Flux Observer for Spool Displacement Sensing in. Self-Sensing Push-Pull Solenoids. Perry Y. Li and Qinghui Yuan. Department of Mechanical Engineering, ...
6th International Conference on Fluid Power Transmission and Control (ICFP 2005).
March 2005.
Flux Observer for Spool Displacement Sensing in Self-Sensing Push-Pull Solenoids Perry Y. Li and Qinghui Yuan Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455 USA, {pli,qhyuan}@me.umn.edu
Abstract Spool position feedback in an electrohydraulic valve typically requires a spool displacement sensing device
for spool feedback even more critical in the new unstable valve designs than conventional valves which are designed to be open-loop stable.
such as a LVDT. Self-sensing is the methodology to obtain spool displacement information directly from the solenoid spool actuators, thus obviating the additional cost and footprint of a LVDT or another displacement sensing device. A self-sensing scheme that uses only the measurement of electrical signals to the pair of push-pull solenoids was first proposed in [1]. It uses the position dependence of electrical inductance in the solenoids to infer the spool displacement. The scheme in [1] relies on a flux observer and several infinite dimensional filters that require explicit resets, and is prone to singularity. The present paper proposes a modified scheme in which the flux observer (and hence the self-sensing) problem is cast in a linear time varying system setting so that standard linear observer design techniques can be brought to bear.
Keywords Self-sensing, observer, push-pull solenoids, spool displacement feedback. electrohydraulic valves
1
To reduce the cost of the additional spool displacement sensing device, a self-sensing scheme was proposed in [1] to obtain spool displacement information by measuring the voltage and current in the solenoids in the push-pull configuration. It uses the position dependence property of the solenoid actuators to determine the spool position. The key element in this scheme is the magnetic flux observer. Unfortunately, because the flux observer in [1] relies on solving the initial flux on a finite moving horizon and utilizes several time delayed (hence infinite dimensional) filters, it is rather complicated and is susceptible to singularity. In this paper, we present a reformulation of the problem into the standard observer design formulation, so that many time varying linear design methodologies can be applied. The resulting observer and spool displacement estimator are easier to implement and less prone to singularity.
2
System Models
Introduction
Spool position feedback is often needed or at least desirable for high performance nonlinear control of electrohydraulic systems. Spool position feedback is especially important in our own research of “unstable valves” where the unstable steady [3] and unstable unsteady [2] flow forces are deliberately utilized to improve the spool agility. This is to reduce the force and power requirements for the solenoid actuators to achieve high flow, high performance in single stage direct acting proportional valves. Since the ``unstable” valves are designed to be open-loop unstable, they need to be stabilized via closed-loop feedback. This makes the need
Fig. 1: Push-pull solenoid configuration.
Consider the push-pull solenoid actuator spool in Fig. 1. We assume that the solenoid inductances are given by [1] [4] [5]:
L1 ( x) =
β1 d1 + x
(1)
6th International Conference on Fluid Power Transmission and Control (ICFP 2005).
L2 ( x ) = where
β2
β1
(2)
d2 − x
x [m] is the spool displacement, β 1 , β 2
L1 ( x)
are decreased ( x decreases for solenoid 1 and increases for solenoid 2). The dynamics of the magnetic flux linkages
λ1 , λ 2
β2 L2 ( x )
= d1 + d 2 =: d
(6)
obtained from (1) and (2)., we have
β1
[ H − m] and d 1 , d 2 [m] are the inductances’ constants. Notice that as inductances increase as the magnetic gaps
+
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i1 (t ) i (t ) + β2 2 =d λ1 (t ) λ 2 (t )
(7)
so that the dependence on x has now been eliminated. Notice that if (7) is linear in
λ1
and
λ 2 , then, assuming
observability condition holds, an observer can be easily
[wb-t] are given by:
designed using (3), (4) and (7). Unfortunately, this is not the case.
λ&1 = − R1i1 + u1
(3) 2.1 Exact Linearization through high pass filtering
λ&2 = − R2 i2 + u 2
It is now shown that an exact linear output equation
(4)
can be obtained by using (7) at two time instances. Let t1
I 1 (t1 , t 0 ) := ∫ [− R1i1 (τ ) + u1 (τ )]dτ
where i1 , i2 [A] and u1 , u 2 [V] are the currents through and the voltages across the solenoids. Since these
t1
can be easily measured in the solenoid driver, they are
I 2 (t1 , t 0 ) := ∫ [− R2 i2 (τ ) + u 2 (τ )]dτ
assumed to be available. The flux linkages and the
;
(9)
t9
currents are related by the inductances via:
λ1 = L1 ( x)i1
(8)
t9
λ2 = L2 ( x)i2
(5)
for
t 0 ≤ t so that λi (t ) = λi (t 0 ) + I i (t , t 0 ) for
i=1,2. Then (7) can be written as: It is easy to see that by properly estimating
λ1
and
λ2 ,
the spool displacement x can be computed from (1), (2) and (5).
