On fuzzy concepts in engineering ppt. ncce

On Fuzzy Concepts in Engineering &

Technology

Surender Singh

Asstt. Prof. , School of Mathematics

Shri Mata Vaishno Devi University , Katra –182320

National Seminar on Engineering Applications of Mathematics (NSEAM)

N.C College of Engineering, Israna

17th March, 2012

Outline

• Introduction

•crisp set

•Fuzzy set

•Fuzzy logic

•Fuzzy logic System

•Example ( To build a fuzzy controller)

•Fuzzy concepts in Engineering

•Some Probabilistic Divergence measures and their fuzzy analogue

•A Model for strategic decision making

•Illustrative Example

•Conclusion

Introduction

The present communication is intended to serve

as introductory material on fuzzy sets and

fuzzy logic . Some contextual usage of

fuzziness in Engineering are presented. Three

divergence measures between fuzzy sets are

introduced and these measures are used to

propose a Model for strategic decision making

environment.

Crisp set

Recall that a crisp set A in a universe of discourse U (which

provides the set of allowable values for a variable) can be

defined by listing all of its members or by identifying the

elements x A. One way to do the latter is to specify a condition

by which x A; thus A can be defined as

A = { x / x meets some condition}.

Alternatively, we can introduce a zero-one membership function

(also called a characteristic function, discrimination function, or

indicator function) for A, denoted µA(x) such that A µA(x) = 1 if

x and µA(x) = 0 if x A. Subset A is mathematically equivalent

to its membership function µA(x) in the sense that knowing

µA(x) the same as knowing A itself.

Fuzzy Set

Definition: Let a universe of discourse X = {x1, x2, x3… an} then a fuzzy

subset of universe X is defined as

A = {(x; µA(x)) / x ε X; µA(x): X [0; 1]}

Where µA(x): X [0; 1] is a membership function defined as follow

0 if x does not belong to A and there is no ambiguity

µA(x) = 1 if x belong to A and there is no ambiguity

0.5 if there is maximum ambiguity whether x belongs to A

or not

Fuzzy set (cont…) In fact µA(x) associates with each x ε X a grade of membership of the set A. Some notions related to fuzzy sets [5].

Example 1

• A car can be viewed as “domestic” or “foreign” from

different perspectives.

Fuzzy Logic

Fuzzy logic is superset of the Boolean logic and it adds degrees between absolute true and absolute false in the sense that some propositions may to more true than others. Like the extension of the crisp set theory to fuzzy set theory, fuzzy logic is an extension of the crisp logic, in which the bivalent membership function is replaced by the fuzzy membership functions. In crisp logic the truth values acquired by the proposition are two valued, namely true as ‘1’ and false as ‘0’ while in the fuzzy logic the truth values acquired by the proposition are multi-valued, as absolutely true , partially true, absolutely false etc. represented numerically as real value between ‘0’ and ‘1’.

Fuzzy Logic System

Fig.2 Fuzzy Logic System

Fuzzy Logic System (Cont…)

• Rules may be provided by experts (you may be such a person) or can be extracted from numerical data. A collection of prepositions containing linguistic variables ; the rules are expressed in the form:

IF x is A and y is B … THEN z is C. where x , z are variables ( e.g. distance , time etc.) and A,B,C are linguistic variables ( e.g. small ,far ,near etc.)

• The fuzzifier maps crisp numbers into fuzzy sets. It is needed in order to activate rules which are in terms of linguistic variables, which have fuzzy sets associated with them.

Fuzzy Logic System (Cont…)

• The inference engine of the FLS maps fuzzy sets into

fuzzy sets. It handles the way in which rules are

combined.

• The defuzzifier maps output sets into crisp numbers.

In a controls application, for example, such a number

corresponds to a control action.

Example 2[1] (To build a fuzzy

controller)

• The temperature of a room equipped with an

fan/air conditioner should be controlled by

adjusting the motor speed of fan/ air

conditioner.

Fig3 describes the control of room temperature. In this

example the goal is to Design a motor speed

controller for fan.

(To build a fuzzy controller)

Fig. 3

(To build a fuzzy controller)

• Step 1: Assign input and output variables

Let X be the temperature in Fahrenheit and Y be the

motor speed of the fan.

• Step 2: Pick fuzzy sets (Fuzzification)

Define linguistic terms of the linguistic variables

temperature (X) and motor speed (Y) and associate

them with fuzzy sets .For example, 5 linguistic terms

/ fuzzy sets on X may be Cold, Cool, Just Right,

Warm, and Hot. Let 5 linguistic terms / fuzzy sets on

Y be Stop, Slow, Medium, Fast, and Blast.

(To build a fuzzy controller)

(To build a fuzzy controller)

(To build a fuzzy controller)

Step 3: Assign a motor speed set to each temperature

set (Rule or Fuzzy controller)

• If temperature is cold then motor speed is stop

• If temperature is cool then motor speed is slow

• If temperature is just right then motor speed is

medium

• If temperature is warm then motor speed is fast

• If temperature is hot then motor speed is blast

(To build a fuzzy controller)

(To build a fuzzy controller)

(To build a fuzzy controller)

(To build a fuzzy controller)

(To build a fuzzy controller)

(To build a fuzzy controller)

Step 4: Defuzzification

In this example crisp

input is X= 63 Fo

and crisp output is

Y= 42%.

Fuzzy concepts in Engineering

• A list of fuzzy terms (see table 1) that are widely used

in control, signal processing and communications.

However we always strive for their crisp values still

these are used in fuzzy control, where they convey

more useful information than would a crisp values.

