research article stabilization and tracking control...

$: Research Article Stabilization and Tracking Control …downloads.hindawi.com/journals/je/2014/752918.pdfis work focuses on the use of fractional calculus to design robust fractional-order$
Research ArticleStabilization and Tracking Control of Inverted Pendulum UsingFractional Order PID Controllers

Sunil Kumar Mishra and Dinesh Chandra

Department of Electrical Engineering Motilal Nehru National Institute of Technology Allahabad 211004 India

Correspondence should be addressed to Sunil Kumar Mishra hariomsunil88gmailcom

Received 7 February 2014 Revised 5 March 2014 Accepted 2 April 2014 Published 23 April 2014

Academic Editor Haranath Kar

Copyright copy 2014 S K Mishra and D ChandraThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This work focuses on the use of fractional calculus to design robust fractional-order PID (PI120582D120583) controller for stabilization andtracking control of inverted pendulum (IP) system A particle swarm optimisation (PSO) based direct tuning technique is usedto design two PI120582D120583 controllers for IP system without linearizing the actual nonlinear model The fitness function is minimizedby running the SIMULINK model of IP system according to the PSO program in MATLAB The performance of proposed PI120582D120583controllers is compared with two PID controllers Simulation results are also obtained by adding disturbances to the model to showthe robustness of the proposed controllers

1 Introduction

The inverted pendulum (IP) system nonlinear and unstablesystem is widely used in laboratories to implement and vali-date new ideas emerging in control engineering The controlof IP system can be broadly divided into three sectionsswing-up control stabilization and tracking control Swing-up control is basically used to swing the pendulum rodfrompending position to stabilization zoneThen a balancingor stabilization control is essential to uphold it in uprightposition for long interval A switching mechanism betweenswinging and stabilization zone is necessary for effectivecontrol [1 2] For swing-up control a technique based onenergy control had been proposed by Astrom and Furuta [3]

There are several different techniques accessible in liter-ature for stabilization and tracking control of IP system forexample linear quadratic regulator (LQR) PID control neu-ral network control fuzzy logic control neural-fuzzy controlslidingmode control and so forthThe LQR an optimal statefeedback controller designed by minimizing a performanceindex is ordinarily used controller for IP systemmodelled instate space form [4] Here the state space model of IP systemis to be inevitably linearized which leads to modelling errorThe PID controller most widely used controller in severalindustrial control problems is one of the favourite controllers

for IP system The comparison of PID controller with othercontrol techniques of IP system was carried out in manystudies [5 6] The major task of PID controller design is theselection of control parameters for desired response Sometuning methods of PID controller for IP system could befound in literature [7ndash9] In [10] the stabilization as well astracking control of IP system with actual nonlinear modelusing PID controllers was investigated but how to choosecontrollers parameters was not clarified Various techniquesother than PID are also existing [11ndash15]

In last two decades the fractional calculus has becomemuch popular among the researchers of different streamsbut its origin is as older as that of classical integer ordercalculus Fractional calculus was not much popular earlierbecause of its highly complex mathematical expressions Butwith the development of computational technologies it hasbecome possible to deal with fractional calculus Fractionalcalculus provides much accurate and generalized solutionas compared to integer order calculus The applications offractional calculus include modelling and control of physicalsystems [16ndash18] One such application is the modelling oftwo-electric pendulum [19]

In the area of control engineering an application of frac-tional calculus is the fractional order PID (PI120582D120583) controllerwhich is an advanced form of PID control In some recent

Hindawi Publishing CorporationJournal of EngineeringVolume 2014 Article ID 752918 9 pageshttpdxdoiorg1011552014752918

2 Journal of Engineering

z

oM

x

m120579

l

l

z998400

o998400 x998400

Fx

Figure 1 Structure of inverted pendulum

studies [20ndash24] the PI120582D120583 controller gives better outcomesthan PID controller Though there are some applications ofPI120582D120583 controller for IP system [25 26] PI120582D120583 controllerhas not received considerable attention for unstable systemssimilar to IP system

Hence in the present work fractional order PID con-troller is designed in time domain to control pendulum angleas well as cart position Fractional order PID controller ischallenging to design because of the use of fractional calculuswith very complex calculations Therefore a direct approachis used for calculating the parameters of both fractionalorder PID controllers with the help of a multiobjectivefitness function (the fitness function consists of the sum ofintegral squares of pendulum angle cart position and controlvoltage) The fitness function is minimized by running themodel according to a particle swarm optimization (PSO)[27ndash29] program in MATLAB PSO is used in this work asit provides greater convergence towards optimal values ascompared to other optimization techniques and it has thesimple algorithm

The rest of the paper is divided into the following sectionsSection 2 gives a description of the inverted pendulum systemand derivation of system equations in state space formSection 3 describes the fractional calculus and structure of thePI120582D120583 controller Section 4 gives details about PSO Section 5gives a complete description of control strategy Section 6gives a comparison of the simulation results for PI120582D120583controller and PID controller with and without disturbancesThe paper ends with the conclusions in Section 7 which isfollowed by the references

