research article genetic algorithm and particle swarm...

Research ArticleGenetic Algorithm and Particle Swarm OptimizationBased Cascade Interval Type 2 Fuzzy PD Controller forRotary Inverted Pendulum System

Mukhtar Fatihu Hamza Hwa Jen Yap and Imtiaz Ahmed Choudhury

Department of Mechanical Engineering University of Malaya 50603 Kuala Lumpur Malaysia

Correspondence should be addressed to Mukhtar Fatihu Hamza mfhamzasiswaumedumy

Received 4 March 2015 Revised 25 May 2015 Accepted 26 May 2015

Academic Editor Roman Lewandowski

Copyright copy 2015 Mukhtar Fatihu Hamza et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents the design of an optimized Interval Type 2 Fuzzy Proportional Derivative Controller (IT2F-PDC) in cascadeform for Rotary Inverted Pendulum (RIP) systemTheparameters of the IT2F-PDC are optimised by usingGenetic Algorithm (GA)and Particle Swarm Optimization (PSO) The goal is to balance the pendulum in upright unstable equilibrium position The IT2F-PDCwhich is the extended version of conventional type 1 fuzzy logic controller improves the control strategy by using the advantageof its footprint of uncertainty for the fuzzy membership functionThe performance characteristics considered for the controller aresteady state error settling time rise time maximum overshoot and control energy Experimental and simulation results indicatedthat the effectiveness and robustness of the proposed GA- and PSO-based controllers on the RIP with respect to load disturbancesparameter variation and noise effects have been improved over state-of-the-art method However the comparative results forsimulation and experiment based on cascade IT2F-PDC indicate that GA-based IT2F-PDC has lower steady state error while PSO-based IT2F-PDC has lower overshoot settling time and control energy but both have almost the same rise time The proposedcontrol strategy can be regarded as a promising strategy for controlling different unstable and nonlinear systems

1 Introduction

Most real industrial systems are nonlinear in nature andexhibit some level of uncertainty [1 2] In the past decadesome modern controls such as nonlinear control adap-tive control variable structure control and optimal controlwere used [3ndash5] Although these control strategies exhibit avery good performance they are also complex and difficultto implement [6] The conventional proportional integralderivative (PID) controller exhibits good performance forlinear system and it is widely employed in industry due toits simple structure and robustness in different operationconditions However the tuning of the parameters of PIDaccurately becomes difficult becausemost of industrial plantsare highly complex and have some issues such as nonlineari-ties time delay and higher order [7] Due to the complexityof most industrial plants and the limitation of PID controlleran unprecedented interest was diverted to the applications

of the fuzzy logic controller (FLC) This is because it usesthe expert knowledge and its control action is described bylinguistic rules Also the FLC does not require the completemathematical model of the system to be controlled and it canwork properly with nonlinearities and uncertainties [1 2 8ndash13] FLC are of two types namely type 1 fuzzy logic controller(T1FLC) and type 2 fuzzy logic controller (T2FLC) In T1FLCthe uncertainty is represented by a precise number in a rangeof (0 1) interpreted as a degree of membership functions(MF) In view of the fact that it is too difficult to knowa precise value for uncertainty working with type 1 modelis more reasonable However some researchers argued thatin case where there is high level of uncertainty T1FLC haslimited ability to handle it because its membership degreefor each input is a crisp number [14] The T2FLC whichuses type 2 fuzzy set (T2FS) was introduced to circumventthe limitations of the T1FLC The main characteristic ofT2FLC is that its MFs are fuzzy Therefore it has more

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 695965 15 pageshttpdxdoiorg1011552015695965

2 Mathematical Problems in Engineering

degree of freedom in designing varieties of systems withuncertainties [1 2 8] The control performance of T1FLCcan be improved by T2FLC because it has the advantage offootprint of uncertainty (FOU) that can be used to improvethe corresponding MF [15]

The major disadvantage of T2FLC is that the process oftuning becomes more difficult andmuch time consuming forincreasing number of inputs and outputs of the controller[16]This problem of tuning can be solved by using optimiza-tion methods [15 16]

Looking at Rotary Inverted Pendulum (RIP) from thecontrol point of view the RIP exhibits many interesting andchallenging properties such as nonlinearities and instabilityThe RIP is in the class of underactuated mechanical sys-tems These features make RIP become known widely asexperimental setup for testing linear and nonlinear controlalgorithms [11] RIP consists of a rotational servomotorsystem which drives the output gear rotational arm anda pendulum It has many important real applications likerobotics aerospace vehicles pointing control and marinevehicles [15]

Four control objectives of RIP as found in the literatureare categorized as follows [17 18] (1) swing-up control con-trolling the pendulum from downward (stable position) toupward (unstable position) (2) stabilization control that isregulating the pendulum to remain at the unstable position(3) switching control that is the switching between swing-up control and stabilization control (4) trajectory controlthat is to control the RIP in such a way that the arm tracks adesired time varying trajectory while the pendulum remainsat unstable position

Recently a lot of research regarding the control of RIP hasbeen published For example energy based compensationPD cascade scheme edge trigger counter sliding modecontrol type 1 fuzzy control and adaptive control have beenused for swing-up control of RIP [18ndash24] To solve thestabilization control problem on RIP multiobjective integralsliding mode controller microcontrollers fuzzy logic regula-tor pole placement technique optimized PID controller andLinear Quadratic Regulator (LQR) have been applied [7 1820 23 25ndash28] Energywas considered for switching control in[18 29] and mode controller was used for the same purposein [20] For trajectory control adaptive PID with slidingmode control linear active disturbance rejection controlhybrid of linear fusion function based on LQR mapping andadaptive control with ANFIS tuning feedback linearizationbased controller and energy based compensation controllerwas used [17 30ndash32] Optimized cascade type 1 fuzzy logiccontroller was used in [33] for controls of pendulum angleand arm angles of RIP

To the best of the authorrsquos knowledge at this momentthere is not any kind of GA and PSO optimised type 2 fuzzylogic control applied to RIP A cascade control method iseffective for a systemwith high level of disturbances and largetime error such as the RIP [24] Also as mentioned earliertype 2 fuzzy control strategy is effective and gives robustcontrol response for systems with high level of uncertaintyandor inaccurate model Putting type 2 fuzzy in cascadetopology will have the advantages of type 2 fuzzy and cascade

structure which will eventually give more robust controllerfor system with uncertainties and large time error [33]

In this paper the IT2F-PDC is designed in cascaded formfor RIPThe parameters of the IT2F-PDC are optimised usingGenetic Algorithm (GA) and Particle Swarm Optimization(PSO) The GA and PSO were chosen in this research inview of the fact that these algorithms are more establishedin the literature than other evolutionary algorithms [3435] In addition the GA and PSO have proven to improveperformance over other algorithms in solving optimizationproblems [36 37] The goal is to balance the pendulum inupright position The servo behaviour (reference tracking)of RIP is analysed Disturbances rejection of the proposedcontroller is analysed by adding the internal noise andexternal disturbance to the system Also the controller isapplied on the real RIP to validate the simulation resultsThe performances of GA and PSO for the optimizationof the parameters of IT2F-PDC were compared Also thedesigning of IT2F-PDC as an optimization dilemma thatslightly altered five performance indices was formulatedwhich includes steady state error settling time rise timemaximum overshoot of the response and control energy ofthe system

The organisation of the paper is as follows Section 2introduces the RIP Section 3 gives brief on fuzzy logicsystem IT2FL-PDC and the design of cascade IT2-PDC andSection 4 defines optimizationmethod and gives some basicson GA and PSOThe formulation of the problem is discussedin Section 5 Sections 6 and 7 present the simulation andexperimental results respectively Finally Section 8 presentsconclusions

2 RIP

The RIP is in the class of underactuated mechanical systemconsisting of a servomotor system that drives the outputgear The rotational arm is attached to the output gear and apendulum is attached at one end of the rotating arm as shownin Figure 1(a) The arm is driven by the gear with the aim ofbalancing the pendulum in an upright position The angle(120601) is the angle of the arm and its direction depends on thedirection of the control voltage (119881

119898) (voltage applied to the

servomotor) In this study the counterclockwise direction isconsidered as positive direction for the arm and the clockwisedirection is the positive direction for the pendulum Theservomotor drives the rotating arm to move in horizontalplane 119883119884 in such a way that it sets the pendulum to theinverted position in the 119883119885 direction perpendicular to thearm (120572) is the pendulum angle Figure 1(b) shows the pictureof the experimental setup using Quanser RIP

21 Nonlinear and Linear Dynamics Model of RIP Thedynamic equations that describe the motion of rotary armand the pendulum with respect to the servomotor voltagewere obtained using Euler-Lagrange equation which is thesystematic way of obtaining the equation of motion [38]Once the kinetic and potential energy are obtained and theLagrangian is found Subsequently the task is to compute var-ious derivatives to get the equations of motion of the system

