameters for solenoid actuators are important for
high performance control and for self-sensing. Due to the nonlinearities in the solenoid actuators, parameter identification procedures that aim to obtain the electro-mechanical property can
be complex and time consuming. In this paper, a self-calibration
procedure for solenoid actuators in push-pull configurations is
proposed. Utilizing the fact that the inductances of the solenoids
share the same parameters as those for the electromagnetic force,
the parameters for the electromagnetic force can be obtained
from the easily obtainable electrical signals such as the voltage
and current signals, and two inexpensive on-off sensors. The
calibration procedure involves only actuating the solenoid actuator back and forth. Simulation study is presented to verify the
method.
flow control valves. In this application, a pair of solenoids in
a push-pull configuration, or a single solenoid acting against a
centering spring, is used to drive a spool whose displacement
in turn meters the flow through the valve. In order to design
high performance control systems that utilize solenoid actuators,
appropriate system models and accurate system parameters that
describe the electro-mechanical properties are needed. Accurate
system parameters are also needed for the design of self-sensing
systems in which the spool displacement information is obtained
from the electrical information [4].
Three methods are typically available for determining the
electro-mechanical properties of solenoid actuators: (1) Finite
Element Method (FEM) analysis of the complete actuator if all
the geometry and material properties are known [5] [6]; (2) Estimating the parameters of the solenoids based on manufacturer
data sheet [7]; (3) Determining the system parameters experimentally [8] [3].
Each method has some deficiencies. The FEM method requires accurate knowledge of the material property and geometric parameters. In addition, many models with different air gaps
need to be constructed and simulated, which can be tedious and
time consuming. Hence, the FEM method is only useful in the
solenoid design stage, rather than for system identification or
calibration. On the other hand, using the manufacturers’ specifications cannot be used to determine an accurate system model
for each product item. Experimentally determining the system
model would be the most reliable. However, because of the nonlinear properties, a large number of experiments are required to
reflect the various combinations of the states and inputs. This is
1 INTRODUCTION
Electromagnetic (EM) actuators are used to provide noncontacting translational force directly. EM actuators are used
in a wide range of applications, such as the control of intraventricular balloons to simulate a beating heart [1], and drug
delivery [2] in bio-engineering, camless engines in automotive
applications [3], and the control of the valve spools in electrohydraulic systems.
Our primary interest in EM actuators is in the context of
solenoid actuators used for controlling proportional hydraulic
∗ THIS RESEARCH IS SUPPORTED BY THE NATIONAL SCIENCE
FOUNDATION ENG/CMS-0088964.
1
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g
Using the principle of virtual work, the magnetic force F in
the direction that opens or closes the air-gap is given by
Coil
Core
F=
Armature
A
∂Wm
B2 A
=
.
∂g
2µ0
(3)
Magnetic flux
The magnetic flux density B is given by:
Figure 1.
Cross section of a commercial solenoid
B=
particularly difficult when extra instrumentations are needed to
measure the mechanical forces. In this paper, we propose a selfcalibration method for solenoid actuators in the push-pull configuration, in which the electrical signals of are utilized to estimate
the parameters in solenoid models. The self-calibration approach
can be accomplished in a short time (a matter of seconds) and requires only a small amount of additional inexpensive hardware
(for monitoring the current or voltages, and for sensing when the
actuator hits the end stops). Moreover, the procedure is almost
identical as to the normal operation of the actuator so that it can
be carried out in-situ and repeated whenever the system parameters are suspected to have varied. Thus robustness of the overall
system can also be enhanced.
The rest of the paper is organized as follows. In Section
2, we formulate a solenoid model for control. In Section 3, the
principle of the self-calibration method is presented. Simulation
study is included in Section 4. Some concluding remarks are
included in Section 5.
λφ Φ λφ Ni
=
A
A R
(4)
where λφ is the flux leakage coefficient, N is the number of turns
in the coil, i is the coil current. R is the reluctance of the solenoid
given by:
g
+
Aµ0
R =
Z
1
dl
A(l)µ(l)
where the first term corresponds to the reluctance of the air-gap,
and the second term corresponds to the reluctance of the rest of
the circuit. In (5), A(·) and µ(·) are the area and the permeability
of the segment along the magnetic circuit.
In typical operating conditions when the air gap is small and
the current does not saturate the core, the second term in (5) is a
constant. Thus, the reluctance can be approximated as the affine
function of g
R = C0 · g +C1
2 SOLENOID MODEL
A typical commercial solenoid is shown in Fig 1. The magnetic flux circuit goes through the armature, the stationary iron
core, and the air gap between them. The density of the stored
energy in the gap is given by [9]:
wm =
1 B2
2 µ0
F=
(1)
B2 Ag
2µ0
(6)
where C0 and C1 are constants.
