Speed Control of Pipeline Pig Using the QFT Method

Speed Control of Pipeline Pig Using the QFT Method M. Mirshamsi and M. Rafeeyan

Department of Mechanical Eng., Faculty of Engineering, Yazd University - Islamic Republic of Irane-mail: [email protected] - [email protected]

Résumé — Contrôle de la vitesse d’un racleur grâce à la méthode de synthèse QFT — Pour qu’uneinspection de pipeline soit efficace, l’opération de raclage doit être réalisée à vitesse constante. Cet articleprésente une méthode simple et efficace fondée sur la théorie QFT (Quantitative Feedback Theory),système de régulation robuste bien connu, permettant de contrôler la vitesse d’un racleur avec bypassdans un oléoduc. Ce régulateur classique commande l’ouverture ou la fermeture d’une soupape installéesur le racleur. Le racleur est ensuite contrôlé grâce au volume de fluide qui le traverse. Pour cela,l’équation dynamique non linéaire du mouvement du racleur est convertie en un ensemble de structureslinéaires instables équivalentes via la méthode Sobhani-Rafeeyan (méthode SR). Puis, un régulateur detype QFT est synthétisé. La méthode présentée est conçue pour des pipelines bidimensionnels dans deuxsituations différentes. Le régulateur créé est simulé numériquement grâce à la boîte à outils Simulink dulogiciel MATLAB. Les résultats de la simulation montrent que le régulateur peut être utilisé pourcontrôler efficacement la vitesse du racleur lorsque celui-ci passe dans des oléoducs.

Abstract — Speed Control of Pipeline Pig Using the QFT Method — To provide an efficient inspectionof pipeline, pigging operations must be executed at a constant speed. This paper presents an efficient andsimple method based on Quantitative Feedback Theory (QFT), which is a well-known robust controllerscheme, for speed control of a pig with bypass flow in a liquid pipeline. This classical-type controllercommands the valve installed in the body of the pig to open or close. Then, the pig is controlled using theamount of bypass flow across its body. For this purpose, the nonlinear dynamic equation of motion of thepig is converted to a family of linear uncertain equivalent plants using Sobhani-Rafeeyan’s method (SRmethod). Then, for this family of uncertain equivalent plants, a QFT-type controller is synthesized. Thepresented method is developed for two-dimensional pipelines in two cases. The designed controllers aresimulated numerically using the Simulink toolbox of the MATLAB software. The simulation results showthat the designed controller can be used for speed control of the pig with good performance when it runsin the liquid pipelines.

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 67 (2012), No. 4, pp. 693-701Copyright © 2012, IFP Energies nouvellesDOI: 10.2516/ogst/2012008

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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 67 (2012), No. 4694

LIST OF SYMBOLS

Ah Area cross-section of the valveD Pipeline diameterg Acceleration of gravitylpig Length of the pigptail Pressure in the tail of the pigpnose Pressure in the nose of the pigVpig Velocity of the pigz(x) Height of the selected positiond Internal diameter of pipelinedvalve Bypass valve diameter

Greek symbols

ρ Density of fluidμ Coefficient of friction

INTRODUCTION

Pigging operations are common and frequently used in the oiland gas transportation industries since it is the only way tomonitor the conditions of the inside walls of the buriedpipelines for inspection purposes. There are several types ofpigs and each type is designed for a desired purpose. All ofthe pigs act better if they run at a near constant speed. Mostpigging operations such as batching, cleaning and liquidremoval in gas pipelines are done at normal operatingvelocities with the regular flow of the product. This velocityis generally in the range of 1-5 m/s in liquid pipelines and2-7 m/s in gas pipelines [1]. The smart pigs (pipelineinspection gauges) are usually equipped with a number oftransducers which are installed on it circumferentially fordetection of surface defects such as cracks, corrosion, etc.Frequency response limitations of these transducers, whichare often MFL (Magnetic Flux Leakage) type, lead to keepingtheir speeds nearly constant during data acquisition. Themore accurate the data acquisition, the greater the decrease incost and time of the maintenance and future dangerousaccidents. To keep the velocity of the pig constant, a bypassflow pig type can be used. The speed of this pig is controlledusing the amount of bypass flow across its body. The amountof bypass flow is regulated by a valve installed on it. This pigis driven by injected fluid flow behind its tail and expelledfluid flow in front of its nose.