Mathematical Problems in Engineering 3

120572

z

x

Pendulum

120601

y

LP

lp

Lr

Arm Jr

mpJp

g

(a) (b)

Figure 1 (a) Convention of RIP (b) Quanser RIP experimental setup

After going through the process the nonlinear equations ofmotion for the RIP can be found [38] Considering the totallength of the pendulum (119871

119901) viscous damping coefficient

is seen at the pivot axis for arm and pendulum (119861119903) and

(119861119901) respectively and moment of inertia about the centre of

mass for arm and pendulum (119869119903) and (119869

119901) respectively The

nonlinear equations of motions of RIP are found as follows

(1198981199011198712119903+141198981199011198712119901minus141198981199011198712119901cos (120572)2 + 119869

119903) 120601

minus (12119898119901119871119901119871119903cos (120572))

+ (121198981199011198712119901sin (120572) cos (120572)) 120601

+ (12119898119901119871119901119871119903sin (120572)) 2 = 120591minus119861

119903

120601

(minus12119898119901119871119901119871119903cos (120572)) 120601 + (119869

119901+141198981199011198712119901)

minus (141198981199011198712119901cos (120572) sin (120572)) 1206012

minus(12119898119901119871119901119892 sin (120572)) = minus119861

119901

(1)

Let (1) be called nonlinear 1 Alternatively the nonlinearequation of RIP can be found in (2) when the length of thependulum from its centre of mass (119897

119901) equivalent moment

of inertia as seen at the load (119869eq) and equivalent viscous

damping coefficient as seen at the load (119861eq) are consideredLet (2) be called nonlinear 2 Consider

(119869eq +1198981199011198712

119903) 120601 minus (119898

119901119897119901119871119903cos (120572))

+ (119898119901119897119901119871119903sin (120572)) 2 = 120591minus119861eq 120601

(431198981199011198972119901) minus (119898

119901119897119901119871119903cos (120572)) 120601 minus119898

119901119897119901119892 sin (120572) = 0

(2)

The torque (120591) at the load gear is generated by servomotor andis described by

120591 =120578119892119870119892120578119898119896119905(119881119898minus 119870119892119896119898

120601)

119877119898

(3)

The linearizedmodel of the nonlinear equations of RIP can befound by substituting sin(120572) = 120572 and cos(120572) = 1 in nonlinearequation [33] and presented in matrix form in (4)ndash(7) Thesummary descriptions of the parameters and their values aregiven in Table 1 Consider

[[[[[

[

120601

120601

]]]]]

]

=

[[[[[[[[[

[

0 0 1 0

0 0 0 1

0119896211989641198965

minus119896311989671198965

0

0119896111989641198965

minus119896211989671198965

0

]]]]]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+

[[[[[[[[[

[

00

1198963120578119892119870119892120578119898119896119905

1198771198981198965

1198962120578119892119870119892120578119898119896119905

1198771198981198965

]]]]]]]]]

]

119881119898

(4)


Table 1 The summary of the parameters values of RIP

Symbol Description Value Unit119898119901

Mass of pendulum 0127 kg119871119901

Total length of pendulum 0337 m119869119901

Pendulum moment of inertia about its centre of mass 00012 kgsdotm2

119861119901

Pendulum viscous damping coefficient as seen at the pivot axis 00024 Nsdotmsdotsrad119897119901

Length of pendulum centre of mass 0156 m119871119903

Rotary arm length 0168 m119861119903

Rotary arm viscous damping coefficient as seen at the pivot axis 00024 Nsdotmsdotsrad119869119903

Rotary arm moment of inertia about its centre of mass 0000998 kgsdotm2

119877119898

Motor armature resistance 26 Ω

119896119905

Motor current-torque constant 000768 N-mA119896119898

Motor back-emf constant 000768 V(rads)119870119892

High-gear total gear ratio 70120578119898

Motor efficiency 69 120578119892

Gearbox efficiency 90 119869eq Equivalent moment of inertia as seen at the load 00036 kgsdotm2

119861eq Equivalent viscous damping coefficient as seen at the load 0004 Nmsrad

where 1198961 = 119869eq + 1198981199011198712

119903 1198962 = 119898119901119897119901119871119903 1198963 = (43)119898119901119897

2

119901 1198964 =

119897119901119898119901119892 1198965 = 11989611198963 minus 119896

2

2

1198967 =1205781198921198701198981205781198981198961199051198962119892+ 119861eq119877119898

119877119898

(5)

[[[[[

[

120601

120601

]]]]]

]

=

[[[[[

[

0 0 1 0

0 0 0 1

0 3491 minus1621 0

0 7544 minus1312 0

]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+[[[[

[

00

28562307

]]]]

]

119881119898

(6)

[120601

120572] = [

1 0 0 0

0 1 0 0][[[[

[

120601

120572120601

]]]]

]

(7)

The open loop poles for the proposed RIP system are 119908 =[0 7491 minus1804 minus56596] The system has one pole at theright hand side of 119904-plane which confirms that the proposedRIP system is unstableTherefore before continuing with anycontrol action the test for controllability of the system has tobe done in order to check for the rank of the matrix119882 in (8)If119882 has full rank then the system is controllable [39]

119882 = [119861 119860119861 1198602119861 119860

3119861 119860

119899minus1119861] rank (119882) = 4 (8)

The matrix119882 has full rank then the system is controllablemeaning that the RIP system has complete state controllabil-ity

22 Open Loop Response of Linear and Nonlinear Model ofRIP The open loop responses for nonlinear 1 nonlinear 2and linear dynamic of RIP are shown in Figures 2(a) and 2(b)Initially the pendulum is positioned in an inverted positionwith very small displacement (0005∘) and then it is allowedto fall by applying a pulse signal to the modelThe simulationresults show that nonlinear 1 and nonlinear 2 are similarand their behaviours are the same for both pendulum angleand arm angle This shows that nonlinear 1 and nonlinear2 can be considered as nonlinear model of the RIP becausenonlinear 1 has already been proved and used in [33] Thelinear model depicts the nonlinear pendulum motion for thefirst 13 seconds until it attains 21∘ then it began to deviatefrom the actual motion The response shows that the wholesystem is nonlinear and unstable

3 Fuzzy Logic Controller

A concise overview of FLC was presented in this section withthe intention of providing the basic knowledge needed tounderstand the basic idea and formulation of interval type 2fuzzy logic controller

31 Interval Type 2 Fuzzy Logic System (IT2FLS) The idea offuzzy logic systems and T2FS was pioneered by Zadeh in 1965and 1975 respectively [40 41] The uncertain knowledge isused to build the fuzzy logic rules which leads to uncertainantecedents and consequents of the rule to be uncertainwhich can not be handled by conventional type 1 fuzzy setMFThis leads to the introduction of type 2 fuzzy logic whichcan handle the issue of uncertainty by using the advantage ofFOU [2 12] All the secondary grades of the IT2FS are equalto 1 and it is completely described by upper MF and lowerMF (UMF and LMF) Figure 3(a) shows the triangular MF of


0 025 05 075 1 125 15 175

LinearNonlinear 1 Nonlinear 2

Time (s)

120572(d

eg)

300

200

100

0

(a)120601

(deg

)

0

60

0

30

025 05 075 1 125 15 175Time (s)

minus30

minus60


(b)

Figure 2 Open loop response for RIP (a) pendulum angle (b) pivot arm angle

IT2FLS and its associated quantities Also the correspondingsecondary MF is shown in Figures 3(b) and 3(c)

T2FLS and T1FLS are similar in terms of their normalarchitecture The main difference between them is in theirstructure The defuzzifier block in type 1 fuzzy is substitutedwith the output processing in type 2 fuzzy comprised of typereduction and defuzzifier blocks [15] The block diagram ofT2FLS is shown in Figure 4

32 Fuzzification The fuzzifiers in T1FLS and T2FLS aredoing the same work which is transforming numeric vectorentries 119909 = (119909

1 119909

119901)119879 isin 119883

1lowast 1198831lowast sdot sdot sdot lowast 119883

119901equiv 119883 into 119860

119909

(type 2 fuzzy set) defined in119883 Giving the singleton numericinputs the mapping can be performed as follows [16]

120583119860119909

(119909) = 11 with 119909 = 1199091015840

120583119860119909

(119909) = 10

for forall119909 isin 119909 with 119909 = 1199091015840

(9)

Equation (9) shows that 120583119909119894

(119909119894) = 11 when 119909

119894= 1199091015840119894and

120583119909119894

(119909119894) = 10 when 119909

119894= 1199091015840119894for all 119894 = 1 119901

33 Rules Both T1FLS and T2FLS use IF-THEN rules Intype 2 the antecedents and consequent MF are representedby T2FS The 119894th rule can be expressed as

