fuzzy self tuning of pid controllers

7/25/2019 Fuzzy Self Tuning of PID Controllers

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F u z z y S e t s a n d S y s t e m s 5 6 ( 1 9 9 3 ) 3 7 - 4 6 3 7

N o r t h - H o l l a n d

Fuzzy se l f tun ing o f PID contro l l ers

S h i - Z h o n g H e 1 S h a o h u a T a n

F e n g - L a n X u :

Department o f Electrical Engineering, National University o f

Singapore, 10 Kent Ridge Crescent, Singapore 0511

P e i -Z h u a n g W a n g 3

Institute of System Science, National University of Singapore,

Heng Mu i Keng Terrace, Kent Ridge, Singapore 0511

R e c e i v e d A u g u s t 1 99 2

R e v i s e d O c t o b e r 1 9 9 2

Abstract. T h i s p a p e r p r e s e n t s a n o v e l f u z z y s e l f - t u n i n g P I D

c o n t r o l s c h e m e f o r r e g u l a t i n g i n d u s t r i a l p r o c e s s e s . T h e

e s s e n t i a l i d e a o f t h e s c h e m e i s t o p a r a m e t e r i z e a

Z i e g l e r - N i c h o l s - l i k e t u n i n g f o r m u l a b y a s i n gl e p a r a m e t e r a ,

t h e n t o u s e a n o n - l i n e f uz z y i n fe r e n c e m e c h a n i s m t o s e l f -t u n e

t h e p a r a m e t e r . T h e f u z z y t u n i n g m e c h a n i s m , w i t h p r o c e s s

o u t p u t e r r o r a n d e r r o r r a t e a s i ts i n p u t s , a d j u s t s c~ i n s u c h a

w a y t h a t i t s p e e d s u p t h e c o n v e r g e n c e o f t h e p r o c e s s o u t p u t

t o a s e t - p o i n t Y r, a n d s l o w s d o w n t h e d i v e r g e n c e t r e n d o f t h e

o u t p u t f r o m Y r. A c o m p a r a t i v e s i m u l a t i o n s t u d y o n v a r i o u s

p r o c e s s e s , i n c l u d i n g a s e c o n d - o r d e r p r o c e s s , p r o c e s s e s w i t h

l o n g d e a d - t i m e a n d n o n - m i n i m u m p h a s e p r o c e s s e s , s h o w s

t h a t t h e p e r f o r m a n c e o f t h e n e w s c h e m e i m p r o v e s

c o n s i d e r a b l y , i n t e r m s o f s e t - p o i n t a n d l o a d d i s t u r b a n c e

r e s p o n s e s , o v e r t h e P I D c o n t r o l l e rs w e l l - t u n e d u s in g b o t h t h e

c l a s s i c a l Z i e g l e r - N i c h o l s f o r m u l a a n d t h e m o r e r e c e n t

R e f i n e d Z i e g l e r - N i c h o l s f o r m u l a .

Keywords:

F u z z y s e l f - t u n i n g ; f u z z y c o n t r o l ; a d a p t i v e c o n t r o l .

1 I n t r o d u c t i o n

Despite the advent of many sophisticated

control theories and techniques, the majority of

industrial processes nowadays are still regulated

by PID controllers. This status quo not just

indicates the cautious attitude of the practicing

Correspondence to: S h a o h u a T a n , D e p a r t m e n t o f E l e c t r i -

c a l E n g i n e e r i n g , N a t i o n a l U n i v e r s i t y o f S i n g a p o r e , 1 0 K e n t

R i d g e C r e s c e n t , S i n g a p o r e 0 5 1 1 .

1 O n l e av e f r o m D e p a r t m e n t o f A u t o m a t i o n , T s i n g h u a

U n i v e r s i t y , B e i j i n g 1 0 00 8 4, C h i n a .

2 O n l e a v e f r o m D e p a r t m e n t o f E l e c t r ic a l E n g i n e e r i n g ,

T s i n g h u a U n i v e r s i t y , B e i j i n g 1 0 0 0 8 4 , C h i n a .

3 O n l e a v e f r o m D e p a r t m e n t o f M a t h e m a t i c s , B e i j i n g

N o r m a l U n i v e r s i t y , B e i j i n g 1 0 0 0 8 8 , C h i n a .

0 1 6 5 - 0 1 1 4 / 9 3 / 0 6 . 0 0 © 1 9 9 3 - - E l s e v i e r S c i e n c e P u b l i s h e r s B . V . A l l

world towards the new invention, it does reveal

the rich potential of this extremely simple

almost primitive, perhaps, in the eyes of some

control theorists) control strategy for meeting

various specifications for a vast variety of

industrial processes.

A crucial issue in the PID control is the setting

of the controller parameters, the so-called tuning

problem. The conventional way to do the tuning

is to study the mathematical models of processes,

and try to come up with a simple tuning law that

will establish a set of constant PID parameters

based on the models.

It is not hard to show theoretically that the

PID is adequate for the processes modelled

perfectly by linear first or second order systems.

Tuning laws can easily be established in these

cases. Unfortunately, real industrial processes

can never be modelled perfectly as simply as the

linear first and the second order systems. They

may have such marked characteristics as

high-order, dead-time, nonlinearity, etc., and

may be affected by noise, load disturbance and

other ambient conditions that cause parameter

variation and sudden model structural change.

The existing theories can no longer provide

systematic and robust tuning laws for these

complex situations. Thus, many of the PID

tuning laws actually incorporate empirical

evidences to compensate for the model com-

plexity and variation. As can be expected, these

tuning laws are often ad hoc in nature, and may

only be useful for a certain class of processes or

under certain conditions. A typical example of

the model-based tuning laws is the famous

Ziegler-Nichols tuning formula [15]. Apart from

the fact that it may completely fail to tune the

processes with, for example, relatively large

dead-time, its tuning will have to be supple-

mente d with purely experience-based fine-tuning

to meet the response requirements.

Along the line of empirical investigation and

approximate analysis, the work on improving the

PID tuning has been going on especially in the

past decade. There has been attempt to revise

r i g h t s r e s e r v e d



8

Shi Zh ong He e t a l. / Fu zzy se l f tun ing o f PID con tro lle rs

t h e h a l f -c e n t u r y -o l d Z i e g l e r - N i c h o l s f o r m u l a t o

e n h a n c e i t s p e r f o r m a n c e a n d a p p l i c a b i l i t y ,

r e s u l t i n g i n t h e s o - c a l l e d R e f i n e d Z i e g l e r -

N i c h o l s t u n i n g f o r m u l a [ 6 ] . T h e s e r e f i n e m e n t s ,

u s e f u l a s t h e y m a y b e i n i m p r o v i n g c e r t a i n

a s p e c t s o f t h e r e s p o n s e s f o r c e r t a i n p r o c e s s e s,

m a y p e r f o r m w o r s e f o r c e r t a i n o t h e r p r o c e s s e s .

