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Multi-objective optimized fuzzy-PID controllers for fourth ordernonlinear systemsM.J. Mahmoodabadi *, H. JahanshahiDepartment of Mechanical Engineering, Sirjan University of Technology, Sirjan, Iran

A R T I C L E I N F O

Article history:Received 13 July 2015Received in revised form28 December 2015Accepted 26 January 2016Available online

Keywords:Fuzzy-PID controllerMulti-objective optimizationGenetic algorithmParticle swarm optimizationBall–beam systemInverted pendulum system

A B S T R A C T

In this paper, the Multi-objective Genetic Algorithm (MOGA) is used to obtain the Pareto frontiers of con-flicting objective functions for the fuzzy-Proportional-Integral-Derivative (fuzzy-PID) controllers. The ball–beam and inverted pendulum fourth order nonlinear systems are regarded as nonlinear benchmarks. Theconsidered objective functions for the ball–beam system are the distance error of the ball, the angle errorof the beam, and the control effort. For the inverted pendulum system, the objective functions are thedistance error of the cart, the angle error of the pendulum, and the control effort, which must be mini-mized simultaneously. The Pareto fronts are compared with those obtained by Multi-objective ParticleSwarm Optimization (MOPSO). Four points are chosen from nondominated solutions of the obtained Paretofronts based on the three conflicting objective functions and used for illustration of the state variablesof the controlled systems. Obtained results elucidate the efficiency of the proposed controller in orderto control nonlinear systems.

Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of KarabukUniversity. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

1. Introduction

Zadeh originally proposed the fuzzy logic and the fuzzy set theory[1,2]. Fuzzy systems are knowledge-based or rule-based systemsformed via human knowledge and heuristics. They have been appliedfor a wide range of researching fields, such as control, communi-cation, medicine, management, business, psychology, etc. The mostsignificant applications and studies about fuzzy systems have con-centrated on the control area [3–10]. The development of fuzzy-PID controllers for various engineering problems has been a majorresearch activity in recent years. Duan et al. proposed an inherentsaturation of the fuzzy-PID controller revealed due to the finite fuzzyrules [11]. Karasakal et al. applied fuzzy PID controllers based onan online tuning method and rule weighing in [12]. Boubertakh et al.proposed new auto-tuning fuzzy PD and PI controllers usingreinforcement-learning algorithm for single-input single-output andtwo-input two-output systems [13]. In this way, the heuristic pa-rameters of fuzzy-PID controllers have to be determined via anappropriate approach. A very effective way to choose these param-eters is the use of evolutionary algorithms [14], such as the GeneticAlgorithm (GA) [15] and particle swarm optimization (PSO) [16],

etc. In [17], a constrained optimization of a simple fuzzy-PID systemwas designed for the online improvement of PID control perfor-mance during productive control runs. Oh et al. developed a designmethodology for a fuzzy PD cascade controller for a ball–beamsystem using particle swarm optimization (PSO) [18]. Mahmoodabadiet al. designed fuzzy controllers for nonlinear systems using MOPSObased on the Lorenz dominance method [19]. Sahib proposed a typeof controller consisting of proportional, integral, derivative, andsecond order derivative terms optimized using the PSO algorithmfor an automatic voltage regulator system [20].

In this paper, a novel optimal fuzzy-PID control strategy is pro-posed and implemented on two nonlinear benchmark systems.Governing equations for ball–beam and inverted pendulum systemstransformed to the state-space forms. Two fuzzy inference engines areutilized. Due to having some different objective functions, MOGA andMOPSO are applied and three and two dimensional Pareto front figuresare shown. The conflicting objective functions for ball–beam systemare the distance error of the ball, the angle error of the beam, and thecontrol effort. For inverted pendulum system, those are the distanceerror of the cart, the angle error of the pendulum, and the control effort.The simulation results corresponding to the optimum points demon-strate that the designed controller has the superior performance incomparison with reported results in published literature.