119877119894

IF 1199091 is 1198601198941 and sdot sdot sdot 119909119901 is 119860119894

119901

THEN 119884119894

= 1198620 + 119862119894

11199091 + sdot sdot sdot + 119862119894

119903119909119901(10)

where 119894 = 1 119898 119862119894119895(119895 = 0 1 119901) are the consequent

type 1 fuzzy set119860119894119896(119896 = 1 119901) are type 2 antecedent fuzzy

set

34 Inference The inference mechanism in T2FLS is like theone in type 1 fuzzy It is a rule combination to produce amapping from input T2FSs to output T2FSs It is necessaryto calculate the intersection union and composition of type2 relations in order to realise this mapping [15] The T2FS119860119909whose MF is 120583

119860119909

= cap119901

119894=1120583119909119894

(119909119894) where 119909

119894(119894 = 1 119901)

are level of fuzzy sets describing the input The result of type2 input and antecedent process which are in the firing setcap119901

119894=1120583119865119894119894

(1199091015840119894equiv 119865119894(1199091015840)) is an interval type 1 fuzzy set as in (11)-

(12) [9]

119865119894

(1199091015840

) = [119891119894

(1199091015840

) 119891119894

(1199091015840

)] equiv [119891119894

119891119894

] (11)

where

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840

1) lowast sdot sdot sdot lowast 120583119865119894

119901

(1199091015840

119901)

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


(1199091015840

119901)

(12)

35 Output Processing The output processing constitutes thetype reduction that generates type 1 fuzzy set and defuzzifierthat converts the generated type 1 fuzzy set to the crispoutput [15] In this paper theWu-Mendel uncertainty boundmethod was used to approximate the type-reduced set and


1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)

Figure 3 (a) FOU of Triangular MF of type 2 fuzzy logic and its associated quantities (b) and (c) the corresponding secondary MF

yry1

min

min

Crisp input

Fuzzification

Inference

Type reduction Defuzzification

Crisp output

Output processing

Output rule calculation

Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)

Figure 4 Type 2 fuzzy logic system

Karnik and Mendel (KM) algorithm was used for calculatingthe two end points of centroids of 119910

119897and 119910

119903[14] presented in

1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)

Once 1199101 and 119910119903are found then the average of 1199101 and 119910

119903

is taken so that the output of the defuzzifier of an intervalsingleton type 2 fuzzy is as follows

119910 (119909) =1199101 + 119910119903

2 (14)

36 Cascade Interval Type 2 Fuzzy Logic PD Controller Inthis paper the cascade controller was designed for RIP TheRIP is a single input multiple output (SIMO) system InSIMO systems change of one output by some disturbancesaffects the control of the other output [33] Considering

nonlinearities behaviour of RIP system it is difficult toachieve the past settling time Also it has high level ofdisturbances and large time constant For system like this thebest control strategy is cascade because it has the advantageof attenuating the effect of disturbances and improving thedynamics of entire control loop [24] The structure of thecascade interval type 2 fuzzy logic PD controller is shown inFigure 5 Both controllers have two inputs and one outputThe overall architecture comprises the outer and the innercontrollersThe input of the inner controller is the error119864

2(119905)

and change in errorΔ1198642(119905) its output is the control voltage to

the servomotor119881119898(119905) and three gains 119892

4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V

119898(119905) respectively as shown in

1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)

where 119905 is the instance sampling 119880(119905) is the control signalfrom the outer loop and120601(119905) is the arm angleThe input of the


Rotary Inverted Pendulum plant

Servomotor PendulumType 2fuzzy logic

Inner fuzzy controllerOuter fuzzy controller

Optimization of gains or controlparameters using GA or PSO

Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt

Figure 5 Overall architecture of IT2FL-PDC in cascade form

Table 2 Fuzzy rules of IT2FL-PDC

Δ119890119890

NL NS ZO PS PLNL PL PL PL PS ZONS PL PL PS ZO NSZO PL PS ZO NS NLPS PS ZO NS NL NLPL ZO NS NL NL NL

outer controller is the error 1198641(119905) and change in error Δ119864

1(119905)

its output is the control signal to the inner loop 119906(119905) and threegains 119892

1 1198922 and 119892

3are used to scale 119890

1(119905) Δ119890

1(119905) and 119906(119905)

respectively as shown in

1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)

where 120572119903and 120572 are reference pendulum angle and pendulum

angle respectively These scaling gains (1198921 1198922 119892

6) are

adjustable parameters just like in any PID controller andthey are used to calibrate input and output [11] The excellentperformance of the controller depends on the values of thesegains [16] The trial-and-error approach can be used to findappropriate values of the gains but it is not feasible so thereis a need of using a systematic procedure which is easier forfinding the optimised values of these gains In this paperGA and PSO are used (Table 3) The triangular MF of T2FSwas used and it is defined in the input and output spaces asshown in Figures 6(a) and 6(b) These MFs are the same forboth two input variables (119890 and Δ119890) and they are applied toboth controllers (inner and outer) The fuzzy rules of IT2F-PDC are shown in Table 2 Where NL NS ZO PS and PLare negative large negative small zero positive small andpositive large respectively

4 Optimization Method

Optimization refers to a class of soft computing techniquesthat relate to obtaining the optimal or satisfactory or best

Table 3 Optimised gains of cascade IT2FL-PDC

Gains 120572119903= 0 120572

119903= 180

GA PSO GA PSO1198921

minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094

solution for a particular problem and the solution may beabsolutely best out of some other possible solution [42]

GA and PSO are stochastic global optimization tech-niques Over the past years GA has received a considerableattention and it is applied for searching the optimal fuzzyparameters which lead to the genetic fuzzy system [9 10]Although GA is successful in so many applications some-times it has a problem of getting trapped before it reaches theoptimal region of the search space especially for multimodaland highly dimensional problems [43 44] The advantagesthat make PSO attract more attention are as follows theparticles in PSO have memory all the particles retained theknowledge of good solution so far and there is a constructivecooperation among the particles meaning the particles in theswarm share the information among themselves [45]

41 Genetic Algorithm The basic foundation of GAs wasproposed in 1975 by JohnHolland [46] It is based onDarwinrsquosideas Darwinrsquos stated that in a competing environment thestronger individuals are more likely to be the winners [47]GA is a metaheuristic search algorithm based on naturalselection and genetic process [46 47] In GA the potentialsolution to a problem is an individual which can be repre-sented by the set of parameters These parameters are justlike a gene of a chromosome and can be represented by thestring of values in binary form [46] The fitness value is usedto test the degree of goodness of the chromosome for solvinga problem that is directly related to the objective valueThe operators employed in a simple GA include selectioncrossover and mutation [46] GAs are often regarded as


NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)

Figure 6 MFs of the (a) premise input variables (b) input variables

function optimizers and they have been applied in manyoptimization problems such as energy consumption [48] Inparticular the use of GAs for fuzzy systems design equipsthem with the adaptation and learning capabilities whichbring about genetic fuzzy systems [10]

42 Particle Swarm Optimization The PSO was introducedin 1995 by Eberhart and Kennedy [49] PSO is metaheuristicsearch algorithm based on social and population behaviourjust like flocking of bird or fish schooling The population inPSO is called swarm that can contain many particles Eachparticle in PSO updates its velocity based on (17) [49 50]Consider

119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2

10038161003816100381610038161003816100381610038162 minus 120573 minus radic1205732 minus 4120573

1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)

where 119875best is the best position attained for the individualparticle and 119892best is the best position attained for the particleamong all the population119875

119894is the position of particle 119894 119903

1and

1199032are randomnumbers between 0 and 1 119888

1and 1198882are position

constants learning rate and119883 is the constriction factor Eachparticle changes its position toward the 119875best and 119892best basedon the updated velocity as given in (18) [51] Consider

119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)

The velocity is restrictedwithin [minus119881max +119881max] If the velocitydeviates from this range it has to be forced to be within therange [50 51]

The following steps are used for the implementation ofPSO algorithm

Step 1 Specify the upper and lower bounds of the controllerparameter and generate the particles randomly the valuesof the performance criteria in time domain are calculatediteratively by sending each controller parameter (particle) toMatlab Simulink after that the cost function is evaluatedfor each particle according to these performance criteriaThen evaluate each particle in the initial population using theobjective function and search for 119901best and 119892best

Step 2 Calculate the velocity and the constriction factorfor the particles and check for the maximum velocity thenupdate the velocity and position of each particle

Step 3 For each particle 119901best is reset in comparison with theprevious 119901best through fitness of objective function then 119892bestis updated in comparison with best 119901best

Step 4 If one of the terminating conditions is satisfied thenstop else go to Step 2

Step 5 The particle that has the latest 119892best is an optimalparameter

The trial-and-error method commonly used in the lit-erature [52] was used for the selection of PSO and GAparameters For PSO method number of particles = 25position constant learning rate 119888

