Control Dc Servo

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Multicriteria Fuzzy Control and

Its

Application

t o DC

Servomotor Position Control

Kwan g - C h u n

Kim

an d J o n g - Hwan Kim*

Abstract- In this paper, we present

a

multicriteria fuzzy

controller(MFC) based on fuzzy m easur es and fuzzy inte-

grals. Th e basic idea underlying this approach is based

on analyzing the source of attributes of the system out-

put responses and applying the fuzzy measure and in-

tegral theory to the existing fuzzy controllers. By this

scheme, we can tune the three attributes including rise

time, overshoot, and settling time of th e output responses.

We demonstrate that MFC have superior control perfor-

mance compared to conventional fuzzy controller by com-

puter simulations as well as via exper iments performed

on

a DC servomotor position control. Moreover, our scheme

can be easily implemented

i n

practice siiiiply by adding

weight terms to an existing fuzzy rules and assigning the

weight to each rule by employing the fuzzy measure and

integral.

I(eyWords-multicriteria fuzzy contr oller(M FC), fuzzy mea-

sure, fuzzy integral, DC servomotor position control.

I .

I N T R O D U C T I O N

Recently, Grabisch[l] presented a survey paper on fuzzy

measures and integrals , where he mentioned that some

application papers using fuzzy measures and integrals

could be found for the prediction of wood strength[2],

the evaluation of printed color images[3], the design of

speakers[4], and the human reliability analysis[5], etc.

Since Sugeno’s thesis on fuzzy integrals[6], some theoret-

ical work

has

been done to show the usefulness of fuzzy

measures an d integrals by M urofushi and Sugeno[7], and

Grabisch e2

a1.[8;91.

However, in spite of their theoreti-

cal work, fuzzy m easure an d integral theory has not been

widely used in a n application field, at least until recently.

A multicriteriafuzzy controller(MFC) using fuzzy mea-

sures and fuzzy integrals

w a s

first for control by K im

[lo].

The idea underlying that scheme is based on analyz-

ing the source of attributes of the sys tem output re -

sponses and applying the fuzzy measure and integral the-

ory to the conventional fuzzy controllers. As already

well known, since Zadeh’s paper[ll], fuzzy logic-based

controllers have received considerable interest in rec,ent

years. T hese fuzzy controllers are based on hum an rules.

In the rules, three attr ibu tes are defined such

as rise l i m e ,

overshoot,

a nd

se t t l ing t ime

to be evaluated t o get desired

output responses. By assigning three partial evaluations

to each rule and expressing grade of importance by a

fuzzy measure, three partial evaluations can be aggre-

gated using a fuzzy integral. Thr ough the scheme, the

three attributes of the output responses can be tuned

‘Dept. of Electrical Engineering, Korea Advanced Institute of Sci-

ence and Technology (KA IST) ,

373-1

Kusung-dong, Yusung-gu, Taejon-

shi 305-701,

Republic

of

Korea. Corresponding author’s e-mail address

: johkimOvivaldi.kaist.ac.kr

under the same given fuzzy rules.

By computer simula-

tions, it is shown that the control scheme has an excel-

lent tuning performance, compared to the existing fuzzy

controlle rs wi thout mult ic r i te r ia funct ion.

And also we

demonstra te the performance of our scheme v ia experi-

ments performed on

a

DC servomotor position control.

Another adva ntage of

o u r

scheme is that it can be easily

implemented in practice simply by add ing weight terms

to an existing fuzzy rules and assigning the weight to

each rule by employing the fuzzy measure and integral.

11. BASICCONCEPTS

As we

will

deal only witch f i n i t e spaces in the subse-

quent applic ation, we restrict original definitions to a fi -

nite space

X

=

( 2 1 ,

. .

.

,z,}.

Let

u s

define fuzzy measures

on the power set of X , denoted P ( X ) .

Definition 1.

A fuzzy m easure

g

defined on

( X ,P ( X ) )

is a set function

g :

P ( X ) -+

[0,

13 verifying the following

axioms

:

(1)

boundary condi t ion

( 2 )

monotonicity

g ( d )

=

0 , g(.V

= 1

If A , B E P ( X )

a nd

A

B ,

thcn

g ( A ) 5 g( B)

Fuzzy measures include belief measures, plausibility

measures, prob ability m easures, possibility and necessity

measures, etc. Another widely used fuzzy measure intro-

duced by Sugeno

is

the A-fuzzy measure, which satisfies

the following axiom:

for every A , B E P ( X ) a nd A fl

B = 4

where is a real

number in ( - 1,

+CO .

Now let

us

define Ch oque t fuzzy integral.

Defiiiitioii

2.

