The Responsibility of a Fuzzy Controller

Copyright © IFAC Control Science and T ech nology (8th Triennial World Congress) Kyoto . Japan. 1981

THE RESPONSIBILITY OF A FUZZY CONTROLLER P. Z. Wang* and S. P. Lou** *Department of Mathematics, Beijing Normal University, Beijing, China **Multi-logic and Fuzzy System Research Section, Shanghai Railway Institute, Shanghai, China

Abstract. In this paper, '.4e will give a mat nematical description for the si mple fuzzy control lers.A definiti on 01 responsibility and some sufficient condi ti ons or it will be given. Keywords. Cybernet ics ; Fuzzy control; con troll er; responsi bility; observers; discre te systems; have

1. Introduction

The area of fuzzy controller has been initiated by E.H.Mamdani and other authors s i nc e 1974 ~ 1-10 ) . Some 6eneral and unified mathematical description of their excellent works is required. In t his paper, we are going to do a few fo r this. Let

see Figure 1 )

be the set of al l fu zz y subsets of~. In abbr evia tion, we d o not distinguish the difference between a fuzzy subset and its membership functicn, so that a fuzzy subset A of V can be d efined immediat e ly by the map

A:

V - [ O . i)

o

~

Ca)

~X cA

is

~ d~_3''''''

_cl"oO

X (b)

cA

is not

A

-normal

Figure 1 For example, in a f uzzy contro ller, the linguistic variables of ERROR ( or ChANGE OF ff.RROR)

a fuzzy subset
t N8 , NIoI,NS,ZE, PS ,

~=

is always designed to be Let

(1.4) ct

_'LL __d.t.(X)_

o

(1.3)

The kernel of

-normal

( 1. 2)

Therefore, we wil J wri t e A(~J instead of the general symbol of membership grade flA'U), When t he universe of discourse X is a subset of the real field,

2.

A

PfII,

rs}

A -normal.

is not empty set, i.e., we call J, the regi on of c/., • Obviously, the c l ass {J, I is uniquely dete rmined by cl a nd it satisfies that

Let

J, (\ Jj

'" .UJj=X-{Td .

be a se t of li no'Uistic variables on X., we call 01. .x -normal if there exist '11\<1 real numbers ( belong or do not bel ong to X ) 1.

=

l=1

(.L . 8) (1 . 9)

The set {1",\ is called a net of cl • Suppose ttat there are many universes

< "(, < ... < '(..

such t r.at for arbitrary fixed \ ll~,~m) we

779

P. Z. Wang and S. P. Lou

780 XL S

and for every t variables on XL

"- set of linguistic

:

\ 1. 11)

we have

x . r;l

X'xX'x·· XX' = lx=(:(',···,:;(.'l! X'EX.', .. ·,:;(.·~X·} {tOlL

J "

'J

al$) J C\~IEa.I..

. .... c(~JEcfs)

(1.12)

which is a fuzzy subset of X havini'; the membership function

=

'lTlin

l

p=

\1

(Xl)) ••. )

ol~ t

X'))

(1.15)

~

where

J:,. X· .. X J.',. .

N~e

( I. 24)

pM e l'Be

NB&

ZE. NS.

NS.

NM. Nil.

NM. NS,

NMe

pe.

NB.

NB. NB.

NM.

Nf~.

NS.

N,s. NS. Z!:. NM. NM. NB.

1I1l.

U.

P~.

PS.

NM. zE. PS.

ZE.

pe.

PB.

PB.

1'15. ZE.

ZE.

PS. PS.

PS,

NB.

PB.

PM.

r5.

ps.

zE. PS.

NB.

N El. NB. foI8. ZE. PB.

PB.

PB.

cI" ... i..

Table 1 (1.16)

and

. U."s J., ... ,,=X-N.

(I. 19)

"1.1 " ',

Unfortunately, the linguistic control rule in some controllers, for example, in the controller of Pap pis ami 11amdani in 1977 ( see [3J ). is not represented by (1.24). The general representation of the linguistic control rule is the map

where 'j' :

N~ (( )(""',0(,') \ (3 \,,,;;, S)«3lL~ JIlL)(XL=T,~»}

\

1.20)

...A~~,

\ 1.25)

S: S'(ci),

(1.26 )

= t Q. ~ ci}

(1.2 T)

where

•

Ji

e

and

1;

J ..