1= 1198882= 2 maximum

iteration = 150 and constriction factor = 04 For GAmethodnumber of individuals = 25 crossover rate = 05 mutationrate = 01 maximum generation = 150 also the type ofoperators used for each population in GA is linear rankingselection algorithm simple crossover and uniformmutationBoth algorithms are implemented in Matlab and each of themethods is tested in 60 independent runs with 60 distinctinitial trial solutions All these were conducted in MatlabR2013a 24GHz processor with 8GB RAM


10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)

Figure 7 Tendency of convergence of (a) mean value of cost function (b) standard deviation of cost function

Table 4 Best results of the cascade IT2FL-PDC for RIP system obtained by GA and PSO in 60 runs

Optimization method 120572119903(deg) Condition for disturbances 119872

119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost

GA0 No disturbances 277 00009 146 00089 18375 18210

With disturbances 308 00021 169 00086 18973 19531

180 No disturbances 284 00018 152 00086 21485 19997With disturbances 328 00028 283 00089 22032 21903

PSO0 No disturbances 214 00029 131 00088 17094 15551



5 Problem Formulation

In this research the performance index which consists ofcontrol energy (119864

119906) steady state error (119864ss) settling time

(119905119904) rise time (119905

119903) and overshoot (119872

119901) was considered

The appropriate IT2FL-PDC parameters that minimize theperformance index were searched and the cost functionproposed is given in

cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)

In this study the weighing factor (120574) was considered to beequal to 1 The standard deviation (120590) and the mean value(120582) of cost value of each individual were examined in orderto measure the dynamic and convergence characteristic ofthe proposed methods The standard deviation is used tomeasure the convergence speed while the mean value is usedto measure the accuracy of the algorithm Equations (20) and(21) show the formula for calculating 120590 and 120582 respectively[53] Consider

120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)

where 119899 is the population size and cost119901119894

is the individual costvalue

6 Results and Discussion

For us to see the convergence characteristics of the con-trollers two simulations are performed (mean and standarddeviation) Figure 7(a) shows that both controllers (GA-based and PSO-based) secure stable mean cost value usingthe same simulation conditions and cost function HoweverPSO-based controller has best mean value and cost valuewhich indicate that it can achieve better accuracy than GA-based Similarly Figure 7(b) shows that in the tendency ofconvergence of standard deviation of cost values PSO-basedcontroller is faster than GA-based controller This indicatesthat PSO method has the best convergence efficiency Therun time in 150 iterations for PSO is 676839 sec and forGA is 1449619 sec The summary of the best simulationresults in 60 runs under different operating condition isshown in Table 4 It can be observed that the percentageovershoot is lower for PSO-based method compared toeach correspondent GA-basedmethodThe improvements inovershoot are 2275 (for 120572

119903= 0 no disturbances) 201

(for 120572119903= 0 with disturbances) 222 (for 120572

119903= 180 no

disturbances) and 180 (for 120572119903= 0 with disturbances)

The settling time is smaller in GA-based method in all the


minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)

Figure 8 Simulated results for 120572119903equal to zero (a) pendulum angle (b) arm angle (c) output of the outer controller and (d) output of inner

controller (control voltage to the servomotor)

simulations but the rise time is almost the same for both PSOand GA in all simulations

61 Reference Tracking In this section the servo behaviour(reference tracking) has been analysed by using two differentvalues of reference pendulum angle (120572

119903) First 120572

119903was set

to be equal to zero that is the pendulum is at equilibriumupright unstable position and we want to control it to goto a stable position Figures 8(a) 8(b) 8(c) and 8(d) showthe best results of the pendulum angle arm angle output ofouter controller and the control voltage to the servomotorrespectively

Second 120572119903was set to be equal to 120587 (180∘) that is the

pendulum is at stable position and we want to control it togo equilibrium upright unstable position Figures 9(a) 9(b)9(c) and 9(d) show the best results of the pendulum anglearm angle output of outer controller and the control voltageto the servomotor respectively

62 Disturbances Rejection Analysis The internal noise andexternal disturbance was added to the system in order to testfor the robustness of the proposed controllers as follows Aload of 0052m height and 0045 kg mass was added to theend of the pendulum also the white noise of 000634 powerand 5 parameter value changes was added to the processoutput as shown in Figure 10 The simulation results shownin Figure 11 indicate the effectiveness and robustness of theproposed controllers

7 Validation and Comparison

The experiments were performed on Quanser SRV02 RIPsetup The US Digital S1 single-ended optical shaft encoderthat can offer a high resolution of 1024 lines per revolution(4096 counts per revolution in quadrature mode) was usedfor measuring the pendulum angle and arm angleThe poweramplifier used was VoltPAQ-X1 The data acquisition device


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)

Figure 9 Simulated results for 120572119903equal to 120587 (a) pendulum angle (b) arm angle (c) output of the outer controller and (d) output of inner


used was Quanser Q2-USB The experimental results thatcompared the proposed controllers and the conventionalenergy based controller for pendulum angle arm angle andcontrol voltage to the servomotor are shown in Figures 12(a)12(b) and 12(c) respectively The time starts from zero inorder to show the performances from the swing-up to thebalance mode Looking at the real plantrsquos results some initialoscillations were noticed due to the swing motion neededto bring the pendulum from stable position to the verticalunstable position Also the two controllers (PSO-based andGA-based) manifest the considerable level of robustness Onthe other hand the conventional energy based controllermanifests a lot of oscillations before it becomes stable Thetime taken to reach the steady state is higher for conventionalenergy based compared to type 2 fuzzy (for both PSO- andGA-based) controllers The experimental results agreed withthe simulation results as shown in Figures 8 9 and 10 whichjustifies the availability of the system models and conforms

the performance of the proposed methods Experimentaland simulation results indicated that the effectiveness androbustness of the proposed controllers with respect to loaddisturbances parameter variation and noise effects havebeen improved over state-of-the-art method In summary itcan be concluded based on the evidence emanated from theexperiment results that the GA- and PSO-based controllershave advanced the performance of the conventional energybased controller on the RIP

8 Conclusions

In this study a cascade type 2 fuzzy logic PD controller wasdesigned with the aid of optimization methods realised byGA and PSO The application of the proposed controller wastested for controlling the pendulum angle and arm angleof RIP system which is highly nonlinear and unstable The


120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum

Figure 10 Cascaded type 2 fuzzy PD controller with source of noise

130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)

Figure 11 Simulated results for systemwith disturbances (a) pendulum angle (b) arm angle (c) output of the outer controller and (d) outputof inner controller (control voltage to the servomotor)

model of the RIP was also realised in two different ways andthe architecture of type 2 fuzzy PD controller was discussed

The comparative analysis was done in order to test theperformance of the proposed controllers in terms of referencetracking and the disturbances rejection Both the simulations

and experimental results show that both PSO and GA canbe used effectively for optimization of parameters of theproposed controller Moreover based on this study PSO andGA optimized cascade type 2 fuzzy PD controllers showsome certain level of robustness when subjected to noise


1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)

Type 2 fuzzy cascade controller (PSO-based)Type 2 fuzzy cascade controller (GA-based)Conventional energy base controller

10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080

Set pointType 2 fuzzy cascade controller (PSO-based)Type 2 fuzzy cascade controller (GA-based)Conventional energy base controller

Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)

Figure 12 Experimental results comparing the conventional energy based controller and IT2F-PD controller (PSO-based and GA-based)(a) pendulum angle (b) arm angle and (c) control voltage to the servomotor

and disturbances also they outperformed the conventionalenergy based controller in terms of the performance criteriasuch as steady state error settling time rise time and max-imum overshoot However in case of GA and PSO in someperformance criteria PSO is better than GA for examplemaximum overshoot (PSO has some improvement over GAwhich is between 180 and 2275) settling time (PSOhas lower settling time between 131 sec and 234 sec) andcontrol energy (PSO requires a low control energy between17094 J and 20995 J) For other performance criteria GAis better than PSO for example steady state error (GA hassteady state error between 00009 and 00028 while PSOhas steady state error between 00029 and 00099) In case

of rise time both GA and PSO have almost the same risetime of 00089 sec in all the simulation In summary it canbe concluded based on the evidence emanated from theexperiment results that the GA- and PSO-based controllershave advanced the performance of the conventional energybased controller on the RIP

The proposed control strategy can be regarded as apromising strategy for controlling different complex systemswhich are unstable and nonlinear

General type 2 fuzzy logic controller which has moredegree of freedom than interval type 2 fuzzy logic controlleris recommended for future work on RIP control Also morepowerful evolutionary optimization algorithms like hybrid


optimization method are recommended for adjusting theparameters associated with type 2 fuzzy logic controller forRIP

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by Ministry of Higher EducationMalaysia with the Grant no UMCHIRMOHEENG35(D000035-16001)