I n a se n s e , t h e y h it u p o n t h e d e l i c a t e b o u n d a r y

o f p e r f o r m a n c e - p r o c e s s t r a d e o ff , a n d t h e c o m -

p l e x i t y m a y n e v e r a l l o w a c l e a r c u t . T a k i n g , a s

a n e x a m p l e , t h e p r o c e s s e s w i t h l o n g d e a d - t i m e , i f

t h e y a r e c o n t r o l l e d b y t h e P I D p l u s t h e

Z i e g l e r - N i c h o l s t u n i n g , t h e f i r s t o v e r s h o o t i n t h e

s e t - p o i n t r e s p o n s e w i l l b e e x c e s s i v e l y h i g h , w h i c h

i s c o n s i d e r e d u n a c c e p t a b l e f o r m a n y a p p li c a -

t io n s . T h e R e f i n e d Z i e g l e r - N i c h o l s f o r m u l a c a n

b e e m p l o y e d i n t h i s c a s e t o r e d u c e t h e

o v e r s h o o t . I n d o i n g s o , h o w e v e r , t h e r e s p o n s e

t i m e w i l l b e s l o w e d d o w n , s o m e t i m e s c o n -

s i d e r a b ly . F u r t h e r , i n t h e c a s e o f m i l d n o n -

l i n e a r i t i e s , i t i s h a r d t o t e l l f o r a p a r t i c u l a r

p r o c e s s i f t h e o r i g i n a l f o r m u l a o r i t s r e f i n e m e n t

s h o u l d b e u s e d . A l l t h e s e m a y b e c o n t r i b u t e d t o

t h e s i n g le fa c t o f r e q u i r i n g t h e P I D p a r a m e t e r s

t o b e f i x e d t h r o u g h o u t t h e c o n t r o l . I n o t h e r

w o r d s , w i t h f ix e d p a r a m e t e r s , t h e c o n t r o l l e r s o f a

s i m p l e s t r u c t u r e , s u c h a s t h e P I D , c a n n o t g o

b e y o n d a c e r t a i n l i m i t i n h a n d l i n g t h e m o d e l

c o m p l e x i t y a n d u n c e r t a i n t y . I f w e s ti ll i ns i st o n

u s i n g t h e s a m e c o n t r o l l e r s t r u c t u r e , t h e c o n -

t r o l l e r p a r a m e t e r a d a p t a t i o n w i t h t i m e s e e m s t o

b e t h e o n l y w a y to e x t e n d b e y o n d t h e l i m i t.

C a r r y i n g o n w i t h t h e t h o u g h t , a n a t u r a l s t e p

a h e a d i s t o c o n s i d e r

self tuning

P I D c o n t r o l ,

w h i c h t u n e s t h e P I D p a r a m e t e r s o n - l i n e t o a d j u s t

t h e c o n t r o l l e r a c t i o n s f o r m e e t i n g t h e r e a l - t i m e

n e e d . T h i s i s p r e c i s e ly t h e d i r e c t i o n p u r s u e d b y a

n u m b e r o f r e s e a r c h e r s [ 3, 10 , 1 ]. T h e i d e a o f t h e

e x i s t i n g P I D s e l f - t u n i n g s c h e m e s e s s e n t i a l l y

f o l lo w s t h a t o f t h e c o n v e n t i o n a l s e l f - t u n i n g

c o n t r o l l e r s , i . e . , t h e t u n i n g a t a n y t i m e i n s t a n c e i s

b a s e d o n a s t r u c t u r a ll y - f ix e d p a r a m e t e r - e v o l v i n g

p r o c e s s m o d e l p r o d u c e d b y a n o n - l i n e i d e n t i f i c a -

t i o n p r o c e d u r e . T h e m o m e n t a r y t u n i n g i t s el f w i ll

s t i l l h a v e t o b e d o n e b y u s i n g s o m e d e s i g n

f o r m u l a , o r ju s t t h e Z i e g l e r - N i c h o l s f o r m u l a .

T h u s , t h e s e s c h e m e s c a n b e s e e n a s tr y i n g to d e a l

w i t h t h e m o d e l c o m p l e x i t y a n d u n c e r t a i n t y

p r o b l e m b y lo c a l iz i n g o n t h e t i m e s c al e ) t h e

c o n v e n t i o n a l t u n i n g m e t h o d s . A s i d e f r o m t h e

o f t e n - c it e d p r o b l e m o f h ig h c o m p u t a t i o n a l

d e m a n d , a m a j o r d i f f ic i e n cy f o r t h e s c h e m e s

s e e m s t o b e t h a t i t d o e s n o t c h a n g e t h e

m o d e l - b a s e d n a t u r e o f n o n - a d a p ti v e P I D c o n -

t r o l l e r s . B y a s s u m i n g a m o d e l , t h e c o n s e q u e n t

r o b u s t n e s s i s s u e s t il l n e e d s t o b e s e t t l e d e v e n a t

l o c a l i z e d t i m e i n s t a n c e s ) , w h i c h p r o v e s t o b e

d i f fi c u lt . F u r t h e r , b e c a u s e a n a p r i o r i a s s u m e d

m o d e l w i l l h a v e t o b e n e c e s s a r i l y s i m p l e , i t o f t e n

c a n n o t a c c o m m o d a t e t h e s t r u c t u r a l d i s t u r b a n c e ,

s u c h a s l o a d d i s t u r b a n c e o n p r o c e s s e s . I f s u c h a

d i s t u r b a n c e h a p p e n s , t h e i d e n t i f i e d m o d e l w i ll b e

h i g h l y i n a c c u r a t e , l e a d i n g t o t h e m o m e n t a r i l y

d e g r a d e d c o n t r o l l e r p e r f o r m a n c e s [7 ].

A g a i n s t t h i s b a c k d r o p , w e p r o p o s e t o u s e a

f u z z y i n f e r e n c e b a s e d s e l f -t u n i n g s c h e m e f o r P I D

c o n t r o l le r s . T h e e s s e n c e o f t h e s c h e m e i s t h a t a t

e v e r y t i m e i n s t a n c e , t h e c o n t r o l l e r e v a l u a t e s t h e

t r e n d o f t h e c o n t r o l l e d p r o c e s s o u t p u t t o d e t e c t

t h e p o s s i b l e d e v i a t i o n f r o m a p r e s c r i b e d c o u r s e .

I f a d e v i a t i o n i s f o u n d , a n a p p r o p r i a t e c o n t r o l

a c t i o n a c c o r d i n g t o t h e n a t u r e o f t h e d e v i a t io n

w i l l b e g e n e r a t e d i n s t a n t a n e o u s l y t o c o r r e c t i t .

C o m p a r e d t o t h e e x i st i ng m o d e l b a s e d s e lf -

t u n i n g s c h e m e s , o u r s c h e m e i s e m p i r i c a l b a s e d ,

a n d a c t s m o r e l i k e w h a t w e d o w h e n w e , f o r

e x a m p l e , t r y t o s t e e r a c o n t r o l l e r m a n u a l l y t o

k e e p t h e o u t p u t o f a p r o c e s s o n a f ix e d c o u r se .

O u r o w n e x p e r i e n c e t e l l s t h a t a p r e c i s e

d e s c r i p t io n o f th e p r o c e s s i n th e f o r m o f a

d y n a m i c a l m o d e l ) i s o f t e n i r r e l e v a n t f o r o u r

s t e e r i n g a c t i o n s . W h a t i s m o r e i m p o r t a n t i s t h e

i n s t a n t o b s e r v a t i o n s o f t h e e r r o r , a n d s u b s e q u e n t

r a t i o n a l a c t i o n s f o r b r i n g i n g i t b a c k t o c o u r s e .