Few studies have been done on the motion of pigs, especiallyconcerning their speed control in pipelines. Some of thesestudies are experimental research or have a commercial basis.Dynamic modeling of various pigs is one of the current subjectsthat several interesting investigations have been conductedon. It seems that the first one was introduced by [2]. Thismodeling was modified and improved by removing somelimitations in [3]. The first pigging model based on full two-phase transient flow formulation was proposed by [4]. This

model is composed of correlations for pressure drop acrossthe pig, slug holdup, pigging efficiency, a pig velocity modeland a gas and liquid mass flow boundary condition applied tothe slug front. Some other complementary research was alsoreported for pigging simulation in two-phase flow straightpipelines [5-8]. Transient pig motion through gas and liquidpipelines was presented by [9]. Modeling and simulation forpig flow control in a natural gas pipeline was studied by [10].This paper solved the governing partial differential equa-tions of the pig using the method of characteristics.Another modeling with a similar solution method but withmore accurate equations was studied by [11]. It seems that thefirst investigation which deals with the speed control of abypass flow pig in a natural gas pipeline was [1]. In thisresearch, a simple nonlinear controller was proposed forcontrolling the pig velocity when it moves in a natural gasstraight pipeline. Also, to provide an efficient tool to assistin the control and design of pig operations through pipelines,a numerical code has been developed by [12]. The resultsobtained with the code in this research were compared withexperimental results and a good agreement between the twowas obtained. In all these studies mentioned, researchersassumed that the pig moves in a straight line in the plane.Simulation of a small pig in space pipeline was studied firstby [13]. In this research, the effect of the flow field on thepig’s trajectory was ignored. This effect was considered in[14] for a bypass flow pig in space pipeline.

The objective of the present work is to synthesize a linearclassical feedback controller for speed control of the bypassflow pig to keep its velocity near a constant value. To do this,the nonlinear equation of motion of the pig is converted to afamily of linear uncertain equivalent plants using the SRmethod [15]. Then, for two different case studies, QFT-typerobust controllers are synthesized for the family of linearuncertain plants. The designed controllers are simulated onthe main system (nonlinear system). The simulation resultsshow that these controllers can control the speed of the pigperfectly.

1 GOVERNING EQUATION OF THE PIG MOTION

Figure 1 shows a typical small pig moving inside a two-dimensional pipeline and its free body diagram. The weight ofthe pig, mg, dry friction, Fμ = μN, normal force by the pipewall, N, upstream acting force of the fluid, F1 and downstreamacting force of the fluid, F2, are the forces acting on the pig.The dynamic equations of the pig, derived from Newton’ssecond law along the tangential and normal directions, are asfollows:

N – mg cos θ = man (1)

F1 – F2 – mg sin θ – sgn(x·)Fμ = mat (2)

where θ is the angle of the tangent to the centerline curve ofthe pipeline with respect to the x-axis at any point; i.e. if f(x)


M Mirshamsi and M Rafeeyan / Speed Control of Pipeline Pig Using the QFT Method 695

is assumed to be the function of the centerline of the pig, thuswe can write:

(3)

If s measures along the pig’s path and the radius of curvatureof the path is R, then we can derive both accelerations of thepig as follows:

(4)

(5)

(6)

The term F1 – F2 on the left side of Equation (2) can bederived as follows [13]: see Equation (7).

a

d s

dt

f x f x

f xx x f xt = =

′′ ′

+ ′+ + ′

2

2 2

2 2

11

( ) ( )

( )(� �� ))2

aV

R

f x

f xxn

pig= =′′

+ ′

2

2

2

1

( )

( )�

V s f x xpig = = + ′� �1 2( )

θ θ= ′ =+ ′

−tan ( ), cos( )

1

2

1

1f x

f x

In general, the pressure difference between the tail andthe nose of the bypass hole in the pig (ptail – pnose) isestablished from three parts; i.e. pressure losses from asudden contraction at the tail (KSC), the valve inside the hole(KV) and a sudden expansion at the nose (KSE). Thefollowing relations are suggested for the pressure differenceand the loss coefficients in the fluid mechanics books andpapers, e.g. [1]:

where:

(8)K K K K

K

total SC V SE

Kd

dSC

valve

= + +

⎛

⎝⎜

⎞

⎠⎟= −0 42 1

2

2.

