Particle Swarm Optimization Based Intelligent Tuning Of PID Controller For DC
Servo Motor Control
1
Aishwarya Yadav , 2 Dr. Sulochana Wadhwani, 3Dr. A. K. Wadhwani
Electrical Engineering Department, MITS, Gwalior, MP-India, yadav.aishwarya@gmail.com
2
Electrical Engineering Department, MITS, Gwalior, MP-India, sulochana_wadhwani1@rediffmail.com
3
Electrical Engineering Department, MITS, Gwalior, MP-India, wadhwani_arun@rediffmail.com
1
Abstract— This paper proposes that the PID controller
process output. Therefore, efficient design and tuning
be tuned using the Particle Swarm Optimization (PSO)
methods leading to an optimal and effective operation of the
algorithm. The proposed technique in this is compared to
PID controllers in order to regulate the different parameters
PID controller tuned by conventional Zeigler-Nichols
of the plant are economically vital for process industries.
technique. The technique was implemented and analysed
The main goal of this paper is to analyse the implementation
on a third order plant model of a DC servomotor with the
intelligent technique viz. Particle Swarm Optimization
aim of developing a position controller. The PID
(PSO)algorithm for optimal tuning of PID controllers
controller is by far the most commonly used controller
parameters and enumerate their advantages over the
strategy in the process control industry due to ease of
conventional
implementation and robust performance .Conventional
Optimization (PSO) was inspired from the social behavior of
Tuning method of PID parameters such as Ziegler-
bird
Nichols method do not provide optimal PID tuning
Algorithms are effective and intelligent choice at finding the
parameters and usually results in closed-loops responses
best solution among the space of all feasible solutions. PSO
characterized by oscillations and a large overshoot. It was
algorithms were used to evaluate the optimum PID controller
found that the PID controller tuned using Particle Swarm
gain values where performance index Mean Of The Square
Optimization
Of The Error (MSE) was used as the objective functions.
(PSO)
exhibited
better
steady-state
flocking.
tuning
methodology.
Particle
Swarm
Particle
Optimization
Swarm
(PSO)
response. Tuning techniques are validated by MATLAB
1
simulations of a DC motor
2
𝑀𝑆𝐸 = ∑𝑛𝑖=1(𝑒(𝑡))
𝑛
Keywords—Particle Swarm Optimization (PSO),PID
controller, DC Servo motor, PID Tuning, Ziegler Nichols
(ZN)
The proposed methodology was verified using a third-order
physical plant (Armature-controlled DC servomotor position
control system)
I.
INTRODUCTION
In process control industry, majority of control system loops
are
(1)
based
on
Proportional-Integral-Derivative
(PID)
controllers. PID controllers are being widely used in industry
due to their well-grounded established theory, simplicity,
maintenance requirements, and ease of tuning. The basic
structure of the PID controllers makes it easy to regulate the
II.
DC SERVO MOTOR MODELLING
As a reference we consider armature controlled DC
servomotor as shown in Figure 1. In the point of control
system, DC servo motor can be considered as linear SISO
plant model having third order transfer function. The DC
servo motors are found to have an excellent speed and
position control. A simple mathematical relationship between
kb back emf constant
0.1 V/rad/sec
the shaft agular position ‘  ’ and voltage input ‘ Va (s) ’ to the
k m motor torque constant
0.1 N.m/Amp
Ra electric resistance
DC motor may be derived from physical laws.
La
electric inductance
2.0 Ohm
0.5 H
the overall transfer function of the system is given as:
 (s)
0.1
(4)

Va ( s ) 0.01s 3  0.09 s 2  0.21s
III.
PID CONTROLLER
The PID controller, represented by Figure 3, is well known
and widely used to improve the dynamic response as well as
to reduce or eliminate the steady state error. The Derivative
controller adds a finite zero to the open loop plant Transfer
function and improves the transient response. The Integral
Figure.1 Schematic Diagram of armature controlled DC
controller adds a pole at the origin, thus increasing system
type by one and reducing the steady state error due to a step
Servo motor
function to zero. PID controller consists of three types of
control Proportional, Integral and Derivative control .
