automatic tuning and adaptation for pid controllers - a survey
TRANSCRIPT
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ControlEng. Practice
Vol. 1, No. 4, pp. 699-714, 1993 0967-0661/93 $6.00+ 0.00
Printed in Crib.at Britain. All rights ~served. © 1993 Pergamon Press Led
A U T O M A T I C T U N I N G A N D A D A P T A T I O N F O R P I D
C O N T R O L L E R S - A S U R V E Y
K.J. A, tr6m*, T. H/igglund*, C.C. Hang** and W.K . Ho**
Department of A utomatic Control, Lurid Institute o f Technology, S-221 O0 Lurid, Swed en
Department of Electrical En gineering, National U niversity of Singapore, Singapore
A b s t r a c t . Adaptive techniques such as gain scheduling, automatic tuning and
continuous adaptation have been used in industrial single-loop controllers for about ten
years. This paper gives a survey of the different adaptive techniques, the underlying
process models and control designs. An overview of industrial products is also
presented, which includes a fairly detailed investigation of four different adaptive
single-loop controllers.
K e y w o r d s . Adaptive control, automatic tuning, gain scheduling, PID control.
1. INTRODUCTION
Single-loop controllers have been used in indus-
try for a long time. For example, it was about
fifty year s ago that derivative action was intro-
duced in pn eumati c controllers by Taylor Instru-
ments. Industrial single-loop controllers have
gone through an interesting change in tech-
nology, from pneumatic via analog electronics
to microprocessor implementation. In spite of
these changes in technology, the functional ity of
the controllers has remained the same, namely
essentially an implementation of the standard
PID algorithm. The controllers are currently
going through an interesting phase of devel-
opment, where features like auto-tuning, gain
scheduling and adaptation are added. There is
hardly any new product announced that does
not incorporate some of these features. The
single-loop controller is therefore rapidly becom-
ing a test bench for control theory. Since the
controllers are made and used in large num-
bers, much industrial experience of using so-
phisticated control is also accumulating. One
reason for this development is the advances in
microelectronics which give cheap microproces-
sors whose computational power is continuously
increasing. Another reason is the pressure from
users and applications, and a third is the grow-
ing experience of using advanced control.
It may be asked why a similar development is
not taking place in distributed control systems.
O n e r e a s o n i s t h a t t h e d i s tr i b u t e d c o n tr o l s y s -
t e m s a r e q u i t e c o m p l i c a t e d a n d t h a t t h e i r d e v e l -
o p m e n t c y c l e i s q u i t e l o n g . T h e r e f o r e i t t a k e s a
l o n g t i m e f o r n e w f e a t u r e s t o b e i n t r o d u c e d i n
s u c h s y s t e m s . T h e r e a r e h o w e v e r i n d ic a t i o n s
t h a t t h e e x p e r i e n c e s f r o m s i n g l e - l oo p c o n t ro l l e rs
a r e m i g r a t i n g i n t o l a r g e r s y s t e m s .
I t i s t h e r e f o r e t i m e l y t o t a k e s t o c k o f t h e d e v e l -
o p m e n t t o a s s e s s t h e c u r r e n t s t a t e o f t h e a r t
a n d t o l o o k a t t h e f u t ur e . T o m a k e s u c h a n e v a l -
u a t i o n i s t h e p u r p o s e o f t h i s p a p er . T h e p a p e r
i s o r g a n i z e d a s f o l l o w s. A r e v i e w o f i n d u s t r i a l
n e e d s i s g i v e n i n S e c ti o n 2 w h i c h a l s o p r e s e n t s
s o m e a d a p t i v e t e c h n i q u e s a n d m a t c h e s t h e m t o
t h e n ee d s . M o r e d e t ai l s o n t h e t e c h n i q u e s a r e
g i v e n i n S e c ti o n s 3 a n d 4 t h a t d e a l w i t h m o d -
e l i n g a n d c o n t r o l m e t h o d s . S e c t i o n 5 g i v e s a n
o v e r v i e w o f s o m e e x i s t i n g i n d us t r i a l p r o d u c ts .
I n S e c t io n 6 f o u r p r o d u c t s w i t h d i f fe r e n t a d a p -
t i v e a p p r o a c h e s a r e d e s c r i b e d i n m o r e d e t ai l.
2. ADAPTIVE TECHNIQUES
The first general purpose adaptive controller
for industrial process control was introduced
around 1983. There are now many brands of
adaptive controllers for industrial process con-
trol and many loops being controlled by s u c h
controllers, see ( D u m o n t e t a l . , 1 9 8 9 ; Kaya and
Titus, 1988; Hess
e t a l . ,
1987; Hagglund and
Astr6m, 1991). As a result of this there is
a growing experience of using adaptive tech-
699
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700 K.J. ,~str6met al
niques. Unf ortuna tely there is also some con-
fusion in terminology. Therefore we will intro-
duce an appropriate terminology at the same
time as we describe some of the adaptive tech-
niques that have been useful.
A u t o m a t i c T u n i n g
By automatic tuning (or auto-tuning) we mean
a method where a controller is tuned automati-
cally on demand from a user. Typically the u ser
will either push a button or send a command to
the controller. Industr ial experience has clearly
indicated that this is a highly desirable and
useful feature. Automatic tuning is sometimes
called tuning on demand or one-shot tuning.
Automatic tuning can also be performed using
external equipment. These products are con-
nected to the control loop only during the tun-
ing phase. When the tuning experiment is fin-
ished, the products suggest controller parame-
ters. Since these products are supposed to work
together with controllers from different manu-
facturers, they must be provided with quite a lot
of information about the controller structure in
order to give an appropriate parame ter sugges-
tion. Such information includes controller struc-
ture (series or parallel form), sampling rate,
filter time constants, and units of the differ-
ent controller paramete rs (gain or proportional
band, minu tes or seconds, time or repeats/time).
G a i n S c h e d u l i n g
By gain scheduling we mean a system where
controller paramet ers are changed depending on
measured operating conditions. The scheduling
variable can, for instance, be the measurement
signal, controller output or an external signal.
For historical reasons the word gain scheduling
is used even if other pa ramet ers like derivative
time or integral time are changed. Gain schedul-
ing is a very effective way of controlling systems
whose dynamics change with the operating con-
ditions. Gain scheduling has, however, not been
used much because of the effort required to im-
plement it. When combined with auto-tuning,
gain scheduling is, however, very easy to use.
A d a p t i v e Co n t r o l
By adaptive control we mean a controller whose
parameter s ar e continuously adjusted to accom-
modate changes in process dynamics and distur-
bances. Adaptation can be applied both to feed-
back and feedforward control parameters. It ha s
proven particularly useful for feedforward con-
trol. The reason for this is tha t feedforward con-
trol requires good models. Adaptation is there-
fore almost a prerequisite for using feedforward
control.
J o n ~ a N u t
unknown dyn mics
I
J o n s t a n t o n t ro l le r
g ~ ~ ~ n m ~
P r K l l c t ~ l e
p a r m r c h an ge s
npc~mb~
J U n p m d c U ~ a l e
p r meter
c h a n g ~ J
Fig. 1. When to use different adaptive
techniques
The notation
a d a p t i v e t e c h n i q u e s
will be used to
cover auto-tuning, gain scheduling and adapta-
tion. Although research on adaptive techniques
has almost exclusively focused on adaptation,
experience has shown that auto-tuning and gain
scheduling have a much wider industrial inter-
est. Figure 1 illustrates when it is appropriate
to use the different techniques.
The first issue to consider is controller perfor-
mance. If the requirements are modest, a con-
troller with constant parameters and conserva-
tive tuning can be used. With higher demands
on performance other solutions should be con-
sidered. If the process dynamics are constant, a
controller with constant parameters should be
used. The parameters of the controller can be
obtained using auto-tuning. If the process dy-
namics or the nature of the disturbances are
changing it is useful to compensate for these
changes by changing the controller. If the vari-
ations can be predicted from measured signals,
gain scheduling should be used as it is simpler
and it gives superior and more robust perfor-
mance than the continuous adaptation. Typi-
cal examples are variations caused by nonlin-
earities in the control loop. Auto-tuning can be
used to build up the gain schedules. There are
also cases where the variations in process dy-
namics are not predictable. Typical examples
are changes due to unm easur able variations in
raw material, wear, fouling etc. These variations
cannot be handled by gain scheduling but must
be dealt with by adaptation. An auto-tuning
procedure is often used to initialize the adap-
tive controller. It is then sometimes called pre-
tuning or initial tuning.
Feedforward control deserves special mention-
ing. It is a very powerful method for dealing
with measurable disturbances. Use of feedfor-
ward control requires however good models of
process dynamics. It is difficult to tune feed-
forward control loops auto matically on demand,
since the operator often cannot manipul ate the
disturbance used for the feedforward control.
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Autom atic Tuning and P ID Controllers
701
ontroller
des ign Model
° - , ' - - ' F -
~ . _ K p
0.63 Kp
~ - y
/
B C
L
v
F ig. 2. Bloc k diagram of indirect
s y s t e m s
T o t u n e t h e f e e d f o r w a r d c o n t r o l l e r i t i s t h e r e -
f o re n e c e s s a r y t o w a i t f or a n a p p r o p r i a t e d i s t u r-
b a n c e . A d a p t a t i o n i s t h e r e f o r e p a r t i c u l a r l y u s e -
f u l f o r t h e f e e d f o r w a r d c o n t r o ll e r .
T h e t e c h n i q u e s u s e d f o r a u t o - t u n i n g a n d a d a p -
t a t i o n a r e v e r y s im i l a r. I n b r e a d t e r m s , o n e c a n
d i s t i n g u i s h b e t w e e n d i r e c t a n d i n d i r e c t m e t h -
o d s . I n a d i r e c t m e t h o d c o n t r o l l e r p a r a m e t e r s
a r e a d j u s t e d d i r e c t l y f r o m d a t a i n c l o s e d - l o o p
o p e r a t i o n . I n i n d i r e c t m e t h o d s a m o d e l o f t h e
p r o c e s s i s f i r s t d e v e l o p e d f r o m o n - l i n e d a t a . T h e
c o n t r o ll e r p a r a m e t e r s a r e th e n d e t e r m i n e d f r om
t h i s m o d e l .
T h e r e i s a la r g e n u m b e r o f m e t h o d s a v a i l a b l e
b o t h f o r d i r e c t a n d i n d i r e c t m e t h o d s . T h e y c a n
c o n v e n i e n t l y b e d e s c r i b e d in t e r m s o f t h e m e t h -
o d s u s e d f o r m o d e l i n g a n d c o n t r o l d e s i g n .
I n t h e d i r e c t m e t h o d s t h e k e y i s su e s a r e t o
f i nd s u i t a b l e f e a t u r e s t h a t c h a r a c t e r i z e r e l e v a n t
p r o p e r t i e s o f t h e c l os e d - lo o p s y s t e m a n d a p p r o-
p r i a t e w a y s o f c h a n g i n g t h e c o n t r o ll e r p a r a m e -
t e r s s o t h a t t h e d e s i r e d p r o p e r t i e s a r e o b t a i n e d .
