8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

1/7

Automatica

Vol. 20, No. 5, pp . 645 651 , 1984

Printed in G reat Britain.

0005 1098/84 3.00 + 0.00

Pergamon Press Ltd.

(c; 1984 International Federatio nof AutomaticControl

Automatic Tuning of S imple Regulators with

Specif ications on Phase and Amplitude Margins

K . J. A S T R O M a n d T. H ~ G G L U N D

Simple robust estimation techniques provide new methods fo r automatic tuning of P ID

regulators which easily can be incorporated in single lo op controllers

Key Wo rds--Adaptive control; co ntrol nonlinearities; describing function; identification; imit cycles;

Nyquist criterion; PID control; relay control.

A l ~ t r a e t - - T h e p a p e r d e s c r i b es p r o c e d u re s f o r a u t o m a t i c t u n i n g

o f r e g u la t o r s o f t h e P I D t y p e t o s p e c if i ca t i on 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 . T h e k e y i d e a i s a s i m p l e m e t h o d f o r

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

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

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

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

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

1 . I N T R O D U C T I O N

T H E

MAJORITYof t he r egu l a t o r s u s ed i n i ndus t ry a re

o f t he P I D t ype. A l a rge i ndus tr i a l p l an t m a y have

hundre ds o f r egu la t o r s. M any i n s t rum en t eng i nee r s

and p lant personnel are used to se lect , ins ta l l and

operate such regulators . Several d i f ferent methods

have been p ropos ed fo r t un i ng P ID regu l a to r s . The

Z i eg l e r -N i cho l s 1943 ) m e t hod is one o f t he m ore

pop ular schemes . In sp i te of th is , i t i s co m m on

exper i ence t ha t m any regu l a t o r s a re i n p rac t i ce

poo r l y t uned . One r eas on i s t ha t s i m p l e robus t

m e t hods fo r t un i ng t he r egu l a t o r s have no t been

avai lab le . This paper addresses the problem of

f i nd i ng au t om at i c t un i ng m e t hods . The m e t hods

p ropos ed a re s i m p l e t o i m p l em en t u s i ng m i c ro -

proce ssors. Th ey offer the possibi li ties to pro vide

au t om at i c t un i ng t oo l s fo r a l a rge c la ss o f com m o n

con t ro l p rob l em s .

The m e t ho ds a re bas ed on a s i m p le i den ti f ica t i on

method which g ives cr i t i ca l poin t s on the Nyquis t

cu rve o f t he open l oop t r ans fe r func ti on . The key

i dea i s a s chem e wh i ch p rov i des au t om at i c

exci ta t ion of the process which i s near ly opt imal for

es t imat ing the des i red process character i s t i cs .

* R e c e iv e d 1 6 No v e m b e r 1 9 83 ; r e v i se d 4 Ap r i l 1 9 84 . T h e o r ig in a l

v e r s io n o f t h is p a p e r w a s p r e s e n t e d a t t h e I F A C W o r k s h o p o n

A d a p t i v e S y s t e m s in C o n t r o l a n d S i g n al P r o c e s s i n g w h i c h w a s

h e l d i n S a n F r a n c i s c o , C a l i f o r n i a , U . S .A . d u r i n g J u n e 1 98 3. T h e

p u b l i s he d p r o c e e d i n g s o f t h is I F A C M e e t i n g m a y b e o r d e r e d f r o m

P e r g a m o n P r e s s L td , H e a d i n g t o n H i ll H a l l, O x f o r d , O X 3 0 B W ,

U . K . T h i s p a p e r w a s r e c o m m e n d e d f o r p u b l ic a t i o n i n r e v is e d fo r m

b y g u e s t e d i t o r L . L j u n g .

t D e p a r t m e n t o f A u t o m a t i c C o n t r o l , L u n d I n s t it u t e o f

T e c h n o l o g y , L u n d , S w e d e n .

645

The m et hods p ropos e d a re p r i m ar i l y i n t ended t o

t une s i m p l e r egu la t o r s o f t he P I D t ype . In s uch

appl icat ions they wi l l o f course inher i t the

l imi ta t ions of the P ID algor i thms. The y wil l no t

work we l l fo r p rob l em s where m ore com pl i ca t ed

regu l a t o r s a re r equ i red . The t echn i que m ay ,

however , a l s o be app l i ed t o m ore com pl i ca t ed

regu l a t o r s and t he exper i ences ob t a i ned s o f a r f rom

exper i m en t a t i on , i n l abo ra t o ry and i ndus t ry ,

indicate that th e s imple vers ions of the a lgo r i thms

work ve ry we ll and i n add i t i on t ha t t hey a re robus t .

The p ropos ed a l go r i thm s m ay be u s ed in s eve ral

d i f fe ren t ways . They m ay be i nco rpo ra t ed i n s i ng le

l oop con t ro l l e rs t o p rov i de an o p t i on fo r au t om at i c

t un ing . The y m ay a l s o be u s ed t o p rov i de a s o l u t ion

to the long -s tanding pro blem of safe in i t ia l i zat ion of

m o re com pl i ca t ed adap t i ve o r s e l f- t un ing s chem es .

W hen com b i ned wi t h a bandw i d t h s e l f- t une r i t is ,

for example , poss ib le to obta in an adapt ive

regulator which may set a su i tab le c losed loop

bandw i d t h au t om at i ca l l y .

There a re o t he r a l t e rna ti ves fo r t un i ng r egu l a t o r s

automat ical ly . Sel f - tuning regulators based on

m i n i m um var i ance , po l e p lacem en t o r LQ G des ign

m et hods m ay be con f i gu red t o g i ve P ID con t ro l .

Such approaches have, e .g . , been cons idered by

W i t t e n m a r k a n d A s tr 6 m 1 9 8 0 ) a n d G a w t h r o p

1982) . These regu lators have the d i sadv antage tha t

s om e i n fo rm a t i on abou t t he t i m e s ca l e o f t he p roces s

m us t be p rov i ded a priori to g ive a reasonable

es t imate of the sampl ing per iod in the re gulator .

