Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

A COMPARATIVE STUDY OF PID CONTROLLER TUNING

USING GA, EP, PSO AND ACO
Submitted 9th August 2010; accepted 22nd February 2011.

B. Nagaraj, P. Vijayakumar

Abstract: to a set point change. A number of tuning techniques that

Proportional – Integral – Derivative control schemes take into consideration the nature of the dynamics present
continue to provide the simplest and effective solutions to within a process control loop have been proposed [4]. All
most of the control engineering applications today. How- these methods are based upon the dynamical behavior of
ever PID controller are poorly tuned in practice with most the system under either open-loop or closed-loop condi-
of the tuning done manually which is difficult and time tions.
consuming. This article comes up with a hybrid approach In this paper, heuristic approach to optimally design
involving Genetic Algorithm (GA), Evolutionary Pro- a PID controller, for a DC motor is proposed. A compari-
gramming (EP), Particle Swarm Optimization (PSO) and son between the results obtained by the heuristic methods
Ant Colony Optimization (ACO). The proposed hybrid and conventional methods via simulation of the DC motor
algorithm is used to tune the PID parameters and its per- is presented in results and comparison section. The pa-
formance has been compared with the conventional me- rameters of a DC motor used in this paper are listed in
thods like Ziegler Nichols and Cohen Coon method. The Table 1.
results obtained reflect that use of heuristic algorithm
based controller improves the performance of process in Table 1. Parameters of the DC Motor.
terms of time domain specifications, set point tracking,
and regulatory changes and also provides an optimum Parameters Values & Units
stability. Speed control of DC motor process is used R 21.2 W
to assess the efficacy of the heuristic algorithm (Resistance of the stator)
methodology. Kb 0.1433 Vs rad-1
(Back electromotive
Keywords: Ant colony algorithm, evolutionary program- force constant)
ming, genetic algorithm particle swarm optimization and D 1*10-4 kg m s/rad
soft computing. (Viscous coefficient)
L 0.0524 H
(Inductance of the stator)
1. Introduction Kt 0.1433 kg m/A
PID controller is a generic control loop feedback me- (Motor torque constant)
chanism widely used in industrial control systems. It cal- J 1*10-5 kg m S2/rad
culates an error value as the difference between measured (Moment of inertia)
process variable and a desired set point [3]. The PID con-
troller calculation involves three separate parameters pro- The characteristic equation of DC motor can be repre-
portional integral and derivative values .The proportional sented as,
value determines the reaction of the current error, the inte-
Ri(t)
gral value determines the reaction based on the sum of re- vapp(t) = Ldi(t) + + Vemf (t) (1)
dt
cent errors, and derivative value determines the reaction
based on the rate at which the error has been changing the Vemf = Kb × w(t) (2)
weighted sum of these three actions is used to adjust the
process via the final control element. T(t) = Kt × i(t) (3)
The goal of PID controller tuning is to determine para-
dw(t)
meters that meet closed loop system performance specifi- T(t) = J + D × w(t) (4)
dt
cations, and the robust performance of the control loop
over a wide range of operating conditions should also be Where Vapp(t) is the applied voltage, w(t) is the motor
ensured. Practically, it is often difficult to simultaneously speed, L is the inductance of the stator, i(t) is the current of
achieve all of these desirable qualities. For example, if the the circuit, R is the resistance of the stator, Vemf (t) is the
PID controller is adjusted to provide better transient res- back electromotive force, T is the torque of the motor, D is
ponse to set point change, it usually results in a sluggish the viscous coefficient. J is the moment of inertia, Kt is the
response when under disturbance conditions [11]. motor torque constant, and Kb is the back electromotive
On the other hand, if the control system is made robust force constant.
to disturbance by choosing conservative values for the From the characteristics equations of the motor, the
PID controller, it may result in a slow closed loop response transfer function is obtained,

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Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

