An improved design procedure for fuzzy control systems

Int. J. Mach. Tools Manufact. Vol. 31. No. 1, pp.107-122. 1991. Printed in Great Britain

AN

IMPROVED

08th~955/9153.(KI + .IX) PergamonPressplc

DESIGN PROCEDURE CONTROL SYSTEMS I. KOUATLI*

FOR

FUZZY

and B. JONES*

(Received 22 December 1989; in final form 26 June 1990) A ~ t r a c t - - T h e purpose of this paper is to introduce simpler and more effective method in designing fuzzy controllers. A scale for partitioning the universe of discourse and choosing the appropriate fuzzy set shapes for the control variables is introduced and is termed "fuzzimetric arcs". Knowledge engineering techniques are used to obtain process information which is interpreted using systematic analysis as an aid to the design of fuzzy logic controller. This has been illustrated by a manufacturing system example in the form of a welding application which provides a guide to the reader who is unfamiliar with such techniques.

1.

INTRODUCTION

Fvzzv control theory can be applied to any process in which a human being plays an important role which depends on his subjective assessment skills. A lathe operator, for example, decides when a tool is worn and needs to be exchanged based upon vague variables such as the chip formation and the colour change, the noise of cutting and surface finish. All of these variables have a gradual transition from an acceptable to unacceptable condition of the tool which characterises the basic concept of a fuzzy variable, e.g. the noise of the cutting tool increases gradually until an unacceptable level is reached. An assembly worker can adjust the gripping force of his hand to assemble objects of brittle, soft and hard materials. A welding expert can similarly adjust the speed of his hand while welding if a cavity exists in the path of the weld; the larger the cavity the slower the speed of his hand should be. This welding example will be considered in more detail to illustrate the method of establishing a fuzzy control strategy which imitates the welding expert in performing an argon arc welding process. Argon arc welding is one type of TIG welding (tungsten electrode, inert gas shielded welding), that is usually applied to steel and a wide range of ferrous alloys without using flux where shielding is produced by argon gas. Figure l(a) shows the system set up for TIG welding in which the arc burns between a tungsten electrode and the work piece within a shield of inert agron gas (Fig. 1). In many applications, the edges of the metals to be welded are well prepared before starting, however, in this example it is assumed that an area of accidental damage is to be welded without previous preparation (Fig. 2). The objective is to control the speed of a robot arm to carry out the weld in the same manner as the human welding operator. Such application is useful in hostile and dangerous environments such as a nuclear plant where a mobile robot is required to perform the welding. The robot should have the intelligence to identify the type of weld required, based on an assessment of the damaged material type and thickness. A vision system is needed to provide a feedback to the robot in order to determine the welding path and the area and depth of any cavity along the path of the weld. The information generated allows the speed of the robot arm to be varied while welding. 2.

KNOWLEDGE ENGINEERING AND THE SYSTEM CONCEPT

Knowledge engineering is the process of determining the reasoning process that must take place and the data required to form a conclusion or to take an action. Several difficulties arise when attempting to define all the variables that the individual is *Department of Mechanical Engineering, University of Birmingham, Box 363, B15 2TT, Birmingham, U.K. 107

108

I. KOUATLIand B. JONES

Combinedwelding cableand gas tube

,

I/Coupling",,I I Regulator

I

Wire I Contacttu...Li I Nozzle I Shieldingggas

High frequencyI High voltage Spark unit I

I

Workpiece]

l

%T/

Workpiece

l Argon gas (a)

(b)

FIG. 1. (a) Set up of TIG welding. (b) Torch/work contact of argon arc welding.

l

Flo. 2. Fractured part to be welded.

required to take into account before deciding upon an action. In the case of welding processes, considerable differences will be apparent between successful welds produced by different individuals. This is due to the different interpretation of the welding process which is based largely upon an operator's own unique individual skills. It is the role of the knowledge engineer to identify the factors that a skilled operator takes into account in order to arrive at a course of action that can be employed by a machine system.