The rest of this paper is organized as follows. Section 2 gives abrief description on the fuzzy-PID controller. Section 3 presents themulti-objective optimization genetic algorithm. In Section 4, the

* Corresponding author. Tel.: +98 34 423 36901; fax: +98 34 423 36900.E-mail address: mahmoodabadi@sirjantech.ac.ir (M.J. Mahmoodabadi).Peer review under responsibility of Karabuk University.

http://dx.doi.org/10.1016/j.jestch.2016.01.0102215-0986/Copyright © 2016, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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dynamic models of the ball–beam and inverted pendulum systemsare recalled. Furthermore, their optimal fuzzy-PID controllers,simulation results and comparison studies to verify the capabilityof the proposed controller are shown in this section. Finally, Section5 concludes the paper.

2. Fuzzy-PID controller

The PID controller has a long history in the control engineeringand is accepted in a lot of real applications due to the simple struc-ture. Hence, this controller is widely used still in so many industrialapplications despite offering several new techniques. Consider afourth order nonlinear system with Equation (1).

��x f b F= +1 1

��θ = +f b F2 2 (1)

where f1, f2, b1, and b2 are nonlinear functions and F is the controlinput. The state-space formulation of the system can be written asEquation (2).

�x x1 2=

�x x2 3=

�x f b F3 1 1= + (2)

�x x4 5=

�x x5 6=

�x f b F6 2 2= +

where x x x x x x x x x x= [ ] = ⎡⎣ ⎤⎦∫ ∫1 2 3 4 5 6, , , , , , , , , ,� �θ θ θ is the state vectorwith desired value x x x x x x x x x xd d d d d d d d d d d d d= [ ] = ⎡⎣ ⎤⎦∫ ∫1 2 3 4 5 6, , , , , , , , , ,� �θ θ θ .The PID controller with inputs e t x xx

d( ) = − and e t dθ θ θ( ) = − and

output F tpid ( ) is commonly defined as Equation (3).

F K e t K e d Kde t

dt

K e t K e d

pid px x ix x

t

dxx

p i

= ( ) + ( ) + ( )

+ + ( )( )

∫ τ τ

τθ θ θ θ

0

ττ θθ

0

t

dKde t

dt∫ + ( )(3)

where Kp, Ki, and Kd are the proportional, integral, and derivativegains, respectively. The adjustment and determination of these designparameters are key issues to design PID controllers. Hence, the fuzzylogic approach is applied to calculate the gains adaptively.

F K f K f K f K f K f K fFuzzy pid ix px dx i p d= + + + + +ˆ ˆ ˆ ˆ ˆ ˆ1 2 3 4 5 6θ θ θ (4)

where FFuzzy pid is the fuzzy-PID control action. f ii, , , ,= 1 2 6… are thefuzzy variables with inputs ∫ ∫xdt x dtdx

dtddt, , , ,θ θ θand , respectively, and

should be obtained by Single Input Fuzzy Inference Motor (SIFIM).Furthermore, in Equation (4), the variables ˆ ˆ ˆ ˆ ˆ ˆK K K K K Kix px dx i p d, , , , andθ θ θ

are calculated by Equation (5).

K K K Wix ixb

ixr= + Δ 1

K K K Wx xb

xr= + Δ 2

K K K Wdx dxb

dxr= + Δ 3

K K K Wi ib

ir

θ θ θ= + Δ 4

K K K Wb rθ θ θ= + Δ 5

K K K Wd db

dr

θ θ θ= + Δ 6(5)

where ΔW ii, , , ,= 1 2 6… are the fuzzy variables with inputs∫ ∫xdt x dtdx

dtddt, , , ,θ θ θand , respectively, and should be obtained by

Preferrer Fuzzy Inference Motor (PFIM). K K K K K Kixb

xb

dxb

ib b

db, , , ,θ θ θand are

the base variables and K K K K K Kixr

xr

dxr

ir r

dr, , , ,θ θ θand are the regulation

variables. The base and regulation variables can be obtained by thetry and error process. However, one of the best solutions to find theseto have an optimal controller is the use of optimization approachessuch as evolutionary methods, such as the genetic algorithm.

3. Optimization

The genetic algorithm is an approach for solving optimizationproblems based on biological evolution via modification of a pop-ulation of individual solutions, repeatedly. At each level, individualsare chosen randomly from the current population (as parents) thenemployed to produce the children for the next generation. In thispaper, toolbox optimization of MATLAB (R2012a) with the follow-ing operators is implemented for optimal design of the fuzzy-PIDcontrollers.

