Journal of Scientific & Industri al Research Vol. 60, November 200 I, pp 85 1-859

Linguistic Fuzzy Modeling for Industrial Applications Nirmal Singh ·. Renu Vig #and J K Sharma

Beant Col lege of Engineering & Technology, Gurdaspur 143 52 1 (India) Received: 27 February 200 I; accepted: 25 June 200 I

The extraction of fuzzy informati on from raw data is important and contai ns savings potential in industri al applications. A general approach to linguistic modeling based on fuzzy logic has been presented. A fu zzy inference algorithm or mechani sm is necessary to use lingui sti c model. Such a mechani sm enables computation of output value, given some input values. The max­min (Mamdani ) inference algorithm for linguistic modeling has been considered, and its implementation is illustrated , using MATLAB package. The lingui stic fuzzy models can be used for diffe rent purposes because of their transparency and can also be used in industrial applications, which are partl y described by lirst pri nciple models and partl y by ex peri ence contained in designers. operators and other workers. An example of linguisti c fuzzy modeling for si mple industrial appli cation of heating power of a gas burner has been presented to illustrate the proposed model and method of computation.

Introduction

Work on fuzzy modeling can be di vided in two groups-the first deal s with modeling of a system itself or a fuzzy modeling for simulation J. 2 and the second deals with fuzzy modeling of a plant for control'- 4

. Just as with the modern control theory, one could des ign a fuzzy controller, based on a fuzzy model of a plant if such a fuzzy model can be identified. The control engineering community developed a complete framework for linear systems, which resulted in many universal analysis and des ign methods for closed, and open loop systems. Although many attempts were made during the last a few decades to develop analysis and des ign methods for non-linear systems, no uni versal method is yet ava il able for these systems. In many cases, one relies on process model s, which are linear around many of operating points.

Development of mathematical models is essential for many di sciplines of engineering and science. Models have been used for simulati on, design and analysis of systems, process control , monitoring and supervision, etc. The traditi onal approac h is based on th orough understanding of the nature and behav iour of the actual system, which upon a suitable mathemat ical treatment leads to the development of a model. For incompletely

*Author for correspondence #Department of Computer Science, ITT! , Sector-26, Chandigarh

14352 1

understood processes, however, thi s approac h may become laborious and inefficient. A large amount of process knowledge is qualitati ve and imprecise and as such cannot be readily transformed into trad iti onal mathematical models based on differential and algebra ic equ ation s. Formal method s to incorporate su c h information in the development of models have been developed. Alternative representat ion schemes, using, e

g, natural language rul es , se manti c ne tw ork s, or qualitative models have been ex plored. Techniques, based on fuzzy sets and fuzzy logic, represent a promising a lte rn ate approac h, which has been deve loped considerably in recent years.

Fuzzy models can also be seen as logica l models which use " if- then" rules and log ica l opera tors to establish qualitat ive relationships among variables in a model. Fuzzy sets serve as a smooth interface between qualitati ve va riables involved in the rules and numerical domains of inputs and outputs of a model. The rule-based nature of fuzzy models permits use of information ex pressed in natural language statements, and makes mode ls amenab le to analys is and interpretati on. At the same time, at the computational level, fuzzy models could be rega rded as fl ex ible mathematica l structures so that process data could be translated in a model and analysed in a manner very similar to what people are acquainted with 5

· 7

.

This paper concentrates on the I i ngu isti c fu zzy techniques for modeling for industri al app li cati ons. It is

852 J SCIIND RES VOL 60 NOVEMBER 200 1

shown that these can, not only be used as black box models, but can also be made transparent and relatively simple to explore main features of the system. A linguistic modeling based on fuzzy logic has been di scussed on a MATLAB platform. It reports on a lingui st ic model as a system model based on linguistic description just as it is used in soc iology or psychology. Such a modeling is necessary for industrial application for modern process control and production methods which are confronted with numerou s requirements posed by in creasi ng competition , environmental regulations, inc rease in energy and raw material costs and an increas in g demand for hi gh qual ity tailored products.

