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Automatica Vol. 28, No. 1, pp. 1-9, 1992Printed n Great Britain. 0005-1098/92 $5.00 + 0.OBPergamon Presspie~0 1991 International Federationof AutomaticControl

T o w a r d s I n t e l l i g e n t P I D C o n t r o lK. J. /~STROM,'~, C. C. HAN G,, P. PERSSON~ and W. K. HO,D i m e n s i o n l e s s n u m b e r s c h a r a c t e r i z i n g o p e n - l o o p p r o c e s s d y n a m i c s a n dc l o s e d - lo o p P I D b e h a v i o u r a r e i n tr o d u c e d a n d u s e d i n a s s es s in g p e r f o r -m a n c e a n d t u n i n g r e la t io n s b e t w e e n th e s e n u m b e r s a r e d e r i v e d u s i n ga n a l y t i c a l m e t h o d a n d s i m u l a t i o n s

ey Words--Automatic tuning; conventional control; expert systems; industrial control; knowledgeengineering; PID control; process control; threeterm control.

Almtm et--Au totu ners for PID controllers have beencommercially available since 198 1. These controllersautomate some tasks normally performed by an instrumentengineer. The autotuners include methods for extractingprocess dynamics from experiments and control designmethods. They may be able to decide when to use PI or PIDcontrol. To make systems with a higher degree of automationit is desirable to also automate tasks normally performed byprocess engineers. To do so, it is necessary to provide thecontrollers with reasoning capabilities. This seems technicallyfeasible with the increased computing power that is nowavailable in single-loop controllers. This paper describessome features that may be included in the next generation ofPID controllers.

1. INTRODUCTIONIs THE DESIGN of a knowledge-based feedbackcontroller (/~,strrm et al. 1986;/~rz~n, 1989), itis desirable to incorporate the expert knowledgeof design engineers so that the controller canmake decisions on the choice of controlalgorithm and provide diagnostics on theeffectiveness of the control system. For real-timeimplementation, it is desirable to have as muchdeep knowledge as possible. It is also desirablethat the controller to a limited degree couldexplain its own reasoning, e.g. why derivativeaction was used. It should also be able to tell ifPID control is appropriate and if not, suggestalternative control schemes.

This paper attempts to develop tools to assesswhat can be achieved by PID control of two

* Received 1 Sept ember 1990; revised 28 April 1991;received in final form 19 June 1991. The original version ofthis paper was presented at the IFAC Workshop on AI inReal Time Control which was held in Shenjang, People'sRepublic of China during September 1989. The publishedproceedings of this IFAC Meeting may be ordered from:Pergamon Press pie, Headington Hill Hall, OxfordOX30BW, U.K. This paper was recommended forpubfication in revised form by Editor A. P. Sage~~ Department of Automat ic Cont rol, Lurid Institute ofTechnology, Box 118, S-221 00 Lurid, Sweden.~t Departmen t of Electrical Engineering , National Univers-ity of Singapore, 10 Kent Ridge Crescent, Singapore 0511.Author to whom all correspondence should beaddressed.

broad classes of systems with a Ziegler-Nichols(1942) like tuning formula. Based on empiricalstudies and approximate analysis, we introducetwo parameters, namely, the normalized dead-time and the normalized process gain tocharacterize the open-loop process dynamics.Two more parameters, the peak load error andthe normalized rise time are also introduced tocharacterize the closed-loop response. Simplemethods of measuring these parameters areproposed.

It is shown that the normalized deadtime andthe normalized process gain can be used topredict the achievable performance of PIDcontrollers tuned by the Ziegler-Nichols for-mula. Using these relations, the controller caninteract with the operator and advise him on thechoice of control algorithms. If desired, it canalso make the choice automatically and explainits reasoning.

Useful relations, which can be used to assesswhether the PID controller is properly tuned arealso established. The relations give significantinsight into the properties of PID control. Thesimplicity of the relations allows the develop-ment of a first generation of intelligentcontrollers using current technology.

The paper is organized as follows. Two classesof processes are introduced in Section 2. Someuseful dimensionless numbers are introduced inSection 3. In Sections 4 and 5, some relationsbetween the features are derived based onapproximate analysis and simulations. Theresults are used in Section 6 to predict theperformance that may be achieved with PIDcontrol based on Ziegler-Nichols tuning. Concl-usions are presented in Section 7.

2. PROCESS CHARACTERISTICSIt is assumed that process dynamics can bedescribed with sufficient accuracy by linear

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2 K . J . ~ STROM t al.m o d e l s . A d d i t i o n a l c o n s t r a i n t s o n s y s t e m d y n a -m i c s b o t h i n t h e t i m e a n d t h e f r e q u e n c y d o m a i n sa r e a l s o i n t r o d u c e d . T h i s w i l l l e a d t o t w o c l a s s e so f s y s te m s t h a t a r e c o m m o n i n p ro c e s s c o n t r o l .Time domain characterization

I t w il l b e a s s u m e d t h a t t h e s t e p r e s p o n s e i sm o n o t o n e o r e s s e n t ia l l y m o n o t o n e , i . e . m o n o -t o n e e x c e p t f o r a s m a l l i n i t i a l p a r t . S u c hp r o c e s s e s c a n b e d i v i d e d i n t o t w o b r o a d c l a s s e s .T h e f i r s t c l a s s c o r r e s p o n d s t o s t a b l e p r o c e s s e s .T h e i r d y n a m i c s c a n b e c h a r a c t e r i z e d b y t h ep a r a m e t e r s Kp, L a n d T , w h e r e Kp i s the s ta t i cp r o c es s g a in , L i s t h e a p p a r e n t d e a d t i m e a n d Ti s t h e a p p a r e n t t i m e c o n s t a n t . T h e s e p a r a m e t e r sc a n b e o b t a i n e d f r o m a s t e p r e s p o n s e e x p e r i m e n t( F i g . 1 ) . T h e t r a n s f e r f u n c t i o n

Kp e_sL (1)G s) = 1 + sTi s a c r u d e a n a l y t i c a l a p p r o x i m a t i o n o f s t a b l epr oces ses .

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

KG(s) - e - 'L (2)s ( 1 + sTo)i s a c r u d e a n a l y t i c a l a p p r o x i m a t i o n o f s u c h ap r o c e s s . A n e v e n s i m p l e r m o d e l i s

G(s) = K~ e_~L (3)sN o t i c e t h a t t h e t r a n s f e r f u n c t i o n ( 3 ) m a y b er e g a r d e d a s t h e l i m i t o f ( 1 ) w h e n Kp a n d T g o t oi n f i n it y i n s u c h a w a y t h a t K p / T = c o n s t a n t = K o .

T h e s y s t e m s c o n s i d e r e d w e r e u s e d i n t h ec l a ss i ca l w o r k s o n Z i e g l e r - N i c h o l s t u n i n g . I nt r a d i t i o n a l p r o c e s s c o n t r o l l i t e r a t u r e t h e c l a s s e so f s y s t e m s a r e c a l l e d p r o c e s s e s w i t h s e l f -r e g u l a t i o n ( s t a b l e p r o c e s s ) a n d p r o c e s s e s w i t h o u ts e l f - r e g u l a t i o n ( p r o c e s s e s w i t h i n t e g r a t i o n ) .N o t i c e t h a t s y s t e m s w i th r e s o n a n t p o l e s a r e n o tc o n s i d e r e d , s u c h p r o c e s s e s d o n o t h a v e e s s e n -t ia l ly m o n o t o n e s t e p r e s p o n s e .