3.1. Population size

Increasing the population size enables the genetic algorithm tosearch more points and thereby obtain a better result. However, thelarger the population size, the longer it takes for genetic algo-rithm to compute each generation.

3.2. Crossover options

Crossover options specify how the genetic algorithm combinestwo individuals, or parents, to form a crossover child for the nextgeneration.

3.3. Crossover fraction

The crossover fraction specifies the fraction of each popula-tion, other than elite children, that is made up of crossover children.

3.4. Selection function

Selection options specify how the genetic algorithm choosesparents for the next generation.

3.5. Migration options

Migration options specify how individuals move between sub-populations. Migration occurs if you set population size to be a vectorof length greater than 1. When migration occurs, the best individu-als from one subpopulation replace the worst individuals in anothersubpopulation. Individuals that migrate from one subpopulation toanother are copied. They are not removed from the sourcesubpopulation.

3.6. Stopping criteria options

Stopping criteria determine what causes the algorithm toterminate.

In this paper, the configuration of the genetic algorithm is setas the values given in Table 1.

Furthermore, the multi-objective optimization of the proposedfuzzy-PID controller would be done with respect to twelve designvariables and three objective functions. The base values [ Kix

b , K pxb , Kdx

b ,Ki

bθ , K p

bθ , Kd

bθ ] and regulation values [ Kix

r , K pxr , Kdx

r , Kirθ , K p

rθ , Kd

rθ ] are

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the design variables. For ball–beam system, the distance error ofthe ball, the angle error of the beam and the control effort, and forinverted pendulum system, the distance error of the cart, the angleerror of the pendulum and the control effort, are the objective func-tions. In other words, for both systems, the objective functions canbe written as Equations (6) to (8).

Objective function 1= ∫ x dt (6)

Objective function 2 = ∫ θ dt (7)

Objective function 3 = ∫ F dt (8)

4. Optimal fuzzy-PID controller design

In this section, the optimal fuzzy-PID controller would be de-signed for the ball–beam and the inverted pendulum systems.

4.1. Ball–beam system

In the following, we consider the ball–beam system depicted inFig. 1. The state vector is the system observable state vectorx x x x x x x x x x= [ ] = ⎡⎣ ⎤⎦∫ ∫1 2 3 4 5 6, , , , , , , , , ,� �θ θ θ , including, respectively, theintegral of the ball position, the ball position, the ball velocity, in-tegral of the beam angle, the beam angle and the beam angularvelocity. The dynamic equations of this system in the state-spaceform are expressed by Equation (9).

�x x1 2=

�x x2 3=

�x B x x gsin x3 2 62

5= − ( )[ ]�x x4 5=

�x x5 6=

�xMx x x gMx x

J J Mx6

2 3 6 2 5

1 2 22

2= − − ( )+ +

τ cos(9)

where M is the ball mass, g is the gravity acceleration, J1 is the ballinertia moment, and J2 is the beam inertia moment. The manipu-lated variable F is related with the torque τ by Equation (10).

τ = + + + +( )2 2 3 6 2 5 1 2 22Mx x x gMx cosx J J Mx F (10)

where F is the control input and F FFuzzy pid= . The system param-eters used for simulation are M = 0.05 kg, J kgm1

6 22 10= × − ,J kgm2

20 02= . , g ms

= 9 81 2. , and B = 0.7143.The initial and desired values are regarded as

x x x x x x x0 0 0 0 0 0 0 0 0 5 0 0 301 2 3 4 5 6( ) = [ ] = − °( ) ( ) ( ) ( ) ( ) ( ), , , , , , . , , , , 00[ ] andx x x x x x xd d d d d d d= [ ] = [ ]1 2 3 4 5 6 0 0 0 0 0 0, , , , , , , , , , , respectively.

The block diagram for the stabilization control of the ball–beam system shown in Fig. 2 illustrates that each of the statevariables ∫ ∫x x v, , , , ,θ θ ω relevant to the ball–beam system is fed backand compared with its desired value. Since all of the desired valuesin the stabilization control are zeros, the variables are reversely in-putted into the Norm block for normalization of the state variablesby their scaling factors. The scaling factors and the normalized formof the outputs are given in Table 2.