T h e i n d i r e c t s y s t e m s c a n a ll b e r e p r e s e n t e d b y
t h e b l o c k d i a g r a m i n F i g u r e 2 . T h e r e is a p a r a m -
e t e r e s t i m a t o r t h a t d e t e r m i n e s t h e p a r a m e t e r s
o f t h e m o d e l b a s e d o n o b s e r v a t i o n s o f pr o c e s s
i n p u t s a n d o u t p u t s . T h e r e i s a ls o a d e s ig n b l o c k
t h a t c o m p u t e s c o n t r o l l e r p a r a m e t e r s f r o m t h e
m o d e l p a r a m e t e r s . I f t h e s y s t e m i s o p e r a t e d a s
a t u n e r t h e p r o c e s s i s e x c it e d b y a n i n p u t s ig -
n a l. T h e p a r a m e t e r s c a n e i t h e r b e e s t i m a t e d r e -
c u r s i v e l y o r i n b a t c h m o d e . C o n t r o l le r p a r a m e -
t e r s a r e c o m p u t e d a n d t h e c o n t r o l le r i s c o m m i s -
s io n e d . I f t h e s y s t e m i s o p e r a t e d a s a n a d a p -
t i v e c o n tr o ll e r, p a r a m e t e r s a r e c o m p u t e d r e c u r-
s i v e ly a n d c o n t r o ll e r p a r a m e t e r s a r e u p d a t e d
w h e n n e w p a r a m e t e r v a l u e s a re o b t a in e d .
3 . M O D E L I N G
A m o d e l o f a s y s t e m c a n b e a n y t y p e o f a b s t r a c t
d e s c r i p t i o n t h a t c a p t u r e s u s e f u l r e l e v a n t f e a -
t u r e s o f a p r o c e ss . M o d e l i n g c a n m e a n m a n y d if -
f e r e n t t h i n g s , f r o m t h e e x t r a c t i o n o f so m e s i m p l e
Fi g . 3 . D e t e r m i n i ng a f ir s t o r de r p l us de a d- t ime mo d e l
f r o m a s t e p r e s p o ns e . T i me c o ns t a n t T c a n be
o b t a i n e d e it h e r a s t h e d i s t an c e A B o r t he d i s t a nc e
AC
f e a t u re s o f a t r a n si e n t r e s p o n s e t o t h e d e v e l o p-
m e n t o f a t r a d it i o n a l c o n t r o l m o d e l i n t e r m s o f a
t r a n s f e r f u n c t i o n o r a n i m p u l s e r e s p o n s e . S o m e
m o d e l s t h a t a r e u s e d i n a d a p t i v e P I D c o nt r o l le r s
w i ll n o w b e d e s c r ib e d .
3 .1 T i m e - D o m a i n M o d e l s
T y p i c a l t i m e - d o m a i n f e a t u r e s a r e s t a t i c g a i n ,
d o m i n a n t t i m e c o n s t a n t
a n d d o m i n a n t
d e a d
t im e . T h e y c a n a l l b e d e t e r m i n e d f r o m t h e s t e p
r e s p o n s e o f t h e p r o c e s s , s e e F i g u r e 3 . S t a t i c g a i n
Kp,
d o m i n a n t t i m e c o n s t a n t T a n d d o m i n a n t
d e a d t i m e L c a n b e u s e d t o o b t a i n a f i r st o rd e r
p l u s d e a d - t i m e m o d e l f o r t h e p r o c e s s a s g iv e n i n
E q u a t i o n ( 1 ) .
Kp e_,L
(1)
G p s) = 1 + s T
T h e c o n s t r u c t i o n s i n F i g u r e 3 a r e s e n s i t i v e
b e c a u s e t h e y a r e b a s e d o n a f e w v a l u e s o f t h e
s t e p r e s p o n s e o n l y .
A n o t h e r m e t h o d , w h i c h i s b a s e d o n d e t e r m i n a -
t io n o f a r e a s c a n b e u s e d , s e e F i g u r e 4 . I n t h i s
m e t h o d , g a i n K p i s f i r s t d e t e r m i n e d f r om t h e
s t e a d y- s t a t e v a l u e o f t h e s t e p r e s p o ns e . A r e a 4 o
i s t h e n d e t e r m i n e d . T h e a v e r a g e r e s i d e n c e t i m e
o f t h e s y s t e m i s t h e n
L + T = A °
o
2 )
g .
i i i i i i i i i i i i i i i l i i i i i i A : : : i i i i i i i i i i i i i i i i ...
L+T
Fi g . 4 . D e t e r m i n i ng a f i r s t o r de r p l us de a d- t i me mo de l
from a step response
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702 K.J./~.str6met al.
Process
P I I
F i g 5 T h e r e l a y a u t o t u ne r I n th e t u n i n g m o d e t h e
p r o c e s s i s c o n n e c t e d t o r e l a y f e e d b a c k
Area A1, which is the area under the step
response up to time L + T, is then determined
and T is given by
/
N a )/G i )
Im
Re
F i g . 6 . T h e n e g a t i v e i n v e r s e d e s c r i b i n g f u n c t io n o f a
relay with hysteresis - 1I N a ) and a Nyquist
c u r v e
G iw)
A 1 e l
T = ~ (3)
This method is less sensitive to high-frequency
disturbances, because it is based on area deter-
mination. More details can be found in (./kstr~m
and H~igglund, 1988; Nishikawa
e t a ] . ,
1984).
Highe r order models can also be obtained from a
step response method, see, for instance, (Seborg
e t a ] . ,
1989).
A d i s cr e t e t i m e m o d e l a s g i v e n i n E q u a t i o n 4 )
c a n a l s o b e u s e d t o d e s c r i b e t h e p r o c es s .
b o + b ] z + . . . + b n _ l z n -1
Gp(z) = ao + alz +. . . + a . z
( 4 )
There are m any methods available to determine
the p aram eter s in Equation (4), for instance the
recursive least squares method.
3 .2 F r e q u e n c y - D o m a i n M o d e ls
Typical frequency-domain characteristics are
amplitude marg in, phase margin, Mp-value, ul-
timate gain, and ultimate period. These quan-
tities are all related to simple properties of the
Nyquist curves of the open-loop transfer func-
tion or of the loop transfer function.
l
R e l a y Exc i t a t i on
Frequency-domain characteristics can be deter-
mined from experiments with relay feedback in
the following way. When the controller is to be
tuned, a relay with hysteresis is introduced in
the loop, and the PID controller is temporar-
ily disconnected, see Figure 5. For large classes
of processes, relay feedback gives an oscillation
with period close to t he ult ima te f requency w~s0.
The gain of the transfer function at this fre-
quency is also easy to obtain from amplitude
measu rements. Details and conditions are given
in (~str~m and H/igglund, 1984; /~str6m and
H a g g l u n d , 1 9 8 8 ; H a n g a n d A s t r 6 m , 1 9 8 8 ) . D e -
s c r i b i n g f u n c t i o n a n a l y s i s c a n b e u s e d t o de t e r -
m i n e t h e p r o c e s s c h a ra c te r i st i cs . h e d e s c r i b i n g
f u n c t i o n o f a r e l a y w i t h h y s t e r e s i s i s
where d is the relay amplitude, e the relay
hysteresis and a is the amplitude of the input
signal. The negative inverse of this describing
function is a straight line parallel to the real
axis, see Figure 6. The oscillation corresponds
to the point where the negative inverse describ-
ing function crosses the Nyquist curve of the
process, i.e. where
1
G iw ) = - N a )
Since
N a )
is known,
G i w )
can be determined
from the amplitude a and the frequency w of the
oscillation.
Cor re la t i on M ethod
In this method, a small pseudo random binary
sequence (PRBS) test signal u t ) is injected and
the resultant process output
y t )
is logged. The
cross correlation, ~.y(V), between
u t )
and
y t )
is then used to compute the impulse response
g v )
of the process (Hang and Sin, 1991) as
f o l l o w s
hffi0
where A is the amplitude of the PRBS signal,
h is the sampling interval and
N h
is the pe-
riod of the PRBS signal. The impulse response
computed is then numerical ly transformed into
its frequency response from which the ultimate
gain, ultimate period, static gain and normal-
ized dead time of the process can be determined.
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AutomaticTuning and PID Controllers
3 . 3 C o n v e r s i o n
to Time- Domain Models
The es timated frequency-domain model in terms
of the full frequency response or at least two
points on the Nyquist curve may be used to de-
rive a corresponding time-domain model. This is
useful if the controller design is done in the time
domain (pole-placement design, Ziegler-Nichols
step response design, IMC design, etc. to be dis-
cussed in Section 4). The following formulas (Ho
et a]. ,
1992), for instance, may be used to derive
useful models from knowledge of the ultimate
gain (Ku), ultimate period (T.), and the static
gain
( K p ) :
Kp e_sL 1
G p ( s ) = I + sT 1
o r
where
Kp e_sL2
Gp s)
i + s T 2 ) 2
Tu ~/ KuK p) 2 1
~= ~
T . ( 2 z T z ~
L I= ~ z - t an -1 - -~ - ]
a n d
T.
T 2 = ~ z v / K ~ K p - 1
T: ( 2zT2~Lz = ~ z- 2t an -z--~-- ]
7 0 3
o f t e n g i v e s e t - p o i n t e s p o n s e s t h a t a r e t o o os c il -
l at or y. O n t h e o t h e r h a n d , t u n i n g t o g i v e g o o d
s e t - p oi n t r e s p o n s e s o f t e n g i v e s s l u g g i s h l o a d -
d i s t u r ba n c e s r e s po n s e . T h e f a c t o r f l w h e n s e t
t o l e ss t h a n o n e c a n r e d u c e t h e s e t - po i n t r e-
s p o n s e o v e r s h oo t f o r t u n i n g t h a t o p t i m i ze 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 . I n s o m e l i te r at u re ,
t h is f o r m o f t h e P I D c o n tr o l l a w i s k n o w n a s
a t w o - d e g r e e - o f - f r e e d o m P I D c o n t r o l le r S h i g e -
m a s a e t a ] . 1 9 8 7 ; T a k a t s u e t a l . 9 9 1 ) . S e v e r a l
i n d u s t r i a l c o n t r o l l e r s a v e f l = 0 .
T h e s e r ie s f o r m o f t h e P I D c o n t r ol l a w , w i t h o u t
f il te r n t h e d e r i v a t i v e t e r m , i s g i v e n i n E q u a -
t i o n 8 ) .
G, (s) = K 1 + (1 + s T y ) (8)
It is not possible to obtain complex zeros from
this form of the controller. However, it is some-
times claimed that the series form is easier to
tune.