Fuzzy System

A static or dynamic system, which makes use of fuzzy sets or fuzzy log ic and of the co rrespondin g mathematical framework, is called afuzzy system. Fuzzy sets can be involved in a system through:

I In the description of a system-A system may be defined, e g, as a collect ion of if-then rules wi th fuzzy predicates, or as a fuzzy re lati on. A fuzzy rul e describing the relationship between heating power and the temperature trend in a room may be described as :

If the heating power is high then the temperature wi ll increase fast.

2 In the specification of a system~- parameters- A system may be defined by an algebraic or differen­tial equation, in which the parameters could be fuzzy

- -numbers in stead of real numbers . y = 3x

1 + 5x2 ,

where 3 and 5 are fuzzy numbers "about three" and "about five", respectively, defined by member­ship functions is one such example. Fuzzy numbers express the uncertain ty in parameter va lues.

3 The input, output and state variables of a system may be fuzzy sets-Fuzzy inputs could be readings from unreliable sensors ("noisy" data) , or quanti ties related to human perception, such as comfort, beauty, etc. Fuzzy systems process such information, that is not the case with conventi onal (crisp) systems.

A fuzzy system can simultaneously have several of the above attributes. Table I gives an overview of the relationships between fuzzy and crisp syste m

11\ h.:rval (lr tuff~ :•n::urw.: m

----·'~--- )~-----

' - --- ' : --: : ' ' ' ' ---. ' ' -----.

l l l ll'l\ ;lll ii: lel h lfl

>.. X

flt.t/~ tun .. : t1o~n

Figure !-Evalu at ion of a crisp, interval and fu zzy function for crisp interval and fuzzy arguments

Table !-Crisp and fuzzy information in systems

System In put da ta Output data Mathemati cal description framework

Crisp Crisp Crisp Functional anal ys is Cri sp Fuzzy Fuzzy Ex tension principle

(Zadeh) Fuzzy Crisp/Fuzzy Fuzzy Fuzzy re lational

calcu lus

descriptions and variables. This paper focuses on the last type of systems, i.e., fuzzily described systems with fuzzy inputs.

Fuzzy systems may be regm·ded <L a general ization of interva l- va lued systems, wh ich, in turn, may be generalized crisp systems. It is depicted in Figu re I , which gives an example of a function and its interva l and fuzzy forms. The evaluat ion of the function for cri sp, interva l and fuzzy data is schematicall y depicted as wel l. A function f: X -7 Y can be regarded as a subset of the Cartes ian product X x Y i e, as a relation . The evaluation of the function for a given input proceeds in three steps:

Extend the given input into the product space X x Y (vertical clashed lines in Figure I).

2 Find the intersection of this ext nsion with the re lation.

3 Project this intersection onto Y (horizontal clashed lines in Figure I ) . Thi s view is independent of the

SINGH eta/.: LINGUISTIC FUZZY MODELING X 53

nature of both the function and the data (crisp,

interval, or fuzzy).

Linguistic Fuzzy Models

Two common sources of information for making fuzzy models are the prior kno wledge and data (measurements) . The prior knowledge can be of a rather approximate nature (qualitative knowledge, heuristics) , which usually originates from "experts", i.e. , process designers, operators, etc. For many processes, data are

available as records of the process operation or special identification experiments can be des igned to obtain the relevant data. Building fuzzy models from data involves methods based on fuzzy logic and approximate reasoning, data analysis and conventional systems identification. The acquisition or tuning of fuzzy model s by means of data is usually termedfuzzy identifica tion.