For ball–beam system, the Single Input Fuzzy Inference Motor(SIFIM) has only one input, and for each normalized variable (Normblock output) an SIFIM is defined. Since there are 6 Norm blockoutput items, 6 SIFIMs would be created. For each input itemX ii; , , ,= 1 2 6… , there is an SIFIM-i ( f ii; , , ,= 1 2 6… ). The PreferrerFuzzy Inference Motor (PFIM) represents the control priority orderof each Norm block output. The PFIM blocks for X ii; , ,= 1 2 3 take theabsolute values of the input items X2 and X5 as the antecedent vari-ables, and the PFIM blocks for X ii; , ,= 4 5 6 take the absolute valueof the input item X5 as their antecedent variable. The membershipfunctions of SIFIMs are shown in Table 3 and Fig. 3, and their rulesare mentioned in Tables 4 and 5.

The output of f ii, , , ,= 1 2 6… for the ball and beam could be cal-culated using Equations (11) and (12).

Table 1Genetic algorithm configuration parameters.

Parameter Value

Crossover fraction 0.8Population size 200Selection function TournamentMutation function Constraint-dependentCrossover function IntermediateMigration direction ForwardMigration fraction 0.2Migration interval 20Stopping criteria Maximum generation 200

Fig. 1. Structure of the ball–beam system.

Table 2State variables and the associated scaling factors and normalized forms for the ball–beam system.

Variable Scaling factor Normalized form

xdt∫ 1 X1

x 1 X2

dxdt 1 X3

θdt∫ 1 X4

θ 45° X5ddtθ

100°/s X6

Table 3Membership functions of SIFIMs for the ball–beam system.

If Then

Xi ≤ −1 VBi = 1POi = 0ZBi = 0

−1 ≤ Xi ≤ 0 VBi = − Xi

POi = Xi+1ZBi = 0

0 ≤ Xi ≤ 1 VBi = 0PO Xi i= − + 1ZBi = Xi

1 ≤ Xi VBi = 0POi = 0ZBi = 1

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fVB f PO f ZB f

VB PO ZBii = × + × + ×

+ +=i i i

i i i

, ,ˆ ˆ ˆ1 2 3 1 2 3 (11)

fVB f PO f ZB f

VB PO ZBii = × + × + ×

+ +=i i i

i i i

, ,ˆ ˆ ˆ

4 5 6 4 5 6 (12)

where variables VB, PO, and ZB are given in Table 3 and Fig. 3. Themembership functions of PFIMs are illustrated in Table 6 and Fig. 4,and their rules are shown in Tables 7 and 8.

Fig. 2. The block diagram of fuzzy-PID control for the ball–beam system.

Fig. 3. Membership functions of SIFIMs for the ball–beam system.

Table 4Fuzzy rules of SIFIMs for the ball.

If X ii =( )1 2 3, , Then

VBi f1 1= −POi f2 0=ZBi f3 1=

Table 5Fuzzy rules of SIFIMs for the beam.

If X ii =( )4 5 6, , Then

VBi f4 1=POi f5 0=ZBi f6 1= −

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The outputs of PFIMs for ball and beam, Δ Δ ΔW W W1 2 3, , ,Δ Δ ΔW W W4 5 6, and , can be calculated by Equations (13) and (14).

ΔWW HS HS W HM HS W HB HS W HS HM W HM HM

i =× × + × × + × × + × × + × ×1 1 3 2 1 3 3 1 3 4 1 3 1 35 ++ × ×

× + × + × + × + × + ×W HB HM

HS HS HM HS HB HS HS HM HM HM HB HM6 1 3

1 3 1 3 1 3 1 3 1 3 1 33 1 3

7 1 3 8 1 3 9 1 3

1 3 1

+ ×+ × × + × × + × ×

+ × + ×

HS HBW HS HB W HM HB W HB HB

HM HB HB H…

BBi

3

1 2 3; , ,=

ΔWW HS W HM W HB

HS HM HBii = × + × + ×

+ +=10 3 11 3 12 3

3 3 3

4 5 6; , , (14)

(13)

After calculation of fi and ΔW ii; , , ,= 1 2 6… , it is possible to definethe fuzzy-PID controller via Equation (4).