This section describes some methods for deter-
mining the param eters of a PID controller. They
can be classified broadly into direct and indirect
methods. The direct control design techniques
are simply prescriptions that tell how the con-
troller parameters should be changed in order
to obtain the desired features. The indirect con-
trol design methods give controller paramet ers
in terms of model parameter s.
4. CONTROL
D E S I G N
The PID control law can be implemented in ei-
the r the parallel or the series form. The parallel
form with derivat ive action on filtered measure-
ment signal to avoid derivative kick is given
b e l o w .
1 i
= K e + ~i e d t - T d
e = y , - y (6 )
d y f N
d t = ~ ( y - y f )
The controller output, process output and set
point are u, y and y,, respectively. A weighting
factor fl can be placed on the set point as shown
in Equation (7).
u = K ( f l y , - y ) + ~ e d t - T d d t }
(7)
The factor fl when set to less than one can
reduce the set-point response overshoot with-
out affecting the load-disturbance response. In
a conventional PID controller with fl = 1, tun-
ing to give good load-disturbance responses will
4 .1 D i r e c t M e t h o d s
A v a s t m a j o r i t y o f t h e P I D c o n t r o ll e r s i n t h e
i n d u st r i e s a r e t u n e d m a n u a l l y b y c o n t r ol e n g i -
n e e r s a n d o p er a to r s . T h e t u n i n g i s d o n e b a s e d
o n p a s t e x p e r i e n c e s a n d h e u r i st i c s. y o b s e r v -
i n g t h e p a t t e r n o f t h e c l o s e d - l o o p r e s p o n s e t o
a s e t - p o i n t c h a n g e , t h e o p e r a t o r s u s e s h e u r i s -
t ic s to d i r ec t l y a d j u s t t h e c o n t ro l l er p a r a m e -
t er s. T h e h e u ri s t ic s a v e b e e n c a p t u r e d i n t u n -
i n g c h a r ts t h a t s h o w t h e r e s p o n s es o f t h e s y s -
t e m f or di f fe r e nt p a r a m e t e r v a l u e s . A c o n s id -
e r a b l e i n s i g h t i n t o c o n t r ol l e r t u n i n g c a n b e
d e v e l o pe d b y s t u d y i n g s u c h c h a r t s a n d p e r -
f o r m i n g s i m u l a t io n s . T h e h e u r i s ti c r u l e s h a v e
a l s o b e e n c a p t u r e d i n k n o w l e d g e b a s e s i n t h e
f o r m o f r u l es , b o t h c r i s p a n d f u z z y, s e e A n -
d e r s o n e t a ] . 1 9 8 8 ; B r i s t o l , 1 9 7 7 ; B r i s t o l , 9 8 6 ;
H i g h a m , 1 9 8 5; K l e in e t a ] . 1 9 91 ; K r a u s a n d
M y r o n , 1 9 8 4; M a r s i k a n d S tr ej c, 1 9 8 9; P a g a n o ,
1 9 9 1; Y a m a m o t o , 1 9 9 1 ) . R u l e s o f t h i s t y p e a r e
u s e d i n s e v er a l c o m m e r c i a l t u n e r s . M o s t p r o d -
u c t s w i l l a i t f o r s e t - p oi n t h a n g e s o r m a j o r l o a d
d i s t ur b a n ce s . W h e n t h e s e o cc u r , p r o p e rt i e s i k e
d a m p i n g , o v e r s h oo t , p e r i o d o f o sc i ll at i on s n d
s t at i c a i n s a r e e s t i m a t e d . B a s e d o n t h e s e p r o p -
e rt ie s, u l e s f o r c h a n g i n g t h e c o n tr o l l e r a r a m e -
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704 K.J. ~,str6m et al
T a b l e
1. Controller parameters o btained by the Ziegler-
Nichols step response method.
Controller K Ti Td Tp
P 1 / a 4 L
PI 0.9a 3L 5.7L
PID
1 2 / a 2 L L / 2
3.4L
ters to meet desired specifications are executed.
An example will be described later in Section 6.
4 .2 I n d i r e c t M e t h o d s
Z i e g l e r . N i c h o l s S t e p R esp o n se M e t h o d
This method is based on a registration of the
open-loop step response of the system, which
is characterized by two parameters. The point
where the slope of the step response has its max-
imum is first determined, and the tangent at
this point is drawn. The intersections between
this tangent and the coordinate axes give the
two parameters a and L, see Figure 3. A sim-
ple model of the process which can be derived
from these parameters is given by the transfer
function
G p s ) = a - , Le . (9)
Ziegler and Nichols have given PID parameters
directly as functions of a and L, see (Ziegler
and Nichols, 1942). These formulas are given
in Table 1. An estimate of the period Tp of the
dominant dynamics of the closed-loop system
is also given in the table. The formulas will
roughly give quarter amplitude damping ratio
for load-disturbance response for processes that
can be modeled by Equation (9). In practice
some form of fine tuni ng has to be done.
There are several design methods which are
similar to the Ziegler-Nichols step response
method in the sense that they are based on a
step response experiment combined with a tab le
that relates the controller parameters to the
characteristics of the step response, see (Hang
eta]., 1979; Hazebroek and van der Waerden,
1950; Miller e t a L , 1967). The most common
method is the Cohen-Coon method (Cohen and
Coon, 1953) based on a first order plus dead-
time model.
O p t i m i z a t i o n Techn iques
There are other tuning formulas derived to opti-
mize certain criteria such as the integrated ab-
solute error (IAE), the in tegrat ed time absolute
error (ITAE), the integr ated square error (ISE),
etc. They are mos tly done for the first-order plus
dead-time model as given in Equation (1). A
word of caution is in order: Formulas derived
for the first-order plus dead-time model may not
° I
.
0
rocess output and s t point
Control signal
~ 2 ~ io l~
Fig. 7. IAE criteria optimized for first order plus dead-
t i m e process applied to a higher order process .
K = 4.3, T~ = 2.0
give good results for higher order systems. An
example is given in Figure 7. A PI controller for
a process with the tran sfer function
e -o.3
G p s )
-
s
+ 1) 2 i0)
is determined by first finding parameters L
and T by the maxi mum slope method and then
applying Shinskey's closed-loop tun ing formula
for minimum IAE, see (Shinskey, 1988). It is
evident that the tuning performance as shown
in Figure 7 is too oscillatory.
P o l e
P l a c e m e n t
If the process is described by a low-order trans-
fer function, a complete pole-placement design
can be performed. Consider the process de-
scribed by a second-order model in Equation
(11).
a s) = Kp i i )
1 + sT1)(1 + s T 2 )
This model has three parameters. By using a
PID controller, which also has three paramete rs,
it is possible to arbitrarily place the three poles
of the closed-loop system. The t ran sfe r function
of the PID controller on parallel form can be
written as
G o s ) =
g(1 +
s T i + s 2 T i T d )
(12)
s Z
The characteristic equation of the closed-loop
system becomes
s 3 + s 2 1 1 K p K T d ~
T II + ~22 + T 1 T 2 ] +
( 1 x . K = o .
s \ T--~ + T1T2} + TiT1---- -~2
1 3 )
A s u i t a b l e c l o s e d - l o o p c h a r a c t e r i s t i c q u a t i o n o f
a
t h i r d - o r d e r s y s t e m i s
(s + am)( s 2 + 2~'~os + 0)2) = 0 (14)
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AutomaticTuning and PID Controllers
w h i c h c o n t a i n s t w o d o m i n a n t p o l e s w i t h r e la t iv e
d a m p i n g (~ ) n d f r e q u e n cy ( w ) , a n d a r e a l p o l e
l o c a t e d i n - s e a . I d e n t i f y i n g t h e c o e f fi c i en t s n
t h e s e t w o c h a r a ct e r i s t ic q u a t i o n s g i v e s
1 1 K p K T d = w ( a + 2~')
T-xl + T22 +
T1----~2
1 K p K
- - + = w 2 (1 + 2 ~ a ) ( 15 )
7 , 7 2 7 1 7 2
K p K = a W 3
Ti T 1T2
These three equations
rameters K, Ti and
T d.
K =
=
T d =
d e t e r m i n e t he P I D p a -
T h e s o l u t i o n s
T1T2eo2(1 + 2~'a) - 1
K p
T 1 T 2 w 2 ( 1 +
2~a) - 1
16)
T 1 T 2 a w 3
T1T2oJ a + - 7 , -
T 1 T 2 ( 1 +
2~'a)co 2 - 1
N o t i c e t h a t p u r e P I c o n t r o l i s o b t a i n e d f o r
7'1+7 2
W c = T1T2(~ + 2~') (17)
Notice also tha t the choice of w may be critical.
The derivative time is negative for w < o)c.
The frequency w~ thus gives a lower bound
to the bandwidth. Also notice that the gain
increases rapidly with co. The upper bound
to the bandwidth is given by the validity of
the simplified model. Automatic tuning of PID
controllers based on pole placement design is
discussed in (Gawthrop, 1986).
C a n c e l l a t i o n o f P r o c e ss P o l e s
The PID controller has two zeros. A popular
design method is to choose these zeros so that
they cancel the two dominant process poles.
The method is popular since it is simple and
gives good response to set-point changes, see
(Haalman, 1965). The method will, however,
often give poor response to load disturbances
(/~,str~im and H~igglund, 1988). An exception to
this is in the case of large dead time, in which
case the settling time is already quite long
relative to the time constant being cancelled (Ho
e t a l . ,
1992).
When pole-zero cancellation is safe to apply,
the simple IMC (Internal Model Control) based
tuning formulas (Morari and Zafiriou, 1989;
Seborg e t a] . , 1989) based on a first order plus
dead-time model--as shown in Table 2--may be
used.
The controll er gain may be chosen to be aggres-
sive or conservative by va rying the value of 2,
which is equivalent to the desired closed-loop
time constant.
705
T a bl e 2 .
IMC based PID controller uning formulas.
C o n tr o l le r K T i T d S u g g e s t e d
1 2 T + L
P I
7 ~ , 2 - N Y - T + L / 2
2 > 0 . 2 T
~t > 1.7L
1 2T+L TL
PID K~p 2-'l -E T + L /2 ~ Jt >
0.2T
>
0.25L
Z i e g l e r . N i c h o l s F r e q u e n c y R e s p o n ~ M e t h o d
This method is based on a very simple charac-
teriza tion of the process dynamics. The design is
based on knowledge of the point on the Nyquist
curve of the process tran sfer function G where
the Nyquist curve intersects the negative real
a x is . F o r h i s to r i ca l e a s o n s t h i s p o i n t is c h a r a c -
t e r i z e d b y t h e p a r a m e t e r s K . a n d T u , w h i c h a r e
c a l l e d t h e u l t i m a t e g a i n a n d t h e u l t i m a t e p e -
r io d. T h e s e t w o q u a n t i ti e s c a n b e o b t a i n ed f o r
m a n y s y s t e m s b y u s i n g p r o po r t io n a l f e e d b a ck
a n d c h o o s i n g s u f f i ci e n tl y i g h g a i n u n t i l c o n -
t i n u o u s c yc l i n g c c u rs . T h e u l t i m a te g a i n K , i s
t h e n g i v e n b y t h e p r o p o rt i o n a l g a i n a n d t h e u l-
t i m a t e p e r i o d T,. i s g i v e n b y t h e p e r i o d o f t h e
o s ci l la t io n s. t i s, h o w e v e r , m o r e p r a c t i c a l o o b -
t a i n t h e s e p a r a m e t e r s u s i n g t h e r e l a y f e e d b a ck
e x p e r i m e n t o r t h e c o r re l a ti o n e t h o d p r e s e n t ed
i n S e c t i o n 3.
