design of a hybrid controller using differential evolution and mit rule for magnetic levitation...

8/20/2019 Design of a Hybrid Controller using Differential Evolution and MIT Rule for Magnetic Levitation System

http://slidepdf.com/reader/full/design-of-a-hybrid-controller-using-differential-evolution-and-mit-rule-for 1/7

www.ijsret.org

International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN 2278 – 0882,

Volume 3 Issue 1, April 2014

Design of a Hybrid Controller using Differential Evolution and MIT Rule for

Magnetic Levitation System

Priyank Jain*, M. J. Nigam**

*Department of Electronics and Communication Engineering, Indian Institute of Technology, Roorkee, India

**Department of Electronics and Communication Engineering, Indian Institute of Technology, Roorkee, India

ABSTRACTIn various industrial systems, parameters variation is one

of the major problems faced by control engineers now

days. To overcome the problem of parameter variations,this paper proposes the hybridization of MIT rule based

online tuning of classical PID controllers with

Differential Evolution algorithm. The hybridization of

two techniques results in the offline as well as online

tuning of PID controllers at the same time. The

developed hybrid controller is then applied on magneticlevitation system using MATLAB and Simulink. The

paper also describes the basic steps involved in

Differential Evolution (DE) algorithm and a comparison

between designed hybrid controller and the simple DEalgorithm based controller has been carried out. The

results depict that the performance of the hybrid

controller is better than the offline tuned PID controller

in terms of transient parameters such as peak overshoot

and settling time.

Keywords – Differential evolution, hybridization, MIT

rule, PID controller, soft computing.

I. INTRODUCTION

Inherent disturbances and inaccuracies lead to

parameter variations in any physical system which may

result in degradation in the performance and sometimes

damage the system. To solve the problem of parameter

variation, one needs to design the control system with

more powerful and advanced techniques so as to

maintain the satisfactory performance of the overall

system. 3Adaptive Control is one of the widely used

advanced control strategies, in which one needs todesign an adjustment mechanism to alter the adjustable

parameters of controller [1]. Gradient theory based MIT

rule is one of them, which uses the concept of altering

the adjustable parameters of conventional PID controller

in the direction so that the error between plant output

and reference input can be minimized [1-3]. This type of

control is also called online tuned PID control. Online

tuning results in non-linear behavior of the overall

control system which results in good performance where

nonlinearities and disturbances are inherent part of the

system [4].

Another approach to automatically tune the parameters

of conventional controller is Differential Evolution Soft

Computing Algorithm widely known as DE algorithm

[5]. This technique uses a population based search

algorithm to estimate the controller parameters so as to

minimize the integral square error (ISE) and thisestimation is done automatically through writing a

program using MATLAB [5-8]. DE based PID tuning

comes under offline tuning methods which sets the

values of PID parameters based on a performance index

[6].Literature suggests that the performance of DE based

controller is better than MIT based controller. This pape

proposes the hybridization of above two techniques and

the performance of the hybrid controller has been

evaluated on magnetic levitation system. Section II-IV

explains strategies, simulations and results withnecessary graphs.

II. MIT RULE

There are various approaches used to design the

adjustment mechanism for an adaptive controller and

gradient theory based MIT rule is one of them. MIT rule

uses the alteration of controller parameters in the

negative direction of gradient of a cost function [4]. This

cost function is defined in terms of the error between the

actual behavior and ideal behavior of the plant.

Defining the cost variable,

J(k) = e2 / 2 (1)

In (1), e is the error between plant output and reference

model output, and k is the adjustable parameter of the

controller.

Applying gradient theory [4],dk

dt ∝ −

∂J

∂k (2)



www.ijsret.org



Using the gradient theory, one will get the following

equation which depicts the relationship between the

change in parameter k with error e(t) [3].

dk/dt = −γ′ e y (3)

Hence the adjustment law developed by the equation

given above using Laplace transform will be,

k (s) = −γ′

s

ℒ {e(t ).y (t)} (4)

In (4), ℒ represents the Laplace transformation and ‘s’represents the Laplace variable. The adjustment

mechanism developed by the eq. (4) will be as shown in

Fig. 1 [4].