Two main approaches to the integra tion of knowl edge and dat a in a fuzzy mod e l can be

di stingui shed: The expert knowl edge ex pressed in a verbal form is translated into a collection of if- then rules. In thi s way, a certain mode l structure is created . Parameters in thi s structure (membership functions , consequent sing letons or parameters) can be fine-tuned using input-output data. No prior knowledge about the system under study is initially used to formulate the rules, and a fuzzy model is constructed from data. It is expected that the ex tracted rules and membership functions can provide a poste riori inte rpretation of th e sys te m 's behavior. An expert can confront such an information

based on knowledge and experience, or modify the rules, supply new ones, or even design additional experiments to obtain more informat ive data . The process is ca lled rule extraction a nd techniques can be combined, depending on a particular application. To des ign a (lingui sti c) fuzzy mode l based on avai labl e expert knowledge, the steps g iven below can be fo llowed :

Select input and output vari ab les, structure of rules , and inference and defuzzification methods .

2 Decide number of linguistic terms for each variable and define the correspond ing membership functions.

3 Formulate the availab le knowledge in terms of fuzzy

if-then rules.

4 Validate the model (typically by using data). If the model does not meet expected performance, iterate the design steps accounted above.

It may be noted that the success of thi s method

heavily depends on the problem at hand , and the ex tent and quality of available knowledge. In some problems, the knowledge-based design may lead fast to useful model s, yet in others, it may be a very time-con suming and inefficient procedure (especially manual fin e-tun ing of the model parameters) .

Different methods have been developed using fuzzy set theory to model systems, such as rule-based fuzzy systems and fuzzy linear regression methods . Thi s paper deals with rule-based linguistic fuzzy models. In rul e­based fuzzy systems, the relationships between variables are represented by means of fuzzy if- then rules of the following general fonn:

If antecedent proposition then consequent proposition.

The antecedent propos iti on is a lways a fu zzy

proposition of the type " x is A" where x is a lingui stic va riabl e and A is a linguisti c constant ( te rm). The proposition 's truth value (a real number between zero and one) depends on the degree of match (similarity)

be tw ee n x a nd A. Depending o n the form of the co nsequ e nt , rul e -based fuzzy lin gui stic model is distinguished in which both the a ntecedent and the consequent are fuzzy propositions . The lingui sti c fuzzy mode l has been introduced as a way to capture ava ilable semi-qualitative knowledge in the form of if- then rulesx :

R; : If X is Ai then y is B; , i=l , 2 , ... , K

... (I)

Here, x is the input (antecedent) linguistic variable. and A; are the antecedent linguist ic terms (constants).

Similarly, y is the output (consequent) linguistic vari able

and 8; are the consequent lingu ist ic terms. The values of

x , y and the linguistic terms A;, 8; are fuzzy sets defined

in the domains of their respective base variables: .r E X c 9\1' and y E Y c 9\1' . Base variab le is the domain

variable in which fuzzy sets are defined. The membership functions of the antecedent and consequent fuzzy sets are the mapp ings: ~(x): X~[O , I] and ~(y): Y~[O. I] , respectively. Fuzzy sets A; define fuzzy regions in the antecedent space, for which the respective consequenr propositions ho ld . The lingui stic terms A and 8 are usually selected fro m sets of predefined te;ms, su~h as low, high, etc . By denoting th ese sets by A and 8

854 J SCI IND RES VOL 60 NOVEMBER 2001

respectively, we haveAi E A and Bi E B. The rule base R = {RJ i =I , 2, ... , K) and the sets A and B constitute the

I

knowledge base of the linguistic model.

An algorithm is required to use the linguistic model, which permits computation of the output value, for a given input va lue. Such algorithms are called fuzzy inference algorithms (or mechanisms). For the linguistic model, the inference mechanism can be derived through fuzzy relational calculus, as described in the subsequent secti on.

Relational Representation of Linguistic Models

Each rul e in Eq. ( I ) can be regarded as a fuzzy re lation (fuzzy res triction on th e s imultaneo us occurrences of values x andy): R;: (X x Y)---j[O , I] . That can be computed in two basic ways-by using fuzzy conjunctions (Mamdani method) and by usin g fuzzy implications (fuzzy logic method). Fuzzy implications are used when the if-then rule (Eq. I ) is strictly regarded as an implication A---jB, i e, "A implies B.". In class ica l

I I I I

logic this means that if A holds, B must hold as well fo r the implicati on to be tme. Nothing can. however, be said about B when A does not hold, and the relati onship also cannot be inverted.