T h e Z i e g le r - N i ch o l s o r m u l a s a r e g i v e n i n T a -
b l e 3 . A n e s t i m a te o f t h e p e r i o d T p o f t h e d o m i -
n a n t d y n a m i c s o f t h e c l os e d- l o op y s t e m i s a l s o
g i v e n i n t h e ta b le . h e s e f o r m u l a s w e r e d e r i v e d
t o g i v e qu a r te r a m p l i t u d e d a m p i n g f or l o a d-
d i s t u r b a n c e r e s p o n s e , s e e ( Z i e g l e r n d N i c h o l s ,
1 9 4 2 ) . B o t h T p a n d t h e d a m p i n g o b t a i n e d w i t h
t h e Z i e g l er - N i c ho l s t u n i n g r u l e c a n , h o w e v e r ,
d i ff e r s i g n i f i ca n t l y f r o m t h e s e e x p e c t e d v a l -
u e s . R e f i n e d Z i e g l e r - N i c h o l s o r m u l a s t o r e d u c e
t h e s e d i s c r e pa n c e s a n d t o i m p r o v e t h e t u n i n g
p e r f o r m a n c e a r e g i v e n i n C H a n g e t a ] . , 1 9 9 1 ) .
P h a s e a n d A m p l i t u d e M a r g i n s
Phase and amplitude margins design methods
give a measur e of the robustness of the system.
It is known from classical control that
t h e
damping of the system is related to the phase
margin. The phase margin is given by
¢ m
: z + a r g G p i w g ) G c i w g )
( 1 8 )
T a b l e 3. C o n t r o l le r a r a m e t e r s b t a i n e d y t h e Z i eg l er -
N i c h o ls r e q u e n c y e s p o n s e e t h o d .
C o n t r o l l e r
K T~ Ta Tp
P 0 . S K . T ,
PI OAK, 0.8T, 1.4T,
PID
0 . 6 K. 0 . 5T u 0 . 1 2T . 0 . 8 57 .
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706 K.J. ,/kstr6m
et al.
w h e r e
IGp(ico )Gc(iCag)l
= 1 a n d w g i s t h e g a i n
c r o ss - o v e r fr e qu e n cy . T h e a m p l i t u d e m a r g i n o r
g a i n m a r g i n i s g i v e n b y
1
Am = iGp(iO)u)Vc(iO), ,) I
(19)
w h e r e o ), i s t h e u l t i m a t e f r e q u e n c y o r t h e p h a s e
c ros s -ove r f r e que nc y .
I f t h e u l t i m a t e g a i n ( K ,.) a n d u l t i m a t e p e r io d
( Tu ) a r e k n o w n , t h e f o ll o w i n g s i m p l e d e s i g n
f o r m u l a ( / ~ s t r ~ m a n d H ~ i g g l u n d , 1 9 8 8 ) m a y b e
u s e d t o a c h i e v e a p r e - s p e c if i ed p h a s e m a r g i n era:
K = K , c osC m
7 , , ( ~ /1 + t an 2 e r a ) ( 2 0 )
~ = ~
t a n C m +
T ~ = T d 4 .
T h e a b o v e d e s i g n w o r k s w e l l fo r p r o c es s e s w i t h
r e l a t i v e l y s m a l l d e a d t i m e . O t h e r w i s e , e s p e -
c ia ll y w h e n t h e d e a d t i m e i s d o m i n a n t , th e a m -
p l i t u d e m a r g i n m a y b e p o o r a l t h o u g h t h e p r e -
s p e c if ie d p h a s e m a r g i n i s a c h i e v e d . I f a f i r s t o r-
d e r p l u s d e a d - t i m e p r o c e s s m o d e l ( E q u a t i o n 1 )
i s o b t a i n e d , t h e f o l l o w i n g d e s i g n f o r m u l a ( H o
e t a / . , 1 9 9 2 ) m a y b e u s e d t o a c h i e v e a p r e s p e c i-
f le d a m p l i t u d e m a r g i n Am:
z T
I : o -
2 A m K p L (21)
T I = T
F o r t h i s m e t h o d , t h e p h a s e m a r g i n is c o n s i s te n t
w i t h t h e a m p l i t u d e m a r g i n a s g i v e n b y t h e
fo l l owi ng re l a t i o n (Ho e t M. , 1992):
z ( 1 - 1 l A i n ) . (22)
T h e a b o v e s i m p l i f i e d d e s i g n i s a c h i e v e d u s i n g
p o l e- z e ro c a n c e l l a t io n f o r a f i r s t o r d e r p l u s d e a d -
t i m e m o d e l . A c o r r e s p o n d i n g s e t o f f o r m u l a c a n
a l s o b e o b t a i n e d f o r a s e c o n d - o r d e r p l u s d e a d -
t i m e p r o c e s s m o d e l . T h e y w o r k w e l l w h e n t h e
p r o c e s s d e a d t i m e i s s u b s t a n t i a l . F o r a l a g
d o m i n a n t p r o c e s s , p o l e - z e r o c a n c e l l a t i o n i s n o t
r e c o m m e n d e d a n d h e n c e s u i t a b l e m o d i fi c at io n s
s h o u l d b e m a d e ( H o e t a l. , 1 9 9 2 ) .
5 . O V E R V I E W O F I N D U S T R I A L
P R O D U C T S
C o m m e r c i a l P I D c o n t r o l l e r s w i t h a d a p t i v e t e c h -
n i q u e s h a v e b e e n a v a i l a b l e s i n ce t h e b e g i n n i n g
o f t h e e i g h t i e s . I n t h i s s e c ti o n , t h e s e p r o d u c t s
wi l l be c l a s s i f i e d wi t h r e spe c t t o t he i r a pp l i c a -
t i o n s a n d a d a p t i v e t e c h n i q u e s u se d . I n t h e n e x t
s e c t io n , s o m e o f th e p r o d u c t s w i ll b e d e s c r i b e d
i n m o r e d e t a i l .
I n t h e f o l l ow i n g w e w i l l d i s t i n gu i s h b e t w e e n
t e m p e r a t u r e c o n t r o l l e r s a n d p r o c e s s c o n t ro l l e rs
T e m p e r a t u r e c o nt r ol l er s r e p r i m a ri l y de s i g n e d
f or t e m p e r a t u r e c o n t r ol w h e r e a s p r o c e s s c o n -
t ro ll er s a r e s u p p o s e d t o w o r k i n n o t o n l y t e m -
p e r a t u r e c o n t r o l l o o p s b u t a l s o i n o t h e r l o o p s
i n t h e p r o c e s s i n d u s t r y s u c h a s f lo w p r e s s u r e
l e v el a n d p H c o n t r o l l o op s .
T e m p e r a t u r e C o n t r o l l e r s
A l a r g e n u m b e r o f P I D c o n t r o ll e rs a r e d e -
s i g n e d s p e c if i ca l ly f or t e m p e r a t u r e c o n t r o l a p p li -
c a t io n s . T h e s e c o n t r o l l e r s a r e n o r m a l l y c h e a p e r
t h a n p r o c es s c o n t r ol le r s . A u t o m a t i c t u n i n g a n d
a d a p t a t i o n i s e a s i e r to i m p l e m e n t in t e m p e r -
a t u r e c o n t ro l le r s , s in c e m o s t t e m p e r a t u r e c o n -
t r o l l o o p s h a v e s e v e r a l c o m m o n f e a tu r e s . T h i s
i s t h e m a i n r e a s o n w h y a u t o m a t i c t u n i n g h a s
b e e n i n t r o d u c e d m o r e r a p i d l y i n t e m p e r a t u r e
c on t ro l l e r s .
I n t e m p e r a t u r e c o n t r o l l o o p s , a s e r i o u s n o n l i n -
e a r i t y m a y r e s u l t i f t h e h e a t i n g a n d c o o li n g d y -
n a m i c s a r e d i f f e r en t . T h i s n o n l i n e a r i t y s h o u l d
b e t r e a t e d b y g a i n s c h e d u l i n g . T h e t e m p e r a t u r e
c o n t r o l l e rs h a v e s o m e t i m e s a f a c i l it y to s h i f t b e -
t w e e n d i f fe r e n t c o n t r o ll e r p a r a m e t e r s w h e n t h e
c o n t r o l a c t i o n s h i f t s b e t w e e n h e a t i n g a n d c o o l -
ing.
P r o c e s s
C o n t r o l l e r s
S i n c e t h e p r o c e s s e s t h a t a r e c o n t r o l l e d w i t h
p r o c e s s c o n t r o l le r s m a y h a v e l a r g e d i f fe r e n c e s i n
t h e i r d y n a m i c s , t u n i n g a n d a d a p t a t i o n b e c o m e s
m o r e d i f fi cu l t c o m p a r e d t o t h e p u r e t e m p e r a t u r e
c o n t r o l l o o ps . O n e o f t h e f i r s t a d a p t i v e p r o c e s s
c o n t r o l le r s w a s r e l e a s e d f r o m L e e d s & N o r t h r u p
C o i n 1982 , s e e (Ha wk, 1983) .
I f a t e m p e r a t u r e c o n t r o l l e r i s a p p l i e d t o , f o r
i n s t a n c e , a p r e s s u r e l o o p , t h e b u i l t - i n t u n i n g
p r o c e d u r e m a y b e v e r y p o o r , s i n c e p r e s s u r e
c o n t r o l l o o p s n o r m a l l y a r e m u c h f a s t e r a n d
m o r e s e n s i ti v e t h a n t e m p e r a t u r e c o n tr o l l oo p s.
T h e f o l lo w i n g p r e s e n t a t i o n w i ll f o c u s o n p r o c e ss
c on t ro l l e r s .
I n s e c t i o n 2 , d i ff e r e n t a d a p t i v e t e c h n i q u e s w e r e
d i s cu s s e d . T h e t e c h n i q u e s a r e : A u t o m a t i c t u n -
i n g , g a in s c h e d u l i n g a n d a d a p t i v e f e e db a c k a n d
fe e dforw a rd c on t ro l . In T a b l e 4 , a c o l l e c t i on o f
p r o c e s s c o n t r o l l e rs i s p r e s e n t e d ~ t o g e t h e r w i t h
i n f o r m a t i o n a b o u t t h e i r a d a p t i v e t e c h n i q u e s .
