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