2 Inverted Pendulum System

As shown in Figure 1 an IP system has a cart which canmovehorizontally One end of the pendulum rod is connected tothe centre of the upper surface of the cart which is called thepivot point while the other end is free to move in verticalplane (119909119911-plane) This pendulum rod is stable in extremedownwards position and known as normal pendulum Butwhen the pendulum rod remains in upright position it isknown as IP system This is an unstable condition whichneeds a continuous balancing force (119865119909) on cart in order toremain upright

In Figure 1 horizontal force is used as control action todisplace the pivot around 119909 axis and the total kinetic energy

Table 1 Inverted pendulum parameters

119872 (Kg) 119898 (Kg) 119897 (m) 119892 (ms2)1 01 03 98

(119870) due to mass of the pivot (119872) in 119909 direction mass ofpendulum rod (119898) in 119909 and 119911 directions and potential energy(119875) of the IP system [10] which are

119870 =

1

2

1198722+

1

2

119898 (2

119901+ 2

119901) 119875 = 119898119892119911119901

(1)

where

119909119901 = 119909 + 119897 sin 120579 119911119901 = 119911 + 119897 cos 120579 (2)

119897 = the distance from the pivot to the mass centre of thependulum (119909 119911) = the position of the pivot in the 119909119900119911coordinate ( ) = the speed in the119909119900119911 coordinate (119909119901 119911119901) =the position in the 119909101584011990010158401199111015840 coordinate (119901 119901) = the speed inthe 119909101584011990010158401199111015840 coordinate and 119892 = the acceleration constant dueto gravity It is assumed that the inertia of the pendulum isnegligibleNumerical values of all the parameters of IP systemare given in Table 1

The Lagrangersquos equations of the IP system are given asfollows

119889

119889119905

(

120597119871

120597119909

) minus

120597119871

120597119909

= 119865119909 (3a)

119889

119889119905

(

120597119871

120597120579

) minus

120597119871

120597120579

= 0 (3b)

where 119871 = 119870 minus 119875 By putting the expression of 119871 in (3a) and(3b) and after solving it Lagrangersquos equations of the IP systemcan be expressed as

(119872 + 119898) + 119898119897 cos 120579 120579 minus 119898119897 sin 120579 1205792 = 119865119909 (4a)

cos 120579 + 119897 120579 minus 119892 sin 120579 = 0 (4b)

where

minus05 le 119909 le 05 (5)

According to (4a) and (4b) the state equations of the IPsystem can be expressed as

1 = 1199092 (6a)

2 =

minus119898119892 cos1199093 sin1199093 + 119898119897 sin11990931199092

4+ 119865119909

119872+119898 sin21199093+ 1198891

(6b)

3 = 1199094 (6c)

4 =

minus119898119897 cos1199093 sin11990931199092

4minus cos1199093119865119909

119872119897 + 119898119897 sin21199093

+

(119872 + 119898) 119892 sin1199093119872119897 + 119898119897 sin21199093

+ 1198892

(6d)

Journal of Engineering 3

Derivative action

Proportional action

Integral action

KD

KP

KI

+

++Σ

s120583

E(s)

1s120582

U(s)

Figure 2 FOPID controller structure

where 1199091 = 119909 1199092 = 3 = 120579 1199094 = 120579 and 1198891 1198892 are the

external disturbancesAs it is known that IP system is a highly nonlinear

and unstable system Therefore accurate modelling of asystem having nonlinear dynamics cannot be obtained usingstandard linearization techniques Hence in this paper (6a)(6b) (6c) and (6d) are considered as it is without using anylinearization technique

3 Fractional Calculus and Fractional OrderPID Controller

31 Fractional Calculus Fractional calculus [16ndash18] is abranch of mathematics which deals with integration anddifferentiation operators that have fractional number powersThough these types of operators are complex in natureas compared to integer order operators they provide ageneralization which also includes integer order operators

The three most frequently used definitions are Riemann-Liouville Grunwald-Letnikov and Caputo [22] The mostcommon definition is known as Riemann-Liouville

119886119863120572

119905119891 (119905) =

1

lceil(119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

(7)

The second one is the Grunwald-Letnikov given as

119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

(119905minus119886)ℎ

sum

119895=0

(minus1)119895(

120572

119895)119891 (119905 minus 119895ℎ) (8)

where ( 120572119895 ) = lceil(120572 + 1)lceil(119895 + 1)lceil(120572 minus 119895 + 1)Finally Caputo expression is defined as

119886119863120572

119905119891 (119905) =

1

lceil(120572 minus 119899)

int

119905

119886

119891119899(120591) 119889120591

(119905 minus 120591)120572minus119899+1

(9)

where 119899 minus 1 lt 120572 lt 119899 119899 is an integer number 119886 is initialcondition

32 Fractional Order PID Controller PID controller [30 31]is one of the most extensively used controllers but in the lasttwo decades the advancement in fractional calculus has intro-duced the fractional order PID controller in control appli-cations PI120582D120583 controller is a generalized form of PID con-troller The PI120582D120583 controller structure is shown in Figure 2in which the introduction of two extra parameters 120582 and 120583

Integration

Derivative

PID

PD

PI

P

(0 1)

(0 0) (1 0)

