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Adaptive Control of Rotary Drum Driers
ADAPTIVE CONTROL OF ROTARY DRUM DRIERS P. E. Moden and T. Nybrant Department of Automatic Control and Systems Analysis, Institute of Technology, Uppsala University, Box 534, S-751 21 Uppsala, Sweden
Abstract. The process of drying green crops in a high-temperature rotary drum dehydrator is difficult to control. It is time-varying and there are stochastic disturbances. Adaptive control of the resulting moisture content is considered. The control laws discussed are based on linear quadratic gaussian optimal control with finite time horizon. They enable a suitable compromise between the minimal output variance wanted and the limited inputs available, and they are easy to apply adaotively. There will be a steady state error unless integral action is included. Different methods to add integral action are discussed. The resulting adaptive controllers were implemented on a microcomputer and applied to a drum drier. The practical results are presented and compared. Keywords. Adaptive control; stochastic control; steady state error; agriculture; drying process; temperature control; moisture control. INTRODUCTION
excessive comsumption of oil.
High-temperature drying of green crops is a way to conserve forage (~owarth,1978). In the process grass is dried and then pressed into cobs. It is very important to keep the moisture content of these cobs within certain limits for several reasons. The high-temperature drying process is known to be difficult to handle. This fact combined with high energy consum~tionof the process explains the need for a good controller. The conventional three-term controllers now in use give in many cases unsatisfactory results. (Carlsson, Mod& and Sandberg, 1978)
gross Input elevglor
--
press cooler A project "Adaptive Control of High-temperature Drum ~riers"has been performed in cooperation between the Institute of Technology at Uppsala Fig. 1. General view of a high-temperature drum drier for green crops. It is University and the Swedish University of Agrifurther described in the text. cultural Sciences. The goal has been to develop an adaptive controller in a microcomputer. In A direct measurement of the moisture in the this paper theoretical aspects as well as cobs seems to be the natural signal to use practical results are presented. for control. However, humidity sensors have not been reliable enough for dried fcrage and instead different temperatures in the proPROCESS DESCRIPTION cess could be used as output signal.
A view of a high-temperature drum drier is The chopped grass is fed in- These temperatures and other signals given in Fig. 1 . interest are: to the slowly rotating drum where it is dried by the hot gases from the oil burner. The grass u fuel valve (oil flow) M is separated in the cyclones from the hot gases v feed rate of grass and it is formed into cobs by the press. Before yl temperature in the oven they can be stored they have to pass a cooler. y2 temperature after the drum M y~ temperature of the dried w Too high a moisture content in the cobs would grass make them impossible to store. If too dry, they y ~ +temperature in the exhaust M would easily fall apart, their nutrition value pipe would be reduced and it would also cause an
-
of 0 , I L/s 1 ,60kg/s 8 0 0 ~ ~ 130-C 85°C 1 1 0 ~ ~
P.E. Mod6n an~d T. Nybrant E x i s t i n g systems use u o r v a s c o n t r o l i n p u t and y2, y3 o r YI, a s o u t p u t s i g n a l . y l i s u s e d t o check t h a t t h e t e m p e r a t u r e i n t h e oven i s n o t t o o h i g h . O t h e r r e l e v a n t d a t a of t h e p l a n t used i n t h e experiments a r e :
The i d e n t i f i c a t i o n e x p e r i m e n t s a l s o c l e a r l y indicated t h a t t h e coefficients i n t h e A, B, C and D polynomials were v a r y i n g w i t h t i m e and working l e v e l . I t i s concluded t h a t t h e system i s n o n - l i n e a r and timev a r i a b l e . F i g u r e 2 shows s t e p r e s p o n s e s o f models o b t a i n e d from d i f f e r e n t e x p e r i m e n t s .