The re are som e possibi li ti es to tune the samp l ing

per i od au t om at i ca l ly . D i f fe ren t s chem es have been

p r o p o s e d b y K u r z 1 97 9) a n d A s t r 6 m a n d Z h a o y i n g

1981) . These m etho ds wi ll , howe ver , only w ork fo r

m odera t e changes i n t he p roces s t i m e cons t an t s .

The m e t hod p ropos ed i n th i s pape r does no t s u f fer

f rom th i s d i sadvantage. I t may be appl ied to

processes ha ving widely d i f ferent t ime scales .

Convent ional se l f - tuning regulators based on

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

2/7

6 4 6 K . J .

/ ~ S T R O M

and T. H,g,GGLUND

re c u r si v e e s t i m a t i o n o f a p a ra m e t r i c m o d e l r e q u i r e s

a c o m p u t e r c o d e o f a f ew k i lo b y t e s . T h e a l g o r i t h m s

p ro p o s e d i n t h i s p a p e r wh i c h a r e b a s e d o n

d e t e rm i n a t i o n o f z e ro -c ro s s in g s a n d p e a k d e t e c t i o n

m a y b e p r o g r a m m e d i n a f ew h u n d r e d b y t es .

T h e p a p e r i s o rg a n i z e d a s fo l l o ws : t h e e s t i m a t i o n

m e t h o d i s d e s c r i b e d i n S e c t i o n 2 a n d a n a l y s e d i n

S e c t i o n 3 . S i m p l e a l g o r i t h m s fo r a u t o m a t i c t u n i n g

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

spec i f ica t ions a re g iven in Sec t ions 4 and 5 . Resu l t s

f ro m l a b o ra t o ry a n d i n d u s t r i a l e x p e r i m e n t s w i t h t h e

a l g o r i t h m s a r e p r e s e n t e d i n S e c t i o n 6 . I n S e c t i o n 7 ,

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

c o n v e n t i o n a l a d a p t i v e c o n t ro l l e r s i s d i s c u ss e d .

2 . T H E B A S I C I D E A

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

re g u l a t o r s i s b a s e d o n t h e o b s e rv a t i o n t h a t t h e

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

k n o w l e d g e o f o ne p o i n t o n t h e N y q u i s t c u r v e o f t h e

o p e n l o o p s y s t e m . T h i s p o i n t i s t h e i n t e r s e c t i o n o f

t h e N y q u i s t c u rv e w i t h t h e n e g a t i v e r e a l a x i s, wh i c h

is t rad i t ion a l ly descr ibed in t e rm s o f the c r i t i ca l ga in ,

kc , and the c r i t i ca l per iod ,

t,. .

In t h e o r i g in a l Z i e g le r N i c h o l s s c h e m e , d e s c ri b e d

in Z ieg ler and N icho ls 1943) , the c r i t ica l ga in an d

t h e c r i t i c a l p e r i o d a r e d e t e rm i n e d i n t h e fo l l o wi n g

wa y : a p ro p o r t i o n a l r e g u l a t o r i s c o n n e c t e d t o t h e

s y s t e m ; t h e g a i n i s g r a d u a l l y i n c re a s e d u n t i l a n

o s c i l l a ti o n is o b t a i n e d ; t h e g a i n w h e n t h i s o c c u r s i s

t h e c r i t ic a l g a i n a n d t h e p e r i o d o f th e o s c i l l a t i o n is

the c r i t i ca l per iod . I t i s d i f f i cu l t to au tom at iz e th i s

e x p e r i m e n t , a n d p e r fo rm i t i n s u c h a wa y t h a t t h e

a m p l i t u d e o f t h e o s c i l l a ti o n is k e p t u n d e r c o n t ro l .

A n o t h e r m e t h o d f o r a u t o m a t i c d e t e r m i n a t i o n o f

spec i f ic po in t s on the Nyqu is t cu rve i s therefo re

p ro p o s e d .

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

sys tem w i th a phase l ag o f a t l eas t rr a t h igh

f r e q u e n c ie s m a y o s c i l l a t e w i t h p e r i o d tc u n d e r r e l a y

c o n t ro l . T o d e t e rm i n e t h e c r i t i c a l g a i n a n d t h e

cr i t i ca l per iod , the sys te m i s con nec te d in a feedba ck

loop w i th a re lay as i s show n in F ig . 1 . The e r r ro r e is

then a per io d ic s igna l wi th the p er iod t~ . I f d i s the

re lay ampl i tude , i t fo l lows f rom a Four ie r se r ies

F t ~ . 1 . B l o c k d i a g r a m o f th e a u t o - t u n e r . T h e s y s t e m o p e r a t e s a s

a r e la y c o n tr o l l e r in t h e tu n i n g m o d e t ) a n d a s a n o r d i n a r y P I D

r e g u l a t o r in t h e c o n t r o l m o d e c ).

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

h a s t h e a m p l i t u d e

4d/Tr .

I f the p roc ess o u tp u t i s u , the

c r i ti c a l g a i n i s th u s a p p ro x i m a t e l y g i v e n b y

4 d

k~ = - -. 1)

T h i s r e s u l t a l s o fo l l o ws f ro m t h e d e s c r i b i n g fu n c t i o n

a p p ro x i m a t i o n . No t i c e t h a t t h e d e s c r i b i n g fu n c t i o n

N ( a )

fo r an idea l re lay i s g iven by

4d

N ( a = . 12~

~

I t m a y b e a d v a n t a g e o u s t o u s e o t h e r n o n -

l i n e a r i t i e s t h a n t h e p u re r e l a y . A r e l a y w i t h

hys te res i s g ives a sys tem which i s l es s sens i t ive to

m e a s u re m e n t n o i se . T h i s c a s e is d i s c u ss e d i n m o re

de ta i l be low.

