Design and implementation of FOPID controllers by PSO, GSA and ...

2015

International Symposium on Advanced Computing and Communication (ISACC)

Design and Implementation ofFOPID Controllers by PSO, GSA and PSOGSA for MagLev System Prasanta Ro/, Manashita Borah2,

Lalbahadur Majhe, Nikita Singh4

Department of Electrical Engineering, NIT Silchar

Department of Electrical Engineering, NIT Silchar

Silchar, India

Silchar, India

[email protected], [email protected]

[email protected] , [email protected]

Abstract-This paper puts forward the design of PID and FOPID controllers

along

with

its

validation

through

hardware

implementation in the Maglev system. The purpose of this work is not only to stabilize a ferromagnetic ball but also to control its position to track a reference signal. The designs have been carried

out

using

Gravitational

the

Search

optimizing Algorithm

algorithms (GSA),

namely,

Particle

the

Swarm

Optimization (PSO) and a hybrid of both the algorithms i.e. PSOGSA. The experimental set up is manufactured by Feedback Instruments

(Model

No

33-210).

The

experimental

results

obtained using a wide variety of test signals prove that the hybrid algorithm PSOGSA is better than its individual counterparts with satisfactory transient and steady state responses and also the performance of FOPID is an improved one compared to that of PID. Keywords-Magnetic levitation; P1D; FOP1D; Particle Swarm

I.

INTRODUCTION

optimization based

fuzzy logic [6, 7], neural adaptive control

[8]

mode control

[10, 11], [13],

[5],

have been used

in Maglev system. Whereas, feedback linearization [9], sliding

control

[12],

identification

[10],

back-stepping control

[14],

FOPID control

quantitate

feedback

theory

[15]

H infinity real

time

etc. are some of the

advanced control techniques that have been applied too.

Literature review revealed that fractional order controllers are

relatively less explored in controlling Maglev system. This

motivates the authors to propose a new FOPID controller tuned by PSOGSA to control the Maglev System.

D. Tuning Algorithms (PSO, GSA and PSOGSA)

ferromagnetic

object

against

gravitational

forces

in

the

presence of an electromagnetic field. This concept has found

versatile use in various modern applications like superfast

magnetic trains, high- precision platforms, magnetic lift etc. Its advantage lies in the fact that it can eliminate losses arising

due to mechanical friction. Based on the measured and desired

position of the levitating object, controllers are designed to

control the current through electromagnetic coil to generate the required force to control its position

[1].

B. Brief description about proposed controllers The PID controller is undoubtedly the most popular

controller in the industries owing to its convenience in design,

cost effectiveness, and acceptable robustness. However its

performance can still be improved by making the order of derivative and integration fractional terms.

Fractional order

controllers provide more alternatives and tlexibilities in design

but also introduce some challenges. A brief modern history of fractional calculus is given in

[2].

In

[3]

a FOPD controller is

designed using Firefly Algorithm and compared its improved

performance with that of a PID.

literature

for

Maglev

subsequently Jiang dispatch problem in

et.al [20].

20 I 0 [19],

and

used PSOGSA in economic load

E. Prime Objectives of this Work Prime objective of this work is to design PID and FOPID controllers considering the exact nonlinear model of Maglev system. The fractional order controller is difficult to design by classical method because of complexity of fractional order calculus. So controllers are designed using three meta heuristic algorithms

namely

PSO,

GSA and

PSOGSA.

Controller

implementation is done through Hardware in Loop (HIL) configuration. Step change, square wave signal, and sine wave signal is used as reference signals to test the effectiveness of the proposed controllers. II.

MAGNETIC LEVITATION SYSTEM DESCRIPTION

The basic setup of the Maglev system manufactured by

Feedback Instruments (Model No

33-210

is shown in Fig. l.

The main components are an optoelectronic position sensor,

electromagnetic actuator coil and a suspended ferromagnetic (HIL) configuration putting the desktop computer in the loop

Various control strategies and tuning algorithms have been in

with the combination of PSO and GSA in

ball. Controller is implemented through Hardware in Loop

Literature Review for Maglev System

reported

based evolutionary computation algorithm proposed by Kennedy and Eberhart [16, 17], inspired from social behaviour of bird flocking. GSA is a meta heuristic optimization method proposed by E. Rashedi et al in 200 9 [18], inspired from Newton's gravitational law. A novel hybrid population-based algorithm (PSOGSA) was proposed

Basic principle of the System The principle of the Maglev system is levitation of a

e.