Water e v a p o r a t i o n c a p a c i t y 4 500 kg/hour O i l consumption t y p 0,36 m3/hour or 0,3 L/kgofcobs Cob p r o d u c t i o n t y p 1 200 kg/hour Moisture content of t h e input g r a s s i s about 80 p e r c e n t and t h e c o r r e s p o n d i n g v a l u e o f t h e cobs i s a b o u t 10 p e r c e n t . Drying o f f i e l e w i l t e d g r a s s r e d u c e s t h e o i l c o s t s by 30-35 p e r c e n t ( C a r l s s o n , 1 9 7 6 ) . On t h e o t h e r hand i n t h i s c a s e t h e c o n v e n t i o n a l three-term c o n t r o l l e r o f t e n g i v e s t o o g r e a t a moisture content v a r i a t i o n i n t h e cobs. A good c o n t r o l l e r t h a t c o u l d h a n d l e t h i s c a s e b e t t e r c o u l d , even i f r a t h e r s o p h i s t i c a t e d , e a s i l y be m o t i v a t e d w i t h r e s p e c t t o t h e current o i l prices.
MODELLING Research done a t t h e Swedish U n i v e r s i t y o f A g r i c u l t u r a l S c i e n c e s showed t h a t ys ( g r a s s t e m p e r a t u r e i n t h e c y c l o n e ) was t h e temperat u r e t h a t had t h e s t r o n g e s t c o r r e l a t i o n w i t h t h e m o i s t u r e c o n t e n t ( B e r g l u n d , Carlstrijm and Svensson, 1 9 7 5 ) . It was t h e r e f o r e chosen a s t h e output signal. Off-line i d e n t i f i c a t i o n e x p e r i m e n t s were p e r formed on t h e a c t u a l p l a n t . They showed t h a t t h e p r o c e s s c o u l d b e w e l l des c r i b e d by t h e f o l l o w i n g l i n e a r model s t r u c ture:
Time (minutes) Fig. 2.
S t e p r e s p o n s e s o f l i n e a r models o b t a i n e d from d i f f e r e n t e x p e r i ment s
.
S i n c e u was found t o i n t r o d u c e l e s s t i m e d e l a y t h a n v , u was chosen a s t h e c o n t r o l i n p u t and v was r e g a r d e d a s a known d i s t u r b a n c e t h a t c o u l d b e manually o p e r a t e d .
ADAPTIVE CONTROL The p u r p o s e o f t h e c o n t r o l c a n b e d i v i d e d i n t o two t a s k s . The v a r i a n c e o f t h e c o n t r o l e r r o r s h o u l d b e s m a l l , and t h e mean v a l u e should be equal t o zero. A conventional t h r e e - t e r m c o n t r o l l e r c o u l d manage t h e mean value, but not t h e variance. Since t h e system i s t i m e - v a r y i n g , t h e r e q u i r e m e n t on t h e v a r i a n c e makes a d a p t i v e c o n t r o l n e c e s s a r y . I n p u t and Output V a r i a n c e s
k=0
t i n e d e l a y from
u
R=l
t i n e d e l a y from
v
g(t ! o u t p u t u( t ) o i l v a l v e
vl t d
:
feed r a t e d e s c r i b e s working l e v e l
e ( t ) w h i t e n o i s e w i t h z e r o mean q
-1
-1 backward s h i f t o p e r a t o r , q y ( t ) = y ( t - l )
A s t r a i g h t f o r w a r d method t o g e t a s m a l l o u t put variance i s t h e self-tuning regulator (Rstrdm and co-workers , 1 9 7 7 ) . I t combines r e c u r s i v e l e a s t s q u a r e s i d e n t i f i c a t i o n (RLS) w i t h minimum v a r i a n c e c o n t r o l (MV), a p p l y i n g t h e c e r t a i n t y equivalence p r i n c i p l e . A forg e t t i n g f a c t o r makes it p o s s i b l e t o compens a t e f o r time-varying system dynamics.
I t i s w e l l known t h a t i n o r d e r t o a c h i e v e t h e minimal o u t p u t v a r i a n c e , a h i g h i n p u t v a r i a n c e i s needed. The models f o r t h e drum d r i e r o b t a i n e d by o f f - l i n e i d e n t i f i c a t i o n i n d i c a t e t h a t t h e physical l i m i t s of t h e inp u t would b e r e a c h e d q u i t e o f t e n , i f MV c o n t r o l was a p p l i e d .
