Modeling of a Pneumatic System for High-Accuracy
Position Control
Deyuan Meng, Guoliang Tao, Jianfeng Chen, Wei Ban
State Key Laboratory of Fluid Power Transmission and Control
Zhejiang University
Hangzhou, China
mengdeyuan_8207@163.com
Abstract—In this paper, a detailed model was developed for a
rodless pneumatic cylinder controlled by a proportional
directional control valve. The dynamic of the valve spool was
firstly investigated and an equation was introduced to describe
the mass flow through the valve’s variable orifice. The
thermodynamics in cylinder chambers was carefully considered
and the heat transfer coefficient between the air in the chamber
and the inside of the barrel was identified experimentally. In
addition, the friction force of the pneumatic cylinder seals was
described using the LuGre model, and several experiments were
conducted to estimate the friction parameters. The proposed
system model was validated by comparing experimental and
simulated open loop step input responses, and can be used to
develop a controller for high accuracy positioning.
Keywords—modeling, proportional directional control valve,
heat transfer coefficient, seal friction
I.
INTRODUCTION
Pneumatic actuators typically are clean, easy to work with
and low cost, in addition, they have a high power-to-weight
ratio. These properties can make them favorable for servo
application. Unfortunately, due to the compressibility of air,
highly nonlinear flow through pneumatic system components
and significant friction, accurate force and position control of
the actuators are somehow difficult. In other words, these
nonlinear behaviors of a pneumatic servo system preclude good
control performance through PID or linear control methods
based on simplified models. Thus, many researchers resort to
advanced control algorithms, for example, adaptive controller
in [1, 2], sliding mode controller in [3, 4] have been tested,
which demonstrated that nonlinear control schemes can obtain
a higher accuracy than conventional linear controller and the
key to a high performance pneumatic servo system is a good
model. In the meantime, although pneumatic servo systems
have been extensively researched, the achievable performance
is far from perfect, especially in the case of continuous path
control or motion control. Besides the defective design of the
controller itself, we believe this is mainly owing to the
deficiencies of the models used in these controllers. As a result,
there is room for further research into modeling of pneumatic
systems.
The most notable works dealing with the modeling of
pneumatic servo systems in the past decade are two papers by
Richer and Hurmuzlu [5, 6], in which they developed a detailed
978-1-4244-8452-2/11/$26.00 ©2011 IEEE
nonlinear model, incorporating not only the nonlinear dynamics
of the cylinder itself but also propagation delays and losses in
the air lines connecting valve and cylinder and the dynamics of
the valve. Their model is targeted to develop force controller,
therefore, they assumed piston seal friction is Coulomb friction,
which is apparently different from reality. And they
characterized the mass flow through the valve by the standard
equation for mass flow through an orifice, with very little
information on how to obtain accurate values for the model
parameters. In addition, their assumption about the thermal
characteristics of charging and discharging process had been
proven inappropriate in [7]. Reference [8] developed a
nonlinear dynamic model for a servo pneumatic positioning
system. They assumed the charging and discharging process
was both adiabatic, piston seal friction is a combination of
Coulomb friction and viscous friction, and neglected the valve
dynamics as well as losses of lines. Their model depends
heavily on curve fitting using experimental values, making it
difficult to apply to other systems.
In this paper, a pneumatic cylinder controlled by a
proportional directional control valve is investigated. Our
objective is to provide a proper model for high accuracy
position control. The rest of this article is organized as follows.
In section II, the pneumatic system we considered is introduced
briefly. Section III is dedicated to the derivations of system
models. The obtained model is validated in section IV through
experiment. Finally, conclusions are drawn in section V.
II.
SYSTEM DESCRIPTION
A schematic representation of the pneumatic system we
consider is shown in Fig. 1, with variables of interest specified
for each component. The system consists of a pneumatic
cylinder (DGPIL-25-500-6K-KF-AU by FESTO) controlled
with a proportional directional control valve (MPYE-5-1/8-HF010B by FESTO). The valve is positioned near the cylinder,
thus, the effects of time delay and attenuation caused by the
connecting tubes are negligible. In addition, a tank is used to
maintain the pressure during charging. The pressures of
chamber A, chamber B and tank, and the position and velocity
of the piston are measured by pressure sensors (SDET-22TD10-G14-I-M12 by FESTO) and position sensor
(RPS0500MD601 V810050 by MTS) respectively (not shown
in Fig. 1 for brevity).