2
[β i (t) − dI (t,t )]λ (t) + [β i (t) − dI (t,t )]λ (t) 2 2
2
0
1
11
1
0
2
+ dI1(t, t0 )I2 (t, t0 ) = dλ1(t0 )λ2 (t0 ) λi (t ) = λi (t 0 ) + I i (t , t 0 )
where
has
been
(10)
used
Observer design formulation repeatedly. Notice that the left hand side of (10) is linear
2.1 Nonlinear output equation
in
λ1
and
λ2 .
The right hand side of (10), although
Each of the equation in (5) can be thought of as the measurement (output) equation for the state equations (3) and (4) for designing an observer for
λ1
and
λ2 .
If
L1 ( x) and L2 ( x ) are known, then the observer design is quite trivial. The difficulty lies in the spool displacement
x being unknown. In this case, the push-pull solenoid configuration allows us to combine the two equations in (5) to form a new nonlinear output equation [1]. Using (5) and the fact that
nonlinear in the unknown initial states
λ 2 (t 0 ) ,
λ1 (t 0 )
and
is not a function of time t. So it can be
eliminated by some form of high pass filtering. In particular,
[ [
] ]
⎫⎪ d ⎧⎪ β 2i2 (t ) − dI 2 (t, t0 ) λ1 (t ) + ⎬=0 ⎨ dt ⎪⎩ β1i1 (t ) − dI1 (t, t0 ) λ2 (t ) + dI1 (t, t0 )I 2 (t, t0 )⎪⎭ (11) A similar expression to (10) was obtained in [1]
6th International Conference on Fluid Power Transmission and Control (ICFP 2005). where (7) is expressed solely in terms of
λi (t 0 ) and
I i (t , t 0 ) but not in terms of λi (t ) . The approach there was to solve for
λi (t 0 )
first using data in the
March 2005.
voltages i1 (t ), i 2 (t ), u1 (t ), u 2 (t ) will not accumulate in
y (t ) .
Remark: Although the Box-car filter is simple to understand, it has the disadvantage that it is an infinite
immediate previous finite horizon. This poses a
dimensional filter. A finite dimensional low pass filter
singularity problem when the data during the horizon is
may also be applied to (11). This will also have the
not sufficiently exciting enough (e.g. when the signal is
similar effect of gradually forgetting the past currents and
not changing) to provide a unique solution for λi (t 0 ) . In
voltages.
this paper, we develop an output equation in terms of
λi (t 0 )
3
Kalman Filter for Flux Reconstruction
so that standard observer design techniques can Eqs. (3)-(4), and (12) with the “measurement”
be applied. The resulting observer updates the state estimates continuously via a set of differential equation without solving for
λi (t 0 )
explicitly, thus avoiding the
y (t ) given by (13)-(14) is in a standard time varying linear state space form. Thus, we can design a Kalman Filter
singularity issue.
[5]
to
reconstruct
the
unknown
states
λ1 (t ) , λ 2 (t ) :
2.1 Box-car filter
[ ]
The integration of (11) over the period t 0 , t , or
d ⎡ λˆ1 ⎤ ⎡ − R1i1 + u1 ⎤ ⎢ ⎥=⎢ ⎥ − L(t )( yˆ (t ) − y (t )) dt ⎣λˆ2 ⎦ ⎣− R2i2 + u2 ⎦
[
]
⎡ β (i (t ) − i2 (t − T )) − d I 2 (t , t − T )⎤ yˆ (t ) = λˆ1 (t ) λˆ2 (t ) ⎢ 2 2 ⎥ ⎣ β1 (i1 (t ) − i1 (t − T )) − d I1 (t , t − T ) ⎦
equivalently, taking the difference of (10) evaluated at t
(15)
and at t 0 gives:
where L(t) is the Kalman filter gain obtained by solving
[λ1 (t )
⎡ β 2 (i2 (t ) − i2 (t 0 )) − d I 2 (t , t 0 )⎤ ⎥ = yt0 (t ) ⎣ β1 (i1 (t ) − i1 (t 0 )) − d I 1 (t , t 0 ) ⎦
λ2 (t )]⎢
(12) where yt0 (t ) = − d I1 (t , t0 ) I 2 (t , t0 ) − β 2i2 (t0 ) I1 (t , t0 ) − β1i1 (t0 ) I 2 (t , t0 )
the Riccati Differential Equation (RDE):
P& (t ) = − P(t )C T (t ) R −1C (t ) P(t ) + Q L(t ) = P(t )C T (t ) R −1
(16)
⎡β (i (t ) − i2 (t − T )) − d I 2 (t , t − T )⎤ C (t ) = ⎢ 2 2 ⎥ ⎣ β1 (i1 (t ) − i1 (t − T )) − d I1 (t , t − T ) ⎦
(13)
T
R ∈ ℜ + and Q ∈ ℜ 2×2 are a positive scalar and a
We can interpret (12) and (13) as the output equations for
Here,
the state equations (3)-(4), with (13) furnishing the
positive definite matrix respectively, that specify the
“measurement”. Since Eqs. (12)-(13) are valid for any
importance of process noise and measurement noise.
t0 ≤ t , we can take t0 = t − T where T is a fixed time The stability of the Kalman filter ensures that in the interval. In this case, we denote
y (t ) := yt −T (t )
absence (14)
This makes (12) the result of filtering (11) using a Box-car or a moving average filter of length T. This has the advantage that measurement errors in currents and
of
measurement
model noise,
uncertainty, we
have
process
noise
λˆ1 (t ) → λ1 (t )
or and
λˆ2 (t ) → λ2 (t ) . The spool displacement estimate can then be
6th International Conference on Fluid Power Transmission and Control (ICFP 2005). obtained by solving x 10
March 2005.