A d a p t i v e Control of R o t a r y Drum D r i e r s
Not o n l y t h e l i m i t s o f t h e i n p u t s i g n a l p u t a r e s t r i c t i o n o n ' t h e i n p u t v a r i a n c e . The f u e l v a l v e (which i s t h e c o n t r o l i n p u t ) i s a c t u a t e d by a r e l a y c o n t r o l l e d AC-motor w i t h c o n s t a n t r a t e o f change. T h i s means t h a t when t h e i n p u t i s changed t h e r e i s a time-delay which i s p r o p o r t i o n a l t o t h e i n c r e m e n t . The models used a r e v a l i d f o r r a t h e r s m a l l i n c r e m e n t s . With t o o l a r g e i n c r e m e n t s t h e d e l a y c o u l d make t h e system non-minimum-phase, and t h u s t h e c l o s e d system u n s t a b l e i n t h e c a s e o f MV c o n t r o l . A s m a l l e r i n p u t v a r i a n c e a l s o makes t h e i n p u t i n c r e m e n t s s m a l l enough. I n order t o decrease t h e input variance l i n e a r q u a d r a t i c o p t i m a l c o n t r o l c a n be a p p l i e d , choosing u ( t ) t o minimize
i s ~ o s i t i v er e a l ( ~ o d g nand b Q 8 ( p 1) / A ( < ' SoderstrGm, 1979)
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Here p i s a p e n a l t y on t h e c o n t r o l i n p u t (~20)T . h i s c r i t e r i o n w i l l b e c a l l e d L%.
2
1
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1
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Having a known t i m e - i n v a r i a n t system and a n i n f i n i t e t i m e h o r i z o n , L&m would g i v e a timei n v a r i a n t c o n t r o l l e r . Applying L&m adapt i v e l y would l e a d t o much more computation t h a n MV. However, v e r y simple computations a r e obtained with a f i n i t e time horizon j u s t k+l s t e p s a h e a d , LQk+l. I n t h i s c a s e u ( t ) minimizes ( 2 ) w i t h N=k+l a s w e l l a s
F i g . 3. The p o s s i b l e combinations o f i n p u t and o u t p u t v a r i a n c e s when LQk+l o r L&m c o n t r o l i s a p p l i e d t o a model o f t h e has drum d r i e r . The open system (p*) t h e o u t p u t v a r i a n c e 19. With MV cont r o l (p=O) t h e i n p u t v a r i a n c e i s 7 4 , and t h e o u t p u t v a r i a n c e i s 1 . (The variances a r e normalized.)
d e s c r i b e d by ( 1 ) , t h i s i s For a system e q u i v a l e n t t o a p p l y i n g MV t o a n augmented system which h a s t h e o u t p u t ( C l a r k and cow o r k e r s , 1975)
From F i g . 3 it i s e a s y t o chose a c o n t r o l l e r which g i v e s a n a c c e p t a b l e i n p u t v a r i a n c e and a n o u t p u t v a r i a n c e which i s c l o s e t o t h e minimal one. For example LQk+l w i t h p 0 . 0 1 w i l l r e d u c e t h e i n p u t v a r i a n c e by a f a c t o r o f 3 and i n c r e a s e t h e o u t p u t v a r i a n c e by o n l y 10 p e r c e n t compared t o MV. S i m i l a r r e s u l t s a r e obt a i n e d f o r a l l t h e o f f - l i n e i d e n t i f i e d drum d r i e r models.