FPM 2011
505
x, x, x
Pb Vb Tb
Pa Va Ta
Sa
Sb
m a
m b
2
3 P0
coincide, and are approximately symmetrical. As a result, we
assume the MPYE-5-1/8-HF-010B valves are symmetrical for
simplicity. Furthermore, it has been confirmed experimentally
that the valve is not matching. Based on steady-state
measurements of mass flow rate through the valve’s variable
orifice as shown in Fig. 5 which has a constant supply pressure
7 bar, an appropriate equation governing mass flow from the
supply port to work port, and from work port to exhaust port
will be introduced.
4
1
P0 5
Ps , Ts
III.
SYSTEM MODELING
A. Model of Proportional Directional Control valve
The proportional 5/3 –way valve MPYE-5-1/8-HF-010B
has an internal control loop for the spool displacement which
can modify the steady-state and dynamic performance
considerably. The valve input accepts 0-10 volt signal, its ports
and their connections are shown in Fig. 1. The model of the
control valve can be divided into a mechanical part that is
responsible for the movement of the spool and a pneumatic part
that describes the flow through the valve as a function of the
valve’s input signal or spool position. The valve’s steady-state
spool displacement is measured as a function of the valve’s
input signal by laser sensor in our lab, which has a quasi-linear
characteristic, see Fig. 2. In addition, it has low hysteresis. The
measured frequency response of the valve is shown in Fig. 3
for four different input amplitudes. The input signal is the
commanded spool position and the output signal is the actual
spool position measured by laser sensor. The non-linear
behavior of the valve can be clearly seen because the gain and
the phase depend on the input amplitude. Since the bandwidth
of a pneumatic servo system is typically not more than 10 Hz,
which is much lower than the bandwidth of the valve, the
dynamics of the mechanical part of the valve can be neglected.
If the controller design makes it necessary to include the
dynamic of the valve spool, then it is sufficient to model it with
a first or second order system.
In the following, this section will focus on modeling the
pneumatic part of the valve. For a constant supply pressure 7
bar and a constant load pressure 5 bar at the work port, three
MPYE-5-1/8-HF-010B valves’ “mass flow rate vs. input
signal” characteristics are obtained in our lab (see Fig. 4). The
tested three valves’ pneumatic null voltage are not 5 V as
expected, they have offsets. In addition, their pneumatic null
biases are different from each other. After removing the null
biases, we found in the range of 2-8V these three curves almost
Phase (deg)
Figure 1. Schematic diagram of pneumatic servo positioning
system
Magnitude (dB)
Figure 2. Measured spool displacement vs. input signal
Figure 3. Measured frequency response of the valve
Figure 4. Measured mass flow rate vs. input signal
Figure 5. Measured mass flow rate vs. input signal and pressure ratio
506
­
°
°
P
° A (u ) C d C 1 u
Tu
°
°
2
°
§ Pd
·
−
b
°
¨ P
¸
P
°
¸
m
= ® A (u ) C d C 1 u 1 − ¨ u
1
−
T
b
¨
¸
°
u
¨
¸
°
©
¹
°
Pd ·
§
°
1−
2
°
P ¨
Pu ¸
§ λ−b·
¸ 1− ¨
° A (u ) C d C 1 u ¨
¸
Tu ¨ 1 − λ ¸
© 1−b ¹
°
¨
¸
°̄
©
¹
γ § 2 ·
¨
¸
R © γ +1 ¹
§p ·
§p ·
Cd = 0.8153 + 0.0933 ¨ d ¸ − 0.1038 ¨ d ¸
© pu ¹
© pu ¹
(2)
Figure 6. Input and exhaust valve areas vs. input signal
0.025
Experimental Values
Port1-Port2 theoretical flow
Port2-Port3 theoretical flow
1V
0.02
2V
0.015
0.005
3V
4V
0
Pd
≤b
Pu
-0.005
-0.01
P
b< d <λ
Pu
2
Fig. 7 gives several measured relationship between mass
flow rate and the pressure ratio for both input and exhaust path
and the calculated values according to (1), with upstream
pressure held constant at 6 bar. The error between measurement
and model is acceptable. Therefore, the proposed model of the
valve is adequate.
