optimal pid controller design of an inverted pendulum dynamics a hybrid pole-placement and firefly...

8/18/2019 Optimal PID Controller Design of an Inverted Pendulum Dynamics a Hybrid Pole-placement and Firefly Algorithm A…

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Optimal PID controlA Hybrid Pole-

N. Surendranath Reddy and Srinivas

Department of Electronics & CommunicSMIT, Majitar, Sikkim, Indi

[email protected] , srinivasa.sake

Abstract — The solution of control for inverchallenging problem and many control tectested on this bench mark problem to testdesigned control law. In this note an attemptstabilize a linear dynamics of inverted penduloptimal PID controller that guarantees theperformance specifications. To accomplishpole-placement and firefly optimization conbeen evolved. To apply the control algorithm,bound of the controller gain is computed uscriteria and later the lower bound of the contout using pole-placement technique. Onccontroller gains are obtained, the optimized vafound using firefly optimization technique.

Keywords- Inverted Pendulum Firefly optimiza PID control

I. I NTRODUCTION An inverted pendulum system consists

which is free to oscillate around a fixed pivoa movable cart. The objective is to maintaithe vertically upright position. However, thunstable and non-linear in nature; thereby ocontrol challenges. Different approaches hsolve this problem. In [1], a Bang-Bang stalgorithm is used to swing up the pendulu

position and Linear Quadratic control metthus. A high speed fuzzy controller that cinverted pendulums simultaneously by usinrules is presented in [2]. In [3], a fuzzy-louses vision feedback for guiding the controllmotion of the cart to stabilize the system isneural network is trained using reinforcedifference techniques to balance the invertEnergy control method which utilizes a serv

to control the pivot acceleration for swinginverted pendulum is presented in [5]. A notheorem is used to show that non-linear, feecan be controlled and is applied to the inv[6]. The architecture in [7] employs a f compensates the plant’s nonlinearities a

pendulum in inverted position.

To compensate for errors in the fuzzy systeinput is used. PID control is anotherapproach; however it involves a challengecontroller parameters. The self-tuning

er design of an Inverted Pendululacement & Firefly Algorithm A

Saketh M

ation Engg.

[email protected]

Pikaso Pal aDept. Electri

NIT Agartala [email protected] ,

ted pendulum is aniques have beenhe efficacy of thehas been made tom by designing aniven time-domainhe task a hybridrol algorithm hasinitially the upperng Routh-Hurwitzoller gain is found

e the ranges oflue of the gains are

ion Pole-placement

of a pendulum,t point attached to

the pendulum insystem is highly

fering a variety ofave been used toate type feedbackm to the invertedod to maintain ituld stabilize two

g a set of controlgic controller thater in manipulating

presented. In [4],aent and temporaled pendulum. An

design technique

up control of an-linear small gain

d forward systemsrted pendulum in

uzzy system thatd stabilizes the

a sliding controlcommonly usedf optimization ofethod for PID

controllers proposed in [8] utistability. In [9], a potentiomecontroller, tuned by trial andcontrol signal. In the comparaPID control methods in [10](ZN) approach is used to tunetuning method of PID and theare compared; FA tuned syresponse and performance inthe PID control design tech

placement technique and the ntechnique of Firefly Algorithcontroller parameters. The polachieve pre-defined perfor considered system. The FA op

by the natural biological phenoan act of signaling, intendeexplained in [12-17].

The paper is organizeddynamics of the inverted pendetail. In Section III, PID cochallenges are discussed. Sectiof PID controllers using FA anSection V concludes this work

II. LINEARIZEDIn this work we consider th

pendulum of Feedback make [1[19] is shown in Fig. 1. Weupright position as highlightedthe dynamics of this experimforces acting on the system un[18] and [19].

