Good-Comparison of Friction Models Applied to a Control Valve

7/29/2019 Good-Comparison of Friction Models Applied to a Control Valve

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Control Engineering Practice 16 (2008) 12311243

Comparison of friction models applied to a control valve

Claudio Garcia

Laboratory of Automation and Control, Department of Telecommunications and Control Engineering,

Polytechnic School of the University of Sao Paulo, 05508-900 Sao Paulo, Brazil

Received 27 June 2006; accepted 24 January 2008

Available online 9 June 2008

Abstract

Eight different models to represent the effect of friction in control valves are presented: four models based on physical principles andfour empirical ones. The physical models, both static and dynamic, have the same structure. The models are implemented in Simulink/

Matlabs and compared, using different friction coefficients and input signals. Three of the models were able to reproduce the stick-slip

phenomenon and passed all the tests, which were applied following ISA standards.

r 2008 Elsevier Ltd. All rights reserved.

Keywords: Control valves; Stiction; Stick-slip; Static friction models; Dynamic friction models; Data-driven models

1. Introduction

Performance assessment of control loops is an important

research theme, and there are many tools to detectvariability in control loops. These tools are employed to

diagnose different causes of variability, such as friction in

the control valve, oversized valves, improperly tuned

controllers, disturbances coming from other control loops,

and so on. Data extracted from real processes is usually

used to test the performance assessment tools. An easier

way to perform the initial tests of the performance

assessment techniques might be to use simulators, in which

the cause of variability is simulated. After these preliminary

tests, the tool can be applied to diagnose real situations

with data collected from existing plants.

Control valves are the most common final control elementsin industry. One of the main factors that affect the behavior

of the control loops is friction in control valves. Among the

variability causes previously mentioned, the most difficult one

to model is friction, and in particular static friction (stiction).

The purpose of this paper is to implement and test different

friction models applied to control valves. The idea is to

analyze the behavior of the models with the valve operating

in open loop, simulating a valve installed in a bench.

It is necessary to take into account that valve behavior

changes significantly as friction increases. Consider, forinstance, an ideal frictionless pneumatic valve with a full

stroke of 0100%. If this same valve is affected by friction,

it will not move until a certain pressure is applied to its

actuator. Besides, when a valve is affected by stiction, the

behavior of the control loop presents variability, since the

valve does not respond instantaneously to the control

signal. What happens is that the signal that comes from the

controller has to reach a value high enough to overcome

the stiction and move the stem. When this occurs, the stem

slips and the valve position normally goes to a point

beyond the desired value, causing oscillations and varia-

bility in the control loop.Models based on physical principles as well as empirical

or data-driven ones have been proposed to simulate valve

friction. Physical models describe the friction phenomenon

using balance of forces and Newtons second law of

motion. The main disadvantage of these models is that they

require knowledge of several parameters such as mass of

the moving parts, spring coefficient, and various friction

coefficients (viscous, Coulomb and static), which are not

easily estimated (Garcia, 2007; Romano & Garcia, 2007,

2008). On the other hand, the data-driven models simplify

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the simulation of a sticky valve and have been used to

study valve stiction (He, Wang, Pottmann, & Qin, 2007).

Many papers on friction modeling in control valves have

been published in the last few years (Choudhury, Jain, &

Shah, 2006; Choudhury, Thornhill, & Shah, 2004;

Choudhury, Thornhill, & Shah, 2005; Eborn & Olsson,

1995; He et al., 2007; Jain, Choudhury, & Shah, 2006;Kano, Maruta, Kugemoto, & Shimizu, 2004; Kayihan &

Doyle, 2000; Stenman, Gustafsson, & Forsman, 2003), but

a full comparison of different models to describe the

behavior of control valves affected by friction has not been

presented. In Eborn and Olsson (1995) the authors

compare some friction models, but the results are presented

with the valve inserted in a control loop, in such a way that

it is difficult to visualize how the isolated valve responds

when submitted to different kinds of input signals. In He et

al. (2007) the authors present one figure comparing some

data-driven models, considering just the case when the

valve is ideal, that is, with no friction.

In this work, the simulated valves are modeled with three

different levels of friction and are submitted to tests that

are recommended in ISA standards for real control valves

(ISA, 2000, 2006). This form of testing the models is a

contribution of this work, since there is not any other

related paper that performs tests in simulated valves

according to international standards.