0.01
(1)
6V
7V
-0.015
-0.02
8V
9V
-0.025
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pressure Ratio Pd/Pu
λ≤
Pd
≤1
Pu
where m is the mass flow rate in kg/s, A(u) is the area of the
valve’s orifice in m2, Cd is the discharge coefficient, pu is the
upstream pressure in Pa, Tu is the upstream temperature of air
in K, pd is the downstream pressure in Pa, b is the critical
pressure ratio and
C1 =
measuring the mass flow rate under different input signals and
output pressure (see Fig. 5), the relation between input signal
and orifice area (input and exhaust paths) could be obtained as
shown in Fig. 6 (remove the null biases). The discharge
coefficient Cd is introduced to account for flow reduction
caused by contraction and losses. According to [9, 10], Cd
depends on the pressure ratio. A mathematical model of this
effect is identified experimentally as follows.
Orifice Area(m)
Reference [9] presents a thorough discussion on the models
of the mass flow rate through a pneumatic component, and
shows the most detailed description of a restriction is given by
the ISO model. For the MPYE-5-1/8-HF-010B valve, the
values of the critical pressure ratio and the sonic conductance
depend on the spool position in a non-linear way and are not
published numerically by the manufacturer. Therefore, these
two parameters need to be calculated from the measured mass
flow rate and the pressure differential with the methods detailed
by the ISO 6358 standard. This procedure is time-consuming
and will introduce computational errors. Besides, the value of
critical pressure ratio is smaller than the published value of
0.29 for large valve opening, this leads to difficulties in
measuring the parameter. In view of the above facts, a
combination of the ISO model and the theoretical model of
compressible flow through an orifice will be considered in this
paper. The MPYE-5-1/8-HF-010B valve is positiveoverlapped, as it is shown in Fig.4. However, when the valve is
in the dead zone, a considerable amount of leakage is present
due to the spool design which favors low friction forces over
zero leakage. Furthermore, the valve would work around the
middle position when the position control of actuator remains
at a constant value. Therefore, accurate modeling of leakage
flow rate within the dead zone is essential to further improving
the positioning accuracy and reducing control input chattering.
Although the annular model is always used to describe the flow
within the dead zone in literature, in order to facilitate the
following design of controller, we also use nozzle model
discussed above to describe this situation, which implies a
curve fitting in the sense of leakage throttling area. The
following equation for the mass flow through the valve’s
variable
orifice
will
be
adopted.
( γ +1) / ( γ −1)
is constant for air (ratio of specific heats =1.4, gas constant
R=287N·m/(kg·K)). For keeping the segment function
continuous, a laminar flow mode is added to the model. is the
minimal pressure ratio to have a laminar flow, and takes a
value close to 1.
To reduce the complexity of the model, a constant will be
used for the critical pressure ratio b in this article. Through
Figure 7. Measured and computed mass flow rate
B. Thermodynamics Model of the Cylinder Chambers
In this section we seek to develop a thermodynamic model
of gas in cylinder chamber. Assuming that the gas obeys the
equation of state of an ideal gas, the pressures and temperature
within each chamber are homogenous. Kinetic and potential
energy terms as well as cylinder leakage will be neglected.
Applying the ideal gas law, the conservation of mass equation
and the first law of thermodynamics to the gas in each chamber
gives the following mathematical models of the gas
temperature and pressure (see [11]).
p dV
R
R
γ −1 ­ dp
°° dt = −γ V dt + γ V m inTs − γ V m outT + V Q
®
2
° dT = T dV (1 − γ ) − m out RT ( γ − 1) + m in RT ( γ Ts − T ) + ( γ − 1) T Q
°̄ dt V dt
pV
pV
pV
(3)
where p is gas pressure in Pa, V is the volume of the chamber
in m3, m in and m out are the mass flows entering and leaving the
507
(4)
§1
·
Vi = V0i + Ai ¨ L ± x ¸
©2
¹
where i = a, b is the cylinder chambers index, V0i is the
dead volume at the end of stroke and admission ports in m3,
which were obtained in [12], Ai is the piston effective area in
m2, L is the piston stroke in m, and x is the piston position in m.
Convection is assumed as mode of the energy transfer between
the air in the chamber and the inside of the barrel. Therefore,
Q can be determined by
(5)
Q = hS h ( x )(Ts − T )
where h is the heat transfer coefficient in Watt/(m2·Kelvin),
Sh(x) is the heat transfer surface are in m2, which is a function
of piston position. Because of the low heat capacity of the air
and the high heat capacity of the surrounding material of the
barrel, the temperature of the metallic parts can be regarded the
same as ambient temperature.