A s i m p l e r e l a y c o n t ro l e x p e r i m e n t t h u s g i ve s th e

i n fo rm a t i o n a b o u t t h e p ro c e s s wh i c h is n e e d e d i n

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

t h e a d v a n t a g e t h a t i t i s e a s y t o c o n t ro l t h e

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

c h o i c e o f t h e r e l a y a m p l i t u d e . No t i c e a l s o t h a t t h e

e s t i m a t i o n m e t h o d w i ll a u t o m a t i c a l l y g e n e r a te a n

inpu t s igna l to the p rocess wh ich has a s ign i f ican t

f requ ency c on t en t a t coc =

2zr/tc.

T h i s e n s u re s t h a t

t h e c r i t i c a l p o i n t c a n b e d e t e rm i n e d a c c u ra t e l y .

W h e n t h e c r i t i c a l p o i n t o n t h e Ny q u i s t c u rv e i s

k n o wn , i t i s s t r a i g h t fo rwa rd t o a p p l y t h e c l a s s i c a l

Z i e g l e r -N i c h o l s t u n i n g ru le s . I t is a l s o p o s s ib l e t o

d e v is e m a n y o t h e r d e s i g n s c h e m e s t h a t a r e b a s e d o n

t h e k n o wl e d g e o f o n e p o i n t o n t h e N y q u i s t c u rv e .

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

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

cr i t e r ia wi l l be g iven in Sec t ions 4 and 5 .

I t i s p o s s ib l e t o m o d i fy t h e p ro c e d u re t o

d e t e rm i n e o t h e r p o i n t s o n t h e Ny q u i s t c u rv e . An

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

re l a y t o o b t a i n t h e p o i n t wh e re t h e Ny q u i s t c u rv e

i n t e r se c t s t h e n e g a t i v e i m a g i n a ry a x is . O t h e r p o i n t s

o n t h e Ny q u i s t c u rv e c a n b e d e t e rm i n e d b y

re p e a t i n g t h e p ro c e d u re w i t h l i n e a r s y s t e m s

i n t ro d u c e d i n t o t h e l o o p . Ne w d e s i g n m e t h o d s

wh i c h a r e b a s e d o n s u c h d a t a a r e d e s c r i b e d i n

As t r6 m a n d H / i g g l u n d 1 98 4b ).

D e t e r m i n a t i o n o J a m p l i t u d e a n d p e r i o d

M e t h o d s f o r a u t o m a t i c d e t e r m i n a t i o n o f th e

f r e q u e n c y a n d t h e a m p l i t u d e o f th e o s c i l l a t i o n w i ll

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

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

e a s i l y b e d e t e rm i n e d b y m e a s u r i n g t h e t i m e s

b e t we e n z e ro -c ro s s i n g s . T h e a m p l i t u d e m a y b e

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

o f t h e o u t p u t . T h e s e e s t im a t i o n m e t h o d s a r e e a s y t o

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

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

3/7

A u to ma t i c t u n in g o f s imp le r e g u l a to rs 6 4 7

c o mp a r i so n s o n ly . S in c e t h e d e sc r ib in g f u n c t i o n

a n a ly s is i s b a se d o n t h e f i rs t h a r mo n ic o f th e

osc i l la tion , the s imple e s t im a t ion tech niques requ i re

tha t th e f ir s t ha rm onic dom ina tes . I f th is is no t the

case , i t may be necessa ry to f i l te r the s igna l be fore

measur ing .

Mo r e e l a b o r a t e e s t ima t io n s c h e me s l i k e l e a s t

sq u a r e s e s t ima t io n a n d e x t e n d e d K a lma n f i l t e r i n g

ma y a l so b e u se d t o d e t e r min e t h e a mp l i t u d e a n d

the f requency of the l imi t cyc le osc i l la t ion .

S imu la t i o n s a n d e x p e r ime n t s o n i n d u s t r i a l p r o -

cesses have ind ica ted th a t l i tt le is ga ined in prac t ice

b y u s in g mo r e so p h i s t i c a te d m e th o d s f o r d e t e r min -

in g t h e a mp l i t u d e a n d t h e p e r io d .

3 A N A L Y S I S

The reasoning in Sec t ion 2 i s pure ly heur is t ic .

A n a ly s i s i s n e e d e d t o u n d e r s t a n d w h e n th e me th o d

w o r k s a n d w h e n i t d o e s n o t w o r k . N a tu r a l q u e s t i o n s

a re : W hen wi ll the re be l imi t cyc le osc i l la t ions?

W h e n a r e t h o se o sc i l l a ti o n s s t a b l e ? H o w a c c u r a t e i s

t h e d e sc r ib in g f u n c t i o n a p p r o x ima t io n ? W h a t

h a p p e n s i f t h e N y q u i s t c u r v e i n t e r s e ct s t h e n e g a t i v e

rea l ax is a t seve ra l po in ts ? Pa r t ia l answe rs to these

ques t ions a re g iven be low.

Exac t ex press ions for the pe r iod o f osc i l la t ion

w e r e o r ig in a ll y d e r iv e d b y H a m e l a n d T sy p k in . A n

expos i t ion o f the re su l t s a re a lso g iven in the tex t -

books by Tsypkin (1958) , Gi l le , Pe legr in and

Decaulne (1959) , Ge lb and Vander Be lde (1968) ,

and A the r to n (1975). Co ndi t io ns for osc il la t ions a re

g iven be low.

T h e o r e m 1 . C o n s id e r t h e d o se d l o o p sy s t e m

o b ta in e d b y a f e e d b a c k c o n n e c t i o n o f a li n e a r

sys tem having the t r ansfe r func t ion

G ( s )

with a r e lay

having hys te resis . L e t H(T, z ) be th e p ulse t r ansfe r

f u n c t i o n o f t h e se r ie s c o mb in a t i o n o f a s a mp le a n d

h o ld w i th p e r io d ~ a n d

G ( s ) .

