On Using Unstable
Electrohydraulic Valves
for Control
Kailash Krishnaswamy
Perry Y. Li
Department of Mechanical Engineering,
University of Minnesota,
111 Church St. SE,
Minneapolis, MN 55455
e-mail: 兵kk,pli其@me.umn.edu
High bandwidth, high flow rate electrohydraulic valves typically have two or more stages.
Most multi-stage valves are expensive, require meticulously clean fluid, and introduce
higher order dynamics. On the other hand, single-stage spool valves are cheaper and
more reliable. However, a majority of them are not suitable for high bandwidth, high flow
rate applications due to limitations of the electromechanical/solenoid spool-stroking actuators. In this paper, we investigate the feasibility of reducing this limitation by exploiting the transient flow forces in the valve so as to achieve spool dynamics that are intrinsically open-loop unstable. While conventional valves are designed to be open-loop
stable, the unstable valve design has to be stabilized via closed-loop feedback. Simulation
case studies are conducted to examine the potential dynamic and energetic advantages
that an unstable valve may offer. These studies indicate that unstable valves provide faster
response than the stable counterparts when stroking forces are limited. Moreover, unstable valves tend to require less positive power and energy to operate.
关DOI: 10.1115/1.1433801兴
Keywords: Flow Instability, Solenoid Actuator, Electrohydraulic Valves, Transient
Forces, Unstable Flow Forces, Unstable Valves
I
Introduction
High bandwidth, high flow rate electrohydraulic valves typically have two or more stages, one of which is usually a nozzle
flapper pilot valve. Although highly popular, multi-stage electrohydraulic valves are more expensive than single-stage valves and
a majority of them have the drawbacks that 1兲 they require meticulously clean fluid, as dirt deposition will cause the pilot valve
to malfunction; 2兲 they increase the order of the dynamics of the
system, thus potentially introducing undesirable time lags, and
making control design more challenging.
Single-stage, direct-acting control valves are valves in which
the spools are directly stroked by an electromechanical or solenoid actuator. They are less expensive and less sensitive to dirt.
They are also easier to manufacture and have lower order dynamics than multistage valves. Proportional control valves are examples of this type of valve. Unfortunately, most of the commercially available single-stage direct-acting valves are not suitable
for high performance, high flow rate applications. It is because at
high bandwidth and large flow rate, the force and power required
of the electromechanical actuator to stroke the spool become very
significant, thus limiting the performance of the single-stage
valve.
The discussion above indicates that single-stage direct-acting
control valves may become more practical for high performance,
high flow rate applications if it is possible to reduce the force
and/or power demand on the electromechanical actuator that
strokes the spool. With advances in control theory and technologies, one idea is to design the valve spool so that they are openloop unstable and to utilize the flow forces associated with the
instability advantageously. Spool valves can be made unstable by
appropriately manipulating the transient flow forces. The unstable
valve will be stabilized subsequently via closed loop feedback.
The idea is similar to the design of high performance ‘‘fly-bywire’’ fighter aircrafts in which the aerodynamics are sometimes
Contributed by the Dynamic Systems and Control Division for publication in the
JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript
received by the Dynamic Systems and Control Division February 9, 2001. Associate
Editor: N. Manring.
deliberately designed to be open-loop unstable so as to enhance
their agility. Past studies on valve instability have been restricted
to ensuring that instabilities do not occur 关1–3兴. In this paper, we
investigate whether a single-stage valve which is designed to be
open-loop unstable offers any advantages in terms of performance
improvements or requirements on the electromechanical/solenoid
actuator.
The rest of the paper is organized as follows. In Section II, we
discuss how flow forces determine the stability of a four way
directional spool valve. In section III we present two numerical
simulation experiments to quantify the potential benefits of unstable valves. Performance is quantified in terms of step responses, and in terms of power and effort required to track sinusoidal signals. Section IV contains discussion and some
concluding remarks.
II
Flow Forces and Spool Stability
A four way directional flow control valve is shown in Fig. 1.
We assume that it is matched and critically centered. The displacement x v of the spool controls the flow into a hydraulic device
connected to the two ports on the right. In addition to the stroking
force u provided by the electromechanical/solenoid actuator, the
spool of the valve experiences both pressure forces and flow induced forces 共or Bernoulli forces兲 关1兴. Since the same pressure
acts on the opposing surfaces of the lands of equal areas, they
have no net effect on the dynamics of the spool. Flow induced
forces are of two types, 1兲 steady-state flow forces and 2兲 transient
flow forces. Inviscid, incompressible flow is assumed in the following derivations.