0.1433
G(S) = (5) sover and mutation to arrive at the best solution [1]. By
5.2e - 007 s2 + 0.000217 s + 2.265
starting at several independent points and searching in pa-
The purpose of this paper is to investigate an optimal rallel, the algorithm avoids local minima and converging
controller design using the evolutionary Programming, to sub optimal solutions.
Genetic Algorithm, Particle Swarm Optimization and Ant
Colony Optimization. The block diagram of a control sys- A. Objective Function of the Genetic Algorithm
tem with unity feedback employing soft computing PID This is the most challenging part of creating a genetic
control action is shown in Figure 1[7]. algorithm is writing the the objective function. In this pro-
The general equation of PID controller is, ject, the objective function is required to evaluate the best
t PID controller for the system. An objective function could
d(e)
Y(t) = [kpe(t) + Kd + Ki e(t)d(t)] (6) be created to find a PID controller that gives the smallest
d(t)
0 overshoot, fastest rise time or quickest settling time. How-
ever in order to combine all of these objectives it was deci-
ded to design an objective function that will minimize the
performance indices of the controlled system instead [2].
Fitness Function
Each chromosome in the population is passed into the ob-
jective function one at a time. The chromosome is then
EP/GA/PS/ACO v evaluated and assigned a number to represent its fitness,
the bigger its number the better its fitness [3]. The genetic
e
U Y algorithm uses the chromosomes fitness value to create
PID PROCESSS a new population consisting of the fittest members. Each
YSP chromosome consists of three separate strings constitu-
Measured y
SENSOR ting a P, I and D term, as defined by the 3-row bounds de-
claration when creating the population [3]. When the
Fig 1. Block diagram of Intelligent PID controller. chromosome enters the evaluation function, it is split up
into its three Terms. The newly formed PID controller is
The initial values of PID gain are calculated using con- placed in a unity feedback loop with the system transfer
ventional Z – N method. Being hybrid approach, optimum function. This will result in a reduce of the compilation ti-
value of gain are obtained using heuristic algorithm. me of the program. The system transfer function is defined
The advantages of using heuristic techniques for PID in another file and imported as a global variable. The
are listed below, controlled system is then given a step input and the error is
i. Heuristic Techniques can be applied for higher order assessed using an error performance criterion such as
systems without model reduction [7]. Integral square error or in short ISE.
ii. These methods can also optimize the design criteria
such as gain margin, Phase margin, Closed Loop Band ¥

Width (CLBW) when the system is subjected to step & ISE = e2(t)dt (7)
load change [7]. 0
Heuristic techniques like Genetic Algorithm (GA),
Evolutionary Programming (EP), Particle Swarm Optimi- The chromosome is assigned an overall fitness value
zation (PSO) and Ant Colony Optimization (ACO) me- according to the magnitude of the error, smaller the error
thods have proved their excellence in giving better results larger the fitness value. Initializing the values of the para-
by improving the steady state characteristics and perfor- meters is as per Table 2. The flowchart of the GA control
mance indices. system is shown in Figure 2.

2. GAbased tuning of the controller Initialize Population

The optimal value of the PID controller parameters Kp,
Ki, Kd is to be found using GA. All possible sets of con-
troller parameters values are particles whose values are Evaluate Fitness
adjusted to minimize the objective function, which in this
case is the error criterion, and it is discussed in detail. For Select Fitness
the PID controller design, it is ensured the controller set-
tings estimated results in a stable closed-loop system [1].
Genetic Algorithms are a stochastic global search me- Mutation
thod that mimics the process of natural evolution. It is one
of the methods used for optimization. John Holland for-
mally introduced this method in the United States in the Cross over/Production
1970 at the University of Michigan. The continuing per- Non
formance improvement of computational systems has ma- Optimum
de them attractive for some types of optimization. The ge- Optimum Solution Solution
netic algorithm starts with no knowledge of the correct so-
lution and depends entirely on responses from its environ- Fig. 2. Flowchart of GA.
ment and evolution operators such as reproduction, cros-

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 Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