Fig. 1: Adjustment Mechanism of Model Reference

Adaptive Controller using MIT rule

III. DIFFERENTIAL EVOLUTION

Differential Evolution (DE) is a recently introduced

population search based soft computing algorithm which

uses heuristic optimization. DE algorithm was first

introduced by Storn and Price in 1996 to solve the

Chebychev Polynomial fitting problem used in filter

designing [13]. The decisive idea behind DE is an

arrangement for producing trial parameter vectors andthe selection of these vectors is based on heuristic

optimization [14-15].

Typical parameters used in DE are listed below;

• D – problem dimension

• N – No. of Population

• CR – Crossover Probability

• F – Scaling Factor

• G – Number of generation/stopping condition

• L,H – boundary constraints

Storn and Price have shown some rules in selecting thecontrol parameters in their first research paper published

in 1996 [5 & 8]. Theses rule are described with details in

[13] and shown below with brief description;

Step 1. Initialization:(i) Define upper and lower boundaries [L, H] and

initialize all DE parameters

(ii) Initialize parameter vectors over lower and upper

limits

xi,G = [x1,i,G , x2,i,G , . . . xD,i,G] ;i = 1, 2, . . . ,N.

Step 2. Mutation:

(i) For a given parameter vector xi,G randomly selec

three vectors xi,p, xi,q and xi,r such that the indices ‘i’, ‘p’‘q’ and ‘r’ are distinct.

(ii) Add the weighted difference of two of the vectors

to the third,

vi,G+1 = xp,G + F(xq,G – xr,G) (5)

The mutation factor F is a constant from [0, 2] and vi,G+

is called the donor vector.

Step 3. Recombination:

(i) Recombination incorporates successful solutions

from the previous generation. The trial vector ui,G+1 is

developed from the elements of the target vector, xi,G

and the elements of the donor vector, vi,G+1

(ii) Elements of the donor vector enter the trial vector

with probability CR

ui,G+1 =v , , ; rand , <

x , , ; rand , >(6)

i = 1, 2, . . . ,N; j = 1, 2, . . . ,D

Step 4. Selection:

The target vectorx , is compared with the trial vector

vi,G+1 and the one with the lowest function value is

admitted to the next generation.

xi,G+1 =u , ; f( u , ) < (x , )

x , ; otherwise(7)

IV. RESULTS AND DISCUSSIONS

Linear model of Magnetic Levitation system has been

used in the paper taken from [9] for simulations on

MATLAB and the transfer function of the system is

shown below, where Ex(s) is the position of the ball in

terms of voltage and Ei(s) is the applied input voltage

[9].

G(s) = Ex(s) ⁄ Ei(s)

G(s) = 77.8421(0.0311s2 − 30.52)

(8)

Table 1 shows the performance of magnetic levitation

system using Differential Evolution based offline tuned

PID controller in terms of transient performance

parameters along with integral square error (ISE) for

various trials. The performance of the overall system isvery satisfactory while considering linear approximated

model of the system in simulations as shown in Table 1.



www.ijsret.org



From table 1, one can observe that in every trial there

is an overshoot of around 25-35% present in the

response. Although the settling time is much lesser for

every trial but such a large overshoot in the system is

undesirable and may cause the actuator breakdown.

Integral square error (ISE) is well under the desirable

range and the overall performance is satisfactory.

Fig. 3 shows the Simulink model of the proposed

hybrid controller, in which a PID controller is generating

the control input for the system. The integral gain of PID

controller is an adjustable gain which is being adjusted

by MIT rule based adaptive mechanism as shown in Fig.

4. The performance of the designed controller for

various trials, in terms of transient parameters and ISE,

is shown in table 2 and the corresponding responses areshown in Fig. 5. These trials have been performed to

calculate the values of PID parameters in offline mode

using DE algorithm and fine tuning in the value of

parameter KI has been carried out by MIT mechanism

with adaptation gain of 23.5 in online mode.