Steady-state flow forces are the reaction forces on the spool due
to the changes in the momenta of the fluid entering and leaving
the valve chamber. Referring to Figs. 2共a兲 and 2共b兲, as the spool
meters flow into 共out of兲 the valve chamber, the vena contracta in
which the fluid enters 共leaves兲 the chamber is at an angle ␪ to the
spool axis. The fluid however leaves 共enters兲 perpendicular to the
spool axis. Thus, fluid entering and leaving the valve chamber can
have different lateral and axial momenta. This necessitates reaction forces on the spool in both lateral and axial directions. By
locating the ports symmetrically on the circumference of the valve
Journal of Dynamic Systems, Measurement, and Control
Copyright © 2002 by ASME
MARCH 2002, Vol. 124 Õ 183
sleeve, net lateral forces can generally be eliminated. The axial
force is however, significant and contributes to net steady state
flow forces on the spool. If the flow rate through an orifice is Q,
then the steady-state flow force for one metering orifice is given
by
␳
Q2
cos ␪
Ac
(1)
where ␳ is the fluid density, A c is the area of the vena contracta
and ␪, which ranges between 21 deg to 69 deg 关1兴, is the angle of
the vena contracta. Notice that the direction of this reaction force
acts to close the orifice, regardless of whether the spool is metering into or metering out of the valve chamber.
Let the area gradient of the valve be w 共in2/in兲. Then the orifice
size is given by A o ⫽w•x v . Assuming Q 1 ⫽Q 2 ⫽Q in Fig. 1, it
can be shown that for a matched, critically centered, symmetric
valve,
Fig. 1 A single-stage critical-centered spool valve connected
to a double-ended actuator
Q⫽Q 1 ⫽Q 2 ⫽C d wx v
冑冉
1
␳
P s⫺
xv
兩 x v兩
PL
冊
(2)
where C d is the discharge coefficient, P s is the hydraulic supply
pressure, and P L ⫽ P 1 ⫺ P 2 is the load pressure, i.e., the differential pressure between the two valve chambers 共or the work ports兲.
Since the area of the vena contracta is proportional to the orifice
area, then in 共1兲, A c ⫽C c A 0 ⫽C c •w•x v for some C c ⬎0. Hence,
the total steady state flow force for both the meter-in and meterout orifices 共in the direction of positive x v 兲 is given by 关1兴:
F st ⫽⫺2C d C v w 共 P s ⫺ P L 兲 cos共 ␪ 兲 •x v
(3)
where C v ⫽C d /C c . Notice from 共3兲 that the steady-state flow
force acts like a spool-centering spring and is proportional to x v .
Hence the magnitude of the steady-state flow force will be large
when the valve operates in applications that require high flow rate.
The transient flow forces are reaction forces on the spool as the
fluid in the annular valve chamber accelerates in response to
variation in flow rate. To derive the transient flow forces, we
utilize the principle of conservation of momentum and a quasisteady analysis, i.e., we assume that the expression for flow rate in
共2兲 holds for time varying flow. Consider the control volumes
CV 1 and CV 2 enclosed in the valve chambers in Figs. 2共a兲 and
2共b兲, respectively. The longitudinal momenta 共in the direction of
positive x v 兲 of the incompressible fluid in chambers i⫽1, 2 are
given by:
Mi ⫽
冕
共 ␳ V x 兲 dV
CV i
where V x is the longitudinal fluid velocity of the fluid element.
Notice that in Fig. 2共a兲, V x is negative for positive Q(t) as shown.
Let the flow rate be given by Q(t)⫽Q in(t)⫽Q out(t). Then, by the
continuity of incompressible flow, for any cross sectional area
normal to the spool axis, we have
Q 共 t 兲 ⫽⫺
冕
V x dA 1 ⫽
A1
冕
V x dA 2
(4)
A2
where A 1 is any normal cross-sectional area in chamber 1 and A 2
is any normal cross-sectional area in chamber 2. Therefore, the
longitudinal momentum of the fluid in the control volumes CV 1
and CV 2 are given by:
Fig. 2 Valve configuration and transient flow forces when
dQ Õ dt Ì0 and Q Ì0. „a… Unstable transient flow force occurs
when flow is metered into the valve chamber; „b… stable transient flow force occurs when flow is metered out of the valve
chamber. Figures „a… and „b… correspond to the upper and lower
chambers in the spool configuration in Fig. 1
184 Õ Vol. 124, MARCH 2002
M1 ⫽
冕 冕
冕 冕
L1
␳ V x dA 1 dx⫽⫺
x⫽0
M2 ⫽
A1
L2
x⫽0
A2
冕
冕
L1
␳ Qdx⫽⫺ ␳ L 1 Q
x⫽0
␳ V x dA 2 dx⫽
L2
␳ Qdx⫽ ␳ L 2 Q
x⫽0
Transactions of the ASME
冉
where L 1 and L 2 are the lengths of the fluid columns for chambers
1 and 2 measured from the port centers. Conservation of momentum dictates that:
M s ẍ v ⫹B f ẋ v ⫹K f x v ⫽u.