J,.

J ..

J'..

J ••

,

J', ..

J,.

J""

:T_

:T••

··

:To>

1.,

J ..

-:1"

J ..

J Il

J ..

J ..

J••

Ju

J"

Ji,

~,

J.,

J..

'Plot)

For example,

f'

:

·. T'

':

T'

e

-Pt .. )~{ 4 ;

(Ne,Nel}, " ' ,

"

tt Pe , Pel};

(\Pe,NeJ,(Pe,PeJ};

t (N., Ne), (/ole, Pi l,l P., Pe>}, • ", {(Ne, Pel, (Pe, Ne), (P.,Pel}; et}

l. ,

X·= (e,

•

{lNe,Nel,(l1e,Pe.l}, . •

Figure 2 s=

=

rJ..L =

= l'I

( Ne, Pe}. [Ne, Pe}, r;J. ~ cX'Xo(' { (Ne .Ne ), , N€ , Pe), ( Pe, Ne), (Pe, Pel} cI-'

T

Usually,

P.

NMe NSe ZEe PSi

PM e NM.

( I. IT)

And we call J., ... i.. the region of Obviously, we have

is called the net of..

----+-

from d

"

(x>-, x')

d."", "

{i.,,·",i.s):\P (j" ... ,j,) ,

( 1.2.~)

For example, from Table 1, we can find what the map is.

(1.16)

provided

et

'j:

(X',···.x·) E J~, ... ~. ~

d.~, ... ~. ex." .. ·, X');;;;. },>

\ NB", NM., NS., :lE., PS" PM., pe.)

'J

If cJ.',"" ",' are all A -normal, d- I have the net NI = {T'~} «.=1,.. ·....) and the class of regions {J~.} ( i . = I,·", _,) , then we have x=

( 1.2")

A linguistic control rule is a map 9' to (3 :

x.')= cl' (x')k··Ad.'ct!) d ..'1

(~"'''' ~",) l ~; E 'FlY)

For example, ( I. 14)

(X.' ... ) I

=

(1.1;3)

For fixed ( ~" ~""', i..), let

eX·",'" "a .

(1.2 I )

and
~

.. ",'Xc/.·X···XoI.'

=

e. -- CHAN .... OF iRROR 1

(1.10)

(-00, ... . . " ,

ther( is

,

e>l

~-EI'IROft,

...d,=

\Cl."Cl."Cl.3}, Q,={(Ne,Ne l),

0..= (( fIo. , P,,), (Pe,li e ))'

781

The Responsibility of a Fuzzy Controller

0.,=

{(Pe,P e )}

=

IN., ZE., P.}

~

'p=p•• lIJ (fine)

'j:$-~

:l'

P.

(0.,)=

'j (a.,.)=ZE.

'f ( a . ) = N. We c.an ,;et a general definition for a simple fuzzy controller as following. DEFINITION 1.1 A simple fuzzy controller is a triplet C = C(<:A, (?>,
I ( not fine) .ft=P(3)

cr=oI.'x···Xol.' c/-

(oI.:" .. ,ot~,1

( 1,28)

(1=1,···,s),

I

(l.'29)

and for any fixed l , o(l is a set of observed linguistic variables on an universe X l = (x'}

]<'ie;'Ure 3

f;; (-oo,+ao)

and

If Fx. is fine, tnen we can choose a point and 1. is nearest from the mid-point of 1. The point 1. is a determinate response of fuzzy controller for :10 • • DEF'INITION 1.3 A simple fu7.zy controller C = (oL~, 9') is said to be A -responsible i f for any x E X the fuzzy response Px is fine, and •• E Y

( 1.:5 0)

is a set of control linguis tic variables on an universe Y=[1\S(-oO,+oo) , and 'J is the linsuis tic control rule, which is a map (l.;'

Il

where

(1.36)

..i.~ 1'(ci)={~IQ.o;;;;otJ.

