Parameter identification of a pneumatic proportional pressure valve

TECHNICAL PAPER

Parameter identification of a pneumatic proportional pressurevalve

Rodrigo Trentini • Alexandre Campos •

Guilherme Espindola • Antonio da Silva Silveira

Received: 8 September 2012 / Accepted: 15 April 2013

� The Brazilian Society of Mechanical Sciences and Engineering 2014

Abstract In this paper, the internal parameters that define

the dynamic behavior of a pneumatic pressure valve are

identified based on theoretical models and experimental

testbed observations. Pneumatic systems became an usual

solution in many industrial environments in the last years.

Their mainly advantages are the low cost, maintenance

facility and security, besides they are clean, renewable and

abundant. However, their mainly disadvantages are the

nonlinearities due to friction and air compressibility, which

difficult their modeling and control. Proportional valves

play an important role in servopneumatic applications,

although hardly ever their internal components are known,

which may cause significant errors in modeling and sim-

ulations. The theoretical valve model is stated from phys-

ical representation of pneumatic phenomena using Mass

and Energy Conservation Theory, where an eighth order

linear model is obtained. Besides, using experimental

observations and the Least Squares method, an eighth order

black-box model with known parameters is determined.

The unknown internal valve constants determination is

carried out through mathematical and black-box models

comparison, where it is found that the identified model is

able to depict the real valve behavior with errors of less

than 10 %.

Keywords Parameter identification � Pneumatic control

valves � Modeling of pneumatic systems

1 Introduction

It is well known that compressed air is one of the oldest

forms of energy, but its application in industry took place

only since 1950, where it replaced the human power in

repetitive tasks. Nowadays, compressed air became indis-

pensable in several industrial sections. Pneumatic systems

work with high efficiency, performing repetitive opera-

tions, saving time, tools and materials. Furthermore, it is a

renewable, clean, abundant and cheap energy form. Devi-

ces moved by compressed air have a high power/weight

relation, low cost, maintenance facility and flexibility in

installation. Currently, the main applications in pneumatic

field are related to linear motion with final course stopping,

among of cutting, drilling, thinning and finishing.

Lately, servopneumatics are being used when interme-

diate positions are required. These systems use actuators,

electronic systems and control valves to reach high-end

results. It is important to notice that servopneumatic

applications reduce the air consumption by up to 30 % in

comparision to standard systems [8].

However, servopneumatic systems presents several dif-

ficulties in control, as air compressibility and friction

effects. Such characteristics make the mathematical model

more complex with the inclusion of nonlinear behaviour on

Technical Editor: Alexandre Abrao.

R. Trentini (&)

Institute fur Regelungstechnik (IRT), Leibniz Universitt

Hannover (LUH), Appelstr. 11, Hannover, Germany

e-mail: [email protected]

A. Campos � G. Espindola � A. S. Silveira

Center of Technological Sciences (CCT), University of the State

of Santa Catarina (UDESC), R. Paulo Malschitzki s/n, Joinville,

Brazil


G. Espindola


A. S. Silveira


123

J Braz. Soc. Mech. Sci. Eng.

DOI 10.1007/s40430-014-0144-0

it. Ali et al. [1] state that friction is the major difficulty to

obtain a satisfactory model. Pneumatic friction effects, like

Coulomb, static, Stribeck and viscous, cause undesired

effects as steady state and tracking errors, stick-slip

movements and limit cycles around the desired position

[5].

Due to these drawbacks, electric systems are most

commonly used in servopositioning systems than pneu-

matic ones, whereas accurate positions are not easily

reached in pneumatics [14]. Therefore, several researches

aim to minimize the nonlinearities influence using different

control strategies reaching accuracies up to 5 lm [5].

Besides the friction problem, it is important to notice

that an accurate valve model is required to improve com-

puter simulations and control design in pneumatic systems.

However, internal parts of pneumatic proportional valves

are hardly known, hence standard models are often used in

simulations. Nevertheless, researches which aim to repre-

sent the dynamical behaviour of different control valves are

being presented over the last years. For instance, Sorli et al.

[15] show a second order nonlinear model of an ordinary

proportional pressure valve using Mass and Energy Con-

servation Theory. Carneiro and Almeida [4] present a

proportional directional valve static model through artifi-

cial neural networks, reaching errors of less than 5 %.