2.1. Systems engineering concept Systems engineering is a modern concept which allows integration of various engineering and social disciplines. The variables in process control may be vague, especially when dealing with man-machine systems where environmental factors can affect human performance. The system designer has to consider both the direct system elements as well as the environmental aspects. Checkland [1] describes the designer of an industrial plant as a system thinker. He has to consider not only the individual reactor vessels, heat exchangers, pumps, etc., which make up the plant but also, at a different level of consideration, the plant as a whole whose overall performance has to be controlled to produce the desired product at the required rate, cost and quality. The designer has to ensure that there exists

Fuzzy Control System

109

means by which information about the state of the process can be obtained that can be used to initiate action to control a reaction within predefined limits by knowing the variability of the base materials and the possible environmental disturbances to which the process will be subjected. A control strategy is then established automatically to maintain the required plant condition. Much has been written of the different aspects of system analysis used in industry. For example, Bekey [2] considered the human operator as an element in the control system and attempts to develop a mathematical model of the human operator. Singleton [3] studied the man-machine systems and suggested a system methodology for manufacturing systems. The conclusion reached was that there was no equivalent to the functional block diagram as a technique for defining and communicating what is required at all levels. Parnaby [4] introduced an input/output analysis block diagram in a study of the concept of the manufacturing system and concluded that there was no clear concept acceptable to everyone, but that some properties may be regarded as fundamental. Adopting such input/output analysis diagrams can illustrate the operator/process interactions in specific processes to be controlled and thus can be used as a tool by knowledge engineers. The inputs are any factors affecting the efficiency of the operator (heat, humidity, social factors, etc.); and the materials used, assets and tools. The outputs are mainly the performance of the process, the next step, profit, etc. The aim of such input/output analysis is to aid the knowledge engineer in gathering information relating to the skills of the operator as well as the information needed to automate the system. Thus, the knowledge engineer faces difficulties when attempting to gather the information required to control a process and systematic analysis may be used to simplify the extraction of the knowledge of a skilled operator and to clarify the control system requirements and demands. The information gathered about the automated welding example can be organized using the knowledge engineering with the aid of systematic analysis which can help in determining control variables. A typical example is that of a welding expert who has to adjust the gas flow, speed of his hand and the current or voltage based upon several variables such as the type of material to be welded, its thickness, the electrode type and the cavity size that exists in the path of the weld. A system input/output analysis of this process is shown in Fig. 3 which is mainly information gathered from a skilled operator. However, since the main objective is to illustrate the fuzzy control theory in modelling the hand movement of the welding operator, only the single input (cavity size)-single output (hand speed) case will be considered in the interest of simplicity.

H / V Weld Cavity size Thickness

Welding current Welding speed

Argon

Material type

Gas flow

Robot type

Tig welding Power supply Argon gas Flow metre Voltage regulator Skilled labour - -

Arc

- -

Welding Process

Welded part

Material handling Temperature - Humidity Hostile environment

Experience Operator morale Quality check

Heat - -

Robot application

FIG. 3. System input/output diagram of TIG welding process.

110

I. KOUATLIand B. JONES 3. THE CONCEPT OF THE FUZZY SET SHAPES

Different applications of the fuzzy control technique use a specific shape of the fuzzy set which is dependent on the system behaviour identified by the knowledge engineer. So far there is no standard method of choosing the proper shape of the fuzzy sets of the control variables. The scale suggested for the fuzzy variables of the control systems are termed "fuzzimetric arcs". Since the main interest in this paper is the application of fuzzy control to manufacturing processes, the welding example is selected to demonstrate the use of fuzzimetric arcs and the application of fuzzy control theory. The first step in establishing the fuzzy control system is the selection of the proper shape of fuzzy sets of the control variables based upon observation of the system behaviour. The fuzzy set shape is an influencing factor on the performance of the controller and may be altered to obtain the most suitable form of the fuzzy variables. In other words, in order to tune the system, an alteration of the control rules can result from a change of the fuzzy variables. Since a fuzzy control algorithm is one means of imitating the human's performance, the shape of the fuzzy sets of the control variables should be logical and acceptable to individuals. For example, for three fuzzy variables, specified as Large, Medium and Small in a universe of discourse U, e.g. the output universe of discourse in the welding process would be all the possible values of the arm speed. An overlap between Large

._o_ t-t/) Q) ..O

t/) ~D

E

Universe of discourse

Universe of discourse

(a)

(b)

FIG. 4. Differencebetween unacceptable (a) and acceptable (b) fuzzyset shape.

Membership value

NB

NM

NS NO' 'PO PS

PM

PB I.-

D-

-6-54-3-2-1

O

1

2

345

6

Universe

FIG. 5. Fuzzysets used by Assilian as assembled from his table [5].