[4],

intelligent control schemes such as the modified PSO

PSO is a swarm

Optimization; Gravitational Search Algorithm

A.

controllers like PID was designed in

system.

978-1-4673-6708-0/15/$31.00 ©2015 IEEE

Conventional

with actual hardware. Optoelectronic sensor determines the

vertical position of the ferromagnetic ball and passes it to

controller through an interface of Advantech card. Based on the

difference

between

desired

and

measured

ku2 .. mx=mg-- x2

output,

controller sends current to the actuator. Actuator consists of an

electromagnet wrapped up by copper wire 012850 turns on a

V = 142.86x-135.86

high permeability cylindrical iron core to generate upward attractive force on the ferromagnetic ball for levitation against

IV.

gravity. A suitable controller is needed to be designed to adjust current through the actuator to stabilize the levitated

ball and to make it follow a reference trajectory. The levitated

object is a hollow ball with mass of 20 g and diameter of 50

mm [4].

(1) (2)

DESIGN AND IMPLEMENTATION OF CONTROLLERS

A. PlD and FOPID Controller dynamics The dynamic equations of PID & FOPID controller are

given in (3) and (4) respectively with usual notations. The parameter

Ie

and I-t stands for the fractional order of integration

and derivative, respectively.

de(t) Kre(t) + KJ 'fe(t)dt + KD = u (t) dt "

(3)

--

B. Performance Index of Controller Performance Index (PI) is a quantitative measure to depict the performance of a controller. The PI adopted in this paper is ISE for all optimization algorithms, given in (5).

f

�

J=ISE= e2(t)dt; C.

(5)

Tuning of PID and FOPID Controller using PSO

The objective in PSO-based optimization is to seek a set of PID and FOPID parameters such that performance index minimized. The particles in PSO are a set of K JJ for PID and

particle's Fig. 1: Magnetic Levitation system (Feedback 33-210) III.

MODELLING BY MAGLEVSYSTEM

The dynamics of the Maglev system is governed by eqn. 1 where

m,x,g,k and u are mass of the ball,

position of the ball

measured from the electromagnet, gravitational acceleration,

electromagnetic coil constant and coil current respectively.

V are

Vi (t) and

A,

andf.1

position

K J and

for FOPID. A

Xi (t) are

updated

according to (6) and (7) respectively. A particle's new velocity

(XPi(t))

of its current position from its own best experience and the group's best experience

(XGb (t)) to

determine the

next direction of search, thereby narrowing the search space

Vi(t + I)=W,Vi(t) + c, .randO· (XPi(t) -Xi(t) ) + c .randO· (XGb (t) -Xi(t)) 2 Xi(t + 1) =Xi(t) + Vi(t + 1)

[16, 17, 19, 20].

position of the ball

and the corresponding sensor voltage respectively. The whole

Maglev control system can be represented by the simple block

diagram as shown in Fig. I.

velocity

K J' K JJ'

,

is

is calculated based on its previous velocity and the distances

Sensor characteristic is shown in eqn. 2 which is provided by the manufacturer [4], where x and

K J"

K J'

J

The function

randO generates

number in (0,

1).

(6)

(7)

a uniformly distributed random

The inertia constant w

takes care of local

and global search in range (0, I).Constants c and c represent 1 2 cognition and social acceleration constants, respectively, in

range (0, 2). Best values are given in Table I. However effect of variation of w, cl and c2 on the results is beyond the scope of Fig 2: System with PD controller The physical equations of the system is given by (1) and (2) [4]

this

paper.

Position

Xi (t) represents

the

controller

parameters. Fitness function used to update X (t) and V (t) is i i

given in (5). The optimized controller parameters are listed in

Table II. Step responses of the system using PID and FOPID tuned by PSO are shown in Fig. 3.

