A Linearized Electrohydraulic

Servovalve Model for Valve

Dean H. Kim
Department of Mechanical Engineering,
Dynamics Sensitivity Analysis
Bradley University,
Peoria, IL 61625 and Control System Design
Tsu-Chin Tsao This paper presents the derivation of a linearized model for flapper-nozzle type two-stage
Mechanical & Aerospace
electrohydraulic servovalves from the nonlinear state equations. The coefficients of the
Engineering Department,
linearized model are derived in terms of the valve physical parameters and fluid proper-
University of California, Los Angeles, ties explicitly, and are useful for valve design and sensitivity analysis. When using this
Los Angles, CA 90095-1597 model structure to fit experimental frequency response data, the results render closer
agreement than when using existing low order linear models. This model also suggests
important servovalve dynamic properties such as the nonminimum phase zero and the
transfer function relative degree, and how they relate to the valve component arrange-
ment. Because of the small modeling errors over a wide frequency range, a high band-
width control system can be designed. A robust performance controller is designed and
implemented to demonstrate the utility of the model. 关S0022-0434共00兲03401-8兴

1 Introduction unique applications which require bandwidth close to the valve

bandwidth. For example, in noncircular turning for camshaft ma-
The design and analysis of electrohydraulic control systems re-
chining 关9,10兴, a typical cam profile can be approximated by its
quires the development of a control-oriented model. While the first ten to twenty Fourier harmonics. Under a nominal work
nature of hydraulic systems is nonlinear, it is often desirable to speed of 1200 rpm 共20 Hz兲, the 20th Fourier harmonic is at 400
have a linear model for linear control design approach. Therefore, Hz. To design such high bandwidth control system, a model that
an accurate linear model for the electrohydraulic system would be is accurate beyond the desired closed loop bandwidth must be
useful for valve design in tailoring the valve dynamics from con- used. This point will be demonstrated by a robust performance
trol standpoint as well as for high performance control system control design example. Controllers are said to provide robust
design. performance for a system if they maintain stability and achieve a
Previous literature on electrohydraulic control systems has in- certain performance in the presence of modeling uncertainty. In
corporated the servovalve dynamics to various extents. Some this paper the robust performance problem is formulated in the
authors ignore the servovalve dynamics 关1,2兴. Other authors who framework of H ⬁ optimization theory 关11–14兴 solved via the
consider servovalve dynamics have used either an assumed mixed sensitivity problem 关15兴, and is verified by experiment.
second-order model 关3–5兴 or a third-order model 关6兴. By far, The organization of the paper is as follows. The nonlinear
the third-order model is the most accurate one. Nonlinear servo- model is presented and is linearized around its equilibrium state,
valve models have been presented in several sources 关6–8兴. How- resulting in the cascaded servovalve linear model and actuator
ever, their usage for control system design and analysis are very linear model, respectively. Various features of the derived model
limited because of lack of nonlinear tools other than numerical are discussed, and this improved model is compared with existing
simulation. linear models. Robust performance controllers are synthesized
In this paper, an improved linearized model of a commonly based on the derived model and the existing second-order model
used two-stage servovalve is derived from a nonlinear state equa- and third-order model, respectively. The experimental results
tion model. The coefficients of this derived model are expressed demonstrate that the control system designed based on the more
in terms of the system parameters, so that servovalve performance accurate model achieves better robust performance.
potentially can be improved by the prudent choice of these param-
eters. The derived model for a valve with a cantilever feedback
spring, the type experimentally tested in this paper, suggests the
2 Dynamic Models of Electrohydraulic Actuators
existence of a nonminimum phase zero. Another type, the spool The electrohydraulic servo system shown in Fig. 1 consists of a
spring feedback valve, is shown to be minimum phase but with an two-stage flow control servovalve and a double-ended actuator.
additional relative degree. The servovalve has a symmetrical double-nozzle and a torque-
Compared with the existing linear models in the literature, this motor driven flapper for the first stage, and a closed center four-
derived model fits closer to the experimental data, especially for way sliding spool for the second stage. Figure 1 displays two
high frequencies. Control system design based on the more accu- types of feedback spring commonly used: a cantilever spring con-
rate model is expected to achieve higher performance because of necting the flapper and spool, and a spring directly acting on the
the reduced unmodeled dynamics. While many industrial electro- spool. The system nonlinear model and linearized model are pre-
hydraulic servo applications demand closed loop bandwidth far sented for either type of feedback spring.
below the servovalve bandwidth and therefore do not require a The nonlinear dynamic model presented next is a compilation
high fidelity model for servo control design, there are some of results from Merritt 关6兴, Watton 关7兴, and Lin and Akers 关8兴. The
system has ten state variables and two input variables as defined
Contributed by the Dynamic Systems and Control Division for publication in the
in the nomenclature and depicted in Fig. 1.
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript The torque-motor stage dynamics are given by
received by the Dynamic Systems and Control Division February 2, 1998. Associate
Technical Editor: R. Chandran. ␪˙ ⫽⍀ (1)

