International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 1191

ISSN 2229-5518

Design of Intelligent PID Controller using Particle Swarm Optimization with Different Performance Indices

Neha Kundariya, Jyoti Ohri

Abstract— PID control scheme is still providing simple and effective solutions to the many control engineering problems. Today many real world problems are nonlinear and time varying, control of such problems are very difficult with the conventional designed PID controller. This paper presents a new designing technique of PID controller, which can effectively control the nonlinear optimization problems. The technique which has been introduced in this research is particle swarm optimization. It is population based stochastic optimization algorithm derived from human behavior and animal behavior as well. Conventional gain tuning schemes such as Ziegler Nichols method usually produces big overshoot therefore modern approach has been used in this paper to tune the parameters of PID controller. For designing of PID controller different performance indices have been used here for different plant transfer functions.

Index Terms—Fitness function, Particle Swarm Optimization, PSO parameters, PID tuning, Social behavior,Time varying inertia weight, Ziegler NicholsTuning method.

—————————— ——————————

Particle swarm optimization has been used here for the tuning

ID control is the most ancient and the strongest control method in process industries. With the advancement in technology control systems are becoming more and more complex day by day. Conventional PID control is not able to solve such complex problems. In recent years many intelligent controllers have been introduced such as fuzzy PID controller, neural network and so on. The intelligent PID controllers hav- ing the properties such as self adaptability, self learning ability and self organization are able to control complex systems [11]. PID controller is widely used in industrial control systems. PID controller calculates the error between set point value and measured response. The objective of PID controller is to min- imize the generating error. PID controller calculation involves three terms proportional, derivative and integral [12]. The purpose of proportional term is to determine the reaction of current error, integrating term determines the reaction of sum of current error and derivative term determines the rate of error generating. The objective of PID controller tuning is to design such a controller which meet the desired closed loop performance. A PID controller improves the transient re- sponse of the system by reducing the overshoot in the step response, and by reducing the settling and rise time. Standard methods of PID tuning involve Ziegler Nichols [4], Cohen- coon’s [6], Astrom and Hagglund [5] and many other tech- niques. This paper presents soft computing technique for de-

signing an intelligent PID controller.

of PID controller. PSO is a population based stochastic optimi- zation algorithm which is first proposed by Eberhart and Kennedy in 1995 [3]. This technique is derived from research on biological organism such as bird flocking and fish school- ing. Craig Reynolds (1987) [1] showed that flock is simply the result of the interaction between the behaviours of individual birds. To simulate a flock we simulate the behaviour of an in- dividual bird. He concluded that to build a simulated bird flock model following three simple rules must be followed: Velocity Matching, centring of bird flock and avoid collisions. Work of Kennedy and Eberhart was influenced by Heppner and Grenander’s (1990) work on simulated behaviour of bird [2].

This section presents introduction about PID controller and proposed scheme of PID tuning, section II presents model of the plant, section III contains model intelligent PID controller, section IV presents particle swarm optimization, section V contains PSO based controller, section VI presents simulation results and comparison and finally section VII contains coclu- sion.

Three plants have been used here for the study. Mathematical model of plants are:

1) Mathematical model of DC shunt motor is given by

[12]:

————————————————

• *Neha Kundariya is currently pursuing masters degree program in electrical*

𝐺1 (𝑠) =

𝑌(𝑠)

𝑈(𝑠)

= 0.005𝑠2

0.01

+0.006𝑠+0.1001

(1)

engineering in NIT Kurukshetra, India.

• *Jyoti Ohri is currently Associate Professor in electrical engineering in NIT Kurukshetra, India.*

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2) Mathematical model of electric DC motor is [13]

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International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 1192

ISSN 2229-5518

𝐺2 (𝑠) =

0.9

0.00105𝑠3 +0.2104𝑠2 +0.8913𝑠

(2)

In PSO a number of particles are placed in the search space of some problem. Each particle in the swarm evaluates the objec- tive function at its current location. Each particle then move through the search space according to the history of its own

3) Mathematical model of time delayed process[14]

𝐺 (𝑠) = 𝑒 −1𝑠

(𝑠+1)

(3)

current and best location of neighbourhood in the swarm on each iteration. The next iteration takes place after all particles have been moved. In PSO swarm moves like a bird flock searching for food. Each individual in the swarm is composed of three d-dimensional vectors, where d is the dimension of the search space. Three vectors are the current position xi , the previous best position pi , and the velocity vi [9].

Objective of tuning method is to find a set of controller pa-

rameters which gives better results. They provide a control

signal which is given by:

de(t)

The PSO algorithm based on the concept that individual member refine their knowledge about the search space by so- cial interaction. In PSO each member is called particle and population is called swarm. The term ‘swarm’ means irregular

u(t) = kp e(t) + ki ∫ e(t)dt + kd

(4)

movement of particles.