(6)
By Newton’s third law, a longitudinal force of magnitude given by
共6兲 must act on the valve in the direction opposite to that of 共6兲.
For inviscid flow, all of this force acts on the spool. For the valve
chamber which is metering flow out of the valve 共Fig. 2共b兲兲, the
transient flow force acts to close (dQ/dt⬎0) or open (dQ/dt
⬍0) the orifice so as to resist dQ/dt. On the other hand, for a
valve chamber that is metering flow into the valve 共Fig. 2共a兲兲, the
transient flow force acts to open (dQ/dt⬎0) or close (dQ/dt
⬍0) the orifice so as to encourage dQ/dt. Hence, the transient
flow forces are stabilizing when flow is being metered out, and it
is destabilizing when flow is being metered in.
If Q is given by the orifice equation 共2兲, we see that in general
dQ/dt and hence the transient flow force, depend on d P L /dt
and dx v /dt. However, as demonstrated experimentally in 关6兴,
the effects of pressure variation are small, and are normally neglected 关1兴.
The four way directional valve in Fig. 1 has both a meter-in and
a meter-out chamber, therefore, combining 共2兲 and 共6兲, the net
transient flow forces acting on the spool 共in the direction of the
positive x v 兲 is:
F tr ⫽⫺
d 共 M1 ⫹M2 兲
dt
⬇ 共 L 1 ⫺L 2 兲 C d w
dQ
⫽ 共 L 1 ⫺L 2 兲 ␳
冑冉
␳ P s⫺
dt
xv
兩 x v兩
冊
PL •
dx v
dt
,
(7)
where L 1 is the distance between the center of the supply pressure
port and the center of the outler load port; and L 2 is the distance
between the center of the return port and the center of the input
load 共see Fig. 1兲. Let us define LªL 2 ⫺L 1 to be the damping
length of the valve. L determines whether the transient flow forces
are stabilizing (L⬎0) or destabilizing (L⬍0). Note that the magnitude of the transient flow force increases as the spool moves
quickly, such as in high bandwidth applications.
In the absence of viscous damping effects and external centering springs, the spool dynamics are given by:
M s ẍ v ⫽F st ⫹F tr ⫹u
(8)
where M s is the mass of the spool, u is the stroking force
produced by the electromechanical/solenoid actuator, F st is the
steady state flow force in 共3兲, and F tr is the unsteady flow force
in 共7兲.
If we define the effective spring rate and damping coefficient
to be:
B f ªLC d w
冑冉
␳ P s⫺
xv
兩 x v兩
PL
冊
(10)
(11)
(5)
Thus, we see that the force acting on the fluid consists of a steady
component, due to the momentum flux, which is exactly the
steady state flow forces already considered, and the transient component due to (d/dt)(M 1 ⫹M 2 ).
Following 关1,4,2,5,3兴, and applying a quasi-steady assumption,
i.e. 共4兲 is valid even for transient flow, the transient flow force
acting on the fluid is
d
dQ
.
共 M1 ⫹M2 兲 ⫽ ␳ 共 L 2 ⫺L 1 兲
dt
dt
冊
xv
P ,
兩 x v兩 L
then, Eq. 共8兲 becomes:
d 共 M1 ⫹M2 兲
⫺net longitudinal momentum flux
dt
⫽longitudinal force applied by the valve.
K f ª2C d C v w cos ␪ P s ⫺
(9)
Journal of Dynamic Systems, Measurement, and Control
Because of the stabilizing effect of the steady-state flow force, the
spring rate K f is always positive. However, the sign of the damping coefficient B f depends on the damping length L. In a conventional single-stage flow-control valve, L⬎0 by design so that B f
⬎0. The electromechanical/solenoid actuator must be powerful
enough to overcome both the steady state and the transient
flow forces to move the spool. As remarked earlier, these forces
become more significant as both flow rate and bandwidth increase. Because the force and power capabilities of the electromechanical/solenoid spool actuator are limited, the performance
in terms of flow rating and bandwidth of single stage valves may
be limited.