The GSA initialization parameters are given in Table III and controller parameters obtained are listed in Table IV. Step responses of the system using PID and FOPID tuned by GSA are shown in Fig. 5. TABLE III: GSA Parameters

TABLE I: PSO Parameters. Number of Swarm=50

1

w

1

=0.9

1

c1 =l .2

c2 =l .2

1

Initial

4.9656

9.4912

0.1765

4.4178

2.4697

0.1727

2

o. a

� Ql

06

o

,

I I

rC'-•

o.2

_.-

,

T

Time ISec]

Controller

K I'

l .8

0.2313

FOPID

3.9822

l .9989

0.1671

--

. - - - - --- . -. .- .-- .

-

10

GSA

i

M and

a

for

.

.

:

:

.

°O�--�--�--�.--�,---7---7lO T i me [Sec]

Fig 5: Simulated step responses using PID & FOPID tuned by GSA

Generate initial popt arion.

worst of the

Calculate wand

No

I

eaclil agent

velocity and

0.1201

. _.- _.-

fi lne!>s fo r each age llt

population.

U pdate

.

illitial p opulat ion

Updale the G, best and

Calculate

0.0712

-----

- --- - - - .... --- -- - - .-----------�-------.-- .. : : : :

--------�-----------.--- . . -

described in the flow chart as shown Fig. 4 [18].

the

-----

£ 1 ���--�--�--�--�--�--�--�======Y � 0 a -- - � - - - � . � --. -..-.+- .------- �--..------+.--..- .-�.-! 0.6 -- --: : :::::::::r, :::::::::. :::::::::::::::::::::::::::::::::r:::, ::::::: :::::::::T::::::::::::, :: :::::::::::::::: 0..

of GSA may be found in [18]. The algorithm may be

Eval u ate

f.L

/L

K IJ

4.9893

GSA is inspired from Newton's law of gravitation. Details

Gellerate

K/

u

PSO

Tuning ofPlD and FOPID Controller using

5

(max_it) =

No. of generations (N) = 20

PID

•

Fig.3 Simulated step responses of PID & FOPID tuned by

D.

3

TABLE IV' Controller parameters obtained by GSA

0.4996

:

100

Constant (a)=21

-----

--

:

Maximum iteration

Gravitational

-PSO.PID -PSO·FOPID -STEP

•

o.4

0

0.4999

'

""

.� 0..

-----

of

value

I I

constant(Go)=90

f.L

/L

K IJ

PID

«i c:J Q) £ '0

c

K/

FOPID 4

oS

K I'

For FOPID

N)

TABLE II: Controller parameters obtained by PSO. Controller

For PID

Dimension of search space (

G

calculat,c j'oJ MId

posi tio n

a

for each ilerntiml, for all paI1idc�

M eeting end of criterion ? o

Return best wl utioll

Fig. 4 Flow chart for GSA Fig. 6 Flow chart for PSOGSA

E. Tuning ofPlD and FOPID Controller using PSOGSA

FOPID tuned by PSOGSA is working better than all other

PSOGSA is basically a hybrid algorithm by incorporating both

controllers considered in this paper in terms of tracking of a X 10 -3 16[--"---,-,---

PSO and GSA. Details of PSOGSA may be found in [20]. The flow chart of the PSOGSA is given in Fig. 6. Optimized

,

controller parameters are shown in Table V. Step responses using PID and FOPID tuned by PSOGSA are shown in Fig. 7.

Kp

KJ

KD

A

JL

PID

3.6451

9.6108

0.3421

-----

-----

FOPID

4.9235

2.1405

0.3650

0.4989

0.5

"

GSA- PID 11- PPSSOOGS A- FOPID

� 12

[]J Q)

-STEP

, :5 a O.8

Qj

Q)

Qj oS 0.6

�

. ... .. . ' r 'r ' , ,

" � --� L --� L --�,L --�,L5 --�5 '"�--�� L --�, L5--� L --� ,O 2O 25 JO J 5 0 0 O Time [Sec]

Fig. 9 Real time system response with PID tuned by PSO using square wave as reference input (only PD duringl st 15 sec)

O.02,-- --- - - --:---:-= --: --: : : r _ IC ..D == ..::c e=: c=s ==a=: 1I == SiICred tio=J:= n PO= Si:: =0.018 -Maglev Ball Position '" []J Q.) 0.016

:5

'00.014

gOA c:

'i::'

20_012-

'iii

Q)

�O.2

E

0

...... 0.01 0

,

,

J

,

c:

5

T ime [Sec]

6

7

8

10

9

Fig 7: Simulated step responses using PID & FOPID tuned by PSOGSA

F.

