Received December 28, 2018, accepted January 21, 2019, date of publication February 15, 2019, date of current version March 8, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2897992

Flow Control of Fluid in PipelinesUsing PID ControllerSINA RAZVARZ 1, CRISTÓBAL VARGAS-JARILLO1,RAHELEH JAFARI2, AND ALEXANDER GEGOV31Departamento de Control Automático, CINVESTAV-IPN, Mexico City 07360, Mexico2Centre for Artificial Intelligence Research (CAIR), University of Agder, 4879 Grimstad, Norway3School of Computing, University of Portsmouth, Portsmouth PO13HE, U.K.

Corresponding author: Sina Razvarz (srazvarz@yahoo.com)

ABSTRACT In this paper, a PID controller is utilized in order to control the flow rate of the heavy oil inpipelines by controlling the vibration in a motor pump. A torsional actuator is placed on the motor pumpin order to control the vibration on a motor and consequently controlling the flow rates in pipelines. Thenecessary conditions for the asymptotic stability of the proposed controller are validated by implementingthe Lyapunov stability theorem. The theoretical concepts are validated utilizing numerical simulations andanalysis, which proves the effectiveness of the PID controller in the control of flow rates in pipelines.

INDEX TERMS Fluid flow control, control engineering, PID control, feedback.

I. INTRODUCTIONClassic PID approaches as well as controllers are updatedand expanded during the years, from the primary controllerson the basis of the relays as well as synchronous electricmotors or pneumatic or hydraulic systems to current micro-processors. Currently, many techniques for the tuning aswell as design of PI and PID controllers are proposed [1].The method proposed in [2] is the most widely utilizedPID parameter tuning methodology in chemical industryand is considered as a conventional technique. Basilio andMatos [3] suggested a new method with less complexity inorder to tune the parameters of PI controllers of the plant withmonotonic step response. The methodology of internal modeprinciple is utilized in [4] and [5] in order to extract the gainsof PID and PI controllers. Exhaustive investigation [6]–[9]revealed that the outcomes of P control are very sensitive tothe sensing location as well as the quantity of phase shift.By suitable selections of these variables, the P control canbe completely efficient in annihilating the vortex sheddingor minimizing its strength. Furthermore, it is demonstratedthat the increment in the proportional gain can results in thedecrement of the velocity fluctuations in the wake and thestrength of vortex shedding. Nevertheless, a large gain causesinstability in the system [6], [7], [10].

The associate editor coordinating the review of this manuscript andapproving it for publication was Auday A. H. Mohamad.

In order to implement the control law, the primary step isto determine a desired output response of a particular systemto an arbitrary input over a time interval, that can be carriedout by system identification [11]. Generally, it is feasible togenerate a model on the basis of a complete physical illus-tration of the system. Nevertheless, this model contains com-plexity, also has high calculation costs [12]. The secondarystep is to define the parameters of the PID controller. Thereexist various literatures associated with the methodologiesfor tuning of PID controllers applied in various controllerstructures [13].

Flow control is a major rapidly evolving field of fluidmechanics. There have been various concepts of flow controlin drag reduction, lift enhancement, mixing enhancement,etc. [14]–[16]. Fadlun et al. [17] implemented the conceptof [18] to a finite-difference methodology where a staggeredgrid is used. In [19] a digital pulse feedback flow controlsystem utilizing microcontroller as well as feedback sensingelement is developed. Surprisingly, even though the flowcontrol methods are widely spread, investigating the stabilityof the control system is very rare. In [7], [9], and [20] theP, PI and PID controls are proposed for flow over a cylinderwith a Reynolds number below the 200. The aim of controlin these studies is the attenuation or annihilation of vortexshedding behind a bluff body. The only investigation on theimplementation of PI and PID controls to the flow over a bluffbody is carried out in [9].

VOLUME 7, 20192169-3536 2019 IEEE. Translations and content mining are permitted for academic research only.