I f the re i s a pe r iod ic

osc i l la tion , then the pe r iod T i s g iven by

U

E

F IG 2 C h a r a c t e r i s t i c s o f a re l a y w i t h h y s t e r e s i s

- ~ k e v e n

u ( k T / 2 ) = k o d d

T h e s t e a d y s t a t e t r an smis s io n o f th e s e q u e n c e

{ u ( k T / 2 ) } t h r o u g h th e s a mp le d sy s t e m i s c h a r a c -

te r ized b y the ga in H ( T / 2 , - 1 . Th e cond itio n (3) is

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

w a v e s ig na l s a r o u n d th e c lo se d l o o p .

A f o r ma l p r o o f o f T h e o r e m 1 i s f o u n d in A th e r to n

(1982) and in Ast r6m and H / igglund (1984a), which

a lso g ives cond i t ions for the s tab i l i ty o f the l imi t

cycle. The la t te r pap e r a lso cov ers the m ore g ene ra l

case of a symmetr ic osc i l la t ions .

I t fo l lows f rom we l l -known se r ie s expans ions of

the pulse t r ansfe r func t ion tha t

H ( z , - 1 ) = ~ 4 ( ~ + 2 n z t )

, : o r t [ 1 + 2 n ] l i n G c i . (4)

The desc r ib ing func t ion approxima t ion (2) i s

obta ined s imply by us ing the f i rs t te rm in th is se rie s

expans ion . The va l id i ty of the desc r ib ing func t ion

a p p r o x im a t io n ( 2) c a n t h u s b e e v a lu a t e d f r o m th i s

formula . In m any cases i t g ives the pe r io d o f the

osc i l la t ion wi th an e r ror of a f ew pe r cen t , which i s

accura te enough for the in tended purpose . I t i s

eas i ly shown tha t the desc r ib ing func t ion approxi -

ma t ion g ives the exac t pe r iod for an in tegra tor w i th

t ime de lay . Anothe r example i l lus t ra te s the pre -

c is ion tha t i s typ ica l ly ob ta ined .

8

H ( T / 2 , - 1) - d (3)

whe re e i s the hy s te res is wid th of the re lay and d is

t h e r e la y o u tp u t . [ ]

The cha rac te r i s t ic s of the re lay wi th hys te res is a re

show n in F ig . 2 . The re su l t o f Th eorem 1 i s eas i ly

u n d e r s to o d b y a s su min g th a t t h e r e e xi s ts a p e r io d i c

osc i l la t ion wi th pe r iod T . Sam pl ing the sys tem wi th

p e r io d T / 2 a t sampl ing ins tan ts which a re

synchronized to the re lay swi tches then g ives the

sa mp le d i n p u t a n d o u tp u t s i g n a l s

k even

y ( k T / 2 ) = - e k o d d

E x a m p l e

1. Cons ide r the l inea r sys tem

G s ) =

s s + 1 ) ( s + a )

F r o m th e d e sc r ib in g f u n c t i o n a p p r o x ima t io n , t h e

pe r iod of osc i l la t ion i s

2n 1

T = ~ 6 . 3 x / ~ .

The va lue of the pulse t r ansfe r func t ion for z = - 1

b e c o m e s

h 1 [ ~ - e - ~ 1 1 - e -a t ]

H ( z , - 1 ) = - 2 - ~ + a - - ~ + e -~ a 2 1 ~ e ~a ~J

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

4/7

648 K .J . ASTROM an d T. H) i,GGLUND

F o r l a rg e v a l u e s o f a, t h e p e r i o d o f o s c i ll a t i o n is

a p p r o x i m a t e l y g i ve n b y

T ~ 4 x / N 6 .9 1

[ ]

T h e d e s c r i b i n g f u n c t i o n a p p r o x i m a t i o n m a y ,

however , g ive mis lead ing resu l t s as i s seen by the

fo l l o w i n g e x a m p l e .

E x a m p l e 2 . C o n s i d e r a l i n e a r s y s t e m w i t h t h e

t r a n s f e r f u n c t i o n

b

G s ) . . . . . e ~ ,o a , b , t o > O .

s + a

S i n c e th e N y q u i s t c u rv e i n t e r s e c t s t h e n e g a t i v e r e a l

a x is a t m a n y p o i n t s , t h e d e s c r i b i n g fu n c t i o n a n a l y s is

p red ic t s severa l poss ib le l imi t cyc les . The va lue o f

t h e p u ls e tr a n s f e r f u n c t i o n fo r z = - 1 i s

b e - a ~ (2 e ' ° - 1 ) - 1

H (z , - 1 ) = - .

a l + e

r

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

T = 2 r = 2 I b d - a e .

- a l n b d 2 e ~ 1 ) + a e

5 )

I t i s s h o w n i n A s t r6 m a n d H / i g g l u n d (1 9 8 4 a ) t h a t

the l imi t cyc le i s s t ab le . [ ]

Th e t r a n s f e r f u n c t i o n G s ) i n E x a m p l e 2 b e c o m e s

s t r i c t ly pos i t ive rea l i f the t im e de lay goes to zero .

T h e d e s c r i b i n g f u n c t i o n a p p r o x i m a t i o n t h e n p r e -

d ic t s tha t there wi l l no t be any osc i l l a t ion . The

sys tem wi l l however exh ib i t a s t ab le per iod ic

s o l u t i o n u n d e r r e l a y c o n t ro l . Th e p e r i o d i s o b t a i n e d

by le t t ing to in equa t ion (5 ) go to zero .

S t a b l e p e r i o d i c s o l u t i o n s w i ll n o t b e o b t a i n e d fo r

a l l s y s t e m s . A d o u b l e i n t e g ra t o r u n d e r p u re r e l a y

con t ro l wi l l g ive , fo r example , per iod ic so lu t ions

w i t h a n a rb i t r a ry p e r i o d .