When using a conjunction, Ai 1\ Bi, the interpretati on of th e if-the n rul es is " it is tru e th a t Ai and Bi simultaneously hold". The relationship is symmetrical and can be inverted. For simplicity, thi s paper confines to the Mamdani (conjunction) method. The relati on R; is computed by the minimum(/\) operator:

Note that the minimum is computed on the Cartesian product space of X and Y, i.e., for all possible pairs ofx and y. The fuzzy relation R representing the entire model (Eq. I ) is given by the disjuncti on (u ni on) of the K individual rule's relations R.:

I

Now the entire rule base is encoded in the fuzzy rel ation Rand the output of the linguistic model can be computed by the relational max-min compos ition (o):

y = x oR. 000 (4)

As the relat ion R can be visualized as ajitzzy gmph in X x Y, the lingui st ic fuzzy model is ometimes called a fuzzy graph. The relational compo it ion (Eq. 4) can be regarded as a function evaluat ion on the fuzzy graph .

Max-min (Mamdani) Inference

It is apparent fro m the previous section that a rule base can be represented as a fuzzy relation. The output of a rule-based fuzzy model is computed by the max­min relational compos ition . In this s ction, it will be shown that the relati onal calculus can be by-passed. This is advantageous, as the discretization of domains and storing of the relation R can be avoided. To show thi s,

assume an input fuzzy value x = A · , for which the

output values· is given by the re lational composition :

00 0 (5)

After substituting for J.iR (x, y), from (Eq . J), the foll owing expression is obtained :

J.1B . (y) = max [J.IA · (x) A max (J.IA (x) .A. f.LB ( v ))] X 1Si$ K I I .

... (6)

As the max and min operati on are taken ove r different domains, their order can be changed as fo ll ows:

!LB · ( y) = max [max (!LA · (x) " ,uA (x )) 1\ pB ( y) I l ~ i$ K X I I

00 0 (7)

Denote [3 ; = max (J.IA . (x) /\ J.1A 1 (x)) the deoree X ' ~,_..,

of.fit(fillment of the ith rule's antecedent. The output fu zzy set of the lingui stic mode l is thus:

000 (8)

The algorithm, called the max-min or Manl(/ani

inference, is summarized below and can be visuali zed

as in Figure 2.

Compute the degree of fu lfi ll ment by

f3 . = max ~LA · (x) " JlA (x) J i = 1,2, .. . , K I X I

2 Derive the output fuzzv sets B; :

SINGH eta/.: LING UISTIC FUZZY MODELING RSS

II n n,

X

{

if\" IS , \ !he'll 1 j, /1.

mndl~l: 1 r \ i' ,, : tth.' ll .\ i' u. lf .1 is A. th,•n ' i> II

Figure 2- A schematic representation of the Mamdani inference algorithm.

f.lB; (y) = /3; A f.lB; (y), y E Y, i = 1,2, ... , K ·

3 Aggregate the output fuzzy sets s: into a single I

A singleton fuzzy set A. (step I ) of the above

algorithm simplifies to /3; ={LA; (x ·). A si ngleton fuzzy set is a fuzzy representation of a crisp number.