T h e r e a r e t w o k e y i d e a s i n o u r s c h e m e . F i r s t ,

t h e Z i e g l e r - N i c h o l s f o r m u l a i s p a r a m e t e r i z e d b y

a si n g l e p a r a m e t e r a . T h i s a i s a r r a n g e d s o t h a t

i ts i n c r e a s e d e c r e a s e ) w i l l l e a d t o th e i n c r e a s e

d e c r e a s e ) i n t h e p r o p o r t i o n a l t e r m a n d t h e

d e c r e a s e i n c r e a s e ) i n b o t h t h e in t e g r a l a n d

d i f f e r e n t i a l t e r m s i n t h e P I D c o n t r o l l e r . S u c h a n

a r r a n g e m e n t i s i n t e n d e d t o d i v e r t t h e t r e n d o f

t h e p r o c e s s o u t p u t u s i n g t h e k n o w l e d g e o f t h e

q u a l i t a t i v e r e l a t i o n s h i p b e t w e e n t h e p r o p o r t i o n s

o f th e P I D f e e d b a c k s a n d t h e p r o f il e s o f t h e

p r o c e s s o u t p u t . S e c o n d l y , t h e o n - l i n e t u n i n g

f o r m u l a f o r a i s a d i s c re t e d y n a m i c a l e q u a t i o n

d r i v e n b y a f u z z y i n f e r e n c e p r o c e d u r e . A s i m p l e

f u z z y m a p is f o r m e d i n s u c h a w a y t h a t i t u p d a t e s

a i n a c c o r d a n c e w i t h th e c u r r e n t r e g u l a t i o n e r r o r

a n d e r r o r r a t e . S p e c i f i c a ll y , i t s p e e d s u p t h e

c o n v e r g e n c e o f t h e p r o c e s s o u t p u t t o a s e t - p o in t



S h i -Z h o n g H e e t a l. / F u z z y s e l f - t u n i n go f P I D c o n t r o ll e rs

39

Yr, and slows down the divergence trend of the

output from Yr. This is, in fact, the fuzzy

adaptation mechanism we have used successfully

in one of our early works on adaptive fuzzy

controller [9].

Many forms of adaptive fuzzy control schemes

exist, see [12,13,9]. Virtually all of these

schemes are genuine fuzzy control schemes in

the sense that the controllers are actually fuzzy,

although the adaptation mechanisms may some-

times employ non-fuzzy tuning laws. The

proposed fuzzy auto-tuning PID is different in

nature from all these schemes in that it is a

non-fuzzy controller tuned with a fuzzy inference

mechanisms. Further, because of its connection

with the Ziegler-Nichols formula, the proposed

control scheme is not completely model-free. A

simple initialization procedure will have to be

used to obtain the ultimate gain and ultimate

period of the process to be controlled in order to

start the fuzzy adaptation. This limited degree of

model dependence reflects the consideration that

if certain information on the process can be

acquired easily and directly, the control scheme

should be able to make use of it. This is in sharp

contrast to a genuine fuzzy controller, which are

completely model-free and all the control rules

are supposed to come directly from the

experiences. Another rationale behind our

scheme is that it is often hard to directly acquire

the knowledge or possess direct human ex-

periences for controlling a complex process.

However, if the controller structure is fixed to be

the PID control, then the experiences are

narrowed down to more specialized experiences

of choosing a few PID parameters. This latter

problem has been under scrutiny for so long a

time that there have been a great amount of

knowledge and experiences accumulated on the

subject. In this context, it seems more meaning-

ful to keep the PID structure and let the

self-tuning part be handled by the fuzzy logic

approach.

The main objective of the present paper is to

propose this new type of fuzzy self-tuning

control scheme, provide the details of the design

procedure, and conduct a simulation analysis to

compare the scheme with two tuning schemes,

namely, the Ziegler-Nichols tuning and the

Refined Ziegler-Nichols tuning. The general

conclusion of the simulation analysis is that the

new fuzzy self-tuning PID controller out-

performs the PID controllers tuned by the two

fixed tuning laws.

The paper is organized as follows. In Section

2, the new fuzzy self-tuning PID controller is

described in detail, the exposition covers the

basic structure of the controller, fuzzy tuner, as

well as the initialization of the controller. The

simulation analysis is carried out in Section 3

followed by Section 4, which contains further

discussions and conclusions.

2 Th e contro l l er and the fuzzy adaptat ion

B a s i c s t r u c t u r e

To begin with, we assume that the process to

be controlled has single input u t ) and single

output y t ) , and the control objective is to bring

the process output y t ) to a prescribed set-point

Yr. The scheme can actually be extended to the

tracking problems where y~ is a time-varying

target output. However, this extension will not

be discussed here to keep our exposition concise.

As mentioned in the previous section, the

fuzzy self-tuning PID controller consists of a

standard PID controller and a fuzzy tuning

mechanism used for the on-line adaptation o f the

PID parameters Figure 1). The PID controller,

which generates a control

u t )

based on the

closed-loop error e t ) = Y r - y t ) , has the follow-

ing standard form

u t ) = K c [ e t ) + T d d e ~ t ) + ~ f e t ) d t ] ,

1)

where Ko Td, T~ are, respectively, the propor-

tional gain, the derivative time and the integral

time of the controller, which are to be adjusted

on-line.

One of the key ideas of the control scheme is

to parametrize the three PID parameters by a

single parameter c~ as shown below

Kc-- 1.2c~ku,

1

T~ = 0.75 1--~-a t°, Ta = 0.25T~, 2)

where ku, tu are, respectively, the ultimate gain

and the ultimate period of the underlying

process, which will be determined shortly.



40

Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

Yr

f

I

I

I

I

I

[ m _

Fuzzy Self-tuning Mechanism

]FuzzyAd ap ta t i o ~ -- -- -~ DPasrgm oerrmZ da

Controller

_

P r oce s s

Fig. 1. The basic structure o f the fuzzy self-tuning PID controller.

T h e f o r m o f t h e p a r a m e t e r i z a t i o n is i n s p ir e d

b y t h e Z i e g l e r - N i c h o l s f o r m u l a , a n d i n f a c t

r e d u c e s t o i t w h e n a = ½. T h u s w e c a n t h i n k o f

t h e c o n t r o l l e r a s c o m p e n s a t i n g t h e b a s i c c o n t r o l

o f t h e Z i e g l e r - N i c h o l s b y b i a si n g a ll th e

p a r a m e t e r s o n - l i n e i n o r d e r t o a d j u s t t h e p r o c e s s

o u t p u t t o a p r e s c r i b e d c o u r s e . I n w h a t f o l l o w s ,

b o t h t h e f u z z y a d a p t a t i o n a n d t h e i n i t ia l i z a t io n

w i l l b e d i s c u s s e d i n d e t a i l .

F u z z y a d a p t a t i o n

A s s h o w n i n F i g u r e 1 , t h e f u z z y s e l f -t u n i n g

m e c h a n i s m w i ll g e n e r a t e a n

a t )

g i v e n t h e

i n s t a n t v a l u e s o f

e t )

an d O t ) a t t i m e t. I t i s

c o m p o s e d o f tw o p a r t s: a f u z z y c o r e a n d a

c o n d i t i o n a l u p d a t i n g f o r m u l a f o r a . T h e f u z z y

c o r e s t a r t s w i t h f u z z i f y i n g e t ) a n d O t) i n t o t w o

f u z z y v a r i a b l e s E , r e s p e c t i v e l y , R . T o e n s u r e a

s p e e d y f u z z y in f e r e n c e , b o t h t h e r a n g e o f

i n t e r e s t f o r

e t )

a n d t h a t f o r ~ t ) a r e c o v e r e d b y

s e v e n d i f f e r e n t f u z z y se t s a s s h o w n b e l o w

E = { N L , N M , N S , Z O , P S , P M , P L } 3 )

R = { N L , N M , N S , Z O , P S , P M , P L } ,

w h e r e , a s us u a l, t h e m e a n i n g s o f th e a c r o n y m s

u s e d i n 3 ) a r e , r e s p e c t i v e l y , P L f o r p o s i ti v e

l a r g e , P M f o r p o s it i v e m e d i u m , P S f o r p o s it i v e

s m a l l, Z O f o r z e r o , N S f o r n e g a t i v e s m a l l , N M

f o r n e g a t iv e m e d i u m , a n d N L f o r n e g a t i v e la r g e .