The paper is organized as follows: in Section 2, the eight

valve friction models applied to a pneumatic spring-

diaphragm sliding stem valve are presented. In Section 3,

the applications of the valve friction models analyzed in

this paper are listed. In Section 4, the tests applied to

control valves according to ISA standards are presented. InSection 5, the characteristics of three valves with different

friction coefficients are presented. In Section 6, the

responses of the model simulations, with valves with

different friction coefficients, applying the ISA recom-

mended testing, are shown and an evaluation table is

presented. Finally, in Section 7, the conclusions are drawn.

2. Control valve friction models

As the main purpose of this paper is to compare friction

models applied to a control valve, eight different models of

friction in pneumatic sliding stem control valves are

presented, starting from simple models, with just one

parameter, and moving to more complex ones, with seven

parameters: Classical (Olsson, 1996), Karnopp (Karnopp,

1985), Seven Parameters (Armstrong-He louvry, Dupont, &

Canudas de Wit, 1994), Lugre (Canudas de Wit, Olsson,

A stro m, & Lischinsky, 1995), Stenman (Stenman et al.,

2003), Choudhury (Choudhury, Jain et al., 2006; Choudh-

ury, Thornhill et al., 2004; Choudhury et al., 2005), Kano

(Kano et al., 2004) and He (He et al., 2007). The first four

are physical models, the first two (Classical and Karnopp)

being static models and the next two (Seven Parameters and

Lugre) dynamic ones. The last four are empirical models.

Notice that the more recent models are all data driven.

2.1. Force balance on the components of a pneumatic sliding

stem valve

The function of the valve actuator is to move the valve

stem to modulate its opening. Pneumatic control valves are

still the most used in the process industries, due to their low

cost and simplicity. In order to model a sliding stem valve,it is assumed that the input variable is the signal that comes

from the controller, converted to a pressure signal, and that

the stem position is the output variable. In that way, the

force balance equation is as follows (Choudhury et al.,

2005; Kayihan & Doyle, 2000):

m x Fpressure Fspring Ffriction Ffluid Fseat, (1)where m is the mass of the valve moving parts (stem and

plug); x is the stem position; Fpressure Sa P is the forceapplied by the actuator, Sa being the diaphragm area and P

the air pressure; Fspring k x is the spring force, k beingthe spring constant; Ffriction is the friction force; Ffluid

a DPis the force due to the fluid pressure drop across thevalve, with a the plug unbalanced area and DPthe pressure

drop; and Fseat is the extra force required for the valve to

be forced into the seat. Following Choudhury et al. (2005)

and Kayihan and Doyle (2000), the contributions of Ffluidand Fseat are negligible in practical situations. Ffluid is

disregarded because it is two orders of magnitude smaller

than the friction and spring forces, which means that the

valve is modeled as if there was no fluid in the line. Fseat is

not considered for simplicity.

The main issue is how to model the friction force in

Eq. (1). This will be done in the following sections through

different friction models.

2.2. Static friction models

According to Olsson (1996), friction models can be

classified as static and dynamic. The classical friction

models are static, which means that the friction is modeled

as a static function of velocity. In the dynamic models there

are time-varying parameters. This classification does not

agree with what is normally defined as static or dynamic

systems, but it has been kept in this work, to be in

agreement with the related published papers.

In the static models, three components are normallyconsidered:

static friction or stiction; viscous friction and Coulomb friction.

The total friction force can be calculated as follows:

Ffrictionv Fc Fs Fc ev=vs2

h isgnv Fv v, (2)

where Fc is the Coulomb friction coefficient, Fs is the static

friction coefficient, v is the stem velocity, vs is the Stribeck

velocity and Fv is the viscous friction coefficient.

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C. Garcia / Control Engineering Practice 16 (2008) 123112431232


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velocity of the surfaces in contact, s0 is the stiffness

coefficient, Ffriction is the friction force, and s1 is the

damping coefficient.

The calculation of the friction force is inserted in the part

of the Karnopp model which is in charge of calculating the

sliding friction force. Therefore, the structure of the

Karnopp model is also applied to simulate the Lugre

model.

2.4. Data-driven models

A detailed physical model has many unknown para-meters, so it is often difficult to estimate them. Besides,

complex models are much slower to run in a computer.

A data-driven model, on the other hand, is useful because it

has only a few parameters to identify, and can be run

faster. This is the reason why, in recent years, the models

presented in the technical literature are all data-driven, as

can be observed next.