Kp

I_ TF IG 1 G r a p h i c a l d e t e r m i n a t i o n o f m a t h e m a t i c a l m o d e l f o r as t a b l e p r o c e s s

Frequency domain characterizationA f r e q u e n c y d o m a i n c h a r a c t e r i z a t io n o f

p r o c e s s d y n a m i c s w i ll al s o b e i n t r o d u c e d . I t isa s s u m e d t h a t t h e N y q u i s t c u r v e i s m o n o t o n e o re s s e n ti a l ly m o n o t o n e , i . e . b o t h t h e p h a s e a n dt h e a m p l i t u d e a r e m o n o t o n e f u n c t i o n s o ff r e q u e n c y . T h i s c o n d i t i o n a l s o e x c l u d e s p r o c -e s s e s w i t h r e s o n a n c e s . T h e m a i n d i f f e r e n c eb e t w e e n t h e t w o c l a s s e s o f p r o c e s s e s i s t h a t a tz e r o f r e q u e n c y , s t a b l e p r o c e s s e s h a v e f i n i t e g a i nw h e r e a s p r o c e s s e s w i t h i n t e g r a l a c t i o n h a v ei n f i n i t e g a i n . B o t h s y s t e m s c a n b e c h a r a c t e r i z e db y t h e f i r st in t e r s e c t i o n o f t h e N y q u i s t c u r v e w i t ht h e n e g a t i v e r e a l a x i s . T h i s d e f i n e s t h e u l t i m a t eg a i n , K u , i . e . t h e c o n t r o l l e r g a i n t h a t m a k e s t h ep r o c e s s u n s t a b l e u n d e r p r o p o r t i o n a l f e e d b a c k ,a n d t h e u l t i m a t e p e r i o d T u . L a c k o f m o n o t o n i -c i ty c an b e a c c e p t e d a t f r e q u e n c i e s w h e r e t h epha se l ag i s l a r ge r tha n 180

3 F E A T U R E SD i m e n s i o n - f r e e p a r a m e t e r s , l ik e R e y n o l d sn u m b e r s , h a v e f o u n d m u c h u s e i n m a n y

b r a n c h e s o f e n g i n e e r i n g . T h e y h a v e , h o w e v e r ,n o t b e e n m u c h u s e d i n a u t o m a t i c c o n t r o l . T h i ss e c t i o n a t t e m p t s t o i n t r o d u c e s o m e d i m e n s i o n -f r e e n u m b e r s t h a t a r e u s e f u l i n a s s e s s i n g c o n t r o ls y s t em p e r f o r m a n c e .

Normal i zed dead t imeA n o r m a l i z e d d e a d t i m e c a n b e d e f i n e d f o r

s t a b l e p r o c e s s e s a s t h e r a t i o o f t h e a p p a r e n td e a d t i m e t o t h e a p p a r e n t t i m e c o n s t a n tL0 1 : ~ (4)

T h i s n u m b e r c a n b e e s t i m a t e d f r o m a r e c o r d o ft h e s t e p r e s p o n s e . I t h a s b e e n k n o w n f r o mp r a c t i c a l e x p e r i e n c e t h a t t h e n o r m a l i z e d d e a d -t i m e c a n b e u s e d a s a m e a s u r e o f t h e d i f f ic u l t y i ncont r o l l ing a p r oces s . P r oces ses wi th a sm a l l 01a r e e a s y t o c o n t r o l a n d p r o c e s s e s w i t h a l a r g e 0 1a r e d i f fi c u l t t o c o n t r o l . F e r t i k ( 1 97 5 ) i n t r o d u c e dt h e n a m e p r o c e s s c o n t r o l l a b i l i t y f o r t h e r e l a t e dq u a n t i t y 0 1 / ( 1 + 0 l ) . P a r a m e t e r 0 1 w a s a c t u a l l yc a ll e d t h e c o n t r o ll a b i l it y ra t i o b y D e s h p a n d e a n dA s h ( 1 98 1 ) . T o a v o i d p o s s ib l e c o n f u s i o n w i t h t h es t a n d a r d t e r m i n o l o g y o f m o d e r n c o n t r o l t h e o r y ,t h e w o r d s n o r m a l i z e d d e a d t i m e w i ll b e u s e d .A r a t i o a n a l o g o u s t o e q u a t i o n ( 4 ) c a n b ei n t r o d u c e d f o r p r o c e s s e s w i t h i n t e g r a t i o n . L e tG(s ) b e t h e t r a n s f e r f u n c t i o n o f s u c h a p r o c e s s .T h e t r a n s f e r f u n c t i o n s G ( s ) b e l o n g s t o t h e c l a s so f st a b l e p r o c e s se s . I t s b e h a v i o u r c a n t h e n b ec h a r a c t e r i z e d b y a n a p p a r e n t d e a d t i m e , L a n da n a p p a r e n t t i m e c o n s t a n t , T ~ . T h e n o r m a l i z e d

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T o w a r d s i n t e l li g e n t P I D c o n t r o l 3d e a d t i m e f o r p r o c e s s e s w i t h i n t e g r a t i o n

L 5 )c a n t h e n b e i n t r o d u c e d .

S i n c e t h e t r a n s f e r f u n c t i o n ( 3 ) m a y b er e g a r d e d a s t h e l im i t o f t h e t r a n s f e r f u n c t i o n ( 1 ) ,p r o c e s s e s w i t h i n t e g r a t i o n m a y b e c o n s i d e r e d a sa s p e c i a l c as e o f s t a b l e p r o c e s s e s w i t h v e r y s m a l lv a l u es o f n o r m a l i z e d d e a d t i m e .N o r m a l i z e d p r o c e s s g a in

P a r a m e t e r K p i s t h e p r o c e s s g a i n f o r s t a b l ep r o c e s s e s . I t i s n o t n e c e s s a r i ly d i m e n s i o n - f r e e . I tc a n h o w e v e r b e m a d e d i m e n s i o n - f r e e b ym u l t i p l y i n g w i t h a f a c t o r . T h e u l t i m a t e g a i n , K u ,i s a s u i t a b l e n o r m a l i z a t i o n f a c t o r . T h e n o r m a l -i z e d p r o c e s s g a i n , r l , i s d e f i n e d a s

G ( O )r~ = IG( / ~ o , ) l - K p K , (6 )

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

a r g G i ~ o , ) = - : r .T h i s n u m b e r ~ 1 i s e a s i l y o b t a i n e d f r o m t h e

N y q u i s t c u r v e . I t a l s o h a s a p h y s i c a l i n t e r -p r e t a t i o n a s t h e l a r g e s t p r o c e s s l o o p g a i n t h a tc a n b e a c h i e v e d u n d e r p r o p o r t i o n a l c o n t r o l . T h i sn u m b e r i s u s e f u l f o r a s s e s s i n g c o n t r o l p e r f o r -m a n c e . R o u g h l y s p e a k i n g , a l a rg e v a l u ei n d i c a te s t h a t t h e p r o c e s s i s e a s y t o c o n t r o l w h i l ea s m a l l v a l u e i n d i c a t e s t h a t t h e p r o c e s s i s d i f f i c u l tt o c o n t r o l .