The basic assumptions used in the derivation of the transient
flow force, i.e., incompressible flow, quasi-static analysis are also
used by past authors 关1,4,2,5,3兴. The model that uses these assumptions and the orifice equation 共2兲 has been used to predict
spool instability in 关3兴. In 关5兴, quasi-static analysis is also used to
study the relative contribution of transient and steady-state flow
forces, except that CFD techniques are used to model, more accurately, the steady-state flow/spool displacement relationship for
the pilot stage of a relief valve. It should be noted that given the
quasi-steady assumption, the information provided by a detailed
flow pattern due to complex valve geometry plays the role of
modifying 1兲 the steady flow force in 共1兲 and 共3兲, and hence the
equivalent spring constant K f in 共10兲; and 2兲 the flow/spool displacement relationship 共2兲. As long as the flow/spool displacement
relationship is monotonic, the effect of transient flow force will
still take the form of a damping term. An orifice equation that
takes into account transient flow is proposed in 关7兴. This model is
compared to the usual orifice equation 共2兲 for an electrohydraulic
valve in 关8兴. However, it is found in 关8兴 that the transient modification has negligible effect on the overall performance of that
particular valve.
In the derivation above, the flow is assumed to be non-viscous.
Fluid viscosity plays two roles in the above analysis. First, in 共5兲,
the spool sleeve in addition to the lands contributes to the force on
the RHS. If the viscous force on the spool sleeve is modeled to be
proportional to the flow rate 共which via 共2兲 is in turn proportional
to the spool displacement兲, then fluid viscosity will increase or
decrease the effective spring constant K f in 共10兲 depending on the
damping length. Second, viscosity contributes to the drag on the
lands as the spool moves in the sleeve. This effect contributes to
an increase in the damping coefficient B f in 共9兲. Interested readers
may therefore interpret the simulation results below by mentally
offsetting the values for K f and B f accordingly in order to understand the effects of complex flow pattern, and of fluid viscosity.
In the next section, we present simulation case studies to investigate the potential performance improvement if the single-stage
valve is made unstable 共by choosing L⬍0, B f ⬍0 in 共9兲兲. The
hypothesis is that the transient flow forces associated with the
instability can be utilized to overcome the steady state flow forces,
thus alleviating the limitation of the electromechanical/solenoid
actuator. This will help expand the use of the inherently cheaper,
and more reliable single-stage valves into higher performance and
higher flow rate applications.
III
Simulation Case Studies
To investigate the hypothesis above, simulation case studies are
conducted in which the force exerted by a double-ended cylinder
actuator is to be controlled. We consider the situation in which the
MARCH 2002, Vol. 124 Õ 185
cylinder is constrained 共e.g., pushing against a rigid wall兲. The
setup is illustrated in Fig. 3. A single-stage direct-acting control
valve, configured either in the stable configuration (L⬎0,B f ⬎0)
or in the unstable mode (L⬍0,B f ⬍0) is used to control the actuator. The dynamics of the system are given by
Ṗ L ⫽
␤
␤ C dw
Q共 t 兲⫽
V cyl
V cyl
冑
P s ⫺sign共 x v 兲 P L
xv
␳
M s ẍ v ⫹B f ẋ v ⫹K f x v ⫽u
(12)
(13)
where ␤ is the compressibility, V cyl is volume of the cylinder, M s
is the mass of the spool, P L is the load pressure, B f and K f are the
flow force induced effective damping and spring rate given in 共9兲
and 共10兲. In these studies P s ⫽20.6 MPa 共3000 psi兲, ␤⫽1.03 GPa
共150,000 psi兲 and ␳ ⫽872 kg m⫺3 共0.0315 lbm in.⫺3兲. Coefficients
in 共9兲–共10兲 are w⫽0.034 m 共1.35 in.兲, C d ⫽0.6, C c ⫽1. These are
typical values in applications. The damping lengths of L⫽
⫾0.0254 m 共⫾1 in.兲 were tested. We assume that the input to the
system is the stroking force u(t) for the valve provided by an
electromechanical actuator.
Since the cylinder is constrained, the piston force is F p
⫽A p P L 共where A p is the cylinder area兲 so that the control problem is equivalent to controlling the differential pressure P L across
the two valve chambers. The stable and unstable valve configurations are compared in two settings:
• What is the optimal step response that the system is capable
of achieving when the electromechanical actuator u(t) is limited?