In Fuzzy Sets theory, the linguistic control rule is represented by a fuzzy relation

R

E

r( X'x",xX' xY)

f(XXY) =

the membership function of which is that

Given a simple fuzzy controlle PROBLEM 1 is it responsible? In Definition 1.1, if Xl (L=I,"',s) and are all real intervals, then we call C continuous. If

Y

xl={.<~, ... ,x~,l (",;;"1) ( l = I , .. ·,.), (1.37) (1.32)

where & = (X.""',:x:.') and d.""", have been defined in (1.15). For arbitrary observation 0 E J'(X) , the fuzzy response of controller is

f = O·1ft

(I. '3 3)

(E ]'i(Y)) ,

whose membership function is as follows:

'lO·cR.1(1J =

V

ItEX

(O(><)J\cR,(JO, 4))

and (1.37')

then we call C a discrete controller. DEFINITION 1.4 A simple fuzzy controller C'= (cA', ~:
rny = See Figure 3

[

11

fx.(1) =

3~:Y

f'xyn L

(1.58)

A subcontroller of a continuous controller is called the discrete-outcome of C if it is discrete. PROBLEM 2 Given a continuous simple fuzzy controller, has it a >. -responsible discrete-outcome?

c

2. S1J}'FICIENT CONDITIONS FOR HESPOKSIBILITY DEFINITION 2.1 Given a simple fuzzy controller C=(cJ,~,go). the linguistic contrQl rule ~ is said to be consistent if for arbi trary (1" (1, E ft and arbi trary (O(:,;",CI:)E ~ we have

P. Z. Wang and S. P. Lou

782

If f is co nsis tent, then we car. equa te wi th :f, :

(2 . 15)

M (t)

[ ~(Q)

,
...1 ' =

~

If
L.EMU 2.2

~

cp

else

(2.))

p . 16)

a nd

is consistent, then

(2,11)

Note that 'f is c omplet e, so that where (2 . 4)

a nd

~ ~, . \>:lp
(.2 . 18)

Y ,

i.e., it

ar;a

Kerl~~I ' ''~')= (~I (3., ... iJo)=1}~4>. .

.

l

J . 19)

Obviously, tnere is an interval I.s;;; t-"', suc n that L.EMr~

a fixed x.

rule

2.3 Eo

s: ,

If r is co nsistent, then for we have

DEFIlHTION 2 .2 The linguis tic control ~ is said to be complete if

v4=cI ~.A.

ctl,)~ cl

(<<:" "',

i. e. , for any

(2.7)

there is a n

1 () Y =

{, I

=

{'31

=

{1\ P 0)= a'.Y lII ..t P t"n} ...

'00) .

1l!1 "'~'W~ ci~, .. ~,(x.)J j(1)=

d~I'''~s(~'») (220)

:!C,

Le., (1.38 ) is true. I t follows tnot PlO. is fine c.(Hl ( 2.21)

Cl.

such t hat (2.8)

DEFI] ITION 2.3 Given a simple fU 1l.zy controller C=(d, @,'Y) , C is said to be A -norma l if oi', , " , 0/' are all .A -normal sets of lin[.uistic variables and 'iJ is consis ten t and complete. THEOIEM 2.1 Given a A -normal simple fuzzy cOIt roller C= Clot,(3,'l') , C is; A -responsi ble i f there exists a net }{ of d which is disj oint wi th :x , Le.,

Nnx=q,. Proof. Suppose that satisfyirg (2.9), then

(2.9)

N is a net o f

d

X-J{= X

The proof is finished . THEOREJli 2 . 2 Gi ven dn A -norma l CO!ltinuous silOpl e fuz zy controller C = C (Q, (l , "l' ) , there is one of ics discrete outcome wnich is A -responsible. Proof. Let N be a net of d for whi ch 0(', "', ",' are all 11 -norma l. For every fixed l (I ~ ( ~ s) , choose a ' I , d enote d b y (x')', fini te subse t of X L - {r,. (1 . 21)

. ( ,)L and let ~d. ,)Li be the restriction of 0( 1i In X • Uno ose a finite subset of Y aenoted by y' (2 .23 )

(210)

and let Let

From (1.19), we have (2.11)

Therefore, for any fi xed x, ~I, .. " ~s such that

x.