The r e s u l t i n g
L%+l
Remembering t h a t LQk+l i s e q u i v a l e n t t o MV c o n t r o l of t h e introduced v a r i a b l e f ( t ) , t h e combination w i t h RLS t o form a n a d a p t i v e c o n t r o l l e r i s s t r a i g h t f o r w a r d ( ~ l a r k e ,Cope and Gawthrop, 1 9 7 5 ) -
controller is
S e t - p o i n t and S t e a d y S t a t e E r r o r where t h e p o l y n o m i a l s F ( q - l ) o f d e g r e e and ~ ( q - ') o f d e g r e e one l e s s t h a n t h e system a r e o b t a i n e d from t h e i d e n t i t y
k
The i n p u t p e n a l t y p i s t r e a t e d a s a d e s i g n v a r i a b l e which s h a l l b e chosen t o g i v e a c c e p t able variances. The p o s s i b l e combinations o f i n p u t and o u t p u t v a r i a n c e s , when L&k+, o r L& i s applied t o a t y p i c a l model o f t h e drum d r i e r , i s shown There i s a d i f f e r e n c e between i n Pig. 3 , LQk+l and L&m b u t it i s t o o s m a l l t o b e v i s i b l e i n t h i s f i g u r e . This i s not t h e case f o r a l l s y s t e m s , b u t good performance w i t h LQk+l i s g u a r a n t e e d i f t h e t r a n s f e r f u n c t i o n
The mean v a l u e o f t h e o u t p u t i s t o be k e p t a t t h e d e s i r e d l e v e l , t h e s e t - p o i n t ~ , ( t )T.h i s r e f e r e n c e v a l u e may be i n c l u d e d i n t h e LQk+l c o n t r o l l e r i n d i f f e r e n t ways. The s i m p l e s t method i s t o s u b s t i t u t e y ( t ) - y r ( t ) f o r y ( t ) i n t h e c o n t r o l law ( 5 ) . T h i s may, however, g i v e bad t r a n s i e n t r e s p o n s e s t o s e t - p o i n t changes. An a l t e r n a t i v e which i s b e t t e r i n t h i s r e s p e c t i s t o s u b s t i t u t e ~ ( i ) - ~ ,f(o tr ) y ( i ) i n t h e c r i t e r i o n ( 2 ) . Also a n i n p u t r e f e r e n c e u r ( t ) may b e i n c l u d e d by s u b s t i t u t i n g u ( i ) - u r ( t ) f o r u ( i ) i n t h e same c r i t e r i o n ( 2 ) . Then w i t h N=k+l ( f i n i t e t i m e h o r i z o n ) u ( t ) minimizes
P . E . Moden and T . Nybrant
I f t h e system S i s g i v e n by ( I ) , t h e c o r r e s ponding augmented system h a s t h e o u t p u t
It i s shown i n F i g . 4 . The r e s u l t i n g L Qk+1 controller with reference signals i s [ B ( ~ - ' I F ( ~ I- +'
$ c(q-'
)IU(~)=
0
The polynomials same a s i n ( 5 ) .
F(q-' )
and
~ ( q - )l
are the
T h i s means t h a t t h e r e i s a p e n a l t y on t h e i n c r e m e n t s ( M ( ~)=1 - ~) o r f i l t e r e d i n c r e m e n t s , and ~ ( l ) = l g u a r a n t e e s z e r o o f f s e t i f t h e c l o s e d system i s s t a b l e ( C l a r k e and Gawthrop, 1979). The above mentioned methods t o e l i m i n a t e t h e s t e a d y s t a t e e r r o r do n o t work u n l e s s t h e system p a r a m e t e r s a r e e x a c t l y known. When combining t h e c o n t r o l law w i t h a r e c u r s i v e p r e d i c t i o n e r r o r i d e n t i f i c a t i o n method ( e . g . RLS) t o an a d a p t i v e c o n t r o l l e r , t h e e s t i m a t e s o f t h e working l e v e l p a r a m e t e r d and t h e s t a t i c g a i n become r a t h e r poor and z e r o o f f s e t cannot be guaranteed. To a v o i d t h e o f f s e t , a f e e d b a c k from t h e integrated control e r r o r has t o be included i n t h e c o n t r o l l e r . The e x c e p t i o n i s when t h e c o n t r o l l e r ( 5 ) , o p e r a t i n g on t h e e r r o r y(t)-yr(t) instead of y ( t ) , i s applied t o a n i n t e g r a t i n g system. The system can b e made i n t e g r a t i n g , by cons i d e r i n g t h e input increments t o be t h e c o n t r o l i n p u t . Then t h e LQk+l c o n t r o l l e r w i l l have a z e r o which c a n c e l s t h e i n t e g r a t o r p o l e . T h i s z e r o c a n b e d e l i b e r a t e l y moved away from 1 , which l e a d s t o t h e c o n t r o l l e r i n F i g . 5 . (Kcrz , Isermann and Schumann, 1978)
.