In the following, the multi-objective optimization of the pro-posed fuzzy-PID controller would be done by MOGA and MOPSO[19] (with the same settings) with respect to the base and regula-tion parameters as design variables, and three objective functionsas the distance error of the ball, the angle error of the beam andthe control effort. The optimum points of the objective functionsare illustrated in Figs. 5–8. Points A, B and C are the best points forobjectives 1, 2 and 3, respectively, and point D is a trade-off point.The time responses of the ball position, beam angle and control inputfor these optimum points are depicted in Figs. 9–11.

Although the complete stabilization occurs and all the state vari-ables converge to zero, by comparison of method proposed by Yiet al. [21] and this work, the superiority of this work from view-points of distance and angle error is obvious. In [21], as shown inFigs. 9 and 10, the ball position and beam angle reached the finalstate almost at 6 and 5.5 seconds, respectively, while this work canachieve almost 3 seconds for the ball position, 3.8 seconds for thebeam angle and the maximum absolute of the control input is about28.7 (point D). The values of design variables relative to point D andobjective functions relative to points A, B, C, and D are given inTables 9 and 10, respectively.

Table 6Membership functions of PFIM for the ball–beam system.

If Then

X2 0 5≤ . HS X1 22 1= − +HM X1 22=HB1 = 0

0 5 2. ≤ X HS1 = 0HM X1 22 2= − +HB X1 22 1= −

X5 0 5≤ . HS X3 52 1= − +HM X3 52= −HB3 = 0

0 5 5. ≤ X HS3 = 0HM X3 52 2= − +HB X3 52 1= −

Fig. 4. Membership functions of PFIM for the ball–beam system.

Table 7Rules of PFIMs for the ball.

If Then

X2 HS1 W1 = 0X5 HS3

X2 HM1 W2 = 0.5X5 HS3

X2 HB1 W3 = 1X5 HS3

X2 HS1 W4 = 0X5 HM3

X2 HM1 W5 = 0X5 HM3

X2 HB1 W6 = 0.5X5 HM3

X2 HS1 W7 = 0X5 HB3

X2 HM1 W8 = 0X5 HB3

X2 HB1 W9 = 0X5 HB3

Table 8Fuzzy rules of PFIM for the beam.

If Then

X5 HS3 W10 0=X5 HM3 W11 0 5= .X5 HB3 W12 1=

Table 9Design variables of optimum point D for fuzzy-PID control of the ball–beam system.

Design variable Value

Kixb 2.45

K xb 6.98

Kdxb 7.75

Kibθ 17.40

K bθ 32.09

Kdbθ 18.59

Kixr 1.24

K xr 7.07

Kdxr 4.86

Kirθ 5.79

K rθ 9.22

Kdrθ 8.71

Table 10Objective functions of points A, B, C, and D for fuzzy-PID control of the ball–beamsystem.

Point Value

A Objective function 1 = 1.79Objective function 2 = 38.13Objective function 3 = 6.98

B Objective function 1 = 3.97Objective function 2 = 27.40Objective function 3 = 6.07

C Objective function 1 = 2.47Objective function 2 = 32.65Objective function 3 = 4.41

D Objective function 1 = 1.91Objective function 2 = 34.75Objective function 3 = 6.04

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4.2. Inverted pendulum system

In the following, we consider the inverted pendulum system de-picted in Fig. 12. x x x x x x x x x x= [ ] = ⎡⎣ ⎤⎦∫ ∫1 2 3 4 5 6, , , , , , , , , ,� �θ θ θ , including,respectively, integral of the cart position, the cart position, the cartvelocity, integral of the pendulum angle, the pendulum angle andthe pendulum angular velocity. The dynamic equations of this systemin the state-space form are expressed by Equation (15).