(1 1)

Figure 3 Generalization of the PID controller

makes it complex as compared to PID controller because ofintroduction of fractional calculus in it

The differential equation of the PI120582D120583 controller [23] isdescribed as

119906 (119905) = 119870119875119890 (119905) + 119870119868119863minus120582119890 (119905) + 119870119863119863

120583119890 (119905) (10)

The generalization of PID controller is shown in Figure 3which can be obtained using different values of 120582 and 120583 in(10) PID (120582 = 1 120583 = 1) PI (120582 = 1 120583 = 0) PD (120582 = 0 120583 = 1)or P (120582 = 0 120583 = 0) are the special cases of PI120582D120583 controller

Taking Laplace transform of (10) the controller expres-sion in s-domain is obtained as

119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904120582+ 119870119863119904

120583 (11)

By putting 120582 = 120583 = 1 in (11) the expression of PIDcontroller can be written as

119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904

+ 119870119863119904 (12)

Hence in the present work PID controller is also studiedalong with the study of PI120582D120583 controller In addition theperformance comparisons with the PID controller basedon the same design specifications to show that the PI120582D120583controller has better performance in terms of performancesindex are carried out

In (11) 119904120582 and 119904120583 have fractional orders which are notdirectly compatible with MATLAB and it becomes difficultto realize hardware of PI120582D120583 controller Therefore there areseveral integer order approximation methods available forfractional order elements [32ndash34] The 5th order Oustalouprsquosinteger order approximation [32] in the frequency range(10minus2 102) rads is used in this work In MATLAB fractional

order PID controller is implemented using FOMCOM tool-box [35] where Oustalouprsquos approximation is realized

4 Particle Swarm Optimization

While solving complex optimization problems having largesearch space the population based swarm intelligencemethodis the widely accepted alternatives to find the optimalsolution The particle swarm optimization (PSO) was firstproposed by Kennedy and Eberhart [27] The PSO method[27ndash29] is population based search techniques used for


Start

Initialized the particle position and

Iter = 1

Evaluate the fitness

No

No

No

If

If

If

fitnesspbest(i) lt fitnesspbest (i minus 1)

Yes

Yes

Yes

Update the velocity and position

End

velocity

Evaluate gbest

Iter = Iter +1

pbest(i) = pbest(i minus 1)

pbest(i) = pbest(i)

gbest(iter) = gbest(iter)gbest(iter) = gbest(iterminus1)

iter = itermax

fitness = fitnesspbest

fitnesspbest(iter) lt fitnesspbest(iterminus1)

Figure 4 Flow diagram of basic PSO algorithm

solving the optimization problems having a large searchspaceThis techniquemimics the behaviour of bird flocks andfish schools and their collision-free and synchronized movesIndividual bird or fish is known as particle in a PSO systemand each particle has its position and velocity Now particlemoves inmultidimensional search space according to its ownexperience and the experience of the neighbouring particlesDuring the movement in search space the position andvelocity of the particles are updated There are three factorsinertia cognitive and social upon which the velocity andposition update of particles dependThe complete procedure

of PSO algorithm can be understoodwith the help of Figure 4which indicates that the PSO algorithm has three steps

(1) to evaluate the fitness value of each particle(2) to update individual best positions (pbest) and global

best positions (gbest) according to best or minimumfitness values

(3) to update velocity and position of each particle in eachiteration

The above steps are repeated until some stopping criteriaare met


Form

Referencer(t)

PI120582D1205832

+

+

+

+ +

+

PI120582D1205831

Control

u(t) Inverted pendulumsystem

W3

W1

W2

Cart positionx(t)

Pendulum angle120579(t)

workspace

workspace Form

To

Integrator

force

minus

Fx

workspace

Figure 5 Basic block diagram of closed loop control system using two PI120582D120583 controllers

Table 2 Parameters of fitness function

119879 1199081 1199082 1199083

30 1 1 005

Table 3 Values of PSO parameters

PSO Parameters ValuesNumber of particles 10Number of iterations 251198881 21198882

2119882max 09119882min 011198771 011198772 01

Table 4 Range of controller parameters

Controller parameters Range1198701198751 1198701198681 1198701198631 minus1198701198752 minus1198701198682minus1198701198632 0 501205821 1205831 1205822 1205832 0 1

Themodified velocity and position of each particle can becalculated using the current velocity and position as follows

119881119894

119896+1= 119882 times 119881119894

119896+ 1198881 times rand () times (119901best 119894 minus 119883119894

119896)

+ 1198882 times rand () times (119892best 119894 minus 119883119894119896)

(13)

Position update equation is given by

119883119896+1

119894= 119883119896

119894+ 119881119896+1

119894 (14)

where 119896 = iteration number119881119894 = velocity of 119894th particle119882 =inertia weight factor 1198881 1198882 = cognitive and social accelerationfactors respectively rand () = random numbers uniformlydistributed in the range (0 1) and119883119894 = position of 119894th particle

The expression for119882 is given by

119882 = 119882max minus [119882max minus119882min119896max

] lowast 119896 (15)

where119882max119882min = maximum and minimum values of119882respectively 119896max = maximum number of iterations