Taghizadeh et al. [16] demonstrate a nonlinear dynamic

model of a pneumatic fast switching valve to be used with

PWM control applications. Additionally, some researchers

such as Guenther and Perondi [10], Ritter [13] and Barreto

[2] do not consider the dynamical features of pneumatic

servovalves stating that their natural frequency are much

higher than the natural frequency of the pneumatic actua-

tor. It is worth highlighting that this consideration may be

important aiming controllers design, where the goal is to

obtain a reduced model that owns the main system features.

Nevertheless, concerning to servopneumatic system simu-

lations, the valve dynamic equationing plays an important

role on its analysis.

In spite of the cited researches, the valve internal con-

stants remain unidentified, since hardly ever the valve

suppliers inform these parameters in their datasheets,

which justifies the use of computational tools to address

this issue. Hence, this paper presents a generic method for

the parameter identification of a pneumatic proportional

pressure valve using Least Squares (LS) method. A testbed

is developed, where a pressure sensor, a data acquisition

system and an electronic proportional valve are used.

In this paper, the valve description is present in Sect. 2.

The system mass conservation is analysed in Sect. 3, as

well as its mass flow rate and forces balance. Section 4

shows the valve black-box identification using LS methods,

whereas in Sect. 5 the internal parameter determination and

identification methods comparison is carried out. The study

conclusions are presented in Sect. 6.

2 Valve description

The electronic proportional pressure valves, from now on

pressure valves, are a specific case of pressure regulator

valves. This kind of valve operates by the force balance

principle, and the pressure regulation is carried out through

a plunger in opposition to a spring force, spool displace-

ment and internal pressure output [7]. Curatolo et al. [6]

state that these valves operate regulating their output

pressure proportionally to an input reference. A generic

schematic drawing of a pressure valve is shown on Fig. 1.

All of its internal components are considered with cylin-

drical shape.

The dark area on Fig. 1 shows the valve internal control

volume. The plunger, which is separated of the spool, has a

leaked part where the pressure is relieved through the

exhaustion port.

In order to understand the pressure valve operation, it is

considered a null input current/voltage on its solenoid, so

that all the internal components are positioned as on Fig. 1.

When an input signal is applied, the plunger is displaced

from top to bottom. If the produced force on the plunger

spring is higher than the spool spring force, the spool,

which at this moment is in contact with the plunger, moves

downwards so that a upstream mass of air fills the control

volume.

The air volume increment into the control volume

increases the valve internal pressure, being measured by

the pressure sensor. When the measured pressure is equal

to the reference input value, the valve internal controller

decreases the solenoid voltage until the spring force is

higher than the force applied on the spool by solenoid

itself, so that the input port is blocked. Hence, the pressure

valve stays at the position shown on Fig. 1.

Fig. 1 Electronic generic pressure valve schematic drawing. Based

on [6]


123

However, if the force due to the control volume pressure

is higher than the reference input, the solenoid voltage

decreases and the valve diaphagm deforms itself upwards.

Then, the spool reaches its mechanical limit and an air

volume passes through the leaked plunger, exiting the

valve through the exhaustion port until the force due to the

control volume pressure and the force due to the solenoid

are the same, so that the components return to the position

shown on Fig. 1.

An important feature of these kind of valves is the dead

band on the beginning of their operation range, which may

varies according to their internal components and

manufacturers.

In next section the mathematical modeling using Mass

and Energy Conservation Theory of a generic pressure

valve is present.

3 Mathematical model

In mathematical modeling, it is often required to reach a

trade off between model simplicity and accuracy, also

called Principle of Parsimony. However, in many cases the

simplifications are not easy to be done, and some a priori

system knowledge is required, since if some relevant

dynamics is neglect, the mathematical model may not have

the needed accuracy. In this work the main used simplifi-

cations are:

– the compressed air kinetic energy is neglected;

– the environment and air temperature are constant;

– the air is a perfect gas;

– the specific pressure and volume heat are constant;

– there are no leakages;

– the system entropy is constant.

3.1 Mass conservation

Considering the simplifications, the equating proposed in

this Section concerns to the Mass and Energy Conservation

theory used for compressed fluids presented by Fox and

McDonald [9].

Being the control volume depicted by the dark area on Fig.

1, the system mass conservation indicates that the mass

variation into the control volume is equal to the total mass

flow that enter or exit throughout the control surface [12], so:

o

ot

Z

VC

q dV þZ

SC

qv dA0 ¼ 0 ð1Þ

where dV is the volume differential, v is the fluid speed, qis the density, A0 is the control surface area and VC and SC

are the control volume and surface, respectively.