Fuzzy Control System

111

and Medium or an overlap between Medium and Small is logical and acceptable. However an overlap between Large and Small is not acceptable to individuals. Figure 4 shows the difference between acceptable and non-accepted overlapping. Assilian [5] allowed such overlapping between Large and Small (Fig. 5) which may be applied to applications of fuzzy sets to topics such as psychology, medical diagnosis, etc. However, when dealing with numeric applications of the type considered in manufacturing such overlapping would not be allowed. Most of the applications of fuzzy control use this type of classification of fuzzy variables. No work so far has resulted in a general method for selection of the shape and scale of the fuzzy sets. For example, Sutton and Towill [6] in their application to modelling a helmsman action in ship steering, used a trapezoidal shape which was found to be the best shape for that specific application where five fuzzy variables were defined as NB, NS, ZO, PS and PB. Overlapping was set only between any two adjacent variables (Fig. 6). Yamaguchi et al. [7] chose a triangular shape of the fuzzy variable when applying the technique to estimate and predict elevator system control and again only the adjacent variables overlapped (Fig. 7). Adoption of this principle, enables a general method to be identified for selection of the optimum fuzzy set shape. 4.

FUZZIMETRIC ARCS

From the observations so far it is clear that a standard format for the fuzzy variables is needed and that an analogy may be drawn between partitioning the universe of discourse and the structure of a trigonometric circle. Three quarters of a circular arc may be introduced as a partitioning scale which has a radius of one and which carries the fuzzy variables Positive Zero (PO), Positive Small (PS), Positive Medium (PM) and Positive Big (PB) where infinity is assumed to be any number that exceed the limit of the universe (Fig. 8). Fuzzy variables will be defined as:

1

-20

-I0

0

10

20

0.5

1.0

15

30

Eo

(a)

I

-I .0

-0.5

0

~de~s (b) 1

-30

-I 5

0

8°

(e) FIG. 6. The shape of fuzzy sets as used by Sutton and Towill [6].

112

I. KOUATLI and B. JONES

ZO

PM

PB

t-

~

E~

0.5

0

it

a

b

c

NB NM NS

0

1

2

3

Z

4

PS PM PB

5

6

7

8 D-

e, Ae (a) Membership functions pA, pB

(b) Membership function pC

FIG. 7. The shape of fuzzy sets as used by Yamaguchi et al. [7].

PS

PO

PB

PS

PM

; PB

Universe

(a)

(b)

FIG. 8. (a) Positive fuzzimetric arc, (b) spread of fuzzy variables on the membership-universe axes.

PO = [sin(rr/2-x)[ = 0 PS = [sin(x)[ = 0 PM = [sin('tr/2-x)[ = 0 PB = [sin(x)[ = 0

for 0 < x < "rr/2 otherwise for 0 < x < ~r otherwise for rr/2 < x < 3~/2 otherwise for rr < x < 3-tr/2 otherwise.

Infinity is assumed at any value greater than the maximum value of Positive Big, and this arc may be termed "a fuzzimetric arc" because of its analogy with the trigonometric circle. Similarly the negative fuzzy variables may be defined in the same manner, but with a different direction i.e. counter-clockwise for the positive direction and clockwise for the negative direction. The zero level may then be shared between the two arcs (Fig. 9). Discretization of the universe in this way will assign certain degrees on the arcs for each member. As an example, assuming a universe of real numbers in the range of { - 9 , +9}, the universe should be partitioned in segments of 30°/member (+270/9 and - 2 7 0 / - 9 ) . Table 1 shows the result of such discretization. Linguistic hedges can be used to alter the shape of the fuzzy set. For example, the fuzzy set "Small" can have the linguistic hedges of "very small" or "more or less small" defined as; 2 gLvcry small ---- ~small 1/2 gt'morc or less small --~ ~ s m a

Fuzzy Control System

113

PS NS PM~

NM Universeofdiscourse

PB NB

FIG. 9. Fuzzimetric arcs, negative and positive.

TABLE 1. VALUES OF UNIVERSE PARTITIONING USING "FUZZIMETRIC ARCS"

-9 NB NM NS NO PO PS PM PB

-8

-7

1 0.86 0.5 0 0.5 0.86 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-6

-5

-4

0 0 0 1 0.86 0.5 0 0.5 0.86 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-3 0 0 1 0 0 0 0 0