VVvalve

SEvalvef

h

dK

d

D=

⎛

⎝⎜

⎞

⎠⎟ = −, 1

2

2

p pK V V

tail nosetotal h pig− =

−ρ( )2

2

y

x g

NP

tn F1

Fμ

θ

F2

mg

p tail

p nose

Valv

e

Figure 1

Schematic view of a pig inside a planer pipeline.

(7)F F p p A fl x l l x

D

Vtail nose h

pig1 2

25

2− = − −

+ −( )

( ) ( )ρ ++ + −

⎡

⎣⎢

⎤

⎦⎥ −

′g Z x l gZ x A gAl

fpig pigρ θ ρ ρ( cos ) ( )5 4

(( )

( )

x

f x1 2+ ′



(9)

Substitutions (3) and (5) in (1) give the normal force N asfollows:

(10)

The final equation of the pig can be derived by substitutingall the terms in (2) as follows: see Equation (11).

2 QFT CONTROLLER DESIGN

General Formulation

Numerical simulation of Equation (11) shows that the pigspeed varies with respect to time. In order to control the pigspeed, the pig’s nonlinear system is converted to a family oflinear uncertain equivalent systems using the SR method. Inorder to demonstrate the remarkable performance of theproposed control approach, two different case studies areconsidered and QFT controllers are designed.

Using the open-loop transmission, each controller shouldbe synthesized such that the closed-loop system is stable andalso satisfies the following conditions.

Robust Stability

The stability margin can be specified in terms of a phasemargin, a gain margin or the corresponding ML contour onthe Nichols Chart (NC) using the associated magnitude indecibels (dB) [16]. If any one of the three stability require-ments is specified, the remaining two can be calculated. TheML contour is the stability specification used directly for theQFT design technique, placing an upper limit on the magni-tude of the closed-loop frequency response:

(12)

The ML contour on the NC therefore forms a boundarywhich must not be violated by a plot of the open-looptransmission L(s) = G(s)P(s). Throughout the design, ML ischosen to be 1.58 dB.

Robust PerformanceThe following design constraint is used to ensure adequatetracking performance:

1

10

+≤ ≥

L jM PL( )

,ω

ω for all

N mf x

f xx

f xg=

′′

+ ′+

+ ′

⎡

⎣⎢⎢

⎤

⎦⎥⎥

( )

( ) ( )1

1

12

2

2�

min( ) .Kd

D

d

Dtotalvalve valve= −

⎛

⎝⎜

⎞

⎠⎟ −1 1 42

2

2

2

2

⎛⎛

⎝⎜

⎞

⎠⎟ (13)

Equation (13) implies that the system’s response to thestep input should be placed in a predefined region specifiedby upper and lower bounds, denoted as TU(jω) and TL(jω),respectively. Suppose that:– the system’s response to the step input is required to settle

in ts ≤ 2.5 s;– an overshoot of ML = 1.4 is appropriate.

The following transfer functions can be selected so as todefine the desired tracking bounds with the aforementionedspecifications:

(14a)

(14b)

These two specifications generate robust bounds on L0(jω)which is a nominal loop transmission at selected frequencies,and the bounds are plotted on the NC. The synthesized L0(jω)must lie on or just above the bound at each frequency tosatisfy the required performance. The proposed compensatorG is designed by adding appropriate poles and zeros to thenominal loop function so that it lies inside the acceptableregions.

The designed compensator, however, guarantees only thatthe variation in ⏐L(jω)/(1+L(jω))⏐ is less than or equal to thatallowed. Therefore, it is necessary that a prefilter be designedto ensure robust performance in the tracking problem.