3. EP based tuning of the controller 4. PSO based tuning of the controller

EP is generally used to optimize real-valued continuo- PSO is one of the optimization techniques and kind of
us functions. EP uses selection and mutation operators and evolutionary computation technique. The technique is de-
does not use the crossover operator. The focus is on the ob- rived from research on swarm such as bird flocking and
served characteristics of the population. The selection ope- fish schooling. In the PSO algorithm, instead of using evo-
rator is used to determine chromosomes for mating in or- lutionary operators such as mutation and crossover to ma-
der to generate new chromosomes [22]. nipulate algorithms, for a d-variable optimization Pro-
There are two important ways in which EP differs from blem, a flock of particles are put into the d-dimensional
GAs. Search space with randomly chosen velocities and posi-
First, there is no constraint on the representation. The tions knowing their best values [8].
typical GA approach involves encoding the problem solu- The algorithm proposed by Eberhart and kennedy
tions as a string of representative tokens, the genome. In (1995) uses a 1-D approach for searching within the solu-
EP, the representation follows from the problem. A neural tion space. For this study the PSO algorithm will be app-
network can be represented in the same manner as it is im- lied to a 2-D or 3-D solution space in search of optimal tu-
plemented, for example, because the mutation operation ning parameters for PI, PD and PID control [21].
does not demand a linear encoding [7]. Consider position Xi,m of the I-th particle as it traverses
Second, the mutation operation simply changes as- a n-dimensional search space: The previous best position
pects of the solution according to a statistical distribution for this i-th particle is recorded and represented as Pbesti,n.
which weights minor variations in the behavior of the off- The best performing particle among the swarm population
spring as highly probable and substantial variations as in- is denoted as gbesti,n and the velocity of each particle with-
creasingly unlikely. in the n-dimension is represented as Vi,n. The new velocity
The steps involved in creating and implementing evo- and position for each particle can be calculated from its
lutionary programming are as follows: current velocity and distance, respectively [18].
• Generate an initial, random population of individuals So far (p best) and the position in the d-dimensional
for a fixed size (according to conventional methods Kp, space [7]. The velocity of each particle, adjusted accor-
Ki, Kd ranges declared). dingly to its own flying experience and the other particles
• Evaluate their fitness (to minimize integral square flying experience [7].
error). For example, the i th particle is represented, as
¥

ISE = e2(t)dt (7) xi = (Xi,1, Xi,2,..., Xi,d)

0
• Select the fittest members of the population. In the d-dimensional space. The best previous position
• Execute mutation operation with low probability. of the i th particle is recorded as,
• Select the best chromosome using competition and
selection. Pbesti = (Pbesti,1, Pbesti,2,..., Pbesti,d) (8)
• If the termination criteria reached (fitness function)
then the process ends. If the termination criteria not The index of best particle among all of the particles in the
reached search for another best chromosome. group in gbestd.The velocity for particle i is represented as

Initializing the values of the parameters is as per Table Vi = (Vi,1, Vi,2,..., Vi,d) (9)
2. The flowchart of the EP control system is shown in Fig. 3.
The modified velocity and position of each particle
can be calculated using the current velocity and distance
Initialize by random
from Pbesti,d to gbestd as shown in the following formulas

Fitness Evaluation
(t+1)
Vi,m = W×Vi,m
(t)
+ c1*rand()*(Pbesti,m - xi,m
(t)
)+
+ c2*Rand()*(gbestm - xi,m)
(t)
(10)
Mutation (t+1)
xi,d (t)
= xi,m + vi,m
(t+1)
i = 1,2,...,n; m = 1,2,...,d (11)

where
Competition
and selection n - number of particles in the group
d - dimension
t - pointer of iterations (generations)
(t+1)
Vi,m - velocity of particle I at iteration t
Terminati on
W - inertia weight factor
criteria reached
c1, c2 - acceleration constant
rand(n) - random number between 0 and 1
(t)
xi,d - current position of particle i at iterations
End Pbesti,m - best previous position of the ith particle
gbestm - best particle among all the particles in the
Fig. 3. Flow Chart of EP. population

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Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

In the proposed PSO method each particle contains the ants do not communicate directly but indirectly by ad-
three members P, I and D. It means that the search space ding pheromone to the environment. Based on the specific
has three dimension and particles must ‘fly’ in a three di- problem an ant is given a starting state and moves through
mensional space. Initializing the values of the parameters a sequence of neighboring states trying to find the shortest
is as per Table 2. The flowchart of the PSO – PID control path. It moves based on a stochastic local search policy
system is shown in Fig. 4. directed by its internal state, the pheromone trails, and
local information encoded in the environment. Ants use
this private and public information inorder to decide when
START
and where to deposit pheromone. In most application the
amount of pheromone deposited is proportional to the
Generate Initial Population quality of the move an ant has made. Thus the more phero-
mone, the better the solution found. After an ant has found
a solution, it dies; i.e.it is deleted from the system [13].
Run The Process Model
ACO uses a pheromone matrix t={tij} for the construc-
tion of potential good solutions. The initial values of t are
Calculate the Parameter Kp, Ki, Kd at PID controller