•• -.. , r " ,

'

r(

�

,

2.,10

I' �c: 6

f

,

-

TABLE V: Controller parameters obtained by PSOGSA Controller

,-----,-----,-----, '-----,--' --;=_= = = d = ire C:: =s=all= p C::os=iti=on=il Des : : -:--. : -- -Maglev Sail Position ---- --------- . :

o

�O.008

o a.. 0.006

Time [Sec]

implementation of proposed controllers in real system

Implementations of the controllers have been done in Matlab

Fig. 10 Real time system response with PID tuned by PSO using sine wave as reference input (only PD during 1st 15 sec)

Simulink environment through HIL configuration. FOPID controller

is

realized

with

the

_

4 D_ DD �--� ----'� -: 0 --�,0-5 -----:0, -0 -----:,0-5 -----:J�O --�J :' ':- 0 --' :'-: 5 -----:'50 � 5 --0

aid

of

Fractional

0.02,--,----,---,--;=====';l

Order

Modelling and Control Toolbox (FOMCON). Because of its

�O_016

inherently unstable nature of the system, integral action is

£0.014

turned on after 15 seconds. To take care of negative gain of the plant, output of the controller is inverted and then applied to the plant. Real time results are shown in Fig. 8 to Fig.19

Q)

o O.01 21l----i------i----i

�

Ci) 0.01 E ...... 0.008 C o

;;:; 0.006 'iii

=

'"

�O_004

14

. O- 002�� '----------� -f ----f.c -- ----!o -- -o-c - -- + ----!'50 25-- � JO -- ---:!:- ----+ 0

[]J

Time [Sec]

Fig.l l Real time system response with PID tuned by GSA with step change as reference input (only PO during15t 15 sec)

2�--�--�--���-- � '5---� --�--�--�--� 50 0 J0

Time [Sec]

.::: . ; . '. ;lli+' I' ••

-Desired Ball Position -Maglev Ball Position

Fig. 8 Real time system response with PID tuned by PSO using step change as reference input (only PO during 1st 15 sec) I.

RESULIS AND DISCUSSIONS

Among step responses shown in Fig. 3, 5, and Fig. 7, FOPID

c

I:::, ,: T OJJ02

0

5

10

15

20

25

Time [Sec]

- - - - I III:-- 1 T

30

J5

40

45

50

tuned by PSOGSA shows the best result (overshoot 0.498%

Fig.12 Real time system response with PID tuned by GSA

and settling time 0.34 sec). Real time results in Fig. 8 to Fig.

using square wave as reference input (only PO during 15t 15

19 indicate that controllers tuned by PSOGSA perform better

sec)

than that of GSA and PSO individually. Results also show that

reference signal. Feedback Maglev 33-210 model being new, has little research work done on it. Hence, comparisons have

been done with similar contemporary set ups. Results in this paper are better than

[10]

and the system inbuilt PID controller

in terms of tracking of a reference signal. In

[4]

[9]

results

show transient and steady state behaviors similar to this paper but it shows chattered behavior during initial 0.3 seconds. Back stepping and high gain observer based controller in

[ I I]

has higher overshoot (12%) and settling time (3.5 sec) than this paper. Hw controller in [12] gives less settling time but

higher overshoot (4%) and the steady state error is not completely zero. QFT controller in [13] gives settling time nearly 1 sec which is higher compared to this paper. In [15] a fractional

order

PID

controller

is

designed

by

classical

0.016 ,----,----_,____---,--c====il - Desired Ball Position - Maglev Ball Position aJ 0.014 ro

'"

.t::

:: 0.012 o

� 0.01 W .s

�o.ooa 'iii

� 0.006 0.004 �-;--" :C- S --'''' 's-----O,so : ----c;:,-30 --c3 0' ----f; 1"O ----f;'S----:!;20;------:'5,---O

Time [Sec]

Fig.16 Real time response with PID tuned by PSOGSA using sine wave signal as reference input (only PD during 1st 15 sec)

optimization method but the real time results contains higher overshoot (10 to 15 % in various cases) and settling time

(8

to

10 sec. in various cases) than those obtained in this paper.