2.4.1. Stenman model

The basic concept behind this model is to try to imitate

the jump that occurs in the stem position, when stiction is

overcome. The Stenman friction model (Stenman et al.,2003) is parameterized by one parameter d and may be

expressed by the following equation:

xt xt 1 if jut xt 1jpd;ut otherwise;

((9)

ARTICLE IN PRESS

0 10 20 30 40 50 60 70 80 90 100

12

24

36

48

60

72

84

Input pressure P (%)

Stemp

ositionx(%)

J

J

J

J

Slip jump J

S

Deadband Stickband

Fig. 1. Example of valve signature showing the parameters J and S

(Choudhury, Jain et al., 2006).

Input pressurex(k)

no

no

no

no

no

no

no

no

yes

yes

yes

yes

yes

yes

yes

yes

xss = xss

y(k) = 0x(k) > 0

x(k) < 100

|x(k) - xss| > J

|x(k) - xss| > S?

y(k) = x(k)

xss = xss

y(k) = 100

xss = x(k-1)

y(k) = y(k-1)

y(k) = y(k-1)

y(k) = x(k) sing (v_new) (SJ) / 2

sign(v_new) = 0

I= 1

I= 0

I= 1?I= 0

Remain stuck

Value slips and moves

Value positiony(k)

Value sticks

sign(v_new) = sign (v_old)

v_new = [x(k) -x(k-1)]/t

S = 0 & J = 0?

Fig. 2. Flowchart for the Choudhury model (Choudhury, Jain et al., 2006).



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where x(t1) and x(t) correspond to past and present stempositions, respectively, u(t) is the present controller output

and d is the valve stiction band.

2.4.2. Choudhury model

Successive versions of the Choudhury model were

presented in Choudhury, Thornhill et al. (2004), Choudh-ury et al. (2005) and Choudhury, Jain et al. (2006). It

requires just two parameters (J and S), as shown in Fig. 1,

which are extracted from signature experiments.

S represents the amplitude of the input signal (pressure)

during the time in which the stem is stuck (stickband+

deadband). Jrepresents the size of the stem slip (slip jump).

This model is described in the flowchart (Choudhury,

Jain et al., 2006) presented in Fig. 2.

2.4.3. Kano model

The Choudhury model is not able to deal with both

deterministic and stochastic signals (Kano et al., 2004). The

Kano model is an extension that requires the same two

parameters used in the Choudhury model. Its algorithm is

presented in a flowchart (Kano et al., 2004) shown in Fig. 3.

2.4.4. He model

The He model (He et al., 2007) is less complex than the

Kano model. It requires two parameters: fS (static friction

band) and fD (kinetic friction band). Jis defined as fSfDand S as fSfD, as shown in Fig. 4 (Kano et al., 2004).

The central line in Fig. 4 presents the situation of an air-

to-open valve with no friction at all. As the friction

increases, the static (fS) and the dynamic (fD) friction bands

also increase. In Fig. 4 the controller output and the valve

position are normalized and both vary in the range

0100%, so all the lines indicating the valve movement

have a slope of 45%.

The flowchart of the He model can be seen in Fig. 5 (He

et al., 2007).

3. Applications of the control valve friction models

Friction is a common problem in spring-diaphragm type

valves. Bialkowski (1993) reported that about 30% of the

ARTICLE IN PRESS

Fig. 3. Flowchart for the Kano model (Kano et al., 2004).

Input pressure u(t)

yes no

cum_u = ur+ [u(t) - u(t-1)]

abs(cum_u) > fs?

uv(t) = u(t) - sign(cum_u fs) fDur= sign(cum_u fs) fD

uv(t) = uv(t-1)

ur= cum_u

Fig. 5. Flowchart for the He model (He et al., 2007).

0 10 20 30 40 50 60 70 80 90 100

12

24

36

48

60

72

84

Input pressure P (%)

Stemp

ositio

nx(%)

JJ

J

J

Slip jump J

fD fS

S

Fig. 4. Relation between controller output and valve position (Kano et

al., 2004).

C. Garcia / Control Engineering Practice 16 (2008) 12311243 1235


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loops are oscillatory due to control valve problems. About

the same numbers are obtained in another investigation

(Ender, 1993). Quantifying the valve friction coefficients

may help to reduce control loop variability. It is part of

what is known as control performance monitoring/assess-

ment, an important asset-management technology to

maintain highly efficient operation performance of auto-mation systems in production plants (Jelali, 2006; Thorn-

hill & Horch, 2006, 2007).