Combining (3), (4), and (6) gives:
λ2φ N 2 i2
2µ0 A(C0 g +C1 )2
.
(7)
Notice that F is a nonlinear function of the air gap length g and
the current i. Eq. (7) can be further simplified as:
where B is magnetic flux density, and µ0 is the permeability in
free space. When the small air-gap is small, B can be assumed to
be uniform so that the total energy stored in the gap is given by:
Wm = wmV = wm Ag =
(5)
F=
i2
β
2 (g + d0 )2
(8)
where
(2)
β=
where V , A and g are the volume, area and width of the gap.
2
λ2φ N 2
µ0 AC02
d0 =
C1
.
C0
(9)
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Now β and d0 are all the parameters that are necessary for specifying the model.
Consider two identical solenoids (with subscripts i = 1 or 2)
are set up in the push-pull configuration as in Fig. 2. Let the
input voltages across the coils be u1 and u2 , and the resulting
currents through the coils be i1 and i2 . The dynamics of the flux
linkages λ1 , λ2 are given by
λ̇1 = −Ri1 + u1
λ̇2 = −Ri2 + u2
Let the armatures of the two solenoids be rigidly connected
to the spool in a push-pull configuration as in Fig. 2. Assume
that two on-off touch sensors A and B are mounted between the
armature and the stationary core of each solenoid to detect when
the spool reaches either end stop. The touch sensor is in the
off state when the metal pad mounted on its armature does not
contact the other metal pad mounted on the core. It is turned on
when the two metal pads contact each other.
Let x ∈ [−xmax , xmax ] be the spool displacement from the
middle position where ±xmax are the locations of the right and
left end-stops. We assume that xmax has been physically measured in advanced. Signals from the touch sensors A and B can
be used to detect to the times when x = −xmax or x = xmax respectively.
Notice that g1 + g2 = 2xmax , g1 = xmax + x and g2 = xmax − x,
so that if we introduce the constant
(10)
where R is the resistance of each solenoid. By definition, the flux
linkages are related to the currents via the inductances:
λ1 = L1 (g1 )i1
λ2 = L2 (g2 )i2
d := d0 + xmax ,
(11)
then the inductances in (13) can be expressed in terms of the
spool displacement:
where L1 (g1 ) and L2 (g2 ) are the inductances of solenoid 1 and
solenoid 2 when their air-gaps are g1 and g2 respectively. The
energies stored in the solenoids are given by [9]:
β
d +x
β
L2 (x) =
d −x
L1 (x) =
1
w1 = L1 (g1 )i21
2
1
w2 = L2 (g2 )i22
2
(12)
We assume that both the currents i1 , i2 and voltages u1 , u2
in (10) can be measured. We also assume that the resistance R in
(11) has also been measured beforehand.
Integrating (11) for each solenoid i = 1, 2, over the time interval t ∈ [t0 ,t f ],
Equating (12) with (1), and using (9), we have:
β
g1 + d0
β
L2 (g2 ) =
g2 + d0
(15)
L1 (g1 ) =
λi (t f ) − λi (t0 ) =
(13)
Z tf
t0
Ii =
Z tf
t0
(−Rii + ui ) dt =: Ii
(16)
Now,
so that the mechanical force of solenoid i = 1, 2 can be expressed
completely in terms of the electrical variables:
Fi =
Li (gi ) 2
i .
2β i
β
i1 (t)
d + x(t)
β
λ2 (t) = L2 (x(t))i2 (t) =
i2 (t)
d − x(t)
λ1 (t) = L1 (x(t))i1 (t) =
(14)
(17)
(18)
(19)
3 Auto-calibration
We now propose a method in which only the electrical signals of the dual-solenoid actuator, such as the voltages and the
currents, and a minimal amount of displacement information are
utilized to estimate the parameters β and d0 in the solenoid models. The estimation procedure can be implemented similarly to
the normal operation of the solenoids.
Therefore,
·
¸
i1 (t f )
i1 (t0 )
λ1 (t f ) − λ1 (t0 ) = β
= I1
−
d + x(t f ) d + x(t0 )
¸
·
i2 (t f )
i2 (t0 )
= I2
−
λ2 (t f ) − λ2 (t0 ) = β
d − x(t f ) d − x(t0 )
3
(20)
(21)
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+u2-
+u1i1
d-x-d0
d+x-d0
i2
Spool
x
On-off position
sensor A
Figure 2.
On-off position
sensor B
A dual-solenoid configuration for self-calibration.