For a particular trial, different responses andcorresponding adjustable parameter for various values of

adaptation gain have been obtained and are shown in the

Fig. 6 and Fig. 7. From Fig. 6, one can observe that for

smaller values of adaptation gain, response of the systemis sluggish with large settling time. As one increases the

value of adaptation gain, the response of the system

becomes better. But after a certain limit of adaptation

gain the response of the system starts deteriorating.

Hence, selection of a range for adaptation gain is verycritical factor.

For a particular value of adaptation gain, responses

obtained for various trials are shown in Fig. 5 and theperformance of the system has been tabulated as shown

in table 2. On comparing the performances of the systemwith DE controller and DE+MIT hybrid controller, one

can observe that the performance of the hybrid controller

is better in terms of peak overshoot and settling time.

The overshoot and settling time has been reduced to

15% and 0.30 sec respectively. Results given in the

paper verify that the performance of the system has been

improved to a large extent with the proposed

hybridization of DE algorithm and MIT rule.

V. CONCLUSION

A brief overview on MIT rule and Differential

Evolution soft computing algorithm has been carried out

in this paper along with the hybridization of two

techniques.DE algorithm is very simple and effective

population based generation algorithm which takes very

less computation time than other soft computing

algorithms.The performance of DE based controller with

appropriately chosen range for various parameters is

satisfactory as shown by the results. Selection of this

range is very critical in DE algorithm and is carried out

carefully. The limitation of DE based controller is that i

does not have the ability to alter the controller

parameters during the run time. To overcome this

limitation, hybridization of DE algorithm with MIT rule

has been proposed in the paper.

Performance of the proposed hybrid controller is

evaluated on MATLAB and the designed controller has

been applied on magnetic levitation system. The

performance of hybrid controller very much depends

upon the value of adaptation gain and selection of this

gain is very critical in designing the controllerEffectiveness of the hybrid controller also depends upon

the efficient programming for DE algorithm carried ou

by an expert. Results carried out and shown in the paper

clearly shows that the performance of hybrid controller

is much better than the DE based controller. Results also

depict the reduction in peak overshoot from 30% to 15%and in settling time from 0.4 sec to 0.3 sec, which proves

better performance of proposed hybrid technique.

REFERENCES[1] K. J. Astrom and B. Wittenmark, Adaptive contro

(2nd edition), Dover Publications Inc., New York

1995, ch. 5, pp. 185 – 234.

[2] P. Swarnkar, S. K. Jain and R. K. Nema, “Effect ofadaptation gain on system performance for model

reference adaptive control scheme using MIT rule”presented at International Conference of World

Academy of Science, Engineering and TechnologyParis, pp. 70-75, 2010.

[3] P. Swarnkar, S. Jain, R. K. Nema,“Application ofModel Reference Adaptive Control Scheme To

Second Order System Using MIT Rule”, presentedat International Conference on Electrical Power

and Energy Systems (ICEPES-2010), MANIT

Bhopal, India, 2010.

[4] Priyank Jain and M. J. Nigam, “Real Time Contro

of Ball and Beam System with Model Reference

Adaptive Control Strategy using MIT Rule”presented at 2013 IEEE International Conference

on Computational Intelligence and Computing Research (ICCIC), Madurai, pp. 305-308, 2013.

[5] R. Storn, K. Price, “Differential evolution-a simple

and efficient adaptive scheme for globa

optimization over continuous spaces," Journal of

Global Optimization, Vol.11, pp. 341-359, 1997.

[6] Rainer Storn, “Differential Evolution forContinuous Function Optimization



www.ijsret.org



"http://www.icsi.berkeley.edu/storn/code.html,

2005.

[7] K. Price, “An introduction to differentialevolution," New Ideas in Optimization, eds. D.

Corne, M. Dorigo, and F. Glover, McGraw-Hill,

London (UK), pp. 79-108, 1999.

[8] R. Storn, “On the Usage of Differential Evolution

for Function Optimization”. in Proceedings of the

Fuzzy Information Processing Society, Berkeley,

CA, USA, 1996.