The control valve friction models described here have

been applied to stiction detection and quantification in

several papers.

The Stenman model is used in Stenman et al. (2003) for

stiction detection. In Kano et al. (2004) the authors

propose an algorithm for valve stiction detection. In Rossi

and Scali (2005) and Singhal and Salsbury (2005) the

Choudhury model is used to analyze the proposed valve

stiction detection method. In He et al. (2007) the authors

consider the He model to detect stiction.

Concerning quantification of control valve stiction, the

papers Garcia (2007) and Romano and Garcia (2007, 2008)

focus on techniques to estimate the parameters of the

Karnopp model (m, k, Fv, Fc and Fs). The idea of Detection

and quantification of control valve stiction is to detect if

the control valve presents stiction and to find out the friction

parameters. In Srinivasan, Rengaswamy, Narasimhan, and

Miller (2005) the goal is to find the parameter d of the

Stenman model. In Choudhury, Shah, Thornhill, and Shook

(2006), and Choudhury, Shah, and Thornhill (2004) and

Ha gglund (2007) the idea is to estimate the parameter S,

whereas in Choudhury, Jain and Shah (2008), Choudhury,

Jain et al. (2006), Jain et al. (2006) and Schoene and Qin(2005) the concern is with J and S. These parameters are

related to the Choudhury and Kano models.

Papers on Friction compensation for control valves

are concerned with developing friction compensators to

deal with the control valves that are affected by this

problem. The authors Kayihan and Doyle (2000) use the

Classical model to develop their compensator. The

compensator in Srinivasan and Rengaswamy (2005, 2006,

2008) employs the Stenman model.

4. Tests recommended by ISA to be applied to control valves

The models presented in Section 2 were implemented in

Matlab/Simulinks and tested with different friction

coefficients and input signals. In order to perform the

simulated tests according to international standards related

to control valves, the choice was to apply the tests that are

defined in the ISA standards (ISA, 2000, 2006). The aspects

of these documents that are relevant to this paper are

discussed next.

4.1. Control valve diagnostic tests

The purpose of the two tests recommended in ISA (2006)

is to provide control valve users with a uniform means of

acquiring and reporting data used for diagnosing valve

operability. All data must be acquired with the valve out of

service, with no internal dynamic forces acting on the

closure member. For both tests, data must be acquired at

equally spaced intervals of input signal specified over the

travel in the opening and closing directions. The recom-

mended minimum number of data points in each directionis 101 for a full signal span. Each method is briefly

summarized as follows:

Dynamic test method: Starting with the user definedminimum signal specified, ramp the input signal (control

signal to a positioner or pressure signal to an actuator)

to the maximum signal specified, wait for the pause time

and ramp down to the minimum signal. Then wait for

the pause time again. The format of the input signals

corresponds to a trapezoidal wave. During the travel,

record the input signal and valve position. This test

generates what are known as signature curves of thevalve, which are produced by operating a valve through

its signal range which, according to ISA (2006), are

plotted with stem travel in the vertical axis and actuator

pressure in the horizontal axis, with both scales ranging

from 0% to 100%. Thus, the signature curves are

derived by plotting the valve position versus the input

signal in both directions, over the minimum to the

maximum input signal specified for the application. The

dynamic test response obtained in this paper in shown in

Fig. 6 and the signature curves can be seen in Figs. 711.

Ramp and pause test method: Starting at the user definedminimum input signal specified, ramp the input signal at

a slow rate. Wait for the ramp and pause time specified.Repeat the procedure up to the maximum input signal

specified. Record the input signal and valve position.

Repeat the preceding process in the opposite direction.

ARTICLE IN PRESS

0 2 4 6 8 10 12 14 16 18 20

0

10

20

30

40

50

60

70

80

90

100

Time t (s)

ActuatorpressureP(%)andstemp

ositionx(%)

Input signal

Classical model

Karnopp model

Fig. 6. Dynamic test response with two input cycles applied to the rough

valve with trapezoidal input.



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In the sketch shown in ISA (2006) there are five pause

times, at 20%, 40%, 60%, 80% and 100% of the input

signal. See this test result in Fig. 12.

4.2. Control valve response measurement from step inputs

In ISA (2000), three alternative environments are defined

to perform the tests: bench, laboratory and in-process

testing. The bench test, which is performed without process

fluid, is the one carried out in this work. In Eq. (1) it is

assumed that the force due to the fluid pressure drop across

the valve Ffluid is null. This is a characteristic of bench

testing. The input and output signals to be measured in

bench testing are the pressure input signal and the stem

position.