S t a b l e p r o c e s s e s h a v e a s t e a d y - s t a t e e r r o ru n d e r p r o p o r t i o n a l f e e d b a c k . T h e e r r o r o b t a i n e df o r a s t e p c o m m a n d o f s i z e So i s

1 1es = 1 + KpK ~ So > 1 + ~ 1 SO (7 )w h e r e K c i s t h e p r o p o r t i o n a l g a i n u s e d . T h ei n e q u a l i t y f o l l o w s b e c a u s e Kp K~ < r x . T h en u m b e r x l c a n t h u s b e u s e d t o e s t i m a t e t h em i n i m u m s t e a d y -s t a te e r r o r u n d e r p r o p o r t i o n a lc o n t r o l a n d a l so t o d e t e r m i n e i f in t e g r a l a c t i o n i sr e q u i r e d i n o r d e r t o s a t i s f y s p e c i f i c a t i o n s o ns t a t i c e r r o r .

F o r p r o c e s s e s w i t h i n t e g r a t i o n , t h e p r o d u c tK~Ko h a s d i m e n s i o n f r e q u e n c y . T h e d i m e n s i o n -f r e e p r o c e s s g a i n t h e r e f o r e h a s t o b e d e f i n e dd i f f e r e n t l y . T h e n o r m a l i z e d p r o c e s s g a i n x 2 f o rp r o c e s s e s w i t h i n t e g r a t i o n i s d e f i n e d a s

l i m s G s ) -,o KoK u KoKuT~r 2 = = 8 )w~ IG(i~o~)l w , 2zrw h e r e G s ) i s t h e p l a n t t r a n s f e r f u n c t i o n a n d w ,

is th e s m a l l es t f r e q u e n c y s u c h t h a ta r g G R o ~ ) = - m

W i t h c o n s t a n t s e t - p o i n t a n d n o l o a d d i s t u r b a n c e ,p r o c e s s e s w i t h i n t e g r a t i o n w i l l n o t h a v e as t e a d y - s t a t e e r r o r u n d e r p r o p o r t i o n a l f e e d b a c k .W i t h a r a m p s e t - p o i n t o f v e l o c i ty v 0, t h es t e a d y - s t a t e e r r o r i s1 1ev = v0 > Vo.K v K c r2 .O uLo a d d i s tu rb a n ce er ro r

T h e r e s p o n s e t o s t e p l o a d d i s t u r b a n c e i s a ni m p o r t a n t f a c t o r w h e n e v a l u a t i n g c o n t r o l s y s t -e m s . T h e e f f e c t o f a l o a d d i s t u r b a n c e d e p e n d s o nw h e r e t h e d i s t u r b a n c e a c t s o n t h e s y s t e m . I t w i l lb e a s s u m e d t h a t t h e d i s t u r b a n c e a c t s o n t h ep r o c e s s i n p u t . W i t h p r o p o r t i o n a l c o n t r o l a s t e pl o a d d i s t u r b a n c e o f m a g n i t u d e lo g i v e s th e s t a ti ce r r o r 11 = K plo Kplo> (9)l + Kp Kc l + r lf o r s t a b l e p r o c e s s e s a n d

12 1o K ~o , lo Kil o K~ Tflo- - - > - - 1 0 1K c K ~ W u K ~ ~our2 2z~r2

f o r p r o c e s s e s w i t h i n t e g r a t i o n . T h e q u a n t i t i e sl l / Kp lo ) a n d 12~ou/ Kolo) a r e t h e r e f o r ed i m e n s i o n - f r e e .W h e n a c o n t r o l l e r w i t h i n t e g r a l a c t io n i s u s e dt h e s t a t i c e r r o r d u e t o a s t e p l o a d d i s t u r b a n c e i sz e r o . A m e a n i n g f u l m e a s u r e i s t h e n t h em a x i m u m e r r o r . T o o b t a i n a d i m e n s i o n - f r e eq u a n t i ty f o r s ta b l e p r o c e ss e s , t h e m a x i m u m e r r o ri s d i v i d e d b y Kp . T h u s t h e p e a k l o a d e r r o r f o rs t a b l e p r o c e s s e s , ) . 1 , c a n b e d e f i n e d a s

Z , = _ . 1 . / m a x ( 1 1 )K i l ow h e r e 10 i s t h e a m p l i t u d e o f t h e s t e p l o a dd i s tu r b a n c e a n d l ~ x t h e m a x i m u m e r r o r d u e tot h e s t e p l o a d d i s t u r b a n c e . F o r p r o c e s s e s w i t hi n t e g r a t i o n t h e c o r r e s p o n d i n g q u a n t i t y i s

~ o . 2~~,2= ~vl~ lmax= l~vTulo lmax. ( 1 2 /

No rm a l i zed c lo sed - lo o p r i s e t im eT h e c l o s e d - l o o p r i s e t i m e , tr, i s a m e a s u r e o f

t h e r e s p o n s e s p e e d o f th e c l o s e d - l o o p s y s te m . T oo b t a i n a d i m e n s i o n - f r e e p a r a m e t e r i t w i ll b en o r m a l i z e d b y t h e a p p a r e n t d e a d t i m e , L , o f t h eo p e n - l o o p s y s t e m . T h u s t h e n o r m a l i z e d r is et i m e , z , f o r b o t h c l a ss e s o f p r o c e s s e s a r e d e f i n e ds tr

~ .v L (13)

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o4 K . J . A ST RO M e t a l .4 . E M P I R I C S

T h e Z i e g l e r - N i c h o l s c l o s e d - l o o p t u n i n g p r o -c e d u r e h a s b e e n a p p l i e d t o a l a r g e n u m b e r o fd i f fe r e n t p r o c e ss e s a n d i t h a s b e e n a t t e m p t e d t oc o r r e la t e t h e o b s e r v e d p r o p e r t ie s o f t h e o p e n -l o o p a n d c l o s e d - l o o p s y s t e m s t o t h e f e a t u r e si n t r o d u c e d i n S e c t i o n 3 . R e s u l t s f o r p r o c e s s e sw i t h t h e t r a n s f e r f u n c t i o n s

e SLG a ( s ) ( l + s ) z 0 . 1 - < L - < 3 ( 1 4 )1G z ( s ) = ( l + s ) 3 - - n - < 2 0 ( 1 5 )

1 - t r sG 3 ( s ) = (1 + s) 3 0 -< a~ -< 2 (16 )w i l l b e u s e d t o i l l u s t r a t e t h e f e a t u r e s o f s t a b l ep r o c e s s e s a n d

e-S/~G , ( s ) = 0 < L (17)e-SLG ~ (s ) - - - 0 . 05 -< L - 0 . 8 ( 18)s ( s + 1)