Personal use is also permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

25673

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

This paper deals with the modeling and control of flowrate in heavy-oil pipelines. For this aim, the PID control algo-rithm is utilized to control the flow mechanism in pipelines.A torsional actuator is placed on the motor-pump in orderto control the vibration on motor. The stability of the PIDcontroller is verified using Lyapunov stability analysis. Thestability analysis of the controller results in a theorem whichvalidate that the system states are bounded. The theoreticalconcepts are validated using numerical simulations and anal-ysis, which proves the effectiveness of the PID controller inthe control of flow rates in pipelines. Further, it is the firstattempt to place a torsional actuator on the motor-pump inorder to control the vibration on motor and hence control theflow rates in pipelines.

This paper is structured as follows. Firstly, in Section II thepumpmodel system is established. The PID control method isdescribed in Section III. In this section, sufficient conditionsfor the controller under the Lyapunov stability theorem aredesigned. A numerical example is presented in Section IV toillustrate the results. Finally, the conclusions are provided inSection V.

FIGURE 1. Scheme of open loop model.

II. MATERIALS AND METHODS FORMODELLING OF THE SYSTEMFlow control loop system is basically a feedback controlsystem. The structure of the pump model system is shownin Figure.1 which is an open loop system. If there is unwantedvibration in the motor, the stability of the flow rate willhamper. Therefore, it is important to change the open loopsystem to a closed loop system by implementing a controllerso as to control the stability of flow rate by controlling thevibration in the motor.

A. MODELLING OF THE PIPELINEThe proposed model consists of induction motor, whichcauses a rotation in pump and consequently can lead to flowof heavy-oil in pipelines as shown in (Figure.1). This flowmodel can be illustrated in the form of partial differentialequation (PDE) [21].

The linear global theory associated with flow stability isrooted in eigendecompositions of the linearized flow opera-tors. The direct as well as adjoint eigendecompositions asso-ciated with these kinds of operators generate informationrelated to the stability of the operator, the acceptance of initial

conditions as well as external forcing, also the sensitivity tospatially localized disturbances.

From the viewpoint of incompressible, constant-density,constant-viscosity flows associated with Newtonian fluids,the nonlinear Navier–Stokes equation is defined for a nondi-mensional velocity field (x, t) : Rn

×R→ Rn, pressure fieldp (x, t) : Rn

× R → Rn and Reynolds number Re > 0 asbelow equation,

∂u∂t= −∇pρ− u.∇u+ Ff (1)

where,ρ is the density in kg

m3

u is the flow velocity in ms ,

∇ is the divergence,p is the pressure in kg

m.s2,

t is time in s,Ff is termed as the summation of external force and body

forcesBy implementing the mass balance into the Equation (1)

the following is concluded [22],

∇ · u = 0 (2)

Equation (1) can be rewritten as,

∂u∂t= −

1ρ

∂p∂x+ Ff (3)

Let ∂p∂x be the change of pressure in two different points,

moreover for achieving a numerical stability of computation,it is essential to partition pipeline into various segment (gen-erally identical), hence the flow in pipeline can be stated as,

∂ui∂t= −

1ρL

(pi − pi−1)+ Ff , i = 1, . . . , n (4)

where L is taken to be the distance between two sections. Nowlet,

αpi = pi−1, i = 1, . . . , n (5)

where α is termed as the coefficient of pressure changes insections

∂ui∂t= −

1ρL

(1− α) pi + Ff , i = 1, . . . , n (6)

The loss of the friction under the conditions of laminar flowconforms with the Hagen–Poiseuille equation [23], [24]. Fora circular pipe having a fluid of density (ρ) and Kinematicviscosity υ, the hydraulic slope Ff can be described as,

Ff =64Re

u2

2gD+ Fb =

64υ2g

uD2 + Fb (7)

where g is the gravity,D is the diameter of the pipes and Fb isthe shape force vector in pipes. By substitution of (7) in (6),the following equation can be extracted,

∂ui∂t= −

1ρL

(1− α) pi +64υ2g

uD2 + Fb, i = 1, . . . , n

(8)

25674 VOLUME 7, 2019

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

The pressure p (x, t) in the pipeline can be described as,

p =FA

(9)

where F is the force inside the pipeline, also A is the crosssection in the pipe.