Journal of Dynamic Systems, Measurement, and Control MARCH 2000, Vol. 122 Õ 179
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 where

x s ⬎0: f r ⫽2C q Wx s cos ␪ f 共 P s ⫺p a1 ⫹p a2 兲 (7)

q 1 ⫽C q Wx s 冑 2 共 P s ⫺p a1 兲
␳
(8)

q 2 ⫽C q Wx s 冑 2p a2
␳
(9)

x s ⫽0: f r ⫽0 (10)

q 1 ⫽0 (11)

q 2 ⫽0 (12)

x s ⬍0: f r ⫽2C q Wx s cos ␪ f 共 P s ⫺p a2 ⫹p a1 兲 (13)

q 1 ⫽C q Wx s 冑 2p a1
␳
(14)

q 2 ⫽C q Wx s 冑 2 共 P s ⫺p a2 兲
␳
(15)
Fig. 1 Electrohydraulic servo actuator
The first two terms in Eq. 共6兲 are the forces from the spool pres-
sures and the spool damping, respectively. The next two terms in
Eq. 共6兲 are, respectively, the flow reaction force, which depends
1 on the actuator pressures, and the transient flow force, which de-
⍀̇⫽ 兵 K i⫺ 共 K a ⫺K m 兲 ␪ ⫺K 关 x s ⫹ 共 R⫹B 兲 ␪ 兴 ⫺B v ⍀⫺ 共 p s1 ⫺p s2 兲 pends on the actuator flow rates. These two terms show the de-
J t
pendence of the servovalve on the actuator dynamics given below.
⫻A n R⫺4 ␲ C 2qn R 关共 X f m ⫺R ␪ 兲 2 p s1 ⫺ 共 X f m ⫹R ␪ 兲 2 p s2 兴 其 (2) The last term in Eq. 共6兲 is the restoring force from the feedback
spring, where
The first term in Eq. 共2兲 is the driving moment from the torque-
motor. The last five terms are restoring moments on the armature K 关 x s ⫹ 共 R⫹B 兲 ␪ 兴
from the net rotational stiffness, the cantilever feedback spring F sp⫽ ⫹2K s x s (16)
stiffness 共if this spring exists兲, the damping in the torque-motor, R⫹B
the pressure difference across the nozzle, and the dynamic flow Note that K s ⫽0 for a servovalve with a cantilever feedback
force on the flapper, respectively. spring, and K⫽0 with a direct feedback spring.
The flapper-nozzle stage dynamics are given by The flow continuity through actuator is given by

ṗ s1 ⫽
␤e
V s1
再 冑
C qo A o
2 共 P s ⫺p s1 兲
␳
⫺C qn a cx 冑 2p s1
␳
⫺A s v s 冎 ṗ a1 ⫽
␤e
共 q ⫺A 1 v a 兲
V a1 1
(17)
(3)

ṗ s2 ⫽
␤e
V s2
再 冑
C qo A o
2 共 P s ⫺p s2 兲
␳
⫺C qn a cy 冑 2p s2
␳
⫹A s v s 冎 ṗ a2 ⫽
␤e
共 A v ⫺q 2 兲
V a2 1 a
(18)

(4) where V a1 ⫽V a0 ⫹A 1 x a and V a2 ⫽V a0 ⫺A 1 x a .

where Finally, the force balance on actuator is given by
a cx ⫽curtain area⫽ 共 X f m ⫺R ␪ 兲 ␲ D n ẋ a ⫽ v a (19)
a cy ⫽curtain area⫽ 共 X f m ⫹R ␪ 兲 ␲ D n
1
V s1 ⫽V so ⫹A s x s v̇ a ⫽ 关共 p a1 ⫺p a2 兲 A 1 ⫺B a v a ⫺ f d 兴 (20)
Mt
V s2 ⫽V so ⫺A s x s The actuator flow rates in Eqs. 共8兲, 共9兲, 共14兲, and 共15兲 depend on
The first two terms in Eq. 共3兲 refer to the volume flows on the the spool position.
‘‘p s1 ’’ side, through the first orifice and through the nozzle, re- The derivation of the linearized model with respect to an equi-
spectively. The first two terms in Eq. 共4兲 refer to the volume flows librium state from the above nonlinear model is tedious but
on the ‘‘p s2 ’’ side, through the second orifice and through the straightforward. The equilibrium states are derived for zero inputs,
nozzle, respectively. Lin and Akers 关8兴 include a leakage volume i.e., i⫽0 and f d ⫽0:
flow term q L ⫽K L (p s1 ⫺p s2 ), which is omitted for this paper.
Watton 关7兴 also ignores this leakage flow for this stage.
␪ e ⫽0, ⍀ e ⫽0, x se ⫽0, v se ⫽0, x ae ⫽0, v ae ⫽0,

冋 册
The force balance on spool is given by
C 2qo A 2o
ẋ s ⫽ v s (5) p s1e ⫽p s2e ⫽ Ps
C 2qo A 2o ⫹C 2qn X 2f m ␲ 2 D 2n
1
v̇ s ⫽ 兵 共 p s1 ⫺p s2 兲 A s ⫺B s v s ⫺ f r ⫺ 关 ␳ L 2 q̇ 1 ⫺ ␳ L 1 q̇ 2 兴 ⫺F sp其 , p a1e ⫽p a2e ⫽0.5P s
Ms
(6) The equilibrium actuator pressures are computed by assuming that