We now describe the computational procedure when the
spool is stroked multiple times. Suppose that the spool is stroked
from one end stop to another N times, i.e. x = ±xmax alternately
N times. Let t = tk for k = 0, 1, 2, 3 . . . N be the times when either
sensor A or B turns from off to on. That is, we have the displacement information x(tk ) = −xmax or x(tk ) = xmax , depending on
which sensor is on.
λ̂1
λˆ1 (t k )
∆λ1,k
First of all, consider the flux linkage of solenoid 1. At the
time interval t ∈ [tk ,tk+1 ], we have
λˆ1 (t k +1 )
t k +1
tk
time
∆λ1,k := λ1 (tk+1 ) − λ1 (tk )
Figure 3. The estimated flux linkage for solenoid 1 at the discrete time
tk for k = 1, 2, 3 · · · N .
=
Z tk+1
tk
Eliminating β, we have the following nonlinear algebraic equation
I2
½
i1 (t f )
i1 (t0 )
−
d + x(t f ) d + x(t0 )
¾
− I1
½
i2 (t f )
i2 (t0 )
−
d − x(t f ) d − x(t0 )
¾
(u1 − i1 R)dt
(23)
In addition, let β̂ and dˆ denote the parameter estimates. Since we
know x(tk ) and x(tk+1 ), Eq. (11) gives the estimate of the flux
linkages
= 0,
(22)
from which d can be solved if x(t0 ) and x(t f ) are known. The
touch sensors can be used to determine t0 and t f when the actuator is at x(t0 ) or x(t f ) = ±xmax . Once d has been solved, β can
be solved from (20) or (21) as well.
The accuracies of the estimated β and d depend on the accuracy of the resistance R. For this reason, R should be measured
right before the identification procedure to minimize the effect
due to temperature variation.
Although in theory, we can formulate (22) (and solve for d
and β) by stroking the spool once (i.e. moving the spool from
one end to another), in practice, the spool should be stroked back
and forth multiple times in order to reduce uncertainties and the
effects of noise.
λ̂1 (tk ) =
λ̂1 (tk+1 ) =
β̂
i1 (tk )
ˆ
d + x(tk )
β̂
i1 (tk+1 )
dˆ + x(tk+1 )
(24)
As shown in Fig. 3, this is a continuous time process with discrete measurement. Likewise, we can obtain ∆λ2,k , λ̂2 (tk ) and
λ̂2 (tk+1 ) for solenoid 2. It is clear that if the parameters are
estimated correctly, then λ̂1 (tk+1 ) − λ̂1 (tk ) = ∆λ1,k , λ̂2 (tk+1 ) −
λ̂2 (tk ) = ∆λ2,k . The parameters can be obtained by minimizing
4
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VDD
the following objective function
ˆ
J(β̂, d)
(25)
N−1
=
∑ [λ̂1 (tk+1 ) − λ̂1 (tk ) − ∆λ1,k ]2 + [λ̂2 (tk+1 ) − λ̂2 (tk ) − ∆λ2,k ]2
M2
M1
k=0
Solenoid
ˆ T , and take the derivative of Eq. (25) with
Define ξ = [β̂, d]
respect to ξ
M3
M4
£
¤
∂J
= 2 ∑ ∑ Dm,k Am,k , β̂Dm,k Bm,k
∂ξ
k=0 m=1
N−1 2
g(ξ) :=
(26)
GND
Figure 4. H-bridge circuit for solenoid drive. M1, M2, M3, M4 are
MOSFETs. M1, M3 pair and M2, M4 pair operate complementarily.
where for m = 1, 2,
Dm,k = λ̂m (tk+1 ) − λ̂m (tk ) − ∆λm,k ,
Vdd
im (tk )
im (tk+1 )
−
,
dˆ + x(tk+1 ) dˆ + x(tk )
im (tk )
im (tk+1 )
+
.
Bm,k = −
2
ˆ
ˆ
(d + x(tk+1 ))
(d + x(tk ))2
Coil
D1
Am,k =
M1
Vc
+
VR
Op Amp
Then we will use Newton’s method to solve for ξ so that g(ξ) =
0:
R
Gnd
ξi+1 = ξi − ε
µ
∂g
∂ξ
¶−1
g(ξi ) for i = 0, 1, 2, 3, · · ·
Figure 5. Electric circuit for driving solenoid.
control the current through the solenoid.
(27)
where ε < 1, and
is the control signal to
the displacement limits being ±xmax = ±6.4 × 10−3 m, the system parameters to be identified are:
#
"
N−1 2
β̂Am,k Bm,k + Dm,k Bm,k
A2m,k
∂g
,
=2 ∑ ∑
2 2
∂ξ
k=0 m=1 β̂Am,k Bm,k + Dm,k Bm,k β̂ Bm,k + β̂Dm,kCm,k
Cm,k = 2
Vc
β = 2.6386 × 10−4 NA−2 m2 , d = 7.76 × 10−3 m.