1 - t ~ sG 6(s) = - - 0 .0 5-< a~-< 0 .7 5 (1 9)s ( s + 1)f o r u n s t a b l e p r o c e s s e s . T h e s e m o d e l s c o v e r aw i d e r a n g e o f d y n a m i c c h a r a c t e r i s t i c s f o u n d i nt y p i c a l 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 e P I D c o n t r o l l e r u s e d h a d t h e f o r m1 T. dye] (20)u ~ = K c ( [ 3 y , - y + ~ f e d t - a d t /

e = y ~ - y ( 2 1 )1Y r ( s ) = 1 + s T a / N Y ( s ) (22)

w h e r e u~, y a n d y ~ a r e t h e c o n t r o l l e r o u t p u t ,p r o c e s s o u t p u t a n d s e t - p o i n t r e s p e c t i v e l y . T h en o i s e f i lt e r in g c o n s t a n t , N , i s u s u a l l y i n t h e r a n g eo f 3 t o 2 0 . T h e v a l u e N = 1 0 w a s u s e d i n t h es i m u l a t i o n s . I n c o n v e n t i o n a l P I D c o n t r o l l e r sp a r a m e t e r f l = 1 . I t i s o f t e n a d v a n t a g e o u s t oh a v e a s m a l l e r v a l u e o f f l t o r e d u c e t h eo v e r s h o o t t o s e t p o i n t c h a n g e s . T h i s i s c a l l e d s e tp o i n t w e i g h t i n g ( / ~ s t r 6 m a n d H i i g g l u n d , 1 9 8 8 ) .T h e v a l u e f l = 1 w a s u s e d i n a l l s i m u l a t i o n s .P a r a m e t e r s o f t h e P I D c o n t r o l l e r w e r e d e t e r -m i n e d b y s t r a i g h t f o r w a r d Z i e g l e r - N i c h o l sc l o s e d - l o o p t u n i n g .T h e r e s u l ts o b t a i n e d a r e s u m m a r i z e d i n T a b l e s1 - 3 f o r s t a b l e p r o c e s s e s a n d T a b l e s 4 - 6 f o rp r o c e ss e s w i t h i n t e g r a t io n . T h e o v e r s h o o t o sa n d u n d e r s h o o t u s , o f t h e c l o se d - l o o p s t e pr e s p o n s e a r e a l s o g i v e n .

5 . R E L A T I O N SN o r m a l i z e d d e a d t i m e s , 0 1 a n d Oz, a n d t h e

n o r m a l i z e d p r o c e s s g a i n s , t q a n d x z , h a v e b e e ni n t r o d u c e d t o c h a r a c t e r i z e t h e o p e n - l o o p d y n a -m i c s . P e a k l o a d e r r o r s , ~.a a n d ~ . z , a n dn o r m a l i z e d c l o s e d - l o o p r i s e t i m e , r , h a v e b e e ni n t r o d u c e d t o c h a r a c t e r i z e t h e c l o s e d - l o o pr e s p o n s e . R e l a t i o n s b e t w e e n t h e s e n u m b e r s w i l ln o w b e e s t a b l i s h e d . I n d o i n g s o a n i n t u i t i v ei n t e r p r e t a t i o n o f t h e n u m b e r s w i l l a l s o b ed e v e l o p e d .N o r m a l i z e d d e a d t i m e a n d n o r m a l i z e d p r o c e s sg a i nF r o m T a b l e s 1 - 3 , t h e r e a p p e a r s t o b e ar e l a t i o n b e t w e e n t h e n o r m a l i z e d d e a d t i m e , 0 1 ,a n d t h e n o r m a l i z e d p r o c e s s g a i n , X l . F o r s p e c i f i cp r o c e s s e s i t is p o s s ib l e t o f i n d t h e r e l a t i o n e x a c t l y( s e e A s t r 6 m e t a l . , 1988) . F or exam ple , f o r f i r s to r d e r p r o c e ss e s w i t h d e a d t i m e w e h a v e

L : r - a t a n ~ - ~ - 10 1 - - - ~ ~ ~ 1 - - 1 ( 23 )

S i m i l a r e x p r e s s i o n s c a n a l s o b e d e r i v e d f o r t h ep r o c e s s e s g i v e n b y e q u a t i o n s ( 1 4 ) , ( 1 5 ) a n d ( 1 6 ).T h e r e l a t i o n s a r e s h o w n i n F i g . 2 . I t c a n b ea p p r o x i m a t e d b y t h e e x p r e s s i o n :

= 2 1 1 1 + -/ (' 1 \ ' 3 - - ~ 1 (24)( S e e H a n g e t a l . , 1 9 9 1 . ) T h i s f o r m u l a i si m p o r t a n t b e c a u s e i t m e a n s t h a t t h e n o r m a l i z e dd e a d t i m e , 0 1, a n d t h e n o r m a l i z e d p r o ce s s g a i n ,r l , c a n b e u s e d i n t e r c h a n g e a b l y to a s s es sp r o c e s s d y n a m i c s .A r e l a t i o n a n a l o g o u s t o e q u a t i o n ( 2 3 ) c a n b ed e r i v e d f o r p r o c e s s e s w i t h i n t e g r a t i o n . L e t G ( s )

16~141210

~:1 864200 0 . s i x . s0 l

FIo. 2. Relation between 01 and x~. -- - - - - First orderproce~ with dead0me --- process of 04 L -- - - process of(15), process of 00 ), - - approximation.

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Towards intelligent PID control 5

K 2

4.5\ ~

3 . 5 .~{:,3

2.5:,1.5

0.5

~:> ......:::~?~??.~?.7. .. .. .. .. .. .. .. .

0 . 6 0 : 80 2

F IG . 3 . R e l a t i o n b e t w e e n 0 2 a n d x 2 . - - -- - . - - p r o c e s s o f ( 1 9 ). p r o c e s s o f ( 1 8 ) ,

be the transfer function of a process withintegration. The transfer function s G ( s ) is of theclass of stable processes and can be characterizedby an apparent deadtime, L and an apparenttime constant, To. For a transfer function givenby (2), it can be shown thatL : r / 2 - a tan ~2- 102 = ~ = X/-~-2 1 (25)

The calculation is the same as in Astr6m et al.(1988). This means that r2 can be used as ameasure of the normalized deadtime, 02 (Fig. 3).N o r m a l i z e d p e a k l o a d e r r o r

Consider the closed-loop system obtained withthe controller G c( s ) and the process Gp(s) .Assume that the disturbance enters at the plantinput. The transfer function from the loaddisturbance to the output is1 G g ( s ) G c( s )Gd(S) = G~(s-~ 1 + Gp (s)Gc (s) (26)

A P I D controller with Ziegler-Nichols tuninghas the transfer functionG~(s) = K~(s + o022o~s

and4~ _ _ ~ m2Td T,

This choice gives good rejection of loaddisturbances as discussed in Hang (1989). WithZiegler-Nichols tuning the closed-loop systemhas a bandwidth o9-~7.4/TO (/~str6m andH~igglund, 1988). For frequencies less than o~,

the transfer function (26) can be approximatedby1 2trsa d s ) = G c( S ) Kc( s + a ) z

The response to a step of size lo is2 a'tlo ~al ( t ) =--e - .g

It has a maximum2/0 0.74/0

/ m a g ~ eKc Kcat t = 2Ta.The normalized p6ak load for stable processes isthus

1 0.74 1.23 (27)~ '1 = / - ~ v l m ax = K c ~ / 1The product x13,1 can thus be expected to beconstant. An analogous calculation for processeswith integral action shows that r23.2 can also beexpected to be constant. This is supported by theexperimental results given in Tables 1-6.Consider the range where Ziegler-Nicholstuning is applicable, i.e. 0.15