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 the double-ended actuator utilizes a matched and symmetrical ser- As
vovalve 关6兴, i.e., z⫽2 D ⫺D 2 (27)
K 1
p a1e ⫹p a2e ⫽ P s (21) where D 1 ,D 2 ⬎0. For large spring values, the zero is stable. As
the spring constant reduces the zero migrates to become nonmini-
The linearized equations are derived for the two cases of mum phase. This has a profound effect on feedback control design
x s ⬎0 and x s ⬍0 and are evaluated at the equilibrium states. It is and the achievable closed loop performance. If such nonminimum
found that the two cases converge to the same linearized equation. phase is too close to the imaginary axis, i.e., the feedback spring is
It is also found that the servovalve dynamics do not depend on the not weak enough, the achievable control bandwidth will be fairly
actuator pressures and flow rates as they do in the nonlinear small. Transfer functions from inputs to other system variables
model. Therefore the linearized model can be represented as a also are given in Appendix A.
servovalve model cascaded with an actuator model through the
spool position. This nice property may not be preserved if the
linearization is conducted around nonzero inputs. 3 Experimental System
The linearized actuator model derived herein is identical to the The experimental system consists of a Moog servovalve utiliz-
existing results 关6,16兴: ing a cantilever feedback spring and a 0.9 kg double-ended actua-
tor with an effective area of 3.613 e⫺4 m2 共0.56 in2兲. The supply
x a共 s 兲 c1 pressure is 18.6 MPa 共2700 psi兲. Mobil DTE Light oil is used at
⫽ (22)
x s 共 s 兲 s 3 ⫹c 2 s 2 ⫹c 3 s an average temperature of 90°F. The actuator and spool are con-
nected to linear variable differential transducers 共LVDT兲 with ⫾3
where the constants c 1 ,c 2 ,c 3 ⬎0. This third-order model has one mV noise level. The range of the actuator LVDT is ⫾2.5 V for the
less state than expected from the four state equations 共17兲–共20兲. total stroke of ⫾0.0254 m 共⫾1 in兲. While the servo valve physical
This is because only the actuator pressure difference ‘‘p a1 ⫺p a2 ’’ parameters are not available for model verification, the experi-
is used, instead of the individual pressures. mental frequency responses for the servovalve and the actuator
The relationship between the actuator position and the distur- are measured with a signal analyzer, using a ‘‘swept sine’’
bance force is method that generates fixed-amplitude sine waves of varying fre-
quencies. This is then used to validate the model order and struc-
x a共 s 兲 ⫺c 4 ture. It is desired to measure this data near the equilibrium actua-
⫽ (23)
f d 共 s 兲 s 2 ⫹c 2 s⫹c 3 tor position x ae ⫽0. Since the actuator is of type 1 as Eq. 共22兲
indicates, it is difficult to keep the actuator near the equilibrium
The derivation of these actuator transfer functions is given in Ap- position in open loop. Therefore, a proportional control loop
pendix A, in which the coefficients (c 1 ,...,c 4 ) are expressed in (gain⫽1) is used to create a stabilized plant, as shown in Fig. 2.
terms of system parameters. With different input amplitude levels 共15 mV, 30 mV, and 50
The linearized model of the two-stage servovalve, derived in mV兲, the frequency responses of the servovalve and actuator are
Appendix A, has the form measured and shown in Figs. 3 and 4, respectively. Using the
frequency responses from the three input amplitudes, an averaged
x s共 s 兲 ⫺b 0 s⫹b 1 frequency response is computed for the servovalve and the actua-
⫽ 5 (24)
i共 s 兲 s ⫹a 1 s 4 ⫹a 2 s 3 ⫹a 3 s 2 ⫹a 4 s⫹a 5 tor, and nominal models are fitted through the use of an equation-
error method 关17兴.
For a servovalve with a direct feedback spring on the spool,
b 0 ⫽0. The transfer function coefficients are expressed in terms of
the servovalve parameters. This fifth-order model has one less
state than expected from the six state equations 共1兲–共6兲. This is
because only the spool pressure difference ‘‘p s1 ⫺p s2 ’’ is used,
instead of the individual pressures.
The derived servovalve model of Eq. 共24兲 is different from
those models used in the previous literature. Some literature 关3–5兴
has assumed a second-order model of the form

x s共 s 兲 K2
⫽ 2 (25)
i共 s 兲 s ⫹2 ␰␻ n s⫹ ␻ 2n

Merritt 关6兴 has derived a third-order model of the form Fig. 2 Control system block diagram

x s共 s 兲 K3
⫽ 3 (26)
i共 s 兲 s ⫹h 1 s ⫹h 2 s⫹h 3
2

by neglecting the spool valve resonance, pressure feedback on the

flapper and the flow forces on the spool.
The derived servovalve model of Eq. 共24兲 suggests a model
structure of relative order of four or five, depending on the type of
spring feedback. The model coefficients expressed explicitly in
terms of the system’s physical parameters in the Appendix can be
used for sensitivity analysis or design of the servovalve dynamics
in various ways. For example, if only some servovalve parameters
are known, this model can identify the unknown parameters from
experimental data. As another example, this new model can test
the effect on the servovalve dynamics of a given parameter, such
as the cantilever spring constant K. The zero of the derived model
共24兲, as shown in Appendix A, has the form Fig. 3 Frequency responses of the servovalve