-2

-1

0 0 0 0 0.86 0.5 0.5 0.86 0 0 0 0 0 0 0 0

0

1

2

0 0 0 0 0 0 0 0 0 1 0 0 1 0.86 0.5 0 0.5 0.86 0 0 0 0 0 0

3

4

5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.86 0.5 0 0.5 0.86 0 0 0

6

7

8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.86 0.5 0 0.5 0.86

9 0 0 0 0 0 0 0 1

where Ix is the m e m b e r s h i p value of a certain m e m b e r in a set. Some trigonometric f o r m u l a e m a y be applied to "fuzzimetric arcs" to define fuzzy variables that lies b e t w e e n two adjacent sets, for example: + B Z S = P O 2 + P S 2 w h e r e P O 2 + P S 2 = 1 f o r 0 ° < x < 90 ° + B S M = P S 2 + P M 2 where PS 2 + P M 2 = 1 for 90 ° < x < 180 ° + B M B = P M 2 + PB 2 w h e r e P M 2 + PB 2 = 1 for 180 ° < x < 2 7 0 °. Similarly: -BZS -BSM -BMB where

where N O 2 + N S 2 = 1 for 0 ° < x < - 9 0 ° w h e r e N S 2 + N M 2 = 1 for - 9 0 ° < x < - 1 8 0 ° = N M 2 + N B 2 w h e r e N M 2 + N B 2 = 1 f o r - 1 8 0 ° < x < - 2 7 0 °.

= NO 2 + NS 2

+BZS +BSM +BMB -BZS -BSM -BMB

= NS 2 + NM 2

= b e t w e e n zero and Positive Small = b e t w e e n Positive Small and Positive M e d i u m = b e t w e e n Positive M e d i u m and Positive Big = b e t w e e n zero and Negative Small = b e t w e e n Negative Small and Negative Medium. = b e t w e e n Negative M e d i u m and Negative Big.

Thus, with the help of fuzzimetric arcs, it is possible to assemble a n o t h e r type of linguistic hedge f r o m two adjacent sets. If a trapezoidal shape was considered appropriate for the control of the process, then the following relationship should be applied to the fuzzy variables:

arcsin (fuzzy variable) IX = --- 1 for t < arcsin (fuzzy variable)

31tI-H

114

I. KOUATL! and B. JONES

where the fuzzy variable could be any of PO, PS, etc., and t is a variable in the range of 0 < t -< 90. A special case is when t = 90 which produces a triangular shape. If the value was greater than 90°, a triangular shape of the fuzzy set results in which maximum value of membership is less than unity (Fig. 10). In this way t can be used as a fuzzy variable in order to determine the optimum shape of the fuzzy set based upon the past performance of the system. Using this principle, fuzzy set theory itself may help in determining a suitable shape for the fuzzy variables. It may be that when partitioning the universe a certain member lies at an angle of 45 ° of the arc. In this case the member has a maximum fuzziness value simply because that member has an equal membership value in two different sets. The maximum fuzziness value exists on the positive fuzzy arc at angles 45° , 135° and 225 °. The degree of fuzziness may be measured using the formula;

d(e) =

Isin 201 for 0 <

0 < 270°. 5.

FUZZY ALGORITHMS

The fuzzimetric arc scale described in the previous section, may now be applied to the welding process. The knowledge extracted from the operator may be organized in a logical control rules format which describes the behaviour of the skilled operator. A fuzzy algorithm can then be set based upon observation, and discussion with the operator. In this case only four rules are employed for the fuzzy controller as follows. Rule Rule Rule Rule

1: 2: 3: 4:

IF IF IF IF

Cavity Cavity Cavity Cavity

Size Size Size Size

is is is is

Tiny, THEN Speed is Fast. Small, THEN Speed is Regular. Medium, THEN Speed is Slow. Large, THEN Speed is Minimum.

These rules are in the form of an expert system carrying fuzzy expressions like Large, Fast, etc., which allows the machine to imitate the skills of the welding operator. The input to the system is the cavity size and the output is the movement of the hand.

5.1.

Universe partitioning

After establishing the control rules and developing the control algorithm, the second step is to partition the universe of the input and the output using the fuzzimetric arc which in this case consists of only the positive arc. The input universe cavity size should be partitioned according to the maximum value allowed to control the process which is approximately three times the diameter of the electrode which is usually 3 mm diameter rod. If the cavity size is larger than this, welding will be required separately (a} .........

(b)

..... (c) ................. (d)

I

0

=

9O Universe

Fro. 10. Fuzzy shape of the fuzzy sets: (a) Sine wave; (b) trapezoidal (p,:arcsin (sin x)/t where t < 90); (c) special case when t = 90; and (d) t > 90.