By taking into consideration F = ma = m d2xd2t

, andu = dx

dt , also by substitution of (9) in (8) the followingequation is extracted,

∂

∂t∂xi∂t= −

(1− α)ρAL

∂2xi∂t2+

64υ2gD2

∂xi∂t+ Fb (10)

Therefore,

−(1− α)+ ρAL

ρAL∂2xi∂t2+

64υ2gD2

∂xi∂t+ Fb = 0 (11)

If an external force, fp generates by pump, (11) can berewritten as follows,

0x +8x + fb = fp (12)

where x ∈ R2, 0 ∈ R2×2,8 ∈ R2×2, fb = [fb1fb2]T ∈R2×1, fp = [fp1fp2]T ∈ R2×1.

Since in this work two pipelines are used, (12) can berewritten as,

γ1x1 + ϕ1x + fb1 = fp1γ2x1 + ϕ2x + fb2 = fp2 (13)

Since the pump is supplying pressure to the pipes for themaintaining the flow rates, so the pipes will have the sameexternal force, fp1 = fp2.

B. MODELLING OF THE ACTUATORIn order to reduce the vibrations of the motor caused by theexternal forces (fp) a torsional actuator is placed on the motor,see Figure. 2.

FIGURE 2. Torsional actuator with motor-pump arrangement.

The motor and the pump are interconnected with the helpof a shaft. Themain purpose of the motor is to drive the pump.The pump with the help of the motor initiate a flow of fluidin the pipe. Any unwanted vibration in the motor will resultin the vibration in the pump, which will result in improperflows in the pipe. Therefore, it is important to control thevibration in the motor, so as to control the vibration in thepump for making a stable flow of fluid in the pipelines.

FIGURE 3. Structure of system.

For this purpose, a torsional actuator having a motor and diskarrangement as shown in figure 3 is placed on the top base ofthe pump. The main intention of the torsional actuator is tocontrol the vibration on motor and consequently controllingthe flow rates in pipelines.

The inertia moment of the torsional actuator is defined as,

Jt = mtr2t (14)

where mt is considered to be the mass of the disc, and rt isthe radius of the disc. The torque produced by means of thedisc is defined as

uθ = Jt (θt + θ ) (15)

where θ is taken to be the angular acceleration of the motorand θt is taken to be the angular acceleration of the torsionalactuator.

In order to reduce the torsional response, the directions ofθt as well as θ are taken to be different. The friction of thetorsional actuator is defined as [25],

fd = cθt + Fctanh(βθt ) (16)

where c is taken to be the torsional viscous friction coeffi-cient, β is the motor constant, Fc is taken to be the coulombfriction torque, also tanh is considered to be the hyperbolictangent which depends on β and motor speed. The finaltorsion control is expressed as,

uθ = Jt(θt + θ

)− fd (17)

C. MODELLING OF THE PUMPThe general equation of the pump supplying pressure to thepipe for flow control can be demonstrated as [26],

T ω = τ − (τp − nω) (18)

where,ω is angular velocity,τ is motor torque,τp is frictional torque of the motor,n is load constant,T is rotations inertia time constant.The equation (18 ) can be modified as follows,

xp =τ − (τp − nxp)

T(19)

VOLUME 7, 2019 25675

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

where xp is the flow acceleration of the pump. Since τ,T , τp,and n are known quantities of pump so xp can be estimated.The external force generated by the pump is

fp = mpxp (20)

where xp is the acceleration of the motor andmp is volumetricmass of the pump.

The shape force vector fb can be modeled as a linear or anonlinear model.

From (12), by considering shape force vector fb asnon-linear model, the following analysis is illustrated:

In simple non-linear case, (12) becomes

0X +8X + fb = fp (21)

where fb is taken to be non-linear.

III. THE TUNING METHOD BASED ON PID CONTROLLERSince 1700’s the control of continuous process has beencarried out by utilizing feedback loop. System with feedbackcontrol contains drawback which is related to the instabilityof the system. In order to resolve this problem an appropriatecontroller should be chosen and also it must be ideal forthe monitoring system. The proportional feedback controlis uncomplicated and relatively easy to implement. Never-theless, its outcome is completely sensitive to the sensinglocation as well as feedback gain. It is concluded from thecontrol theory that these drawbacks of the P control should beovercome by adopting I as well as D controls. Nevertheless,there exist very few studies investigating the application ofthe PID control for fluid-mechanics problems. In addition,there are a very limited number of studies dealing with theP control and PI controller for pipeline. Due to this lack ofinvestigations, this paper aims to develop a PID control forflow rate in the pipeline.