5 PI120582D120583 Controller Design Strategy

Themain objectives of controller design are as follows

(1) to stabilize the pendulum at its upright position(2) to uphold the cart position at the origin(3) tracking of desired position by pendulum cart(4) to use minimum control effort required to control the

pendulum angle and cart position

To achieve the abovementioned control objective twoPI120582D120583 controllers are used as shown in Figure 5 There aretwo feedback paths from the two outputs (pendulum angleand cart position) of the IP system and this feedback is given


Table 5 Values of controller parameters

Controller 119870119875

119870119868

119870119863

120582 120583 119869

PI120582D1205831

224734 18185 31337 03579 09038 17131PI120582D1205832 minus16749 minus07636 minus18977 04764 08908PID1 239288 12548 32143 mdash mdash 18449PID2 minus26622 minus13422 minus27675 mdash mdash

to PI120582D1205831and PI120582D120583

2 The output of each PI120582D120583 is added and

given as control input to the IP systemThe fitness function 119869 to beminimized using PSO is given

as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)

where 120579(119905) 119909(119905) and 119906(119905) are pendulum angle cart positionand control input respectively and 1199081 1199082 and 1199083 are theweights to give equal weightage to all parameters 119879 is thesimulation time used for running model in SIMULINK Thefitness function given by (16) has three terms The first termis the integral square of pendulum angle 120579(119905) which is usedto stabilize pendulum angle The second term is the integralsquare of cart position 119909(119905) which is used to stabilize cartposition Finally the third term is the integral square ofcontrol input 119906(119905) which is used to minimize the requiredcontrol force

Basic block diagram of closed loop control system asshown in Figure 5 is prepared in MATLABSIMULINKThismodel has ten unknown parameters of two PI120582D120583 controllersThese parameters are supplied by PSO program Initiallyparameters 1198701198751 1198701198681 1198701198631 1205821 1205831 and 1198701198752 1198701198682 1198701198632 1205822 1205832are generated randomly but later in terms of pbest and gbestby updating the velocity and position of particles in eachiteration After generating controller parameters SIMULINKmodel is executed according to PSO program When thismodel is executed the fitness value (as given in (16)) of theSIMULINK model is saved in MATLAB workspace whichis further utilized by PSO program for evaluating the mini-mum fitness value and corresponding controller parametersThe whole process is repeated until maximum number ofiteration is reached At the end of the process the values ofparameters 1198701198751 1198701198681 1198701198631 1205821 1205831 and 1198701198752 1198701198682 1198701198632 1205822 1205832are obtained which provides the desired performance of theIP system Also by considering 120582 = 120583 = 1 in both the PI120582D120583controllers integer order PID controllers have been designedusing the same specifications and comparative study has beencarried out to show the validity of the proposed work

6 Simulation and Results

Parameters of (16) are given in Table 2 As 120579(119905) and 119909(119905) lie inthe ranges minus05 05 rad and minus05 05m respectively andare given equal weight but 119906(119905) is given very less weightagebecause of its high range minus20 20N Therefore all threeterms of (16) are minimized equally

With the help of PSO parameters and controller param-eters given in Tables 3 and 4 the SIMULINK model shown

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)

Figure 6 Stabilization of inverted pendulum (without disturbances1198891 = 1198892 = 0)

in Figure 5 is executed by PSO program to obtain finalparameters for both PI120582D120583 and PID cases as given in Table 5

From Table 5 it can be concluded that based on the samespecifications as given in Tables 2 3 and 4 the fitness value 119869using PI120582D120583 controller is less as compared to PID controllerController parameters are calculated for stabilization control(without disturbances) but these values are also applicable inother cases Simulation results as shown in Figures 6ndash9 areobtained for stabilization and tracking control of IP systemwith and without disturbances All the simulation results areparticularized in next two subsections titled as stabilizationand tracking control of IP system

61 Stabilization of Inverted Pendulum For stabilizationreference cart position 119903(119905) = 0 In Figure 6 waveforms for120579(119905) 119909(119905) and 119906(119905) settle to steady state approximately atthe same time for both PI120582D120583 controller and PID controllerbut during transient period PI120582D120583 performs better than PIDcontroller Now to check the robustness of the designed


04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)

Figure 7 Stabilization of inverted pendulum (with disturbances1198891 = 1198892 = 20 sin(20120587119905))

PI120582D120583 and PID controllers disturbances are 1198891 = 1198892 =20 sin(20120587119905)

Figure 7 shows the simulation results with disturbanceswhich are still valid and proves the effectiveness of proposedPI120582D120583 andPID controllersThewaveforms for 120579(119905) and119906(119905) inFigure 7 in case of PI120582D120583 are better than PID controller butfor 119909(119905) PID perform slightly better than PI120582D120583 controllerAlso in the case of control input 119906(119905) PI120582D120583 provides lessdeviation during steady state as compared to PID controller