Considering that the density is invariant with the vol-

ume, the first term on Eq. 1 become:

o

ot

Z

VC

q dV ¼ qdV

dtþ V

dqdt ð2Þ

By being a compressible fluid, q is function of pressure p

and time t, or in other words, q ¼ f ðpðtÞÞ, therefore:

qdV

dtþ V

dqdt¼ q

dV

dtþ V

dqdp

dp

dtð3Þ

The term q dVdt

represents the control volume mass accu-

mulation rate due to the volume variation itself, and the

term V dqdp

dpdt

represents the control volume mass accumu-

lation due to the air compressibility [12].

The second term of Eq. 1 corresponds to the liquid fluid

flow rate, and it could be written considering the mass flow

rate as:Z

SC

qv dA0 ¼ �Dq ð4Þ

where Dq is the liquid difference between the mass flow

rates that enters and exits the system.

Therefore, replacing Eqs. 3 and 4 in Eq. 1,

Dq ¼ qdV

dtþ V

dqdp

dp

dtð5Þ

being p the pressure into the valve control volume.

The air compressibility coefficient b at a constant tem-

perature is given by the applied pressure rate and the vol-

umetric rate of a given control volume (b ¼ �Vdp=dV)

[9], so:

b ¼ qdp

dqð6Þ

Replacing Eqs. 6 in 5:

Dq ¼ qdV

dtþ Vq

bdp

dtð7Þ

For perfect gases q ¼ p=ðRTÞ, where R is the universal gas

constant and T is the absolute temperature. In addition, the

compressibility coefficient for reversible adiabatic pro-

cesses is b ¼ cp, being c the rate between the gas specific

heat for constant pressure and volume (c ¼ Cp=Cv). Then,

neglecting the control volume variation, the mass flow rates

for the control volume pressurization qm and exhaustion qe

are given respectively by:

qm ¼V

cRT_p ð8Þ

qe ¼V

cRT_p ð9Þ


123

3.2 Mass flow rate

The mass flow rates given by Eqs. 8 and 9 are analysed

using the mass flow equating for compressible fluid shown

also by Fox and McDonald [9]. Some considerations are

taken into account as well:

• the process is adiabatic, reversible and has high speed,

featuring an isentropic process;

• the flow is Unidirectional;

• the fluid speed is uniform;

• the stagnation pressure psup is kept constant.

Generalizing, the valve input may be considered as a

convergent nozzle. Therefore, according to Fox and

McDonald [9] there is a limit for the mass flow rate through

the nozzle, featuring that the flow is blocked and mass flow

rates at supersonic conditions are not reached.

Figure 2 shows the relation between the mass flow rate q

with the absolute stagnation down- and upstream pres-

sures—pm and pj respectively—in the pressure valve.

It is noted that from 52:8 % of the upstream pressure pm,

the mass flow rate has nonlinear behavior. Therefore, taking

into account only the linear range shown on Fig. 2, being the

absolute supply pressure psup ¼ 8 � 105 Pa and the atmo-

spheric pressure patm ¼ 1 � 105 Pa, the absolute pressure pa

into the control volume must be subjected to the restriction

1:894 � 105 Pa � pa� 4:224 � 105 Pa ð10Þ

As the pressure valve operates with manometric pressure,

the restriction become,

0:894 � 105 Pa � p� 3:224 � 105 Pa ð11Þ

which represents the restriction that the pressure p must

have so that the mass flow rate has linear feature.

Hence, the expression which represents the blocked

mass flow rate for a compressible fluid in a convergent

nozzle is [9]:

q ¼ Apm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

RT

2

cþ 1

� �cþ1c�1

sð12Þ

where A is the control port area, which for the analised

valve depends on the spool displacement when there is no

mass flow rate through the exhaustion port. On the other

hand, whenever there is mass flow rate through the

exhaustion port, the area A depends only on the plunger

displacement. Then, it may be state that:

A ¼ k1xc for qm ð13Þ

A ¼ k2xe for qe; ð14Þ

being xc the spool displacement, xe the plunger displace-

ment, and k1 and k2 are proportionality constants between

xc and xe and the analised control port cross-sectional area,

respectively. Therefore, Eq. 12 may be written as:

qm ¼ k1kaxcpsup ð15Þ

qe ¼ k2kaxep ð16Þ

which are the final expressions for the mass flow rate, being

qm and qe the flow rate for the valve pressurization and

exhaustion, respectively, and ka ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic

RT2

cþ1

� �cþ1c�1

r.

