Model Reference Fuzzy Adaptive Control

COj>H igIH © I F:\C ~lth T riellnial \\"011<1 Congrt·.; ~ Budapt'\ I . 1I ung;ll"\. I q~q

MODEL REFERENCE FUZZY ADAPTIVE CONTROL We i-min Cheng. Huang Shen- yue and Ho Hua-ko n

Abstract. Based on the experience of the operators i n tuning the parameters of the widely used PI controller, a PI model reference fuzzy adaptive control system (MRFAC) is studied . In case the controlled plant is fuzzy , MRFAC is a simple and practi cal method. For the designing of the controller , controlling matrix and connecting matrices are defined. Using the nossibility measures , the fuzzy control can be represented explicitly. Keywords . Fuzzy control; adaptive control; control systems .

INrRODucrION

FUZZY CONTROL AND CONVENTIONAL

PO CONTROL PID controllers (proportional plus integral and deri vati ve control) are wi dely used in process control SYstems . At the desiqninq staqe, the parameters of the PlO controller are determined by enpi r i cal formulas or by the methods mentioned in the control theory . But usually the dynamic response characteri stics of the process under control i s only known approximately or fuzzy , therefore the tuning of the PID parameters by the operating engi neer is always needed when a control system i s put into operati on . In addition, the dynamic characteristics of the process under control may vary s l owly from time to time , thus re- tuning of the parameters is desi rable.

In the fuzzy control, a set of control statements , such as "If x is NL and x ' is NL , then C is NL" or else . etc ., can be represented by a table. As an example , an illustrating fuzzy control table is shown in Fig . L , where L,M, S represent large , medium , small and P , N represent positive, negati ve , respectively. The error x, error derivati ve x ' , and the control statement C in the table are all lingui stic vari abl es , expressed as PL , PM , PS , 0, NS , NM, or NL . In the case of the conventi onal PO controller , for example C=2x ' +x , the discrete control table is shown in Fig . 2 . . Comparing the fuzzy control table \, ith the conventional PO discrete control table, it can be seen that the fuzzy controller is essentially a PO controller , but it is coarse and highly non-liner .

To simulate the experience of the operating engineer in adjusting the PID parameters , we can onl y use linguisti c or fuzzy control rul es . Based on this idea , a PI model reference fuzzy adaptive control system has been studied. The principle and the canputation of fuzzy control are given in sect i on 2 to 5 usi ng the possibility measure theory and f uzzy logic operati ons. The simulati on study i n secti on 6 shows the applicabi l i ty of such a MRFAC system .

CONI'ROLLING MATRIX AND

CONNEX::TING MATRICES The fuzzy control table can be expressed as a (pxr) matrix which is called controlling matrix CM .

x'

B

n

NL

N'-'

NS

0

PS

PM

PL

~

-3

-2

-1

0

1

2

3

PL

PS

PM

PM

PL

PL

PL

PL

3

-3

-1

1

3

5

7

9

PM

0 I PS

P'1

PL i PL

2

·4

-2

0

2

4

6

8

1

-5

-3

-1

1

3

5

7

0

-6

-4 -2

0

2

4

6

-1

-7

-5

-3

-1

1

3

5

-2

-8

-6

-4 -2

0

2

4

-3

-9

-7

-5

-1

1

3

PS

PS , PM

i PS

PM

NS : o

i PS

PS

PS ' PM

N."1 I NS , 0 I PS I

1

1

PM

,

X

' PH

0

Nl1 ! N1-'1

NS

NL

l NM

NS

NS

!I 0

N'~

NL NL

N"~

N'1

i NM i NS I 0

NL

'lL

~

NL

NL

NM

I

I

NS

l

0

Fig . L

-3

Fig. 2 .

889

890

Hei - min Cheng, Huang She n- yue a nd Ho Hua-kon

CM=

wher e C

C ll

C - - Clr lk

C il

C - - C, lr ik

CpI -

C - - C pk pr

ik

given Aj and B , is naturally defined in the l follCMing: (1)

where t\ represents min operation.

, i=l, ... p , k=l , ... r, is a control state-

ment corresponding to inputs Ai and

~.

(2)

where n is the number of different control statements and ' + ' means union. ~j ' corresponding to Cj , i s a (p , r) matrix only with entries ' 1 ' and ' 0 '. For example , =

CCMPUTATION OF THE OUTPUT OF FUZZY CONTROLLER

Actually ,

the number of different control statements i s less than the nurrber of elements in the controlling matrix. Probably , there are all together only seven different statements , such as :m , 'l", :;S, '), :'S , PM, PL, which can be assigned as Cl " " ,C . Then 7 the controllin<:r matrix can be decaT1!X>sed as fol lCMs:

CM

(8)

Pjl(i ,k) = P(Ai)/\ P(~)l

Fran the control table , \"e kncJ.V that if the fuzz ified inputs are Ai and ~, then the control statement is C . ,')hen the fuzzified inputs are Aj ik and B , we can calculate the composite possibility l measure P jl (i , k) of Ai and ~ given Aj and Bl fran~ . 8 . Thus it is reasonable to define the component output C as fol l CMs : jl C (i,k) = P (i , k) t\ C jl ik jl

As a consequence , the total output , which i s the union of the component outputs , equals Cjl =i~k( Pjl(i , k) t\ Cik )

[ ~~ ~~]=[~ ~]Cl +[~ ~lC2= ~t{1+M2C2

=i ~k ( P (Ai) j " P (~)

Matrices Ml .. . M , are called connecting matrices n in which the entry ' 1 ' means that there is a connection to a certain statement and the entry ' 0 ' means no connection . Obvious l y , the union of connecting matrices is an n by n matrix E that has every entry equal to 1 , i.e. , (3)