-3
⎡ i1 / λˆ1 ⎤ ⎡1 / L1 ( x) ⎤ ⎡ β1 / d1 ⎤ ⎡1 / d1 ⎤ ⎢ ˆ ⎥=⎢ ⎥=⎢ ⎥+⎢ ⎥⋅x ⎣i1 / λ2 ⎦ ⎣1 / L2 ( x)⎦ ⎣ β 2 / d 2 ⎦ ⎣1 / d 2 ⎦
Spool displacement - m
from (1)-(2) and (5) using the least squares method.
4
x estimated x
3
Simulations
2 1 0 -1 -2
Simulation results for the self-sensing spool displacement -3
scheme is presented in this section. The solenoid and parameters are: β1 = β 2 = 2.68e −4 H − m
0
1
2
3
4
5
6
7
8
9
10
Time - s
d = 0.0078 m. The time horizon T for the Box-car filter in (14) is 0.2s. The spool dynamics are given by a
Fig. 3: Actual and estimated spool displacements
mass-spring-damper system with sufficiently large spring constant so that the valve is open-loop stable. In the simulation, the spool is commanded to move according to corrupted by white noise of amplitude about 10% of the voltage signals themselves (Fig. 2). Using
R = 1e − 5, Q = 1e8 ⋅ I 2×2 in the Kalman
Solenoid 1
λ1 and estimate [Wb-t]
a 0.1Hz to 5Hz chirp signal. The input voltages are
0.1
λ1 0.05 Estimate 0
0
1
2
3
4
show that the self-sensing scheme is effective in estimating the unknown flux linkage and the spool displacement.
λ2 and estimate [Wb-t]
Filter design (16), the results are shown in Figs. 3-6. They 0.1
5 6 Time - s Solenoid 2
7
8
9
10
7
8
9
10
Estimate
0.05
λ2 0
0
1
2
3
4
1.3
5 Time - s
6
1.2
Fig. 4: Actual and estimated flux linkages Flux linkage estimate errors - Wb
Volts - V
1.1
1
0.9
0.8
0.7
0.5
1
1.5
2
2.5
3 3.5 Time - s
4
4.5
5
0.03 0.02 Solenoid 1
0.01 0
Solenoid 2
-0.01 1
2
3
4
5.5
5 Time - s
6
7
8
9
Fig. 5: Flux linkage estimation error Fig. 2: Input voltages (with process noise) to solenoids
-3
x estimate error - m
1
x 10
0
-1
-2
0
1
2
3
4
5 Time - s
6
7
8
9
Fig. 6: Spool displacement estimation error
10
6th International Conference on Fluid Power Transmission and Control (ICFP 2005).
5. Conclusions The proposed self-sensing spool displacement scheme uses the position dependence of the inductances in the solenoids to estimate the spool position without the use of a LVDT. By reformulating the nonlinear observer problem into a linear time varying system, time varying Kalman filter can be used to design a flux observer and a displacement estimator. Simulation shows that it is effective. The scheme however relies on a simple theoretical model. It would be interesting to see how the approach can be extended to empirically obtained solenoid inductance model.
Acknowledgement: This research is supported by the National
Science
Foundation
under
grant
ENG/CMS-0088964.
References [1] Yuan, Q.-H., Li, P. Y. Self-sensing actuators in electrohydraulic valves. Proceedings of the 2004 ASME-IMECE,
Anaheim,
CA,
USA,
Paper
#62104, 2004. [2] Krishnaswamy, K., Li, P. Y. On using unstable electrohydraulic valves for control. ASME Journal of Dynamic Systems, Measurement and Control, Vol 124(1), pp. 182-190, 2002. [3] Yuan, Q.-H., Li, P. Y. Using steady flow force for unstable valve design. ASME Journal of Dynamic Systems, Measurement and Control, (to appear). [4] Xu, Y., Jones, B. A simple means of predicting the dynamic response of electromagnetic actuators. Mechatronics, Vol. 7(7), pp. 589-598, 1997. [5] Koch, C. R., Lynch, A. F., Chladny, B. R., Modeling and
control
of
solenoid
combustion engines. 2
nd
valves
for
internal
IFAC Conference on
Mechatronic Systems, Berkeley, CA., USA. pp. 213-218, 2002. [6] Anderson, B. D. O., Moore, J. B. Optimal Control – Linear Quadratic Methods, Prentice Hall, 1989.
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