The three valve tests recommended in ISA (2000) can be

summarized as follows:

Baseline test: Used to evaluate measurement noise, thepresence of limit cycles, and the baseline response time

T86, that corresponds to the interval of time between

initiation of an input signal step change and the moment

at which the response reaches 86.5% of its full steady-

state value. In the example given in ISA (2000), two

steps up are applied in the input signal, from 50% to

52% and from 52% to 54%; and two steps down, from

54% to 52% and from 52% to 50%. Each step lasts

1 min. This test result is presented in Fig. 13.

Small step test: Used to determine dead band and

resolution. The dead band is the range through which an

ARTICLE IN PRESS

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Actuator pressure P (%)

S

temp

ositionx(%)

Vendor valve

Nominal valve

Rough valve

Fig. 8. Signature curves of vendor, nominal and rough valves employing

the Karnopp and Lugre models.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100


Stemp

ositionx(%)

Vendor valve

Nominal valve

Rough valve


the Stenman model.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100


Stemp

ositionx(%)

Vendor v alve

Nominal valve

Rough valve


the Choudhury and Kano models.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100


Stemp

ositio

nx(%)

Vendor

Nominal

Rough


the Seven Parameter model.



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input signal may be varied, with reversal of direction,

without initiating an observable change in output signal.

In ISA (2000) it is defined as a percentage of input span.

The resolution is the smallest step increment of input

signal in one direction for which movement of the

output is observed, expressed as percentage of input

span. In ISA (2000) the test given as example wasperformed with incremental steps of 0.1%, starting at

50% of the input signal and reaching 50.6% and coming

back to 50%, each step lasting 30 s. The whole cycle

must be repeated twice. It is necessary to assure that

there is at least one step in addition to the step causing

initial movement. This test result is presented in Fig. 14.

Response time test: Consists of an increasing sequence ofstep sizes. Start the test with a step size of 0.1% and end

with 10%, assuming the following values: 0.1%, 0.2%,

0.5%, 1%, 2%, 5% and 10% of input signal span.

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0 4 8 12 16 20 24 28 32 36 40

0

10

20

30

40

50

60

70

80

90

100

Time t (s)

Actuator

pressureP(%)andstemp

ositionx(%)

Input signalKarnopp modelChoudhury model

Fig. 12. Response of the Karnopp and Choudhury models to the ramp

and pause test method employing nominal valve.

0 4 8 12 16 20 24 28 32 36 40

50

54

58

62

66

70

74

78

82

86

90

Time t (s)


ositionx(%)

Input signal

Karnopp model

Fig. 13. Results of baseline test with nominal valve.

0 6 12 18 24 30 36 42 48 54 60 66 72

50

50.2

50.4

50.6

50.8

51

51.2

51.4

51.6

Time t (s)


ositionx(%)

Input signal

Karnopp model

Fig. 14. Result of small step test with vendor valve.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100


Stemp

ositio

nx(%)

Vendor valve

Nominal valve

Rough valve


the He model.

0 3 6 9 12 15 18 21 24 27 30 33 36

0

10

20

30

40

50

60

70

80

90

100

Time t (s)


ositionx(%)

Input signal

Karnopp model

Fig. 15. Results of response time test applied to the nominal valve.



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Larger steps sizes such as 20% and 50% may be used if

desired. In the example given in ISA (2000), each step

lasts 1 min. In Fig. 15 this test result is presented.

5. Characteristics of the simulated tested valves

It was selected a sliding stem globe valve, withdiaphragm actuator, provided with a spring that moves

the actuator stem in a direction opposite to the movement

caused by the diaphragm pressure. Its nominal size is 4 in.

and its full stem stroke is 4 in. (0.1016 m).

So as to simulate the behavior of a control valve, it is

necessary to define the model parameters. The mass m and

the friction force depend on the valve. The parameter

values were extracted from Fitzgerald (1995) and Kayihan

and Doyle (2000). In this last paper, the authors have

defined three valves with different friction coefficients:

vendor: it is the valve with smaller friction;

nominal: it is the valve with mean friction; rough: it is the valve with higher friction.

The vendor valve has only the Coulomb friction

component and corresponds to a new valve with Teflon

stem packing. The nominalvalve presents stiction and may

correspond to a new valve but with graphite packing

operating at ambient temperature. Finally, the rough valve

corresponds to a rather worn out one. The features of each

valve are shown in Table 1.