�x x1 2=

�x x2 3=

�xF m l x x m gsin x x

m m m cos

p p p

c p p

3

62

5 5 5

2

43

43

=+ ( )[ ]− ( ) ( )

+( ) −

sin cos

xx5( )

�x x4 5=

�x x5 6=

�xm m gsin x F m l x x x

m m m c

c p p p

c p p

65 6

25 5

43

=+( ) ( ) − + ( )[ ] ( )

+( ) −

sin cos

oos x lp2

5( )⎡⎣⎢

⎤⎦⎥

(15)

where mc is the mass of the cart, mp is the mass of the pendulum,g is the gravity acceleration, and F is the control input andF FFuzzy pid= ×11 . lp is the length from the center of the pendulumto the pivot and equals to the half-length of the pendulum.For simulation, the following specifications are usedm kg m kg l m and gc p p

ms

= = = =1 0 1 0 5 9 8 2, . , . , . .The initial and desired conditions are

x x x x x x x0 0 0 0 0 0 0 0 2 0 0 0 01 2 3 4 5 6( ) = [ ] = [ ]( ) ( ) ( ) ( ) ( ) ( ), , , , , , , , , , and

x x x x x x xd d d d d d d= [ ] = [ ]1 2 3 4 5 6 0 0 0 0 0 0, , , , , , , , , , , respectively. The blockdiagram for the stabilization control of the inverted pendulum systemillustrated in Fig. 13 shows that each of the state variables∫ ∫x x v, , , , ,θ θ ω relevant to inverted pendulum system is fed backand compared with its desired value. Since all of the desired valuesin the stabilization control are zero, the variables are directly in-putted into the Norm block. The normalization of the state variablesbased on their scaling factors and creating input itemsX X X X X X1 2 3 4 5 6, , , , , from ∫ ∫x x v, , , , ,θ θ ω , respectively, is done bythe Norm block. The scaling factors and the normalized form of theNorm block outputs are given in Table 11.

Here, similar to fuzzy-PID control of the ball–beam system, twofuzzy inference engines SIFIM and PFIM are utilized. Each input itemX ii; , , ,= 1 2 6… is guided to the SIFIM-i, and f ii; , , ,= 1 2 6… are itsoutput corresponding to the input item Xi. PFIM represents thecontrol priority order of each Norm block output. All of the PFIMblocks take the absolute value of the input item X5 as their ante-cedent variable. The membership functions of SIFIMs are depictedin Table 12 and Fig. 14. The rules of the PFIMs are given in Tables 13and 14.

11.5

22.5

33.5

44.5

2628

3032

3436

3840

4

4.5

5

5.5

6

6.5

7

7.5

8

Objective function 1Objective function 2

Obj

ectiv

e fu

nctio

n 3

A

B

MOGA

MOPSO

C

D

Fig. 5. Three-dimensional Pareto fronts of objective functions 1, 2 and 3 for the ball–beam system.

Table 11Scaling factors and normalized forms of the state variables for the inverted pendu-lum system.

Variable Scaling factor Normalized form

xdt∫ 1 X1

x 2.4 m X2

dxdt 1 X3

θdt∫ 1 X4

θ 30° X5

ddtθ

100°/s X6

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The output of SIFIM-i (fi) for the cart and pendulum is calcu-lated by Equations (16) and (17), respectively.

fVB f PO f ZB f

VB PO ZBii = × + × + ×

+ +=i i i

i i i

; , ,ˆ ˆ ˆ1 2 3 1 2 3 (16)

fVB f PO f ZB f

VB PO ZBii = × + × + ×

+ +=i i i

i i i

; , ,ˆ ˆ ˆ

4 5 6 4 5 6 (17)

The membership functions of PFIMs are shown in Table 15 andFig. 15. The rules of the PFIMs are given in Tables 16 and 17.

The outputs of PFIMs for ball and beam, ΔW1, ΔW2, ΔW3, ΔW4,ΔW5 and ΔW6, can be calculated by Equations (18) and (19).

ΔWW HS W HM W HB

HS HM HBii = × + × + ×

+ +=1 1 2 1 3 1

1 1 1

1 2 3; , , (18)

1 1.5 2 2.5 3 3.5 4 4.526

28

30

32

34

36

38

40

Objective function 1

Obj

ectiv

e fu

nctio

n 2

D

C

B

A

MOGA

MOPSO

Fig. 6. Pareto fronts of objective functions 1 and 2 for the ball–beam system.