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 Fig. 4 Frequency responses of the actuator „dot line: 15 mV
input; dash line: 50 mV input; solid line: model…
Fig. 5 Comparison of servovalve models „solid line: average
of experimental data; dot line: second order model; dash line:
third order model; dot-dash line: derived model…
The nominal model for the averaged actuator frequency re-
sponse is found to be
x a共 s 兲 7.46e 8 not vary much with respect to input amplitude. The averaged ser-
⫽ 3 . (28) vovalve frequency response is shown in Fig. 5 with the best curve
x s 共 s 兲 s ⫹ 共 1.19e 3 兲 s 2 ⫹ 共 1.57e 7 兲 s
fits for the models of Eqs. 共24兲–共26兲. The derived servovalve
Figure 4 demonstrates that the actuator frequency responses do model of Eq. 共24兲 is

x s共 s 兲 ⫺ 共 2.09e 14兲 s⫹1.25e 18

⫽ 5 (29)
i共 s 兲 s ⫹ 共 9.90e 3 兲 s 4 ⫹ 共 6.89e 7 兲 s 3 ⫹ 共 3.09e 11兲 s 2 ⫹ 共 6.57e 14兲 s⫹3.39e 17
where ‘‘x s ’’ is measured in terms of LVDT voltage. This model contains a nonminimum phase zero as predicted in Eq. 共24兲. The best
curve fit for the second-order model in Eq. 共25兲 is
x s共 s 兲 6.79e 6
⫽ 2 (30)
i共 s 兲 s ⫹ 共 2.89e 3 兲 s⫹1.83e 6
and the third-order model in Eq. 共26兲 is
x s共 s 兲 2.06e 10
⫽ 3 . (31)
i共 s 兲 s ⫹ 共 6.72e 兲 s ⫹ 共 1.29e 7 兲 s 2 ⫹7.01e 9
3 3

It is observed from Fig. 5 that the derived servovalve model fits the experimental data more closely than the standard models.
Specifically, the frequency response of the derived model overlays the experimental data almost exactly for frequencies up to 1000 Hz.
In contrast, the frequency responses of the commonly used second-order model and third-order model start to deviate from the
experimental data at 40 Hz and 150 Hz, respectively.
For control system design, the second-order model and the third-order model are chosen for comparison with the derived fifth order
model because recent literature uses these models for the servovalve dynamics 关4,5兴. Adding the actuator model of Eq. 共28兲 and the
analog proportional feedback in the power amplifier, the plant model based on the fifth-order servovalve model, hereafter called the
‘‘eighth-order model,’’ is
⫺1.88e 23s⫹1.21e 27
P o8 共 s 兲 ⫽ . (32)
s ⫹1.27e s ⫹9.62e s ⫹5.79e s ⫹2.03e 15s 4 ⫹5.89e 18s 3 ⫹1.07e 22s 2 ⫹4.79e 24s⫹1.21e 27
8 4 7 7 6 11 5

The third-order servovalve model results in a ‘‘sixth-order model:’’

1.94e 19
P o6 共 s 兲 ⫽ (33)
s ⫹7.91e s ⫹3.66e s ⫹1.28e 11s 3 ⫹2.11e 14s 2 ⫹1.11e 17s⫹1.94e 19
6 3 5 7 4

and the second-order servovalve model results in a ‘‘fifth-order model:’’

5.06e 15
P o5 共 s 兲 ⫽ (34)
s ⫹4.08e s ⫹2.10e s ⫹4.76e 10s 2 ⫹2.87e 13s⫹5.06e 15
5 3 4 7 3

The stabilized plant models 共共32兲, 共33兲, and 共34兲兲 are considered as the nominal models for the subsequent digital control system design.

4 Robust Performance Control System Design troller design is to compare the achievable robust performance for
Controllers are said to provide robust performance for a system the three nominal plant models.
if they maintain stability and achieve performance in the presence The system dynamics are modeled with multiplicative uncer-
of modeling uncertainty. The main objective of the following con- tainty in the form

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 P 共 s 兲 ⫽ 关 1⫹⌬ 共 s 兲 W r 共 s 兲兴 P o 共 s 兲 (35)
where P(s) represents the experimental frequency response data,
P o (s) is the nominal model of Eqs. 共32兲, 共33兲, or 共34兲, and W r (s)
is a fixed stable transfer function which bounds the modeling un-
certainty. The function ⌬(s) is from the set of stable transfer
satisfying
储 ⌬ 共 s 兲储 ⬁ ªsup兩 ⌬ 共 j ␻ 兲 兩 ⭐1. (36)
␻