I t w o u l d b e h i g h l y d e s i r a b l e t o g i v e a c o m p l e t e

c h a ra c t e r i z a t i o n o f t h e s y s t e m s fo r w h i c h t h e r e w i l l

b e a u n i q u e s t a b l e l im i t c yc l e. Th e o re m 1 a n d t h e

s t a b i l i t y c o n d i t i o n s i n A s t r6 m a n d H ~ i g g l u n d

(1 9 8 4 a ) g i v e s o m e g u i d a n c e , b u t t h e g e n e ra l

c o n d i t i o n s a r e s ti ll u n k n o w n . C o n s i d e r , f o r e x a m p l e ,

s t a b le s y s t e m s . I t f o l l ow s f ro m s a m p l e d d a t a t h e o ry

t h a t

l im

H z , - 1 ) = - G 0 ) = - K

~ ~ c

where G(0)

F u r t h e r m o r e ,

fo l l o w s t h a t

i s the D C g a in o f the p rocess .

i f G s ) goes to zero as s --* oo i t also

H ( 0 , - 1 ) = 0 . P r o v i d e d t h a t e K i s

l a rg e r t h a n d , e q u a t i o n (3) t h u s a l w a y s h a s a t l e a s t

o n e s o l u t i o n . Th e re m a y o f c o u r s e b e s e v e ra l

s o l u t io n s . I t f o ll o w s f ro m Th e o re m 3 o f A s t r6 m a n d

H ~ i g g lu n d (1 98 4 a) a n d T h e o re m 2 o f A s t r6 m et al.

(1984) tha t the pe r iod ic so lu t ion i s s t ab le a t l eas t i fe

i s su f fi c ien t ly l a rge , I t i s con je c tu re d th a t there i s a

un ique s tab le l imi t cyc le fo r s t ab le sys tems .

4. A M P L I T U D E M A R G I N A U T O - T U N E R S

When the c r i t i ca l po in t i s known, i t i s

s t r a i g h t fo rw a rd t o f i n d a r e g u l a t o r w h i c h g i v e s a

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

a p ro p o r t i o n a l r e g u l a t o r w i t h t h e g a i n

k = A c / A

6 )

whe re AM is the des i re d am pl i tud e ma rg in and kc i s

the c r i t i ca l ga in .

S o m e t i m e s t h i s s o l u t i o n i s n o t s a t i s f a c t o ry

b e c a u s e i n t e g ra l a c t i o n m a y b e r e q u i r e d . S i n c e th e

f r e q u e n c y re s p o n s e o f a P I D r e g u l a t o r c a n b e

w r i t t e n a s

G R i c o ) = k l + i o J T i 1 - o o 2 T i T a ) ) (7)

i t f o l lo w s t h a t a n y P ID r e g u l a t o r w i t h t h e g a i n g i v e n

by (6 ) and

1

T ~ - 2 (8)

o9~ T/

wh ere ~oc = 2 n / 6 a l s o g i v e s t h e d e s i r e d a m p l i t u d e

m a rg i n . Th e i n t e g ra t i o n t i m e c a n t h e n b e c h o s e n

arb i t ra r i ly , and the der iva t ive t ime i s g iven by

equat ion (8 ) .

5. P H A S E M A R G I N A U T O - T U N E R S

C o n s i d e r a s i t u a t i o n w h e n o n e p o i n t o n t h e

N y q u i s t c u rv e fo r t h e o p e n l o o p s y s t e m i s k n o w n .

W i t h P I , P D o r P I D c o n t ro l i t i s p o s s i b le t o m o v e

t h e g i v e n p o i n t o n t h e N y q u i s t c u rv e to a n a rb i t r a ry

pos i t ion in the com plex p lane , as i s ind ica ted in F ig .

3 . Th e p o i n t A m a y b e m o v e d i n t h e d i r e c t i o n o f

I ~ I r n G

iu2 O iu2)

F I G . 3 . S h o w s t h a t a g i v e n p o i n t o n t h e N y q u i s t c u r v e m a y b e

m o v e d t o a n a r b i t r a ry p o s i t i o n in t h e G - p la n e b y P I P D o r P I D

c o n t r o l . T h e p o i n t A m a y b e m o v e d i n t h e d i r e c t i o n s G ioJ),

G ioJ)/i~

a n d

ioJG io))

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

d e r iv a t iv e g a in r e s p ec t iv e ly .

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

5/7

Au t o m a t i c t u n i n g o f s i m p l e r e g u l a t o r s 6 49

G ( i o )

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

d i r e c t i o n b y c h a n g i n g t h e i n t e g ra l o r t h e d e r i v a t iv e

ga in . I t is thus poss ib le to mo ve a spec i f ied po in t on

t h e Ny q u i s t c u rv e t o a n a rb i t r a ry p o s i t i o n . T h i s i d e a

c a n b e u s e d t o o b t a i n d e s i g n m e t h o d s . S y s t e m s w i t h

a p re s c r i b e d p h a s e -m a rg i n a r e o b t a i n e d b y , e . g .

m o v i n g A t o a p o i n t o n t h e u n i t c i rc le . An e x a m p l e i s

g iven be low.

E x a m p l e

3 . C o n s i d e r a p ro c e s s w i t h t h e tr a n s f e r

fu n c t i o n

G ( s ) .

T h e l o o p t r a n s f er f u n c t io n w i t h P I D

con t ro l i s

k ( l + s T a + s l ~ i i ) G ( s ) •

As s u m e t h a t t h e p o i n t w h e re t h e Ny q u i s t c u rv e o f G

in tersec t s the nega t ive rea l ax i s i s known. Le t th i s

p o i n t c o r r e s p o n d t o c o = c o c . T h e fo l l o wi n g c o n -

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

a rg u m e n t o f t h e l o o p t r a n s f e r fu n c t i o n a t c ot is

] m - - 7 ~ .