The control mechanism is demonstrated in Figure. 3 whichshows the entire control process of the flow rate of theheavy-oil in pipelines.

FIGURE 4. PID controller.

The PID control is considered as a control law in which theexistence of output for feedback is essential. This practicalcontrol method is widely utilized in the control society. In thePID control, the controller is made of a simple gain (P con-trol), an integrator (I control), a differentiator (D control) orsome weighted composition of these possibilities [27], seeFigure 4.

The PID control is expressed as,

9 (t) = −κpe (t)−κ i

∫ t

0e(t)dτ − κd e (t) (22)

where kp, ki, as well as kd are positive definite and ki is theintegration gain. For the flow control, Xd is desired referenceand also Xd = Xd = 0. Hence, equation (22) is rewritten asbelow,

9 (t) = −κpX−κ i

∫ t

0Xdτ − κd X (23)

For analyzing the PID controller, equation (22) can bestated as below,

9 (t) = −κpX − κd X − ϑ ϑ = κi

∫ t

0Xdτ , ϑ (0) = 0

(24)

The closed-loop system of eq. (12) along with the PIDcontrol (eq. (23)) is demonstrated as below,

0x +8x + Fg = −κpX − κd X − ϑϑ = κiX (25)

In matrix form, the closed-loop system is defined as,

ddt

ϑXX

= κiX

X−0−1(8X + Fg + kpX + kd X + ϑ)

(26)

Here the stability of the PID control demonstrated byeq. (23) is analyzed. The equilibrium of eq. (26) is presentedby[ϑ X X

]=[ϑ 0 0

]. As at equilibrium point X = 0 as

well asX = 0, the equilibrium is [ f (0) , 0, 0 ]. For movingthe equilibrium to the origin, the following is defined,

ϑ = ϑ − f (0) (27)

Therefore, the final closed-loop equation is defined as,

0X +8X + Fg = −κpX − κd X − ϑ + f (0)ϑ = κix

(28)

For analyzing the stability of equations (28), the followingproperties are required,Property 1: The positive definite matrix 0 should satisfy

in the below condition

0 < λmin(0) ≤ ‖0‖ ≤ λMax(0) ≤ γ (29)

such that λmin (0) as well as λMax(0) are considered asthe minimum and maximum eigenvalues of the matrix 0,respectively also γ > 0 is taken to be the upper bound.Property 2: f is taken to be Lipschitz over x and y if

‖f (x)− f (y)‖ ≤ � ‖x − y‖ (30)

As Fg is first-order continuous functions also satisfies inLipschitz condition, Property 2 is hereby established.

The lower bound of Fg can be calculated as below,∫ t

0fdx =

∫ t

0Fgdx +

∫ t

0dudx (31)

25676 VOLUME 7, 2019

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

The lower bound of∫ t0 Fgdx is stated as −Fg also

∫ t0 dudx

as -Du. Therefore, the lower bound of � is defined as,

� = −Fg − Du (32)

The stability analysis of PID control approach is given bythe below mentioned theorem.Theorem: By taking into consideration the structural sys-

tem of equation (12) controlled by the PID control approachof equation (22), the closed-loop system of equations (28)is taken to be asymptotically stable at the equilibriums[ϑ − f (0) ,X , X

]T= 0, if the following gains are satisfied,

λmin (κd ) ≥14

(13λmin (0) λmin

(κp))1/2 [

1+ke

λMax (0)

]− λmin(8)

λMax (κi) ≤16

(13λmin (0) λmin

(κp))1/2

[λmin

(κp)

λMax (0)

]

λmin(κp)≥

32[�+4] (33)

Proof: The Lyapunov function can be stated as below,

V =12XT0X +

12XT κpX +

σ

4ϑT κ−1i ϑ + XTϑ

+σ

2XT κdX +

σ

4XT κdX +

∫ t

0Fgdx−� (34)

where V (0) = 0. For representing that V ≥ 0, we divide itinto three parts, V = V1 + V2 + V3, where,

V1 =16XT κpX +

σ

4XT κdX +

∫ t

0Fgdx −�

≥ 0, κp > 0, κd > 0 (35)