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o6 K .J . AST ROM et a l .TABLE 1. EXPERIMENTAL RESULTS FOR A SYSTEM WITH G s ) = e s L / s + 1) 2

L T . o s u s 0 1 r I ~ r l ~ t0.1 1.4 68 26 0.14 20.5 0.81 0.08 1.60.2 2.0 61 16 0.18 10.5 1.04 0.15 1.60.4 2.9 49 6 0.25 5.7 0.95 0.27 1.50.6 3.6 41 2 0.32 4.0 0.93 0.36 1.41 . 0 4.8 30 5 0.47 2.7 0.88 0.51 1.41.5 6.1 22 12 0.66 2.1 0 .82 0.64 1.32.0 7.3 16 18 0.84 1.74 0.78 0.73 1.32.5 8.5 13 23 1.02 1.55 0.75 0.80 1.23.0 9.5 12 26 1.21 1.44 0.71 0.86 1.2

TABLE 2. EXPERIMENTAL RESULTS FOR A SYSTEM WITH G s) = 1 / S + 1) nn Tu os us 01 r t r Z~ r t~ ~3 3.6 52 14 0.22 8.0 1.16 0.19 1.54 6.3 38 7 0.32 4.0 1.22 0.35 1.46 10.9 23 10 0.49 2.4 1.11 0.54 1.38 15.2 17 16 0.64 1.88 1.01 0.65 1.2

1 0 19.3 13 20 0.77 1.65 0.93 0.73 1.215 29.5 9 27 1.05 1.39 0.82 0.85 1.220 39.7 6 31 1.29 1.28 0.76 0.91 1.2

T A B L E 3 . E X P E R I M E N T A L R E S U L T S F O R A S Y ST EM W I T H G s)= 1- t rs) /( S + 1 ) 3f T u os us 01 / (1 ~ /~1 g ' l~ ' l

0 3.6 52 15 0.22 8.0 1.16 0.19 1.50.1 4.1 48 10 0.25 6.2 1.09 0.24 1.50.25 4.6 42 6 0.29 4.6 1.09 0.31 1.40.5 5.3 33 3 0.38 3.2 1.16 0.44 1.41.0 6.3 18 4 0.57 2.0 0.98 0.67 1.31.5 6.9 4 12 0.78 1.45 0.89 0.90 1.32.0 7.4 -7 22 1.0 1.14 0.84 1.08 1.2

TABLE 4. E X P E R I M E N T A L R E S U L T S F O R A S Y ST E M W I T HG s ) = e - s L / sT u o s u s 0 2 ~ 2 ~ ~ 2 K 2 ~ 24L 71 12 ~ 1 0.69 1.82 1.8N o t i c e t h a t c h a n g e s i n L o n l y c h a n g e t h e t i m e s c a l e .

H e n c e e x c e p t f o r T t h e re s t o f th e v a lu e s in th et a b l e d o n o t c h a n g e w i t h L.

ha l f the appar e nt de adt im e for pr oc e s s e s w i thinte gr ation . For s tab le pr oc e s s e s Z ie g le r -Nic ho l s tun ing g ive s a c lo s e d- loop r e s pons e w i ththe r i s e t ime e qua l s to the de adt ime . T hi se xpla ins why the tuning r u le doe s no t wor k we l lfor pr oc e s s e s w i th nor mal i z e d de adt ime s lar ge rthan one .

6. ZIEGLE R-NIC HOLS TUNINGT h e r e s u l t s o b t a i n e d w i l l n o w b e c o m b i n e dw i t h p r e v i o u s e x p e r i e n c e t o e v a l u a t e Z i e g l e r -Nic ho l s tun ing . F ir s t no t i c e tha t Z ie g le r -Nic ho l stuning i s ve r y s imple . I t i s bas e d on a s implec har ac te r i z a t ion o f the pr oc e s s dynamic s fr omthe s te p r e s pons e or fr om the c r i t i c a l po int ont h e N y q u i s t c u r v e . T h e Z i e g l e r - N i c h o l s t u n i n gr ule s or modi f i c a t ions o f the m ar e a l s oc ommonly us e d in the indus tr y .Whe n c an Z i e g l e r N i c h o l s t u n i n g b e u s ed ?

T h e r e s u l t s o b t a i n e d s h o w t h a t Z i e g l e r -Nic ho l s tun ing g ive s good r e s u l t s unde r c e r ta inc ondi t ions whic h c an be c har ac te r i z e d by 01 orr l for s tab le pr oc e s s e s and 02 or r 2 for

T A B L E 5 . E X P E R I M E N T A L R E S U LT S F O R A S YS TE M W I T H G s ) = e - s L / s S + 1)L T. os us 02 = L r 2 r ~ K ~

0.05 1.4 79 44 0. 05 4.6 0.31 0.39 1.80 . 1 2.0 78 37 0.1 3.3 0.39 0.54 1.80.2 2.9 76 30 0.2 2.4 0.42 0.74 1.80.4 4.2 74 21 0.4 1.80 0.63 0.98 1.80.6 5.3 73 16 0.6 1.54 0.68 1.14 1.80.8 6.4 72 13 0.8 1.41 0.69 1.25 1.8

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T o w a r d s i n t e l l ig e n t P I D c o n t r o lTABLE6. EXPERIMENTAL ESULTSFOR A SYSTEMWITHG s) = 1 - as)/s 1 + s)

T . = 2 ~ V ~ o s u s 0 2 = m r 2 =~-T 7_ r ~ 2 r 2 ~ 2v v ~0 . 0 5 1 . 4 8 0 4 4 0 . 0 5 4 . 5 0 . 2 6 0 . 4 0 1 . 80 . 1 2 . 0 7 9 3 8 0 . 1 3 . 2 0 . 3 4 0 . 5 7 1 . 80 . 2 5 3 . 1 7 9 2 9 0 . 2 5 2 . 0 0 . 4 5 0 . 9 0 1 . 80 . 5 4 . 4 7 9 2 3 0 . 5 0 1 . 4 1 0 . 5 0 1 . 2 8 1 . 80 .75 5 .4 79 21 0 .75 1 .15 0 .51 1 .56 1 .8

TABLE 7 . CHO ICE OF CONTROLLERT i g h t c o n t r o l i s r e q u i r e d

L o wT i g h t m e a s u r e m e n tc o n t r o l H i g h n o i s e a n d h i g hi s n o t m e a s u r e m e n t L o w s a t u r a t io n s a t u r a t io nr e q u i r e d n o i s e l i m i t l i m i t1 . (0 1 > 1 ; r i < 1 .5 ) I I + B + C P I + B + C P I + B + D2 . ( 0 .6 < 0 1 < 1 ; 1 .5 < r l < 2 . 25 ) I o r P I I + A P I + A P I + A + C o rP I D + A + C3 . ( 0 . 15 < 0 ~ < 0 . 6 ; 2 . 2 5 < r l < 1 5) P I P I P I o r P I D P I D4 . ( 0 ~ 1 5