A u t o m a t i c tu n i n g i s t h e m o s t c o m m o n a d a p t i v e
t e c h n i q u e i n i n d u s t r i a l p r o d u c t s . T h e u s e fu l -
n e s s o f t h i s t e c h n i q u e is a l s o o b v i o u s f ro m F i g -
u r e I . T h e a u t o - t u n i n g p r o c e d u r e s a r e n e c e s s a r y
t o o b t a i n a r e a s o n a b l y c o m f o r t a b l e o p e r a t i o n o f
t h e o th e r a d a p t i v e t e c h n i qu e s . o s t a u t o - t un i n g
p r o c e d u r e s a r e b a s e d o n s t e p r e s p o n s e a n a ly s i s .
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Au t o m a t i c Tu n i n g a n d P I D Co n t r o l l e rs
T a b l e 4 . E x a m p l e s o f i n d u s t r i a l a d a p t i v e p r o c e s s c o n t r o ll e r s .
707
M a n u f a c t u r e r C o n t r o l l e r A u t o m a t i c G a i n
t u n i n g s c h e d u l i n g
A d a p t i v e A d a p t i v e
f e e d b a c k f e e d f o r wa r d
Ba i l e y Co n t r o l s CLC0 4 S t e p Y e s
C o n t r o l T e c h n i q u e s E x p e r t c o n t r o l le r R a m p s -
F i s h e r C o n t r o l s D P R g 0 0 R e l a y Y e s
DPR9 1 0 Re l a y Y e s
F o x b o r o E x a c t S t e p -
F u j i CC-S : PN A 3 S t e p s Y e s
H a r t m a n n & B r a u n P r o t r o n ic P S t e p -
Di g i t r i c P S t e p -
Ho n e y we l l UDC 6 0 0 0 S t e p Y e s
Sa t t C o n t r o l ECA4 0 Re l a y Y e s
ECA 4 0 0 Re l a y Y e s
S i e m e n s S I P A R T D R 2 2 S t e p Y e s
To s h i b a TOSD I C-2 1 5 D PRBS Y e s
EC3 0 0 PRBS Y e s
T u r n b u l l C o n t r o l S y s t e m s T C S 6 3 5 5 S t e p s -
Y o k o g a wa SLPC-1 7 1 ,2 7 1 S t e p Y e s
SLPC-1 8 1 ,2 8 1 S t e p Y e s
M o d e l b a s e d
M o d e l b a s e d
M o d e l b a s e d M o d e l b a s e d
R u l e b a s e d
R u l e b a s e d
M o d e l b a s e d M o d e l b a s e d
M o d e l b a s e d
M o d e l b a s e d
M o d e l b a s e d
R u l e b a s e d
M o d e l b a s e d
Gain scheduling is often not available in the
controllers. This is surprising, since gain sched-
uling is found to be more useful than continuous
adaptation in most situations. Furthermore, the
technique is much simpler to implement than
automatic tuning or adaptation.
It is interesting to see that many industrial
adaptive controllers are rule-based instead of
model-based. The research on adaptive control
at universities has been almost exclusively fo-
cused on model-based adaptive control.
Adaptive feedforward control is seldom provided
in the industrial controllers. This is surprising,
since adaptive feedforward control is known
to be of great value. Furthermore, it is easier
to develop robust adaptive feedforward control
than adaptive feedback control.
6 . S O M E P I D C O N T R O L L E R S W IT H
ADAPTIVE TECHNIQUES
In this section, some industrial adaptive PID
controllers will be presented. The controllers
presented are the Foxboro EXACT (760/761),
which uses step response analysis for automatic
tuning and pattern recognition technique and
heuristic rules for its adaptation; the SattCon-
trol ECA400 controller, which uses relay auto-
tuning and model based adaptation; the Hon-
eywell UDC 6000 controller, which uses step
response analysis for automatic tuning and a
rule base for adaptation, and finally the Yoko-
gawa SLPC-181 and 281, which use step re-
s p o n s e a n a l y s i s f o r a u t o t u n i n g a n d a m o d e l
b a s e d a d a p t a t i o n .
6.1 Fox bor o EXACT
The single-loop adaptive controller EXACT was
announced by Foxboro in October 1984. The
reported application experience in using this
controller has been favorable, see, for instance,
(Callaghan
e ta] .
1986). The adaptive features
are now also available in their DCS products.
Process M ode l ing
The Foxboro system is based on the determi-
nation of dynamic characteristics from a tran-
sient, which resul ts in a sufficiently large error.
If the controller parameters are reasonable, a
transient error response of the type shown in
Figure 8 is obtained. Heuristic logic is used to
detect that a proper disturbance has occurred
and to detect peaks el, e2, e3, and oscillation pe-
riod T p
Contro l Des ign
The idea behind the tuning method is pattern
recognition as described in (Bristol, 1977). The
user specifications are given in terms of maxi-
mum overshoot and maximum damping. They
are defined as
damping
= e3 - e2
e l - - e 2
l e 2 1
overshoot =
2 3 )
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708 K.J. Astr6m et a l
Error
T
e
Error
e I
I e 2 I
~ m V l
P
t
F i g . 8 . R e s p o n s e t o a s t e p c h a n g e o f s e t p o i n t u p p e r
c u r v e ) a n d l o a d t o w e r c u r v e )
f o r s e t - p o in t c h a n g e s a s w e l l a s l o a d d i s tu r -
b a n c e s . N o t e t h a t th e d ef i ni t io n f d a m p i n g h e r e
i s d i ff e re n t f r o m t h e d a m p i n g f a c t o r a s s o c i at e d
w i t h a s t a n d a r d s e c o n d - o rd e r s y s t e m.
T h e c o n t r o l l e r s t r u c t u r e i s o f t h e s e r i e s f o r m .
F r o m t h e r e s p o n s e t o a s e t- p oi n t c h a n g e o r l o a d
d i s t u r b an c e , t h e a c t u al d a m p i n g a n d o v e r s h o o t
p a t t e r n o f t h e e r r o r s i g n a l i s r e c o g n i z e d a n d t h e
p e r i o d o f o s c i ll a t i o n
T p
m e a s u r e d . T h i s i n fo r m a -
t i o n is u s e d b y t h e h e u r i s t i c r u l e s t o d i r e c t l y a d -
j u s t 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 g i v e t h e s p e ci -
f i ed d a m p i n g a n d o v e r s h oo t . It i s t h e r e fo r e a d i-
r e c t m e t h o d . E x a m p l e s o f h e u ri s t ic s a r e d e c r e a s-
i n g p r o p o r t io n a l b a n d
P B ,
i n t e g ra l t i m e T i a n d
d e r i va t i v e t i m e
T d
i f d i s t i nc t p e a k s a r e n o t d e -
t e ct ed . I f d i st i nc t p e a k s h a v e o c c u r r e d a n d b o t h
d a m p i n g a n d o v e r s h o ot a r e l e s s t h a n t h e m a x i -
m u m v a l u es ,
P B
i s d e c r e a s e d .
P r i o r I n f o r m a t i o n a n d
P r e t u n i n g
T h e c o n t r o l le r h a s a s e t o f r e q u i r e d p a r a m e t e r s
t h a t h a v e t o b e s e t e i t h e r b y t h e u s e r f ro m p r i o r
k n o w l e d g e o f th e l o o p o r e s t i m a t e d u s i n g t h e
p r e t u n e f u n c t i o n . ( P r e t u n e is F o x b o r o s ' n o t a t i o n
f or a u t o - t u n i n g ) . T h e r e q u i r e d p a r a m e t e r s a r e
( i) I n i t i a l v a l u e s o f
P B , T i
a n d
Td.
( i i ) N o i s e b a n d
N B ) . T h e
c o n t r o l l e r s t a r t s
a d a p t a t i o n w h e n e v e r t h e e r r o r s i g n al e x -
c e e d s tw o t i m e s
N B .
( ii i) M a x i m u m w a i t t i m e ( W m ~x ). T h e c o n t r o l l e r
w a i t s f o r a t i m e o f W m ax f o r t h e o c c u r r e n c e
o f t h e s e c o n d p e a k .
I f t h e u s e r i s u n a b l e t o p r o v i d e t h e s e t o f r e -
q u i r e d p a r a m e t e r s a p r e t u n e f u n c t i o n c a n b e a c-
t i v a t e d t o e s t i m a t e t h e s e q u a n t i ti e s . T o a c t i v a t e
t h e p r e t u n e f u n c t i o n , t h e c o n t r o l l e r m u s t f ir st
b e p u t i n m a n u a l . W h e n t h e p r e t u n e f u n ct i o n i s
a c ti v at e d, a s t e p i n p u t i s g e n e r a t e d a n d f r o m a
s i m p l e a n a l y s i s o f t h e p r o c e s s r e a c t i o n c u r v e a s
s h o w n i n F i g u r e 3 , v a l u e s f or t h e P I D p a r a m e -
t e r s a r e c a l c u l a t e d u s i n g a Z i e g le r - N ic h o l s- l i k e
f o r m u l a :
P B = 1 2 0 K p L / T
Ti = 1.5L (24)
T d = T d 6 .
M a x i m u m w a i t t i m e , W m a~ i s a ls o d e t e r m i n e d
f r o m t h e s t e p r e s p o n s e :
Wm~= = 5L
T h e n o i s e b a n d i s d e t e r m i n e d d u r i n g t h e l a st
p h a s e o f t h e p r e t u n e m o d e . T h e c o n tr o l s i g na l
i s f i r st r e t u r n e d t o t h e l e v e l b e f o r e t h e s t e p
c h a n g e . W i t h t h e c o nt r o ll e r st ill n m a n u a l a n d
t h e c o n t r o l s i g n a l h e l d c o n s t a n t , t h e o u t p u t i s
p a s s e d t h r o u g h a h i g h - p a s s f il te r w i t h a c u t -
o ff f r e q u e n cy h i g h e r t h a n t h e c u t -o f f r e q u e n c y
o f t h e c l o s e d - l o o p s y s t e m . T h e n o i s e b a n d i s
c a l c u l a te d a s a n e s t i m a t e o f t h e p e a k - t o - p e a k
a m p l i t u d e o f t h e o u t p u t f r o m t h e h i g h - p a s s
filter.
T h e e s ti m at e d n o i s e b a n d
N B )
i s u s e d t o
i n it i al i ze h e d e r i v a t i v e t e r m . D e r i v a t i v e a c t i o n
s h o u l d b e d e c re a s e d w h e n t h e n o i s e l e v e l i s
h i g h t o a v o i d l a r g e f l u c t u a t i o n s i n t h e c o n t r ol
s i g na l . T h e d e r i v a t i v e t e r m i s i n i ti a l iz e d s i n g
t h e f o l l o w i n g l og ic :
C a l c u l a t e a q u a n t i t y Z = 3 . 0
2 N B ) / 2 . 5 ;
i f Z > 1 t h e n s e t
T d = T J 6 ;
i f Z < 0 t h e n s e t T d = 0 ;
i f 0 < Z < l t h e n s e t T d = Z . T d 6 .