The Over all transfer function of DC servo motor for position
control is
 ( s)
Va ( S )

km
s[( La s  Ra )( Js  b)  (km kb )]
(2)
Or
km
 (s)

Va ( S ) La Js 3  ( La b  Ra j ) s 2  ( Ra b  k m k b ) s
(3)
Here the angular displacement  (s ) is considered the output
and the armature voltage Va (s) is considered the input. The
Figure.3 Block diagram of conventional PID controller
block diagram representation is shown in figure 2.
The PID controller output relating the error can be described
by,
t
k de(t )
u (t )  k pe(t )  ki  e(t )dt  d
dt
0
(5)
Disturbance 𝑇𝑑
Armature
𝟏
(𝑳𝒂𝒔 + 𝑹𝒂)
𝑽𝒂(𝒔)
+
Load
𝑰𝒂(𝒔)
𝒌𝒎
𝑻𝒎
𝑻𝒍
𝟏
(𝑱𝒔+ 𝑩)
𝑽𝒃(𝒔)
𝑆𝑝𝑒𝑒𝑑 𝟏
𝒔
𝝎(𝒔)
Where
𝜽(𝒔)
plant) can be expressed by the following transfer function:
Various parameters associated with the motor are:
b
moment of inertia of the rotor
motor viscous friction constant
ki and
are the Proportional, Integral and Derivative
gains.
In the frequency domain, the relation between the PID
controller input e (error signal) and output u (input to the
𝒌𝒃
Figure.2 Block Diagram representation of a DC Servo
motor
J
e(t ) is the error, u (t ) the controller output and kp
kd
0.02 kg.m2
0.1 N.m.s
k
U (S )
(6)
 k p  i  kd s
E(S )
s
The closed loop transfer function is given by,
G PID ( S )G( S )
Y (S )
(7)
GCL 

R( S ) 1  G PID ( S )G( S )
The tuning of a PID controller consists of selecting gains Kp,
G PID ( S ) 
Ki and Kd so that performance specifications are satisfied.
IV.
TUNING OF PID CONTROLLER USING
CONVENTIONAL APPROACH
A. Conventional Approach - Ziegler Nichols Method
Ziegler-Nichols (ZN) method for tuning of PID controllers,
though a classic method, has been widely used for the design
of various controllers. Ziegler and Nichols presented two
methods, a step response method and a frequency response
method. In this paper we have employed the frequency
response method for tuning of the PID controller.
Figure 4 Step Response of DC Motor with ZN Tuned PID
B. Implementation of ZN based PID controller
In this method, the integral time Ti will be set to infinity and
V.
PARTICLE SWARM OPTIMIZATION
A. Overview
the derivative time Td to zero. This is used to get the initial
PID setting of the system. Thus the proportional control is
selected alone. Increasing the value of the proportional gain
until the
point of instability is reached
(sustained
oscillations), gives the critical value of gain, Kc. Thereafter
measurement of the period of oscillation of the response is
used to obtain the critical time constant, Tc.
Once the values for Kc and Tc are obtained, the PID
Figure.5 PSO based PID Controller
parameters can be calculated, according to the design
specifications, as given in Table 1. Further the values of the
PID gain coefficients Kp, Ki and Kd for the system described
by equation (6), obtained after simulation in MATLAB are
Particle swarm optimization (PSO) algorithm is a populationbased evolutionary computation technique developed by the
inspiration of the social behaviour in bird flocking or fish
schooling. It attempts to mimic the natural process of group
given in Table II.
communication of individual knowledge, to achieve some
TABLE I Ziegler-Nichols PID tuning parameters
optimum property. In this method, a population of swarm is
initialized with random positions Si and velocities Vi. At the
CONTROLLER
Kp
Ti
Td
PID
0.6Kc
0.5Tc
0.125Tc
beginning, each particle of the population is scattered
randomly throughout the entire search space and with the
guidance of the performance criterion, the flying particles
TABLE II Ziegler-Nichols PID tuning values
dynamically adjust their velocities according to their own
flying experience and their companions flying experience.
Gain
Coefficients
Kp
Ki
Kd
In PSO, each single solution is a “bird” in the search space;
Values
113.4
165.41
19.43
this is referred to as a “particle”. The swarm is modelled as
Here,
𝐾𝑖 =
𝐾𝑝
𝑇𝑖
and
𝐾𝑑 = 𝐾𝑝 × 𝑇𝑑
particles in a multidimensional space, which have positions
and velocities. These particles have two essential capabilities:
From the above formulation the step response of the overall
their memory of their own best position and knowledge of
system with conventionally Zeigler-Nichols tuned PID
the global best [14].
controller is shown in Figure.4.