set tij = t0"(i,j), where t0 > 0

Calculate The Fitness Function

The probability PijA(t) of choosing a node j at node I is

Calculate The Pbest of Each Particle
and gbest of Population defined in the equation (12). At each generation of the
algorithm, the ant constructs a complete solution using
(12), starting at source node.
Update The Velocity, Position,
gbest and Pbest of Particles [tij(t)]a[hij]b
PijA(t) = if i,j Î TA (12)
å Atij(t)]a[hij]b
i,j ÎT

Maximum Iteration 1
number reacted
where hij = , j = [p,i,d]:
NO kj
representing heuristic functions.
YES
a and b are constants that determine the relative influence
STOP
of the pheromone values and the heuristic values on the
decision of the ant.
TA: is the path effectuated by the ant A at a given time.
Fig. 4. Flowchart of PSO. The quantity of pheromone DtijA on each path may be
defined as
5. ACO based tuning of the controller
ACO’s are especially suited for finding solutions to Lmin
different optimization problems.Acolony of artificial ants Dt = LA
A
ij if i, j ÎTA (13)
cooperates to find good solutions, which are an emergent 0
property of the ant’s co-operative interaction. Based on else
their similarities with ant colonies in nature, ant algo-
rithms are adaptive and robust and can be applied to dif- where:
ferent versions of the same problem as well as to different LA - is the value of the objective function found by the
optimization problems [23]. The main traits of artificial ant A.
ants are taken from their natural model. These main traits Lmin - is the best solution carried out by the set of the ants
are (1) artificial ants exist in colonies of cooperating until the current iteration.
individuals, (2) they communicate indirectly by deposi-
ting pheromone (3) they use a sequence of local moves to The pheromone evaporation is a way to avoid unlimi-
find the shortest path from a starting position, to a destina- ted increase of pheromone trails. Also it allows the forget-
tion point they apply a stochastic decision policy using fulness of the bad choices.
NA
local information only to find the best solution. If neces- tij(t) = ptij(t-1) + å DtijA (t)
A=1
sary in order to solve a particular optimization problem, (14)
artificial ants have been enriched with some additional
capabilities not present in real ants [16]. where:
An ant searches collectively foe a good solution to NA - number of ants
a given optimization problem. Each individual ant can P - the evaporation rate. 0 < p < =1.
find a solution or at least part of a solution to the optimi-
zation problem on its own but only when many ants work A. Implementation algorithm
together they can find the optimal solution [4]. Since the Step 1
optimal solution can only be found through the global co- Initialize randomly a potential solutions of the para-
operation of all the ants in a colony, it is an emergent result meters (Kp, Ki, Kd) by using uniform distribution. Initialize
of such this cooperation. While searching for a solution the pheromone trail and the heuristic value.

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 Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

