Research Journal of Applied Sciences, Engineering and Technology 7(7): 1116-1122, 2014
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2014
Submitted: April 16, 2013
Accepted: May 21, 2013
Published: February 20, 2014
Optimization of PID Controller for Brushless DC Motor by using
Bio-inspired Algorithms
1
Sanjay Kr. Singh, 2Nitish Katal and 3S.G. Modani
Anand International College of Engineering, Jaipur, Rajasthan, India
2
Department of Electronics and Communication Engineering,
Amity University, Rajasthan, India
3
Malaviya National Institute of Technology, Jaipur, Rajasthan, India
1
Abstract: This study presents the use and comparison of various bio-inspired algorithms for optimizing the response
of a PID controller for a Brushless DC Motor in contrast to the conventional methods of tuning. For the optimization
of the PID controllers Genetic Algorithm, Multi-objective Genetic Algorithm and Simulated Annealing have been
used. PID controller tuning with soft-computing algorithms comprises of obtaining the best possible outcome for the
three PID parameters for improving the steady state characteristics and performance indices like overshoot
percentage, rise time and settling time. For the calculation and simulation of the results the Brushless DC Motor
model, Maxon EC 45 flat ф 45 mm with Hall Sensors Motor has been used. The results obtained the optimization
using Genetic Algorithms, Multi-objective Genetic Algorithm and Simulated Annealing is compared with the ones
derived from the Ziegler-Nichols method and the MATLAB SISO Tool. And it is observed that comparatively better
results are obtained by optimization using Simulated Annealing offering better steady state response.
Keywords: Brushless DC motor, controller tuning, genetic algorithm, PID controllers, PID optimization, simulated
annealing, ziegler nichols

INTRODUCTION
The generic dc motors have high efficiency and
have a high starting torque versus falling speed
characteristics, which helps to counter the sudden rise
in load and thus find their application in industries since
ages (Katal et al., 2012). Since dc motors suffer from
the deficiencies like:





The lack of periodic maintenance
Mechanical wear-outs
Acoustic noise
Sparkling
Brushes effect, etc., so, the current focus has
adapted the development of brushless direct current
models
The Brushless Direct Current (BLDC) motors are
typically dc voltage driven permanent synchronous
motors and are gaining grounds in aeronautics,
medicine, consumer and industrial automation
applications. The BLDC motors have better:




Speed versus torque characteristics
High efficiency
High dynamic response
Noiseless operation
Low maintenance and many more (Padmaraja,
2003; Krause et al., 2002) and the best advantage
in terms of higher ratio of torque obtained to the
size of motor
Proportional,
Integral
and
Derivative-PID
controllers are playing an imperative role in the
industrial control applications. Because of their
simplicity and wide acceptability, they are still the best
solutions for the industrial control processes (Åström
et al., 2001). Modern industrial controls are often
required to regulate the closed-loop response of a
system and PID controllers account for the 90% of the
total controllers used in the industrial automation. The
simple block level representation of the PID controller
based system can be obtained as in Fig. 1.
The general equation for a PID controller for the
above figure can be given as Norman (2003):
C ( s )  Kp .e ( s )  Ki  e ( s ) dt  Kd
de ( s )
dt
where,
Kp, Ki and Kd = The controller gains
C (s)
= Output signal
e (s)
= The difference between the desired
output and output obtained
Corresponding Author: Sanjay Kr. Singh, Anand International College of Engineering, Jaipur, Rajasthan, India
1116
Res. J. Appl. Sci. Eng. Technol., 7(7): 1116-1122, 2014
Fig. 1: Block diagram of a PID control based system with unity feedback
Fig. 2: Schematic diagram of a BLDC Motor
In this study, the optimization of the PID controller
gains has been carried out using Genetic Algorithms
(GA), Multi-Objective Genetic Algorithm (MOGA) and
Simulated Annealing (SA) in contrast to the ZieglerNichols (ZN) method and the automated tuning
provided in MATLAB viz. SISO Tool. Then, these gain
parameters can be optimally tuned with respect to the
objective function, stated as “Sum of the integral of the
squared error and the squared controller output deviated
from its steady-state” (Goodwin et al., 2001).