Application to the Proposed SystemThe dynamic equation of motion of the pig can be written as:

(15)

where:

(16)W tK A

mAtotal

h

( ) =ρ 2

2

��

x W tV x c

cf

l

Dm c

Vpig=− +( )

+−

+

+

( )1

1

5

1 2

4

22

2 2

2

ρ

ρggAlc

m cg

c

c

bd

cx

g

c

pig ( )1 1

1 1

2

2

2

22

2

+−

+

−+

++

⎛

⎝⎜μ �

⎞⎞

⎠⎟ −

+cd

cx

1 22�

T js s s sL ( )

( )( )( )( )ω =

+ + + +8 400

3 4 10 70

T js

s jU ( ). ( )

.ω =

++ ±4 938 4

2 3 969

T jL j

L jT jL U P( )

( )

( )( ) ,ω

ωω

ω ω≤+

≤ ≥1

0 for all

(11)

K A V x f x

Af

l

D

Vg

total

h

pigρ

ρ ρ

2 22

21

2

5

24

− + ′( )− +

� ( )AAl

f x

f xmg

f x

f x

x

pig

′

+ ′−

′

+ ′

−

( )

( )

( )

( )

sgn( )

1 12 2

� mmf x

f xx

g

f xm

fμ

′′

+ ′+

+ ′

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=′( )

( ) ( )

(

1 12

2

2� xx f x

f xx x f x

) ( )

( )( )

′′

+ ′+ + ′

⎡

⎣⎢⎢

⎤

⎦⎥⎥1

12

2 2� ��



b = sgn(x·) (17)

c = f ’(x), d = f ”(x) (18)

Since Ah is time-variant, W(t) can be considered as controlinput. b, c and d can be considered as system uncertainties.Now, the nonlinear system governed by Equation (15) isreplaced with a family of linear time equivalent systemsusing the SR method. This method uses an arbitrary family ofoutput functions for construction of this equivalent family,substitutes these functions in dynamic equations, obtains theLaplace transform of the equation and obtains a family oflinear equivalent uncertain transfer functions for a nonlinearsystem [14]. In this study, a set of step functions with uncertainheight is considered arbitrarily, as follows:

Vpig = x· = a.u(t) (19)

where u(t) is a step function. Substituting this function in thedynamic equation of the pig (15) and getting the Laplacetransform of the equation, the transfer function of linearequivalent plants is obtained as follows:

(20)

where:

(21)

Now two different case studies are considered and QFTcontrollers and prefilters are designed for each one.

3 NUMERICAL EXAMPLES

Case 1For the first case study, let us assume a 2D pipeline with thelinear equation as y(x) = x. The numerical values for the massof the pig, length of the pig, pipe diameter and bypassdiameter are assumed to be 50 kg, 0.3 m, 0.25 m and 0.12 m,

k fl

mD c

VgAl

c

m c

gc

c

pigpig= −

++

+

−+

ρ ρ5

1 24

1

1

2

2

2

2

( )

−−+

bg

cμ

1 2

PV s

W s

aV a V c c a

a c s

pig= =− + + +

+ + −

( )

( )

( )2 2 2 2 3

2

2 1 1

1 kkb d cd

ca c+

++

⎛

⎝⎜

⎞

⎠⎟ +

μ1

12

2 2

-350

-40

-30

-20

-10

0

Mag

nitu

de (

dB)

1

22

1

5

5

10

10

50

80

100

5080100200

400

800

10

20

30

40

-300 -250 -200Phase (°)

-150 -100 -50 0

Figure 2

Nominal loop-shaping for case 1.

c f x d f x= ′ = ′′( ), ( )



Mag

nitu

de (

dB)

100

-400-90

-100

0

-200

-300

100 108102 104 10610-2

Pha

se (

°)

-270

-225

-180

-135

Frequency (rad/s)

Bode diagram

Figure 3

The Bode plot of G1(s) for case 1.

Mag

nitu

de (

dB)

1

1

22

5

5

10

5080

10050

80

100

200

200

400

800

10

40

30

20

10

0

-10

-20

-30

-40

-200-250-300-350 -150 -100 -50 0Phase (°)

Figure 4

Nominal loop-shaping for case 2.

respectively. The fluid is water with density of 1000 kg/m3

and velocity of 6 m/s. The dynamic and static coefficients offriction are assumed to be 0.2 and 0.3, respectively. The pigposition at initial time is x(0) = y(0) = 0. Since f”(x) = 0 andf’(x) = 1 in this example, c = 1 and d = 0. It is assumed thatb ∈ [-1 1], a ∈ [1 5]. QFT compensators and prefilters aredesigned for the family of uncertain linear time-invariantequivalent plants (20) using the numerical values above. Theprefilter H1 and compensator G1 are designed as:

(22)

(23)

In Figure 2 colored lines show the resulting QFT designbounds (stability bounds and performance bounds) for thepre-determined set of design frequencies ω = {1, 2, 5, 10, 50,80, 100, 200, 400, 800} and the black line shows the loopshaping L(jω) for these frequencies. It is noted that the plot at

G ss s s

125 24

137 1 2 33 137 1 12( )

.