E

J ~I

• ..

~

~3

Because 'i (2. 3) a.nd.

==;.

•

is consisteT)t and complete

J

frOIl

(aI" .. . i,(x.l;\(>" , .:,{I»)(2.13) (aI;'/"loltl)~

M ( Px. ) = M( i) ,

I , , ~ (to('\,,"', «0,,)=

["'(

J

I

,

)"

0(,,,"',0(,, Jy '

BI BLIOGHAPHY

1.

(2.7), we have

From (2.12), it is clear that where

/

(2.12)

~x (3)= !RP:,, ') = V o

~

o/;""il(.>:)~>.>oIi,··· .P)

.

be tile r estriction of ~, in y' . ,

where L... J~ , is the r es t rH.: tion of * in Y: Obviously, the subcontroller C' = C:Co!:t'-~) satisfies the conditions stated in Theorem 2.1. Consequently, C ' is A -respons ible anci. the d iscrete- outcome of C . The proof is finished.

X , there are

From (1.16), we have (;,, " ',") ~ (~I, . . . , ~.)

(3:

(2 . 14)

2.

Marndani, E.H., Applica tions of fuzzy algorithms for control of simple dynamic plant , Proc. lEE, 121, 158 ) -1 586 (1974) Mamdani, E. h ., Assilian, S., A }uzzy Logic Controller for a Dynamic Plant, Int. J. Man-Machine Stud., 7, 1-13 (1975) Pappis, C.P., Marnaani, E.H., A Fuzzy Logic Controller for a Traffic Juncti on , IEEE Trans. Syst. Man & Cybern ., SKC-7, Ho.lO,

The Responsibility of a Fuzzy Controllet

4. 5.

6.

707-717 (1977) Rutherford, D.A.~ Bloore! C.G., The Implementa~ion 01 ruzzy algori~hms fo r COl.trol, IEEE,64, 572-573 (1976) Kikert, ~/ .J .~. • , Van Naufa Lemker, H.R., Appl. of FUzzy Sets T~ eory to Control a Warm V,ater Plant, Automatica, 12, 301-

3GB (1976)

Tong, R.~:., Analysis of FUzzy Control Algo rit hms using the Relation Matrix, Int. J. Man-Machine Studies, 8, FUZ,CON (1976) 7. King, P.J., Marndani, E.H., The Appl. of FUzzy Control Systems to Industrial Process, J.A., 17 (1976) '3 . Li Bao -shau, Liu Zhi-jun, The Appl. of FUzzy Sets Theory to the Design of a Class Controller, Acta Automat i ca Sinioa, 6 No.l, 25- 32 (1980) 9. Zheng Wei-min, Ren Shou-chu, Wu Chiu-feng • Tsuei Tsehsing, Intersection of FUzzy Subsets and the Robustness of FUzzy Control (going to publish) 10. Long Sheng-zho, He Kai-yuan, Zhaa Xu, Zhao Yun-long, Determination of Man-Machine Systems, (going to publish).

783

Discussion to Paper 21 . 4 M.M. Gupta (Canada): The authors are to be congratulated for their fine work that they are doing in China. I would lik e to hear from Dr. Hang about the applications of some of this work . P.Z. \'iang (China): The main application of this paper is in constructing a fuzzy controller by means of a parameter method which wi ll be presented in S.Z. Long and P.Z. Wa ng , " sel f-t urning on the f uzzy control rule". E. Sanchez (France): Are your conditions of v-responsibility easily satisfied in practice, and are linguistic variables usually normal? P.Z. l'iang (China): Th e condition necessary f or responsiveness is very weak , so that it i s easily satisfied in practice. The linguistic variables can be made in the normal form in practice. In the paper by Long ShengZho et al., the set of linguistic values ob tainea-from experiment are all normal .