[9] AbhishekRawat and M. J. Nigam, “Comparisonbetween adaptive linear controller and radial basis

function neural neurocontroller with real time

implementation on magnetic levitation system”,

presented at 2013 IEEE International Conference

on Computational Intelligence and Computing

Research (ICCIC), Madurai , pp. 515-518 , 2013.

[10] Adrian-VasileDuka, StelianEmilianOltean, and

MirceaDulău, “Model Reference Adaptive vs.

Learning Control for the Inverted Pendulum”,

presented at the International Conference onControl Engineering and Applied Informatics

(CEAI), vol. 9, pp. 67-75, 2007.

[11] M. S. Ehsani, “Adaptive Control of Servo Motor by

MRAC Method”, presented at IEEE International

Conference on Vehicle, Power and Propulsion

Arlington, TX, pp. 78-83, 2007.

[12] K. S. Narendra and A. M. Annaswamy, Stable

Adaptive Systems, Prentice-Hall, Englewood Cliffs

New Jersey, 1989.

[13] Mohd S. Saad, HishamuddinJamaluddin, Intan Z

M. Darus “PID Controller Tuning UsingEvolutionary Algorithms, WSEAS Transactions on

System and Control, Vol. 7, 4, pp. 139-149, 2012.

[14] K. J. Astrom and T. Hagglund, “Automatic Tuning

of PID Controllers”, Instrument Society of America

1988.

[15] J. G. Ziegler and N.B. Nichols, “Optimum Settingsfor Automatic Controllers”, Trans. ASME . Vol. 64

8: pp. 759 – 768, 1942.[16] B. C. kuo and F. Golnaraghi, Automatic Contro

Systems (9thedition), John Wiley & Sons, Inc.

2010, ch. 4, pp. 147-252.

[17] G. Saravanakumar, R. S. D. Wahidhabanu, and V

I. George, “Robustness and Performance of

Modified Smith Predictors for Processes withLonger Dead-Times”, International Journal on

Automatic Control and System Engineering

Fig. 2 Simulink diagram of magnetic levitation system with DE based control

TABLE 1: Various values of PID parameters, transient parameters and respective value of ISE on different trials of

DE based controller for Magnetic Levitation system

TrialPID Parameters Overshoot

(%)

Settling

Time

(sec)

Integral

Square Error

(ISE)KP KI KD

I 2.9900 0.0443 15.7818 25.5 0.478 0.0498

II 2.8700 0.0685 18.0636 29 0.32 0.0227



International Journal of

III 1.9500 0.0395 1

IV 2.7650 0.0673 1

V 2.1150 0.0422 1

Fig. 3 Simulink diagram of magn

Fig. 4 Si

TABLE 2: Various values of PID param

hybrid

TrialPID Parameters

KP KI

I 2.9732 0.0899 1

II 2.8635 0.2234 1

III 2.9989 0.4981 1

IV 2.9006 0.3649 1

www.ijsret.org

cientific Research Engineering & Technology (IJS

V

.7300 32 0.37 0.031

.8495 30 0.31 0.028

.0800 33 0.35 0.033

etic levitation system with DE and MIT rule based

ulink diagram of adjustable PID controller

ters, transient parameters and respective value of I

controller for Magnetic Levitation system

Overshoot

(%)

Settling

Time(sec)

Integr

Square E(ISEKD

.3613 16.6 0.23 0.008

.6812 15.49 0.30 0.009

.2816 14.74 0.34 0.008

.2532 15.33 0.26 0.008

ET), ISSN 2278 – 0882,

lume 3 Issue 1, April 2014

2

ybrid controller

SE on different trials of

l

rror



www.ijsret.org



Fig. 5 Response of Magnetic Levitation system with DE-MIT hybrid controller for different trials with adjustable

parameter k=23.5

Fig. 6 Response of Magnetic Levitation system with DE-MIT hybrid controller for different values of adaptation gain:

Position of the ball



www.ijsret.org



Fig. 7 Response of Magnetic Levitation system with DE-MIT hybrid controller for different values of adaptation gain:

Adjustable parameter

design of a hybrid controller using differential evolution and mit rule for magnetic levitation...

Documents