• What are the force and power requirements of the electromechanical actuator in order for the piston force to track the
desired sinusoidal profiles?
The first situation aims to illustrate the differences of the capabilities of the stable and unstable valves when the
electromechanical/solenoid actuators have limited capabilities.
The second situation aims to illuminate the differences in the requirements for the electromechanical/solenoid actuators for the
two valves to achieve the same level of bandwidth performance. It
is emphasized that the simulation studies are intended to illustrate
only the potential performances of the two valves and their requirements for electromechanical actuators. The actual closedloop performances, which would depend on the controller design,
is not considered here.
A Step Responses Under Input Constraints. Suppose that the
stroking force u(t) in 共13兲 is constrained. We wish to compare the
optimal feasible responses for the system given by 共12兲, 共13兲 for
various stable and unstable configurations 共specified by damping
Fig. 3 Actuator-valve setup considered in the simulation
studies
186 Õ Vol. 124, MARCH 2002
Fig. 4 Optimal step responses for the stable „L Ä0.0254 m
„¿1 in…… and unstable configuration „ L ÄÀ0.0254 m „À1 in……
with upper actuator constraint ū of 222.4 N „50 lb… „top… and
889.6 N „200 lb… „bottom…
lengths L兲 to achieve a desired pressure of P des⫽6.89 MPa 共1000
psi兲 quickly, from an initial pressure of 0 Pa. The optimal control
problem is to find u(t), t⫽ 关 0,T f 兴 so that
min J u ª
冑冕
Tf
兩 P des⫺ P L 共 t 兲 兩 2 dt
0
subject to u⭐u 共 t 兲 ⭐ū
(14)
where uគ and ū are, respectively, the imposed lower and upper
saturation limits on the stroking force. All initial conditions are
assumed to be 0. Minimizing J minimizes indirectly the response
time. Other cost functions that reflect the general idea of a good
step response can also be used.
To compute the optimal solutions, the optimal control problem
was discretized using a sampling time of 2.5 ms and converted to
a parameter optimization problem. The final time T f ⫽1.5 s in 共14兲
was judged adequate for this problem. Matlab’s 共Ver 5.3.1 共R11兲
Mathworks Inc., MA兲 constrained optimization function, fmincon
was then used to compute the 1.5/0.0025⫽600 optimal control
values. Note that the optimal responses are open-loop solutions
which are intended only to illustrate the capabilities of the valves.
To achieve this actual optimal response, close-loop control is
needed.
The optimal controls were computed for both the stable
(L⫽0.0254 m 共1 in兲兲 and unstable (L⫽⫺0.0254 m 共⫺1 in兲兲
configurations under two different actuator saturation limits
(uគ ,ū)⫽(⫺1.3 kN (⫺300 lbf), 222.4 N (50 lbf))
and
(uគ ,ū)
⫽(⫺1.3 kN (⫺300 lbf), 889.6 N (200 lbf)). A less severe negative constraint on the electromechanical actuator is imposed to
emphasize the cost of positive energy generating actuation. For
this particular task, negative actuation corresponds to braking,
which may have potentially less limiting ways of implementation.
The respective optimal step responses are shown in Fig. 4. With
the unstable valve, P L rises to the desired level P des faster than
with the stable valve. The 100 percent rise times are 17 percent
and 24 percent shorter for the unstable valve than for the stable
valve for the two sets of saturation limits 共Table 1兲.
The optimal stroking forces u o pt (t) are shown in Fig. 5. For
both the stable and unstable configurations, the optimal control is
initially saturated at the upper limits (ū). This is followed by a
period of deceleration. In the case of the unstable configuration,
the maximum allowable braking force occurs at some times for
both sets of upper actuator limits ū⫽222.4 N 共50 lbf兲 and 889.6 N
Transactions of the ASME
Table 1 100% use time for the stable and unstable valve for
two sets of ū
Fig. 7 Power Consumed by the valves
Fig. 5 Optimal control effort for stable and unstable valve
configurations
共200 lbf兲. The spool’s phase portraits of (x ␯ ,ẋ ␯ ) 共Fig. 6兲 reveal
that the spool indeed attains higher velocities in the unstable case
than in the stable case. This is especially the case when the actuator limitation is more severe (ū⫽222.4 N (50 lbf)). In this situation, the spool speed is not high enough to require full braking.
The higher achievable spool speeds contribute to the faster responses. This confirms the expectation that the spool is more agile
in the unstable configuration than in the stable configuration.