Though (3) can describe the temperature and pressure
evolution in a chamber accurately, it is not suited for control
design, model reduction will be required in most occasions. In
previous works [2, 3, 5] the authors always neglected the gas
temperature dynamics and assumed that the charging and
discharging processes were isothermal, adiabatic or polytropic.
Their simplified models had been proven inappropriate for high
accuracy position control in [7]. Furthermore, early researchers
assumed the gas temperature T to be constant, however, there is
no doubt that the temperature in a cylinder chamber changes
when the piston is moving, charging or discharging takes place,
taking temperature fluctuations into account can improve the
quality of pressure predictions. Reference [7] recommended a
simplified model as follows
( n −1)
­
n
°T = T § p ·
¨
¸
s
°
© ps ¹
®
° dp
γ −1
p dV
R
R
hS h ( x )(Ts − T )
+ γ m inTs − γ m outT +
° = −γ
V dt
V
V
V
¯ dt
(6)
where ps is gas pressure entering the chamber in Pa, n is the
polytropic index. For expansion a value of n=1.11 and for
compression a value of n=1.18 are recommended (see [9]).
The heat transfer coefficient h is identified experimentally
by positioning the piston at the end of its stroke, filling the
chamber and then emptying it, recording the temperature and
pressure evolution in the chamber during charging and
discharging, and finally fitting the theoretical thermodynamics
equation (3), to the recorded values. Since the gas temperature
in chamber is difficult to measure directly, we estimate it using
stop method introduced as follows (see Fig. 8): Position the
piston at the end of its stroke. Terminate charging or
discharging at the time we want to know the
temperature T ( t ) and close all the solenoid valves. Record the
gas pressure p(t) in chamber immediately. Measure the gas
pressure p∞ when the temperature recovers to ambient level.
Finally, the average temperature at the time t can be estimated
using the Law of Charles.
T (t ) =
p (t )
Ts
p∞
(7)
By changing the time to stop charging or discharging, the
average temperature in chamber at any time could be estimated.
Reference [13] confirmed that the uncertainty of this method is
within 3%.
Solenoid
Valve
Pneumatic
Cylinder
A/D
Hand
Air
Pressure Valve
D/A
Supply Regulator
PC
Figure 8. Measurement set-up for the gas temperature in chamber
Heat Transfer Coefficient (W/(m ·K) )
chamber in kg/s, Ts is the temperature of the entering air in
Kelvin, assumed to be ambient temperature in practice, T is gas
temperature inside the chamber in Kelvin, Q is the heat transfer
between the air in the chamber and the inside of the barrel in
Joule/s. Choosing the origin of piston displacement at the
middle of the stroke, see Fig. 1, the volume of each chamber
can be expressed as
Figure 9. Measured values of heat transfer coefficient
Fig. 9 shows the obtained heat transfer coefficient for the
cylinder DGPIL-25-500-6K-KF-AU. Heat transfer in
pneumatic cylinders is a complex phenomenon. The values of
heat transfer coefficient vary significantly during charging or
discharging. In practice, a constant value can be set to the
coefficient as a first step (see [11]). For example, a value of 50
Watt/(m2·Kelvin) for charging process and a value of 20
Watt/(m2·Kelvin) for discharging process will be chosen for
model validation in section IV.
C. Piston-Load Dynamics and Seal Friction Model
The movement of the piston-load assembly can be
described by
(8)
Mx = ( pa Aa − pb Ab ) − Ff − FL
where x is the piston position in m, M is the lumped mass
including piston, slider and external load in kg, pa and pb are
pressures in chamber A and B in Pa, Aa and Ab are the piston
effective areas in m2, FL and Ff are the external force and seal
friction force in N.
We describe the friction force of the pneumatic cylinder
seals using the LuGre model (see [14]), as follows
508
­ dz
x
z
° = x − σ 0
g ( x )
° dt
®
dz
°
°̄ Ff = σ 0 z + σ 1 dt + σ 2 x
(9)
where the friction internal state z (m) describes the average
relative deflection of the contact surfaces during the stiction
phases, and is not measurable, 0 (N/m) and 1 (N·s/m) can be
understood as being the stiffness coefficient and the damping
coefficient of the cylinder seals, 2 (N·s/m) is the coefficient of
the viscous friction. The term g ( x ) is a finite function which
and xs =5.03 mm/s. Furthermore, the influence of the pressure
in the cylinder chambers on the seal friction has been
investigated. Fig. 12 presents the dependence of the parameters
on the pressure in the cylinder chambers. Fig. 13 shows the
dependence of the parameters on the pressure difference
between the cylinder chambers. The conclusion from Fig. 12
and Fig. 13 is that there are almost linear relationships between
the parameters 2, FC and FS and the pressure in chambers or
pressure difference between the two chambers, while the
Stribeck velocity may be considered as independence of
pressure variation in chambers.