F i g . 4 . Block scheme showing t h e c o n s t r u c t i o n o f t h e h e l p v a r i a b l e f ( t) Minimum v a r i a n c e c o n t r o l o f f ( t ) i s e q u i v a l e n t t o LQ control (with r e f e r e n c e s i g n a l s y A f t h e system S .
.
With t h i s c o n t r o l l e r ( 8 ) a p p l i e d t o a system ( 1 ) w i t h e x a c t l y known p a r a m e t e r s , t h e mean v a l u e o f t h e o u t p u t becomes
F i g . 5 . Block scheme showing a method t o i n c l u d e i n t e g r a l a c t i o n i n t h e cont r o l l e r . No feed-forward t e r m s have been i n c l u d e d .
if
y r , ur and v a r e c o n s t a n t . The o f f s e t i s z e r o i f e i t h e r p=O o r t h e p a r e n t h e s i s i s e q u a l t o z e r o . Thus w i t h p > 0 t h e i n p u t r e f e r e n c e u r ( t ) must b e chosen c o r r e c t l y . I t c a n a l s o be u p d a t e d by f i l t e r i n g o l d v a l u e s of t h e input. ur(t)=q-'hl(q-l
)u(t)
(10)
Here ~ ( q - )' i s a t r a n s f e r f u n c t i o n w i t h I n t h i s case t h e help variable i s M( 1 ) = l
.
A p e n a l t y on t h e i n t e g r a t e d e r r o r can be i n c l u d e d by t a k i n g t h e h e l p v a r i a b l e f ( t )t o be
-1 where P ( q ) i s i n t e g r a t i n g ( C l a r k e and Gawthrop, 1 9 7 9 ) . F i g 6 shows t h e r e s u l t i n g c o n t r o l l e r when ~ ( q - )l = p N ( q - l ) / (1-q-l )
.
E a r l i e r i n t h i s s e c t i o n one way o f u p d a t i n g t h e i n p u t r e f e r e n c e u r ( t ) was mentioned. It c a n a l s o b e u p d a t e d u s i n g t h e i n t e g r a t e d error. I n order t o r e t a i n t h e integration f o r t h e c a s e p=O, one c a n s u b s t i t u t e w ( t )
Adaptive Control of Rotary Drum D r i e r s PRACTICAL RESULTS During t h e autumn o f 1979 a d a p t i v e c o n t r o l was a p p l i e d t o t h e r o t a r y drum d r i e r . Some d i f f e r e n t s e l f - t u n i n g c o n t r o l l e r s based on LQk+l were t e s t e d . The d i f f e r e n c e was t h e method t o handle t h e s t e a d y s t a t e e r r o r , t h a t i s , how t h e i n t e g r a l a c t i o n was added. A l l of t h e c o n t r o l l e r s i n c l u d e d RLS i d e n t i f i c a t i o n of t h e c o n t r o l l e r parameters d i r e c t l y . I n t h e RLS t h e U-D-factorization a l g o r i t h m was used (Bierman, 1975). The c o n t r o l l e r s were implemented on a comp u t e r system f o r experiments i n automatic c o n t r o l . The system c o n s i s t s of a 2-80 microcomputer and a programpackage f o r o p e r a t o r s communication (Nybrant and ~ o d 6 n ,1979). The F i g . 6. Block scheme showing t h e c o n t r o l o f c o n t r o l l e r programs were w r i t t e n i n Z i l o g ' s t h e system S, when t h e LQk+l c o n t r o l h i g h - l e v e l language PLZ and t h e t y p i c a l s i z e l e r i s extended t o i n c l u d e a p e n a l t y of t h e c o n t r o l l e r programs was i n t h e o r d e r on t h e i n t e g r a t e d e r r o r . The h e l p v a r i of 8 kbyte. S i n c e f a s t a r i t h m e t i c o p e r a t i o n s a b l e f ( t ) i s i n t h i s case f ( t ) = p N ( q - l) I (1-q-I ) [ ~ ( t ) - ~ ~ ( t)I+- ~ - l were d e s i r e d , f l o a t i n g - p o i n t hardware was i n c l u d e d i n t h e system. This i n c r e a s e d t h e +p/bou(t-k-1 ) . The polynomials F and G computation speed about 60 times compared a r e given be t h e i d e n t i t y with arithmetic i n software.
