Approach to an adaptive method of fuzzy control systems

ZESZYTY NA UKOWE POLITECHNIKI ŚLĄSKIEJ Seria: A U T O M A T Y K A z. 61

________ 1981 Nr kol. 701

Ernest CZ OGALA W i t o l d PE DRYCZ

AP PR OA CH TO AN ADAP TI VE ME TH OD OF FUZZY CONTROL SYSTEMS

S u m m a r y . From the point of vi ew of the control theory a great amount of papere on fuzzy controllers have been concerned with one- -level structure of the controller.

In the paper a two-level structure of fuzzy controller is being pr oposed and discussed, where, the first one consists of a fuzzy lo­

gic controller and the second one has two components: an Identifi­

cation Block and Fuzzy Optimizer, which makes it possible to tune the controller in a formal way.

1. Introduct ion

In many work s on fuzzy control systems in the application area, a fuz­

zy logic controller containing a set of control rules of a complex, ill- -defined process, forms a precise, formal description of the actions of a human operator, [l] , [2], [3]. From the point of v i e w of the control theory a great amount of papers have been concerned with one-level struc­

ture of the controller (fig. l). Because of the ease of computation, fuz­

zy control algorithms are implemented mainly on real time minicomputers.

Fuzzy control rules, their structure, the definition of basic sets of er­

ror, change of error, control etc. are defined at the design 3tage of controller. A t t e mp ts have been made to improve such kind of algorithms, modifying properly the control rules discussing their influence on the dynamical characteristics (e.g. unit step response, rise time, damping factor of the process) [lj. Se l f - o r g a n i z i n g , adaptive algorithms have been discussed, too [2 3 , [3], The w a y of obtaining the final proper conditions of a fuzzy controller is in its main idea rather heuristic.

In this paper a two-level structure of fuzzy controller is being pro­

posed. The first one consists of a fuzzy logic controller, the second one which consists of a Fuzzy Optimizer and an Identification B l o c k , makes it possible to tune the controller in a formal way. Thus the second level enables us to analyse the properties of the controlled process, as well.

A similar two-level structure of fuzzy controller, i.e. a self-organi­

zing controller has been described by Procyk and Mamdari [33. where the second level consists of two blocks: a performance measure and a i n c r e m e n ­ tal model, which according to assumption, is more precise than a fuzzy

80 E. Czogaia, W. Pedrycz

Fig. 1 . One-level structure of a fuzzy logic controller

Fig. 2. Two-level structure of a fuzzy logic controller

Fie. 3. General structure of e self -o rg an iz in g controller

logic controller. When the concrete values of error and change of error are given, the incremental va lu e r k+1 of control is computed a.s follows

where U k+1 denotes the valu e of control computed by means of maxmin o peration of input fuzzy sets and the controller s matrix. Fuzzy sets F-k , C k ' U k+1 ’ U k + 1 sre c r e a t e d by means of fuzzification [6j :

A p p r o a c h to an a d a p t i v e me t h o d . 81

E k - F ( V c k = F ( c k >

= Ffu, „

k+1 k+1

k+1 k + 1

The control rules of the controller in the form:

- if error is equal to E^

else

if change of error is equal to , then

control is equal to U. . ,

should be replaced by the following implication:A

- if error is equal to E^

else

if change of error is equal to then

control is equal to + j

Thus the replacement might be expressed as follows:

The matrix R |< + 1 of controller should contain the rule but not the rule R'k , i.e.

R. . ( S . A R ! ) V R"

k+1 k k k (3)

The method is an effective one, under the assumption that the incre­

mental model is an adequate one, and in the case, when the input v a r i a b ­ les i.e. error and change of error are measurable, so that a hierarchical' model can be co ns tructed in an analytical form.

Vie present the method of fuzzy process identification and the al go­

rithm of generation of control rules with respect .to the performance in­

dex. For further discussion we assume that all the spaces in which fuzzy sets of state or error, and control are defined, have a finite number of elements i.e.:

c a r d( X ) =n

c a r d ( U ) =m (4)

where ^ stands for state space and denotes the control space.

82 E. Czogala, W. Pedrycz

2. Identification of a fuzzy system

We take Into account a relational de3criptio n of the process:

X k+ 1 “ Xk 0 U k ° R (5)

where , X k+1 are fuzzy sets of the space of process in discrete time moments, is a fuzzy control and R denotes a fuzzy relation des­

cribing the connections in the process

x k ,xk+i - u k e i ( m ). R e ? (ir * * * u )

where ? ( • ) denotes a family of fuzzy sets defined in the spaces X andt) respectively [4 ] . 0 stands for maxmin operation. Eq (5 ) could be re wr it­

ten in the following equivalent form:

m r n -1

Z *!5 ^{■ V [ V (^} 1 , (yJ )A^R(xi >yj ' Ul ))| (6)

K+1 1-1 j-1 K

V = max x i 'Yj É ^ 'x 2 ••••>x_ j A = nin u^e J) -juj .u2 .. >um |

Our task is to estimate the unknown process relation, where a collection of "measurements" is given. Before describing the procoss idenfification let us introduce some useful definitions.

Def. 1 Fuzzy set X € ? ( ¡?) is called an l-normal fuzzy set if ¿1^ (x.^ )=1.

Def. 2 i-normal and J-normal fuzzy sets X,Y e ? ( ) are called in de pe n­

dent if 1 / J .