Step 2 6. Results and comparisons

Place the Ath ant on the node. Compute the heuristic The transfer function of DC motor has been taken to
value associated in the objective (minimize the error). analyze the performance of various heuristic algorithms.
Transfer function is given by,
Step 3
0.1433
Use pheromone evaporation given by eqn (14) to G(S) =
5.2e - 007 s2 + 0.000217 s + 2.265
avoid unlimited increase of pheromone trails and allow
the forgetfulness of bad choices. The initial values of PID gains are calculated using
conventional Z –N method.
Step 4 In this paper a time domain criterion is used for evalu-
Evaluate the obtained solutions according to the ating the PID controller. A set of good control parameters,
objectives. P, I and D can yield a good step response that will result in
performance criteria minimization in time domain [18].
Step 5 These performance criteria in time domain include the
Display the optimum values of the optimization overshoot, rise time and settling time.
parameters. To show the effectiveness of the heuristic method,
a comparison is made with the conventional designed PID
Step 6 controller with GA, EP, and PSO &ACO method.
Globally update the pheromone, according to the At first method, PID controller design using Z – N
optimum solutions calculated at step 5. Iterate from step 2 method & the values of designed PID controller are
until the maximum of iterations is reached. Kp = 9.3883, Ki = 36.4170, and Kd = 0.6051.
Initialize the values of the parameters EP, GA, PSO &
Initializing the values of the parameters is as per Table ACO is as per table 2. The values of EP, GA, PSO and
2. The flowchart of the ACO – PID control system is ACO designed PID controllers are tabulated in table 3.
shown in Fig. 5. Performance characteristics of DC motor were indicated
& compared with heuristic tuning methods as shown in
START Fig 6.
Simulation shows the performance characteristics of
conventional method of controller tuning lead to a large
Initialize - Number of ants, Pheromone,
Probability selected path. Population of settling time, overshoot, rise time & steady state error,
(Kp, Ki, Kd) GA, EP, PSO & ACO based tuning methods have proved
their excellence in giving better result by improving the
steady state characteristics and performance indices.
Run The Process Model
7. Conclusion
Evaluate the Fitness Function Research work has been carried out to get an optimal
PID tuning by using GA, EP, PSO and ACO. Simulation
Udate Pheromone and Probability results demonstrate the tuning methods that have a better
control performance compared with the conventional
Calculate Optimum of ones. It is possible to consider several design criteria in
Kp, Ki, Kd a balanced and unified way. Approximations that are
typical to classical tuning rules are not needed. Soft com-
puting techniques are often criticized for two reasons:
Maximum Iteration algorithms are computationally heavy and convergence to
number reached the optimal Solution cannot be guaranteed. PID controller
tuning is a small-scale problem and thus computational
complexity is not really an issue here. It took only a couple
of seconds to solve the problem. Compared to conventio-
STOP nally tuned system, GA, EP , PSO and ACO tuned system
has good steady state response and performance indices.
Fig 5. Flowchart of ACO.

Table 2. PSO, GA, EP and ACO Parameters.

PSO PARAMETERS GA PARAMETERS EP PARAMETERS ACO PARAMETERS

Population size:100 Population size:100 Population size:100 Population size:100
Wmax=0.6/ Wmin=0.1 Mutation rate:0.1 Normal distribution No of Ants = 10
C1 = C2 = 1.5 Arithmetic Crossover Mutation rate: 0.01 No. of Path = 15
C1 = C2 = 2
Iteration:100 Iteration:100 Iteration:100 Iteration :100
Fitnessfunction:ISE Fitnessfunction:ISE Fitnessfunction:ISE Fitnessfunction: ISE

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Journal of Automation, Mobile Robotics & Intelligent Systems VOLUME 5, N° 2 2011

Table 3. Comparison result of Z-N and Heuristic methods.

Tuning PID Parameters Dynamic performance specifications Performance

Method Index
Kp Ki Kd Tr Ts M p (%) ISE
(Proportional gain) (Integral gain) (Derivative Gain) (Rise time) (Settling time) (Peak overshoot) (Integral
square error)
ZN 9.3883 36.4170 0.6051 1.27 2.235 1 2.2926
EP 10 100 0.1 0. 468 0.877 0. 0 1.334
GA 3 90 0.001 0. 444 0.781 0. 0 1.4406
PSO 1.5 500 0.02 0. 0761 0.13 0. 0 1.0024
ACO 10 200 0.21 0. 00105 0.429 0. 0 1.174

Step Response
Amplitude

Time (sec)

Fig. 6. Comparison result of Z-N and Heuristic methods.

margins”, Automatica, vol. 20, 1984, pp. 645- 651.

AUTHORS [5] A.A. Khan, N. Rapal “Fuzzy PID controller: design,
B. Nagaraj* - Tamilnadu Newsprint and Papers Ltd, Ta- tuning and comparison with conventional PID control-
milnadu, India. Research Scholar Karpagam University. ler”. In: IEEE International Conference on Engineering
E-mail: nagarajice@gmail.com. of Intelligent Systems, 2006, pp. 1-6, DOI 10.1109/
P. Vijayakumar - Karpagam College of Engineering, ICEIS.2006.1703213.
Tamilnadu, India. [6] S. Saha, “Performance Comparison of Pid base Position
* Corresponding author control system with FLC based position control sys-
tem”, TIG Research Journal, vol. 1, no. 2, Sept. 2008.
[7] J. Lieslehto, “PID controller tuning using Evolutionary
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