According to Eborn and Olsson (1995), it is assumed

that tl

0:02s, g

100s and Fs;1

Fs

Fc for the Seven

Parameters model. In the simulations, it is assumed for thismodel that the time in which the stem was stuck at the start

of the simulation was very long, so that the initial stiction

has reached its maximum value Fs;1 Fs Fc.In the Lugre model, the following parameters were used

(Eborn & Olsson, 1995): s0 108 N=m ands1 9000Ns=m.

To estimate parameter d in the Stenman model, a search

was performed, aiming at best reproducing the expected

signature curve of Fig. 1. The values found are:

Valve Vendor Nominal Rough

d 0.010 0.4666 0.334

The parameters J and S for the Choudhury, Kano and

He models are:

Valve Vendor Nominal Rough

S 0.0167 0.5894 0.9224

J 0 0.0550 0.0860

S (width of the signal pressure during sticking) is very

simple to measure, because the width of the signal pressure

during sticking corresponds to the two horizontal lines in

Fig. 1. The measurement of J (slip jump size) is not as

simple, because it is not so easy to determine where it ends.

The values shown above were calculated in order to

generate similar figures to the Karnopp and Lugre models,

since they had generated an expected signature curve.

6. Simulation of the friction models

In this paper, the eight models are compared through the

five tests described in Section 4. The valve is supposed to

operate in open loop in a bench, eliminating any test

related to observing limit cycling.

6.1. Search for an adequate integration step time

As it has already been mentioned, one of the great

problems to simulate friction is related to the model

behavior close to null velocities. If the integration step of

the numeric integration method is not sufficiently small, the

stem velocity can cross the zero velocity and simply reverse,without stopping. Therefore, to make the stem stop, it is

necessary for the integration step to be sufficiently small.

Fig. 1 shows the presumed signature of a valve with

stiction. It is used as the basis of comparison for defining

the adequate integration step time for the static and

dynamic models. When the stem is stuck, the effect is

named deadband+stickband, and when it is slipping, it is

called slip jump. The slip jump corresponds to the

momentary slipping suffered by the stem when the

externally applied force overcomes the stiction force. At

some instants, the resistance is much reduced, causing a

large acceleration and peaks in the stem velocity. A good

model of the rough and nominal valves, which present

ARTICLE IN PRESS

Table 1

Features of the simulated valves

Parameter Vendor Nominal Rough

Sa, area of the actuator diaphragm (m2) 0.06452 0.06452 0.06452

m, mass of the moving parts of the valve (kg) 1.361 1.361 1.361

k, spring constant (N/m) 52538 52538 52538

Fc, Coulomb friction coefficient (N) 44.48 1423 2224

Fs, stiction coefficient (N) 44.48 1707.7 2668.8

Fv, viscous friction coefficient (N s/m) 612.9 612.9 1226

vs, Stribeck velocity (m/s) 2.54

104 2.54

104 2.54

104



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stiction, should be able to reproduce the stick-slip

phenomenon. The signature curve shown in Fig. 1 was

obtained through a dynamic test method, applied as

described in Section 4.

To find an adequate integration step for the static and

dynamic models, simulations were performed employing

integration steps of 103

, 104

, 105

and 106

s. The bestresults were obtained with 106 s. However, with such a

small step, the running time of the models is very long.

Among the numeric methods with fixed and variable step

tested to minimize running time without affecting the

model accuracy, Eulers fixed step method resulted in lower

running time and responses similar to the others. The

Stenman, Choudhury, Kano and He models are not first

principle ones, so there is no concern about the behavior of

the velocity close to zero. The Euler method with time step

of 103 s was employed for them.

6.2. Dynamic test method and signature curves

In the papers (Choudhury et al., 2005; Kayihan & Doyle,

2000) about friction in control valves, the authors

employed the valve signature curve, with a sinusoidal

actuator pressure input, in order to show the behavior of

the model in open loop. In this paper, following the ISA

standard recommendations, the input signal to be used is

trapezoidal or triangular and not sinusoidal.

To perform this test, simulations were carried out,

varying the input signal that represents the pressure on the

actuator in the range of 012psi (082,737 Pa) and

observing the behavior of the stem position. The ramp

time is 4s and the pause time is null for the triangularsignal or 1 s for the trapezoidal one. In the simulations, two

complete opening and closing cycles are applied.