Fuzzy Control System

115

at each of the edges before welding across the width of the cavity. On this basis the universe of the cavity size should be in the range of [0, 9] with any value above assumed to be infinity and a zero value implying that there is almost no cavity. The fuzzimetric arc then should be split into nine units with each unit equivalent to 30 ° (Fig. 11). The fuzzy variables for the Cavity Size will be defined as:

Isin "tr/2-x[

Tiny

=

Small

= Isin xl

Medium

=

Large

= [sin xl

0 <- x <- ~r/2

0 ~ x ~

Isin ~r/2-xl

"tr/2 <. x <- 3~r/2

~r <-- x -< 3"rr/2

where x = 30 x X x ~/180 ° rad, X = 1, 2, 3 ... maximum cavity size universe limit. In a similar manner the universe of the output, which is hand speed, should be partitioned according to the range of speed required, which is between 20 and 35 cm/min and, the universe should be split to 15 units with each unit representing 18°, Any value above the limit of the universe is infinity and the zero level is assumed to be the minimum speed (20 cm/min) which should be added to the final output of the controller (Fig. 12). The fuzzy variables of the speed universe will be defined as: Minimum

Slow

= =

[sin "rr/2-yl

0 <- y <- 'rr/2

Isin yl

0 -< y -< rr

'rr/2-yl

Average

= [sin

Fast

= [sin y[

"rr/2 <. y <-- 3~r/2

rr --- y --- 3~r/2

Small 3

Medium 6

~

tiny Smell Medium

o TTny

0

9

Large

3 6 Cavitysize

[Large=

9

FK;. 11. Partitioning cavity size universe.

Slow 5

Regular1

0

~

15

Fast

0 Minimum

p

5

10

Speedrange FIG. 12. Partitioning speed universe.

15

D

116

I. KouAru and B. JOr~ES

where y = 18 x Y x ~r/180° Y = 1, 2, 3 ... maximum speed universe limit, i.e. 15. Tables 2 and 3 show the results of such discretization of the input and the output, respectively. 5.2.

Fuzzy relation

The fuzzy relation is the relationship between the input and the output of the control system, such as that between the fuzzy sets "Tiny" and "Fast" in the first rule of Section 5 for example. Then the relationship between the input (Tiny Cavity) and the output (Speed Value) can be found using Cartesian product expressions of the two sets R = input * output where * represents the Cartesian product. In the case of rule 1, the relation would be R1 = (Cavity SiZe)Tiny * (Speed Value)Fast which has a membership function of

I&R1= min{l&Tinycavity(CS), I,I,Fastspeea (S) } • From Table 2, the fuzzy set Tiny is defined as Tiny = 1/0 + 0.866/1 + 0.5/2 + 0/3 + 0/4 + 0/5 + 0/6 + 0/7 + 0/8 + 0/9.

From Table 3, the fuzzy set Fast is defined as Fast = 0/0 + 0/1 + 0/2 + 0/3 + 0/4 + 0/5 + 0/6 + 0/7 + 0/8 + 0/9 + 0/10 + 0.31/11 + 0.587/12 + 0.81/13 + 0.95/14 + 1/15.

The relation then between Tiny and Fast will be as follows Speed universe 0 1 2 3 4 5 6 7 8 910 Cavity universe

RI=

0 1 2 3 4 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000000 00000 0000 0000000

0 0 0 0 0 0 0 0

11

12

0 0.310.587 0 0.31 0.587 0 0.310.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13

14

15

0.810.95 1 0.81 0 . 8 ~ 0 . 8 6 6 0.5 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Fuzzy Control System

117

TABLE 2. DISCRETIZEDUNIVERSEOF CAVITYSIZE

Tiny Small Medium Large

0

1

2

3

4

5

6

7

8

9

1 0 0 0

0.86 0.5 0 0

0.5 0.86 0 0

0 1 0 0

0 0.86 0.5 0

0 0.5 0.86 0

0 0 1 0

0 0 0.86 0.5

0 0 0.5 0.86

0 0 0 1

TABLE 3.

0 Minimum Slow Regular Fast

1 0 0 0

1

2

3

4

0.95 0.81 0.58 0.31 0.31 0.58 0.81 0.95 0 0 0 0 0 0 0 0

6.