1

co~Td -- - - = ta n ~b~. 9)

The re a re m an y Td and T / wh ich sa t i s fy th i s

con d i t ion . O ne poss ib i l i ty i s to cho ose T~ an d Td SO

t h a t

T~ = ~r a. 10)

E q u a t i o n 9 ) t h e n g iv e s a s e c o n d o rd e r e q u a t i o n fo r

Td which has the so l u t ion

ta n q~,. + ~ + tanZq~,,

T~ = 11)

26o~

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

func t ion h as un i t ga in a t co~ i f the re gu la to r ga in i s

c h o s e n a s

k - cosq~,. _ k~cos~ b,. 12)

I G ( i o ¢ ) l

whe re kc i s the c r i t i ca l ga in . The des ign ru les a re thus

g i v e n b y t h e e q u a t i o n s 9 ) - 1 2 ) .

There a re many o ther poss ib i l i t i es , e .g . the

par am ete r T~ m ay be ch osen so th a t ogcT~has a g iven

value . [ ]

A p o i n t o n t h e Ny q u i s t c u rv e wh i c h i s d i f f e r e n t

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

h y s t e re s i s . T h e d e s i g n m e t h o d i n E x a m p l e 3 c a n b e

e x t e n d e d t o c o v e r t h i s c a s e t o o . T h e n e g a t i v e

re c i p ro c a l o f t h e d e s c r i b i n g fu n c t i o n o f a r e l a y w i t h

hysteresis is

1 n ~ - e z . h e 1 3 )

U ( a ) - ~ d x / a - - t ~

wh e re d i s t h e r e l a y a m p l i t u d e a n d e is t h e h y s t e r e s is

w i d t h . T h i s fu n c t i o n m a y b e d e s c r i b e d a s a s t r a i g h t

l ine para l le l to the rea l ax i s , in the complex p lane .

S e e F i g . 4 . B y c h o o s i n g t h e r e l a t i o n b e t we e n e a n d d

i t is t h e r e fo re p o ss i b l e t o d e t e rm i n e a p o i n t o n t h e

Ny q u i s t c u rv e w i t h a s p e c i f ie d i m a g i n a ry p a r t . I n t h e

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

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

sys tem.

E x a m p l e

4 . C o n s i d e r a p ro c e s s w i t h t r a n s f e r

fu n c t i o n G s ), c o n t ro l l e d b y a p ro p o r t i o n a l r e g u -

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

k G ( s ) .

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

l o o p s y s t e m wi t h t h e p h a s e m a rg i n q ~,,. C h o o s e t h e

re l a y c h a ra c t e r i s ti c s s o t h a t t h e n e g a t i v e r e c i p ro c a l

o f th e d e s c r i b i n g fu n c t i o n g o e s t h ro u g h t h e p o i n t P

d e f i n e d in F i g . 4 . T h e p a ra m e t e r s a r e t h e n

/ra*

d = -~ - e = a* s in ~b,,)

wh e re a * i s t h e d e s i r e d a m p l i t u d e o f th e o s c i l l a ti o n s .

T h e d e s ir e d p h a s e m a r g i n i s o b t a i n e d i f t h e N y q u i s t

c u rv e g o e s t h ro u g h t h e p o i n t P i n F i g . 4. S i nc e t h e

i n t e rs e c t io n b e t w e e n - 1 / N a ) a n d

k G ( k o )

c a n b e

d e t e rm i n e d f ro m t h e a m p l i t u d e o f t h e o s c i ll a t io n ,

th i s po in t can be reached , e .g . by i t e ra t ive ly

c h a n g i n g t h e g a i n k . T h e fo rm u l a

k , + 1 = k , - ( a , - a * ) k

k . -

1

an an 1

14)

h a s a q u a d ra t i c c o n v e rg e n c e r a t e n e a r t h e s o l u t i o n .

In t e g ra l a n d d e r i v a t iv e a c t i o n c a n b e i n c l u d e d , u s i n g

t h e m e t h o d s p ro p o s e d i n E x a m p l e 3 .

T h e re a r e m a n y p o s s i b le v a r i a t i o n s o f th e g i v e n

d e s ig n m e t h o d s f o r P I D r e g u la t o rs . A l l m e t h o d s a r e

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

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

p o i n t o n t h e Ny q u i s t c u rv e . T h e p o i n t s wh e re t h e

Ny q u i s t c u rv e i n t e r s e c t s t h e r e a l a x i s o r s t r a i g h t

l ines para l le l to the rea l axes a re s im ple cho ices . The

0 m

N A )

FIG. 4. The negative reciprocal of the describing function

N(a)

and the Nyquist curve of

G(s).

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

6/7

6 5 0 K . J . / ~ S T R O M and T. H) i .GGLUND

d e s i g n m e t h o d s m a y b e m o d i f i e d . O t h e r r e l a t i o n s

be tw een T~ an d Ta tha n tho se g iven by 10) m ay be

u s ed . O t h e r c r it e ri a l ik e d a m p i n g o r b a n d w i d t h m a y

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

m a rg i n s . I t i s a ls o p o s s i b le t o h a v e d e s i g n m e t h o d s

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

N y q u i s t c u rv e . Se e A s t r6 m a n d H / i g g l u n d 1 98 4b ).

6 E X P E R I M E N T S

A l a r g e n u m b e r o f s im u l a t i o n s a n d e x p e r im e n t s

o n l a b o ra t o ry p ro c e s s e s a n d i n d u s t r i a l p l a n t s h a v e

b e e n p e r fo rm e d i n o rd e r t o f i n d o u t i f a u s e fu l a u t o -

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

in the p rev ious sec t ions . The resu l t s o f the

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

Pr ac t i c a l a s pe c t s

Th e re a r e s e v e ra l p r a c t i c a l p ro b l e m s w h i c h m u s t

b e s o l v e d i n o rd e r t o i m p l e m e n t a n a u t o - t u n e r . I t is ,

f o r e x a m p l e , n e c e s s a ry t o a c c o u n t f o r m e a s u re m e n t

n o i se , le v el a d j u s t m e n t , s a t u r a t i o n o f a c t u a t o r s a n d

a u t o m a t i c a d j u s t m e n t o f th e a m p l i t u d e o f t h e

osc i l l a t ion .