In order to reproduce the signature curve of Fig. 1, it is

necessary to have at least two complete cycles of the input

signal, a fact which is not stated explicitly in the ISA

standards. Only the data relative to the second cycle,

corresponding to the situation when the valve reaches its

normal operation cycle, is used. In the first cycle the valve

stem starts at position 0% (completely open), as shown in

Fig. 6, which depicts the response of the Classical and

Karnopp models, applied to the rough valve with

trapezoidal input. It can be noticed that, after the first

input cycle, the stem motion is reduced to a small range of

its full stroke. As stated in item 2.2.1, the main problem

with the Classical model is its behavior around null

velocity, in such a way that the valve is never sticky. As

it is fundamental for the model to reproduce the stick-slip

phenomenon, the Classical model is discarded.

The signature curves presented next employ a trapezoidal

input. Fig. 7 shows the response of the Seven Parameter

model for the three valves with different friction coefficients.

Because the time during which the trapezoidal input

stops is only 1 s, the Seven Parameter model responses in

Fig. 7 has a negligible slip jump with the parameters used.

In order to make the stiction force appear in this model, it

would be necessary for the trapezoidal input to have a

longer pause. Only then would it be possible to observe the

slip jump. The pause time of the input signals to be used

here is not large, so the Seven Parameter model is also

discarded.

The signature curves of the Karnopp and Lugre models

are shown in Fig. 8, in which the stick-slip phenomenonappears for both models, as shown in Fig. 1.

The response of the Lugre model with the assigned

parameters is quite similar to the Karnopp one. Never-

theless, if the parameters are changed, the response of the

model becomes different. For instance, if the values

(Olsson, 1996) s0 105 N/m and s1 2 ffiffiffiffiffis0p N s=m areconsidered, the Lugre model behavior is very different.

The signature curves generated by the Stenman model

are shown in Fig. 9 and they are completely different from

what is expected. Therefore, this model is discarded.

The signature curves derived from the Choudhury and

Kano models are shown in Fig. 10. It indicates that the

responses of both models are identical. Comparing Figs. 8

and 10, it can be noticed that the Karnopp, Lugre,

Choudhury and Kano models present very similar signa-

ture curves for the three valves.

The responses of the He model when submitted to a

trapezoidal input are shown in Fig. 11 and they do not

comply with what is expected, so it is discarded.

Although not being exactly equal, the signature curves of

all the models for the vendor valve are similar, which means

that all of them are able to represent the behavior of valves

with low friction coefficients. Nevertheless, when the

friction coefficients increase, the differences appear. As

the behavior of the Karnopp, Lugre, Choudhury and Kanomodels are the ones that responded as expected, they

continue being analyzed in the next sections.

6.3. Ramp and pause test method

The ramp and pause method was applied with ramp time

of 1s, pause time of 1s and pauses in 20%, 40%, 60%,

80% and 100%. The test is performed twice, although this

is not required in ISA (2006). The responses of the

Karnopp, Lugre, Choudhury and Kano models for the

vendor valve, not shown here, are quite similar, following

closely the input signal. The responses for the nominalvalve

employing the Karnopp and Choudhury models are shown

in Fig. 12. The Lugre and Kano models presented

responses quite similar to the Karnopp one, so they are

not shown in Fig. 12.

Fig. 12 reveals that the Karnopp model responded

similarly, keeping a certain distance from the input signal,

due to the relatively high friction coefficients in this valve,

whereas the Choudhury model did not move at all. Based

on that, this last model is discarded. The behavior of the

models for the rough valve are similar to the one for the

nominal valve, that is, the Karnopp, Lugre and Kano

models responded quite similarly, whereas the Choudhury

model did not move.

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In its first movement in Fig. 12, the valve has to

overcome the fs value, which corresponds to fs J S=2 32:22%. Therefore, the valve moves whenpressure Preaches 32.22%. As the valve movement goes on

in the same direction, without a valve reversion, the second

valve movement occurs when the pressure overcomes

J 5:50%, that is, it has to increase 5.50% above thepressure value when the valve stuck. In Fig. 12 the valve

stuck when the pressure reached its second pause, that is,

40%, so the next pressure value for the valve to slip is

45.50%. When there is a valve direction reversal, the input

signal has to change an amount of S 58:94% for thevalve to move. Analyzing Fig. 12, it is possible to see that

the pressure had to change from 100% to 41.08% for the

valve to move, which is equivalent to S.