A p a r t f r o m t h e s et o f r e q u i r e d p a r a m et e r s , t h e r e
i s a l s o a s e t o f o p t i o n a l p a r a m e t e r s . I f t h e s e a r e
n o t s u p p l i e d b y t h e u s e r t h e n t h e d e f a u l t v a l u e s
w il l b e u s e d . T h e o p t i o n a l p a r a m e t e r s a r e a s
f o l l o w s d e f a u l t v a l u e s i n p a r e n t h e s i s ) :
(i)
( i i)
(iii)
M a x i m u m a l l o w e d d a m p i n g 0 . 3 )
M a x i m u m a l l ow e d o v e rs h o ot 0 .5 )
D e r i v a t i v e f a c t o r ( 1 ). T h e d e r i v a t i v e t e r m i s
m u l t i p l i e d b y t h e d e r i v a t i v e f a c to r . T h i s a l-
l o w s t h e d e r i v a t i v e i n f l u e n c e to b e a d j u s t e d
b y t h e u s e r . S e t t i n g t h e d e r i v a t i v e f a c to r to
z e r o r e s u l t s i n P I c o n t r o l .
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AutomaticTuning and PID Controllers
709
(iv) Change Limit (10). This factor limits the
controller parameters to a certain range.
Thus the controller will not set the P B ,
7 / and
Td
values higher or lower than ten
times the ir initial values if the default of 10
is used for the change limit.
6 .2 S a t t C o n t r o l E C A 4 0 0
This controller was announced by SattControl in
1988. It has t he adaptive functions of automatic
tuning, gain scheduling and adaptive control.
r - -
r
V
P FILTER
T
P FILTER
T
Automat ic Tuning
The auto-tuning is performed using the relay
method in the following way, see (H~igglund and
./~str6m, 1991). The process is brought to a de-
sired operating point, either by the operator in
manua l mode or by a previously tuned controller
in automatic mode. When the loop is station-
ary, the operator presses a tuning button. Af-
ter a short period, when the noise level is mea-
sured automatically, a relay with hysteresis is
introduced in the loop, and the PID controller
is temporarily disconnected, see Figure 5. The
hysteresis of the relay is determined automat-
ically from the noise level. During the oscilla-
tion, the relay amplitude is adjusted so that a
desired level of the oscillation amplitude is ob-
tained. When an oscillation with constant am-
plitude and period is obtained, the relay exper-
iment is interrupted and Gp io)o), i.e. the value
of the transfer function at oscillation frequency
Wo, is calculated using the describing function
analysis.
Contr ol Design
The PID algorithm in the ECA400 controller
is of series form. The identification procedure
provides a process model in terms of one point
Gp iWo)
on the Nyquist curve. By introducing
the PID controller
Gc io))
in the control loop,
it is possible to give the Nyquist curve of the
compensa ted sys tem Gp Gc a des ired location at
the frequency COo. For most purposes, the PID
param eters ar e chosen so that
Gp iwo)
is moved
to the point where
Gp iO)o)Gc icoo) = 0.5e i135~/180 (25)
This design method can be viewed as a combi-
nation of phase- and amplitude-margin specifi-
cation. Since there are thr ee adjustable parame-
ters, K, T: and
Td,
and the design criterion (25)
can be obtained with only two parameters, it is
furthermore required th at
Ti = 4Td
(26)
For some simple control problems, where the
process is approximately a first-order system,
Fig. 9. The adaptive control procedure in ECA400
the D-part is switched off and only a PI con-
troller is used. This kind of processes is auto-
matically detected. For this PI controller, the
following design is used:
K
0 . 5 / I G p i e o o ) l
(27)
Ti = 4/Wo.
There is also another situation when it is de-
mrable to switch off the derivative part, namel y
for processes with long dead time. If the oper-
ator tells the controller that the process has a
long dead time, a PI controller with the follow-
ing design will replace the PID controller.
K = 0.25/IGp iO)o)l
(28)
Ti = 1.6/O)o.
Gain S c h e d u l i n g
The ECA400 controller also has gain schedul-
ing. Three sets of controller paramet ers can be
stored. The parameters are obtained by using
the auto-tu ner three times, once at every oper-
ating condition. The actual va lue of the schedul-
ing variable, which can be the controller output,
the measurement signal or an external signal,
determines which parameter set to use.
A d a p t i v e F eedback
The ECA400 controller uses the information
from the relay feedback experiment to initialize
the adaptive controller. Figure 9 shows
t h e
principle of the adaptive controller. The main
idea is to track the point on the Nyquist curve
obtained by the re lay autotuner. I t is performed
in the following way. The control signal u and
the measurement signal y are filtered through
narrow band-pass filters at the frequency wo,
which is the frequency obtained from the relay
experiment. These signals are then analyzed
in a least-squares es timato r which provides an
estimate of the point
G ieoo) . The
method is
described in more detail in the paper (Htigglund
and Astr6m, 1991).
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A d a p t i v e F e e d f o r w a r d Co n t r o l
A n o t h e r f e a t u r e i s a n a d a p t i v e f e e d - f o r w a r d
a l g o r i t h m . T h e a d a p t i v e f e e d f o r w a r d c o n t r o l
p r o c e d u r e i s in i t ia l i z e d b y t h e r e l a y a u t o t u n e r .
A l e a s t s q u a r e s a l g o r i t h m i s u s e d t o id e n t i f y
p a r a m e t e r s a a n d b i n t h e m o d e l
y t ) = a u t - 4 h ) + b v t - 4 h ) ( 2 9 )
w h e r e y is th e m e a s u r e m e n t s i g na l , u i s t h e
c o n t r o l s i g n a l a n d v i s t h e d i s t u r b a n c e s i g n a l
t h a t s h o u l d b e f e d f o r w a r d. T h e s a m p l i n g i n te r -
v a l h i s d e t e r m i n e d f r o m t h e r e l a y e x p e r i m e n t :
h = T 0 / 8 , w h e r e T o i s t h e o s c i l l a t io n p e r io d . T h e
f e e d f o r w a r d c o m p e n s a t o r h a s t h e s i m p l e s t r u c -
t u r e
A u f f t ) = k f f t ) A v t ) (30)
w h e r e t h e f e e d f o r w a r d g a i n k f f i s c a l c u l a t e d
f ro m t h e e s t i m a t e d p r o c e s s p a r a m e t e r s
/~(t) (31)
k f f t ) = - 0 . 8 a ( t ) '
T h e d e t a i l s a r e g i v e n i n ( H ~ i g g lu n d a n d A s t r ~ m ,
1 9 9 1 ) .
O p e r a t o r I n t e r f a c e
T h e i n i t i a l r e l a y a m p l i t u d e i s g i v e n a d e f a u l t
v a l u e w h i c h i s s u i t a b l e f o r m o s t p r o c e s s c o n t r o l
a p p l i c a t i o n s . T h i s p a r a m e t e r i s n o t c r i t i c a l s i n c e
i t w i l l b e a d j u s t e d a f t e r t h e f i r s t h a l f p e r i o d
t o gi v e a n a d m i s s i b le a m p l i t u d e o f t h e l i m i t
c y c l e o s c i l la t i o n . T h e o p e r a t i o n o f t h e a u t o t u n e r
i s t h e n v e r y s im p l e . T o u s e t h e t u n e r , t h e
p r o c e s s i s s i m p l y b r o u g h t t o a n e q u i l i b r i u m
b y s e t t i n g a c o n s t a n t c o n tr o l s ig n a l i n m a n u a l
m o d e . T h e t u n i n g i s a c t i v a t e d b y p u s h i n g t h e
t u n i n g s w i t c h . T h e r e g u l a t o r i s a u t o m a t i c a l l y
s w i tc h e d t o a u to m a t i c m o d e w h e n t h e t u n i n g
i s c o m p l e t e .
T h e w i d t h o f t h e h y s t e r e s i s i s s e t a u t o m a t i c a l l y ,
b a s e d o n m e a s u r e m e n t o f t h e n o i s e l ev e l i n t h e
p r o c e s s . T h e l o w e r t h e n o i s e le v e l , t h e l o w e r
i s t h e a m p l i t u d e r e q u i r e d f r o m t h e m e a s u r e d
s i gn a l . T h e r e l a y a m p l i t u d e i s c o n t r o ll e d s o t h a t
t h e o s c i l l a t i o n i s k e p t a t a m i n i m u m l e v e l a b o v e
t h e n o i s e l e v e l.
T h e f o l l o w i n g a r e s o m e o p t i o n a l s e t t i n g s t h a t
m a y b e s e t b y t h e o p e r a t o r :
( 1) C o n t r o l D e s i g n [ n o rm a l , P I , d e a d t im e [ .
T h e d e f a u l t d e s i g n m e t h o d [ n o r m a l ] c a n r e -
s u l t i n e i t h e r a P I o r a P I D c o n t r o l l e r d e -
p e n d i n g o n w h e t h e r t h e p r o c e s s is o f f i rs t
o r d e r o r n o t ( s e e t h e d i s c u s si o n a b o u t c on -
t r o l d e s i g n a b o v e ) . T h e o p e r a t o r c a n f o r c e
t h e c o n t r o l l e r t o b e P I b y s e l e c t in g [ P I] .
F i n a l l y , t h e P I d e s i g n s u i t e d f o r 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 s u s e d i f t h e o p t i o n
[ d e a d t i m e ] i s s e l e ct e d . T h e c o n t r o l d e s i g n
c a n b e c h a n g e d e i t h e r b e f o r e o r a f t e r a t u n -
i n g , a n d d u r i n g t h e a d a p t a t io n .
2 ) R e s e t [ Y e s / No ] . A r e s e t o f i n f o r m a ti o n c o n -
c e r n i n g t h e a u t o t u n er , t h e g a i n s c h e d u l -
i n g o r t h e a d a p t i v e c o n t r ol l e r c a n b e d o n e .
S o m e i n f o r m a t i o n f r o m a t u n i n g i s s a v e d
a n d u s e d t o i m p r o v e t h e a c c u r ac y o f t h e
s u b s e q u e n t t u n i n g s , i n c l u d i n g t h e n o i s e
l e ve l , t h e i n it i al r e l a y a m p l i t u d e , a n d
t h e
• e r i o d o f t h e o s ci l la t io n . f a m a j o r c h a n g e
i s m a d e i n t h e c o n t r o l l o o p , f o r i n s t a n c e i f
t h e c o n t r ol l er i s m o v e d t o a n o t h e r l o op ,
t h e
o p e r a t o r c a n m a k e a r e s e t o f
t h e t u n i n g i n
f o r m a t i o n .
A r e s e t o f t h e a d a p t i v e c o n tr o l le r
w i ll r e s e t t h e c o n tr o l l e r a r a m e t e r s t o
t h o s e
o b t a i n e d b y t h e r e l a y au t o t u n e r.