Each particle remembers its best position obtained so far,
which is denoted as pbest (𝑃𝑖𝑡 ). It also receives the globally
best position achieved by any particle in the population,
which is denoted as gbest (𝐺𝑖𝑡 ).
The updated velocity of each particle can be calculated using
the present velocity and the distances from pbest and gbest as
Step 6
Steps 2–5 are repeated until the predefined value of
the function or the number of iterations has been
given by the following equations:
reached. Record the optimized Kp, Ki and Kd
𝑉𝑖𝑡+1 = 𝑊 𝑡 ∙ 𝑉𝑖𝑡 + 𝐶1 ∙ 𝑅1 ∙ (𝑃𝑖𝑡 − 𝑆𝑖𝑡 ) + 𝐶2 ∙ 𝑅2 ∙ (𝐺𝑖𝑡 − 𝑆𝑖𝑡 )
𝑆𝑖𝑡+1 = 𝑆𝑖𝑡 + 𝑉𝑖𝑡+1
𝑊 𝑡 = (𝑊𝑚𝑎𝑥 − 𝐼𝑡𝑒𝑟) × [
values
(8)
(9)
(𝑊𝑚𝑎𝑥 −𝑊𝑚𝑖𝑛 )
𝐼𝑡𝑒𝑟𝑚𝑎𝑥
]
(10)
Step 7
Perform closed-loop test with the optimised values
of controller parameters and calculate the time
The updated velocity and the position are given in (8) and
domain specification for the system.
(9), respectively. Equation (10) shows the inertia weight.
If the values are within the allowable limit, consider
B. PSO-Based PID Controller Optimization
1) PSO Tuning Parameters
The values in the Table III describe the PSO settings used
for this work
TABLE III
PSO Tuning Parameters
the current Kp, Ki and Kd values. Otherwise
perform the retuning operation for Ki, by replacing
the optimized numerical values for Kp and Kd.
Figure.6 Shows the optimized values of Kp,Ki and Kd for
different iterations
PARAMETERS
Lower bound [Kp Ki Kd]
Upper bound [Kp Ki Kd]
Max Iterations
Population Size
Inertial weight [Wmax,Wmin]
VALUES
[0 0 0]
[5 5 5]
10
120
[0.8,0.4]
Acceleration coefficients [c1, c2]
[2 2]
2)
Step 1
Steps in PSO-Based PID Controller Optimization
% Assign values for the PSO parameters %
Initialize: swarm (N) and step size; learning rate (C1,
C2) dimension for search space (D); inertia (W);
Step 2
% Initialize Swarm Velocity and Position %
Step 3
Evaluate the objective function of every particle
and record each particle’s 𝑃𝑖𝑡 and𝐺𝑖𝑡 . Evaluate the
desired optimization fitness function in D-dimension
variables.
Step 4
Compare the fitness of particle with its 𝑃𝑖𝑡 and
replace the local best value as given below.
for i=1: N
If current fitness (i) < local best fitness (i);
Then local best fitness = current fitness;
local best position = current position (i);
end
% same operation to be performed for 𝐺𝑖𝑡 %.
Step 5
Figure .6
Change the current velocity and position of the
The output response of the system tuned by using PSO-based
particle group according to (1) and (2).
PID controller is shown in figure 7.The system exhibits very
less overshoot comparatively conventionally tuned Zeigler-
and setting time are obtained by PSO tuned PID Controller
Nichols Method .
are a bit on higher side but are in acceptable limit.
The comparative output responses of the system tune using
PSO-based PID controller and conventionally tuned PID
controller using Zeigler Nichols (ZN) method is shown in
figure 8. The PSO tuned system shows greatly reduced
overshoot
Figure.7 Step Response of DC Motor with PSO Tuned
PID
VI.
SIMULATION RESULTS
In order to improve the performance of the dc motor under
transient and steady state condition, a PID controller is
Fig.8 Comparitive step responses for PSO and ZN tuned
system
inserted in the forward path as shown in Fig 5. The
parameters of the PID controller are now adjusted by using
conventional method i.e. Ziegler-Nichols method and the
response obtained for the DC servomotor is evaluated.