M e a s u r e m e n t n o i se m a y g i v e e rr o r s i n d e t e c ti o n

of peaks and zero -cross ings . A hys te res i s in the re lay

i s a s imple way to reduce the in f luence o f

m e a s u re m e n t n o i s e . F i l t e r i n g is a n o t h e r p o s s ib i l it y .

Th e e s t i m a t i o n s c h e m e s b a s e d o n l e a st s q u a re s a n d

e x t e n d e d K a l m a n f i l t e r i n g c a n b e m a d e l e s s se n s it i ve

t o n o i s e. S i m p l e d e t e c t i o n o f p e a k s a n d z e ro -

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

re lay has worked very wel l in p rac t i ce . See , e .g .

As t r6m 1982) .

Th e p ro c e s s o u t p u t m a y b e f a r f r o m t h e d e s i r e d

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

swi tched on . In such cases i t wou ld be des i rab le to

h a v e t h e s y s t e m r e a c h i t s e q u i l i b r i u m a u t o m a t i c a l l y .

F o r a p ro c e s s w i t h f i n it e lo w - f r e q u e n c y g a i n t h e r e i s

n o g u a ra n t e e t h a t t h e d e s i r e d s t e a d y s t a t e w i l l b e

a c h i e v e d w i t h r e l a y c o n t ro l u n l e s s t h e r e l a y

a m p l i t u d e i s s u f f ic i e n tl y l a rg e . To g u a ra n t e e t h a t t h e

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

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

reset .

I t i s a l s o d e s i r a b le t o a d j u s t t h e r e l a y a m p l i t u d e

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

t h a t t h e o s c i l l a t i o n is a g i v e n p e rc e n t a g e o f th e

a d m i s s i b l e s w i n g in t h e o u t p u t s i g na l .

D i f f e r e n t e s t i m a t i o n s c h e m e s h a v e b e e n e x p l o re d

b y s i m u l a t i o n s c o v e r i n g w i d e r a n g e s o f p ro ce s s

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

e f fe c t s o f m e a s u re m e n t n o i s e a n d l o a d d i s t u rb a n c e s

h a v e b e e n i n v e s t i g a t e d . Th e e x p e r i m e n t s i n d i c a t e

t h a t t h e s i m p l e e s t i m a t i o n m e t h o d b a s e d o n z e ro -

c ro s s i n g a n d p e a k d e t e c t i o n w o rk s v e ry w e l l . Th e

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

a d j u s t m e n t m e t h o d s o f t e n a r e s a t i s f a c t o ry .

I m p l e m e n t a t i o n s

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

s e v e ra l d if f e r e n t c o m p u t e r s . A D EC LS I 1 1/ 03 w a s

u s e d i n s o m e e a r l y e x p e r i m e n t s . Th e a l g o r i t h m s

w e re c o d e d i n P a s c a l u s i n g a r e a l t i m e k e rn e l. S m a l l

l a b o ra t o ry p ro c e s s e s w e re c o n t ro l l e d . Th e e x p e r i -

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

ro b u s t a n d t h a t t h e y w o rk e d w e ll . Th e a l g o r i t h m s

w e re a l s o c o d e d i n B a s i c u s i n g t h e A p p l e I I

c o m p u t e r . Th i s i m p l e m e n t a t i o n w a s v e ry e a s y t o u se

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

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

u s i n g t h e IB M P C a n d d e d i c a t e d m i c ro p ro c e s s o r s .

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

A n e x p e r i m e n t m a d e w i t h t h e A p p l e I I i m p l e -

m e n t a t i o n w i ll n o w b e p r e s e n t e d . F i g u re 5 s h o w s t h e

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

c o n t ro l i n a t a n k w i t h a p u m p a n d a f re e o u t l e t . Th e

p u m p w a s c o n t r o ll e d f r o m m e a s u r e m e n t s o f th e

w a t e r l e v e l . Th e t u n i n g p ro c e d u re c a n b e d i v i d e d

in to two phases . The f i r s t phase i s an in i t i a l phase

which b r ings the p rocess to equ i l ib r ium, i . e . to the

des i red re fe rence l eve l. The secon d pha se i s the f ina l

t u n i n g p h a s e . Th e t w o p h a s e s a r e d e s c r i b e d i n m o re

de ta i l be low.

P h a s e 1 . W h e n t h e p ro c e s s d y n a m i c s i s t o t a l l y

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

h a l f w a y b e t w e e n t h e c u r r e n t a n d t h e d e s ir e d

s e t p o i n t. A c ru d e e s t i m a t e o f c ri t ic a l g a i n a n d c r it ic a l

p e r i o d i s m a d e b a s e d o n o n e p e r i o d o f o sc i l la t i o n .

Th is i s done in the f i r s t phase . Based on th i s rough

c h a ra c t e r i z a t i o n o f t h e p ro c e ss , a c o n s e rv a t i v e P I

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

e q u i l i b r i u m w i t h a s l o p e d e t e rm i n e d f ro m t h e

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

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

equ i l ib r ium.

P h a s e

2 . When the des i red l eve l i s reached , the

e s t i m a t i o n p ro c e d u re s t a r t s . A r e l a y w i t h a s m a l l

° T 7~ ̂ ^ ^

2 4

fs]

0 2

2 4

Phase I Phase

F IG 5 E x p e r i m e n t s m a d e o n t h e t a n k p r o c e s s

7

Is]

8/17/2019 1984_Automatic tuning of simple regulators with specifications on phase and amplitude margins.pdf

7/7

Automatic tuning of simple regulators 651

hysteresis is introduced in the loop as shown in Fig.

I. The relay amplitude is adjusted automatical ly so

that an oscillation with desired amplitude is

obtained. The amplitude and the frequency of the

oscillation are estimated by peak detection and

determinat ion of the times between zero-crossings

of the control error.