3 ) I n i t i a l r e l a y a m p l i t u d e . I n s o m e v e r y
s e n s i t i v e l o o p s , t h e i n i ti a l n p u t s t e p o f
t h e
r e l a y e x p e r i m e n t m a y b e t o o l a r ge . T h e
i ni ti al s t e p c a n t h e n b e d e c i d e d b y
t h e
o p e r a t o r .
(4 ) G a i n s c h e d u l i n g r e f e r e n c e . T h e
s w i t c h e s
b e t w e e n t h e d i ff e re n t s e t s o f P I D p a r a m e -
t e r s i n t h e g a i n - s c h e d u l i n g t a b l e c a n b e p e r -
f o r m e d w i t h d i f f e r e n t s i g n a l s a s t h e g a i n -
s c h e d u l i n g r e f e re n c e . T h e
schedu l ing can
b e
b a s e d o n t h e c o n t r o l s i g n a l , t h e m e a s u r e -
m e n t s i g na l , o r a n e x t e r n a l s i g n al . T h e g a i n
s c h e d u l i n g i s a c t i v e o n l y w h e n a r e f e r e n c e
s i g n a l i s c o n f i g u r e d .
6 .3 H o n e y w e l l U D C 6 0 0 0
T h e H o n e y w e l l U D C 6 0 0 0 co n t r o ll e r h a s a n
a d a p t i v e f u n c t i o n c a l l e d
A c c u t u n e .
A c c u t u n e
u s e s b o t h m o d e l b a s e d p r o c e d u r e s a n d r u l e -
b a s e d p r o c e d u r e s . I t c a n o n l y b e u s e d o n s t a -
b l e p r o c e s s e s . I n t e g r a t i n g p r o c e s s e s c a n c o n s e -
q u e n t l y n o t b e t r e a t e d .
I n i t i a l T u n i n g
T h e a d a p t i v e p r o c e d u r e i s i n i ti a l iz e d b y a s t e p
r e s p o n s e e x p e r i m e n t . T h e u s e r b r i n g s t h e p ro -
c e s s v a l u e t o a p o i n t s o m e d i s t a n c e a w a y f ro m
t h e d e s i r e d s e t p o i n t i n m a n u a l , a n d w a i t s f o r
s t e a d y s t a t e . S w i t c h i n g t o a u t o m a t i c m o d e w i l l
i n i t i a t e a s t e p r e s p o n s e e x p e r i m e n t , w h e r e t h e
s iz e o f th e s t e p i s c a l c u l a t e d t o b e s o la r g e t h a t
i t i s s u p p o s e d t o t a k e t h e p r o c e s s v a l u e t o t h e
s e t p o i n t .
D u r i n g t h e e x p e r i m e n t , t h e p r o c e s s v a l u e a n d
i t s d e r i v a t i v e a r e c o n t i n u o u s l y m o n i t o r e d . T h e
d e a d t i m e L i s c a l c u l a t e d a s t h e t i m e i n t e r v a l
b e t w e e n t h e s t e p c h a n g e a n d t h e m o m e n t t h e
p r o c e s s v a l u e c r o s s e s a c e r t a i n s m a l l l i m i t .
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Automatic Tuning and PID Controllers
711
I f t h e d e r i v a t i v e o f th e p r o c e s s v a l u e c o n t i n u -
o u s l y d e c r e a s e s f r o m t h e s t a r t , i t i s c o n c l u d e d
t h a t t h e p r o c e s s i s o f f i r s t o rd e r . S t a t i c g a i n
K p
a n d t i m e c o n s t a n t 7'1 a r e t h e n c a l c u l a t e d a s
s t e a d y - s t a t e l e v e l s . T h i s p r o v i d e s t h e p o s s i b il i ty
t o c a l c u l a t e t h e o t h e r t h r e e u n k n o w n s f r o m t h e
t h r e e e q u a t i o n s a b o v e .
T1 = Y2 - Yl
d y l d y 2
(32)
Y2 + dy2 T1
gp AU
w h e r e y x a n d Y 2 a r e t w o m e a s u r e m e n t s o f t h e
p r o c e s s v a l u e ,
d y l
a n d
d y 2
a r e e s t i m a t e s o f t h e
d e r i v a t i v e s o f th e p r o c e s s v a l u e a t t h e c o r r e -
s p o n d i n g i n s t a n t s , a n d A u i s t h e s t e p s i z e o f
t h e c o n t r o l s i g n a l . T h e v a l u e s y i a r e c a l c u l a t e d
a s d i s t a n c e s f r o m t h e v a l u e a t t h e o n s e t o f t h e
s t e p . T h e s e c a l c u l a t i o n s c a n b e p e r f o r m e d b e -
f o r e t h e s t e a d y - s t a t e i s r e a c h e d . I t i s c l a i m e d
t h a t t h e p r o c e s s i s i d e n t i f i e d i n a t i m e l e s s t h a n
o n e t h i r d o f t h e t i m e c o n s t a n t . T h e c o n t r o l l e r is
t h e n s w i t c h e d t o a u t o m a t i c m o d e a n d c o n t ro l le d
t o t h e s e t p o i n t . W h e n t h i s i s d o n e, a f i n e a d j u s t -
m e n t o f t h e p a r a m e t e r s i s d o ne b y c a l cu l a ti n g
t h e g a i n
K p
f r o m t h e s t e a d y - s t a t e l e v e ls .
I f t h e d e r i v a t i v e o f t h e p r o c e s s v a l u e i n c r e a s e s
t o a m a x i m u m a n d t h e n d e c r e a s e s , t h e p r oc e s s
i s i d e n t i f i e d a s a s e c o n d - o r d e r p r o c e s s . T h e s t e p
r e s p o n s e o f a s e c o n d - o r d e r p r o c e ss w i t h t w o t i m e
c o n s t a n t s i s
t t
y t ) = 1 + T , T2
(33)
T h i s e q u a t i o n i s u s e d t o c a l c u l a t e s t a t i c g a i n K p
a n d t i m e c o n s t a n t s T1 a n d 7 '2 . A s f o r th e f i r s t
o r d e r s y s t e m , t h e p r o c e s s i d e n t i f i c a t i o n i s p e r -
f o r m e d i n t w o s t e p s . A f i r s t c a l c u l a t i o n i s d o n e
s h o r t l y a f t e r t h e t i m e o f m a x i m u m s lo p e. T h e
c o n t r o l l e r i s t h e n s w i t c h e d t o a u t o m a t i c m o d e
a n d c o n t r o l l e d t o t h e s e t p o i n t . W h e n s t e a d y
s t a t e i s r e a c h e d , t h e p a r a m e t e r s a r e r e c a lc u -
l a t e d u s i n g t h e a d d i t i o n a l i n f o r m a t i o n o f s t e a d y
s t a t e l e v e l s . T h e e q u a t i o n s f o r c a l c u l a t i n g t h e
c o n t r o ll e r p a r a m e t e r s a r e
Adaptation
T h e U D C 6 0 0 0 c o n t ro l l e r a l s o h a s t h e c o n t i n u-
o u s a d a p t a t i o n f e a t u r e . T h e a d a p t a t i o n m e c h -
a n i s m i s a c t i v a t e d w h e n t h e p r o c e s s v a l u e
c h a n g e s m o r e t h a n 0 . 3 % f r o m t h e s e t p o i n t o r
i f t h e s e t - p o i n t c h a n g e s m o r e t h a n a p r e s c r ib e d
v a l u e . ( M o r e d e t a i l s a r e g i v e n b e l o w . )
T h e d e t a i ls o f t h e a d a p t i v e c o n t r o l le r a r e n o t
p u b l i s h e d , b u t i t c o n t a in s a r u l e b a s e w h e r e
s o m e o f t h e r u l e s w i ll b e p r e s e n t e d . T h e c o n -
t r o l l e r m o n i t o r s t h e p r o c e s s v a l u e . D e p e n d i n g
o n i t s n a t u r e , t h e f o l l o w i n g c o n t r o l l e r a d j u s t -
m e n t s a r e m a d e :
(1)
T h e c o n t r o l l e r d e t e c t s o s c i l l a t i o n s i n t h e
p r o c e s s v a l u e . I f o s c i l l a ti o n f r e q u e n c y w 0
i s l e s s t h a n
1 / T i ,
t h e n t h e i n t e g r a l t i m e i s
i n c r e a s e d t o
Ti = 2/COo.
(2 )
I f t h e o s c i l la t i o n f r e q u e n c y w 0 i s g r e a t e r
t h a n
l I T .
t h e n t h e d e r i v a t i v e t i m e i s c h o -
s e n t o b e
T d = 1 / W o .
(3)
I f t h e o s c i ll a ti o n r e m a i n s a f t e r a d j u s t m e n t
( 1 ) o r ( 2 ) , t h e c o n t r o l l e r w i l l c u t i t s g a i n K
i n h a l f .
(4 )
I f a l o a d d i s t u r b a n c e o r a s e t - p o i n t c h a n g e
g i v e s a r e s p o n s e w i t h a d a m p e d o s c i l l a t i o n ,
t h e d e r i v a t i v e t i m e i s c h o s e n t o b e
T d =
1~COo.
(5)
I f a l o a d d i s t u r b a n c e o r a s e t - p o i n t c h a n g e
g i v e s a s l u g g i s h r e s p o n s e , w h e r e t h e t i m e t o
r e a c h s e t p o i n t i s l o n g e r t h a n L + T 1 + T 2,
b o t h i n t e g r a l t i m e 7 ' / a n d d e r i v a t i v e t i m e
T d
a r e d i v i d e d b y a f a c t o r o f 1 .3 .
y t ma , ) + d y t , n a , ) T 1 +
7'2)
gp = Au
1 - N 2
7 '1 + 7 2 - N l n ( ~ t ,~ax (34)
7 1
N = - -
T2
w h e r e
t ma ,
i s th e t i m e f r o m t h e s t a r t o f t h e
r i s e to t h e p o i n t o f m a x i m u m s lo p e. T h e s e
t h r e e e q u a t i o n s h a v e f o u r u n k n o w n s :
K p , T 1 ,
T 2 a n d N . A t t h e f i r s t t i m e o f i d e n t i f i c a ti o n ,
s h o r t l y a f t e r t h e t i m e o f m a x i m u m s lo p e , i t is
a s s u m e d t h a t N = 6 , a n d K p , T 1 a n d 7'2 a r e
d e t e r m i n e d f r o m t h e e q u a t i o n s . W h e n s t e a d y
s t a t e i s r e a c h e d , g a i n
K p
i s c a l c u l a t e d f r o m t h e
(6)
I f t h e s t a t i c p r o c e s s g a i n
K p
c h a n g e s , t h e
c o n t r o ll e r g a in K i s c h a n g e d s o t h a t t h e
p r o d u c t
K K p
r e m a i n s c o n s t a n t .