+ M = E n

(9)

1'" Cik )

(10)

In this expression , P (i , k) may be interpreted jl as a wei ghting factor or importance factor attached to the control statement Cik . C

can also be expressed in matrix operations by jl ITBking use of the possibility measure vectors and controlling matrix or connecting matri ces , i.e., (11)

POSSIBILITY ~tEASURES OF WPlITS OF FUZZY CONTROLLER

As proposed by Zadeh , if Ai i s a fuzzy subset cor-

C = pT (A) j jl

P(Ai)j= Poss( x is Ail x is Aj Sup ( ""'A{ U) t\ jJ. Aj (U)

)

(4)

whi ch may be interpreted as the possibility that x is Ai given that x is Aj . Simi l arly, if ~ is a fuzzy subset corresponding to x', then the possibility measure of ~ , given that x' i s B , is l P(~)l=

Poss( x' is

~I

Sup ( }) 8. (u) t\ P. 8~ (u) )

(5)

The two i nput s of the fuzzy controller are fuzzy subsets Aj and Bl correspondi ng to x and x ', res-

1p (B) 1

(12)

\')here all matheITBtical operations should be understood as matrix fuzzy l ogic operations , i.e. , mul tiplication means min operation and '+' means un i on . The result of Eq . 12 is same as the result of l1ax- min composite rule , but Eq .12 is mere explicit and inforITBtive. According to Eq . 12, a fuzzy controller can be scheITBtically shown as Fig. 3. , in whi ch the decision block ' d ' will transform C into a deterministic control action , jl 1.e. ,

U jl

x ' is Bl

P(B) 1

pT(A)j [~lCl+" '+ MnCn

responding to x i n a universe of discourse U, then the possibi lity measure of Ai' given that x i s Aj ' is defined as follCMs:

[CM]

= f(

Cjl

(l3)

where f(·) depends on the dec i sion rule . PI MJDEL REFERENCE FUZZY ADAPTIVE CONTROL SYSTEM

pectively. Defi ne two rCM vectors P(A ) "P(A ) ', ... , PtA ) ,) l J 2 J p J

(6)

(7)

Evidently, they are the possibility measure vectors of the two inputs of the fuzzy controller. The composite possibility measure of Ai and ~,

It is well known that t.l'Je ?roportional control is used in the feedback control system to assure the required dynamic performence , while the integral control is used to assure the static performence wi thout ~iring the dynamic response. Such kind of proportional pl us i ntegral control (PI controller) has been applied successfully and wi del y in the f ield of i ndustri a l process =ntrol.

In practice , when a =ntrol system with PI control

Model Refe r ence Fuzzy Adaptive Contr ol

x

x'

89 1

A.

Fuzzifier

t---'--.., Possibili t y

u

01

11easure

Fig. 3.

Fig. 4.

x is put into operation or during operation , tuning or re- ad justing the PI parameters is usually necessary. This is because the dynamic characteristics of the process i s only knavn a?proximately, or fuzzy or varying . Combining the advantages of the PI controller in the process control and the fuzzy control in simulating the tuning experience of the operator, a PI model reference fuzzy adaptive control syster,) is f ormed as shown in Fig. 4 . • In the control system as shown, F , Fa2 , F , Fb2 al bl are fuzzifiers which classify e and e ' i nto linguistic values . FC and FC 2 ar e two fuzzy controllers . l P1 and P2 are pr oportional constants \vhich are ad-

X (\vith adaptive control ) 100 80 60 adapti ve contro] ·'

40 20

~--~2~0--~ 47 0 ---6~0~~8~ 0--~1~0~0~- t

justed to rreet the ranges of change of kl and k2 for a specific system. As in the conventional nI control system, in order to pr event 'vindup , the integral control action is switched in only near the end of stars- up. Such a non- l inear ~vitch SI, is also provided. Fig . 5 . shavs two results of simulation studies of the system in which the determination of the rules of the adaptive loop is based on the tuning rules of some experts and designed according to Eg.12. The transfer function of the plant ~ under control i n Fig . 5. is in the form - Ts ke / Tls+l , where T =3.6-- - 36 and T=0---2. l'/hen l the mathematical model of the plant is fuzzy , consequently the controller can not be designed by analytical rrethod . In this case, the results of the simulation studi es sho.l that the fuzzy adaptive control based on the exnerience of exoerts can be a sirrq"lle and practica·l rrethod. .

Fig. 5 . CONCWSION Fuzzy control is based on the experience and/or knowledge of experts and it can not be designed analytically . Therefore, it is most suitable for the control in the higher level, e.g ., in the adaotive control loop . Taking the advantage of PI control as well as the advantage of fuzzy control , a natural consequence is PI model refer ence fuzzy adaptive control . The result of simulation study is satisfactory . By rreans of the controlling matrix and the possibility rreasure vectors , fuzzy control can be expressed in a more explicit and informati ve way .

892

Wei-min Cheng , Huang Shan-yue and Ho Hua-kon

Mamdani, E. H. (1974) . Application of fuzzy algorithm for control of sirn?le dynamic rlant. Proc . IEEE, 121 , 2 , 1585- 1588 . loJei- min Cheng , Shou-ju-Ren, Chiu- feng Wu, Tse- hsing Tsuei . (1982) . An expression for fuzzy controller. In ~l . M. Gupta and E . Sanchez (Eel. . ) , Fuzzy Information and Decision Processes, North- Holland Publishing Company , Holland. rp. 411-413. Zadeh, L. A. (1981) . Possibility theory and soft data analysis. In L. Cobb and R. M. Thrall (Eel..) , Mathematical Frontiers of the Social and Policy Science, \'iestview Press, Boulder , Colorado. pp. 69-129.