Table 12Membership functions of SIFIMs for the inverted pendulum system.

If Then

Xi ≤ −1 VBi = 1POi = 0ZBi = 0

−1 ≤ Xi ≤ 0 VBi = − Xi

POi = Xi+1ZBi = 0

0 ≤ Xi ≤ 1 VBi = 0PO Xi i= − + 1ZBi = Xi

1 ≤ Xi VBi = 0POi = 0ZBi = 1

Table 13Fuzzy rules of SIFIMs for the cart.

If X ii =( )1 2 3, , Then

VBi f1 1= −POi f2 0=ZBi f3 1=

Table 14Fuzzy rules of SIFIMs for the pendulum.

If X ii =( )1 2 3, , Then

VBi f4 1= −POi f5 0=ZBi f6 1=

Table 15Membership functions of PFIM for inverted pendulum system.

If Then

X2 0 5≤ . HS X1 22 1= − +HM X1 22=HB1 = 0

0 5 2. ≤ X HS1 = 0HM X1 22 2= − +HB X1 22 1= −

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ΔWW HS W HM W HB

HS HM HBii = × + × + ×

+ +=4 1 5 1 6 1

1 1 1

4 5 6; , , (19)

where variables HS1, HM1, and HB1 are given in Table 15 and Fig. 15.After calculation of fi and ΔWi, it is possible to define the fuzzy-

PID controller based on Equation (4).The Pareto fronts obtained via the multi-objective genetic algo-

rithm and particle swarm optimization [19] (with the similarconfigurations) are given in Figs. 16–19. Points A, B, and C are thebest points of objectives 1, 2, and 3, respectively, and point D is se-lected as a trade-off optimum point. Trade-off is the best point thatis obtained by substituting base variables and regulation variablesof all the optimal points achieved of optimization via genetic al-gorithm and acquiring the best condition according to settling time

and overshoot. The time responses of cart position, pendulumangular and derive force for the optimum points are illustrated inFigs. 20–22.

It is observable from Figs. 20–22 that all the state variablesconverge to zero and complete stabilization occurs. Moreover,the superiority of this work in comparison with proposedapproach in [22] is obvious. In [22], the cart position and pendu-lum angle reached the final state almost in 7.2 and 7.5 seconds,respectively, while this work can achieve almost 3 seconds for theball position and 4 seconds for the beam angle and the maximumabsolute of the control input is about 8.02 (Point D). The values ofdesign variables relative to point D and objective functions rela-tive to points A, B, C, and D are given in Tables 18 and 19,respectively.

1 1.5 2 2.5 3 3.5 4 4.54

4.5

5

5.5

6

6.5

7

7.5

8

Objective function 1

Obj

ectiv

e fu

nctio

n 3

A

D

B

C

MOGA

MOPSO

Fig. 7. Pareto fronts of objective functions 1 and 3 for the ball–beam system.

Table 16Fuzzy rules of PFIMs for the cart.

If Then

X5 HS1 W1 = 1X5 HM1 W2 = 0.5X5 HB1 W3 = 0

Table 17Fuzzy rules of PFIM for the beam.

If Then

X5 HS1 W4 = 0X5 HM1 W5 = 0.5X5 HB1 W6 = 1

Table 18Design variables of optimum point D for fuzzy-PID control of the inverted pendu-lum system.

Design variable Value

Kixb 0.029

K xb 0.583

Kdxb 0.110

Kibθ 2.88

K bθ 2.61

Kdbθ 1.92

Kixr −0.030

K xr 0.291

Kdxr 0.152

Kirθ 3.79

K rθ −1.50

Kdrθ 5.00

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26 28 30 32 34 36 38 404

4.5

5

5.5

6

6.5

7

7.5

8

Objective function 2

Obj

ectiv

e fu

nctio

n 3

BA

C

D

MOGA

MOPSO

Fig. 8. Pareto fronts of objective functions 2 and 3 for the ball–beam system.

0 1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Bal

l Pos

ition

(m)

Optimum point AOptimum point BOptimum point COptimum point DProposed method in [21]

Fig. 9. Time response of the ball position for the ball–beam system.