F o r e a s e o f t h e n o t a t i o n , w e s h a ll a s si g n t h e

i n t e g e r s f o r t h e f u z z y s e ts a s

P L = 3 , P M = 2 , P S = I , Z P = 0 ,

N S = - I , N M = - 2 , N L = - 3 ,

Table 1. T he fuzzy map from E and R to H

H R

- 3 - 2 - 1 0 1 2 3

E - 3 - 3 - 3 - 2 - 2 - 1 -1 0

- 2 - 3 - 2 -2 - 1 - 1 0 1

- 1 - 2 - 2 - I - 1 0 1 1

0 -2 -1 -1 0 1 1 2

1 1 1 0 1 1 2 2

2 -1 0 1 1 2 2 3

3 0 1 1 2 2 3 3

a n d d e n o t e t h e f u z z y s e ts b y t h e i r c o r r e s p o n d i n g

n u m b e r s .

T h e s e c o n d p a r t o f t h e f u z z y c o r e i s t h e f u z z y

m a p p i n g f r o m E a n d R t o H , w h e r e H i s a n o t h e r

f u z z y v a r i a b l e w h o s e d e f u z z i f i e d v e r s i o n w i ll b e

u s e d f o r th e l a t e r u p d a t in g e q u a t i o n o f a. T h e

r a n g e o f H s i m i la r l y c o n s is t s o f s e v e n f u z z y s e ts

H = { - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 }, 4 )

a n d i t i s l i n k e d t o E a n d R b y a f u z z y m a p a

p a r t i c u l a r e x a m p l e t h a t w e s h a l l u s e i s g i v e n i n

T a b l e 1 ). T h e f u z z y i n f e r e n c e i s t h e s t a n d a r d

C R I p r o c e d u r e , h e n c e a ll th e f u z z y r u le s , s uc h a s

t h o s e s h o w n i n T a b l e 1 , w i l l b e i n v o l v e d i n e v e r y

s i n gl e i n f e r e n c e .

W e a l s o a s s u m e t h a t t h e m e m b e r s h i p f u n c -

t i o n s f o r a l l f u z z y s e ts h a v e t h e f o l l o w i n g

s t a n d a rd f o r m

2

r e x ) = e ~ ) , 5 )

w h e r e x i , o- d e n o t e , r e s p e c t i v e l y , t h e c e n t r e a n d

t h e s p r e a d o f e a c h f u z z y s e t , a n d t h e i r c h o i c e s

a r e s o m e h o w s u b j e c t i v e .



Shi-Zhong He et al. / Fuzzy self-tuning of PID controllers

41

T h e l a s t p a r t o f t h e f u z z y c o r e i s t h e

d e f u z z i f i c a t i o n o f H i n t o a r e a l v a r i a b l e h t ) . F o r

s m o o t h o p e a t i o n , w e c h o o s e t o u s e t h e c e n t r e o f

g r a v it y m e t h o d .

A f t e r h t ) i s o b t a i n e d , i t i s u s e d i n t h e

f o l l o w i n g r e c u r si v e e q u a t i o n t o u p d a t e a

c~(t + 1)

= { a ( t ) + y h ( t ) ( 1 - a ( t ) ) f o r o l( t) > 0 . 5 ,

a t ) + v h t ) c e t ) fo r c~(t) ~< 0.5, (6)

w h e r e y is a p o s i ti v e c o n s t a n t u s e d t o m o d i f y t h e

c o n v e r g e n c e r a t e o f t h e u p d a t i n g f o r m u l a e . T h e

r a n g e o f 7 is w i d e , a n d i s t y p i c a l l y c h o s e n t o b e

w i t h i n t h e i n t e r v a l [ 0 . 2 , 0 . 6 ] f o r m o s t o f t h e

p r o c e s s e s. N o t e t h a t a ( 0 ) i s n o t a r b i t r a r y a n d h a s

t o b e s e t a t 0 .5 . I t a l so f o l l o w s f r o m ( 6 ) t h a t a ( 0 )

i s a c u t - o ff v a l u e w h e r e b y t h e t w o u p d a t i n g

f o r m u l a e s w i t c h f r o m o n e t o t h e o t h e r . S u c h a n

a r r a n g e m e n t a l o n g w i t h t h e f u z z y m a p g u a r-

a n t e e s t h e s m o o t h a n d b o u n d e d ( b e t w e e n 0 a n d

1 ) v a r i a t i o n o f ~ , w h i c h i n t u r n l e a d s t o s m o o t h

a n d b o u n d e d a d a p t a t i o n o f t h e P I D p a r a m e t e r s .

In i t ia l i z a t ion o f the con t r o l l e r

A s t h e t u n i n g f o r m u l a in v o l v e s t h e u l t i m a t e

g a i n k u a n d t h e u l t i m a t e p e r i o d tu , t h e y w i ll h a v e

t o b e d e t e r m i n e d p r i o r t o t h e u s e o f t h e

c o n t r o l l e r . T h i s p r o c e s s i s c a l l e d i n i t i a l i z a t i o n .

S e v e r a l m e t h o d s f o r i n i t ia l i zi n g t h e P I D

c o n t r o l l e r s a r e a v a i l a b l e . B e c a u s e o f i t s i n t u i t i v e

a p p e a l a n d e a s e o f o p e r a t i o n , t h e r e l a y f e e d b a c k

m e t h o d p r o p o s e d i n [ 3, 5 ] i s u s e d t o i n i t ia l i z e t h e

p a r a m e t e r s o f t h e f u z z y s e l f - tu n i n g P I D

c o n t r o l l e r .

T h e m e t h o d o f re l a y f e e d b a c k i s b a s e d o n t h e

f a c t t h a t d y n a m i c a l p r o c e ss e s t y p i c a l l y e n c o u n -

t e r e d i n p r o c e s s c o n t r o l w i l l e x h i b i t l i m i t c y c l e

o s c i l l a t i o n u n d e r r e l a y f e e d b a c k . T h e f r e q u e n c y

o f t h e l i m i t c y c l e i s a p p r o x i m a t e l y t h e u l t i m a t e

f r e q u e n c y w h e r e t h e p r o c e s s h a s a p h a s e l a g o f

1 80 ° . T h e p e r i o d o f t h e o s c i l l a t i o n tu is e a s i l y

o b t a i n e d b y m e a s u r i n g t h e t i m e b e t w e e n z e r o

c ro s si ng s . T h e a m p l i t u d e m a y b e d e t e r m i n e d b y

m e a s u r i n g t h e p e a k t o p e a k v a l u e s o f t h e o u t p u t .

I f d i s t h e r e l a y a m p l i t u d e a n d a i s t h e p r o c e s s

o u t p u t a m p l i t u d e , t h e u l t i m a t e g a i n i s a p p r o x i m -

a t e l y g i v e n b y

4 d

k . - - . 7 )

n a

T h e r e l a y a u t o - t u n e r p r i n c ip l e i s s h o w n i n F i g u r e

2 .