3.3 Force balance

The force balance on the pressure valve is given by New-

ton’s second Law. However, there are two different situa-

tions that must be analised: whenever the control volume

pressure p is lower than the regulated pressure p*, and

whenever p is higher than p*. The reason is that, whenever

p [ p* the spool displacement is mechanically limited, so

its dynamics does not influence the system pressure. These

conditions indicates the air flow direction through the valve:

when p� p* there is an air mass entering the control vol-

ume, whereas if p [ p* the air mass exhaustion happens.

On Fig. 3 is shown the valve force diagram, considering

an unidirectional diaphragm deformation, and also that the

forces FPaand Fp are uniformly distributed.

3.4 Condition 1: for p � p*

At this condition, the force balance is given by Eqs. 17 and

18:

me€xe ¼ Fsol � Fme � Fatreð17Þ

mc€xc ¼ FP � FPaþ Fme � Fatrc

� Fmc ð18Þ

where me is the plunger mass, xe is the plunger displace-

ment, Fsol is the solenoid force, Fme is the plunger spring

force, Fatreis the plunger friction force, mc is the spool

Fig. 2 Relation between down- and upstream pressures with the mass

flow rate for a compressible fluid through a nozzle. Based on [9]


123

mass, xc is the spool displacement, FP is the force due to

the air pressure into the control volume, Fatrcis the spool

friction force, Fmc is the spool spring force and FPaare the

atmospheric pressure force.

The solenoid force is given by:

Fsol ¼ ksu ð19Þ

being ks the solenoid constant and u the voltage input.

The plunger spring force Fme and the spool spring force

Fmc are given by:

Fme ¼ kme xe � xcð Þ ð20Þ

Fmc ¼ kmcxc ð21Þ

where kme and kmc are the spring elasticity constants of the

plunger and the spool, respectively.

The friction forces Fatreand Fatrc

are considered only

with their viscous component Bme and Bmc, being:

Fatre¼ Bme _xe ð22Þ

Fatrc¼ Bmc _xc ð23Þ

The force due to the control volume pressure FP is given

by:

FP ¼ Adp ð24Þ

where Ad is the diaphragm area.

Similarly, the force due to the atmospheric pressure FPa

is:

FPa¼ Adpatm ð25Þ

Therefore, the force balance on the pressure valve for the

condition when p � p* is given by:

€xe ¼�Bme _xe � kmexe þ ksuþ kmexc

me

ð26Þ

€xc ¼�Bmc _xc � kmc þ kmeð Þxc þ Ad patm � pð Þ þ kmexe

mc

ð27Þ

3.5 Condition 2: for p [ p*

As cited previously, whenever p [ p*, there is a mechan-

ical limit for the spool, so that its dynamics do not affect

the pressure output. Thus, using the same analysis shown

previously, the force balance for the cited condition is:

€xe ¼�Bme _xe � kmexe þ Ad patm � pð Þ þ ksu

me

ð28Þ

3.6 Linear model

The complete model of the pressure valve, taking into

account only the linear range of the mass flow rate, is

depicted by Eqs. 8, 26 and 27—using also Eq. 15—for the

control volume pressurization condition, which are linear.

For the exhausting condition, the mathematical model is

depicted by Eqs. 9 and 28—using also Eq. 16—, which are

nonlinear due to the multiplication of two system states in

Eq. 16, named xe and p.

Using Taylor series for the system linearization, the

equilibrium points for the condition p [ p* are obtained

considering _p ¼ _xe ¼ €xe ¼ 0. Assuming that the control

volume equilibrium pressure is different than zero, it is

obtained:

xe ¼ 0

p ¼ ks

Ad

u:ð29Þ

Hence, expanding Eq. 16 in Taylor series and neglecting

the terms with order higher than one:

qe ¼ k2kapxe ¼ 0

A ¼ oqe

oxe

��p¼p

¼ k2kap

B ¼ oqe

op

��xe¼xe

¼ 0

Then, the linearized equation for the mass flow rate

exhaustion qe is given by,

Fig. 3 Pressure valve force

diagram


123

qe � qe þ A xe � xeð Þ þ B p� pð Þ� k2kapxe

ð30Þ

Considering only the manometric pressure (patm ¼ 0) and

converting the equations to Laplace complex domain, after

some algebraic manipulations the control volume dynamics

for the pressurization and exhaustion conditions are

depicted respectively by:

PðsÞUðsÞ ¼

c1

s5 þ c2s4 þ c3s3 þ c4s2 þ c5sþ c6

ð31Þ

PðsÞUðsÞ ¼

c7

s3 þ c8s2 þ c9sþ c10

ð32Þ

where,

c1 ¼k1kakmekspsupcRT

mcmeðVi þ VaÞ

c2 ¼Bmcme þ Bmemc

mcme

c3 ¼ðkme þ kmcÞme þ kmemc þ BmcBme

mcme

c4 ¼Adk1kamepsupcRT þ ðBme þ BmcÞkme þ Bmekmc½ �ðVi þ VaÞ

mcmeðVi þ VaÞ

c5 ¼AdBmek1kapsupcRT þ kmckmeðVi þ VaÞ

mcmeðVi þ VaÞ

c6 ¼Adk1kakmepsupcRT

mcmeðVi þ VaÞ

c7 ¼k2kapkscRT

meðVi þ VaÞ

c8 ¼Bme

me

c9 ¼kme

me

c10 ¼k2kapAdcRT

meðVi þ VaÞ

4 Black-box model

This Section describes the black-box pressure valve iden-

tification using LS method based on the valve mathemati-

cal model presented in the previous section, in order to

obtain the internal valve parameters values.

Converting Eqs. 31 and 32 to complex discrete time

domain using Euler Method and considering the sampling

input delay, the pressurization and exhaustion conditions

are depicted respectively by:

Pðz�1ÞUðz�1Þ ¼

b0z�1

1þ a1z�1 þ a2z�2 þ a3z�3 þ a4z�4 þ a5z�5for p � p�

ð33Þ

Pðz�1ÞUðz�1Þ ¼

b1z�1

1þ a6z�1 þ a7z�2 þ a8z�3for p [ p�

ð34Þ

being,

b0 ¼ c1t5s a4 ¼ c5t4

s � 2c4t3s þ 3c3t2

s � 4c2ts þ 5

b1 ¼ c7t3s a5 ¼ c6t5

s � c5t4s þ c4t3

s � c3t2s þ c2ts � 1

a1 ¼ c2ts � 5 a6 ¼ c8ts � 3

a2 ¼ c3t2s � 4c2ts þ 10 a7 ¼ c9t2

s � 2c8ts þ 3

a3 ¼ c4t3s � 3c3t2

s þ 6c2ts � 10 a8 ¼ c10t3s � c9t2

s þ c8ts � 1

ð35Þ

where ts is the sampling time.

The experiments are carried out using a testbed

composed by an electronic pressure valve, a pressure

sensor and a data acquisition system, consisted by an

electronic board and computational software. In Table

1 is shown the main characteristics of the used devi-

ces, whereas Fig. 4 presents the testbed schematic

drawing.

An initial step experiment is carried out in order to

verify the acquisition system sampling time. Besides, the

manufacturer datasheet states that each 1 V on the valve

input corresponds to 0:9 � 105 Pa on the valve output (in

steady state), with a linear behavior. Hence, an 1.4 V step

input is applied on the system at t ¼ 0, considering a

system previous manometric pressure of 0:9 � 105 Pa.

Additionally, the used valve allows some external tuning in

order to make faster or slower the output response. In the

performed experiment, the pressure output was empirically

subjected to not have overshoot. The experiment result is

shown on Fig. 5.

It is noted in Fig. 5 that the experimented curve may be

approximated to a first order with transport delay system,

so that its time constant s is determined, neglecting the 0.3

s delay, as being s ¼ 0:3 s, allowing the maximum board

data acquisition time (ts ¼ 0:005 s).

Assuming that the output pressure behavior is linear

within the range of interest where the mass flow rate is also

linear (0:894 � 105 Pa � p � 3:224 � 105 Pa ), the valve

linearity features do not need to be experimented. There-

fore, the system is submitted to the increasing and

decreasing step experiment shown on Fig. 6.

Table 1 Testbed used devices

Component Brand Part number

Electronic pressure

valve

SMC ITV

1050-31F1BL3

Pressure sensor Sensata

technologies

100CP2-74

Data acquisition board National

instruments

NI USB-6009


123

Considering the cited linear pressure valve model, the

parameter identification using Least Squares method is

presented in the following section.