V2 =16XT κpX +

σ

4ϑT κ−1i ϑ + XTϑ ≥

1213λm(κp)‖x‖2

+σλmin(κ

−1i )

4‖ϑ‖2 − ‖x‖ ‖ϑ‖ (36)

In a case that σ ≥ 3(λmin(κ

−1i )λmin(κp))

, the following is

obtained,

V2 ≥12

(√λmin(κp)

3‖x‖ −

√3

4(λmin(κp))‖ξ‖

)2

≥ 0 (37)

Also,

V3 ≥16XT kpX +

12XT0X +

σ

2XT0X (38)

Since

XTAX ≥ ‖X‖ ‖AX‖ ≥ ‖X‖ ‖A‖ ‖X‖ ≥ λMax(A) ‖X‖2

(39)

In a case that σ ≤ 12

√13λmin(0)λm(κp)λMax (8)

, the following isobtained,

V3 ≥12

(13λmin

(κp)‖X‖2 + λmin (0)

∥∥X∥∥2+σλMax (0) ‖X‖

∥∥X∥∥)

=12

√λmin (κp)3

‖X‖ +√λMax (0)

∥∥X∥∥2

≥ 0 (40)

Therefore,

12

√13λmin (0) λm

(κp)

λMax (0)≥ σ ≥

3

(λmin(κ−1i

)λmin

(κp))

(41)

The derivative of equation (26) is obtained as below,

V = XT0X + XT κpX +σ

2ϑT κ−1i ϑ + XTϑ + XTϑ + XTϑ

+σ

2XT0X +

σ

2XT0X + σ XT κdX + XTFg (42)

For matrix the inequality of below equation isvalidated [28],

ATB+ BTA ≤ AT3A+ BT3−1B (43)

The inequality of (42) is valid for any A,B ∈ Rn×m andany 0 < 3 = 3T

∈ Rn×n, hence the scalar variable XTFgcan be stated as,

XTFg =12XTFg +

12FTg X ≤ X

T3Fg X + FTg 3−1Fg Fg (44)

Utilizing equation (42) the following is obtained,

−σ

2XT8X ≤

σ

248

(XT x + XT X

)(45)

where ‖8‖ ≤ 48. Therefore,ϑ = ki, ϑT k−1i ϑ becomes xTϑ

also xTϑ becomes xT ki. Utilizing equation (45) the followingis extracted,

V = −XT[8+ κd −

σ

20 −

σ

24]X

−XT[σ2κp − κi −

σ

24]X

−σ

2XT

[Fg − f (0)

]+ XT f (0) (46)

By applying the Lipschitz condition of equation (30) thefollowing is obtained,

σ

2XT

[f (0)− Fg

]≤σ

2κf ‖X‖2 (47)

−σ

2XT [Fg − f (0)] ≤ XT

σ

2�X (48)

From equation (42) we have,

XT f (0) ≥ −f T (0)3−1f (0) (49)

VOLUME 7, 2019 25677

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

By utilizing equation (38)

V = −XT[8+ κd −

σ

20 −

σ

24]X

−XT[σ2κp − κi −

σ

24−

σ

2κf

]X + XT f (0) (50)

Becomes

V ≤ −XT[λmin (8)+ λmin (κd )−

σ

2λMax (0)−

σ

24]X

−XT[σ2λmin

(κp)− λmin(κ i)−

σ

24−

σ

2�]X (51)

Therefore, V ≤ 0, ‖X‖ diminishes if the followingconditions are held:

λmin (8)+ λmin (κd ) ≥σ

2[λMax (0)+ κc]

λmin(kp)≥

2σλMax(k i)+4+�

By utilizing equation (41) as well as λmin(κ−1i ) = 1

λMax (κ i),

the following is obtained,

λmin (κd ) ≥14

(13λmin (0) λm

(κp))1/2

×

[1+

4

λMax (0)

]− λmin(8) (52)

Again 2σλMax(k i) =

23λmin

(kp). Thus,

λMax (κi) ≤16

(13λmin (0) λmin

(κp))1/2 λmin

(κp)

λMax (0)(53)

Furthermore,

λmin(κp)≥

32[�+4] (54)

This theorem suggests that the closed-loop system isasymptotically stable. �

IV. NUMERICAL RESULTSFor the numerical analysis purpose and for the validation ofthe novel control strategy, the various parameters associatedwith the flow control are described in Table1:

TABLE 1. Parameters associated with the flow control.