Particle swarm optimization is a member of swarm in-

where ‘e’ is the error, kp , ki and kd are controller parameters,

P

input error output

∑ I ∑ Plant

D

Fig.1. Block diagram of PID controller

and ‘u’ is controller output. Block diagram of PID controller is shown in fig.1. and block diagram of Intelligent PID controller is shown in fig.2.

telligence family it has some advantages over other intelligent optimization techniques [10]:

1) It is simple to implement

2) There are fewer parameters to adjust

3) It has more effective memory capability

4) It uses a relatively small population

5) It is fast

6) PSO is more effective in maintaining diversity of swarm and lead to fast convergence

These advantages have given it popularity to solve nonlinear optimization problems in the field of evolutionary computa- tion. PSO have been successfully applied in many areas of sys- tem design, system modeling, system identification, signal processing, pattern recognition, robotic applications. The algo- rithm of PSO include following steps [8]:

1) Initialize the swarm by assigning random position and velocity to each particle.

2) Evaluate fitness function for each particle.

3) Compare the current fitness value with the pbest val-

input error

_

PID Controller

Plant

output

ue of the particle in history.

4) If current fitness value is better than the previous best

value (pbest), then set this value as current pbest.

5) Now best evaluated value of pbest is set as gbest val-

ue.

6) Update the velocity and position of the particles ac-

cording to the equation (5) and (6).

7) Repeat the steps 2 to 6 until sufficiently good stop-

Fig.2. Block diagram of Intelligent PID controller

The objective of PID controller is to adjust parameters like that system perform better in the wide range of operating condi-

ping criterion is met such as maximum number of it- erations or best fitness value.

Update in particle’s velocities and positions are given by fol- lowing equations:

tions. PID controller improves the transient as well as steady

k+1 = w ∗ vk + c1 ∗ r1 ∗ �pbest

− xk � + c2 ∗ r2

state response of the system. PSO tuned PID controller is

vn,d

n,d

n,d

n,d

shown in fig.2 in which parameters are adjusted by PSO algo- rithm [11].

∗ (gbestd − xk ) (5)

k+1 = xk + vk+1 (6)

xn,d

n,d

n,d

In 1995 James Kennedy and Russel Eberhart [3] proposed first simulation which was influenced by Heppner and Grenander’s (1990) work on simulated behaviour of bird [2].

n=1,2,.....,N , d=1,2,........,D and k=1,2,.....,T

where,

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International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 1193

ISSN 2229-5518

N Number of particles

D Dimension of the problem space

T Maximum number of iterations

vk+1 Velocity of nth particle with dimension d at

∞

𝐽 = 𝑒 (𝑡)𝑑𝑡

0

(9)

iteration k+1

(k)

(k)

3) Integral of Time multiplied by Absolute Errors

if vi,j

else if v(k)

> 𝑣𝑚𝑎𝑥 then vi,j = 𝑣𝑚𝑎𝑥

(k)

𝑚𝑎𝑥 i,j 𝑚𝑎𝑥

∞

𝐽𝐼𝑇𝐴𝐸 = ∫0

𝑡|𝑒(𝑡)|𝑑𝑡

(10)

xk Current position of nth particle with

dimension

c1,c2 Acceleration factors

r1,r2 Random numbers between [0,1]

pbestn,d Personal best value of nth particle with dimension d

gbestn,d Global best value of swarm w Inertia weight

Most of the PSO strategies use time varying inertia weight. This inertia weight may be linear or nonlinear and increasing or decreasing in nature.

Simulation results of proposed tuning method for different performance indices in time domain are showm in table 1, 2 and table 3.

TABLE 1

TIME DOMAIN PERFORMANCE SPECIFICATIONS FOR

PLANT 1

Shi and R. Eberhart proposed a new method in 1998 [7]. In this | Peak | Rrise | settling | PI | |

method the value of ‘w’ starting with the value greater than 1 | Overshoot(%) | time(sec) | time(sec) | ||

and decreasing eventually to a value less than 1 so later on it | |||||

was kept linear from 0.9 to 0.4.Inertia weight ‘w’ is given by: | ITAE | 0 | 0.22 | 0.39 | 0.01046 |

w(iter) = wmax − �

wmax −wmin

itermax

� ∗ iter (7)

IAE 2.02 0.03 1.15 0.05948

ISE 0.58 0.008 0.013 0.00585

where wmax = 0.9 and wmin = 0.4

To design PID controller with PSO some parameters and fit- ness function are required:

TABLE 2

TIME DOMAIN PERFORMANCE SPECIFICATIONS FOR

PLANT 2

Particle swarm optimization algorithm is population based

Peak

Overshoot(%)

Rrise time(sec)

settling PI

time(sec)

technique so first of all we have to produce initial swarm of particles in search space represented by a matrix of dimension swarm size x 3. Three parameters are there to be tuned where their values are set in the range of 0 to 100. For this three di- mensional problem position and velocity are represented by matrices of dimension swarm size x 3. Swarm size is the num- ber of particles, 30 is considered here for the problem. Maxi- mum number of iterations 25 has been used here. However more number of iterations produce better results but for study and comparison between different performance indices 25 iterations have been used in this paper.

