AUTOMATIC TUNING AND ADAPTATION FOR PID CONTROLLERS - A SURVEY

8/15/2019 AUTOMATIC TUNING AND ADAPTATION FOR PID CONTROLLERS - A SURVEY

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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


4/16

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 -


6/16

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)


7/16


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 .


8/16

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 .


9/16

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 )


10/16

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 .


11/16


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).


12/16

710 K.J. ~strtim et al.

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 .


13/16

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


14/16

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 .


15/16

Autom atic Tun ing and PID C ontrollers

713

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

ANDERSON, I~ L., G. L. BLANKENSHIP, an d

L . G . LEB OW (1988) : A ru l e -ba se d P ID c on-

t ro l l e r . In P r e c. I E E E C o n f e r e n c e o n D ec i-

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k~STROM, K . J .

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m a t i c t u n i n g o f s im p l e r e g u l a t o r s w i th s p ec -

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

A u t o m a t i c a , 2 0 , p p . 6 4 5 - 6 5 1 .

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m a t i c T u n i n g o f P I D C o n tr o ll er s. I n s t r u -

m e n t S o c ie ty o f A m e r i ca , R e s e a r c h T r i a n g l e

P a r k , N o r t h C a r o l i n a.

BENGTSSON, G. an d B. ECA~DT (198 4): Exp e-

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Congr es s , B u d a p e s t , H u n g a r y .

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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

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