0 2 > 0 . 3 ; r 2 < 2 ) P o r P I P I P I o r P I D P I o r P I D5. ( 0 2 < 0 . 3 ; r 2 > 2 ) P D + E F P D + E P D + EA = F e e d f o r w a r d c o m p e n s a t i o n r e c o m m e n d e d , B = f e e d f o r w a r d c o m p e n s a t i o n e s s e n t ia l , C = d e a d t i m e c o m p e n s a -t io n r e c o m m e n d e d , D = d e a d t i m e c o m p e n s a t i o n e s s e n t i a l, E = s e t p o i n t w e i g h ti n g n e c e s s a r y, F = p o l e p l a c e m e n tt u n i n g .

p r o c e s s e s w i t h i n t e g r a t i o n . T h e c o n d i t i o n s a r es u m m a r i z e d i n T a b l e 7 . F i v e c a s e s a r ei n t r o d u c e d i n t h e t a b le . C a s e s 1 - 4 a r e a p p l i c a b l et o s t a b l e p r o c e s s e s w h i l e c a s e s 4 - 5 a r e a p p l i c a b l et o p r o c e s s e s w i t h i n t e g r a t i o n . T h e y a r e c l a s s i f i e das f o l lows .C a s e 1 S tab le p r oce s ses wi th 01 > 1 , r l 1 5; p r o c e s s e s w i t h i n t e g r a t i o n h a v i n g0 2 > 0 . 3 , / ~ 2 < 2 ) . P r o c e s se s w i t h i n t e g r a t i o n m a yb e v i e w e d a s t h e l i m i t o f s t a b l e p r o c e s s e s w h e nt h e n o r m a l i z e d d e a d t i m e , 0 1 , g o e s t o z e r o . Ap r o c e s s w i t h i n t e g r a t i o n m a y b e r e g a r d e d a s a na p p r o x i m a t i o n o f a s t a b l e p r o c e s s w i t h s m a l la p p a r e n t d e a d t i m e b u t a l a r g e a p p a r e n t t i m ec o n s t a n t . T h e g r o s s b e h a v i o u r o f p r o c e s s es w i t hi n t e g r a t i o n c a n b e c h a r a c t e r i z e d b y 0 2 , t h en o r m a l i z e d d e a d t i m e o b t a i n e d w h e n t h e i n -t e g r a t o r o r t h e l a r g e t i m e c o n s t a n t i s r e m o v e d .T h e b e h a v i o u r o f s u c h p r o c e s s e s c a n a l s o b ec h a r a c t e r i z e d b y t h e n o r m a l i z e d p r o c e s s g a i n r ET a b l e s 4 , 5 a n d 6 i n d i c a t e t h a t Z i e g l e r - N i c h o l st u n i n g g i v e s s y s t e m s w i t h g o o d d a m p i n gp r o v i d e d t h a t 0 2 i s l a r g e o r e q u i v a l e n t l y r E s m a l li .e . t h e d y n a m i c s w i t h t h e i n t e g r a t o r r e m o v e d i sd e a d t i m e d o m i n a n t . T h i s i s n o t s u r p r i s i n g s i n c et h e Z i e g l e r - N i c h o l s r u l e s w e r e d e r i v e d f o rp r o c e ss e s o f th i s t y p e . N o t i c e , h o w e v e r , t h a t t h eo v e r s h o o t i s i n g e n e r a l t o o l a r g e . S e t p o i n tw e i g h t i n g / ~ s t r 6 m a n d H ~ i g g l u n d , 1 9 8 8; H a n g e t

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8 K . J . STROM t al.al. 1991) i s thus e s sen t ia l f o r p r oces ses o f th i sc a t e g o r y .

C o n t r o l l e r s w i t h h i g h g a i n s c a n b e u s e d f o rt h i s c a s e . F o r s y s t e m s w i t h m o d e r a t e r e q u i r e -m e n t s , P I o r e v e n P c o n t r o l m a y b e s u f f i c ie n t .W i t h h i g h g a i n , m e a s u r e m e n t n o i s e b e c o m e s a ni m p o r t a n t i s s u e s i n c e it c a n r e s u l t i n s a t u r a t i o n o ft h e c o n t r o l s i g n a l . I n s o m e c a s e s p e r f o r m a n c ec a n b e i n c r e a s e d s i g n i f i c a n t ly b y u s i n g d e r i v a t i v ea c t i o n o r e v e n m o r e c o m p l i c a t e d c o n t r o l l a w s .T e m p e r a t u r e c o n t r o l w h e r e t h e d y n a m i c s isd o m i n a t e d b y o n e l a r g e t i m e c o n s t a n t i s a t y p i c a lcase .Case 5 ( P r o c e s s e s w i t h i n t e g r a t i o n h a v i n g02 < 0 . 3 , r 2 > 2 ) . Th is case cor r e spo nd s to ani n t e g r a t o r w i t h a n a d d i t i o n a l d y n a m i c s t h a t i s l a gd o m i n a t e d . P D c o n t r o l c a n b e u s e d f o r t h i s t y p eo f p ro c e s se s . P I D c o n t r o l w i t h Z i e g l e r - N i c h o l st u n i n g d o e s n o t g i v e g o o d r e su l t s; b o t h d a m p i n ga n d o v e r s h o o t a r e u n s a t i s f a c t o r y . D e s i g n b a s e do n d i r e c t p o l e p l a c e m e n t i s r e c o m m e n d e d( A s t r 6 m a n d H ~ i g g l u n d , 1 9 8 8 ) . D e r i v a t i v e a c t i o ni s o f t e n e s s e n t i a l f o r g o o d p e r f o r m a n c e .Im plicatio ns fo r intelligent controllers

T h e r e a r e s e v e r a l s i m p l e a u t o t u n e r s t h a t a r eb a s e d o n t h e Z i e g l e r - N i c h o l s t u n i n g p r o c e d u r e .A d r a w b a c k w i t h t h e m i s t h a t t h e y a r e u n a b l e t or e a s o n a b o u t t h e a c h i e v a b l e p e r f o r m a n c e . T h er e s u l t o f t h i s p a p e r i n d i c a t e s t h a t p e r f o r m a n c ec a n b e p r e d i c t e d f r o m k n o w l e d g e o f p a r a m e t e r s0 1 o r r l f o r s t a b l e p r o c e s s e s a n d 2 o r /(2 forp r o c e s s e s w i t h i n t e g r a t i o n . F u r t h e r m o r e , i t i se a s y t o s e l e c t t h e c o n t r o l l e r f o r m : P , P I , P D o rP I D b a s e d o n T a b l e 7 w h i c h a l s o i n d i c a t e s if am o r e s o p h i s t i c a t e d c o n t r o l l a w w o u l d b e u s e f u l .