Generally fuzzy sets A 1 ,A2 ,.,.,An are called mu tually independent if I j ^ i g / i j / . . . ~ i n wh e r e max ¿1^ (x, ) = (x± )

x ^ X 1 1 3

Def. 3 Degree of fuzzines3 of fuzzy set X is a no nn egative real number

* v .

n

max ' 7 '>

X j t X 1=1

Def. 4 An extension of fuzzy relation G induced by the fuzzy singleton jujJ we call a fuzzy relation Ext G e ? ( ) f >Hj) defined as fol­

lows :

A'p o r o a c h t o a d a p t i v e m e t h o d . . ■ 83

^ x t C f y 'V -u) ¿xG (x.y) . u ^ 0, otherwise

We di scuss and identification method, baaing on an active experiment, so the following oq.

Y. - X. o U. o R k k l

holds. For every degenerate control .U , whero (u 1 - <J

1 i u j J l'U 1

lection of fuzzy input and output sets is given: J

N) hi

(8)

a col- give

j « 1.2,

where jXjjJ forms a family of mutu al ly independent normal fuzzy sets. As a the result of identification we take a fuzzy relation R computed a 9 :

m n

u Ext[n xji@yji]

i»i 3-1

( 9 )

nhare @ stands for of-operator defined as [5] :

T = x j i©^-) i 6 ^ ( X *

def

^ T (x r ^ s 5 =

1, if u x ( x _ ) « a Y (y ) 3 i • Y ji s

U y ( y s). if (x ) > ^ Y ( y g )

. ji Ji r Ji

fio)

An active experiment as defined above enables us to identify accurately an un known fuzzy relation. Otherwise if identified fuzzy sets are not chosen as described above, having a higher va lu e of degree of fuzziness, the pe rf or ma nc e index of identification expressed as the Ha mming distance

betwen R and /? has a non-zero value.

3. G e ne ra ti on of the rules of fuzzy logic controller

The task of genera ti ng the rules of a fuzzy logic controller might to stated as follows. The local perf or ma nc e index.of the control

0 ( U k , X k ) = p H (Xo p t , X k + 1 ) +CY pH (Uo p t , (1.1)

64 E. Czogała. W. Pe drycz

and a set of fuzzy sets X 1 ,X2 ,... .X*'1 (specified e.g. by the human op e­

rator of a contro ll ed process) are given.

Th e proc ed ur e of generating a fuzzy control U 1 for each X 1 ,X2 ...,XN deals with the co mp ut at io n of the membership function in the following way For each X 1 ,i ° l , 2 , . . . ,N we compute a valu e of the perf or ma nc e index Q i x 1 , U j ) for a control equal to:

,;: - N i.e. Q(x^ ,Uj )

Thus the fuzzy control U* has a membership function:

Q C x 1 ,u ) . i ( u i > ■

1

( 1 2 )

Repe at in g this pr oc ed ur e for each X'1', we generate a collection of ru­

les of a fuzzy controller:

X 1 ^ U 1

X 2 ^ U 2 (13)

xN - u N

In the case v»hen u,. (u) «cT, and pi H (u) » <J , the Hamming dl- r U opt opt * opt * opt u *u i

stance in eq. 11 is equal to:

P H < V U opt) . j,.U.1.".U °P.tj. (14) max u

Uj6

The matrix of a fuzzy controller 13 formed as a logical sum of each of the rules i.e.

N

G =

(J

i=l

with the me mb er sh ip function:

fl6)

The method d e sc ri be d above may be generalized, if some different co m p e t i ­ tive criteria are discussed, i.e.:

II A III

Q (17)

with a slight modification, that G is cr eated in a different way:

(18) p-1 1-1

where X i p , U ip stand for i-th fuzzy set of state a n d control respecti- v e l y di sc us se d from the point of v i e w of p-th criterion.

4 . Conclusions

A two-level structure of a fuzzy controller permits to identify an u n ­ known process snd to generate the control rules an alytically, minimizing the pe rf or ma nc e index. The fuzzy re lation ob tained by the Id en tification p rocedure might be useful for the solution of another problem as well, i.e. the pr ed ic ti on of the stat e of the process. The co ntrol rules ge ne­

rated a c co rd in g to the m e th od described in the paper, can form the matrix of a fuzzy controller and be compared with those given by a human op er a­

tor making it po ssible to try to estimate (at least ap pr oximately) the perf or ma nc e index (i.e. its general feature) using a human beinig in the control procedure. The structure of the controller defines also the struc­

ture of the computer system and its working mode, so the opti mi za ti on and identification may be w o rk ed out of f-line and the control algo ri th m can be im pl em en te d on on -l in e minicomputer.

RE FE RE NC ES

[1] Brape M . , R u rh er fo rd D.A. : Selection of pa ra me te rs for a fuzzy logic controller. Fuzzy Sets and Systems 2, 1979.

[2] Ma md an i E.H. : A p p l ic at io n of fuzzy logic to a p p r ox im at e reasoning using linguistic synthesis, IEEE Trans on Computers, 12, 1977.

[ 3 ] Procyk T.3. , M a md an i E.H.: A linguistic s e lf -o rg an iz in g process con­

troller, A u to ma ti cs 1, 1979.

[4 ] Ne go it a C . V . , Ra lescu D . A . : Ap pl ic a t i o n s of fuzzy sets to systems analysis, Birkhauser, Basel, 1975.

[5j Sanchez E . : R e so lu ti on of composite relation equations. Information and Co ntrol 1, 1976.

86 E. C z o g a ł a . A'. F e d r y c z

¡6] Zadeh L . A . : The concept of u linguistic variable and its applications to approximate reasoning, A m e r ic an Elsevier Publishing Comp. ,New York 1973.

Złożono w redakcji 4.12.79 r. Recenzent

W formie ostatecznej 30 .05.80 r. Ooc. dr hab, inż. Andrzej Ty li ko ws ki

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