The implemented software in this paper is Matlab/Simulink. Simulations are presented to show that the motorvibration can be attenuated to a significant level by using the

torsional actuator with the developed controllers thus validat-ing the effectiveness of the proposed control approach usingPID controllers. A simulation period of 20s is considered forevaluation. For the simulation purposes, the weight of thetorsional actuator is considered to be 5% of the motor andpump weight in combination.

The Theorem proposed in this paper generates sufficientconditions for the minimum amounts of the proportional aswell as the derivative gains. This Theorem validates that bothproportional and derivative gain must be positive as negativegains can make the systems unstable. The PID gains areselected within the stable range by the stability analysis inorder to ensure the efficiency.

Since the maximum flow rate of the pipeline is 13m3/s

so the other nonlinear force associated with � has to be lessthan 13m3

/s. Hence, we select � = 13 m3

/s, and

λmin (γ1) = 1.0002, λmin (γ2) = 1.000206, λmin (41)

= 2.09, λmin (42) = 2.09

From Theorem 2, we use the following PID gains

λmin(κp1)≥453, λmin (κd1) ≥ 8, λMax (κi1) ≤ 928 (55)

Also for pipe 2 we have,

λmin(κp2)≥ 453, λmin (κd2) ≥ 8, λMax (κi2) ≤ 928

(56)

From ranges mentioned by eq. (55) and (56), the best valueof gains are

κp1 = 458, κd1 = 110, κi1 = 650, κp2 = 480,

κd2 = 115, κi2 = 645

Two subsystem blocks of model, one in the absence of con-trol mechanism as open loop system and another with controlmechanism are generated for comparing the outcomes. Theflow rate from the pump is the input to the flow model.Numerical integrators are utilized in order to calculate thevelocity as well as the position from the acceleration signal.The control signal from the controller subsystem block isgiven to the Torsional actuator simulation block in order toproduce the essential control forces.

FIGURE 5. Comparison of motor vibration attenuation using PIDcontroller for pipeline 1.

Figure 5, 6 represents the vibration attenuate in motor.From these Figures, it can be concluded that PID controller isperforming good in minimizing the vibration. Figure 7, 8 rep-resents the flow rate in pipeline 1 and pipeline 2. When PID

25678 VOLUME 7, 2019

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

FIGURE 6. Comparison of motor vibration attenuation using PIDcontroller for pipeline 2.

FIGURE 7. Stability of flow rate with using of PID controller in pipeline 1.

FIGURE 8. Stability of flow rate with using of PID controller in pipeline 2.

controller are used, the flowrates initiate from zero and main-taining stable flow rate, which proves the effectiveness of PIDcontroller.

V. CONCLUSIONSIn this paper, a novel active control strategy for the attenuationof motor vibration is proposed which consequently controlsthe flow rate in heavy-oil pipelines. The important theoreticalcontribution associated with the stability analysis for the PIDcontroller is developed. The required stability conditions areobtained for the purpose of tuning the PID gains. By utilizingLyapunov stability analysis, the sufficient conditions for theminimum amounts of the proportional, integrator as well asthe derivative gains are obtained. The numerical simulationand analysis validates the effectiveness of PID controllers inthe minimization of motor vibration to control the flow ratein pipelines. The main contributions of this paper are:

1) In this work, the stability of PID controller is validatedwhich has not been given importance in earlier researchesconsidering the flow rate control.

2) The technique of using torsional actuator on the motor-pump arrangement is entirely a new concept.

Future work is intended towards the development of theexperimental setup for further investigation and the improve-ment of the controller by fuzzy methods

REFERENCES[1] K. J. Astrom and B. Wittenmark, Computer-controlled Systems: Theory

and Design, 2nd ed. Englewood Cliffs, NJ, USA: Prentice-Hall, 1990.[2] J. G. Ziegler and N. B. Nichols, ‘‘Optimum settings for automatic con-

trollers,’’ Trans. ASME, vol. 64, no. 1, pp. 759–768, 1942.[3] J. C. Basilio and S. R. Matos, ‘‘Design of PI and PID controllers with

transient performance specification,’’ IEEE Trans. Edu., vol. 45, no. 4,pp. 364–370, Nov. 2002.