The electromechanical actuator power inputs (u•ẋ ␯ ) and the
work done by the stroking actuator are shown in Fig. 7 and Table
2 respectively. Notice that using the unstable valve, the positive
work performed by the stroking actuator is reduced to less than
1/3 required for the stable valve. In terms of negative 共braking兲
power and work, for both saturation limits, these are larger in
magnitude for the unstable valve than for the stable valve. The
peak positive power for the unstable valve is larger than for the
stable valve when ū⫽222.4 N 共50 lbf兲. This is attributed to the
fact that the peak power for the unstable valve occurs when the
actuator is in saturation and when the spool speed is very high. It
is interesting to note that the peak power for the stable configuration when ū⫽222.4 N 共50 lbf兲 is used for closing the valve. Note
also that both the rise time improvement 共Table 1兲 and the positive
work reduction 共Table 2兲 are more pronounced when the saturation limit is more severe.
The effect of damping length L on the rise times of both the
unstable and the stable valve are investigated next. The optimal
stroking force, u(t) under saturation limits of (uគ ,ū)
⫽(⫺1.3 KN (⫺300 lbf), 222.4 N(50 lbf)) for various damping
lengths were computed 共Fig. 8兲. We observe from Fig. 8 that as
the damping length increases, the valve takes longer to reach the
desired pressure, P des . In the stable configuration (L⬎0), the 100
percent rise times increase rapidly with larger positive damping
lengths. In the unstable configuration (L⬍0), the rise times decrease as the damping length becomes more negative. However, a
lower limit is achieved for very large negative damping lengths
共e.g., compare L⫽⫺0.177 m 共⫺7 in兲 and L⫽⫺0.101 m 共⫺4 in兲兲.
The limited return in benefit is due to the fact that for large negative damping lengths, the optimal control is limited by the lower
共braking兲 capability of the electromechanical actuator.
B Tracking Sinusoidal Force Profiles. We now consider the
task of controlling the system in Eqs. 共12兲–共13兲 to track various
sinusoidal differential pressure 共or equivalently actuator force兲
profiles. The goal here is to compare the force and power requirements for the electromechanical stroking actuators to achieve a
certain level of performance in terms of bandwidth. To do this, we
Table 2 Work done by the stroking actuator
Fig. 6 Phase Plots, x ␯ vÕs ẋ ␯
Journal of Dynamic Systems, Measurement, and Control
MARCH 2002, Vol. 124 Õ 187
共19兲 is true, then we achieve our desired pressure tracking as
S 1 ( P L ) would converge to 0. To achieve this, we introduce a
second sliding surface, S 2 (x ␯ ) as follows:
des
S 2 共 x ␯ 兲 ⫽ 共 ẋ ␯ ⫺ẋ des
␯ 兲 ⫹ ␥ 共 x ␯ ⫺x ␯ 兲 .
(20)
As the dynamics of the spool 共13兲 are second order, it is necessary
to differentiate 共20兲 once to determine the control, u. However,
this is not possible as 共19兲 is discontinuous in x ␯ due to the ‘‘sign’’
function. Therefore, we propose a uniformly bounded C ⬁ smooth
approximation to the sign function, ŝ(x ␯ ), which is given below:
ŝ 共 x ␯ 兲 ª
2
• arctan共 c•x ␯ 兲
␲
(21)
where c⬎0. It can be verified that the approximation error,
兩 ŝ(x ␯ )⫺sign(x ␯ ) 兩 is uniform in x ␯ and can be made arbitrarily
small by choosing c to be sufficiently large. Now, the dynamics of
the desired spool position, x des
␯ in Eq. 共19兲 is modified to be:
x des
␯ ⫽
Fig. 8 Damping length versus rise times
V cyl
共 ⫺␭ 1 S 1 ⫹ Ṗ Ldes兲
␤
C dw
do not assume that the electromechanical stroking actuator is limited, and determine and compare the control u(t) for both the
stable and unstable valves so that
P L 共 t 兲 → P Ldes共 t 兲 ⫽ P̄• sin共 2 ␲ f •t 兲
(15)
for various frequencies f 共Hz兲.
To determine the required control effort, a multiple sliding surface control law 关9兴 was designed to achieve near perfect tracking
of the desired force 共differential pressure兲 trajectories. This is just
a convenient way to obtain the control effort 共or the so called
inverse dynamics兲 required to achieve tracking of 共15兲. In principle, any controller that achieves asymptotic tracking should utilize the same control effort after convergence. The multiple sliding surface controller is designed as follows.