A necessary and sufficient condition for robust stability 共Chen and

Desoer 关18兴兲 is
储 W r T o 储 ⬁ ⬍1 (37)
where
P oK
T oª (38) Fig. 7 Multiplicative uncertainties of the plant models „dash-
1⫹ P o K dot line: W r 5 ; dot line: W r 6 ; dash line: W r 8 …
Robust performance controllers must satisfy the robust stability
condition 共37兲 and performance condition:
储 W p S 储 ⬁ ⬍1 (39) The mixed sensitivity problem has been presented as a standard
problem in the H ⬁ framework 关12兴, and is shown in Fig. 6. The
1 e
Sª ⫽ . (40) standard H ⬁ design goal is to minimize the infinity norm of the
1⫹ PK r f transfer matrix T wz , where the vector w contains the exogenous
Typically, the performance weight W p (s) has larger magnitudes inputs and the vector z contains the outputs to be minimized. The
at lower frequencies, signifying desired tracking at these frequen- design software which minimizes the infinity norm of a transfer
cies. A necessary and sufficient condition 关19兴 for the robust per- matrix is used for controller synthesis 关20兴. The transfer matrix
formance in Eq. 共39兲 is T wz of Fig. 6 is 关 W p S o W r T o 兴 T and its infinity norm is exactly
the quantity to be minimized in the mixed sensitivity problem in
储 兩 W p S o 兩 ⫹ 兩 W r T o 兩 储 ⬁ ⬍1, (41) Eq. 共43兲.
1 The following describes the general procedure for the mixed
S oª (42) sensitivity control system design in this paper. The unmodeled
1⫹ P o K dynamics are computed from the modeled and experimental plant
There is no known procedure to find directly the minimum data, and the corresponding bounds W r are fitted, as shown in Fig.
achievable value of Eq. 共41兲, and the corresponding controller 7. W r 8 has smaller magnitudes than W r 6 and W r 5 for frequencies
K(s). A modified robust performance problem, known as the above 15 Hz, verifying the improved accuracy of the plant model
mixed sensitivity problem, has been suggested 关15兴, where the based on the derived servovalve model. A benefit from the lower
objective is to minimize magnitudes of W r 8 is that more demanding performance weights
W p 共i.e., with higher bandwidths兲 can be expected from the robust
sup冑兩 W p 共 j ␻ 兲 S o 共 j ␻ 兲 兩 2 ⫹ 兩 W r 共 j ␻ 兲 T o 共 j ␻ 兲 兩 2 (43)
␻
performance design. One demonstration of this fact comes from a
necessary condition for robust performance 关17兴,
The solution of the mixed sensitivity problem provides an ap-
proximate solution of the robust performance problem because for min兵 兩 W p 共 j ␻ 兲 兩 , 兩 W r 共 j ␻ 兲 兩 其 ⬍1, for all ␻, (45)
any complex numbers ‘‘a’’ and ‘‘b,’’
which reduces the possible W p bandwidths for controller synthesis
冑兩 a 兩 2 ⫹ 兩 b 兩 2 ⭐ 兩 a 兩 ⫹ 兩 b 兩 ⭐& 冑兩 a 兩 2 ⫹ 兩 b 兩 2 (44) with the fifth-order model. Figure 7 illustrates this fact, since the
frequencies corresponding to 兩 W r 兩 ⫽1 are 800 Hz for the eighth-
order model, 150 Hz for the sixth-order model and 50 Hz for the
fifth-order model. Therefore, better robust performance is ex-
pected from controllers synthesized with the eighth-order 共i.e.,
more accurate兲 model.
The discrete nominal plants P o (z) are computed from the plant
models 共32兲, 共33兲, and 共34兲 using a zero-order-hold transforma-
tion, and then Tustin transformations are performed via the
relation
1⫹ 共 T/2兲 w
z⫽ (46)
1⫺ 共 T/2兲 w
where T is the sampling interval, resulting in the plants P o (w).
The designs are performed in the w-plane because each controller
K(w) can be mapped directly to the discrete controller K(z),
without any approximation. A sampling interval 共T兲 of 0.5 ms is
chosen, which is sufficient because the plant bandwidth is 50 Hz.
The performance weights used for controller synthesis are cho-
sen to have the following low-pass form:
K
W p共 w 兲 ⫽ (47)
Fig. 6 Mixed sensitivity problem as a standard problem ␶ w⫹1

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 order model, since the rise time has been reduced to 1 ms. In
contrast, the rise times of the open loop plant, the fifth-order
model, and sixth-order model, are 9, 10, and 5 ms, respectively.
Using the more accurate eighth-order model, a controller can be
designed with higher sampling rate to further reduce the rise time.
The control parameters resulting from the design process with 0.5
ms sampling time are listed in Appendix B.