Table1. Engineering Terms whose Contextual usages is usually quite fuzzy

Terms Contextual Usage

Alias None , a bit , high

Bandwidth Narrowband, broadband

Blur Somewhat ,quite , very

Correlation Low, medium, high, perfect

Errors Large ,medium, small, a lot of, so

great, very large, very small, almost

zero

Frequency High , low , ultra-high

Resolution Low , high

Sampling Low-rate, medium-rate, high-rate

Stability Lightly damped, highly damped, over

damped, critically damped ,unstable

Fuzzy concepts in Engineering

(cont…) • Correlation is an interesting example, because it can

be defined mathematically so that, for a given set of data, we can compute a crisp number for it. Let’s assume that correlation has been normalized so that it can range between zero and unity, and that for a given set of data we compute the correlation value as 0.15. When explaining the amount of data correlation to someone else, it is usually more meaningful to explain it as “this data has low correlation.”When we do this, we are actually fuzzifying the crisp value of 0.15 into the fuzzy set “low correlation.”

Other fuzzy terms appearing in Table 1 can also be interpreted accordingly.

Applications of Fuzzy logic in Engineering

and interdisciplinary sciences

• A short list of applications of FL includes: Controls

Applications-aircraft control (Rockwell Corp.), Sendai

subway operation (Hitachi), cruise control (Nissan),

automatic transmission (Nissan, Subaru), self-parking

model car (Tokyo Tech. Univ.), and space shuttle docking

(NASA): Scheduling and Optimization-elevator

scheduling (Hitachi, Fujitech, Mitsubishi) and stock

market analysis (Yamaichi Securities); and Signal

Analysis for Tuning and Interpretation - TV picture

adjustment (Sony), handwriting recognition (Sony Palm

Top), video camera autofocus (Sanyol Fisher, Canon) and

video image stabilizer (Matshushita Panasonic). For

many additional applications, see [1], [2], [3], [7] and [8].

Some probabilistic divergence measures

be the set of all complete finite discrete probability distributions. Then for all P,Q ε Гn. Bhatia, Singh and Kumar [6] proposed three probabilistic divergence measures to discriminate between two probability distributions as follow:

Some probabilistic divergence measures

(cont…)

Fuzzy Analogue of Prob. Div.

Measures

Where A and B are fuzzy sets and µA(x), µB(x) are their respective membership functions.

.

.

Model for Strategic Decision making

Let the organization X want to apply m strategies S1, S2,…

Sm to meet a target. Let each strategy has varied degree of

effectiveness if cost associated with it is varied, let

{C1,C2,…Cn} be cost set. Let the fuzzy set X denotes the

effectiveness of a particular strategy with uniform cost.

Therefore

Further, let Cj be a fuzzy set denotes the degree of

effectiveness of a strategy when a it implemented with cost Cj

.

.

where j= 1,2...,n.

Model for Strategic Decision making

(cont…)

Taking A=X and B = Cj in the fuzzy divergence measures

and calculate the value of Then

. Let the minimum value is attained at Ct ,

With this Ct find , let it corresponds

to Sp ,

Thus if the strategy Sp is Implemented with cost Cp

then organization will meet its target in most cost

effective manner.

Determines the suitability of Cj

Illustrative Example Let m = n = 5 in the above model.

The table below shows the effectiveness of strategies at

uniform cost. Table:2

Illustrative Example (cont…) The table below shows the effectiveness of strategies

at particular cost. Table:3

Illustrative Example (cont…) The table below shows the divergence between X

and Cj , j = 1,2,3 ,4 ,5.

Table:4

Illustrative Example (cont…)

According to the divergence measures presented

in the table 4 budget C2 is more suitable and

after examining the table 3 , it is observed that

strategy S1 is most effective. Therefore the

organization will achieve its target in most cost

effective manner if the strategy S1 is

implemented with a budget C2 .

Scope for further research

In this communication the basics of fuzzy set and fuzzy

logic are discussed. There are some advanced

concepts , like fuzzy c-means , Intustic fuzzy valued

sets etc. These concepts can also be applied in certain

areas.The concept of fuzziness can be used in the

research related to digital image registration, image

processing , pattern recognition , genome analysis for

effective gene selection, network and queuing theory.

References

[1]B. Kosko, Fuzzy Thinking: The New Science of

Fuzzy Logic. New York Hyperion, 1993

[2] C. C. Lee, “Fuzzy logic in control systems: Fuzzy

logic controller, part I,” IEEE Trans. Syst., Man, and

Cybern., vol.SMC-20, no. 2, pp. 404-418, 1990.

[3] D. Schwartz, G. J. Klir, H. W. Lewis 111, and Y.

Ezawa, “Applications of fuzzy sets and approximate

reasoning,” IEEE Proc., vol. 82, pp. 482-498, 1994.

[4] G.J Klir And T.A Folger, Fuzzy sets ,Uncertainty

and Information ,Prentice Hall International 1998.

References (cont…)

[5] L. A. Zadeh, “Fuzzy sets,” Information and Control,

vol. 8, pp.338-353 ,1965.

[6] P.K Bhatia, Surender Singh And Vinod Kumar.

Some New Divergence Measures and Their

Properties. Int. J. of Mathematical Sciences and

Applications,1(3), 2011,1349-1356

[7] T. Terano, K. Asai, and M. Sugeno, Fuzzy Systems

Theory and Its Applications. New York Academic,

1992.

[8] J. Yen, R. Langari, and L. Zadeh, Eds., Industrial

Applications of Fuzy Logic and Intelligent Systems.

New York: IEEE Press, 1995.

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