According to the results obtained in this paper,
considerably better results have been obtained in the
case of the Simulated Annealing (SA) when compared
to those obtained by Genetic Algorithm, Multiobjective Genetic Algorithm, Ziegler-Nichols method
and the MATLAB SISO Tool in respective the step
response of the system.
used. The schematic illustration of the considered
system is shown in Fig. 2.
Using Kirchhoff’s Voltage Law (KVL), the
following equation is obtained:
Vs  Ri  L.
(1)
where,
Vs = The DC Source voltage
i
= Armature current
Similarly while considering the mechanical
properties, Newton’s second law of motion gives the
relative dependence of torque of the system as the
product of the inertial load, J and the rate of angular
velocity, ωm, as:
MATERIALS AND METHODS
J
Mathematical model of brushless DC motor: In this
study the model of a BLDC motor has been considered,
unlike the dc motor, the commutation of the BLDC can
only be done by electronic control (Padmaraja, 2003).
The operation of BLDC motor can be realized in many
modes (phases), generally 3 phases. The main
advantage of 3-Phase is better efficiency and quiet low
torque and has best precision in control (Texas
Instruments).
In this study, the use of Maxon EC flat ф 45 mm,
brushless, 30 Watt motor with Hall Sensors has been
di
e
dt
d m
  Ti
dt
Te  k f  m J
(2)
d m
 TL
dt
where,
Te = Electric torque
kf = Friction constant
J = Rotor inertia
ωn = The angular velocity
TL = The supposed mechanical load
1117 (3)
Res. J. Appl.
A
Sci. Eng. Technol., 7(7): 1116-1122, 2014
2
Table 1: Paraameters and units of
o maxon motor
Maxon motorr data
Value at nom
minal voltage
Nominal voltage
No load speedd
No load curreent
Nominal speeed
Nominal torqque (max. continuo
ous torque )
Nominal currrent (max. continuo
ous current)
Stall torque
Starting curreent
Maximum freequency
Characteristiccs
Terminal resistance phase to ph
hase
Terminal induuctance phase to phase
p
Toque constaant
Speed constannt
Speed/torque gradient
Mechanical tiime constant
Rotor inertia
Number of phhases
U
Unit
Valuue
V
R
Rpm
m
mA
R
Rm
m
mNm
A
m
mNm
A
%
12.0000
4370
151
2860
58.0000
2.1440
255
10.0000
77.0000
Ω
m
mH
m
mNm/A
rppm/V
rppm/mNm
M
Ms
g 2
gcm
-
1.2000
0.5660
25.5500
37.4400
17.6600
17.1100
92.5500
3.0000
P
controller caalculated by zieglerTable 2: Paraameters of the PID
nichhols method
Valuue
Ziegler nichools PID parameterss
Kp
0.31660
Ki
31.30000
Kd
0.00008
Table 3: Paraameters of the PID
D controller calcuulated by SISO baased
autoomated designing
SISO tuned parameters
p
Value
Kp
0.2650000
Ki
17.8670000
Kd
-0.0005886
where,
ke = The
T back emf
kt = Torque
T
constannt
Thhus the transferr function can be
b obtained byy using
the ratiio of the anguular velocity, ωm to source voltage
v
Vs, as:
1
ke
G (s) 

Vs
 m . e .s 2   m .s  1
m
(5)
Sinnce the system
m is symmetricaal and a three phase,
thus coonstants used abbove in Eq. (5)) can be given as:
Thhe Mechanicaal, τm (Tim
me Constant) and
Electriccal, τe (Time Constant):
C
m 
R.3 J
ke kt
andd  e  L
3 .R
Thhe mathematiccal model of the Maxon BLDC
B
motor (Sabudin, 20012) can be modelled on the
parameeters listed in Table
T
1.
Thhus by using the above lissted parameterrs, the
value for
fo Ke, τm and τn can be obtainned as:
τe = 155.56 * 100 - 6 Kgm2; τm = 0.0171 andd Ke =
0.7763 v-sec/rad
Therefoore, the transfeer function G (ss) becomes:
G(s) 
13.11
2.66  10 6  s 2  0.0171  s  1
(6)
g
us the open-loop trransfer
Eqquation (6) gives
function for the Brushhless DC Maxoon Motor.