5 Conclusions
A linearized servovalve model has been derived from the non-
linear model for an electrohydraulic system consisting of a linear
actuator piston and a two-stage servovalve. The model coeffi-
cients are explicitly in terms of the system physical parameters
and therefore reveal several model structural properties. First, the
Fig. 8 Performance weights and experimental sensitivites „dot valve model has a relative order of 4 or 5, depending on the type
line: using K 5 from the second order model; dash line: using of spring feedback. Second, there is a possibility of a nonmini-
K 6 from the third order model; solid line: using K 8 from the mum phase zero when a cantilever feedback spring is used. Third,
derived fifth order model… when the previous case exists, a weaker spring is desirable to
drive the non-minimum phase zero away from the imaginary axis.
This improved servovalve model, a third-order model, and a
With a fixed low frequency magnitude K⫽3200 for steady state second-order model have been fitted to experimental data. The
performance, robust performance controllers are synthesized with results demonstrate the improved accuracy of the derived servov-
the three plant models, as the break frequency of W p (w) is in- alve model. Robust performance control system design has been
creased until the limit of the robust performance condition 共41兲 is performed based on these models. As expected and verified by
reached. This procedure results in ␶⫽6 for the eighth-order model, experiment, better robust performance is achieved for the im-
␶⫽16 for the eighth-order model, and ␶⫽25 for the fifth-order proved servovalve model, thus signifying its utility for high per-
model. Figure 8 shows the inverse of each W p function, which formance control design. Indeed in the camshaft turning applica-
provides the bound of robust performance. tion, the proposed model structure was necessary 共as opposed to
The digital controllers obtained from converting the w-domain the lower order models兲 for successful design and implementation
designs to the z-domain were implemented. Each experimental of a repetitive controller to achieve high bandwidth cam profile
sensitivity function of the sampled data system was obtained by tracking performance 关21兴.
using the signal analyzer ‘‘swept sine’’ method for a reference
amplitude of 30 mV. Figure 8 shows that robust performance is
achieved with the three digital controllers, as the experimental Appendix A: Derivation of the Linearized Models
sensitivity functions fall below the inverse of the corresponding
W p function. Servovalve Model. Equations 共3兲 and 共4兲 are linearized, and
Figure 8 clearly shows that better robust performance is the Laplace transforms are taken:
achieved with the more accurate eighth-order model. As one ex-
ample, consider the frequencies for which each experimental sen- 共 s⫹D 2 兲关 p s1 共 s 兲 ⫺p s2 共 s 兲兴 ⫽2D 1 ␪ 共 s 兲 ⫺2D 3 sx s 共 s 兲 (A1)
sitivity function is less than ⫺20 dB, signifying at most 10 percent Equation 共2兲 is linearized, and the Laplace transform is taken:
error magnitude ratio. From Fig. 8, the maximum frequencies that
meet this criterion are 16, 4, and 2 Hz, using the controllers from 关 s 2 ⫹D 5 s⫹D 4 兴 ␪ 共 s 兲 ⫽⫺D 6 关 p s1 共 s 兲 ⫺p s2 共 s 兲兴 ⫺D 7 x s 共 s 兲 ⫹D 8 i 共 s 兲
the eighth-order model, sixth-order model, and fifth-order model, (A2)
respectively. Equation 共6兲 is linearized, and the Laplace transform is taken:
Figure 9 shows the experimental step response using each ro-
bust performance controller for a reference amplitude of 30 mV. 关 s 2 ⫹D 12s⫹D 11兴 x s 共 s 兲 ⫽D 10关 p s1 共 s 兲 ⫺p s2 共 s 兲兴 ⫺D 9 ␪ 共 s 兲
Better system response is achieved with the more accurate eighth- (A3)

D 1 ⫽C qn ␲ D n R
␤e
V so
冑 2p s1e
␳

␤ e C qo A o ␤ e C qn ␲ D n X f m
D 2⫽ ⫹
V so 冑2 ␳ 共 p s ⫺p s1e 兲 V so 冑2 ␳ 共 p s1e 兲
␤e
D 3⫽ A
V so s
共 K a ⫺K m 兲 ⫹K 共 R⫹B 兲 ⫺16␲ C 2qn R 2 X f m p s1e
D 4⫽
J
Bv
D 5⫽
J
A n R⫹4 ␲ C 2qn X 2f m R
D 6⫽
J

Fig. 9 Experimental step responses „dash-dot line: using K 5 ;

K
D 7⫽
dot line: using K 6 ; dash line: using K 8 ; solid line: plant… J

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 Kt p s1 共 s 兲 ⫺p s2 共 s 兲 g 0 s 2 ⫹g 1 s⫹g 2
D 8⫽ ⫽ 5 (A6)
J i共 s 兲 s ⫹a 1 s ⫹a 2 s 3 ⫹a 3 s 2 ⫹a 4 s⫹a 5
4

K g 0 ⫽2D 8 D 1
D 9⫽
Ms g 1 ⫽2D 8 兵 D 1 D 12⫹D 3 D 9 其
As g 2 ⫽2D 8 D 1 D 11
D 10⫽
Ms
Actuator Model. Equations 共17兲 and 共18兲 are linearized and
K the Laplace transforms are taken:

冋 冑 册
⫹2C q W cos ␪ f P s ⫹2K s
D 11⫽
R⫹B
Ms s 关 p a1 共 s 兲 ⫺p a2 共 s 兲兴 ⫽x s 共 s 兲 2
Ps
␳
␤e
⫺x a 共 s 兲 2
C W
V a0 q
␤e
A s
V a0 1 冋 册
B s ⫹ 共 L 2 ⫺L 1 兲 C q W 冑P s ␳ (A7)
D 12⫽ Equation 共20兲 is linearized and the Laplace transform is taken:

冋 册 冋 册
Ms
Ba A1 1
Recall that K s ⫽0 for a servovalve with a cantilever feedback s 2⫹ s x a 共 s 兲 ⫽ 关 p a1 共 s 兲 ⫺p a2 共 s 兲兴 ⫺ f d 共 s 兲 (A8)
spring and K⫽0 for a servovalve with a direct spool feedback Mt Mt Mt
spring. The two equations 共A7兲 and 共A8兲 contain two unknowns: p a1 (s)
The three equations 共共A1兲, 共A2兲, and 共A3兲兲 contain three un- ⫺p a2 (s) and x a (s). These two unknowns are solved as a function
knowns: p s1 (s)⫺p s2 (s), ␪ (s), and x s (s). These three unknowns of the spool position ‘‘x s ’’ and the disturbance force f d . The
are solved as a function of the input current. results are
x s共 s 兲
i共 s 兲
⫽ 5
⫺b 0 s⫹b 1
s ⫹a 1 s ⫹a 2 s 3 ⫹a 3 s 2 ⫹a 4 s⫹a 5
4 (A4) x a 共 s 兲 ⫽x s 共 s 兲 冋 c1
s 3 ⫹c 2 s 2 ⫹c 3 s册⫺ f d共 s 兲 2冋 c4
s ⫹c 2 s⫹c 3 册 (A9)

b 0 ⫽D 8 D 9
b 1 ⫽D 8 兵 2D 1 D 10⫺D 2 D 9 其
p a1 共 s 兲 ⫺p a2 共 s 兲 ⫽x s 共 s 兲 冋 c 5 s⫹c 6
s 2 ⫹c 2 s⫹c 3册⫹ f d共 s 兲 2冋 c7
s ⫹c 2 s⫹c 3 册
(A10)
a 1 ⫽D 2 ⫹D 5 ⫹D 12
a 2 ⫽2D 3 D 10⫹D 2 D 5 ⫹D 2 D 12⫹D 5 D 12⫹D 4 ⫹D 11 c 1 ⫽2
A1 ␤e
C W
M t V a0 q
冑 Ps
␳
a 3 ⫽D 2 兵 D 4 ⫹D 11⫹D 5 D 12其 ⫹2D 1 D 6 ⫹2D 3 D 5 D 10 Ba
c 2⫽
⫹D 4 D 12⫹D 5 D 11 Mt
a 4 ⫽D 2 兵 D 4 D 12⫹D 5 D 11其 ⫹2D 1 D 6 D 12⫹2D 3 D 4 D 10 ␤e
c 3 ⫽2 A2
M t V a0 1
⫹2D 3 D 6 D 9 ⫹D 4 D 11⫺D 7 D 9
1
a 5 ⫽D 2 兵 D 4 D 11⫺D 7 D 9 其 ⫹2D 1 D 6 D 11⫹2D 1 D 7 D 10 c 4⫽
Mt
␪共 s 兲 f 0 s 3 ⫹ f 1 s 2 ⫹ f 2 s⫹ f 3
⫽ 5
i 共 s 兲 s ⫹a 1 s 4 ⫹a 2 s 3 ⫹a 3 s 2 ⫹a 4 s⫹a 5
(A5)
c 5 ⫽2
␤e
C W
V a0 q
冑 Ps
␳
f 0 ⫽D 8
f 1 ⫽D 8 兵 D 2 ⫹D 12其 c 6 ⫽2
Ba ␤e
C W
M t V a0 q
冑 Ps
␳
f 2 ⫽D 8 兵 D 11⫹D 2 D 12⫹2D 3 D 10其 2 ␤e
c 7⫽ A
f 3 ⫽D 8 D 2 D 11 M t V a0 1

Appendix B: Model and Controller Parameters

2.10e ⫺3 w 8 ⫹1.63e 1 w 7 ⫺7.41e 5 w 6 ⫹2.97e 9 w 5 ⫹4.23e 13w 4 ⫹9.26e 16w 3 ⫺8.48e 20w 2 ⫺7.91e 23w⫹5.65e 27
P o8共 w 兲 ⫽
w 8 ⫹2.64e 4 w 7 ⫹3.50e 8 w 6 ⫹2.42e 12w 5 ⫹1.21e 16w 4 ⫹3.83e 19w 3 ⫹5.51e 22w 2 ⫹2.34e 25w⫹5.65e 27

3.27e ⫺4 w 6 ⫹1.33e1w 5 ⫹3.77e 4 w 4 ⫺6.42e 8 w 3 ⫺2.57e 12w 2 ⫹6.92e 15w⫹2.98e 19

P o6共 w 兲 ⫽
w 6 ⫹9.34e 3 w 5 ⫹6.47e 7 w 4 ⫹2.35e 11w 3 ⫹3.49e 14w 2 ⫹1.74e 17w⫹2.98e 19