6.4. Baseline tests

The valve considered in the simulations has no dynamics

associated with the actuator, that is, it responds instantly as

the pressure signal varies, so there is no meaning in

estimating T86, and this test is applied simply to compare

the behavior of the models. The responses of the three

models are very similar for the three valves. Fig. 13

presents the results of the baseline test applied to the

nominal valve. It is presented just the response of the

Karnopp model, since the Lugre and Kano model

responses are practically identical.

The input pressure has to increase an amount of fS 100 S J=2 32:22% in order to give the first jump.Thus, as its initial value is 50%, the valve moves as soon as

the input signal jumps from 82% to 84%. For the secondjump, the input pressure has to overcome the J value

(5.50%), so it moves again when the input pressure jumps

from 88% to 90%. In order to move in the closing

direction, the input signal would have to overcome the S

value (58.94%). However, the input signal only varies 40%,

which is not enough to move the valve again, so the valve

does not return to its initial position.

6.5. Small step tests

This test is carried out on the vendor and nominalvalves,

since the rough valve did not move with the input pressure

varying from 50% to 100%. The results of the three models

are quite similar. Fig. 14 shows the results of the vendor

valve, just presenting the response of the Karnopp model.

To fulfill what is expected of the small step test, it might

be enough to increase the input signal up to 51%,

corresponding to the point where a second step is applied

after the first movement of the valve. Nevertheless, the

input pressure was extended up to 61.7%, to allow

measurement of the dead band. The input pressure has to

increase an amount of fS 100 S J=2 0:835% inorder to give the first jump. Its initial value is 50%, so the

valve moves as soon as it reaches 50.9%. For the second

jump, the input pressure has to overcome the Jvalue (0%),

so in the next step it moves again. In order to move in the

closing direction, the input signal has to overcome the S

value (1.67%). Therefore, when the input signal reversion

reaches 1.7%, the valve moves in the opposite direction. It

keeps oscillating around 50.85% and does not return to the

initial position.

The valve output moved when the input changed from

50.9% to 51%, meaning that the resolution is less than orequal to 0.1%. The measured dead band is between 1.6%

and 1.7%. Its true value is S 1:67%.

ARTICLE IN PRESS

Table 2

Summary of the performed tests

Test type Valve type Classical Karnopp Seven Parameter Lugre Stenman Choudhury Kano He

Dynamic test method (signature curve) Vendor Y Y Y Y Y Y Y Y

Nominal N Y N Y N Y Y N

Rough N Y N Y N Y Y N

Ramp and pause test method Vendor Y Y Y Y Y Y Y Y

Nominal N Y N Y N N Y N

Rough N Y N Y N N Y N

Baseline tests Vendor Y Y Y Y N N Y Y

Nominal N Y N Y N N Y Y

Rough N Y Y Y N Y Y Y

Small step tests Vendor N Y Y Y N N Y Y

Nominal N Y Y Y Y Y Y Y

Rough N Y Y Y Y Y Y Y

Response time test Vendor Y Y Y Y Y N Y Y

Nominal N Y Y Y N N Y Y

Rough N Y Y Y N Y Y N

Y, response as expected; N, response not as expected.



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6.6. Response time test

The responses of the three models for the three valves are

very similar. In Fig. 15, just the response of this test applied

to the nominal valve is presented. In it, the three models

only responded to the input excitation when its size was

50% and their outputs did not return to 50%, even whenthe input did.

6.7. Summary of the testing results

A summary of the testing results employing trapezoidal

input is presented in Table 2.

7. Conclusions

In this work, eight friction models applied to control

valves were implemented and compared. They were tested

with different input signals and with valves with differentfriction coefficients. ISA standards related to real control

valve testing were also presented. All the bench tests

recommended in these standards were applied. The tests

were performed with the valve operating in open loop. The

Karnopp, Lugre and Kano models were able to represent

the expected behavior of the valves, mainly the stick-slip

phenomenon.

It was noticed that the friction values of the rough valve

are so high that in some tests the stem did not even move.

In order to work with more realistic friction coefficient

values, it is proposed, as future work, to deal with model

parameters estimated from real valves in operational

conditions (Romano & Garcia, 2008), with differentfriction statuses, instead of the vendor, nominal and rough

valves used here. Also, the intention is to test the three

selected models operating in closed loop, with plants with

different dynamics, in order to evaluate their behavior in

this condition.

Acknowledgment

The author thanks the support provided by Cenpes/

Petrobras.

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