References

[1] R Martınez-Soto O Castillo and L T Aguilar ldquoType-1 andType-2 fuzzy logic controller design using a Hybrid PSO-GAoptimizationmethodrdquo Information Sciences vol 285 pp 35ndash492014

[2] J M Mendel H Hagras W Tan W W Melek and HYing Introduction to Type-2 Fuzzy Logic Control Theory andApplications John Wiley amp Sons Hoboken NJ USA 2014

[3] R A Krohling J P Rey R A Krohling and J P Rey ldquoDesignof optimal disturbance rejection PID controllers using geneticalgorithmsrdquo IEEE Transactions on Evolutionary Computationvol 5 no 1 pp 78ndash82 2001

[4] M Krstic I Kanellakopoulos and P V Kokotovic Nonlinearand Adaptive Control Design John Wiley amp Sons 1995

[5] J Y Hung W Gao and J C Hung ldquoVariable structure controlA surveyrdquo IEEE Transactions on Industrial Electronics vol 40no 1 pp 2ndash22 1993

[6] Z Zhang and S Zhang ldquoType-2 fuzzy soft sets and their applica-tions in decision makingrdquo Journal of Applied Mathematics vol2012 Article ID 608681 35 pages 2012

[7] I Hassanzadeh and S Mobayen ldquoController design for rotaryinverted pendulum system using evolutionary algorithmsrdquoMathematical Problems in Engineering vol 2011 Article ID572424 17 pages 2011

[8] J M Mendel ldquoGeneral type-2 fuzzy logic systems made simplea tutorialrdquo IEEE Transactions on Fuzzy Systems vol 22 no 5pp 1162ndash1182 2014

[9] F Herrera ldquoGenetic fuzzy systems taxonomy current researchtrends and prospectsrdquo Evolutionary Intelligence vol 1 no 1 pp27ndash46 2008

[10] O Cordon ldquoA historical review of evolutionary learning meth-ods for Mamdani-type fuzzy rule-based systems designinginterpretable genetic fuzzy systemsrdquo International Journal ofApproximate Reasoning vol 52 no 6 pp 894ndash913 2011

[11] AM El-NagarM El-Bardini andNM El-Rabaie ldquoIntelligentcontrol for nonlinear inverted pendulum based on intervaltype-2 fuzzy PDcontrollerrdquoAlexandria Engineering Journal vol53 no 1 pp 23ndash32 2014

[12] O Castillo J R Castro P Melin and A Rodriguez-DiazldquoUniversal approximation of a class of interval type-2 fuzzyneural networks in nonlinear identificationrdquo Advances in FuzzySystems vol 2013 Article ID 136214 16 pages 2013

[13] S Zhao and H Li ldquoThe construction of type-2 fuzzy reasoningrelations for type-2 fuzzy logic systemsrdquo Journal of AppliedMathematics vol 2014 Article ID 459508 13 pages 2014

[14] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzy logicsystems made simplerdquo IEEE Transactions on Fuzzy Systems vol14 no 6 pp 808ndash821 2006

[15] O Castillo Type-2 Fuzzy Logic in Intelligent Control Appli-cations vol 272 of Studies in Fuzziness and Soft ComputingSpringer Berlin Germany 2012

[16] S-K Oh H-J Jang and W Pedrycz ldquoA comparative exper-imental study of type-1type-2 fuzzy cascade controller basedon genetic algorithms and particle swarm optimizationrdquo ExpertSystems with Applications vol 38 no 9 pp 11217ndash11229 2011

[17] C Aguilar-Avelar and J Moreno-Valenzuela ldquoA composite con-troller for trajectory tracking applied to the Furuta pendulumrdquoISA Transactions 2015

[18] S-E Oltean ldquoSwing-up and stabilization of the rotationalinverted pendulum using PD and fuzzy-PD controllersrdquo Pro-cedia Technology vol 12 pp 57ndash64 2014

[19] H Qin W Shao and L Guo ldquoResearch and verification onswing-up control algorithm of rotary inverted pendulumrdquo inProceedings of the 26th Chinese Control AndDecision Conference(CCDC rsquo14) pp 4941ndash4945 Changsha China May 2014

[20] V Nath and RMitra ldquoSwing-up and control of Rotary InvertedPendulum using pole placement with integratorrdquo in Proceedingsof the Recent Advances in Engineering and ComputationalSciences (RAECS rsquo14) pp 1ndash5 IEEE Chandigarh India March2014

[21] P Faradja G Qi and M Tatchum ldquoSliding mode control ofa Rotary Inverted Pendulum using higher order differentialobserverrdquo in Proceedings of the 14th International Conferenceon Control Automation and Systems (ICCAS rsquo14) pp 1123ndash1127October 2014

[22] K Y Chou and Y P Chen ldquoEnergy based swing-up controllerdesign using phase plane method for rotary inverted pendu-lumrdquo in Proceedings of the 13th International Conference onControl Automation Robotics amp Vision (ICARCV rsquo14) pp 975ndash979 Singapore December 2014

[23] Y-F Chen and A-C Huang ldquoAdaptive control of rotaryinverted pendulum system with time-varying uncertaintiesrdquoNonlinear Dynamics vol 76 no 1 pp 95ndash102 2014

[24] S-K Oh H-J Jang andW Pedrycz ldquoOptimized fuzzy PD cas-cade controller a comparative analysis and designrdquo SimulationModelling Practice andTheory vol 19 no 1 pp 181ndash195 2011

[25] M A Fairus Z Mohamed M N Ahmad and W S LoildquoLMI-based multiobjective integral sliding mode control forrotary inverted pendulum system under load variationsrdquo JurnalTeknologi vol 73 no 6 2015

[26] S-E Oltean and A-V Duka ldquoBalance control system usingmicrocontrollers for a rotational inverted pendulumrdquo ProcediaTechnology vol 12 pp 11ndash19 2014

[27] Q V Dang B Allouche L Vermeiren A Dequidt and MDambrine ldquoDesign and implementation of a robust fuzzy con-troller for a rotary inverted pendulum using the Takagi-Sugenodescriptor representationrdquo in Proceedings of the IEEE Sympo-sium on Computational Intelligence in Control and Automation(CICA rsquo14) pp 1ndash6 Orlando Fla USA December 2014

[28] M U Hassan M B Kadri and I Amin ldquoProficiency of fuzzylogic controller for stabilization of rotary inverted pendulumbased on LQR mappingrdquo in Fuzzy Logic and Applications vol8256 of Lecture Notes in Computer Science pp 201ndash211 Springer2013

[29] P M Mary and N S Marimuthu ldquoMinimum time swing upand stabilization of rotary inverted pendulum using pulse step


controlrdquo Iranian Journal of Fuzzy Systems vol 6 no 3 pp 1ndash152009

[30] M Ramırez-Neria H Sira-Ramırez R Garrido-Moctezumaand A Luviano-Juarez ldquoLinear active disturbance rejectioncontrol of underactuated systems the case of the Furutapendulumrdquo ISA Transactions vol 53 no 4 pp 920ndash928 2014

[31] T C Kuo Y J Huang and B W Hong ldquoAdaptive PID withsliding mode control for the rotary inverted pendulum systemrdquoin Proceedings of the IEEEASME International Conference onAdvanced Intelligent Mechatronics (AIM rsquo09) pp 1804ndash1809July 2009

[32] Y Singh A Kumar and R Mitra ldquoDesign of ANFIS controllerbased on fusion function for rotary inverted pendulumrdquo inProceedings of the International Conference on Advances inPower Conversion and Energy Technologies (APCET rsquo12) pp 1ndash5Mylavaram India August 2012

[33] S-K Oh S-H Jung and W Pedrycz ldquoDesign of opti-mized fuzzy cascade controllers by means of Hierarchical FairCompetition-based Genetic Algorithmsrdquo Expert Systems withApplications vol 36 no 9 pp 11641ndash11651 2009

[34] P Civicioglu and E Besdok ldquoA conceptual comparison of theCuckoo-search particle swarmoptimization differential evolu-tion and artificial bee colony algorithmsrdquo Artificial IntelligenceReview vol 39 no 4 pp 315ndash346 2013

[35] O Castillo and P Melin ldquoOptimization of type-2 fuzzy systemsbased on bio-inspired methods a concise reviewrdquo InformationSciences vol 205 pp 1ndash19 2012

[36] H Chiroma S Abdulkareem and T Herawan ldquoEvolutionaryNeural Network model for West Texas Intermediate crude oilprice predictionrdquo Applied Energy vol 142 pp 266ndash273 2015

[37] N M Nawi A Khan M Rehman H Chiroma and THerawan ldquoWeight optimization in recurrent neural networkswith hybrid metaheuristic Cuckoo search techniques for dataclassificationrdquoMathematical Problems in Engineering In press

[38] J Mei W Ren and G Ma ldquoDistributed coordinated trackingwith a dynamic leader for multiple Euler-Lagrange systemsrdquoIEEE Transactions on Automatic Control vol 56 no 6 pp 1415ndash1421 2011