Fig. 1. Cart-pendulu

Dynamics: proach

nd Rajeeb Deycal Engineeringand Silchar, [email protected]

lizes the concept of Lyapunover is used for sensing and PDrror method, for providing theive study of Sliding Mode and, the classical Ziegler-Nicholsthe PID controller. In [11], ZNrecent Firefly Algorithm (FA)

stem had better steady stateices. This paper amalgamates

nique with a traditional poleew heuristic based optimization

(FA) for optimization of PID placement technique is used toance specification of the

timization approach is inspiredmenon of firefly flash, which isd to attract other fireflies as

s follows. In Section II, thedulum system are discussed introller design and optimizationn IV addresses the optimizationd describes the obtained results.

YNAMICS OF SYSTEM

experimental set up of inverted9]. The cart-pendulum set up ofwill operate the pendulum in

in previous section. To computental set up we analyze variouser equilibrium and be found in

experimental set up.

2016 IEEE First International Conference on Control, Measurement and Instrumentation (CMI)

978-1-4799-1769-3/16/$31.00 ©2016 IEEE 305


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pendulum has its CG at the center of the pemovable in one plane only. The equatiohorizontal motion of the cart is:

( ) f t M x b x N = + +

Where ( ) f t is the force applied on the cart,force due to friction, N is reaction force ofis the inertial force due to mass of the cart.the cart is not assumed.The rotational motion of the pendulum abou

cos sin I d N l P l θ + θ + θ = θ The horizontal motion of CG of the pendulu

2cos - sin N m x m l m l= + θ θ θ θ The vertical motion of CG of the pendulum i

2- sin - cos P mg m l m l = θ θ θ θ Substituting for in the equation (1),

( ) ( ) cos - f t M m x b x m l m= + + + θ θ θ

Substituting for N and P in the equation (2)

( )2

cos - sin I ml d m l x mg l + θ + θ + θ θ

A. LinearizationConsidering small angles of deviation θ , lias follows:The Taylor’s series expansion of ( ) f θ is giv

( ) ( ) ( ) ( ) (

( ) ( )

0 0 0

3

0 0

--

-

3!

f f f

f HOD

θθ = θ + θ θ θ +

θ θ θ+

′

′′′+

Neglecting the higher order derivatives, th(9) are obtained:

( ) ( ) f t M m x b x m l= + + + θ

( )2 - 0 I ml d m l x m g l+ θ + θ + θ = The Laplace transform of equations (8) & (9

( ) ( ) ( ) ( )2 F s X s s M m sb s = + + + θ

( ) ( ) ( )2 2 2- s s I ml sd mgl X s s θ + + + B. Transfer Function

In the inverted pendulum system, ( ) f t is t

( )t θ are the outputs. Transfer function analconsidering both outputs independently

( )( )

( )2

3 2

)( ) ( ) ] [

( )] [ ( )]

[({ (

2 2

2 4m ml b

d M m

I + ml s ds - mgl X s=

F s I +

s bd mgl

ml s

M m s

+

+ − +

+ + − +

( )( ) 2)( ) ( ) ]

( ) ] [ (

([{ 2 3

2

s -mls=

F s I + ml sm ml

M m d bd mg s l M m

+ − +

+ + − +

dulum bob and isn describing the

(1)

b x is the restoringhe pendulum, Mx

ertical motion of

its mass is:

(2)is:

(3)s

(4)

2 sinl θ (5),

0= (6)

earize the system

n by:

) ( )20 0''2!

f θ

(7)

equations (8) &

(8)

(9)

yields2 s ml (10)

0ml = (11)

he input, ( ) x t and

ysis is performeds done in [10].

)

glb }

2+ ml

m s

+

−

(12)

[( )

)] glb}

2 I + bml

s m

+

−

(13)

The parameter values of invert be found in [19]. SubstitutionTable I in equation (12) & (13

( )( )

2

4

0.12880.3319 0.019

X s s F s s

+=

+

( )( ) 3 2-4.008 -0 .236 s s

F s s s

θ=

Fig. 2 & 3 show the opemodels (14) & (15) respectisystem is unstable under open-lunbounded.