DISCRETIZED UNIVERSE OF SPEED

5 0 1 0 0

6

7

8

9

0 0 0 0 0.95 0.81 0.58 0.31 0.31 0.58 0.81 0.95 0 0 0 0

10 0 0 1 0

11

12

13

14

15

0 0 0 0 0 0 0 0 0.95 0.81 0.58 0.31 0.31 0.58 0.81 0.95

0 0 0 1

COMPOSITION RULE OF INFERENCE

Due to the difficulty of having a control rule for every possible situation, a composition rule of inference may be used to obtain an output subset which belongs to the output fuzzy set from an input using the fuzzy relationship between the input and the output. Suppose there is an input of cavity size "about two millimeters" which may be expressed in fuzzy terms as 0.5/0 + 0.866/1 + 1/2 + 0.866/3 + 0.5/4 + 0/5 + 0/6 + 0/7 + 0/8 + 0/9.

Using the fuzzy relation R = input * output. For a given input, the output may be obtained using the composition rule of inference written as Fast' = Tiny' o R1 where o denotes the max-rain Cartesian product. Thus the membership function of the output fuzzy subset "Tiny'" is defined as; I~FasC(S) = m a x rain [ Wriny ( C S ) , I~m ( C S , S ) ) .

From an application point of view, it may be required to answer the question "if the cavity size is about two millimetres, what would the speed be?". Consider the first part of the equation where the minimum of the membership values between the input (Tiny') and the relation (R1) are to be determined

118

I. KOUATLI and B. JONES

Speeduniverse 01234567891011 0 1 2 3 4 5 6 7 8 9

Min =

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

13

0 0.310.5 0.5 0 0.31 0.5870.81 0 0.31 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14

15

0.5 0.5 0.8~0.8~ 0.5 0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0

In order to satisfy the second part of the equation, the maximum of those minimums is required which will result in the following fuzzy set Fast' = [ 0 0 0 0 0 0 0 0 0 0 0 0 . 3 1 0 . 5 8 7 0 . 8 1 0 . 8 6 6 0 . 8 6 6 ] . It should be noted that the resulting fuzzy control set "Fast'" is a subset of the original fuzzy set "Fast". Also, since machines do not understand fuzzy variables, averaging of this result will be required and the following formula may be used

Average Value =

E Speed Value x Ix (S) E Ix (S)

For the "Fast" fuzzy control set the average result would be

Speed Value =

0.31 * 11 + 0.587 * 12 + 0.81 * 13 + 0.866 * 14 + 0.866 * 15 0.31 + 0.587 + 0.81 + 0.866 + 0.866

= 13.4 cm/min. Thus for a cavity size "about two millimetres", the speed of the robot arm should be 13.4 + 20 = 33.4 cm/min where 20 cm/min is the original speed which is subtracted before splitting the speed universe. 7. RULES COMBINATION The compositional rule of inference using the relationship of the first rule has been shown in the previous section. In a similar manner the relationships of the second, third and fourth rules may be developed. From Section 5 the second rule states IF Cavity Size is Small THEN Speed is Regular which is represented as a fuzzy relation between the input (Small cavity) and the output (Regular speed) as

119

Fuzzy Control System

R2 = (Cavity Size)smaJl * (Speed Value)aegular i.e. Speed universe

0 1 2 3 4 R2= 5 6 7 8 9

0 1 2 3 4 5 6

7

8

9

10

11

12

13

14

15

0000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000 0000000 0000000 0000000

0 0.5 0.587 0.587 0.587 0.5 0 0 0 0

0 0.5 0.81 0.81 0.81 0.5 0 0 0 0

0 0.5 0.866 0.95 0.866 0.5 0 0 0 0

0 0.5 0.866 1 0.866 0.5 0 0 0 0

0 0.5 0.866 0.95 0.866 0.5 0 0 0 0

0 0.5 0.81 0.81 0.81 0.5 0 0 0 0

0 0.5 0.587 0.587 0.587 0.5 0 0 0 0

0 0.31 0.31 0.31 0.31 0.31 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0.31 0.31 0.31 0.31 0.31

Thus, the second relation (R2) can be combined only when an input is such that it is within the "Small" cavity size classification and allows the regular speed value to be inferred. It is obvious that the first and second relations may be combined to produce one which allows an input to be either "Tiny" OR "Small". The combination operator may be assumed to be "OR" function which is represented as the maximum of the membership values of the two different relations. The fuzzy statement combined from two fuzzy rules R1 and R2 will be

IF Cavity Size is Tiny THEN Speed is Fast OR IF Cavity Size is Small THEN Speed is Regular which is equivalent to; I~RI+R2 = max {~R1, ~R2} and is represented in the following form. Speed universe 0123456 0 1 2 3 R1 4 +R2 = 5 6 7 8 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000 0000000 0000000 0000000