Suppose a crisp number x', the corresponding fuzzy

singleton A is defined by: pA. (x·) =I and

pA . (x) = 0, V x E X ,x :t x ' . The Mamdani inference method does not require discretization of domains and thus can work with analytically defined membership

functions. In many applications, a crisp output Y is desired . To obtain a crisp value, the output fuzzy set must be defuzzified. With the Mamdani inference scheme, the center of gravity (COG) defuzzification method is used

where Y coordinate of the center of gravity of the area

under the fuzzy set s ' is computed as:

... (9)

where K is the number of elements .Y; in Y. Continuous domain Y thus must be discretized to compute the center

Example

Consider a simple fuzzy model , which qualitatively describes how the heating power of a gas burner depends on the oxygen supply (assuming a constant gas supply). There is a scalar input, the oxygen flow rate (x), and a scalar output, the heating power (y). The set of antecedent linguistic terms is defined as: A= {Low, OK, High), and the set of consequent linguistic terms: B = {Low, High}. The qualitative relationship between input and output can be expressed by the following rules :

Rr· ff01 flow rate is Low then heating power is Lml'.

Rr · If 01j low rate is OK then heating power is High.

R r· lfO, jlow rate is High then heating power is Lml'.

The meaning of the linguistic terms in input and output is defined by their membership functions along with fuzzy inference system (FIS), as depicted in Figures 3(a)-3(c). The numerical values along the base variables are selected somewhat arbitrarily. It may be noted that no universal meaning of the linguistic terms can be defined . For the example, it will depend on the type and flow rate of the fuel gas and type of burner. Nevertheless, the qualitative relationship expressed by the mles remains valid.

A algorithm is required to compute the output va lue for given input value under the linguistic model. The algorithm is called the j~tzzy inference algorithm (or mechanism) . For the linguistic model , the inference mechanism can be derived by fuzzy relational calculus . Let us compute the fuzzy relation for the linguistic model. First input and output domains are dicretized, e g, X = [0, I, 2, 3], and the set of consequent linguistic terms: Y

OK

/-\ I \

O.R

0.6 \

\

II \

02 \

0.4

0 -~-'--------~~~-----------\-\~ 0 0.5 1.5 25

856 J SCIIND RES VOL 60 NOVEMBER 2001

0.6

0.4

0.2

0

0

/ //

/ /

/

/

/

10 20 :10

/

/ /

/

40 50

/

/ /

/

60 70 RO <)()

Figure 3(b)- Membership functi on of output

Burner

(mamdan i)

3 rul es

I!Xl

!low-rate (3) heating-power (2)

System Burner: I inputs, I outputs, 3 rul es

Figure 3(c)-Fuzzy Inference Scheme (FIS ) Scheme

= [0, 25 , 50, 75 , I 00]. The (di sc rete) membership functions are given in Table 2 for the antecedent lingui stic terms, and in Table 3 for the consequent terms.

The fuz zy relation s R; correspondi ng to th e individual rule, can now be computed from Eq. (2). For

rule 1?1, R, =Low x Low, and rule R

2, R2 =OK x High,

and fin ally for rule R3

, R1

= High x LolV . The fuzzy relation R is the union (element-wise maximum) of the relations R

I

ri.O

0.6 R =

I 0

1.0 0.6 0 ~01- ' 0.6 0.6 0 ---

0 0 0

Table2-Antecedent membership funct ions

Linguistic 0 I 2 3

Low 1.0 0.6 0.0 0.0

OK 0.0 0.4 1.0 0.4

High 0.0 0.0 0.1 1.0

Table 3-Consequent membership functions

Lingui sti c

term

Low

High

0

0 R, =

0

0

0

0 R, =

0.1

1.0

0

0

0

0

0

1.0

0.0

0 0

0.~ 0.4

0.~ 0.9

0 .~ 0.4

0 0

0 0

0.1 0.1

1.0 0.6

25

1.0

0.0

0

0.4 ~ !? =

1.0

0.4

[) 01 0 0 ~

0 0

0 0

50

0.6

0. 3

['' 0.6

0. 1

1.0

75

0.0

0.9

1.0 0.6 0

0.6 0.6 0.4

0. 1 0 .3 0.9

1.0 0.6 0.4

100

0.0

I .0

() 1 0.4 .

1.0

0.4

Graphical visua li zation of these steps is given in Figure 4 where the relations were computed on a fin er discretization using the member ·hip functions of Figure 3. The MATLAB implementation sc ri pt is as under.