[39] J Lather S Dhillon and S Marwaha ldquoModern control aspectsin doubly fed induction generator based power systems areviewrdquo International Journal of Advanced Research in ElectricalElectronics and Instrumentation Engineering vol 2 pp 2149ndash2161 2013

[40] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[41] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoningrdquo in Learning Systems andIntelligent Robots pp 1ndash10 Springer 1974

[42] S N Sivanandam and S N Deepa Introduction to GeneticAlgorithms Springer Berlin Germany 2007

[43] Y-T Kao and E Zahara ldquoA hybrid genetic algorithm andparticle swarm optimization formultimodal functionsrdquoAppliedSoft Computing vol 8 no 2 pp 849ndash857 2008

[44] C-T Li C-H Lee F-Y Chang and C-M Lin ldquoAn intervaltype-2 fuzzy system with a species-based hybrid algorithm fornonlinear system control designrdquo Mathematical Problems inEngineering vol 2014 Article ID 735310 19 pages 2014

[45] C A C Coello S Dehuri and S Ghosh Swarm Intelligencefor Multi-Objective Problems in Data Mining vol 242 SpringerScience+Business Media 2009

[46] R L Haupt and S E Haupt Practical Genetic Algorithms JohnWiley amp Sons New York NY USA 2004

[47] S N Sivanandam and S N Deepa Introduction to GeneticAlgorithms Springer 2008

[48] H Chiroma S Abdulkareem E N Sari et al ldquoSoft computingapproach in modeling energy consumptionrdquo in ComputationalScience and Its ApplicationsmdashICCSA 2014 vol 8584 of LectureNotes in Computer Science pp 770ndash782 Springer 2014

[49] R Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 October1995

[50] R C Eberhart and Y Shi ldquoParticle swarm optimization devel-opments applications and resourcesrdquo in Proceedings of theCongress on Congress on Evolutionary Computation pp 81ndash86May 2001

[51] S Dehuri S Ghosh and C A C Coello ldquoAn introductionto swarm intelligence for multi-objective problemsrdquo in SwarmIntelligence for Multi-objective Problems in Data Mining SDehuri S Ghosh andC A C Coello Eds vol 242 of Studies inComputational Intelligence pp 1ndash17 Springer Berlin Germany2009

[52] H Chiroma S Abdulkareem A Abubakar and M J UsmanldquoComputational intelligence techniques with application tocrude oil price projection a literature survey from 2001ndash2012rdquoNeural Network World vol 23 no 6 pp 523ndash551 2013

[53] P Melin D Sanchez and L Cervantes ldquoHierarchical geneticalgorithms for optimal type-2 fuzzy system designrdquo in Pro-ceedings of the Annual Meeting of the North American FuzzyInformation Processing Society (NAFIPS rsquo11) pp 1ndash6 IEEE ElPaso Tex USA March 2011

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


degree of freedom in designing varieties of systems withuncertainties [1 2 8] The control performance of T1FLCcan be improved by T2FLC because it has the advantage offootprint of uncertainty (FOU) that can be used to improvethe corresponding MF [15]

The major disadvantage of T2FLC is that the process oftuning becomes more difficult andmuch time consuming forincreasing number of inputs and outputs of the controller[16]This problem of tuning can be solved by using optimiza-tion methods [15 16]

Looking at Rotary Inverted Pendulum (RIP) from thecontrol point of view the RIP exhibits many interesting andchallenging properties such as nonlinearities and instabilityThe RIP is in the class of underactuated mechanical sys-tems These features make RIP become known widely asexperimental setup for testing linear and nonlinear controlalgorithms [11] RIP consists of a rotational servomotorsystem which drives the output gear rotational arm anda pendulum It has many important real applications likerobotics aerospace vehicles pointing control and marinevehicles [15]

Four control objectives of RIP as found in the literatureare categorized as follows [17 18] (1) swing-up control con-trolling the pendulum from downward (stable position) toupward (unstable position) (2) stabilization control that isregulating the pendulum to remain at the unstable position(3) switching control that is the switching between swing-up control and stabilization control (4) trajectory controlthat is to control the RIP in such a way that the arm tracks adesired time varying trajectory while the pendulum remainsat unstable position

Recently a lot of research regarding the control of RIP hasbeen published For example energy based compensationPD cascade scheme edge trigger counter sliding modecontrol type 1 fuzzy control and adaptive control have beenused for swing-up control of RIP [18ndash24] To solve thestabilization control problem on RIP multiobjective integralsliding mode controller microcontrollers fuzzy logic regula-tor pole placement technique optimized PID controller andLinear Quadratic Regulator (LQR) have been applied [7 1820 23 25ndash28] Energywas considered for switching control in[18 29] and mode controller was used for the same purposein [20] For trajectory control adaptive PID with slidingmode control linear active disturbance rejection controlhybrid of linear fusion function based on LQR mapping andadaptive control with ANFIS tuning feedback linearizationbased controller and energy based compensation controllerwas used [17 30ndash32] Optimized cascade type 1 fuzzy logiccontroller was used in [33] for controls of pendulum angleand arm angles of RIP

To the best of the authorrsquos knowledge at this momentthere is not any kind of GA and PSO optimised type 2 fuzzylogic control applied to RIP A cascade control method iseffective for a systemwith high level of disturbances and largetime error such as the RIP [24] Also as mentioned earliertype 2 fuzzy control strategy is effective and gives robustcontrol response for systems with high level of uncertaintyandor inaccurate model Putting type 2 fuzzy in cascadetopology will have the advantages of type 2 fuzzy and cascade

structure which will eventually give more robust controllerfor system with uncertainties and large time error [33]

In this paper the IT2F-PDC is designed in cascaded formfor RIPThe parameters of the IT2F-PDC are optimised usingGenetic Algorithm (GA) and Particle Swarm Optimization(PSO) The GA and PSO were chosen in this research inview of the fact that these algorithms are more establishedin the literature than other evolutionary algorithms [3435] In addition the GA and PSO have proven to improveperformance over other algorithms in solving optimizationproblems [36 37] The goal is to balance the pendulum inupright position The servo behaviour (reference tracking)of RIP is analysed Disturbances rejection of the proposedcontroller is analysed by adding the internal noise andexternal disturbance to the system Also the controller isapplied on the real RIP to validate the simulation resultsThe performances of GA and PSO for the optimizationof the parameters of IT2F-PDC were compared Also thedesigning of IT2F-PDC as an optimization dilemma thatslightly altered five performance indices was formulatedwhich includes steady state error settling time rise timemaximum overshoot of the response and control energy ofthe system

The organisation of the paper is as follows Section 2introduces the RIP Section 3 gives brief on fuzzy logicsystem IT2FL-PDC and the design of cascade IT2-PDC andSection 4 defines optimizationmethod and gives some basicson GA and PSOThe formulation of the problem is discussedin Section 5 Sections 6 and 7 present the simulation andexperimental results respectively Finally Section 8 presentsconclusions

2 RIP

The RIP is in the class of underactuated mechanical systemconsisting of a servomotor system that drives the outputgear The rotational arm is attached to the output gear and apendulum is attached at one end of the rotating arm as shownin Figure 1(a) The arm is driven by the gear with the aim ofbalancing the pendulum in an upright position The angle(120601) is the angle of the arm and its direction depends on thedirection of the control voltage (119881

119898) (voltage applied to the

servomotor) In this study the counterclockwise direction isconsidered as positive direction for the arm and the clockwisedirection is the positive direction for the pendulum Theservomotor drives the rotating arm to move in horizontalplane 119883119884 in such a way that it sets the pendulum to theinverted position in the 119883119885 direction perpendicular to thearm (120572) is the pendulum angle Figure 1(b) shows the pictureof the experimental setup using Quanser RIP

21 Nonlinear and Linear Dynamics Model of RIP Thedynamic equations that describe the motion of rotary armand the pendulum with respect to the servomotor voltagewere obtained using Euler-Lagrange equation which is thesystematic way of obtaining the equation of motion [38]Once the kinetic and potential energy are obtained and theLagrangian is found Subsequently the task is to compute var-ious derivatives to get the equations of motion of the system


120572

z

x

Pendulum

120601

y

LP

lp

Lr

Arm Jr

mpJp

g

(a) (b)









(1198981199011198712119903+141198981199011198712119901minus141198981199011198712119901cos (120572)2 + 119869

119903) 120601

minus (12119898119901119871119901119871119903cos (120572))

+ (121198981199011198712119901sin (120572) cos (120572)) 120601

+ (12119898119901119871119901119871119903sin (120572)) 2 = 120591minus119861

119903

120601

(minus12119898119901119871119901119871119903cos (120572)) 120601 + (119869

119901+141198981199011198712119901)

minus (141198981199011198712119901cos (120572) sin (120572)) 1206012

minus(12119898119901119871119901119892 sin (120572)) = minus119861

119901

(1)