W i t h b o t h k u a n d t u i n p la c e , a n d w i t h a ( 0 ) s e t

t o b e 0 . 5 , t h e i n i t i a l v a l u e s f o r t h e P I D

p a r a m e t e r s n a t u r a l l y f o ll o w f r o m ( 2)

Kc = 0.6 ku , T~ = 0.5tu,

T j

: 0.125tu, (8)

w h i c h i s p r e c i s e l y t h e Z i e g l e r - N i c h o l s f o r m u l a .

I t i s t h u s c l e a r t h a t e x c e p t f o r t h e i n i t i a l i z a t i o n

w h e r e a n i m p l i ci t m o d e l a s s u m p t i o n o n t h e

i n i t i a l s t a t u s o f t h e p r o c e s s i s m a d e , o u r f u z z y

s e l f - t u n i n g P I D c o n t r o l l e r i s a m o d e l - f r e e c o n t r o l

s c h e m e .

A c l o se e x a m i n a t i o n o f b o t h ( 2 ) a n d ( 8 ) s h o w s

t h a t o u r a d a p t a t i o n s c h e m e c a n b e i n t e r p r e te d a s

u s i n g a s i n g l e p a r a m e t e r t o b i a s t h e P I D

c o n t r o ll e r p a r a m e t e r s a w a y f r o m t h e ir Z i e g l e r -

N i c h o l s s e tt i n g s i n o r d e r t o c o m p e n s a t e f o r a n y

i n a d e q u a c y . T h i s i d e a o f a d a p t a t i o n s h o u l d b e

c o n t r a s t e d w i t h t h e c o n v e n t i o n a l P I D s e l f -t u n i n g

s c h e m e s w h e r e b y a t e a c h t i m e i n s t a n c e a

Y~ C )

l

•

uzzy

Self-tuning

• P ID Q ~ . ~

• Process

Fig. 2. Block diagram of the relay auto-tuner.

y



4 2

Shi Zh ong He e t a l. / Fu zzy se l f tun ing o f PID con tro lle rs

p r e - s p e c i f i e d p r o c e s s m o d e l w i l l h a v e t o b e

i d e n t i f i e d , a n d t h e t u n i n g e i t h e r m o d i f i e s k u a n d

tu, o r c ha nge s d i r ec t ly K c , T~, Td ba s ed on the

i d e n t i f ie d p r o c e s s m o d e l . T o h i g h l i g h t t h e i r

d i f f e re n c e , w e c a n t h i n k o f t h e t w o t y p e s o f

a d a p t a t i o n s a s e i t h e r

e l a s t i c

o r

p l a s t i c

T h e

a d a p t a t i o n w e h a v e p r o p o s e d i s e l a s ti c in t h e

s e n s e t h a t it o n l y d e f o r m s f r o m a s t a n d a r d s et

o f c o n t ro l l e r p a r a m e t e r s ( t h e Z i e g l e r - N i c h o l s i n

o u r c a s e ) t o c o u n t e r f o r t h e p r o c e s s c o m p l e x i t y

a n d v a r ia t io n . T h e d e f o r m a t i o n m a y b o u n c e

b a c k a n d f o r t h c a u s i n g a l l t h e c o n t r o l l e r

p a r a m e t e r s t o v a r y a r o u n d t h e s t a n d a r d s e t o f

c o n t r o ll e r p a ra m e t e r s . W h e r e a s t h e c o n v e n t i o n a l

w a y o f a d a p t a t i o n i s p l a s t i c i n t h e s e n s e t h a t t h e

d e f o r m a t i o n o f t h e c o n t r o l le r p a r a m e t e r s a t

e v e r y t i m e i n s t a n c e i s d i c t a t e d b y t h e i d e n t i f i e d

m o d e l a n d t h u s i r r e v e r s i b l e u n l e s s t h e

c o e f f ic i e n ts o f t h e i d e n t i f i e d m o d e l a r e m o v i n g

u p a n d d o w n a r o u n d a s t a n d a r d s e t o f

c o e f f i c i e n t s, w h i c h is h i g h l y u n l i k e l y . I t i s in

g e n e r a l d i ff ic u lt t o t e l l w h i c h f o r m o f a d a p t a t i o n

i s s u p e r i o r . I n t h e p r e s e n t c o n t e x t , h o w e v e r , t h e

m o d e l - f r e e n a t u r e o f o u r e l a st ic a d a p t a t i o n

a p p e a r s t o b e c o n c e p t u a l l y s i m p l e r a n d m o r e

e f f ic i e n t t h a n t h e c o n v e n t i o n a l m o d e l - b a s e d

p l a s ti c a d a p t a t i o n .

W i t h t h e p r e c e d i n g d e s i g n d e ta i l s o f t h e n e w

c o n t r o l s c h e m e , l e t u s p r o v i d e a r o u g h a c c o u n t

o f h o w i t a c t u a l l y w o r k s . N o t e t h a t t h i s f u z z y

s e l f -t u n i n g m e c h a n i s m i s th e o n e t h a t w e

p r o p o s e d i n o n e o f o u r e a r l y w o r k s [9 ]. A s b e i n g

e x p l a i n e d i n [ 9] , t h e e s s e n t i a l i d e a o f t h e

a d a p t a t i o n i s t o p r o v i d e a p p r o p r i a t e c~ f o r

s e v e r a l r e a l - ti m e s c e n a r i o s d e f i n e d o n t h e p r o f i l e

o f y i n r e l a t i o n t o Yr- H e r e w e a r e o n l y i n t e r e s t e d

i n f o u r p o s s i b l e s c e n a r i o s : y a p p r o a c h e s t o Yr

f r o m a b o v e o r b e l o w , a n d y d i v e r t s u p a n d d o w n

a w a y f r o m Yr. W h e n y a p p r o a c h e s Yr f r o m a b o v e

o r b e l o w , t h e c o m b i n e d e f f o r t o f b o t h t h e f u z z y

m a p i n T a b l e 1 a n d t h e u p d a t i n g e q u a t i o n s ( 6)

w i ll i n c r e a s e a , ( 2 ) w i l l c o n s e q u e n t l y e n s u r e t h a t

s uch a n inc reas e w i l l pus h K c h ig her a nd T i, Td

l o w e r , w h i c h i n t u r n s p e e d s u p t h e a p p r o a c h i n g

of y to Yr- S im i la r ly , w he n y d iv e rge s up (o r

do w n) f ro m Y r, c~ w i l l be de c rea s ed , th us

d e c r e a s i n g K c a n d i n c r e a s i n g T~, T d, c o n s e q u e n t l y

c a u s i n g t h e d i v e r g e n c e t o b e s l o w e d d o w n .

B e c a u s e t h e r e i s o n l y o n e p a r a m e t e r c~

i n v o l v e d , t h e r u l e s f o r g e n e r a t i n g a l l t h e c h a n g e s

f o r t h e f o u r p o s s i b l e s c e n a r i o s a r e e a s y t o f o r m .

T a b l e 1 i s o n l y o n e e x a m p l e o f a p p r o p r i a t e f u z z y

m a p s t h a t c a n b e u s e d f o r t h e p u r p o s e . T h e

c o n s t r u c t i o n o f t h e k i n d o f f u z z y m a p r e q u i r e s

t h e a n a l y s i s o f r e s p o n s e p r o f i l e s a n d t h e i r

r e l a t i o n s h i p s t o c ~ , a n d i s a l s o a f f e c t e d b y t h e

r a t e o f c h a n g e w e d e s i r e o n a . T h e f u z z y m a p i n

T a b l e 1 o n l y e ff e c ts a m o d e r a t e a c h a n g e r a t e .