Fig. 2.Angular displacement (pendu

Fig. 3. Displacement of COG (pend

III. P ID CONA PID controller is used tointroduced in the feed forward

Let (

( s

F s

The transfer function of the plone can write,

3 2

sG(s)

-4.008s -0.23s +25=

The transfer function of a PID

( )( )( ) c p

F s= G s = K +

E s

The closed loop transfer functias:

( ) ( ) ( )( ) (( )

c

c

G s G s(s)= T s =

R s 1+ G s G sθ

From equations (16), (17) & (1

ed pendulum in (12) & (13) canf parameter values specified in

) give (14) & (15) respectively,

3 2

0.005 - 0.8122- 2 .136 - 0.0406

s s s s

(14)

25.79 0.490 s+ + (15)

n-loop response of the systemely, to a unit step input. Theoop condition as the response is

lum) on application of unit step input

lum) on application of unit step input

TROLLER DESGN

stabilize this system and is path as shown in Fig. 4 [9].

))

( )G s=

ant shown in Fig.5. From (15),

.79s+0.490 (16)

controller is given by

id + K s s

(17)

on of the system can be written

) (18)

8), we can write:


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Fig. 4 Block diagram of closed loop system wit

( ) (( )

[

]

2 p

3 2d

i

i d K s + K + K sT(s)=- 4.008s - 0.236 - K s -25.

- -0.490- K

−

The characteristic equation obtained from eq

( )

( ) ( )

3

i

2d

p

-4.008s - 0.236 - K s

- -25.79 - K s - -0.490 - K = 0

It is a necessary condition for the stability ocoefficients of its characteristic equatiTherefore, from (21), (22) & (23) we oranges of d p i K ,K & K respectively:

, p i d K < -25.79, K < -0.490 K < 0.236

IV. MAIN RESULTS

A. Firefly OptimizationFA wasformulated by Xin-She Yang in 2007

• All fireflies are unisexual, so thatattracted to all other fireflies.

• Attractiveness is proportional to thfor any two fireflies, the duller on

by (and thus move to) the brighter brightness can decrease as their dist

• If there are no fireflies brighter thawill move randomly.

The brightness should be associated wifunction so that it may be optimized.attractiveness is proportional to the light intadjacent fireflies, the variation of attracti

distance r is given by,2r e−γ οβ = β where οβ is

at 0r = . The distance r or ijr between any t

j at positions i x and j x respectivel

ij i jr x x= − . The movement of a firely

brighter firefly j is dete

( )2

1 ijr t t t t t i i j i t i x x e x x

−γ +ο= + β − + α ε , where th

due to the attraction. The third term is randoas the randomization parameter, and t iε is anumbers drawn from a gaussian distribdistribution at time t . If 0οβ = , it becomeswalk. The γ determines the variation of aincreasing distance from communicatecorresponds to no variation and reduces to aswarm optimization. As t α controls the

PID controller

) p79 - K s (19)

uation (19) is:

(20)

a system that theon be positive.tained following

(21)

by assuming:ne firefly will be

ir brightness, andwill be attracted

one; however, theance increases.a given firefly, it

th the objectiveAs a firefly’s

ensity seen by theveness β with the

the attractiveness

wo fireflies i and

is given by

i , attracted to a

rmined by,

e second term is

mization with t αvector of randomtion or uniforma simple random

ttractiveness withfirefly. 0γ =

variant of particlerandomness, this

parameter can be tuned to vaduring iterations as given by α

B. Proposed Firefly optimized In this paper, among variousused for evaluating the PID cocode for implementation of FAStep1: Initialize the algorithmalpha, beta and gamma).Step2: Specify the lower acontroller parameters. The upthe Routh-Hurwitz criteria,approximated from the poledesign requirements.Step3: Define the transfer funand calculate the error.Step4: Define ITSE a

( ) ( )1 2, , , , d f x x x x x= … .The aim is to minimize ( ) f x

Step5: Generate an initi( )1, 2, ,i x i n= … .Step6: Determine the light intStep7: While ( t < MaximumFor 1:i n= (all n fireflies), Foif ( ) j i I I > Move fireflies i and j accordiEvaluate new solutions and unext iteration. Check whethewithin the limits/bounds.End for j

End for i Sort the fireflies to find the preEnd while. Begin post proceindicated in [11, 13, 14, 17].