/ . . / .=

+ +( )

H ss1

1

29 19 1( )

/ .=

+

f x′′( )f x′( )



Mag

nitu

de (

dB)

100

-300

-45

-200

-100

0

100 108102 104 10610-2

Pha

se (

°) -90

-135

Frequency (rad/s)

Bode diagram

-180

Pig

vel

ocity

(m

/s)

3.5

0

0.5

1.0

1.5

2.0

2.5

3.0

Time (s)9 10876543210

Figure 5

The Bode plot of G2(s) for case 2.

Figure 6

Simulated pig velocity for case 1.

each chosen frequency satisfies the specified bound, that is,L(jω) does not violate the U-contour (for stability) and anypoints of L(jω) are on or above the performance bound curvefor the frequencies. Figure 3 demonstrates the Bode plot ofthe first designed controller, G1(s).

Case 2For the second case, a pipeline with the curve equation asy(x) = –0.5(x + sin x) is selected. The numerical values for themass of the pig, length of the pig, pipe diameter, bypassdiameter, fluid density and velocity are the same as case 1.The pig position at initial time is at x(0) = y(0) = 0. Since– 0.5 ≤ f ”(x) ≤ 0.5 and –1 ≤ f ’(x) ≤ 0 in this example, c ∈ [–1 0] and d ∈ [– 0.5 0.5]. It is assumed that b ∈ [–1 1],a ∈ [1 5]. The prefilter H2 and compensator G2 are designedfor this case using the QFT approach.

(24)

(25)

The designed compensator G2 and its Bode plot for case 2are depicted in Figures 4 and 5.

4 RESULTS AND DISCUSSION

Figures 6 and 7 show the velocity of the pig for cases 1 and2, respectively. These figures also show that the velocity ofthe pig follows the desired velocity (i.e. 3 m/s) very soon. Itis worth noting that the designed controllers were simulated

G ss

s s s2

32 43 35 28 1

196 1 712 196 12( )

. ( / . )

/ . /=

+

+ +( ))

H ss2

1

11 1( )

/=

+on the main nonlinear plant, not on a linear equivalent plant,as obtained in (20). Therefore, we can conclude that thepresent QFT-based controller design can be used for such pigspeed control problems. Figures 3 and 5 show that the designedcontrollers are realizable because their bandwidths are limited.

Figures 8 and 9 show the changes of valve diameter forcases 1 and 2, respectively. As these figures represent, thevalve must be closed approximately at the initial time andrapidly opened some seconds later. This is because of thelarge slope of the pipeline curve at these times. After someseconds, the pig gets a constant velocity and therefore therequired diameter of the valve becomes constant. Simulationdiagrams for both cases are shown in Figures 10 and 11.

Pig

vel

ocity

(m

/s)

3.5

0

0.5

1.0

1.5

2.0

2.5

3.0

Time (s)9 10876543210

Figure 7

Simulated pig velocity for case 2.

f x′′( ) f x′( ),



Val

ve d

iam

eter

(m

)0.13

0.04

0.12

0.11

0.10

0.09

0.08

0.07

0.06

0.05

Time (s)9 10876543210

Figure 8

Changes in valve diameter, for case 1.

Figure10

Block diagram.

Val

ve d

iam

eter

(m

)

0.13

0.05

0.12

0.11

0.10

0.09

0.08

0.07

0.06

Time (s)9 10876543210

1

2Out 2

PIGSaturation

W+

+

++

-C-

V

+–

+–

-C-

u2 sqrt-K-

sqrtd_valve

Scope

0.5

V1

+–

num f1

den f1

PrefilterStep

In

num g1

den g1

Controller

In1 Out1

1Out 1

Figure 9

Changes in valve diameter, for case 2.