O.OJ ,-------,--,--;=====;] rnO.D25 al

'"

I

O.024

::S

_,____ --,-------,-- ----,--�,____ _,____ = = ,r= ===il= =::I:== Desired === Ball Position -=:= 0.022 - Maglev Ball position � 0.02 £0.016 '00.015

:

:

: :;::::: ::::

0.02

�O.015 Qj

E. 0.01

f-

C o

'" "000.005 o a.

° L-�----:�,---�, � --- 20----:� 25-�30-�-�-�-�SO O

�O.014

Time [Sec]

Q)

g O.012

Fig. 17 Real time response with FOPID tuned by PSOGSA

§ 0.01

with step change as reference input (only FOPD during 1st 15

�O.008

o 0... 0.006

sec)

D- OD4 0�--;---f. 1O;------:'!;-5 ---:!;; : S --'---f.c30 ----:!; J S----:----:4!;-0 ----f; 20-----:' :45-----: 5'0:

O.OJ ,------,,--- ,------,----,----,----, r==========;] - Desire d Ball Position - Maglev Ball Position 0.025

Time [Sec]

Fig. 13 Real time system response with PID tuned by GSA using sine wave as reference input (only PD during 1st 15 sec)

1-

0.03

� O_025

n; aJ

'"

0.02

-5

00.015

�

Desired Ball Position ball Position

Qj

E

0.01

c o

Q)

-5 a 0.02

,

S�S=:::j "·

�

M

:

:

10

15

:

-

: .........

-

:

:

:

:

:

,

-

20

25

30

35

40

45

50

Time [Sec]

Fig. 14 Real time response with PID tuned by PSOGSA using

:;:::0:; .005

'iii o a.

° 0:-�L-����� 20�� '� 5 � 30�� 3S,---� 5'O 40;--�'S----:----;

Time [Sec]

Fig. 18 Real time response with FOPID tuned by PSOGSA using square wave as reference input (only FOPD during 1st 15 sec)

:

:

:

- Maglev Ball Position

8 f- .. · ..

step change as reference input (only PD during1st 15 sec)

:

1- Desired Ball Position

0.02 0 ....

_

....

. ......

- Desired Ball Position - Maglev Ball Position

....... ........ ........... , .......... ......... + ......... ......................

, 8

"'r

,

. ... 1 + , ..

0.004

.. .... ..

0.002 :-____::-____:';- ---c'C;-5 --c,;'; o -;;;;'- ' 5 --3C;;-0 ----; ;35 ----;'; 40 ----;'; ; " -----; 50 ; 0 0

Time [Sec]

Fig.15 Real time response with PID tuned by PSOGSA using square wave as reference input (only PD duringl st 15 sec)

0

10

'5

20

25

30

Time [Sec]

35

40

45

50

Fig. 19 Real time response with FOPID tuned by PSOGSA using sine wave as reference input (only FOPD during 1st 15 sec)

CONCLUSIONS AND FUTURE SCOPE In this paper, design and implementation of PID and FOPID controllers have been proposed for the nonlinear Maglev system

using

PSO,

GSA

and

PSOGSA.

It

is

quite

a

challenging task to tune an FOPID controller especially for a nonlinear and inherently unstable plant. The hardware results clearly justify that FOPID controllers exhibit still a better response in terms of transient characteristics when compared to their integer order counterparts i.e. PID controllers. In addition to this, PSOGSA is a suitable technique to optimize the FOPID controller parameters in the Maglev system. Comparisons with contemporary literatures show that the proposed

method

has

better

performance

over

existing

controllers. The future scope lies in running the device as a standalone system without the use of a computer. Exploration with the designs using complex reference signals could be attempted in future. Fractional modelling of the Maglev system could be another future scope of the work. REFERENCES

[I] [2] [3] [4] [5] [6]

[7]

[8] [9]

[10]

[11] [12]

[13]

1.