62 Tracking Control of Inverted Pendulum For track-ing control reference cart position 119903(119905) is considered as03sin(005120587t) Figure 8 shows the tracking control of IPsystem in which PI120582D120583 controller for 120579(119905) provides lessdeviation in transient period and settles earlier to steady stateas compared to PID controller In Figure 8 both PI120582D120583 andPID provide good tracking but in case of PI120582D120583 less controleffort is required In Figure 9 in the presence of disturbancesthe PI120582D120583 controller still outperforms PID controller and inPI120582D120583 case less control effort is required

Simulation results shown in Figures 6ndash9 are furthermoreimportant from real-time implementation viewpoint as inthis simulation study practical conditions have also beentaken into consideration

7 Conclusion

The stabilization and tracking control of IP system areattained successfully using PSO based direct tuning method

05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)

Figure 8 Tracking of inverted pendulum (without disturbances1198891 = 1198892 = 0)

Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)

Figure 9 Tracking of inverted pendulum (with disturbances 1198891=

1198892 = 20 sin(20120587119905))


The use of PSO technique for calculating controller param-eters is very simple and provides good convergence towardsoptimal values Two integer order PID controllers have alsobeen designed by keeping the same specifications A compar-ative study has been carried out and the obtained results arequite acceptable for both PI120582D120583 and PID controllers but thePI120582D120583 controller seems to be more robust The PI120582D120583 couldbe the good replacement for PID in the forthcoming yearsThe real time implementation of PI120582D120583controller might bethe subject of further research

Conflict of Interests

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

References

[1] N Muskinja and B Tovornik ldquoSwinging up and stabilizationof a real inverted pendulumrdquo IEEE Transactions on IndustrialElectronics vol 53 no 2 pp 631ndash639 2006

[2] T Agustinah A Jazidie andM Nuh ldquoHybrid fuzzy control forswinging up and stabilizing of the pendulum-cart systemrdquo inProceedings of the IEEE International Conference on ComputerScience and Automation Engineering (CSAE rsquo11) vol 4 pp 109ndash113 June 2011

[3] K J Astrom andK Furuta ldquoSwinging up a pendulumby energycontrolrdquo Automatica vol 36 no 2 pp 287ndash295 2000

[4] A Ghosh T R Krishnan and B Subudhi ldquoRobustproportional-integral-derivative compensation of an invertedcart-pendulum system an experimental studyrdquo IET ControlTheory amp Applications vol 6 no 8 pp 1145ndash1152 2012

[5] C-E Huang D-H Li and Y Su ldquoSimulation and robustnessstudies on an inverted pendulumrdquo in Proceedings of the 30thChinese Control Conference (CCC rsquo11) pp 615ndash619 July 2011

[6] L B Prasad B Tyagi andHOGupta ldquoModellingamp simulationfor optimal control of nonlinear inverted pendulum dynamicalsystem using PID controller amp LQRrdquo in Proceedings of the 6thAsia Modelling Symposium (AMS rsquo12) pp 138ndash143 2012

[7] M R Dastranj M Moghaddas S S Afghu and M RouhanildquoPID control of inverted pendulum using particle swarmoptimization (PSO) algorithmrdquo in Proceedings of the 3rd IEEEInternational Conference on Communication Software and Net-works (ICCSN rsquo11) pp 575ndash578 May 2011

[8] H Lee J Lee and J Lee ldquoHill climbing algorithm of an invertedpendulumrdquo in Proceedings of the IEEE International Symposiumon Computational Intelligence in Robotics and Automation(CIRA rsquo09) pp 574ndash579 December 2009

[9] S Li C Huo and Y Liu ldquoInverted pendulum system control byusing modified PID neural networkrdquo in Proceedings of the 3rdInternational Conference on Innovative Computing Informationand Control (ICICIC rsquo08) pp 1ndash426 June 2008

[10] J-J Wang ldquoSimulation studies of inverted pendulum based onPID controllersrdquo Simulation Modelling Practice andTheory vol19 no 1 pp 440ndash449 2011

[11] S Jung H-T Cho and T C Hsia ldquoNeural network controlfor position tracking of a two-axis Inverted pendulum systemexperimental studiesrdquo IEEE Transactions on Neural Networksvol 18 no 4 pp 1042ndash1048 2007

[12] S Omatu and S Deris ldquoStabilization of inverted pendulum bythe genetic algorithmrdquo in Proceedings of the IEEE Conference onEmerging Technologies and Factory Automation (ETFA rsquo96) vol1 pp 282ndash287 November 1996

[13] X-H Yang H-S Liu G-P Liu and G-F Xiao ldquoControlexperiment of the inverted pendulum using adaptive neural-fuzzy controllerrdquo in Proceedings of the International Conferenceon Electrical and Control Engineering (ICECE rsquo10) pp 629ndash632June 2010

[14] M-S Park and D Chwa ldquoSwing-up and stabilization control ofinverted-pendulum systems via coupled sliding-mode controlmethodrdquo IEEETransactions on Industrial Electronics vol 56 no9 pp 3541ndash3555 2009

[15] R-J Wai M-A Kuo and J-D Lee ldquoDesign of cascade adaptivefuzzy sliding-mode control for nonlinear two-axis inverted-pendulum servomechanismrdquo IEEE Transactions on Fuzzy Sys-tems vol 16 no 5 pp 1232ndash1244 2008