4.1 Least squares identification

In many science fields, the linear system parameters

determination is often carried out using Least Squares (LS)

method, which was proposed by Karl Friedrich Gauss at

the end of the eighteenth century.

The non-recursive LS method is used in offline batched

analisys, or in other words, it is assumed that all the input

and output data are available for the estimation algorithm.

The estimated parameters vector H is given by [11]:

H ¼ WTW� �1

WTY: ð36Þ

where W is the regressors matrix and Y is the experimental

output data vector, which in this paper are given by,

W ¼

�pðk � 1Þ � pðk � 2Þ � pðk � 3Þ xpðk � 1Þ... ..

. ... ..

.

�pðN � 1Þ � pðN � 2Þ � pðN � 3Þ xpðN � 1Þ

2664

3775

Y ¼

pðkÞ...

pðNÞ

2664

3775

being k the discrete time and N the experimented data

number. It is important to notice that, in order to better fit

the experimented and estimated parameters, the input and

output vector are considered with zero mean.

Two different parameters estimation are performed,

according to the linear valve conditions presented in Eqs.

33 and 34. For the cited conditions, the identified param-

eters using LS method are:Fig. 4 Testbed schematic drawing

Fig. 5 Pressure valve time

constant

Fig. 6 Step experiment


123

b0 ¼ 73:40 b1 ¼ 88:72 a1 ¼ �1:084

a2 ¼ �0:296 a3 ¼ 0:043

a4 ¼ 0:219 a5 ¼ 0:118 a6 ¼ �1:333

a7 ¼ �0:217 a8 ¼ 0:551

ð37Þ

5 Internal parameter determination

The identification of the empirical parameters shown in the

previous sections allows the internal pressure valve

unknown constant determination.

The access to the valve diaphragm is obtained through

four screws, according to Fig. 7, so that the diaphragm area

is measured.

Solving the linear system composed by the equations

shown in (35) using the identified values presented in (37)

in addition to Eq. 29 and considering the viscous friction

components Bme ¼ Bmc ¼ 12 Ns/m according to Carducci

et al. [3], the identified valve parameters are presented in

Table 2. The system constants are shown in Table 3.

In Fig. 8 is shown the model validation using the

experimented parameters obtained with LS method, con-

sidering only the valve linear behavior range (0:894 � 105

Pa � p � 3:224 � 105 Pa). The validation input is given by

random steps.

In order to evaluate the identification method, the fitting

coefficient shown in Eq. 38 is used, being y, y and y the

measured, the estimated and the mean value of the mea-

sured output, respectively.

fit ¼ 100 � 1� jjy� yjjjjy� yjj

� �¼ 91:66 %: ð38Þ

6 Conclusions

Although pneumatics is widely known as an important

alternative way to control mechanical systems, its intrinsic

issues, such as friction, air compressibility and unknown

valve internal parameters, hamper the modeling and con-

trol of such systems.

Fig. 7 Access to the valve diaphragm

Table 2 Identified internal

pressure valve parametersParameter (unity) LS

me (kg) 35:99 � 10�3

mc (kg) 26:68 � 10�3

kme (N/m) 168:45

kmc (N/m) 1434:31

k1 (m) 8:52 � 10�7

k2 (m) 6:51 � 10�7

ks (C/m) 102:92

Table 3 System constants

Parameter Value (Unity) Parameter Value (unity)

c 1:4 [1] R 287:04 [J/(K� kg)]

Ad 1:16 � 10�3 (m2) T 298 (K)

psup 8 � 105 (Pa) ka 2:34 � 10�3 (s/m)

Bme 12 (Ns/m) Bmc 12 (Ns/m)

Vi 6 � 10�6 (m3) Va 140:74 � 10�6 (m3)

u 2:25 (V) ts 0:005 (s)

Fig. 8 Model validation


123

In this paper, it is shown a systematic way to determine

the internal parameters of a proportional pressure valve

working within its linear mass flow rate range. The study is

performed using Mass and Energy Conservation Theory in

conjunction with empirical experiments that use Least

Squares (LS) method, so that the valve internal parameters

are determined through the analytic and empiric models

comparison. The result shows that the identified parameters

are able to depict the system dynamic behavior with an

accuracy of 91.66 %.

In further studies it will be investigated the full range

identification of the used pressure valve through nonlinear

methods, such as Particle Swarm Identification (PSO).

Acknowledgments R. Trentini acknowledges the financial support

of the Brazilian Research Agency (CNPq).

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Parameter identification of a pneumatic proportional pressure valve