The objective function which has been used here is error crite-

ria. Performance of controller is based on error criterion or

performance index. Commonly employed performance indi-

ces are [15]:

1) Integral of Absolute Errors, given by

ITAE 5.63 0.02 0.348 0.00656

IAE 28.3 0.008 0.067 0.01916

ISE 37.2 0.007 0.075 0.0072

TABLE 3

TIME DOMAIN PERFORMANCE SPECIFICATIONS FOR

PLANT 3

∞ IAE 0

|e(t)|dt

(8)

Comparison of different performance indices for plant 1, plant

2 and plant 3 are shown in fig3, fig 4 and fig 5. From the re-

2) Integral of Squared Errors, given by

sults we can see that integral time of absolute error perform-

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International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 1194

ISSN 2229-5518

nace criterion gives better results than other performance cri- terion. It produces less overshoot in all the three case and op- timum value is achieved by it.

1.4

ITAE

A new optimization technique has been implemented here on three different plant transfer functions for obtaining the step response of systems for different performance indices. From the results obtained, it is clear that integral time of absolute error gives good results in time domain compare to other per-

1.2

1

0.8

0.6

0.4

0.2

0

IAE ISE

0 1 2 3 4 5 6 7 8 9 10

Time

formance indices. It does not give big overshoot and also give

less settling time and less rise time.

[1] C.W. Reynolds, “Flocks, herds and schools: a distributed behavioral model”, Computer Graphics, 21(4), p.25-34, 1987.

[2] Heppner and Grenander, “A stochastic nonlinear model for coordinated bird flocks”, In S. Krasner, Ed., The Ubiquity of Chaos. AAAS Publications, Washington, DC, 1990.

[3] J. Kennedy and R.C.Eberhart, “Particle swarm optimization,” in Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948, Dec 1995.

[4] Ziegler, G. and Nichols, N. B “Optimum settings for automatic

controllers”, Trans. ASME, 64,759-768, 1942..

Fig.3. Comparison of different performance indices for plant 1

1.4

ITAE

[5] Astrom, K J.;.Hagglund .T, “Automatic tuning of simple regulators with specifications on phase and amplitude margins”, Automatica, 20,645-651, 1984.

[6] G.H Cohen and G.A Coon, “Theoretical Consideration of

1.2

1

0.8

0.6

0.4

0.2

0

IAE

ISE

0 1 2 3 4 5 6 7 8 9 10

Time

Retarded Contro”, Trans ASME 75,pp.827/834,(1953).

[7] Y.Shi and R.Eberhart, “A modified particle swarm optimizer”, in Proc. IEEE World Congr. Comput. Intell, pp. 69-73, 1998.

[8] Yamille del Valle, Ganesh Kumar Venayagamoorthy,Salman Mohagheghi, Jean-Carlos Hernandez and Ronald G. Harley, “Particle swarm optimization:Basic concepts,Variants and Applications in power systems”, IEEE Transaction on Evolutionary Computation,Vol. 12, No. 2, April 2008.

[9] F. van den Bergh, “An Analysis of Particle Swarm Optimizaers”, PhD thesis, Department of Computer Science, University of Pretoria, South Africa 2002.

[10] R. Mendes .Population Topologies and Their Influence in Particle Swarm Performance. PhD thesis, Escola de Engenharia, Universidade do Minho, 2004.

Fig.4. comparison of different performance indices for plant 3

150

ITAE IAE

[11] Liguo Qu, Yourui Huang, and Liuyi Ling, “Design of Intelligent PID Controller Based on Adaptive Genetic Algorithm and Implementation of FPGA”, F. Sun et al. (Eds.): ISNN 2008, Part II, LNCS 5264, pp. 542–551, 2008.

100

50

0

-50

ISE

[12] Mehdi Ghazavi Dozein, “Speed Control of DC Motor using

Different Optimization Techniques based PID Controller”, Journal of Applied Scientific Research. 2(7) 6488-6494, 2012.

[13] B. Nagraj, S. Subha, and B. Rampriya, “Tuning algorithm for PID controller using soft comuting techniques”, IJCSNS International journal of computer science and network security, Vol.8, No. 4, April 2008.

[14] Dan Chen and Dale E.Seborg, “PI/PID controller design based on direct sysnthesis and disturbance rejection”, Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002.

-100

0 1 2 3 4 5 6 7 8 9 10

Time

Fig.5. comparison of different performance indices for plant 3

[15] S. M. Giriraj Kumar, Deepak Jayaraj, Anoop. R. Kishan, “PSO based tuning of a PID controller for a high performance drilling machine”, International Journal of Computer Applications(0975-8887), Volume

1, No. 19, 2010.

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