The design method was based on a combination

of phase and amplitude margin specification. It was

required that the Nyquist curve intersects the circle

with radius 0.5 at an angle of 225 ° Two step

responses are shown in Fig. 5. The lack of symmetry

depends on the nonl inearity of the pump. The high

frequency disturbance in the control signal is caused

by round-off errors in the AD-converter, eight bits

only.

7. I N I T I A L I Z A T I O N O F A D A P T I V E C O N T R O L L E R S

The new estimation procedure presented in this

paper has been used to derive a technique to tune

simple regulators automatically . Initial ization of

conventional adaptive controllers is another impor-

tant application. Adaptive and self-tuning con-

trollers based on parameter estimation require prior

knowledge of the magni tude of the time delay of the

process. This is needed to select the sampling period.

An upper bound of the time delay is given by tc /2

which follows from Example 2 for first order

systems. It is also true for systems having monotone

step responses. A suitable sampling period for a self-

tuner can thus easily be determined from the upper

bound of the time delay. Having obtained a suitable

sampling period, parameter estimation may also be

applied to the signals obtained during the auto-

tuning to give good initial parameter estimates for a

self-tuner. By combining the auto-tuner with a self-

tuner of the type discussed in Astr6m and

Wittenmark 1973), it is possible to obtain an

adaptive regulator which can work for processes

having a wide range of time delays and time

constants.

Another interesting system is obtained by

combining an au to-tuner with the bandwidth self-

tuner discussed in Astr6m 1983). A reasonable

estimate of the desired bandwidth can be obtained

from the critical period. It is then possible to design

an adaptive regulator which by itself can determine

a reasonable value of the closed loop bandwidth and

then execute a control law which gives this. More

elaborate combinations of control algorithms are

suggested in Astr~Sm and Anton 1984).

8 . C O N C L U S I O N S

Simple methods for tuning PID regulators have

been proposed. The methods have been investigated

theoretically and experimentally. The methods are

robust and easy to use. In contrast to other methods

based on self-tuning control they do not require a

p r i o r i information about time scales. The methods

will of course inherit the limitations of the PID

algorithms. They will not work well in situations

where more complicated regulators are required.

The algorithms may be used in many different

ways. They may be incorporated in single loop

controllers to provide an option for automatic

tuning. They may also be used to initialize more

sophisticated adaptive algorithms.

Ac k nowle dge me nts - -Thi s

work was pa r t ia l ly suppor ted by

research grant 82-3430 from the Swedish Board o f Technical

develo pme nt STU). This support is gratefully acknowledged.

The au thors a re a l so g ra te fu l to Per Hagander who has g iven

useful comm ents on different versions of the man uscript a nd to

Lars B~i~th, Kai Siew Wong, La rs G6 ran Elfgren and Kalle

As t r6m who have ass is ted wi th p rogramming s imula t ions and

experiments.

R E F E R E N C E S

Astr6m, K. J . 1982) . Z ieg le r -N icho ls au to - tuners . Repor t

TFRT-3167 . Dep t o f Au tom at ic Con t ro l , Lund Ins t i tu te o f

Technology, Ltmd, Sweden.

Astr6m , K. J . 1983). Theo ry and applicatio ns of adap tive control.

Automatica,

19, 471-486.

Astr6m, K. J . and J. Anton 1984). Expert control.

Proceedings

IF AC 9th W or ld Congres s ,

Budapest.

Astr6m , K. J . , P . Hagan der and J. Sternby 1984). Zeros of

sampled systems.

Automatica,

20, 31-38.

Astr6m , K. J . and T. H~igglund 1984a). Autom atic tuning of

simple regulators.

Proceedings IFAC 9th World Congress ,

Budapest.

~.str6m, K. J . and T. H~gglund 1984b). A frequency domain

appro ach to analysis a nd design of simple feedback loops.

Proceedings 23rd I EE E ConJerence on Decis ion and Control ,

Las Vegas.

Astr6m, K . J . and B. Witte nm ark 1973). On self- tuning

regulators.

Automatica,

9, 185-199.

Astr6m, K . J . and Z. Zhaoy ing 1981). Self-tuners w ith automa tic

adjustm ent o f the sampling perio d for processes with t ime

de lays. Rep or t TFRT-7229 . D ep t o f Au tomat ic Con t ro l , Lund

Insti tute of Technolog y, Lund, Sweden.

Ather ton , D. P . 1975) .

Nonlinear Control Engineering--

Describing Function Analysis and Design.

V a n N o s t r a n d

Reinho ld , London .

Athe rton, D . P. 1982). Limit cycles in relay systems.

Electronic

Lett.,

15, No. 2l .

Gaw throp , P. J . 1982). Self-tuning P I and P ID regulators, .

Proceedings lE E Con[erence on Applications o[ Adaptive and

Multivariable Control,

H u l l

Gelb, A. and W . E. Van der Velde 1968).

Multiple-input

Describing Functions and Nonlinear Systems Design.

M c G r a w

Hill , New York.

Gille, J . C. , M. J . Pelegrin and P. Decauln e 1959).

Feedback

Control Systems.

McGraw-Hi l l , New York .

Kurz, H. 1979). Digital paramete r-adaptiv e contro l of processes

wi th unknown cons tan t o r t ime-vary ing dead t ime .

Preprints

5th IF A C Symposium on ldent f ication and P arameter

Estimation,

Darmstad t .

Tsypkin, J . A. 1958 ).

Theorie der Relais Systeme der

Automatischen Regelung.

R. Oldenburg , Munich .

Witten mark , B. and K . J. Astr6m 1980). Simple self- tuning

controllers . In H. Un behau en ed.),

Methods and Applications

in Adaptive Control. Springer, Berlin.

Ziegler, J. G. and N. B. Nich ols 1943). Op tim um settings for

autom atic controllers .

Trans . ASME,

65, 433-444.