Design
n of the PID coontrollers:
Tuningg of PID gainss using zieglerr nichols: One of the
most widely
w
used method
m
for thee tuning of the PID
controlller gains is too use the opeen loop responnse as
inferredd by Ziegler-N
Nichols (ZN), yet
y this methodd finds
its in application
a
till the ratio of 4:1
4 for the first two
peaks in the closed loop responsse (Goodwin et al.,
2001), which leads to a oscillatoory response of
o the
system.
Iniitially, the uniit step functioon (Fig. 3 andd 4) is
derivedd and hence as suggested by the Ziegler-Niichols,
the paraameters required can easily be
b estimated ass given
in Tablle 2.
Fig. 3: Openn loop step respo
onse of the systeem
Fig. 4: Clossed loop step ressponse of the syystem with ZN-P
PID
conttrollers
The electrical
e
torqu
ue and the baack emf can be
obtained ass:
e  k e m and Te  k t m
(
(4)
MATL
LAB SISO auttomated design
ning: The SISO tool
accelerrates the PID controller
c
desiggn by its GUI based
interacttive compensattor tuning. Thee tuning can bee done
interacttively by balanncing the poless and zeros onn Bode
or root--locus plots, or can be optim
mized by meetinng the
pre-deffined time andd frequency domain
d
requireements
by the system. The SISO
S
tuned PID
D controllers for
f the
BLDC Motor can be seen in the Figg. 5 (Table 3).
1118 Res. J. Appl.
A
Sci. Eng. Technol., 7(7): 1116-1122, 2014
2
Table 4: Paraameters used in opttimization by geneetic algorithm
Genetic algorrithm optimized paarameters
Kp
Ki
Kd
Vallue
5.00001
0.00039
5e-44
mplementation of the
Thhe steps involvved in the im
Geneticc Algorithms for
f a control syystem are as folllows:




s
with SIS
SOFig. 5: Clossed loop step reesponse of the system
PID controllers

Geenerating the initial,
i
random
m population of the
fixxed numbered individuals foor the declarattion of
thee initial ranges for Kp, Ki andd Kd
Foor the evaluaation of the fitness integgral it
miinimizes the inntegral squaree error, follow
wed by
thee selection of the fittest individuals of
o the
poppulation
Reeproduction am
mong members of population
Crossover of the reproduced chhromosome folllowed
by the mutation operations andd the selection of the
besst individuals i.e.,
i Survival of the Fittest
Loooping the stepp 2 till the pre-ddefined converrgence
is obtained
o
Thhe optimizationn of the system
m has been dessigned
and sim
mulated in MATLAB
M
and Genetic Algoorithm
toolboxx, with popuulation size of 100, scaattered
crossovver and migratiion direction inn both sides. Table
T
4
shows the GA optim
mized PID parrameters and Fig.
F 6
shows the
t GA-PID sttep response off the system.
PID controller opttimization usiing multi-objjective
geneticc algorithm: Optimization
O
of PID’s using multiobjectivve genetic algoorithm aims at using the conttrolled
elitist genetic algoriithm which boosts
b
obtaininng the
better fitness
f
value of
o the individuaals and if the fitness
f
value is
i less, it still favors increassing the diverssity of
the poppulation (Deb, 2001). Diverssity is controllled by
the elite members of the poppulation; elitissm is
controllled by Pareto fraction and at Pareto Fronnt also
bound the number of individuals. Optimization of the
PID controllers using
u
Multi-Objective Genetic
G
Algoritthm aims at im
mproving the objective
o
functtion of
the botth the objectivves used by obtaining
o
an opptimal
Pareto solution. In thhis study, twoo objective funnctions
have beeen used F1 (IT
TSE) and F2 (O
OS %):
Fig. 6: Clossed loop step response of the sysstem with GA-PIID
conttrollers
s
with MoobjFig. 7: Clossed loop step reesponse of the system
GA--PID controllers
ation using gen
netic algorithm
m:
PID contrroller optimiza
Since the designing
d
of the
t PID controollers by ZieglerNichols methods,
m
gives an oscillatory response; hennce
the controoller parameterrs obtained frrom ZN are not
n
optimum for
f the directly
y implementatiion for the plaant,
so their orrganized optim
mization must be
b carried out, so
that the better possible paarameters can be estimated and
a
implementted for the bestt performance of
o the system.