To apply FA its parametersvalues are as shown in Table Irange of d p i K ,K & K to be ch

determined using Routh-Hurw(21). The lower bound will betechnique and further the optiwill be obtained using Firefly o

TABLE I. PAR Number offireflies (n)

Maxgeneration

5 100

C. Hybrid Pole-placement & parameters for Angular co

The upper bound of Kp, Kicontrol has been determined

bound of controller gain is pr pole-placement technique, a

y with the iteration counter t t

t ο= α δ , where 0 1< δ < .

PID Contrller algorithm performance criteria, ITSE isntroller parameters. The pseudois given below,

parameters (number of fireflies,

d upper bounds of the three per bound is determined using

and the lower bound islacement technique as per the

ction of the closed loop system

s the objective function

performance index).

l population of n fireflies

ensity i I of the fireflies.eneration)

r 1: j i= (all n fireflies)

g to their attractiveness.date the light intensity for thethe new updated fireflies lie

sent bestss on best results obtained as

need to be initialized and itsI. Application of FA requires asen. The upper bound has been

itz stability criteria in equationdetermined by pole-placement

mized value of controller gain ptimization algorithm.

METERS OF FAAlpha Beta Gamma

0.5 0.2 0.5

A for optimization of PIDtrolKd values of PID for angularin equation (21). The lower

posed to be determined by thecording to the given design


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specifications as follows. The closed loop characteristicequation is given by:

( ) ( ) ( )3 2 -25.79-0.236 - -0.49 - 04.008 4.008 4.008

pd i K K K

s s s+ + + = (24)

The desired characteristic equation of closed loop system is:

( )3 2 2 2 32 1 2 )( 0n n n s m s m s m+ ζω + ζ ω ζω + + + = (25)

Comparing equation (24) and (25), we get:( )( )

0.236 -2

4.008d

n

K m= + ζω (26)

( ) 2 2-25.79- (1 2 )4.008

p

n

K m= + ζ ω (27)

( ) 3-0.49 -4.008

in

K m= ζω (28)

The design specifications to meet the unit step response of thesystem are assumed as:

• A settling time less than 5s.• An Overshoot less than 25% and

• A Rise time less than 1s.Hence, choosing some desired values of , & n mζ ω , the tunedvalues of , & p i d K K K can be obtained from the equations

(26), (27) & (28), as per pole-placement technique. The valuesthus obtained are shown in Table III. A case study was donewith six different combinations of , &n mζ ω values. Theangular displacement of pendulum for these cases is shown inFig. 5. Values of time response specifications were obtainedfrom the simulation results of the case study and are shown inTable IV. From Table III, two ranges of , & p i d K K K values

were chosen and FA was applied for these ranges byconsidering ITSE the performance criteria. Table V. shows theoptimal values of , & p i d K K K . After optimizing the values

of , & p i d K K K using FA, a unit step input is given to the

closed loop system shown in Fig. 5. The time response plot isshown in Fig. 7.

TABLE II. THE VALUES OF PID PARAMETERS OBTAINED FROMPOLE-PLACEMENT

S N Closed loop parameters p

K i K d K

1 0.8,ζ = 1.5,nω = 9m = -202.90 -97.88-52.67

2 0.8,ζ = 1.5,nω = 5m = -92.52 -11.31-

33.433 0.8,ζ = 1,nω = 15m = -106.75 -48.59

-54.27

4 0.5,ζ = 1.2,nω = 20m = -89.28 -69.75-52.67

5 0.5,ζ = 1.5,nω = 9m = -75.39 -61.36-32.83

6 0.5,ζ = 1,nω = 9m = -47.83 -18.53-21.80

D. Hybrid Pole-placement & FA for optimization of PID parameters for Cart-Position control

The upper bound has been determined using Routh-Hurwitzstability criteria in equation (21). The lower bound can bedetermined by the pole-placement technique, according to thedesign requirements as follows:

0 5 1 0 1 5 2 0 2 5 3 00. 0

0. 2

0. 4

0. 6

0. 8

1. 0

1. 2

1. 4

1. 6

1. 8

2. 0

2. 2

C a s e s t u d y : 1 C a s e s t u d y : 2 C a s e s t u d y : 3 C a s e s t u d y : 4 C a s e s t u d y : 5 C a s e s t u d y : 6

T i m e ( s )

A n g l e

Fig. 5. Angle of deviation for different values of closed loop parameters

( )( )

2 2

2 42 2

3 2

))( ) ( ) ] [ ( )

( )] [ ( )]

({[(

- glb }

X s I ml s F s I ml s ml

ds mgl M m ml b I

d M m bd mgl M m s s m s

+ −+ − + +

+

+

+

+=

+ − + For position control,

( )( ) 2

5.841 X s

F s s= (29)

The closed loop characteristic equation is given by3 25.841 5.841 5.841 0d p i s s K sK K + + =+ (30)

The desired closed loop characteristic equation is given by3 2 2 2 3(2 ) (1 2 ) 0n n nm m s m s sζω ζ ω + + ζω + + + = (31)

Comparing equation (30) and (31), we get(2 ) 5.841n d m K + =ζω (32)

2 2 5.841( 2 )1 n pm K =ζ ω + (33) 3 5.841n im K ζ =ω (34)

TABLE III. TIME DOMAIN PERFORMANCE SPECIFICATIONS FORDIFFERENT DESIRED VALUES FOR THE CLOSED LOOP SYSTEM

ParamCase

Study:1

CaseStudy:2

CaseStudy:3

CaseStudy:

4

CaseStudy:

5

CaseStudy:

6

R.T 0.33 0.53 0.56 0.60 0.42 0.48S.T 4.84 8.74 5.35 6.42 5.24 8.20

%OS 21.62 22.7536.5

1 47.30 57.27 86.86

US 0 0 0 0 0 0Peak 1.23 1.40 1.38 1.48 1.58 1.96P.T 1.04 1.89 1.92 1.75 1.40 1.98


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TABLE IV. TIME DOMAIN PERFORMANCE SPECIFICATIONSOBTAINED USING FA FOR DIFFERENT RANGES

Param

Range: 1-100[ - 25.79, p to K =

-100 -0.49,i to K =-100 0.236]d to K =

Range: 2-200[ - 25.79, p to K =

-100 - 0.49,i to K =-100 0.236]d to K =

p K -99.9987 -190.6896

i K -86.7029 -99.6066

d K -46.3397 -66.6712R.T 0.4553 0.4328S.T 5.2691 4.7834Peak 1.4569 1.2454P.T 1.4530 1.4067%OS 44.8910% 23.9302%

US 0 0Legend: R.T = Rise time, S.T = Settling time, P.T = Peaktime, OS = Overshoot, US = Undershoot.

0 5 1 0 1 5 2 0 2 5 3 00 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

R a n g e : 1 R a n g e : 2

T i m e ( s )

A n g

l e

Fig. 6 Angular displacement of pendulum with for the two ranges of

, & p i d K K K values specified in TABLE V.

Choosing some desired values of , &n mζ ω , the tuned valuesof , & p i d K K K can be obtained from the equations (32), (33)

& (34), as per pole-placement technique. The values thusobtained are shown in Table VI. The simulation results viz.

position of COG of pendulum with application of step input,for the case study of different values of closed loop parametersare shown in Fig. 6. Measurements of time responsespecifications were obtained from the simulation resultsshown in Fig. 7 and are presented in Table IV.

0 1 2 3 4 5 6 7 8 9 100. 0

0. 2

0. 4

0. 6

0. 8

1. 0

1. 2

1. 4

T i m e ( s )

Case s tudy :1 Case s tudy :2 Case s tudy :3 Case s tudy :4 Case s tudy :5 Case s tudy :6 Case s tudy :7

P o s i t i o n

Fig. 7 Position of COG of pendulum for various values of PID parameters

The ITSE is considered as the performance criteria for FA andis applied to two ranges of , & p i d K K K values obtained from

Table VI. The optimal values of , & p i d K K K obtained by

minimizing the ITSE are shown in Table VIII.