H i g h r a t e o f c h a n g e t e n d s t o c a u s e o s c i ll a t o ry

b e h a v i o u r , a n d t h u s i s n o t a l w a y s d e s i r a b l e .

T h e o r e t i c a l v e r i f i c a t i o n o f t h e p r e c e d i n g

s t a t e m e n t s i n t h e c o n t e x t o f m o d e l c o m p l e x i t y

a n d u n c e r t a i n t y i s n o t y e t a v a i l a b l e , a l t h o u g h

s u c h a v e r i f i c a t i o n i s p o s s i b l e f o r a g i v e n s i m p l e

p r o c e s s m o d e l . B u t t h i s l a t t e r t y p e o f v e r i f i c a t i o n

h a s s p e c i a li z e d n a t u r e a n d d o e s n o t i l lu s t r a te o u r

p o i n t . F o r t h e t i m e b e i n g w e m o s t l y re l y o n t h e

e m p i r i c i a l s t u d i e s a n d s i m u l a t i o n a n a l y s i s f o r

s u c h a v e r i f i c a t i o n . F i g u r e 3 i s t y p i c a l o f t h e

s i m u l a t i o n r e s u l t s , w h i c h s h o w s h o w c~ c h a n g e s

g i v e n t h e c o r r e s p o n d i n g p r o f il e o f y . T h i s p l o t

c o n f i r m s t h a t t h e p a r t i c u l a r u p d a t i n g f o r m u l a ( 6 )

a n d t h e f u z z y m a p i n d e e d m o d i f i e s c~ a s

p r e s c r i b e d .

2

1.5

I

0 . 5

0

0

3 4 5 7 8

T i m e v s y p a n d a l p h a, y p - ) , a lp h a - - )

F i g . 3 . T h e t r e n d o f o~ i n r e l a t io n t o t h e p r o c e s s o u t p u t y .



Shi-Z hong H e et a l. / Fuz zy sel f - tun ing of P ID contro llers

43

3 S i m u l a t i o n a n a l y s i s

T o c o v e r t y p i c al k in d s o f c o m m o n i n d u st r ia l

p r o c e s s e s t h r e e g r o u p s o f r e p r e s e n t a t i v e p r o c -

e s s e s a r e c h o s e n f o r o u r s i m u l a t i o n a n a l y si s . F o r

e a c h o f th e l a t t e r t w o t y p e s o f p r o c e s s e s

d i f f e r e n t c o e f f ic i e n t s a r e s e t f o r t h e s a m e p r o c e s s

m o d e l s j u s t t o r e f l e c t t h e s e v e r i t y o f t h e

t i m e - d e l a y o r n o n - m i n i m u m p h a s e c h a r a c t e r is -

t ic s. T o e v a l u a t e t h e p e r f o r m a n c e o f th e n e w

f u z z y s e l f- t u ni n g P I D c o n t r o l l e r w e c o m p a r e it s

s e t - p o i n t a s w e l l a s t h e l o a d d i s t u r b a n c e

r e s p o n s e s w i t h t h e P I D c o n t r o l l e r s t u n e d b y t h e

Z i e g l e r - N i c h o l s f o r m u l a a n d t h e R e f i n e d

Z i e g l e r - N i c h o l s f o r m u l a .

A second order process

T h e f i r s t p r o c e s s i s c h o s e n t o h a v e t h e

f o l lo w i n g s i m p l e s e c o n d - o r d e r c h a r a c t e ri s ti c s

G s )

= k , , 9)

T , s + 1 ) ~ s + 1 )

T h e p a r a m e t e r s o f t h e p r o c e ss a r e k p = 1 , T l = l

a n d ~ = 0 .5 . T h e c h o i c e o f th e s a m p l i n g t i m e t~

i s b a s e d o n t h e p r o c e s s t i m e c o n s t a n t s T ~ T 2

a n d h e r e w e s e t ts t o b e 0 .1 . T h e n e w s e t - p o i n t y

i s 1 a n d a st a t i c l o a d d i s t u r b a n c e i s a l s o

i n t r o d u c e d i n t o t h e p r o c e s s a t t = 2 0 s .

T h i s p r o c e s s is r e g u l a t e d s e p a r a t e l y b y t h e

f u z z y s e l f - tu n i n g P I D c o n t r o l l e r a n d b y t h e t w o

P I D c o n t r o l l e r s t u n e d u s i n g t h e Z i e g l e r - N i c h o l s

f o r m u l a a n d th e R e f i n e d Z i e g l e r - N i c h o l s tu n i n g

f o r m u l a . T h e r e s p e c t i v e s e t - p o i n t r e s p o n s e s a n d

l o a d d i s t u r b a n c e r e s p o n s e s a r e s u m m e r i z e d i n

F i g u r e 4 w h e r e y~ is t h e r e s p o n s e o f t h e f u z z y

s e l f - t u n i n g P I D c o n t r o l l e r Y 2 Y3 a r e t h o s e o f t h e

P I D c o n t r o l l e r s t u n e d u s in g t h e R e f i n e d

Z i e g l e r - N i c h o l s f o r m u l a r e s p e c t iv e l y t h e

Z i e g l e r - N i c h o l s f o r m u l a . W e s h a l l u s e t h e s a m e

n o t a t i o n s f o r t h e r e s t o f th e s i m u l a t i o n e x a m p l e s .

F i g u r e 4 c l e a rl y sh o w s t h e r e m a r k a b l e

s e t - p o in t r e s p o n s e p e r f o r m a n c e o f t h e f u z z y

s e l f - t u n i n g c o n t r o l l e r o v e r t h e o t h e r t w o

c o n t r o l l e r s w i t h s h o r t e r r is e ti m e s h o r t e r s e t tl i n g

t i m e a n d l es s o v e r s h o o t . O b s e r v e t h a t u n l i k e

n o r m a l f u z z y c o n t r o l le r s w h e r e l o w e r i n g th c

o v e r s h o o t is o f t e n a t t h e e x p e n s e o f s lo w i n g

d o w n c o n s i d e r a b l y t h e r is e t i m e t h e n e w c o n t r o l

s c h e m e s e e m s t o r e c o n c i l e th e s e t w o r e q u i r e -

m e n t s . T h e r e a s o n i s r o u g h l y th a t t h e s e l f - t u n i n g

n a t u r e o f t h e c o n t r o l l e r a l lo w s t h e s e l e c t i o n o f

d i f f e r e n t P I D c o n t r o l l e r s f o r c o n t r o l l i n g t h e

p r o c e s s a t d i f f e r e n t s t ag e s o f t h e r e s p o n s e s . T h e

c o n t r o l l e r i s d e s i g n e d s o t h a t i t p i c k s u p a n

a p p r o p r i a t e P I D c o n t r o l l e r a t e a ch s t a ge .

F i g u r e 4 a l s o s h o w s t h a t t h e i m p r o v e m e n t i n

t h e s e t - p o i n t r e s p o n s e o f th e n e w c o n t r o l l e r is

n o t a t t h e c o s t o f t h e l o a d d i s t u r b a n c e r e s p o n s e

a l t h o u g h t h e r e i s n o o b v i o u s i m p r o v e m e n t i n t h e

l o a d d i s t u r b a n c e r e s p o n s e w i t h r e s p e c t t o t h o s e

b y t h e o t h e r t w o c o n t r o l l e r s . F o r c o n v e n t i o n a l

c o n t r o l le r s t h e s e tw o r e s p o n s e s t e n d t o u n d o

e a c h o t h e r l e a d i n g t o a c o m p r o m i s e d d e s i g n in

w h i c h n o n e o f th e r e s p o n s e s i s a t i ts b e s t. T h e

n e w c o n t r o l s c h e m e s e e m s t o h a v e p r o v i d e d y e t

a n o t h e r r e c o n c i l i a t i o n t o w a r d s t h i s p r o b l e m . A s

w e s h a l l s e e a g a i n i n t h e l a t e r s i m u l a t i o n

e x a m p l e s t h e f u z z y s e lf - tu n i n g P I D c o n t r o l l e r

t e n d s t o e n h a n c e t h e s e t - p o i n t r e s p o n s e w h i l e

k e e p i n g t h e l o a d d i s t u r b a n c e a t a n a c c e p t a b l e

l e v e l .