Define the load pressure sliding surface as follows:
S 1 共 P L 兲 ⫽ P L ⫺ P Ldes .
(16)
We determine the desired spool position trajectory, x ␯des , so that
the surface defined in 共16兲 is exponentially convergent. Assume
for the time being that we can directly control the spool displacement x ␯ . This assumption will be removed later by introducing a
second sliding surface. Hence, we need to choose x ␯ so that
S 1 ( P L ) converges to zero, exponentially.
Utilizing the pressure dynamics, 共12兲, we obtain:
d
S 共 P 兲 ⫽ Ṗ L ⫺ Ṗ Ldes ,
dt 1 L
d
␤ C dw
S 共 P 兲⫽
dt 1 L
V cyl
冑
(17)
P s ⫺sign共 x ␯ 兲 P L
x ␯ ⫺ Ṗ Ldes .
␳
(18)
If we choose x ␯ to be:
x ␯ ⫽x des
␯ ⫽
共 ⫺␭ 1 S 1 ⫹ Ṗ Ldes兲
V cyl
␤
C dw
冑
1
␳
,
(19)
共 P s ⫺sign共 x ␯ 兲 P L 兲
where ␭ 1 苸R⫹ is the gain to the sliding surface, it is easy to see
that the surface S 1 ( P L (t)) and hence P L (t)⫺ P Ldes(t) converge
exponentially to 0.
Unfortunately, the actual control is u(t), the spool dynamics
are given by 共13兲, and so x ␯ cannot be controlled directly. Notice
from 共13兲 that if x ␯ is controlled in such a way, using u, so that
188 Õ Vol. 124, MARCH 2002
冑
1
␳
.
(22)
共 P s ŝ 共 x v 兲 P L 兲
Differentiating 共20兲, we have,
d
S 共 x 兲 ⫽ 共 ẍ ␯ ⫺ẍ ␯des兲 ⫹ ␥ 共 ẋ ␯ ⫺ẋ des
␯ 兲.
dt 2 ␯
(23)
Using the spool dynamics 共13兲, we have:
d
1
des
S 共 x 兲⫽
共 ⫺B f ẋ ␯ ⫺K f x ␯ ⫹u 兲 ⫺ẍ des
␯ ⫹ ␥ 共 ẋ ␯ ⫺ẋ ␯ 兲 .
dt 2 ␯
Ms
(24)
Therefore, the control u,
des
u⫽B f ẋ ␯ ⫹K f x ␯ ⫹M s 共 ẍ des
␯ ⫺ ␥ 共 ẋ ␯ ⫺ẋ ␯ 兲 ⫺␭ 2 S 2 共 x ␯ 兲兲 , (25)
where ␭ 2 ⬎0, guarantees that the second sliding surface, S 2 (x ␯ ) is
exponentially convergent. In Eq. 共25兲, the derivatives ẋ ␯des and ẍ des
␯
were analytically determined from Eq. 共22兲. Because the dynamics of the first sliding surface 共16兲 is exponentially stable, by
choosing c in 共21兲 appropriately, it can be shown that the P L (t)
⫺ P Ldes(t) can be arbitrarily close to 0.
Having designed the controller, simulations were performed to
compare the power consumption of the valve in the stable configuration 共damping length L⫽0.0254 m 共1 in兲兲 and in the unstable configuration (L⫽⫺0.0254 m (⫺1 in)) at different frequencies. We emphasize that as long as the desired force
共pressure兲 profile is tracked, the control effort required after convergence should not depend on the particular control law. The
multiple sliding surface controller is used simply because it is
easy to design and implement. Notice from Figs. 9 and 10 that the
multiple sliding surface controller is able to achieve load pressure
tracking at both low and high frequencies for both the stable and
the unstable valves.
The trajectories for control effort and the power input to track a
10 Hz and 140 Hz sinusoidal pressure profile are shown in Figs.
11 and 12. To track the sinusoidal signals, both the stable and the
unstable valve require near sinusoidal control inputs. It is interesting to note that the magnitudes of the control effort are the same
for both the stable and the unstable configurations. The magnitudes of the control effort in both cases increase with frequency.
There is however a phase difference between the stable and unstable configurations. It can be shown that compared to the stable
configuration, the unstable configuration demands a smaller control effort than the stable configuration when x ␯ and ẋ ␯ are of the
same sign, and it demands more force when x ␯ and ẋ ␯ are of
different signs.