[16] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Springer Berlin Germany 2011

[17] R Caponetto G Dongola L Fortuna and I Petras FractionalOrder Systems Modeling and Control Applications vol 72 ofWorld Scientific Series on Nonlinear Science Series A WorldScientific Publishing Singapore 2010

[18] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Publishing Singa-pore 2012

[19] D Baleanu J H Asad and I Petras ldquoFractional-order two-electric pendulumrdquo Romanian Reports in Physics vol 64 no4 pp 907ndash914 2012

[20] I Podlubny ldquoFractional-order systems and PI120582D120583 controllersrdquoIEEE Transactions on Automatic Control vol 44 no 1 pp 208ndash214 1999

[21] I Podlubny I Petras B M Vinagre P OrsquoLeary and LDorcak ldquoAnalogue realizations of fractional-order controllersrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 281ndash296 2002

[22] C Yeroglu and N Tan ldquoNote on fractional-order proportional-integral-differential controller designrdquo IET Control Theory andApplications vol 5 no 17 pp 1978ndash1989 2011

[23] D Maiti A Acharya M Chakraborty A Konar and RJanarthanan ldquoTuning pid and PI120582D120575 controllers using theintegral time absolute error criterionrdquo in Proceedings of the 4thInternational Conference on Information and Automation forSustainability (ICIAFS rsquo08) pp 457ndash462 December 2008

[24] S Das S Das and A Gupta ldquoFractional order modeling ofa PHWR under step-back condition and control of its globalpower with a robust PI120582D120583 controllerrdquo IEEE Transactions onNuclear Science vol 58 no 5 pp 2431ndash2441 2011

[25] F Ikeda and S Toyama ldquoFractional derivative control designs byinhomogeneous sampling for systemswith nonlinear elementsrdquoin Proceedings of the SICE Annual Conference (SICE rsquo07) pp1224ndash1227 September 2007

[26] S K Mishra and D Chandra ldquoStabilization of inverted cart-pendulum system using PI120582D120583 controller a frequency-domainapproachrdquo Chinese Journal of Engineering vol 2013 Article ID962401 7 pages 2013

[27] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[28] R Eberhart Y Shi and J Kennedy Swarm Intelligence MorganKaufmann San Mateo Calif USA 2001


[29] S Yang MWang and L Jiao ldquoA quantum particle swarm opti-mizationrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation (CEC rsquo04) vol 1 pp 320ndash324 2004

[30] K Ogata Modern Control Engineering Prentice-Hall UpperSaddle River NJ USA 2002

[31] K A Astrom and T Hagglund PID Controllers Theory Designand Tuning Instrument Society of America Research TrianglePark NC USA 1995

[32] A Oustaloup F Levron B Mathieu and F M NanotldquoFrequency-band complex noninteger differentiator charac-terization and synthesisrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 47 no 1pp 25ndash39 2000

[33] D Valerio and J S da Costa ldquoTime-domain implementationof fractional order controllersrdquo IEE Proceedings Control Theoryand Applications vol 152 no 5 pp 539ndash552 2005

[34] G E Carlson and C A Halijak ldquoApproximation of fractionalcapacitors (1s) by a regular Newton processrdquo IRE Transactionson Circuit Theory vol 11 no 2 pp 210ndash213 1964

[35] A Tepljakov E Petlenkov and J Belikov ldquoFOMCONfractional-order modeling and control toolbox for MATLABrdquoin Proceedings of the 18th International Conference on MixedDesign of Integrated Circuits and Systems (MIXDES rsquo11) pp684ndash689 June 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



z

oM

x

m120579

l

l

z998400

o998400 x998400

Fx

Figure 1 Structure of inverted pendulum

studies [20ndash24] the PI120582D120583 controller gives better outcomesthan PID controller Though there are some applications ofPI120582D120583 controller for IP system [25 26] PI120582D120583 controllerhas not received considerable attention for unstable systemssimilar to IP system

Hence in the present work fractional order PID con-troller is designed in time domain to control pendulum angleas well as cart position Fractional order PID controller ischallenging to design because of the use of fractional calculuswith very complex calculations Therefore a direct approachis used for calculating the parameters of both fractionalorder PID controllers with the help of a multiobjectivefitness function (the fitness function consists of the sum ofintegral squares of pendulum angle cart position and controlvoltage) The fitness function is minimized by running themodel according to a particle swarm optimization (PSO)[27ndash29] program in MATLAB PSO is used in this work asit provides greater convergence towards optimal values ascompared to other optimization techniques and it has thesimple algorithm

The rest of the paper is divided into the following sectionsSection 2 gives a description of the inverted pendulum systemand derivation of system equations in state space formSection 3 describes the fractional calculus and structure of thePI120582D120583 controller Section 4 gives details about PSO Section 5gives a complete description of control strategy Section 6gives a comparison of the simulation results for PI120582D120583controller and PID controller with and without disturbancesThe paper ends with the conclusions in Section 7 which isfollowed by the references