So, thee Genetic Algo
orithms can bee used along with
w
the param
meters obtain
ned by the Ziegler-Nichools
response, as
a the parametter determinedd by ZN helps in
the determ
mination of the lower and uppper bound lim
mits
to be useed for the esttimation of parameters
p
usiing
Genetic Allgorithms.
where,
ς = Thhe Damping rattio
Firrst objective function
f
ITSE
E i.e., Integral Time
Square Error tries to minimize the larger
l
amplituddes by
supresssing the persistent larger errors (Jean-P
Pierre,
2004) while second objective function oveershoot
percenttage; thus forciing the solution towards the global
best sollution.
Thhe system impplementation annd optimizatioon has
been carried
c
out in MATLAB
B and SIMU
ULINK
1119 Res. J. Appl.
A
Sci. Eng. Technol., 7(7): 1116-1122, 2014
2
Fig. 8: (a) Average
A
distancee between indiviiduals of the gennerated populatioons and, (b) averrage pareto spreaad between indivviduals
of thhe generated populations
Table 5: Paraameters used in op
ptimization by muulti-objective geneetic
algorithm
hm optimized paraameters Value
Multi-objectivve genetic algorith
Kp
5.2938
Ki
0.0042
4.5024ee-4
Kd
Table 6: Paraameters used in opttimization by simuulated annealing
Simulated annnealing optimized parameters
Value
Kp
5.0424
Ki
0.0040
3.0454ee-4
Kd
bal Optimizatioon Toolbox. The
T
environmeent using Glob
populationn size of 45 has been considered,
c
w
with
adaptive feeasible mutatio
on function, heeuristic crossovver
and the selection
s
of individuals on
o the basis of
tournamennt with a tourrnament size of 2. A hybrid
function of
o Fitness Goaal Attain (fgoaalattain) is ussed
which furrther minimizzes the funcction after GA
G
terminates. After the op
ptimization the PID parameteers
ong with the coontroller responnse
are shown in Table 5 alo
in Fig. 7 annd 8.
ntroller optiimization ussing simulatted
PID con
annealing: Simulated annealing is a globbal
optimizatioon algorithm, as
a the name suuggests, the muuse
comes froom metallurgiic annealing, which involvves
relation between
b
the relation
r
betweeen the staticcal
mechanics and multivarriate optimizaation (Berrsim
mas,
f
the technique involvving heating the
t
1993). It follows
material followed
f
by controlled coooling, fetchiing
increased crystal
c
size and
d reduced deforrmities.
Fig. 9: C
Closed loop stepp response of thee system with GA
A-PID
c
controllers
In Simulated Annnealing (SA), at each iterattion, a
new pooint is randomlly generated and
a its distancee from
the cuurrent point is the functtion of probability
distribuution with a scale
s
proportioonal to temperrature.
The raandomly generrated points are
a accepted iff they
lower the
t objective buut in order to help
h the algoritthm to
search for global sollution and to omit
o
the trapping of
the alggorithm in loccal minima, some
s
points are
a so
chosen that they raisee the objectivee. With the advvent of
algorithhm, the tempperature is deecreased leadiing in
reduction of the exxtent of searcch to convergge the
minimaa.
Thhe optimizationn has been carried out in MAT
TLAB
and SIM
MULINK envvironment withh the help of Global
G
Optimization Toolbox using Simulated Annnealing
function. For simullated annealinng, the Boltzzmann
w an exponenntially
Annealling function has been used with
updatinng temperaturee. The parameters obtainedd after
optimizzation are in Table 6 and the response of thhe PID
controlller with SA optimised
o
paraameters is shoown in
1120 Res. J. Appl.
A
Sci. Eng. Technol., 7(7): 1116-1122, 2014
2
Fig. 10: Plot for the best fun
nction value of thhe simulated annnealing optimizaation
Fig. 11: Com
mpared closed lo
oop step responsse of the ZN, SIS
SO and GA-PID controllers
Fig. 12: Com
mpared variation
n in rise and settlling times of thee ZN, SISO and GA-PID controlllers
Table 7: Com
mparison of the resu
ults
Methods of design
d
Overrshoot (%)
Ziegler-nichools method 11.50
MATLAB SIISO tool
5.88
Genetic algorrithm
0.00
Multi-objectivve genetic 0.00
algorithm
Simulated annnealing
0.00
Tr (sec)
0.0086690
0.0062230
0.0006643
0.0005560
Ts (secc)
0.0114400
0.0189900
0.001319
0.0010040
0.0005542
0.000888
Fig. 9. Figure 10 shoows the Plot for
f the Best fuunction
value of
o the Simulated Annealing opptimization.