We assume and model each solenoid as being energized by the
H-bridge circuit in Fig. 4. The advantage of H-bridge configuration over the current driver circuit in Fig. 5 is that by carefully designing the PWM MOSFET drive signals, we can obtain
the bidirectional voltage excitation across the solenoid, thereby
improving our ability to control the current. The H-bridges are
driven by PWM signals with a carrier frequency of 500Hz a duty
ratio between 0% to 100%. The power supply is 24V . Based on
the signals provided by sensors A and B, a controller is designed
to actuate the spool so that it moves toward sensor B if it touches
sensor A, and vice versa. Hence we can achieve multiple strokes
of the spool for self-calibration. The details of the controller will
not discussed in this paper.
Fig. 6 shows the measured signals in simulation during the
im (tk+1 )
im (tk )
−2
.
3
ˆ
ˆ
(d + x(tk+1 ))
(d + x(tk ))3
The initial solution ξ0 is arbitrarily guessed.
4 SIMULATION
The proposed identification algorithm is simulated in the
Matlab/Simulink environment (Mathworks Inc., US). We model
the two solenoids identically modeled using typical parameters
of a commercial solenoids with R = 0.5Ω, d0 = 1.36 × 10−3 m,
5
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0.014
A
B
A
B
A
B
0.012
A
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
β^ (NA−2m2)
Sensor event
2
0.4
0.006
0.004
0.01
Position (m)
0.01
0.008
0.002
0
0
20
40
60
80
100
120
140
0
20
40
60
80
100
120
140
0
0.02
−0.01
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.018
0.4
0.016
d^ (m)
Voltage (V)
50
0
u1
u2
−50
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Current (A)
0.012
0.01
0.008
0.4
0.006
1
Figure 7.
0.5
0
−0.5
0.014
0.05
0.1
0.15
0.2
time (s)
0.25
0.3
0.35
ˆ
[β̂, d]
iteration steps in Newton’s method.
i1
i2
0
Estimated parameters
in Eq.
(25)
as a function of
The solution converges to
2.68 × 10−4 NA−2 m2 , dˆ = 0.0078m.
β̂ =
0.4
electrical signals, and minimal displacement information. The
proposed approach is cost-effective as it shares nearly identical
hardware to that in a single stage proportional valve. The only
extra overhead is two on-off sensors, which can be implemented
inexpensively. The procedure can be completed in a couple of
seconds. The procedure can also be repeated whenever needed.
When combined with a self-sensing scheme [4], the high cost of
LVDTs for spool displacement measurement commonly used in
proportional valves can be eliminated.
Figure 6. Simulation result for dynamic identification. Both solenoids are
modeled using β = 2.6386 × 10−4 NA−2 m2 and d0 = 1.36 × 10−3 m.
Let xmax = 6.4 × 10−3 m, then d = 7.76 × 10−3 m. In the top figure,
symbol ’A’ represents that sensor A is on, symbol ’B’ represents that sensor B is on.
self-calibration process. Within 0.4s, the spool has moved back
and forth N = 6 times. Assume that tk for k = 0 . . . N correspond
to the times when either sensor A or B turns on. For example,
after starting the process, sensor A first turns on at t = 0.0382 s,
and then sensor B turns on at t = 0.086 s, so t0 = 0.0382 s, and
t1 = 0.086 s etc. Hence, from Eq. (24), we know that λ̂1 (tk ),
λ̂2 (tk ). Next, from the measured currents i1 (t), i2 (t) and the voltages u1 (t), u2 (t), we can calculate ∆λ1,k , ∆λ2,k in Eq. (23). Finally, Eq. (27) is utilized to obtain β̂ and dˆ via iteration.
As shown in Fig 7, the estimates from the self-calibration
converge to β̂ = 2.68 × 10−4 NA−2 m2 , dˆ = 0.0078m which differ
from the actual parameters by less than 1.5%. The total consumed time that includes experiment and post processing, would
be just a couple of seconds.
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5 CONCLUSION
We have proposed a self-calibration method for determining
the actuator model for dual-solenoid actuators which are common in electrohydraulic valves. The key idea is to utilize the fact
that the electro-mechanical force model and the electro-magnetic
model of the solenoid share the same system parameters. Selfcalibration is then achieved by measuring the easily obtainable
6
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Copyright °
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[8] Stubbs, A., 2000. Modeling and controller design of an electromagnetic engine valve. Master’s thesis, University of Illinois at Urbaba-Champaign.
[9] Mohan, N., Undeland, T. M., and Robbins, W. P., 2003.
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