1 . 5 r

Closed , -Loo p Resp onse o f p ,rocess

0 5

~ i I i i i i i

0 5 10 15 20 25 30 35 40

y l - ) , y 2 - . ) a n d y 3 - - ) v s t i m e

F i g . 4 . T h e s e t - p o i n t a n d l o a d d i s t u r b a n c e r e s p o n s e s o f t h e s e c o n d - o r d e r p r o c e s s .



44

Shi-Z hon g He et al. / Fu zzy self-tuning o f PID controllers

1.5

, Closed-LoopResponse,ofprocess

s ,

1 ~ V -~ r '~ '''~ - ~ ~ , , ~ , ~ - - , - - - '

0.5

i i i i i i

00 10 20 30 40 50 60 70

yl(-), y2(-.) and y3(--) vs time

Fig. 5. The set-point and load disturbance responses of the proce ss with small dead-time.

T i m e - d e l a y p r o c e s s e s

T h e s e c o n d s i m u l a t i o n c o n c e r n s t i m e - d e l a y

p r o c e s s e s o f t h e f o l l o w i n g g e n e r a l f o r m

- k p e - ° d s

( 1 0 )

G S) Ts ~- 1 ) 2 .

T h e t w o p a r a m e t e r s k p a n d T a r e a l l s e t to 1 , a n d

t h e d e a d - t i m e 0 d , h o w e v e r , w i l l b e s e t t o t w o

d i f f e r e n t v a l u e s t o e x a m i n e t h e a b i l it y o f t h e n e w

c o n t r o l l e r i n h a n d l i n g s m a l l o r l a r g e d e a d - t i m e .

T h e s a m p l i n g r a te f o r t h e s i m u l a t io n r e m a i n s t o

b e 0 . 1 .

0 d i s f i r s t c h o s e n t o b e 2 , c o r r e s p o n d i n g t o a

r e l a t i v e l y s m a l l d e a d - t i m e . Y r i s 1 , a n d a s t a t i c

l o a d d i s t u r b a n c e is i n t r o d u c e d a t t = 3 5. T h e

s i m u l a t i o n r e s u l t s a r e s h o w n i n F i g u r e 5 . N o t e

t h a t t h e n o t a t i o n s f o r th e t h r e e r e s p o n s e c u r v e s

Y l, y2 a n d Y3 a r e t h e s a m e a s t h o s e u s e d i n t h e

f i r s t s i m u l a t i o n e x a m p l e .

0 d i s t h e n s e t t o b e 6 , c o r r e s p o n d i n g t o a

r e l a t i v e l y l a r g e d e a d - t i m e . T h i s t i m e t h e s t a ti c

l o a d d i s t u r b a n c e is i n t r o d u c e d a t t = 5 5 t o a l lo w

t h e p r o c e s s t o s e t t l e d o w n . T h e s i m u l a t i o n

r e s u l t s a r e s u m m a r i z e d i n F i g u r e 6 .

F i g u r e s 5 a n d 6 a l l o w u s t o d r a w s i m i l a r

c o n c l u s i o n s : T h e s e t - p o i n t r e s p o n s e i m p r o v e s

c o n s i d e r a b l y w h i l e t h e l o a d d i s t u r b a n c e r e s -

p o n s e s a r e e i t h e r c o m p a r a b l e t o t h o s e o b t a i n e d

b y t h e o t h e r t w o c o n t r o l l e r s ( in t h e c a s e o f s m a l l

d e a d - t i m e ) , o r o n l y i m p r o v e s m a r g i n a l l y ( in th e

c a s e o f l a r g e d e a d - t i m e ) . I t i s a l s o i n t e r e s t i n g t o

o b s e r v e t h a t i n b o t h c a s e s, t h e f u z z y s e l f- t u n i n g

P I D c o n t r o l l e r a c t s m o r e l i k e a f u r t h e r r e f i n e d

Z i e g l e r - N i c h o l s P I D c o n t ro l , c a p a b l e o f c u r b i n g

t h e e x c e s s i v e o v e r s h o o t .

N o n - m i n i m u m p h a s e p r o ce s se s

T h e l a s t s i m u l a t i o n h a s t o d o w i t h n o n -

m i n i m u m p h a s e p r o c e ss e s . T h e g e n e r a l f o r m o f

t h e p r o c e s s i s g i v e n a s

kp(1 - p s )

G s ) - - ~ + ~ . (11)

F o r t h e f ir st p r o c e s s , t h e p a r a m e t e r s a r e c h o s e n

a s f o ll ow s ; k p = l , T = I a n d p = 1 . 4 . C l e a rl y ,

t h i s p r o c e s s h a s a n o n - m i n i m u m p h a s e z e r o a t

s = l i p = 3. T h e s a m p l i n g r a t e i s c h o s e n t o b e

1 5 C lo se -L o o p a e~ p o n s~ o e p ro cess

; : , / '~ . , , , - ~ _ ~ ' . ' - - ~ . .~_

1

0.5

10 20 30 40 50 60 70 80 90 100

yl(-), y2(-.) and y3(--) vs time

Fig. 6. The set-point and load disturbance responses of the proce ss with large dead-time.



Shi Zh ong H e et a l. / Fuz zy sel f tun ing o f PID contro llers 45

1 . 5 Close -Loop Response of process , ,

1 ' / . . . . . . . . . - ° - -

0 5

-0 .5 --

0 5 10 15 20 25 30 35 40

yl -), y2 -.) and y3 --) vs time

Fig. 7. The set-point and load disturbance responses of the first non-m inimum phase process.

0 .1 , a n d t h e l o a d d i s t u r b a n c e i s i n t r o d u c e d

t = 20 . F i g u r e 7 s h o w s a l l t h e r e s p o n s e c u r v e s .

F o r t h e s e c o n d n o n - m i n i m u m p h a s e p r o c e s s , p

i s s e t t o 2 . 5 , b o t h T a n d p r e m a i n t h e s a m e a s i n

t h e p r e v i o u s p r o c e s s . T h e n o n - m i n i m u m p h a s e

z e r o i n t h is c a s e i s a t s = 3, m u c h c l o s e r t o t h e

o r i g in o f t h e s - p l a n e . A l l t h e s i m u l a t i o n r e s u l t s

f o r t h i s p r o c e s s a r e s h o w n i n F i g u r e 8 .