A n a u t o t u n e r b a s e d o n t h e t r a n s i e n t r e s p o n s em e t h o d c a n g i v e 0 1 f r o m i t s m e a s u r e m e n t s o f Ta n d L . F o r t h e c o r r e l a t i o n - b a s e d a u t o t u n e r( H a n g a n d S i n , 1 9 8 8 ), 1 a n d x l o r 0 2 a n d x 2 a r er e a d il y c o m p u t e d f r o m t h e i m p u l s e r e s p o n s eg e n e r a t e d b y t h e c o r r e l a to r . F o r t h e r e l a y - b a se da u t o t u n e r ( / ~ s t r 6 m a n d H / i g g l u n d , 1 9 8 4 ) o n ea d d i t i o n a l m e a s u r e m e n t h a s t o b e m a d e t od e t e r m i n e r l o r / c ~ , t h e s t a t i c g a i n f o r st a b l ep r o c e s s e s o r t h e i n t e g r a t o r g a i n f o r p r o c e s s e sw i t h i n t e g r a t i o n . T h e s e g a i n s c a n b e d e t e r m i n e di n c l o s e d - l o o p b y i n t r o d u c i n g a s m a l l s e t p o i n tc h a n g e . F o r s t a b l e p r o c e s s e s , t h e s t a t i c g a i n a n dt h e s u m o f d e a d t i m e a n d t i m e c o n s t a n t s c a n b ec o m p u t e d u s i n g t h e m e t h o d o f m o m e n t s(A,s t r ( i m a n d H [ i g g l u n d , 1 9 8 8 ) . T h e n o r m a l i z e dd e a d t i m e , 0 1 , c a n t h e n b e c o m p u t e d . A c h e c ko n t h e c o m p u t e d v a l u e s o f 0 1 a n d r l c a n b em a d e u s i n g e q u a t i o n ( 2 4 ) .H a n g et al. ( 1991) have used 01 and r ~ toi m p r o v e o n t h e Z i e g l e r - N i c h o l s t u n i n g f o r m u l af o r s t a b le p r o c e s s e s . T h e f o l l o w i n g m o d i f i c a -

t i o n s , e x p r e s s e d a s a s i m p l e f u n c t i o n o f 0 1 a n d/('1 , h a v e b e e n r e c o m m e n d e d t o e l i m i n a t em a n u a l f i n e t u n i n g . F o r P I D c o n t r o l , w h e n0 1 < 0 .6 o r r l > 2 .2 5 , t h e m a i n d r a w b a c k o f t h eZ i e g l e r - N i c h o l s f o r m u l a i s e x c e s s i v e o v e r s h o o ta n d h e n c e s e t p o i n t w e i g h t i n g i s r e c o m m e n d e d .W h e n 01>0 6 or t 1 < 2 . 2 5 t h e i n t e g r al t im ec o m p u t e d b y t h e Z i e g l e r - N i c h o l s f o r m u l a i s t o ol a r g e a n d s h o u l d b e r e d u c e d . F o r P I c o n t r o l ,b o t h t h e c o n t r o l l e r g a i n a n d t h e i n t e g r a l t i m eh a v e t o b e m o d i f i ed . T h e r e f in e d t u n i n g f o r m u l ai s g i v e n in A p p e n d i x I . T h e s e m o d i f i c a t i o n s a r ee s s e n t i a l f o r o b t a i n i n g h i g h q u a l i t y P I D o r P Ic o n t r o l w i t h o u t m a n u a l f i n e t u n i n g . T u n i n gf o r m u l a s f o r p r o c e s s e s w i t h i n t e g r a t i o n a r e g i v e nin Ho ( 1990) .On - line assessmen t o f con tro l per formance

T h e r e s u l t s o f t h i s p a p e r c a n a l s o b e u s e d t oe v a l u a t e p e r f o r m a n c e o f f e e d b a c k l o o p s u n d e rc l o s e d - l o o p o p e r a t i o n . C o n s i d e r t h e r e l a t i o n s( 3 1 ) a n d ( 3 2 ) f o r t h e n o r m a l i z e d r i s e t i m e . T h er i s e t i m e c a n b e m e a s u r e d w h e n t h e s e t p o i n t i sc h a n g e d . I f t h e c o n t r o l l e r i s p r o p e r l y t u n e d t h e nt h e c l o s e d - l o o p r i s e t i m e s h o u l d b e e q u a l t o t h ea p p a r e n t d e a d t i m e f o r s t a b l e p r o c e s s e s a n d h a l fo f th e a p p a r e n t d e a d t i m e f o r p ro c e s s e s w i t hi n t e g r a t i o n . I f t h e a c t u a l r i se t i m e i s s i g n i f ic a n t l yd i f f e r e n t , f o r i n s t a n c e 5 0 l a r g e r , i t i n d i c a t e st h a t t h e l o o p i s p o o r l y t u n e d . T h i s i s p a r t i c u l a r l yu s e f u l f o r a s s e s s i n g w h e t h e r t h e c o n t r o l i s t o os l u g g is h . S i m i l a r ly t h e r e l a t i o n s ( 2 8 ) a n d ( 2 9 ) c a nb e u s e d b y i n t r o d u c i n g a p e r t u r b a t i o n a t t h ec o n t r o l l e r o u t p u t . A m a x i m u m e r r o r t h a t i ss i g n i fi c a n t ly l a r g e r t h a n t h a t p r e d i c t e d b ye q u a t i o n ( 2 8 ) a n d e q u a t i o n ( 2 9 ) i n d i c a t e s t h a tt h e l o o p i s p o o r l y t u n e d .

7. CONCLUSIONSI n t h i s p a p e r i t h a s b e e n a t t e m p t e d t o a n a l y z e

s i m p l e f e e d b a c k l o o p s w i t h P I D c o n t r o l l e r s t h a ta r e t u n e d u s i n g t h e Z i e g l e r - N i c h o l s c l o s e d - l o o pm e t h o d . I t h a s b e e n s h o w n t h a t f o r p r o c e s s e sw i t h e s se n t i al l y m o n o t o n e s t e p r e s p o n s e s t h e r ee x i s t q u a n t i t i e s t h a t a r e u s e f u l f o r a s s e s s i n g t h ea c h i e v a b l e p e r f o r m a n c e a n d s e l e c t i n g s u i t a b l ec o n t r o l l e r s . T h e s e q u a n t i t i e s a r e t h e n o r m a l i z e dd e a d t i m e s , 0 1 a n d 0 2 , t h e n o r m a l i z e d p r o c e s sg a i n s , r l a n d r 2 , t h e p e a k l o a d e r r o r s , ~.~ a n d~ .2 , a n d t h e n o r m a l i z e d c l o s e d - l o o p r i s e t i m e , r .S i m p l e m e t h o d s d e t e r m i n e t h e s e p a r a m e t e r sh a v e a l s o b e e n s u g g e s t e d .I t h a s b e e n s h o w n t h a t 0 1 i s r e l a t e d t o r l a n d0 2 t o x ~ a n d t h a t x i a n d 0 i c a n b e u s e di n t e r c h a n g e a b l y t o a s s e ss t h e c o n t r o l p r o b l e m .Th er e a r e a l so r e la t ion s l ike ~ 1 and x1~ ,1 ~ 1 . 4f or s t ab le p r oces ses , and z ~ 0 . 5 a nd x2~ ,2 ~ 1 . 8

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Towards intelligent PID control 9for processes with integration. These relationsmay be used to assess the closed-loop responsetime and the load rejection properties. Theresults indicate that it would be useful todetermine at least 01 or /('1 for stable processesand 0: or/t 2 for processes with integration whentuning controllers because these parameters canbe used to predict the achievable performance.A c k n o w l e d g e m e n t - - T h i s w o rk h a s b e e n su p p o r t e d b y t h eS w e d i s h B o a r d f o r T e c h n i c a l D e v e l o p m e n t ( S T U ) u n d e rcont rac t DUP 85-3084P.