[4] D. E. Rivera, M. Morari, and S. Skogestad, ‘‘Internal model control: PIDcontroller design,’’ Ind. Eng. Chem. Process Des. Develop., vol. 25, no. 4,pp. 252–265, 1986.

[5] J. C. Basilio, J. A. Silva, Jr., L. G. B. Rolim, and M. V. Moreira,‘‘H∞ design of rotor flux-oriented current-controlled induction motordrives: speed control, noise attenuation and stability robustness,’’ IETControl Theory Appl., vol. 4, no. 11, pp. 2491–2505, 2010.

[6] J. E. F. Williams and B. C. Zhao, ‘‘The active control of vortex shedding,’’J. Fluids Struct., vol. 3, no. 2, pp. 115–122, 1989.

[7] K. Roussopoulos, ‘‘Feedback control of vortex shedding at low Reynoldsnumbers,’’ J. Fluid Mech., vol. 248, pp. 267–296, Mar. 1993.

[8] A. Baz and J. Ro, ‘‘Active control of flow-induced vibrations of a flexiblecylinder using direct velocity feedback,’’ J. Sound Vib., vol. 146, pp. 33–45,Apr. 1991.

[9] E. Berger, ‘‘Suppression of vortex shedding and turbulence behind oscil-lating cylinders,’’ Phys. Fluids, vol. 10, no. 9, pp. S191–S193, 1967.

[10] S. Hiejima, T. Kumao, and T. Taniguchi, ‘‘Feedback control of vortexshedding around a bluff body by velocity excitation.,’’ Int. J. Comput. FluidDyn., vol. 19, no. 1, pp. 87–92, 2005.

[11] S. W. Sung, J. Lee, and I-B. Lee, Process Identification and PID Control.Piscataway, NJ, USA: IEEE Press, 2009.

[12] A. Kusters and G. A. J. M. van Ditzhuijzen, ‘‘MIMO system identificationof a slab reheating furnace,’’ in Proc. 3rd IEEE Conf. Control Appl.,Glasgow, U.K., Aug. 1994, pp. 1557–1563.

[13] A. O’Dwyer, Handbook of PI and PID Controller Tuning Rules, London,U.K.: Imperial College Press, 2009.

[14] H. Choi,W.-P. Jeon, and J. Kim, ‘‘Control of flow over a bluff body,’’Annu.Rev. Fluid Mech., vol. 40, no. 1, pp. 113–139, 2008.

[15] S. S. Collis, R. D. Joslin, A. Seifert, and V. Theofilis, ‘‘Issues in active flowcontrol: Theory, control, simulation, and experiment,’’ Prog. Aerosp. Sci.,vol. 40, pp. 237–289, May/Jul. 2004.

[16] M. Gad-el-Hak, A. Pollard, and J.-P. Bonnet, Flow Control: Fundamentalsand Practices. Berlin, Germany: Springer, 1998.

[17] E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, ‘‘Combinedimmersed-boundary finite-difference methods for three-dimensional com-plex flow simulations,’’ J. Comput. Phys., vol. 161, pp. 35–60, Jun. 2000.

[18] J. Mohd-Yusof, ‘‘Combined immersed-boundary/B-spline methods forsimulations of flow in complex geometries,’’ Annu. Res. Briefs, CenterTurbulence Res., NASA Ames Stanford Univ., vol. 1, no. 1 pp. 317–327,1997.

[19] A. Nandy, S. Mondal, P. Chakraborty, and G. C. Nandi, ‘‘Development ofa robust microcontroller based intelligent prosthetic limb,’’ in Proc. Int.Conf. Contemp. Comput., 2012, pp. 452–462.

[20] D. S. Park, D. M. Ladd, and E. W. Hendricks, ‘‘Feedback control ofvon Kármán vortex shedding behind a circular cylinder at low Reynoldsnumbers,’’ Phys. Fluids, vol. 6, no. 7, pp. 2390–2405, 1994.