(119869eq +1198981199011198712

119903) 120601 minus (119898

119901119897119901119871119903cos (120572))

+ (119898119901119897119901119871119903sin (120572)) 2 = 120591minus119861eq 120601

(431198981199011198972119901) minus (119898

119901119897119901119871119903cos (120572)) 120601 minus119898

119901119897119901119892 sin (120572) = 0

(2)


120591 =120578119892119870119892120578119898119896119905(119881119898minus 119870119892119896119898

120601)

119877119898

(3)


[[[[[

[

120601

120601

]]]]]

]

=

[[[[[[[[[

[

0 0 1 0

0 0 0 1

0119896211989641198965

minus119896311989671198965

0

0119896111989641198965

minus119896211989671198965

0

]]]]]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+

[[[[[[[[[

[

00

1198963120578119892119870119892120578119898119896119905

1198771198981198965

1198962120578119892119870119892120578119898119896119905

1198771198981198965

]]]]]]]]]

]

119881119898

(4)







119861119901






119877119898


119896119905







where 1198961 = 119869eq + 1198981199011198712

119903 1198962 = 119898119901119897119901119871119903 1198963 = (43)119898119901119897

2

119901 1198964 =

119897119901119898119901119892 1198965 = 11989611198963 minus 119896

2

2

1198967 =1205781198921198701198981205781198981198961199051198962119892+ 119861eq119877119898

119877119898

(5)

[[[[[

[

120601

120601

]]]]]

]

=

[[[[[

[

0 0 1 0

0 0 0 1

0 3491 minus1621 0

0 7544 minus1312 0

]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+[[[[

[

00

28562307

]]]]

]

119881119898

(6)

[120601

120572] = [

1 0 0 0

0 1 0 0][[[[

[

120601

120572120601

]]]]

]

(7)


119882 = [119861 119860119861 1198602119861 119860

3119861 119860

119899minus1119861] rank (119882) = 4 (8)







0 025 05 075 1 125 15 175


Time (s)

120572(d

eg)

300

200

100

0

(a)120601

(deg

)

0

60

0

30

025 05 075 1 125 15 175Time (s)

minus30

minus60


(b)





1 119909

119901)119879 isin 119883


119901equiv 119883 into 119860

119909


120583119860119909

(119909) = 11 with 119909 = 1199091015840

120583119860119909

(119909) = 10


(9)


(119909119894) = 11 when 119909

119894= 1199091015840119894and

120583119909119894

(119909119894) = 10 when 119909

119894= 1199091015840119894for all 119894 = 1 119901


119877119894


119901

THEN 119884119894

= 1198620 + 119862119894

11199091 + sdot sdot sdot + 119862119894

119903119909119901(10)



set


119860119909

= cap119901

119894=1120583119909119894

(119909119894) where 119909

119894(119894 = 1 119901)


119894=1120583119865119894119894


(12) [9]

119865119894

(1199091015840

) = [119891119894

(1199091015840

) 119891119894

(1199091015840

)] equiv [119891119894

119891119894

] (11)

where

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


119901

(1199091015840

119901)

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


(1199091015840

119901)

(12)



1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)


yry1

min

min

Crisp input

Fuzzification

Inference


Crisp output

Output processing


Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)



119897and 119910


1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)


119903


119910 (119909) =1199101 + 119910119903

2 (14)



2(119905)



4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V


1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)







Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120572

z

x

Pendulum

120601

y

LP

lp

Lr

Arm Jr

mpJp

g

(a) (b)









(1198981199011198712119903+141198981199011198712119901minus141198981199011198712119901cos (120572)2 + 119869

119903) 120601

minus (12119898119901119871119901119871119903cos (120572))

+ (121198981199011198712119901sin (120572) cos (120572)) 120601

+ (12119898119901119871119901119871119903sin (120572)) 2 = 120591minus119861

119903

120601

(minus12119898119901119871119901119871119903cos (120572)) 120601 + (119869

119901+141198981199011198712119901)

minus (141198981199011198712119901cos (120572) sin (120572)) 1206012

minus(12119898119901119871119901119892 sin (120572)) = minus119861

119901

(1)





(119869eq +1198981199011198712

119903) 120601 minus (119898

119901119897119901119871119903cos (120572))

+ (119898119901119897119901119871119903sin (120572)) 2 = 120591minus119861eq 120601

(431198981199011198972119901) minus (119898

119901119897119901119871119903cos (120572)) 120601 minus119898

119901119897119901119892 sin (120572) = 0

(2)


120591 =120578119892119870119892120578119898119896119905(119881119898minus 119870119892119896119898

120601)

119877119898

(3)


[[[[[

[

120601

120601

]]]]]

]

=

[[[[[[[[[

[

0 0 1 0

0 0 0 1

0119896211989641198965

minus119896311989671198965

0

0119896111989641198965

minus119896211989671198965

0

]]]]]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+

[[[[[[[[[

[

00

1198963120578119892119870119892120578119898119896119905

1198771198981198965

1198962120578119892119870119892120578119898119896119905

1198771198981198965

]]]]]]]]]

]

119881119898

(4)







119861119901






119877119898


119896119905







where 1198961 = 119869eq + 1198981199011198712

119903 1198962 = 119898119901119897119901119871119903 1198963 = (43)119898119901119897

2

119901 1198964 =

119897119901119898119901119892 1198965 = 11989611198963 minus 119896

2

2

1198967 =1205781198921198701198981205781198981198961199051198962119892+ 119861eq119877119898

119877119898

(5)

[[[[[

[

120601

120601

]]]]]

]

=

[[[[[

[

0 0 1 0

0 0 0 1

0 3491 minus1621 0

0 7544 minus1312 0

]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+[[[[

[

00

28562307

]]]]

]

119881119898

(6)

[120601

120572] = [

1 0 0 0

0 1 0 0][[[[

[

120601

120572120601

]]]]

]

(7)


119882 = [119861 119860119861 1198602119861 119860

3119861 119860

119899minus1119861] rank (119882) = 4 (8)







0 025 05 075 1 125 15 175


Time (s)

120572(d

eg)

300

200

100

0

(a)120601

(deg

)

0

60

0

30

025 05 075 1 125 15 175Time (s)

minus30

minus60


(b)





1 119909

119901)119879 isin 119883


119901equiv 119883 into 119860

119909


120583119860119909

(119909) = 11 with 119909 = 1199091015840

120583119860119909

(119909) = 10


(9)


(119909119894) = 11 when 119909

119894= 1199091015840119894and

120583119909119894

(119909119894) = 10 when 119909

119894= 1199091015840119894for all 119894 = 1 119901


119877119894


119901

THEN 119884119894

= 1198620 + 119862119894

11199091 + sdot sdot sdot + 119862119894

119903119909119901(10)



set


119860119909

= cap119901

119894=1120583119909119894

(119909119894) where 119909

119894(119894 = 1 119901)


119894=1120583119865119894119894


(12) [9]

119865119894

(1199091015840

) = [119891119894

(1199091015840

) 119891119894

(1199091015840

)] equiv [119891119894

119891119894

] (11)

where

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


119901

(1199091015840

119901)

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


(1199091015840

119901)

(12)



1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)


yry1

min

min

Crisp input

Fuzzification

Inference


Crisp output

Output processing


Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)



119897and 119910


1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)


119903


119910 (119909) =1199101 + 119910119903

2 (14)



2(119905)



4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V


1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)







Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119861119901






119877119898


119896119905







where 1198961 = 119869eq + 1198981199011198712

119903 1198962 = 119898119901119897119901119871119903 1198963 = (43)119898119901119897

2

119901 1198964 =

119897119901119898119901119892 1198965 = 11989611198963 minus 119896

2

2

1198967 =1205781198921198701198981205781198981198961199051198962119892+ 119861eq119877119898

119877119898

(5)

[[[[[

[

120601

120601

]]]]]

]

=

[[[[[

[

0 0 1 0

0 0 0 1

0 3491 minus1621 0

0 7544 minus1312 0

]]]]]

]

[[[[

[

120601

120572120601

]]]]

]

+[[[[

[

00

28562307

]]]]

]

119881119898

(6)

[120601

120572] = [

1 0 0 0

0 1 0 0][[[[

[

120601

120572120601

]]]]

]

(7)


119882 = [119861 119860119861 1198602119861 119860

3119861 119860

119899minus1119861] rank (119882) = 4 (8)







0 025 05 075 1 125 15 175


Time (s)

120572(d

eg)

300

200

100

0

(a)120601

(deg

)

0

60

0

30

025 05 075 1 125 15 175Time (s)

minus30

minus60


(b)





1 119909

119901)119879 isin 119883


119901equiv 119883 into 119860

119909


120583119860119909

(119909) = 11 with 119909 = 1199091015840

120583119860119909

(119909) = 10


(9)