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0 1 2 3 4 5 6 7 8-30

-20

-10

0

10

20

30

Time (s)

Bea

m A

ngle

(deg

)

Optimum point AOptimum point BOptimum point COptimum point DProposed method in [21]

Fig. 10. Time response of the beam angle for the ball–beam system.

0 1 2 3 4 5 6 7 8-20

-15

-10

-5

0

5

10

15

20

25

30

Time (s)

Con

trol I

nput

Optimum point AOptimum point BOptimum point COptimum point DProposed method in [21]

Fig. 11. Time response of the driving force for the ball–beam system.

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Fig. 12. Structure of the inverted pendulum system.

Fig. 13. Block diagram of fuzzy-PID control for the inverted pendulum system.

Fig. 14. Membership functions of SIFIMs for the inverted pendulum system.

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5. Conclusion

In this work, multi-objective optimization algorithms, i.e. MOGAand MOPSO, were successfully used to optimum design the fuzzy-PID controllers for the ball–beam and inverted pendulum systems.An integral term was augmented to the state variables in order toeliminate the steady state errors and decrease the rising time. Theconflicting objective functions for the ball–beam system wereFig. 15. Membership functions of PFIM for the inverted pendulum system.

0

5

1015

20

0

5

10

15

20

250

2

4

6

8

10

12

14

Objective function 1Objective function 2

Obj

ectiv

e fu

nctio

n 3

C B

D

A

MOGA

MOPSO

Fig. 16. Three-dimensional Pareto fronts of objective functions 1, 2 and 3 for the inverted pendulum system.

0 2 4 6 8 10 12 14 16 18 20 220

5

10

15

20

25

Objective function 1

Obj

ectiv

e fu

nctio

n 2

A

D

CB

MOGA

MOPSO

Fig. 17. Pareto fronts of objective functions 1 and 2 for the inverted pendulum system.

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0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

12

14

Objective function 1

Obj

ectiv

e fu

nctio

n 3

A

B

D

C

MOGA

MOPSO

Fig. 18. Pareto fronts of objective functions 1 and 3 for the inverted pendulum system.

0 5 10 15 20 250

2

4

6

8

10

12

14

Objective function 2

Obj

ectiv

e fu

nctio

n 3

C

B

A

D

MOGA

MOPSO

Fig. 19. Pareto fronts of objective functions 2 and 3 for the inverted pendulum system.

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0 1 2 3 4 5 6 7 8-0.5

0

0.5

1

1.5

2

2.5

Time (s)

Car

t Pos

ition

(m)

Optimum Point AOptimum Point BOptimum Point COptimum Point DProposed Method in [22]

Fig. 20. Time response of the cart position for the inverted pendulum system.

0 1 2 3 4 5 6 7 8 9 10-15

-10

-5

0

5

10

15

Time (s)

Pen

dulu

m A

ngle

(deg

)

Optimum Point AOptimum Point BOptimum Point COptimum Point DProposed Method in [22]

Fig. 21. Time response of the pendulum angle for the inverted pendulum system.

0 1 2 3 4 5 6 7 8-10

-5

0

5

10

15

20

Time (s)

Con

trol I

nput

Optimum Point AOptimum Point BOptimum Point COptimum Point DProposed Method in [22]

Fig. 22. Time response of the driving force for the inverted pendulum system.

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considered as the distance error of the ball, the angle error of thebeam, and the control effort. The conflicting objective functions forthe inverted pendulum system were considered as the distance errorof the cart, the angle error of the pendulum, and the control effort.The reported results demonstrated that the proposed methodolo-gy can effectively control the nonlinear systems.

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Table 19Objective functions of points A, B, C, and D for fuzzy-PID control of the inverted pen-dulum system.

Point Value

A Objective function 1 = 2.32Objective function 2 = 23.52Objective function 3 = 12.89

B Objective function 1 = 19.41Objective function 2 = 1.32Objective function 3 = 0.79

C Objective function 1 = 19.18Objective function 2 = 1.60Objective function 3 = 0.64

D Objective function 1 = 3.18Objective function 2 = 13.47Objective function 3 = 3.90

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