[21] A. Herrán-González, J. M. De La Cruz, B. De Andrés-Toro, andJ. L. Risco-Martín, ‘‘Modeling and simulation of a gas distribution pipelinenetwork,’’ Appl. Math. Model., vol. 33, no. 3, pp. 1584–1600, 2009.

[22] R. D.Whitaker, ‘‘An historical note on the conservation of mass,’’ J. Chem.Educ., vol. 52, no. 10, p. 658, 1975.

[23] L. F. Moody, ‘‘Friction factors for pipe flow,’’ Trans. ASME, vol. 66, no. 8,pp. 671–684, 1944.

[24] G. O. Brown, ‘‘The history of the Darcy-Weisbach equation for pipe flowresistance,’’ Environ. Water Resour. Hist., vol. 4, no. 1, pp. 34–43, 2003

[25] C. Roldán, F. J. Campa, O. Altuzarra, and E. Amezua, ‘‘Automatic identifi-cation of the inertia and friction of an electromechanical actuator,’’ in NewAdvances in Mechanisms, Transmissions and Applications (Mechanismsand Machine Science), vol. 17. Dordrecht, The Netherlands: Springer,pp. 409–416. doi: 10.1007/978-94-007-7485-8_50.

[26] L. Wozniak, ‘‘A graphical approach to hydrogenerator governor tuning,’’IEEE Trans. Energy Convers., vol. 5, no. 3, pp. 417–421, Sep. 1990.

[27] K. De Cock, B. De Moor, and W. Minten, ‘‘A tutorial on PID control,’’Katholieke Univ. Leuven, Leuven, Belgium, Rep. ESAT-SISTA/TR, 1997.

[28] A. V. den Bos, ‘‘Appendix C: Positive semidefinite and positive de finitematrices,’’ in Parameter Estimation for Scientists and Engineers. NewYork, NY, USA: Wiley, 2007, pp. 259–263. doi: 10.1002/9780470173862.

VOLUME 7, 2019 25679

S. Razvarz et al.: Flow Control of Fluid in Pipelines Using PID Controller

SINA RAZVARZ received the B.S. degree inmechanical engineering, heat and fluids fromIslamic Azad University, Iran, in 2009, and theM.S. degree in mechanical engineering fromIslamic Azad University, Science and ResearchBranch, Tehran, in 2010. He is currently pursu-ing the Ph.D. degree with the Departamento deControl Automtico, CINVESTAV-IPN (NationalPolytechnic Institute), Mexico City, Mexico. Hisresearch interests include heat transfer, fluid

dynamic, fuzzy equation, neural networks, and nonlinear system modeling.

CRISTÓBAL VARGAS-JARILLO received theB.S. degree from the Instituto PolitécnicoNacional, Mexico, and the M.S. degree in com-puter science from the University of Wisconsin,USA, and the Ph.D. degree in mathematical sci-ences from The University of Texas at Arlington.

He was with the Departamento de ControlAutomatico, CINVESTAV-IPN, Mexico.

RAHELEH JAFARI received the B.S. degree inpure mathematics from Islamic Azad University,Shabestar, Iran, in 2008, theM.S. degree in appliedmathematics from Islamic Azad University, Arak,in 2010, and the Ph.D. degree in automatic controlfrom the CINVESTAV-IPN (National Polytech-nic Institute), Mexico City, Mexico, in 2017. Sheis currently a Postdoctoral Research Fellow withthe Department of Information and Communica-tion Technology, University of Agder, Grimstad,

Norway. Her research interests include fuzzy logic, fuzzy equation, neuralnetworks, artificial intelligence, fuzzy control, and nonlinear system mod-eling. She serves as an Associate Editor for the Journal of Intelligent andFuzzy Systems.

ALEXANDER GEGOV received the Ph.D. degreein control systems and the D.Sc. degree in intel-ligent systems from the Bulgarian Academy ofSciences. He is currently a Reader in computa-tional intelligence with the School of Computing,University of Portsmouth, Portsmouth, U.K. Hehas been a recipient of the National Award for theBest Young Researcher from the Bulgarian Unionof Scientists. He has been a Humboldt GuestResearcher with the Universities of Duisburg and

Wuppertal, Germany, and an EU Visiting Researcher at the Delft Universityof Technology, The Netherlands.

25680 VOLUME 7, 2019