CONCLUSIONS

This paper proposed a new and simple controller designmethod to maintain the speed of a pipeline pig. This method,which is called Sobhani-Rafeeyan’s method, replaces thenonlinear dynamic equation of the pig with bypass flow inliquid pipeline with a family of uncertain linear systems.Then, a robust controller using the Quantitative FeedbackTheory (QFT) method was synthesized for this family ofuncertain linear systems. The designed controller wassimulated on the original nonlinear system and good results

were obtained. The results from the two studies showed thatthe speed of the pig reaches its desired value very soon. Thisresearch is the first application of the QFT method, which isa well-known robust control method, in the area of speedcontrol of the pig. Also, it is the first control method for pigswhose paths are two-dimensional. It is expected that thismethod can be developed for speed control of a pig in anatural gas pipeline. Also, this method can include someuncertainty of parameters such as: density of fluid,coefficient of friction, length of the pig and so on. The finaldesigned controller is always of classical type and realizable.


REFERENCES

1 Nguyen T.T., Yoo H.R., Rho Y.W., Kim S.B. (2001) Speedcontrol of pig bypass flow in natural gas pipeline, InternationalSymposium on Industrial Electronics, Pusan, Korea, June 12-16.

2 McDonald A., Baker O. (1964) Multiphase flow in (Gas)pipelines, Oil Gas J. 62, 24, 68-71; Oil Gas J. 62, 25, 171-175;Oil Gas J. 62, 26, 64-67.

3 Barua S. (1982) An experimental verification and modificationof the McDonald and Baker pigging model for horizontal flow,PhD Thesis, University of Tulsa, Texas.

4 Kohda K., Suzukawa Y., Furukwa H. (1988) A new method foranalyzing transient flow after pigging scores well, Oil Gas J. 9,40-47.

5 Minami K., Shoham O. (1991) Pigging dynamics in two-phaseflow pipelines: experiment and modeling, SPE Prod. Facil. 10,4, 225-231.

6 Taitel Y., Shoham O., Brill J.P. (1989) Simplified transientsolution and simulation of two-phase flow in pipelines, Chem.Eng. Sci. 44, 1353-1359.

7 Scoggins Jr. (1977) Numerical simulation model for transienttwo-phase flow in a pipeline, PhD Thesis, University of Tulsa,Texas.

8 Xiao-Xuan X., Gong J. (2005) Pigging simulation for horizontalgas-condensate pipelines with low-liquid loading, Journal ofPetroleum Science Engineering 48, 272-280.

9 Nieckele A.O., Braga A.M.B., Azevedo L.F.A. (2001) Transientpig motion trough gas and liquid pipelines, Journal of EnergyResources. ASME 123, 260-269.

10 Nguyen T.T., Kim S.B., Yoo H.R., Rho Y.W. (2001) Modelingand simulation for pig flow control in natural gas pipeline,J. Mech. Sci. Tech. 15, 8, 1165-1173.

11 Tolmasquim S.T., Nieckele A.O. (2008) Design and control ofpig operations through pipelines, Journal of Petroleum ScienceEngineering 62, 102-110.

12 Esmaeilzadeh F., Mowla D., Asemani M. (2009) Mathematicalmodeling and simulation of pigging operation in gas and liquidpipelines, Journal of Petroleum Science Engineering 69, 100-106.

13 Saeidbakhsh M., Rafeeyan M., Ziaei-rad S. (2009) Dynamicanalysis of small pigs in space pipelines, Oil Gas Sci. Technol.64, 2, 155-164.

14 Lesani M., Rafeeyan M., Sohankar A. (2012) Dynamic analysisof Small Pig through two and three dimensional liquid pipeline,Journal of Applied Fluid Mechanics. 5, 2, 75-83.

15 Sobhani M., Rafeeyan M. (2000) Robust controller design formultivariable nonlinear uncertain systems, Iran. J. Sci. Technol.24, 3, Transaction B, 345-356.

16 Yaniv O. (1999) Quantitative feedback design of linear andnonlinear control systems, Kluwer Academic Publishers, UnitedStates of America.

Final manuscript received in October 2011Published online in August 2012


Figure 11

Dynamic of pig.

0.4

X

X

-C-

k

8

u2

u2

6

V

+

++

+–

++ 1–s

Integrator

+

k2

k_p

k1

1

-K-

Mu

-K- -K-

-K-

1/k_z k_z

In

1

Out

Sign

–+

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Speed Control of Pipeline Pig Using the QFT Method