⫺7.81e ⫺4 w 5 ⫹1.23e 1 w 4 ⫹9.91e 4 w 3 ⫺7.73e 8 w 2 ⫺1.61e 12w⫹1.00e 16

P o5共 w 兲 ⫽
w 5 ⫹6.37e 3 w 4 ⫹4.56e 7 w 3 ⫹9.83e 10w 2 ⫹5.78e 13w⫹1.00e 16

1.02w 2 ⫹6.44e 2 w⫹3.20e 3

W r 8共 w 兲 ⫽
w 2 ⫹5.04e 3 w⫹2.00e 5

Journal of Dynamic Systems, Measurement, and Control MARCH 2000, Vol. 122 Õ 185

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 2.85e 1 w 2 ⫹6.80e 2 w⫹2.20e 4
W r 6共 w 兲 ⫽
w 2 ⫹2.10e 4 w⫹1.30e 6
6.39e 1 w 2 ⫹1.55e 3 w⫹4.66e 4
w r 5共 w 兲 ⫽
w 2 ⫹2.00e 4 w⫹1.40e 6
2.84e 7 w 8 ⫹7.53e 11w 7 ⫹9.95e 15w 6 ⫹6.90e 19w 5 ⫹3.44e 23w 4 ⫹1.09e 27w 3 ⫹1.57e 28w 2 ⫹6.80e 32w⫹1.61e 35
K 8共 w 兲 ⫽
w ⫹7.98e 4 w 8 ⫹5.78e 9 w 7 ⫹1.54e 14w 6 ⫹1.67e 18w 5 ⫹1.00e 22w 4 ⫹4.69e 25w 3 ⫹1.59e 29w 2 ⫹2.39e 32w⫹3.41e 33
9

7.29e 8 w 7 ⫹2.96e 13w 6 ⫹2.17e 17w 5 ⫹1.35e 21w 4 ⫹4.73e 24w 3 ⫹6.74e 27w 2 ⫹3.13e 30w⫹4.97e 32
K 6共 w 兲 ⫽
w 8 ⫹3.61e 6 w 7 ⫹7.21e 11w 6 ⫹1.52e 16w 5 ⫹1.39e 20w 4 ⫹6.60e 23w 3 ⫹1.62e 27w 2 ⫹1.64e 30⫹1.06e 31
8.69e 5 w 6 ⫹2.29e 10w 5 ⫹1.50e 14w 4 ⫹8.78e 17w 3 ⫹1.76e 21w 2 ⫹1.01e 24w⫹1.74e 26
K 5共 w 兲 ⫽
w ⫹1.01e 5 w 6 ⫹3.08e 9 w 5 ⫹3.96e 13w 4 ⫹2.49e 17w 3 ⫹7.79e 20w 2 ⫹9.78e 23w⫹3.91e 24
7

储 冑兩 W p 8 S 0 兩 2 ⫹ 兩 W r 8 T 0 兩 2 储 ⬁ ⫽0.72 and 储 兩 W p 8 S 0 兩 ⫹ 兩 W r 8 T 0 兩 储 ⬁ ⫽0.98.

储 冑兩 W p 6 S 0 兩 2 ⫹ 兩 W r 6 T 0 兩 2 储 ⬁ ⫽0.72 and 储 兩 W p 6 S 0 兩 ⫹ 兩 W r 6 T 0 兩 储 ⬁ ⫽0.99.

储 冑兩 W p 5 S 0 兩 2 ⫹ 兩 W r 5 T 0 兩 2 储 ⬁ ⫽0.71 and 储 兩 W p 5 S 0 兩 ⫹ 兩 W r 5 T 0 兩 储 ⬁ ⫽0.97.

Nomenclature W ⫽ area gradient 共i.e., flow area/spool displacement兲

关 共m2兲/m兴
Torque-motor stage:
V ao ⫽ enclosed volume on each side of actuator when x a
Kt ⫽ torque-motor gain 关N-m/amps兴 ⫽0 关 m2兴
Ka ⫽ rotational stiffness of flexure tube 关N-m/rad兴
Force balance around actuator:
Km ⫽ electromagnetic rotational stiffness 关N-m/rad兴
An ⫽ area of nozzle 关 m2兴 M t ⫽ mass of actuator 关kg兴
R ⫽ distance from nozzle to the pivot point of the flexure B a ⫽ damping coefficient of actuator 关N/共m/s兲兴
tube 关m兴 A 1 ⫽ effective area of double-ended piston 关 m2兴
K ⫽ net stiffness of cantilever feedback spring connected to System states and input variables:
flapper 关N-m/m兴
B ⫽ distance from the nozzle to the spool 关m兴
␪ ⫽ angular position of armature/flapper
⍀ ⫽ angular velocity of armature/flapper
Bv ⫽ damping coefficient of torque-motor 关N-m/共rad/s兲兴
p s1 ⫽ pressure on one end of spool
J ⫽ moment of inertia of torque-motor 关 kg-m2兴 p s2 ⫽ pressure on one end of spool
Flapper-nozzle stage: x s ⫽ spool position
v s ⫽ spool velocity
Ao ⫽ cross-sectional area of orifice 关 m2兴
p a1 ⫽ pressure on one side of actuator
As ⫽ area of spool valve 关 m2兴 p a2 ⫽ pressure on one side of actuator
␤e ⫽ compressibility of hydraulic oil 关 N/共m2兲兴 x a ⫽ actuator position
C qo ⫽ orifice flow coefficient 关dimensionless兴 v a ⫽ actuator velocity
C qn ⫽ nozzle flow coefficient 关dimensionless兴 i ⫽ input current to torque-motor
Dn ⫽ diameter of nozzle 关m兴 f d ⫽ disturbance force input on actuator
Ps ⫽ supply pressure 关 N/共m2兲兴
␳ ⫽ density of hydraulic oil 关 kg/共m3兲兴
Xfm ⫽ maximum flapper displacement 关m兴
V so ⫽ enclosed volume on each side of spool when x s References
⫽0 关 m2兴 关1兴 Hori, N., Pannala, A. S., Ukrainetz, P. R., and Nikiforuk, P. N., 1989, ‘‘Design
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