Further again the parameters of PID controller are obtained
using Artificial Intelligence based PSO Algoriths and the
system step responses are evaluated.
VII.
CONCLUSIONS
In this paper, two different control techniques are used for
the tuning controller which are, conventional Ziegler Nichols
Method and Intelligent control technique(PSO).Application
of Intelligent technique (PSO) to the optimum tuning of PID
controller led to a satisfactory close-loop response for the
system under consideration. The controller tuned using
The controller gains were computed by using the classical
Zeigler-Nichols rules and Particle swarm optimization. The
controller gains obtained from the methods are listed in Table
IV.
Particle Swarm Optimization (PSO) algorithms resulted in
the most satisfactory performance (very less overshoot,
minimal rise time).The same Intelligent technique(PSO) can
be implemented with the Induction motor drive for analysis
TABLE IV
Comparison of steady state responses
TITLE
ZN_PID
PSO_PID
Rise Time(sec)
0.0813
0.3437
Settling Time (sec)
2.3630
7.1901
% Overshoot
74.40
9.977
Peak Time (sec)
0.2290
1.0998
Kd
19.4353
4.8534
Kp
113.40
3.9353
Ki
165.45
2.1101
of dynamic response .
VIII.
[1] Dorf and Bishop, Modern Control Systems, 9th
Ed., Prentice-Hall, Inc. 2001.
[2] Neenu Thomas, Dr. P. Poongodi, Position Control
of DC Motor Using Genetic Algorithm Based PID
Controller, Proceedings of the World Congress on
Engineering 2009 (ISBN: 978-988-18210-1-0) Vol
II WCE 2009, July 1 - 3, 2009, London, U.K.
[3] Harinath Babu Kamepalli, The optimal basics for
Gas, IEEE Potentials, 0278-6648/01/$10.00 © 2001
IEEE, April/May 2001 pp 25-27.
Figure 8 shows the corresponding step responses of ZN and
[4] Tze-Fun Chan, and Keli Shi, “Applied Intelligent
Control of Motor Drives” IEEE Willey Press, First
edition, 2011.
PSO-based PID systems. It can be clearly seen that PSO
tuned system shows improved response with respect to the
overshoot as compared to that of ZN tuned system. Rise time
REFERENCES
.
[5] P.C. Krause, "Analysis of Electrical Machinery and
Drives System, "IEEE Willey Press, 2000.
[6] J. Kennedy and R. Eberhart, “Particle swarm
optimization,” in Proc. IEEE Int. Conf. Neural
Networks, vol. IV, Perth, Australia, 1995, pp.
1942–1948.
[7] Astrom , K.J. and A _ rz_en, K.E. (1993) Expert
control, in An Introduction to Intelligent and
Autonomous Control(eds P.J. Antsaklis, K.M.
Passino, and MA. Norwell) Kluwer Academic
Publishers, pp. 163–168.
[8] Astrom, K.J. and Bjorn, W. (1995) Adaptive
Control, Addison Wesley Publishing Company,
Reading, MA.
[9] Blashke, F. (1972) The principle of fieldorientation as applied to the new ‘transvektor’
closed-loop control system for rotating-field
machines. Simians Review, 34(5), 21–220.
[10] Bose, B.K. (1997b) Intelligent control and
estimation in power electronics and drives. The
1997 IEEE Internationa lElectric Machines and
Drives Conference, USA, May.
[11] Bose, B.K., Patel, N.R., and Rajashekara, K. (1997)
A neuro fuzzy based on-line efficiency
optimization control ofa stator flux oriented direct
vector controlled induction motor drive. IEEE
Transactions on Industrial Electronics,44, 270–273.
[12] Simoes, M.G. and Bose, B.K. (1995) Neural
network based estimation of feedback signals for a
vector controlled induction motor drive. IEEE
Transactions on Industry Applications, 31, 620–
629.
[13] M. Molenaar, P. Nijdam, Y. Yan, W.A. Klop,
Tuning a PID controller: Particle Swarm
Optimization
versus
Genetic
Algorithms,
http://www.martinm.nl/attachments/028_paper_
tuning_ a_PID_controller.pdf
[14] N. Pillay, “A Particle Swarm Optimization
Approach for Tuning of SISO PID Control Loops”,
Durban University of Technology, 2008.