0 0 0 0 0 0

0 0 0 0 0 0

0 0.31 0.31 0.31 0.31 0.31

7

8

9

10

11

12

13

0 0.5 0.587 0.587 0.587 0.5 0 0 0 0

0 0.5 0.81 0.81 0.81 0.5 0 0 0 0

0 0.5 0.866 0.95 0.866 0.5 0 0 0 0

0 0.5 0.866 1 0.866 0.5 0 0 0 0

0.31 0.5 0.866 0.95 0.866 0.5 0 0 0 0

0.587 0.587 0.81 0.81 0.81 0.5 0 0 0 0

0.81 0.81 0.587 0.587 0.587 0.5 0 0 0 0

14

15

0.95 0.866 0.5 0.31 0.31 0.31 0 0 0 0

1 0.866 0.5 0 0 0 0 0 0 0

120

I. KOUATLI and B. JONES

Similarly, the third rule may be established as follows.

0 1 2 3 4 5 6 7 8 9

R3=

Speed universe 6 7 8

01

2

3

4

5

00 00 00 00 0 0.31 0 0.31 0 0.31 0 0.31 0 0.31 00

0 0 0 0 0.5 0.587 0.587 0.587 0.5 0

0 0 0 0 0.5 0.81 0.81 0.81 0.5 0

0 0 0 0 0.5 0.866 0.95 0.866 0.5 0

0 0 0 0 0.5 0.866 1 0.866 0.5 0

0 0 0 0 0.5 0.866 0.95 0.866 0.5 0

0 0 0 0 0.5 0.81 0.81 0.81 0.5 0

0 0 0 0 0.5 0.587 0.587 0.587 0.5 0

9

1011 12 13 14 15

0 0 0 0 0.31 0.31 0.31 0.31 0.31 0

00 00 00 00 0 0 0 0 0 0 0 0 0 0 00

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

The final rule which states IF Cavity Size is Large THEN Speed is Minimum where the fuzzy sets "Large" and "Minimum" are defined by Large = 0/0 + 0/1 + 0/2 + 0/3 + 0/4 + 0/5 + 0/6 + 0.5/7 + 0.866/8 + 1/9 Minimum = 1/0 + 0.95/1 + 0.81/2 + 0.587/3 + 0.31/4 + 0/5 + 0/6 + 0/7 + 0/8 + 0/9 + 0/10 + 0/11 + 0/12 + 0/13 + 0/14 + 0/15. The fourth relation then can be developed as follows. Speed universe

R4=

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5 6 7 8 9 1011 12131415

0 0 0 0 0 0 0 0.5 0.866 1

0 0 0 0 0 0 0 0.5 0.866 0.95

0 0 0 0 0 0 0 0.5 0.81 0.81

0 0 0 0 0 0 0 0.5 0.587 0.587

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.3100000000000 0.31 0 0 0 0 0 0 0 0.31 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Thus the total combination of the four relations using "OR" operator is the maximum of the memberships. Thus the fuzzy algorithm IF IF IF IF

Cavity Cavity Cavity Cavity

Size Size Size Size

= = -=

Tiny T H E N Speed = Fast OR Small THEN Speed -- Regular OR Medium T H E N Speed = Slow OR Large THEN Speed = Minimum

Fuzzy Control System

121

can be represented in the relation R which has a membership function of IxR = max {IXRI, [,I,R2, ~I,R3, Id,R4} which can be represented by the following relation Speed universe