(Choquet Integral) Let

h

be a ma pping

from

X

to [0,1]. The Choquet fuzzy integral of

h

over a

subset of X E

P ( X )

with respect to a fuzzy measure g is

defined:

J h

0 g =

- h ( z i - I ) ) g ( H i )

i = l

where it

is

assumed without

loss

of generality that

0 5

h(zl) 5 . . . 5 h x,) 5 1, Hi = {zklk

=

i , . . . , I } ,

a nd

h(z*)

= 0.

837

-7803-1896-X/94 $4.00 0199 4 IEEE

~ ~-

-

~

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I l l .

MULTICRITERIA

U Z Z Y

ONTROLLERFC)

A .

Fuzzy Reasoning wi th Weight

fuzzy

rules with weights.

We first describe a fuzzy reasoning for the following

R I : If E ( ' ) and

.

.

and

E : ) ,

Rz :

If

E ')

and . . . and

E : ) ,

then U ( ' ) with

tu( )

then U 2)ith

t

R,

Fact

:

If

e;

Cons : then U

: If Eln)

and

. .

.

and E?,"),

then

U ( )

with tu( )

and..

and

e h

where E?) a n d U(' ) are fuzzy sets, e: is

a

crisp value,

and w ( j ) s a weight.

Th e weight d i ) s a real number in [0, 11. The weight

can

be considered as a cer tainty factor [ l2] , a weighting

factor for emphatic and restrained effects on the fuzzy

inference result[ l3], or a possibility of the rule's correct-

nes s[ 141.

The firing of these rules w i t h crisp input values e ; ,

j

=

I .

.

. , t gives the trutIi values

f ( ) ,

i = 1, . .

, I

of t,he

preiiiises which are obta ined from th e C artesian protliict

x

E t ) x . .

x E:):

The defuzzification process maps the result of fuzzy

logic rule stage to a real nuniber output. I n this

pa-

per, we use the

h e i g h t defu:z?ficatzon t w t h o d

[ i s ] , whicl i

is simple to itnpleiiient and gives relatively good results.

The inference result is tlieii given by

where f i ) is called a height , and d i )

s

the peak value of

U( i ) , which is the do main element on 11 that has degree

of membership 1.

In this fuzzy reasoning, the proble in may be how to

assign weight to each rule. It is hard tlo figure out the

control result,s by changin g the weight,s of

fuzzy

rules.

Therefore, determ inati on of the postulated weight. vi&

ues

is also, i n general, based 011 lieiiristics. 111 the next

section, we

will

describe a consistent, Iiietliotl for assign-

ing the weight to each fuzzy

r u l e

by employing the

fuzzy

measure and integral .

B .

Deszgn of

M F C

Now let us consider a problem where a person subjec-

tively evaluates an ob ject according to his preference. As-

sume that the object in the problem can be divided into

elements or it has attr ibutes . Let

X

=

{

z1,

. . .

,z,} e

a

set of such elements

or

a t t r ibu tes .

Let h

: X

0 ,1 ] be par t ia l evaluat ion of the object .

T h a t is,

h(z , )

mplies a partial evaluation of the object

from the viewpoint of an element z,. n the applications,

a par t ia l evaluat ion

h ( t i )

can be determined either ob-

jectively or subjectively.

NE NM NS U FS FN PB

I

0

I

0 I

Fig. 1. Membership Functions

Taking a fuzzy integral of h with respect to

g ,

we have

w = k h o g

Above equ atio n represents the aggregation of parti al

evaluations, where w can be considered as the overall

evaluat ion of the object .

In the previous fuzzy rules, we choose attr ibu tes and

assign partial evaluations h ( ' ) ( z j ) o rule R, f r om the

viewpoint of an at tr ibu tes z j Th e average value in each

rule c a n be obtained from the fuzzy integral of h ( ' ) ( z j )

with respect to preassigned fuzzy measure

g .

The av-

erage value

is

then used as a weight to each rule. T his

process is summarized as follows:

RI

/ L ( l ) ( = l ) , . . , h ( l ) ( ~ , )

+ w 1 )

=

h( ' )

o g

R2

( ) ( z l ) ,

. ,

-

= o

g

Tliiis,

f inal consequence U*

is

inferred

as

the weighted

average of di) by the degree

where d i Jx

h(')

o g

In this pa per, we choose time-domain specifications in-

cluding

z1

= r i s e t i n e , 2 2 = overshoot , a n d t3 = se t t l ing

l i m e

as

the at tr ibutes of the system output responses .