C o n t r o l D e si g n
T h e U D C 6 0 0 0 c o n t r o l l e r i s w r i t t e n o n s e r i e s
f o r m a n d h a s t h e t r a n s f e r f u n c t i o n
Go s ) = K
( 1 + s T i ) ( 1 +
s T d )
s T i 1 + O .1 2 5 S T d )
T h e d e s i g n g o a l i s t o c a n c e l t h e p r o c e s s p o l e s
u s i n g t h e t w o z e r o s i n t h e c o nt r ol l er . I f t h e r e
i s n o d e a d t i m e i n t h e p r o c e s s t h e c on t r o ll e r
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7 1 2 K . J . / ~ , s t r 6 m e t a l
p a r a m e t e r s a r e c h o s e n i n t h e f o l lo w i n g w a y :
F i r s t o r d e r p r o c e s s : K = 2 4 / K p
Ti = 0.167'1
T j = 0
S e c o n d o r d e r p r o c e s s :
K = 6 / K p
T i T I
T d = T 2
F o r p r o c e s s e s w i t h d e a d t i m e , t h e f o l l o w i n g
c h o ic e o f c o n t r o ll e r p a r a m e t e r s i s m a d e :
F i r s t o r d e r p r o c e s s :
S e c o n d o r d e r p r o c e s s :
3
K =
Kp (1 + 3 L / T i )
T i = T 1
T d = 0
3
K =
Kp (1 +
3 L / T i )
T i = T I + T 2
7 72
T d -
T I + T 2
O p e r a t o r I n t e r f a ce
T h e f o l l o w in g a r e s o m e o p t io n a l p a r a m e t e r s
t h a t m a y b e s e t b y t h e op e ra t o r:
1 ) S e l e c t w h e t h e r a d a p t a ti o n s h o u l d b e p e r -
f o r m e d d u r i n g s e t - po i n t c h a n g e s o n l y , o r
d u r i n g b o t h s e t -p o in t h a n g e s a n d l o a d d is -
t u r b a n c e s .
2 ) S e t t h e m i n i m u m v a l u e o f s e t - p o i n t h a n g e
t h a t w i l l a c t i v a te t h e a d a p t a ti o n . R a n g e :
+ 5 % t o + 1 5 % .
6 .4 Yo k o g a w a S L P C- 1 8 1, 2 8 1
T h e Y o k o g a w a S L P C - 1 8 1 a n d 2 8 1 b o th u s e a
f i r s t o r d e r p l u s d e a d - t i m e m o d e l o f t h e p r o -
c e s s fo r c a l c u l a t i n g t h e P I D p a r a m e t e r s . A n o n-
l i n e a r p r o g r a m m i n g t e c h n i q u e i s u s e d t o o b -
t a i n t h e m o d e l . W i t h t h e f i r s t - o r d e r p l u s d e a d -
t i m e m o d e l , t h e P I D p a r a m e t e r s a r e c a l c u l a t e d
f r om e q u a t i o n s d e v e l o p e d f r om n u m e r o u s s i m u -
l a t i o n s . T h e e x a c t e q u a t i o n s a r e n o t p u b l i s h e d .
T h e P I D c o n t r o ll e r c a n t a k e o n e o f t h e f o ll o w in g
s t r u c t u r e s :
1 / d y f ~
1:
u = K - y + ~ e d t - Td d t }
(35)
e + ~ f e d t - T d ? )
:
u = K
w h e r e
d y f N
- y - y f ) (36)
d t T d
T a b l e 5 . S e t - p o i n t r e s p o n s e s p e c i f i c a t i o n s .
I ~ p e F e a t u r e s C r i t e r ia
1 n o o v e r s h o o t n o o v e r s h o o t
2 5 % o v e r s h o o t I T A E m i n i m u m
3 1 0 % o v e r s h o o t I A E m i n i m u m
4 1 5 % o v e r s h o o t I S E m i n i m u m
S t r u c t u r e 1 i s r e c o m m e n d e d i f l o a d - d i s t u r b a n c e
r e j e c ti o n is m o s t i m p o r t a n t , a n d s t r u c t u r e 2 i s
s e t - p o i n t r e s p o n s e s a r e m o s t im p o r t a n t . T h e s e t
p o i n t c a n a l s o b e p a s s e d t h r o u g h t w o f il t er s i n
s e r i e s :
1 + a i s T i
F i l t e r 1 :
l + sT i
(37)
1 - adST d
F i l t e r 2 :
l + sTd
w h e r e a i a n d a d a r e p a r a m e t e r s s e t b y th e u se r,
m a i n l y t o a d j u s t t h e o v e r s h o o t o f t h e s e t - p o i n t
r e s p o n s e . T h e e f f e c t s o f t h e s e t w o f i l te r s a r e
e s s e n t i a l l y e q u i v a l e n t t o s e t - p o i n t w e i g h ti n g . I t
c a n b e sh o w n t h a t a i = f l , w h e r e fl is t h e s e t -
p o i n t w e i g h t i n g f a c t o r in E q u a t i o n 7 .
T h e u s e r s p e c if ie s t h e t y p e o f s e t- p o i n t r e s p o n s e
p e r f o r m a n c e a c c o r d i n g t o T a b l e 5 . A h i g h o v e r -
s h o o t w i l l o f c o u r s e y i e l d a f a s t e r r e s p o n s e .
T h e c o n t r o l l e r h a s f o u r a d a p t i v e m o d e s :
( 1 ) A u t o m o d e . T h e a d a p t i v e c o n t r o l i s o n . P I D
p a r a m e t e r s a r e a u t o m a t i c a l l y u p d a t e d .
( 2 ) M o n i t o r i n g m o d e . I n t h i s m o d e , t h e c o m -
p u t e d m o d e l a nd t h e P I D p a r a m e t e r s a r e
d i s p l a y e d o n ly . T h i s m o d e i s u s e f u l f o r v a l -
i d a t i n g t h e a d a p t i v e f u n c t io n o r c h e c k i n g
t h e p l a n t d y n a m i c s v a r i a t i o n s d u r i n g o p e r -
a t i o n .
( 3) A u t o - s t a r t u p m o d e . T h i s i s u s e d t o c o m p u t e
t h e i n i t i a l P I D p a r a m e t e r s . A n o p e n - l o o p
s t e p r e s p o n s e i s u s e d t o e s t i m a t e t h e m o d e l .
( 4) O n - d e m a n d m o d e . T h i s m o d e i s u s e d w h e n
i t i s d e s i r a b l e t o m a k e a s e t - p o i n t c h a n g e .
W h e n t h e o n - d e m a n d t u n i n g i s r e q u e s te d , a
s t e p c h a n g e i s a p p l i e d t o t h e p r o c e s s i n p u t
i n c l o s e d l o o p . T h e c o n t r o l l e r e s t i m a t e s t h e
p r o c e s s m o d e l u s i n g t h e s u b s e q u e n t cl o se d -
l o o p r e s p o n s e .
T h e c o n t r o l l e r c o n s t a n t l y m o n i t o r s t h e p e r f o r -
m a n c e o f t h e s y s t e m b y c o m p u t i n g t h e r a t i o of
t h e p r o c e s s o u t p u t a n d m o d e l o u t p u t v a r i a n c e s .
T h i s r a t i o i s e x p e c t e d t o b e a b o u t 1 . I f i t is
g r e a t e r t h a n 2 o r l e s s t h a n 0 .5 , a w a r n i n g m e s -
s a g e f o r r e t u n i n g o f t h e c o n t r o l l e r i s g i v e n . D e a d
t i m e a n d f e e d - f o r w a r d c o m p e n s a t i o n a r e a v a i l -
a b l e f o r t h e c o n s t a n t g a i n c o n t r o l l e r b u t t h e y
a r e n o t r e c o m m e n d e d b y t h e m a n u f a c t u r e r t o
b e u s e d i n c o n j u n c t i o n w i t h a d a p t a t i o n .
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7 . C O N C L U S I O N S
V a r i o u s a d a p t i v e t e c h n i q u e s h a v e b e e n i m p l e -
m e n t e d i n s in g l e- l o o p P I D c o n t r o ll e rs d u r i n g
t h e l a s t d e c a d e. I n t h i s p a p e r , a s u m m a r y o f
t h e s e d i f f e r e n t t e c h n i q u e s h a s b e e n g i v e n t o -
g e t h e r w i t h a n o v e r v ie w o f i n d u s t r i a l p r o d u c ts .
A l m o s t a l l c o n t r o l l e r s t h a t u s e a d a p t i v e t e c h -
n i q u e s h a v e s o m e f o rm o f a u t o m a t i c t u n i n g
f u n c ti o n . T h e a u t o m a t i c t u n i n g f u n c t io n is n o t
o n l y u s e f u l t o h e l p t h e o p e r a t o r i n f i n d i n g s u i t -
a b l e c o n t r o l l e r p a r a m e t e r s . I t i s a l s o u s e fu l t o
b u i l d g a i n s c h e d u l e s a n d t o i n i t i a l i z e a d a p t i v e
c ont ro l l e r s .
G a i n s c h e d u l i n g e ff e c ti v e ly c o m p e n s a t e s f o r
n o n l i n e a r i t i e s a n d o t h e r p r e d i c ta b l e v a r i a ti o n s
i n t h e p r o c e s s d y n a m i c s . S i n c e i t is m u c h e a s i e r
t o i m p l e m e n t t h a n o t h e r a d a p t i v e t e c h n iq u e s ,
i t i s s u r p r i s i n g t h a t m a n y i n d u s t r i a l a d a p t i v e
c o n t r o l l e r s l a c k t h i s f e a t u r e .
T h e a d a p t i v e c o n t r o l l e r s c a n b e d i v i d e d i n t o
t w o c a t e g o r i e s , n a m e l y t h e m o d e l b a s e d a n d
t h e r u l e b a s e d . I t i s i n t e r e s t i n g t o n o t e t h a t
t h e r e a r e m a n y s u c c e s s f u l a p p l i c a t i o n s o f r u l e
b a s e d a d a p t i v e c o n t r o l l e r s , a l t h o u g h a l m o s t a l l
u n i v e r s i t y r e s e a r c h h a s b e e n f o c u se d o n m o d e l
b a s e d a d a p t a t io n .
A d a p t i v e f e e d f o r w a r d c o n t r o l i s r a r e l y u s e d i n
s i n g le - l o o p c o n t r o l le r s . T h i s i s a l s o s u r p r i s i n g ,
s i n c e a d a p t i v e f e e d f o r w a r d c o n t r o l h a s p r o v e n
t o b e v e r y u s e f u l i n o t h e r a p p l i c a t i o n s o f a d a p -
t i v e c o n t r o l , s e e , f o r i n s t a n c e , ( B e n g t s s o n a n d
E g a r d t , 1 9 8 4 ) . A d a p t i v e f e e d f o r w a r d c o n t r o l i s
a l so e a s i e r t o i m p l e m e n t t h a n a d a p t i v e f e ed b a c k
c ont ro l .
8 . R E F E R E N C E S
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a l t e r n a t i v e t o p a r a m e t e r i d e n t if i ca