2 Inverted Pendulum System

As shown in Figure 1 an IP system has a cart which canmovehorizontally One end of the pendulum rod is connected tothe centre of the upper surface of the cart which is called thepivot point while the other end is free to move in verticalplane (119909119911-plane) This pendulum rod is stable in extremedownwards position and known as normal pendulum Butwhen the pendulum rod remains in upright position it isknown as IP system This is an unstable condition whichneeds a continuous balancing force (119865119909) on cart in order toremain upright

In Figure 1 horizontal force is used as control action todisplace the pivot around 119909 axis and the total kinetic energy

Table 1 Inverted pendulum parameters

119872 (Kg) 119898 (Kg) 119897 (m) 119892 (ms2)1 01 03 98

(119870) due to mass of the pivot (119872) in 119909 direction mass ofpendulum rod (119898) in 119909 and 119911 directions and potential energy(119875) of the IP system [10] which are

119870 =

1

2

1198722+

1

2

119898 (2

119901+ 2

119901) 119875 = 119898119892119911119901

(1)

where

119909119901 = 119909 + 119897 sin 120579 119911119901 = 119911 + 119897 cos 120579 (2)

119897 = the distance from the pivot to the mass centre of thependulum (119909 119911) = the position of the pivot in the 119909119900119911coordinate ( ) = the speed in the119909119900119911 coordinate (119909119901 119911119901) =the position in the 119909101584011990010158401199111015840 coordinate (119901 119901) = the speed inthe 119909101584011990010158401199111015840 coordinate and 119892 = the acceleration constant dueto gravity It is assumed that the inertia of the pendulum isnegligibleNumerical values of all the parameters of IP systemare given in Table 1

The Lagrangersquos equations of the IP system are given asfollows

119889

119889119905

(

120597119871

120597119909

) minus

120597119871

120597119909

= 119865119909 (3a)

119889

119889119905

(

120597119871

120597120579

) minus

120597119871

120597120579

= 0 (3b)

where 119871 = 119870 minus 119875 By putting the expression of 119871 in (3a) and(3b) and after solving it Lagrangersquos equations of the IP systemcan be expressed as

(119872 + 119898) + 119898119897 cos 120579 120579 minus 119898119897 sin 120579 1205792 = 119865119909 (4a)

cos 120579 + 119897 120579 minus 119892 sin 120579 = 0 (4b)

where

minus05 le 119909 le 05 (5)

According to (4a) and (4b) the state equations of the IPsystem can be expressed as

1 = 1199092 (6a)

2 =

minus119898119892 cos1199093 sin1199093 + 119898119897 sin11990931199092

4+ 119865119909

119872+119898 sin21199093+ 1198891

(6b)

3 = 1199094 (6c)

4 =

minus119898119897 cos1199093 sin11990931199092

4minus cos1199093119865119909

119872119897 + 119898119897 sin21199093

+

(119872 + 119898) 119892 sin1199093119872119897 + 119898119897 sin21199093

+ 1198892

(6d)


Derivative action

Proportional action

Integral action

KD

KP

KI

+

++Σ

s120583

E(s)

1s120582

U(s)


where 1199091 = 119909 1199092 = 3 = 120579 1199094 = 120579 and 1198891 1198892 are the






119886119863120572

119905119891 (119905) =

1

lceil(119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

(7)


119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

(119905minus119886)ℎ

sum

119895=0

(minus1)119895(

120572

119895)119891 (119905 minus 119895ℎ) (8)


119886119863120572

119905119891 (119905) =

1

lceil(120572 minus 119899)

int

119905

119886

119891119899(120591) 119889120591

(119905 minus 120591)120572minus119899+1

(9)



Integration

Derivative

PID

PD

PI

P

(0 1)

(0 0) (1 0)

(1 1)




119906 (119905) = 119870119875119890 (119905) + 119870119868119863minus120582119890 (119905) + 119870119863119863

120583119890 (119905) (10)



119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904120582+ 119870119863119904

120583 (11)


119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904

+ 119870119863119904 (12)







Start


Iter = 1


No

No

No

If

If

If


Yes

Yes

Yes


End

velocity

Evaluate gbest

Iter = Iter +1


pbest(i) = pbest(i)


iter = itermax











Form

Referencer(t)

PI120582D1205832

+

+

+

+ +

+

PI120582D1205831

Control


W3

W1

W2

Cart positionx(t)


workspace

workspace Form

To

Integrator

force

minus

Fx

workspace



119879 1199081 1199082 1199083

30 1 1 005



2119882max 09119882min 011198771 011198772 01




119881119894

119896+1= 119882 times 119881119894


119896)


(13)


119883119896+1

119894= 119883119896

119894+ 119881119896+1

119894 (14)




] lowast 119896 (15)










119870119868

119870119863

120582 120583 119869

PI120582D1205831


to PI120582D1205831and PI120582D120583



as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)






05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)






04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Derivative action

Proportional action

Integral action

KD

KP

KI

+

++Σ

s120583

E(s)

1s120582

U(s)


where 1199091 = 119909 1199092 = 3 = 120579 1199094 = 120579 and 1198891 1198892 are the






119886119863120572

119905119891 (119905) =

1

lceil(119899 minus 120572)

119889119899

119889119905119899int

119905

119886

119891 (120591) 119889120591

(119905 minus 120591)120572minus119899+1

(7)