H a n g , C . C . , K . J . / ~ s t r fm a n d W. K . H o (1 9 9 1 ) .R e f i n e m e n t s o f t h e Z i e g l e r - N i c h o l s T u n in g F o r m u l a .Proc . lEE, Pt . D, 1~, no 2 , 111-118 .Hang, C. C. and K. K. Sin (1988). On-l ine au to-tun ing ofPID c o n t ro l l e r s b a se d o n c ro ss c o r re l a t i o n . Proc. Int.Conf. on Industrial Electronics, Si n g a p o re , 4 4 1 -4 4 6 .H o , W. K . (1 9 90 ) . Tu n i n g o f P I c o n t ro l l e r s fo r p ro c esse sw i t h i n t e g ra t i o n b a se d o n g a i n a n d p h a se ma rg i nsp ec if ic a ti on s . R e p o r t C O D E N : LU TF D 2 / T FR T -7 4 7 2 .D e p a r t me n t o f A u t o ma t i c C o n t ro l , Lu n d In s t i t u t e o fTe c h n o l o g y , Lu n d , Sw e d e n .K ra u s , T . W. a n d T . J . M y ro n (1 9 8 4 ) . Se l f - t u n i n g PIDc o n t ro l l e r u se s p a t t e rn re c o g n i t i o n a p p ro a c h . ControlEngng, June, 108-111 .Z i e g l e r , J . G . a n d N . B . N i c h o l s (1 9 42 ) . O p t i mu m se t ti n g sfo r a u t o m a t i c c o n t ro l l e r s . T r a n s . A S M E , 64, 759-768 .

R E F E R E N C E S,~ ,rz6n , K. E. (1989). An arch i tec ture for expert systemb a se d fe e d b a c k c o n t ro l . Automatica, 2,5 , 813-827 ./ ~ s t r6 m, K . J . (1 9 91 ) . A sse ssme n t o f a c h i e v a b l e p e r fo rma n c eo f s i mp l e fe e d b a c k l o o p s . Int . J . Adaptive Control SignalProcess, 5, 3 -1 9 .A s t r6 m, K . J . , J . J . A n t o n a n d K . E . / ~ rz 6 n (1 9 8 6 ) . Ex p e r tcont ro l . Automatica, 7,2 , 227-286 ./~st r6m, K. J . and T. H~igglund (1984). Automat ic tun ing of

s i mp l e re g u l a t o r s w i t h sp e c i f i ca t i o n s o n p h a se a n da mp l i t u d e m a rg i n s . Automatica, 2~, 6 4 5 -6 5 1 . s t r6m, K. J . and T. H~igglund (1988). Automat ic tun ing ofP I D c o n t r o ll e r s. I S A , R e s e a r c h T r i a n g l e P a rk , N C , U S A ./ ~ s t r6 m, K . J . , C . C . H a n g a n d P . P e rs so n (1 98 8 ). H e u r i s t i c sfo r a s se s sme n t o f P ID c o n t ro l w i t h Z i e g l e r -N i c h o l s t u n in g .R e p o r t C O D E N : L U T F D 2 / T F R T - 7 4 0 4 , D e p a r t m e n t o fA u t o m a t i c C o n t ro l , Lu n d In s t i t u t e o f Te c h n o l o g y , Lu n d ,Sw e d e n ..~ s t r fm, K . J . , C . C . H a n g a n d P . Pe rs so n (1 9 8 9 ) . To w a rd si n t e l l i g e n t P ID c o n t ro l . In p re p r i n t s I F A C W o r k s h o p o nAI in Real Time Control , Sh e n y a n g , P . R . C h i n a , p p .3 8 -4 3 , Pe rg a mo n Pre s s , O x fo rd .Bris to l , E. H. (1977). Pat tern recogni t ion : an a l ternat ive top a ra me t e r i d e n t i f i c a t i o n i n a d a p t i v e c o n t ro l . Automatica,13, 197-202.C o h e n , G . H . a n d G . A . C o o n (1 9 5 3 ) . Th e o re t i c a lc o n s i d e ra t i o n o f r e t a rd e d c o n t ro l . T r a n s . A S M E , 7 5 ,827-834 .D e sh p a n d e , P . B . a n d R . H . A sh (1 9 8 1 ) . Computer ProcessControl, I S A , R e s e a r c h T r i a n g le P a r k , U . S . A .Fe r t i k , H . A . (1 97 5 ). Tu n i n g c o n t ro l l e r s fo r n o i sy p ro c e sse s .ISA Trans . , 14, 4.Golf , K. W. (1966). Dynamics in d i rec t d ig i ta l cont ro l , part sI and II . I S A J . , 1 3 , 4 4 -5 4 .H f i g g lu n d , T . (1 9 91 ) . A d e a d t i me c o mp e n sa t i n g t h re e - t e rmc o n t ro l le r . 9 t h I F A C / I F O R S S y m p o s i u m o n I d e n ti f ic a t io na n d Sy s t e m Pa ra me t e r Es t i ma t i o n , B u d a p e s t .H a n g , C . C . (1 98 9 ). C o n t ro l l e r Ze ro s , IEEE Contro lSys tems M agazine , 9 , 7 2 -7 5 .

A P P E N D IX : R E F I N E D Z I E G L E R - N I C H O L S T U N I N GF O R M U L AFo r a PID c o n t ro l l e r o f t h e fo rm

uc = KP lY r- Y + ~ f e dt-dyf~dt ] (A 1 )e = y , - y (A2)

Y~ s) 1 + STd/lO Y s) ( A 3 )w h e re u c , Y, Y , a re t h e c o n t ro l l e r o u t p u t , p ro c e ss o u t p u t a n dse t -p o i n t r e sp e c t i v e l y , t h e re f i n e d Z i e g l e r -N i c h o l s t u n i n gformula for s tab le processes i s as fo l lows.P I D : Case 1 (2.25 < x~ < 15; 0.16 < 0~ < 0.57)

fl 1 5 - r ~= ~ ~ ~ fo r 1 0 % se t -p o i n t r e sp o n se o v e rsh o o t (A 4 )36f l = 27 + 5 r I fo r 20 % se t -p o i n t r e sp o n se o v e rsh o o t . (A 5 )

Case 2 ( 1 . 5 < r I < 2 . 2 5 ; 0 . 5 7 < 0 1 < 0 . 9 6 ) . 2 0 % s e t -p o i n tre sp o n se o v e rsh o o tT~ = 0. 5#T . (A 6)U = ~~ (A7)f l = ~ (~ x ~ + 1 ) (A 8 )

w h e re # i s d e f i n e d a s t h e ra t i o o f t h e mo d i f i e d i n t e g ra l t imet o t h e Z i e g l e r - N i c h o l s in t e g ra l t ime .PI: (1.2 < x~ < 15; 0.16 < 01 < 1.4)K ~ = 5 1 2 + r I ~ ( A 9 )K . 6 \ 1 5 + 1 4 r ~ /_ v , = ~ ~ , , + 1 ) . (AIO)L