Ahmad, and M. A. Javaid, "Nonlinear model & controller design for Magnetic levitation system", 9th WSEAS international conference on Signal processing, robotics and automation, 2010, Cambridge, UK. J. T. Machado, V. Kiryakova, and F. Mainardi, "Recent history of fractional calculus", Commun Nonlinear Sci Numer Simulat vol. 16, 2011, pp. 1140-1153. Manashita Borah, L. Majhi, P. Roy , B.K.Roy, "Design of a Fractional Order PD Controller Tuned by Firef1y Algorithm for Stability Control of the Nonlinear Ball and Plate System", ICACCCT, IEEE, 2014,India. Coupled Tanks Control Experiments Manual Model No 33-04IS, 2011, Feedback Instruments Ltd. UK. E. Anene and G. K. Venayagamoorthy, "PSO Tuned Flatness Based Control of a Magnetic Levitation System", IEEE Industry Applications Society Annual Meeting (lAS), 2010, Houston, US, pp. 1-5. M. Gholami, S. Aliyari, M. Shoorehdeli, Z. A. S. Dashti, and M. Teshnehlab, "Design of fuzzy parallel distributed compensation controller for magnetic levitation system", Fuzzy Systems (lFSC), 2013 13th Iranian Conference, Qazvin, Iran, pp.I-5. A. Kumar, M. K. Panda, S. Kundu, and V. Kumar, "Designing of an interval type-2 fuzzy logic controller for Magnetic Levitation System with reduced rule base", Third International Conference on Computing Communication & Networking Technologies, 2012, Coimbatore, India. Z. A. S Dashti, M. Gholami, M. Jafari, M. A. Shoorehdeli, and Hajimani, "Neural Adaptive controller for Magnetic levitation system", Iranian Conference on Intelligent Systems, 2014, Bam, Iran. N. Naz, M. B. Malik, and M. Salman, "Real time implementation of feedback linearizing controllers for magnetic levitation system", IEEE Conference on Systems, Process & Control (lCSPC), 2013, Kuala Lumpur. Malaysia. F. 1. Lin, and S. Y. Chen, "Intelligent integral backstepping sliding mode control using recurrent neural network for magnetic levitation system", Neural Networks (IJCNN), The 2010 International Joint Conference, Barcelona, Spain. H. Katayama, and T. Oshima, "Stabilization of a magnetic levitation system by backstepping and high-gainobservers", SICE Annual Conference (SICE) 2011, Tokyo, Japan. G. Wei, Y. Guo, F. Huang, Y. Wu, "Experimental studies on Hco control for magnetic levitation system with friction compensation" Modelling, Identification & Control (ICMIC), 2012 Proceedings of International Conference, Wuhan, Hubei, China. P. S. V. Nataraj, and M. D. Patil, "Nonlinear control of a Magnetic Levitation system Using Quantitative Feedback Theory (QFT)", Reliability, Safety and Hazard (lCRESH), 2010 2nd International Conference, Mumbai, India.

[14] L. B. Majhi, M. Borah and P. Roy, "Fractional Order System Identification of Maglev Model from Real-Time Data", IEEE ICACCCT, 2014, India. [15] H. Gole, Harshit, P. Barve, A. A. Kesarkar, and N. Selvaganesan,"Investigation of fractional control performance for magnetic levitation experimental set-up", International Conference on Emerging Trends in Science, Engineering and Technology (INCOSET), 2012, Tiruchirappalli, India. [16] J. Kennedy and R. C. Eberhart, "Particle Swarm Optimization",IEEE International Conference on Neural Networks, 1995. Perth, Austrailia. [17] Y. Shi and R. Eberhart, "A modified particle swarm optimizer," Evolutionary Computation Proceedings, 1998, IEEE World Congress on Computational Intelligence, Anchorage, US. [18] E. Rashedi, H. Nezamabadi-pour, S. Saryazdi, "GSA: A Gravitational Search Algorithm", Information Sciences, vol. 179 , 2009, pp. 22322248. [19] S. Mirjalili, S. Zaiton M. Hashim, "A New Hybrid PSOGSA Algorithm for Function Optimization", IEEE,ICCIA 2010, Malaysia, pp.-374-377 [20] S. Jiang, Z. Ji, Y. Shen, "A novel hybrid particle swarm optimization and gravitational search algorithm for solving economic emission load dispatch problems with various practical constraints", International Journal of Electrical Power & Energy Systems, 2014, vol. 55, pp. 628644.