119886119863120572

119905119891 (119905) = lim

ℎrarr0

1

ℎ120572

(119905minus119886)ℎ

sum

119895=0

(minus1)119895(

120572

119895)119891 (119905 minus 119895ℎ) (8)


119886119863120572

119905119891 (119905) =

1

lceil(120572 minus 119899)

int

119905

119886

119891119899(120591) 119889120591

(119905 minus 120591)120572minus119899+1

(9)



Integration

Derivative

PID

PD

PI

P

(0 1)

(0 0) (1 0)

(1 1)




119906 (119905) = 119870119875119890 (119905) + 119870119868119863minus120582119890 (119905) + 119870119863119863

120583119890 (119905) (10)



119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904120582+ 119870119863119904

120583 (11)


119862 (119904) =

119880 (119904)

119864 (119904)

= 119870119875 +

119870119868

119904

+ 119870119863119904 (12)







Start


Iter = 1


No

No

No

If

If

If


Yes

Yes

Yes


End

velocity

Evaluate gbest

Iter = Iter +1


pbest(i) = pbest(i)


iter = itermax











Form

Referencer(t)

PI120582D1205832

+

+

+

+ +

+

PI120582D1205831

Control


W3

W1

W2

Cart positionx(t)


workspace

workspace Form

To

Integrator

force

minus

Fx

workspace



119879 1199081 1199082 1199083

30 1 1 005



2119882max 09119882min 011198771 011198772 01




119881119894

119896+1= 119882 times 119881119894


119896)


(13)


119883119896+1

119894= 119883119896

119894+ 119881119896+1

119894 (14)




] lowast 119896 (15)










119870119868

119870119863

120582 120583 119869

PI120582D1205831


to PI120582D1205831and PI120582D120583



as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)






05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)






04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Start


Iter = 1


No

No

No

If

If

If


Yes

Yes

Yes


End

velocity

Evaluate gbest

Iter = Iter +1


pbest(i) = pbest(i)


iter = itermax











Form

Referencer(t)

PI120582D1205832

+

+

+

+ +

+

PI120582D1205831

Control


W3

W1

W2

Cart positionx(t)


workspace

workspace Form

To

Integrator

force

minus

Fx

workspace



119879 1199081 1199082 1199083

30 1 1 005



2119882max 09119882min 011198771 011198772 01




119881119894

119896+1= 119882 times 119881119894


119896)


(13)


119883119896+1

119894= 119883119896

119894+ 119881119896+1

119894 (14)




] lowast 119896 (15)










119870119868

119870119863

120582 120583 119869

PI120582D1205831


to PI120582D1205831and PI120582D120583



as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)






05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)






04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Form

Referencer(t)

PI120582D1205832

+

+

+

+ +

+

PI120582D1205831

Control


W3

W1

W2

Cart positionx(t)


workspace

workspace Form

To

Integrator

force

minus

Fx

workspace



119879 1199081 1199082 1199083

30 1 1 005



2119882max 09119882min 011198771 011198772 01




119881119894

119896+1= 119882 times 119881119894


119896)


(13)


119883119896+1

119894= 119883119896

119894+ 119881119896+1

119894 (14)




] lowast 119896 (15)










119870119868

119870119863

120582 120583 119869

PI120582D1205831


to PI120582D1205831and PI120582D120583



as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)






05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)






04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119870119868

119870119863

120582 120583 119869

PI120582D1205831


to PI120582D1205831and PI120582D120583



as

119869 = int

119879

0

[1199081 times 1205792(119905) + 1199082 times 119909

2(119905) + 1199083 times 119906

2(119905)] 119889119905 (16)






05

0

minus050 2 4 6 8 10 12 14 16 18 20

Ang

le (r

ad)

Time (s)

(a)

05

0

minus050 2 4 6 8 10 12 14 16 18 20

Time (s)Po

sitio

n (m

)(b)

20100

minus10

PIDFOPID

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Con

trol f

orce

Fx

(c)






04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










04020

minus02

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Ang

le (r

ad)

(a)

05

0

minus05Posit

ion

(m)

0 5 10 15 20 25 30 35 40 45 50

Time (s)

(b)

PIDFOPID

0 5 10 15 20 25 30 35 40 45 50

Time (s)

20

10

0

minus10

Con

trol f

orce

Fx

(c)






7 Conclusion


05

0

minus050 10 20 30 40 50 60 70 80 90 100

Ang

le (r

ad)

Time (s)

(a)

Posit

ion

(m)

0604020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

PIDFOPID

0 10 20 30 40 50 60 70 80 90 100

Time (s)

Con

trol f

orce

Fx

(c)


Ang

le (r

ad) 06

04020

minus02minus04

0 10 20 30 40 50 60 70 80 90 100

Time (s)

(a)

Posit

ion

(m) 05

0

minus050 10 20 30 40 50 60 70 80 90 100

Time (s)

(b)

151050

minus5minus10

0 10 20 30 40 50 60 70 80 90 100

Time (s)

PIDFOPID

Con

trol f

orce

Fx

(c)


1198892 = 20 sin(20120587119905))





References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













References







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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