This is consistent with the power input profiles in Figs. 11 and
12, which show that in the unstable configuration, the power input
Transactions of the ASME
Fig. 9 Sinusoidal tracking at 10 Hz
Fig. 12 Effort and power needed to track a 140 Hz sinusoidal
trajectory
is mainly negative 共electromechanical actuator absorbing energy兲;
whereas for the stable configuration, the power input is mainly
positive 共electromechanical actuator generating energy兲. This observation persists for all frequencies 共Figs. 13 and 14兲. At high
frequencies, the positive power requirement for the unstable valve
is only about 20 percent of that of the stable valve. At low frequency 共e.g., 10 Hz, see Fig. 11兲, negligible positive power is
required for the unstable valve. On the other hand, the magnitude
of the negative power and work by the electromechanical actuator
are larger for the unstable valve than for the stable valve. Thus the
electromechanical actuator acts like a brake 共Figs. 13 and 14兲. As
the frequency increases, such as in high bandwidth applications,
the differences in the positive and negative power requirements
between the stable and unstable configurations become very
prominent.
IV
Discussion and Conclusions
Fig. 10 Sinusoidal tracking at 140 Hz
The simulation case studies presented in Section III indicate
that unstable valves are more agile and can potentially achieve a
significantly faster response than the stable valve under the same
Fig. 11 Effort and power needed to track a 10 Hz sinusoidal
trajectory
Fig. 13 Maximum positive power consumed by the stable and
unstable valves at various frequencies
Journal of Dynamic Systems, Measurement, and Control
MARCH 2002, Vol. 124 Õ 189
be used, closed-loop feedback control is a necessity. With advances in control theory and technologies, and the fact that embedded sensing and feedback control are already prevalent in electrohydraulics, this might not be a significant impediment.
Although, as demonstrated in the sinusoidal responses, the
force magnitude required of the stroking actuator may not necessarily be small, unstable valves tend to require smaller positive
power. When larger stroking forces are required, they are power
absorbing and are associated with braking. These results suggest
that if valve instability is to be exploited, the stroking actuators for
the unstable valves should be designed to act predominantly as
controllable brakes.
References
Fig. 14 Maximum negative power consumed by the stable and
unstable valves at various frequencies
stroke actuator force limitation. This offers the opportunities for
single-stage valves to improve on their performances toward those
offered by multistage servo valves. The unstable valve also requires significantly less positive power input than what a stable
valve does. Moreover, these benefits become more significant at
high frequencies or at low actuator capabilities. These are exactly
the situations where performance improvements are especially
needed of single-stage valves. Of course, if unstable valves are to
190 Õ Vol. 124, MARCH 2002
关1兴 Hebert E. Merritt, 1967, Hydraulic Control Systems, Wiley, NY.
关2兴 D. McCloy and H. R. Martin, 1973, The Control of Fluid Power, Wiley, NY.
关3兴 J. F. Blackburn, G. Reethof, and L. L. Shearer 1960, Fluid Power Control,
MIT Press, Cambridge, MA.
关4兴 Lee, S-Y., and Blackburn, J. F., 1952, ‘‘Contributions to hydraulic control
2-transient flow forces and valve instability,’’ ASME Machine Design Division, Vol. 74, pp. 1013–1016.
关5兴 M. Borghi, M. Milani, and R. Paoluzzi, 1998, ‘‘Transient flow force estimation
on the pilot stage of a hydraulic valve,’’ Proceedings of the ASME-IMECE
FPST-Fluid Power Systems & Tech., Vol. 5, pp. 157–162.
关6兴 D. Wang, R. Dolid, M. Donath, and J. Albright, 1995, ‘‘Development and
verification of a two-stage flow control servovalve model,’’ Proceedings of the
ASME-IMECE FPST-Fluid Power Systems & Tech., Vol. 2, pp. 121–129.
关7兴 Funk, J. E., Wood, D. J., and Chao, S. P., 1972, ‘‘The transient response of
orifices and very short lines,’’ ASME J. Basic Eng., 94, No. 2, pp. 483– 491.
关8兴 Martin, D. J., and Burrows, C. R., 1976, ‘‘The dynamic characteristics of an
electrohydraulic servovalve,’’ ASME J. Dyn. Syst., Meas., Control, 98, No. 4,
pp. 395– 406.
关9兴 Won, M., and Hedrick, J. K., 1996, ‘‘Multiple-surface sliding control of a class
of uncertain nonlinear systems,’’ Int. J. Control, 64, pp. 693–706.
Transactions of the ASME