R=

0 1 2 3 4 5 6 7 8 9

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 0 0 0 0 0 0 0.5 0.86 1

0 0 0 0 0.31 0.31 0.31 0.5 0.86 0.95

0 0 0 0 0.5 0.58 0.58 0.58 0.81 0.81

0 0 0 0 0.5 0.81 0.81 0.81 0.58 0.58

0 0 0 0 0.5 0.86 0.95 0.86 0.5 0.31

0 0 0 0 0.5 0.86 1 0.86 0.5 0

0 0.31 0.31 0.31 0.5 0.86 0.95 0.86 0.5 0

0 0.5 0.58 0.58 0.58 0.81 0.81 0.81 0.5 0

0 0.5 0.81 0.81 0.81 0.58 0.58 0.58 0.5 0

0 0.5 0.86 0.95 0.86 0.5 0.31 0.31 0.31 0

0 0.5 0.86 1 0.86 0.5 0 0 0 0

0.31 0.5 0.86 0.95 0.86 0.5 0 0 0 0

0.58 0.58 0.81 0.81 0.81 0.5 0 0 0 0

0.81 0.81 0.58 0.58 0.58 0.5 0 0 0 0

0.95 0.86 0.5 0.31 0.31 0.31 0 0 0 0

1 0.86 0.5 0 0 0 0 0 0 0

This relation is in fact the model of the action of the welding operator. Combining this relation with any value of the cavity size that lies in its universe [0-9 mm], results in the required speed output for proper welding. Assuming the input is a fuzzy singleton (i.e. a fuzzy set which has a unity membership value at the required input and has a value of zero elsewhere), then averaging the membership values in the above relation will result in the final output of the controller, e.g. if the input was "almost seven" then the speed should be

speed(cavity=7) =

0 . 5 × ( 0 + 1 ) + 0 . 5 8 x ( 2 + 8 ) + 0 . 8 1 x ( 3 + 7 ) + 0 . 8 6 x ( 4 + 5 + 6 ) + 0.31 x 9 0.5 + 0.5 + 0.58 + 0.81 + 0.86 + 0.86 + 0.86 + 0.81 + 0.58 + 0.31

= 4.51 cm/min i.e. total speed value for cavity size of 7 should be; speed = 20 + 4.51 = 24.51 cm/min.

The following table shows the averaged values of speed for a specific input of cavity size. Cavity Size 0 1 2 3 4 5 6 7 8 9

Averaged Speed 13.48 11.28 10.48 10.00 8.04 6.145 5.00 4.51 3.718 1.52

Final Speed 33.48 31.28 30.48 30 28.04 26.145 25 24.51 23.718 21.52

122

I. KOUATLIand B. JONES 8.

CONCLUSIONS

From the above table it may be noticed that the fuzzy logic control in this case can be looked upon as an interface between the qualitative world of human beings and the quantitative world of engineering where it provides a translation of human's brain interpretation of fuzzy variables such as Fast, Slow, Large, etc., into a specific output by means of the fuzzy relations developed. The control engineer does not have to be exact in specifying the speed of the robot arm as long as the boundaries of the universes of the input and the output of the fuzzy variables are specified, i.e. maximum values expected of the input and the output. Thus fuzzy logic provides a methodology and imitation of a human's way of thinking which is very useful in such applications where the mathematical model of the process does not exist or is inaccurate and in the case of the welding example, the mathematical model clearly does not exist. However, since fuzzy control is one type of "intelligent control", the performance of the fuzzy controller will be very much dependent on the knowledge provided to the system. This task of obtaining and organizing the process related human knowledge is difficult and the input-output analysis described in this paper can provide the knowledge engineer as well as the control engineer with a simplified design of the process. The use of the suggested fuzzimetric arcs in the design of fuzzy controllers provides engineer with the following techniques: (i) More systemic and simpler methodology in choosing the fuzzy set shape of the control variables. These can be scaled on the fuzzimetric arcs and where the universe can be partitioned automatically if the maximum expected number of the variable (universe of discourse) is defined. (ii) More powerful automatic selection of the optimum fuzzy set shape of the control variable, which is one factor influencing the performance of the system. The two techniques proposed suggest the possibility of developing an expert system that can be used by the control system engineer as an aid in establishing the strategy of controlling a specific process. The major task of the expert system is to determine the optimum rules of the process as well as the optimum fuzzy set shape of the control variables to be adopted. REFERENCES

[1] P. [2]

[3] [4]

[5] [6] [7]

B. CHECKLAND,Science and system movement. In System Behaviour (Edited by J. BEISHONand G. PETERS), 3rd edn (1976). G. A. BEKEY,The human operator in control systems. In System Behaviour (Edited by J. BEISHONand G. PETERS), 3rd edn (1976). W. T., SINGLETON,Man-machine systems. In System Behaviour (Edited by J. BEISHONand G. PETERS), 3rd edn (1976). J. PARNABY,Concept of manufacturing system. In System Behaviour (Edited by J. BEISHON and G. PETERS), 3rd edn (1976). S. ASSILIAN,Artificial intelligence in the control of real dynamic systems, Ph.D. Thesis, University of London (1974). R. SUTTONand D. R. TOWlLL, An introduction to the use of fuzzy sets in the implementation of control algorithms, J. Inst. Elect Radio Engrs 55, 357-367 (1985). T. YAMAGUCHI,T. ENDO and K. HARUKI, Fuzzy predict and control method and its application. IEE Int. Conf. Control gg, 287-292 (13-15 April 1988).