These three at tr ib utes are the m ain terms of t ime-domain

specification when we design a control sys tem. Th e col-

lection of l inguis f ic va lues is

L =

{

N B , N M ,

N S ,

20,PSIP M , P B )

Figure 1 shows

a

plot of the membership functions of

linguistic values. Th e meaning'' of each lingu istic value

should be clear from its mnemonic.

Fuzzy control rules are given in Table 1.

e ( k )

= yr (k ) -

y k) where y,(k)

is

the reference input,

y k)

is the system

o u t p u t , a n d A e ( k ) = e ( k )- ( k - 1) .

As shown in Table 2 , each rule in Table 1 contains the

three partial evaluations It( )(.) o each at tr ibu te such as

r i se h i e , o ve rs ho o t,

a n d

s et tl in g f i n e .

These three eval-

uat ions are based on hu man reasoning, and the basic idea

of assigning these evaluations is very clear and consistent

from Figure 2

-

.

Figure 2 shows how the pa rtial evaluation s h( ')(z l ) re

obta ined . Since the fuzzy rules with positive

e ( k )

a n d



I I 1

where

a = 1 ,

b

= d 2 ,

c =

b l2 , d

=

cE,e

=

d/2

Table 1 Fuzzy control rules and partlal evaluations

Table 2. Rule

table notat.ion

Fig.

2 .

Partial

evaluation

about

c1

Fig.

3 .

Partial Evaluation about 2

r-

Fig.

4 .

Partial evaluation

ahout

x3

Fig. 5. Armature controlled DC motor

A u ( k ) effect on rise t i m e , these are given higher evalua-

tions. T he rules with larger Au(k) effec t more on t he

r i s e

t i m e .

Thu s , h igher eva lua tion are ass igned to t he rules

with larger A u ( k ) to get a fast

rise

t i m e .

We part i t ion

the grade of evaluation by

5

s t e ps . T he g ra de a =

1

is

the highest score and e = (1/2)4 is the lowest score.

Th e evalua t ion h ( i ) ( z 2 ) re chosen t o give the effect

as

in Figure

3.

Th e rules wi th negat ive

Au(k)

have more

effect on the

overshoot .

And higher evaluation is assigned

to the rule with more negat ive A u ( k ) to decrease the

overshoot.

Figure 4 hows how th e evaluations h(*) ( t3 ) re chosen.

As the arrows show, the higher evaluation are given to

the rules with larger IAu(k)I.

I V . COMPUTERIMULATION

Let us cons ider the arm ature control led D C m otor po-

s i t ion control . T he re la t ions for the arm atu re control led

DC motor a re shown schematica l ly in Figure 5 . We can

obta in the t ransfer funct ion G(S) r o m Figure 5 .

@(SI Km

Va(s) - s [ ( R a+ L a s ) ( J m s

Bm)

+ IcbIi m]

2.2

s(8.959 x 10-6s2 + 7.268 x 10-3s +

0.9449)

G ( s ) =

-

-

where the specifications of motor are

La

= 5 .27 m H

R, =

3 . 9 R

l i b = 22.5 Vlli rpm = 0.215

V s e c

t,, = 14 ms e c

A' ,

=

2 . 2 Kgf-cni lA

J,,, = 0.0017 Ilrgf-cni-sec2

B =

J , , / t ,

=

0.121 A gf-cm-sec

.

Th e ac tua l angle of 360 i s normal ized to 1 in this simu-

la t ion. The condi t ions are

as

follows: y,(k) i s se t to 1.0

which corresponds to 360 scale facto r of e ( k ) is 1.0, scale

factor of Ae(k) is 15.0, scale factor of

Au(k)

s 1.0, a nd

s a mpl ing t ime i s

20 msec.

For the scale factor s, we refer

the readers to [16]. Figure

6

shows control results using

the proposed scheme for four cases. T he fuzzy measure

means the degree of consideration

or

importance of the

set of the attrib utes. Here, probability m easure is used

as

a fuzzy measure, which corresponds t o the case with

X = 0 in A-fuzzy measure. And we use Choquet integral

as

a fuzzy integral

as

defined in section

11.

Comp ared to the wel l - tuned fuzzy outp ut response as

in Figure 6(a ) , we can ge t a n out pu t response with a

fast

rase t ime or

in Figure

6(b)

by assigning g((z1)) =

1 ,

g ( ( 2 2 ) ) = 0, a nd g( (23) )

=

0. Similarly, by changing

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Fig.

7

Rise time

~

the degree of importance, we can get an output response

with a decreased

overshoot

as in Figure 6(c)

or

tha t wi th

a fast s e t t l i n g t i m e as in Figure

G(d).

We call conclude

tha t

o u r

scheme can be used in the fine control problem.