describes part of the ‘steady state’ characteristics of the model
for constant velocity motions. Reference [14] recommended the
following parameterization of g ( x ) to describe the Stribeck
effect.
g ( x ) = FC + ( FS − FC ) e (
− x / xs )
2
(10)
where FC (N) is the Coulomb friction level, FS(N) is the level
of the stiction force, and xs (m/s) is the Stribeck velocity. For
steady-state motion, i.e., when x is constant, it follows [14] that
the relation between velocity and friction force is given by
− x / x
Fss = FC + ( FS − FC ) e ( ) sgn ( x ) + σ 2 x
(11)
2
s
)
Figure 11. Seal friction force vs. piston velocity
σ2
xs
FS
FC
(
Figure 10. Measurement set-up for the seal friction force
σ2
FS
FC
Figure 12. Influence of the pressure in chambers on
the static parameters
xs
The six parameters 0, 1, 2, FC, FS and xs of the seal
friction model for the cylinder DGPIL-25-500-6K-KF-AU can
be determined by system identification experiments described
in [15]. A test rig was built in our lab as shown in Fig. 10. With
the pressure in cylinder chambers held constant at 4 bar, the
piston is moved at a low velocity around the middle position.
We determine the parameters 0 and 1 as 0=9.14h105N/m
and 1=1327N·s/m using the information provided by piston
transient motion due to velocity reversal. Also, we confirmed
experimentally that these two parameters are insensitive to the
pressure variation in cylinder chambers. In order to estimate the
parameters 2, FC, FS and xs , the piston is moved by the
servomotor at different constant velocity levels ranging from 0
to 100mm/s. The external forces and the pressures during
constant velocity motions are recorded, and the seal friction
forces are calculated using (8). With the air ports of the
cylinder disconnected, the steady state seal friction force of the
cylinder is obtained and plotted as a function of the velocity,
see Fig. 11. Fitting the theoretic steady state seal friction
expression (11), to the experimental values, we determined
these four parameters as 2=64.5N·s/m, FC=20.2 N, FS =28.9N
Figure 13. Influence of the pressure difference between chambers
on the static parameters
509
IV.
VALIDATION OF THE MODEL
The complete model for the system we considered consists
of the equation for the mass flow through the valve’s variable
orifice (1), two equations for the chamber pressure time
derivatives as (6), the piston-load equation of motion (8) and
the equation for the friction force of the pneumatic cylinder
seals (9), was verified by comparing experimental and
simulated open loop step input responses. We positioned the
piston at the middle of the stroke, and then applied a step input
of 4V to the control valve. Fig. 14 shows both the experimental
and theoretically computed results for the pressures in cylinder
chambers, and the displacement as well as the velocity of the
piston. The simulated and experimental responses are almost
identical. Further validation of the system model has been
performed under other values of step input, and again the
responses were well predicted by the model. Therefore, the
model can be used as a mathematical model for high accuracy
position control analysis and design.
Figure 14. Experimental and simulated open loop step input responses
V.
CONCLUSIONS
In this paper a detailed model for a rodless pneumatic
cylinder controlled by a proportional directional control valve
was developed. The proposed model can be used to develop a
controller for high accuracy positioning. We divided the model
of the control valve into a mechanical part and a pneumatic
part, investigated the dynamic of the valve spool, and
introduced a new equation to describe the mass flow through
the valve’s variable orifice. The thermodynamics in cylinder
chambers was considered and the heat transfer coefficient
between the air in the chamber and the inside of the barrel was
identified experimentally. Since the seal friction is an important
aspect of the piston-load dynamics, we described the friction
force of the pneumatic cylinder seals using the LuGre model.
Several experiments were conducted to estimate the friction
parameters. We validated the complete system model by
comparing experimental and simulated open loop step input
responses. There was a close agreement between the theoretical
and experimental results. However, this paper omitted the
influence of the tubes for valves are always mounted nearby the
pneumatic cylinder. Otherwise, the effects of the tubes that
connect the pneumatic cylinder with the valve must be taken in
account.
ACKNOWLEDGMENT
This work is supported by National Natural Science
Foundation of China (No. 50775200).
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