TS AND DISCU
USSION
RESULT
In this study, a dynamic
d
modell of a Maxon BLDC
B
Motor has been designed andd implementeed in
1121 Res. J. Appl. Sci. Eng. Technol., 7(7): 1116-1122, 2014
MATLAB along with the optimization using the
various bio-inspired algorithms like Genetic Algorithm,
Multi-objective Genetic Algorithm and Simulated
Annealing. The value of parameters obtained using
Ziegler-Nichols rules (Ziegler and Nichols, 1942) were
used in the formation of the boundary limits for the
intervals for the design parameters in soft-computing
algorithms, to control the controller by minimizing the
error and hence the determination of the optimum
parameters for the plant.
The computation of the PID parameters is done by
the Ziegler-Nichols rules, SISO Design Tool, Genetic
Algorithms, Multi-objective Genetic Algorithm and
Simulated Annealing and their closed loop step
responses are shown in Fig. 4 to 9. Figure 11 shows the
comparative response of all the controllers over a single
plot and Fig. 12 shows the comparative values of the
various steady-state parameters. Table 7 shows the
numeric comparison of the results of various steadystate parameters. From Table 7 and Fig. 11 and 12, it’s
clearly evident that Simulated Annealing solutions
present zero oscillatory response and reduced rise and
settling times in contrast to the Ziegler-Nichols, SISO,
Genetic Algorithm and Multi-objective Genetic
Algorithm. Concluding, Simulated Annealing offers
superior results in terms of system performance and
controller output for the tuning of PID controllers.
CONCLUSION
The use of Simulated Annealing for optimizing the
PID controller parameters as presented in this study
offers advantages of decreased overshoot percentage,
rise and settling times for the Maxon EC flat ф45 mm,
brushless, 30 Watt motor. Results when compared with
the other tuning mythologies as presented in this study,
the Simulated Annealing has proved superior in
achieving the steady-state response and performance
indices.
REFERENCES
Åström, K.J., P. Albertos and J. Quevedo, 2001. PID
control. Control Eng. Pract., 9: 159-1161.
Berrsimas, D., 1993. Tsitsiklis: Simulated annealing.
Stat. Sci., 8(1): 10-15.
Deb, K., 2001. Multi-Objective Optimization Using
Evolutionary Algorithms. John Wiley and Sons,
Ltd., Chichester.
Goodwin, G.C., S.F. Graebe and M.E. Salgado, 2001.
Control System Design. Prentice Hall Inc., New
Jersey.
Jean-Pierre, C., 2004. Process Control: Theory and
Applications. Springer, London, pp: 752.
Katal, N., S.K. Singh and M. Agrawal, 2012.
Optimizing response of PID controller for servo
DC motor by genetic algorithm. Int. J. Appl. Eng.
Res., 7(11).
Krause, P.C., O. Wasynozuk and S.D. Sudhoff, 2002.
Analysis of Electric Machinery and Drive Systems.
2nd Edn., IEEE Press, New York.
Norman, S.N., 2003. Control Systems Engineering,
JustAsk! Control Solutions Companion. 4th Edn.,
Wiley, Hoboken, N.J, pp: 96.
Padmaraja, Y., 2003. Brushless DC Motor
Fundamentals.
Microchip
Technology
Incorporated. Retrieved from: www.microchip.
com/stellent/idcplg?IdcService=SS_GET...1824...
Sabudin, E.N.B., 2012. Development of position
tracking of BLDC motor using adaptive fuzzy logic
controller. M.A. Thesis. Faculty of Electrical and
Electronic Engineering, Universiti Tun Hussein
Onn, Malaysia.
Ziegler, J.G. and N.B. Nichols, 1942. Optimum settings
for automatic controllers. Trans. ASME, 64:
759-768.
1122