I t f o l l o w s f r o m F i g u r e s 7 a n d 8 t h a t w h i l e b o t h

t h e Z i e r g l e r - N i c h o l s t u n i n g a n d t h e R e f i n e d

Z i e g l e r - N i c h o l s t u n i n g c a n n o t p r o v i d e a d e q u a t e

c o n t r o l t o t h e n o n - m i n i m u m p h a s e p r o c e s s e s ,

e s p e c i a l l y f o r t h e s e c o n d o n e , t h e n e w s c h e m e

w o r k s r e m a r k a b l y w e l l i n b o t h c a s e s . O b s e r v e

t h a t t h e l o a d d i s t u r b a n c e r e s p o n s e s a l s o s e e m t o

i m p r o v e c o n s i d e r a b l y f o r t h e s e p r o c e s s e s , t h u s

c o n t r a d i c t i n g t h e g e n r a l p a t t e r n s o b s e r v e d i n t h e

p r e c e d i n g s i m u l a t i o n s . T h i s , p e r h a p s , s h o u l d n o t

b e u n d e r s t o o d in te r m s o f t h e i m p r o v e m e n t

m a d e b y t h e f u z z y s e l f - t u n in g c o n t r o l l e r , i n f a c t ,

i ts l o a d d i s t u r b a n c e is a l w a y s a c c e p t a b l e b u t n o t

s u p e r i o r ) f o r v a r i o u s k i n d s o f p r o c e s s e s . I t

s h o u l d b e u n d e r s t o o d i n t e r m s o f t h e f a i lu r e o f

t h e o t h e r P I D t u n i n g s c h e m e s i n p r o v i d i n g a n

a c c e p t a b l e l e v e l o f c o n t r o l f o r c e r t a i n d i f f ic u l t

p r o c e ss e s . T h e a d v a n t a g e o f th e n e w s c h e m e

b e c o m e s e v i d e n t f o r c o n t r o ll i n g t h e s e p r o c e s s e s .

4 . D i s c u s s i o n s a n d c o n c l u s i o n s

W e h a v e p r e s e n t e d a d e t a i l e d a c c o u n t o f a

f u z z y s e l f - t u n i n g P I D c o n t r o l l e r , i ts b a s i c

p r i n c i p l e , d e s i g n s t e p s a n d s i m u l a t i o n a n a l y s i s .

T h e p e r f o r m a n c e o f t h e n e w c o n t r o l le r i s

c o m p a r e d f a v o u r a b l y t o t h o s e o f t he t w o e x is ti n g

c o n v e n t i o n a l P I D t u n i n g f o r m u l a e .

I n a s e n s e , t h e p r e s e n t w o r k r e p r e s e n t s a n

e x t e n s i o n o f o u r e a r l y w o r k w h e r e e x a c t l y t h e

s a m e a d a p t a t i o n m e c h a n i s m is u s e d t o t u n e a se t

o f f u z z y c o n t r o l r u l e s [9 ]. In b o t h w o r k s , o u r a i m

i s t o s h o w t h a t t h i s s o - c a l l e d e l a s t i c a d a p t a t i o n

c a n b e u s e d t o g r e a t a d v a n t a g e i n p r o c e s s

c o n t r o l .

O n e i n t e r e s t i n g f e a t u r e o f t h e p r e s e n t w o r k i s

t h e c o m b i n a t i o n o f t he c o n v e n t i o n a l P I D

c o n t r o l l e r w i t h t h e f u z z y i n f e r e n c e m e t h o d .

C o m b i n a t i o n s a m o n g o t h e r c o n tr o l le r s a n d

1.5 CloseA,-LoopResponse of process __ ,

0.50

0 5 . . . . . . .

' 0 5 10 15 20 25 30 35 4 0

yl -), y2 -.) and y3 --) vs time

Fig. 8. Thc set-point and load disturbance responses of the second non-minimum phase proc ess.



46

Shi- Zho ng H e e t a l. / Fu zzy se l f - tun ing o f PID con tro l le r s

m e c h a n i s m s h a v e l e d t o e n c o u r a g i n g r e s u l t s [ 1 4 ] .

W e b e l i e v e t h a t t h e i d e a o f f u z z y c o n t r o l s h o u l d

g o b e y o n d t h e p r e s e n t l i m i t o f u s i n g C R I t o p i c k

u p t h e f u z z y c on t r o l r u le s . In f ac t t h e r e i s a

g r e a t p o t e n t i a l i n u s i n g t h e f u z z y i n f e r e n c e

m e t h o d a s p a r t o f a co n t r o l s c h e m e w h i c h a s s is t s

t h e c o n t r o l f o r m a t i o n r a t h e r t h a n g e n e r a t e s t h e

c o n t r o l d i r e c t l y . T h e r e h a v e b e e n c a s e s w h e r e

t h e f u z z y l o g i c i s u s e d a l o n g t h i s l i n e . A

p a r t i c u l a r e x a m p l e w e w o u l d l i k e t o m e n t i o n i s

t o u s e f u z z y l o g i c t o o b t a i n a n e s t i m a t e o f t h e

s o - c a l l e d t r a n s i e n t p e r i o d u s e f u l f o r c e r t a i n

a d a p t i v e c o n t r o l s c h e m e [ 8 ] . T h e p r e s e n t p a p e r

o f f e r s y e t a n o t h e r n o n - t r i v i a l e x a m p l e o f u s i n g

f u z z y i n f e r e n c e m e t h o d a s a s e l f - t u n i n g m e c h a n -

i s m o f a c o n v e n t i o n a l c o n t r o l l e r . A s t h e

a d a p t a t i o n i s a m a j o r p a r t o f a n y a d a p t i v e

c o n t r o l s c h e m e t h e i n v e s t i g a t i o n a l o n g t h is l i n e

i s c e r t a in l y i m p o r t a n t i n i t s o w n r i g ht a n d m a y

l e a d t o h i g h - p e r f o r m a n c e a d a p t i v e c o n t r o l

s c h e m e s .

R e f e r e n c e s

[1] K . J . ~ , s t r6 m an d B . W i t t en m ark ,

A d a p t i v e C o n t r o l

Ad d i so n -W es l ey , Read in g , M A. , 1 9 89 ).

[2] K . J . ,~ s tr0 m an d T . H ag g lu n d , Au to m at i c t u n in g o f P ID

contro l lers ,

In s t r u m e n t S o c ie ty o f A m e r ic a

US A, 1 9 88 .

[3] K.J . ,~st r6m and T. H~igglund , An indust r ia l adap t ive

P ID co n t ro l l e r ,

P r o c . 1 9 8 9 IF A C S y m p . A d a p t i v e

Sys tem in Contro l and S igna l Process ing 1 9 8 9 , p p .

293-298 .

[4] C .C . Ha n g a n d K . J. ~ s t r0 m , Ref in em en t s o f th e

Z ieg l e r -N ich o l s t u n in g fo rm u la fo r P ID a u to - t u n er ,

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Ho u s to n ,

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[5] C.C. Hang and K.J . Ast r6m, Pract ical aspects o f PID

au to - tu n er s b ased o n r e l ay f eed b ack , Proc. Intern.

Syrup . Adapt ive Contro l o f Chemica l Processes

De nm ark , 1988 , pp . 149-154.

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Int . Symp. Adapt ive Sys tems in Contro l and S igna l

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log ic i n to t h e n o rm al -m o d e in ac t i o n ad ap t i v e P ID

contro l ler , Proc. 1992 Singapore Intern. Conf. Intelli.

Control Instru.

Singapore, Vol. 2, 1992, pp. 1357-1362.

[9 ] S .Z . He , S . Tan , C .C . Han g an d P .Z . Wan g , Des ig n o f

an on- l ine ru le-ada pt ive fuzzy co nt ro l system,

Proc.

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[10] H.S . Hoopes, W.K. Hawk and R.C. Lewis , A sel f - tun ing

co n t ro l l e r ,

ISA Trans.

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[11 ] T .W. Krau s an d T . J. May ro n , S e l f - t u n in g P ID

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fuzzy self tuning of pid controllers

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