% The MATLAB script for Figure 4

close al l

set(O, ' DefaultAxesFontName ' , 't i mes');

set ( 0, • Defaul tTextFontName ' , • t i mes' ) ;

set(O, 'DefaultAxesFontSize ', 1 1 );

set(O, 'DefaultTextFontSize' , 11) ;

set ( 0, · Defaul tAxesLineWidth ' , 0 . 2) ;

%set(O,'DefaultAxesPosition' , [.1.3 . 26 . 77 5 . 6 )) ;

figure(l); elf;

s u bplot(221);

mesh(XXi, YYi, Rli'); title( 'Rl = Low and Low'); xlabel ( 'x'); ylabel ( 'y');

view(332 , 45)

subplot(222);

SINGH eta/.: LINGUISTIC FUZZY MODELING

0.5

0 100

R1 =Low and Low

, ..... ····

; . -··· .... ···

X

R3 = High and Low

. . . . . . . . . .

. . : . . . . .

y 0 0 X

4

4

0.5

0 100

R2 = OK and High

b~"~ ~~-~ .

y 0 0 X

R = R1 or R2 or R3

0.5 . . . ·~·· -<~ •

0 100

y 0 0

.. ~.

X

4

4

Figure 4 - Fuzzy relations R"~,R1 corresponding to the individual rules, and the aggregated relation R corresponding to the entire rule base

xlabel ( 'x'); ylabel ( 'y');

view(332, 45)

subplot(223);

mesh(XXi, YYi , R3i'); title( 'R3 = High and Low'); xlabel('x '); y label('y ');

view ( 3 3 2 , 4 5 )

subplot(224);

mesh(XXi, YYi. Ri'); title('R = R1 or R2 or R3'); xlabel('x'); ylabel('y');

view(332, 45)

Figure 5 (a) mesh (b) image shows the fuzzy graph for the example (contours of R, where the shading corresponds to the membership degree) . The relational composition (Eq. 4) can be regarded as a function evaluation on the fuzzy graph. The MATLAB implementation script is as under.

% The MATLAB script for Figure 5 (a) & (b) and calculation o f 'R'

X [0 1 2 3];

y [0 25 50 75 100];

Xi 0 : 0 .1: 3;

Yi 0:4:100 ;

mfsx = [ . ..

1.0000 0.7475 1.3781 2.2347 3.5259

1.0000 1.8935 2.7487 4.0 000 4.0000];

mfsy = [ ...

1 .0 000 0 0 32.9187 74.9 856

1.0000 37.657 3 79 . 1120 110.0000 110.0000];

Low_x = [1 . 0 0.6 0 . 0 0.0];

OK_x = [0.0 0.4 1.0 0.4];

High_x = [0 .0 0.0 0.1 1.0];

Low_y = [1.0 1 . 0 0.6 0.0 0 . 0];

High_y = [0.0 0.0 0.3 0.9 1.0];

R1 and_za(Low_x ', Low_y);

R2 and_za( OK_x', High_y);

R3 and_za(High_x ' , Low_y);

R = or_za(or_za(R1 , R2), R3);

R

Low_xi = mgrade(Xi, mfsx( 1, :)) ;

OK_x i = mgrade( Xi, mfsx( 2, : )) ;

High_xi = mgrade(Xi , mfsx(3, :) ) ;

Low_yi = mgrade(Yi, mfsy (1, : )) ;

High_yi = mgrade(Yi, mfsy( 2, :)) ;

Rli and_za(Low_xi', Low_yi);

R2i and_za( OK_xi', Hi gh_yi);

857

858 1 SCI JND RES VOL 60 NOVEMBER 200 I

R1 =Low and Low R2 =OK and High •' •••• , .·

I .,'[&;:..,_ . .. , \.·· .

0.5 . . ·· .. 0.5

0 0 100 100

4 4

y 0 0 X y 0 0 X

R3 = High and Low R = R 1 or R2 or R3 , . · , .