V. R OBUSTNESS STUDY

Considering the same design specification in section IVC after

(28) we vary the length of the inverted pendulum by 10% thanthat of the nominal value an attempt was made to stabilize thesystem by meeting the desired criteria using the proposedmethod as well as LQR technique. The Table IX below depictsthat the proposed method could meet the desired specificationfrom the ranges of the gain calculated by satisfying the Routh-Hurwitz criteria and pole-placement technique. Also for thiscase we have compared the transient response obtained by the

proposed method as well as LQR technique which is placed inFig. 8.

VI. CONCLUSION In this paper an optimal PID controller is designed adopting ahybrid control structure. The design technique combines thetraditional techniques in control theory and evolutionaryalgorithm to stabilize a cart-inverted pendulum dynamics. Theresult obtained for the system is convincing using the

proposed hybrid control law for a preliminary robustnessstudy. As a future work the control algorithm will be modifiedto develop a robust PID controller capable to tackle parametricuncertainties and inclusion of delay in the system.

TABLE V: CLOSED LOOP PARAMETERS FROM POLE-PLACEMENT

TABLE VI. TIME DOMAIN PERFORMANCE SPECIFICATIONS FORDIFFERENT DESIRED VALUES FOR THE CLOSED LOOP SYSTEM

Par am

CaseStud

y:1

CaseStudy

:2

CaseStud

y:3

CaseStud

y:4

CaseStud

y:5

CaseStudy

:6

CaseStud

y:7R.T 0.67 0.27 0.20 0.2005 0.21 0.17 0.18

S.T 4.51 1.747 1.31 1.35 1.30 1.04 1.58OS 20.0 19.45 19.4 20.7 24.0 23.9 33.3US 0 0 0 0 0 0 0Peak 1.21 1.19 1.19 1.20 1.24 1.24 1.33

P.T 1.86 0.74 0.54 0.54 0.56 0.479 0.51

Sno.

Closed loop system parameters p

K i K d K

1. 0.8,ζ = 1.2,nω = 9m = 3.09 2.13 1.802. 0.8,ζ = 3,nω = 20m = 40.00 73.95 9.039

3. 0.8,ζ = 4,nω = 20m = 72.86175.3

1 12.05

4. 0.8,ζ = 4,nω = 10m = 37.80 87.66 6.575. 0.7,ζ = 4,nω = 10m = 29.58 76.70 5.75

6. 0.7,ζ = 5,nω = 10m = 46.22149.8

0 7.19


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TABLE VII. TIME DOMAIN PERFORMANCE SPECIFICATIONSOBTAINED USING FA FOR DIFFERENT RANGES

Parameters

Range: 1[ 0 100, p K to=

0 100,i K to=0 100]d K to=

Range: 2[ 0 100, p K to=

0 100,i K to=0 10]d K to=

p K 79.8383 60.0933

i K 13.6195 59.1724

d K 97.6556 10RT 2.1426 0.2494ST 8.4746 2.3167Peak value 1.1198 1.1064PT 5.1932 0.7267% OS 5.7259 10.6351US 0 0

TABLE VIII. CONTROLLER GAINS CORRESPONDING TO ROBUSTNESS STUDY Paramet

erRange: Arbitraryrange selection

[ = -100 to -25.8, -100 to -0.4905, = -100 to 0.2185]

Range selection byProposed method

[ = -250 to -25.8, -120 to -0.4905,

= -80 to 0.2185]

p K -99.9975 -192.2073

i K -65.7356 -99.0493

d K -66.1587 -42.2962

R.T 0.6602 0.2782S.T 7.5948 4.4850Peakvalue

1.432 1.2392

P.T 2.046 0.9161

% OS 41.3378 23.2577US 0 0

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0 1 2 3 4 5 6 7 8 9 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

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1.5

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Time (sec)

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A n g

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Fig.8: Comparison of results under robustness study

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