(119909119894) = 11 when 119909

119894= 1199091015840119894and

120583119909119894

(119909119894) = 10 when 119909

119894= 1199091015840119894for all 119894 = 1 119901


119877119894


119901

THEN 119884119894

= 1198620 + 119862119894

11199091 + sdot sdot sdot + 119862119894

119903119909119901(10)



set


119860119909

= cap119901

119894=1120583119909119894

(119909119894) where 119909

119894(119894 = 1 119901)


119894=1120583119865119894119894


(12) [9]

119865119894

(1199091015840

) = [119891119894

(1199091015840

) 119891119894

(1199091015840

)] equiv [119891119894

119891119894

] (11)

where

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


119901

(1199091015840

119901)

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


(1199091015840

119901)

(12)



1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)


yry1

min

min

Crisp input

Fuzzification

Inference


Crisp output

Output processing


Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)



119897and 119910


1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)


119903


119910 (119909) =1199101 + 119910119903

2 (14)



2(119905)



4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V


1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)







Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 025 05 075 1 125 15 175


Time (s)

120572(d

eg)

300

200

100

0

(a)120601

(deg

)

0

60

0

30

025 05 075 1 125 15 175Time (s)

minus30

minus60


(b)





1 119909

119901)119879 isin 119883


119901equiv 119883 into 119860

119909


120583119860119909

(119909) = 11 with 119909 = 1199091015840

120583119860119909

(119909) = 10


(9)


(119909119894) = 11 when 119909

119894= 1199091015840119894and

120583119909119894

(119909119894) = 10 when 119909

119894= 1199091015840119894for all 119894 = 1 119901


119877119894


119901

THEN 119884119894

= 1198620 + 119862119894

11199091 + sdot sdot sdot + 119862119894

119903119909119901(10)



set


119860119909

= cap119901

119894=1120583119909119894

(119909119894) where 119909

119894(119894 = 1 119901)


119894=1120583119865119894119894


(12) [9]

119865119894

(1199091015840

) = [119891119894

(1199091015840

) 119891119894

(1199091015840

)] equiv [119891119894

119891119894

] (11)

where

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


119901

(1199091015840

119901)

119891119894

(1199091015840

119894) = 120583119865119894

1(1199091015840


(1199091015840

119901)

(12)



1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)


yry1

min

min

Crisp input

Fuzzification

Inference


Crisp output

Output processing


Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)



119897and 119910


1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)


119903


119910 (119909) =1199101 + 119910119903

2 (14)



2(119905)



4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V


1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)







Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

0x

x1 x2

120583(x)

UMF(A)

LMF(A)

FOU(A)

120583 (x2)UFM

120583 (x1)UFM

LFM120583(x2)

LFM120583(x1)

(a)

0

1

120583A(x)

120583A(x)UFM120583119860(x1)

(b)

1

0

120583A(x)

120583A(x)

UFM120583119860(x2)LFM120583119860(x2)

(c)


yry1

min

min

Crisp input

Fuzzification

Inference


Crisp output

Output processing


Fired rule

c0f1(x) fr(x)

A(x998400)

A(x998400)

x9984002 x2

FOU(A2)

x1x9984001

FOU(A1)

120583A(x9984002)

120583A1(x9984001)

x2 isin X

x1 isin X

Y = f(x) isin X

120583A1(x9984001)

120583A1(x9984002)



119897and 119910


1199101 =sum119872

119894=1 119891119894

1119910119894

1

sum119872

119894=1 119891119894

1

119910119903=sum119872

119894=1 119891119894

119903119910119894119903

sum119872

119894=1 119891119894

119903

(13)


119903


119910 (119909) =1199101 + 119910119903

2 (14)



2(119905)



4 1198925 and 119892

6are used

to scale 1198902(119905) Δ119890

2(119905) and V


1198642 (119905) = 11989241198902 (119905) = 1198924 (119880 (119905) minus 120601 (119905))

Δ1198642 (119905) = 1198925Δ1198902 (119905) = 1198925 (1198902 (119905) minus 1198902 (119905 minus 1))

119881119898(119905) = 1198926V119898 (119905)

(15)







Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Type 2fuzzy logic

e1 g1

g2

e2

g3

g4

g5

+

minus+

minus

Set 120572

120572

120572

g6U Vm

de1dt

120601

120601

120601

de2dt



Δ119890119890



1(119905)


1 1198922 and 119892


1(119905) Δ119890

1(119905) and 119906(119905)


1198641 (119905) = 11989211198901 (119905) = 1198921 (120572119903 (119905) minus 120572 (119905))

Δ1198641 (119905) = 1198922Δ1198901 (119905) = 1198922 (1198901 (119905) minus 1198901 (119905 minus 1))

119880 (119905) = 1198923119906 (119905)

(16)



6) are





Gains 120572119903= 0 120572

119903= 180


minus95701 minus96046 65632 598251198922

minus13161 minus10527 00119 001021198923

32616 27104 89646 1001091198924

minus13664 minus32773 03679 019981198925

38597 62357 00612 007011198926

08597 06589 03951 05094





NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










NL NS ZO PS PL1

0x

120583A(x)

(a)

yry1

1NL NS ZO PS PL

0 uc0

(b)




119881119894(119898) = 119883 [119881

119894(119898minus 1) + 11988811199031 (119875best

119894

minus119876119894(119898minus 1))

+ 11988821199032 (119892best minus119876119894 (119898minus 1))]

119883 =2


1003816100381610038161003816100381610038161003816

120573 = 1198881 + 1198882 gt 4

(17)



1and




119875119894= 119875119894(119898minus 1) +119881

119894(119898) (18)









1= 1198882= 2 maximum



10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

20

30

40

50M

ean

cost

015 30 45 60 75 90 105 130 145 1600

Generation

PSOGA

(a)

1

2

3

4

5

Stan

dard

dev

iatio

n

15 30 45 60 75 90 105 130 145 1600Generation

PSOGA

0

(b)




119901() 119864ss 119905

119904(s) 119905

119903(s) 119864

119906(J) Cost










(119905119904) rise time (119905




cost = 12(119872119901+119864ss +119864119906) minus

119890minus120574

2(119905119903minus 119905119904+119864ss +119872119901) (19)


120590 = radic1119899

119899

sum119894=1(cost119901119894

minus 120582)2 (20)

120582 =sum119899

119894=1 cost119901119894119899

(21)







119903= 180 no




minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus15

minus05

minus2

minus1

0 1 2 3 4 5

0

05

1

Arm

angl

e (de

g)

PSOGA

Time (s)

(a)

0

025

05

Pend

ulum

angl

e (de

g)

PSOGA

0 1 2 3 4 5Time (s)

minus05

minus075

minus025

minus1

(b)

0 02 04 06 08 1 12 14 16 18 2

0

1

225

15

05

335

4

PSOGA

Time (s)

minus15

minus05

minus2

minus1

Out

put o

f out

er lo

opu(t)

(c)

0

1

2

3

4

Con

trol v

olta

ge to

the s

ervo

mot

or

0 02 04 06 08 1 12 14 16 18 2

PSOGA

Time (s)

minus4

minus3

minus2

minus1

(d)





119903) First 120572

119903was set








130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

Time (s)

PSOGA

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

minus20

minus40

minus60

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0 2

0123456

Con

trol v

olta

ge to

the s

ervo

mot

or

Time (s)02 04 06 08 1 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

(d)





8 Conclusions



120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120572

120601

type 2fuzzy

Outer

logic

type 2fuzzy

Inner

logic

Whitenoise 5

120572r+

+

minus minusΔe1

e1

Δe2

e2 Rotaryinverted

pendulum


130

140

150

160

170

180

190

200

0 1 2 3 4 5 6 7 8 9 10

Pend

ulum

angl

e (de

g)

Time (s)

PSOGA

(a)

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

Arm

angl

e (de

g)

Time (s)

PSOGA

minus60

minus40

minus20

minus80

(b)

0

02

04

06

08

1

0 1 2 3 4 5 6 7 8 9 10Time (s)

PSOGA

minus04

minus02

Out

put o

f out

er lo

opU(t)

(c)

0

1

2

3

4

5

6

Con

trol v

olta

ge to

the s

ervo

mot

or

0

025

Time (s)0 202 04 06 08 1 12 14 16 18

21 12 14 16 18

PSOGA

minus4

minus5

minus6

minus3

minus2

minus1

minus025

(d)






1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 85 6 7 8

90

0

100

200

10

Pend

ulum

angl

e (de

g)


10Time (s)

minus100

minus200

minus10

(a)

Arm

angl

e (de

g)

101 2 3 4 5 6 7 8 90

020406080


Time (s)

minus60

minus40

minus20

minus80

(b)


0 2 4 6 8 10

0

1

2

3

4

5

6

Time (s)

minus4

minus5

minus6

minus3

minus2

minus1

Vm

(V)

(c)










Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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