To show the effect of the fuzzy measure on the sys-

t e m ou tpu t , Figure 7

-

9 are given. Here, r i se t i m e

is defined

as

the 10-90% rise t i m e . T h e

overshool

is

the percent overshoot, P.O. which is defined as P.O. =

Mp- v ) / f w

x

100% where Mp s the peak value of the

t ime response and f v

is

the final value of the response.

T h e

settl ing t ime is

defined

as

the time required for the

sys tem to se t t le w i thin

2

of the input ampli tude . Fig-

ure 7 shows the r i s e t i m e variation, where g((zl}) and

g((z2)) are continuously changed from 0 to 1 with the

condition g((z1))

+

g((z2))

+

g((z3)) = 1. Similarly,

Figure

8

a nd Figure

9

show the

overshoot

variation and

the

sett l ing time

variation according to the change of the

fuzzy measure. From Figure

7

- 9, we can see that

as

g({zi}) becomes larger, the value of the attribute zi de-

creases. T h e effectiveness of the M FC scheme is clear

from these plots.

PWM

CW

Circuk

-

Chapper

Motor

Firing

WM

c

V . EXPERIMENT

Figure 10 shows a diagram of the control sys tem. T he

specif ica t ions of the motor a re the same

as

those for

Fig 8. Overshoot

Fig. 9 Settling time

computer simulation in Section IV. The pos i t ion of the

DC servomotor is controlled

v i a

a n IBM PC /AT wi th a

12MHz Intel 80286 microprocessor an d a 80287 coproces-

sor. The PC/AT is in terfaced to the chopper c i rcui t and

the motor shaft encoder through a custom card contain-

ing a 82C54 programmable in terva l and t ime(P1T) and

a 82C55 programm able periphera l in terface( PP I) . T he

computed control input

is

appl ied to the motor by ad-

jus t ing th e duty r a t io of PW M s ig nal , which

is

fed to the

PWM chopper circuit.

We use Choquet integral as a fuzzy integral and prob-

ability measure

as

a fuzzy measure . Th e ac tua l angle of

360 is normalized to 1 in th is experiment . The condi-

t ions are as follows:

y r(k ) is

s e t t o

1.0

which corresponds

to

360°,

scale factor of e ( k )

is 1.0,

scale factor of

Ae(k)

is

6.7,

scale factor of A u ( k ) is 0.1, and sampling t ime is

20 nisec.

lniedace

Card Motor

Driver

Counter r

Fig.

10 . D C

servomotor position control system

840

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13

1

1 -

0.8.

”

0.6-

0.4.

,--_

-.-

.

-.-

....... ....a

........-

....

@)

................

b)

(C)

Com pared to th e wel l - tuned fuzzy outp ut response as

in Figure l l (a) ,we can get an outp ut response with a fa s t

r i se t ime as

in Figure l l (b) by ass igning g((z1))

=

1,

g((z2)) =

0 ,

a nd g ( ( z g ) ) =

0.

Similarly, by changing

the degree of impor tance , we can get a n out, put response

with

a

decreased

overshoot

as

n Figure 1

( c )

or

tha t wi th

a fas t se t t l ing t ime as i n F igu re l l (d ) .

Figure 12

-

4 show th e effect of th e fuzzy measure o n

the s ys te mou tpu t . B y a s s ign ingg( (z 1 ) ) = 0.5 , g((z2 )) =

0 .5 , a nd g ( (z 3 ) ) = 0, we can get a output response Fig-

ure 12 (b) which lies between Figure 12(a) and (c). Simi-

larly, Figure 13 and Figure 14 show th e ont,pnt, response

varia t ions according to the fnzzy measnre va lues . From

Figure 12 - 14, we can see tha t the M FC can adjus t

sub-

t ly control results by cha nging the fuzzy meas ure values.

V I .

CONCLUSIONS

In this paper, we have proposed

a

multicriteria fuzzy

controller(MFC) using fuzzy measure and integral the-

ory. In thi s controller, thre e item s of time-dom ain speci-

fications including rise t ime, overshoot , a nd settling tiiue

were considered as the a t t r ibutes of the fuzzy measures

and in tegra ls . Good control performance has been ob-

tained by the M FC since this controller can adjust subtJy

1

control results by changing the weights of fuzzy control

rules with th e fuzzy integra l according to the preassigned

fnzzy measure to th e se t of a t t r ibute s . We have demon-

s t ra ted the performance of

M

FC v ia c ompute r s imu la -

t ions and experiments .

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5th

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pp. 841-845, Iizuka, Japan, July 1992.

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945-948,

1 9 9 ~

[I41

Arth ur Ra me r, .John Hiller,

arid

Colette Patlet,

842

c

EST COPY

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