, . ..

0.5 . . ~ .

,,£f~ 0.5 '<ifj} : .. 0 ·'· 0

100 100

4 4

y 0 0 X y 0 0 X

Fi gure 5- (a) A fu zzy graph (image for the lingui stic model. Darker shading corresponds to hi gher me mbership degree. The solid line is a possible cri sp function representing a simil ar relati onship as the lingu istic fu zzy model

SINGH eta/.: LINGUISTIC FUZZY MODELING 859

Ri = or_za(or_za(Rli, R2i}, R3i};

[XXi, YYi) = rneshgrid(Xi', Yi};

figure(l}; elf

rnesh(XXi, YYi, Ri'}, view(332, 65};

colorrnap(hsv}

figure(2}; elf

irnage(Xi, Yi, 10+70*{1-Ri'}}

set(gca, 'ydir', 'normal'}

colorrnap(gray}

Let an input fuzzy set to be the model

A· =[I , 0.6, 0.3, 0] which can be denoted as Somewhat Low flow rate, as it is close to Low but not equal to Low. The result of max-min composition is the fuzzy set

s · = [1 , I, 0.6, 0.4, 0.4], which gives the expected approximately Low heating power, computed by the Mamdani inference method .

Step I yields the following degrees of fulfillment :

{J, =max [JlA ·(x)" JlA, (x)] =max ([1,0 .6,0.3,0] " [1 ,0.6,0,0]) =•I X

{3 , = max[.u,( (x) "J.LA, (x)] =max ([1 ,0.6,0.3,0] " [0,0.4,1,0.4]) = 0.4• • X -

{3 1 = max [J.LA · (x) "J.LA1 (x)] = max ([I ,0.6,0.3.0] "[0.0,0.1.1]) = 0. I. . X

In step 2, the individual consequent fuzzy sets are computed :

B; = {3 , B, =I/\ [1 ,1,0.6,0,0] = [1,1,0.6,0,0],

B~ = /3 2B2 = 0.4 A [0,0,0.3,0.9,1] = [0,0,0.3,0.4,0.4],

B~ = {3 2 8 2 = 0.1/\ [1,1,0.6,0,0] = [0.1,0. 1.,0. 1,0,0].

Finally, step 3 gives the overall output fuzzy set:

s· = maxj.1l3; = [1, I, 0.6, 0.4, 0.4] ·

Similarly for A' = [0, 0.2, I, 0.2] (Approximately OK), B' = [0.2, 0.2, 0.3, 0.9, I], i e, approximately High heating power. Consider the output fuzzy set B' = [0.2, 0.2, 0.3, 0.9, I] where the output domain is Y = [0, 25, 50, 75, I 00]. The defuzzified output obtained by applying formula (Eq. 9) is:

0.2 ·0+0.2. 25+0.3· 50+0.9. 75 + 1· 100 2 2 y = =7 . I

0.2+ 0.2 + 0.3 + 0.9 +I

The heating power of the burner, computed by the fuzzy model, is thus 72.12 W.

Conclusions

The terminology of lingui st ic fuzzy modeling is straightforward, more appropriate and one of the important issues since the beginning of fuzzy theory. In this paper a simple industrial application of burner heating power with flow rate of oxygen is taken up with max-min (Mamdani) inference method and concept of fuzzy graph is explored, using MATLAB package. It is shown that linguistic models can be usefu l for various purposes because of their transparency and can be used in industrial applications, which are partly described by first principle models and partly by experience contained in designers and operators.

It is considered, however, that the linguistic fuzzy modeling approach may be of use, when additional system inputs are included, considering the relatively large computational overheads using different operators. The approach can be investigated further for design of fuzzy logic controllers for industrial applications.

Acknowledgement

The authors acknowledge the help extended by Dr R Babaska, Control Engineering Laboratory, Faculty of Information Technology and Systems, Delft University of Technology, The Netherlands.

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