for let
u ( t ) i n t h e c o n t r o l l e r ( 8 ) and
yr(t)+ bo
Here t h e c o n t r o l l e r s ' a b i l i t y t o reduce t h e o u t p u t v a r i a n c e and t o e l i m i n a t e t h e o f f s e t i s i l l u s t r a t e d i n diagrams showing t h e logged o u t p u t s and s e t - p o i n t s . The sequences shown a r e r e p r e s e n t a t i v e f o r t h e behaviour of t h e d r i e r , under normal o p e r a t i n g c o n d i t i o n s , when t h e d i f f e r e n t a d a p t i v e c o n t r o l l e r s a r e used.
-1 where t h e t r a n s f e r f u n c t i o n H(q ) i s i n t e g r a t i n g and y r o ( t ) i s t h e s e t - p o i n t . This l e a d s t o cascade c o n t r o l . I t i s d e s c r i b e d i n F i g . 7. I n t h e i n n e r loop t h e r e i s t h e LQk+l c o n t r o l l e r w i t h r e f e r e n c e s i n a l s , and i n t h e o u t e r l o o p t h e r e i s t h e ~ ( q -) , which t y p i c a l l y i s a PIc o n t r o l l e r . This can be compared t o t h e cascade c o n t r o l suggested by Clarke and Gawthrop ( 1979). Note however, t h a t t h e i n t e r p r e t a t i o n of ( 13) labove may g i v e some h e l p i n t h e choice of ~ ( q) .
7
90
F i g . 7. Block scheme showing cascade c o n t r o l , with a PI-controller i n t h e outer l o o p . The s i g n a l w( t ) r e p l a c e s y r ( )t+ ~ / b o u , ( t) i n t h e LQk+l c o n t r o l l e r (8). Figures 5, 6 and 7 r e p r e s e n t t h r e e d i f f e r e n t methods t o i n c l u d e i n t e g r a l a c t i o n by some ext e n s i o n of t h e LQk+l c o n t r o l l e r . There a r e of course o t h e r methods t o o .
100 110 minutes
120
Fig. 8 . S e t p o i n t and o u t p u t from t h e drum d r i e r . The c o n t r o l l e r i s a d a p t i v e and i n c l u d e s a p e n a l t y on t h e i n t e g r a t e d e r r o r , c f . F i g . 6 . The sequence shown i s t h e l a s t 40 minutes of t h e r u n . I n Fig. 8 t h e controller includes a penalty on t h e i n t e g r a t e d e r r o r , a s d e s c r i b e d by ( 1 2 ) and F i g . 6 , w i t h ~ ( q - lkpN(q-l ) / ( 1-q-l ) The performance i s r a t h e r poor. The v a r i a n c e i s t o o l a r g e , and t h e r e a l s o seems t o be an o f f s e t . One p o s s i b l e e x p l a n a t i o n i s t h a t t h e
.
P.E.
Mod6n and T. Nybrant
model s t r u c t u r e used f o r t h e RLS i d e n t i f i c a t i o n i s not s u i t a b l e f o r t h i s c o n t r o l method. With t h i s s t r u c t u r e t h e i d e n t i f i c a t i o n does not t a k e advantage of t h e f a c t t h a t t h e t r a n s f e r f u n c t i o n ~ ( ~ - i ln ) ( 1 2 ) i s known. Adaptive c o n t r o l without i n t e g r a l a c t i o n was a l s o t e s t e d . This was t h e c a s e i n F i g . 9. The s h o r t term v a r i a n c e i s a c c e p t a b l e from 40 minutes and on, but t h e r e i s an o f f s e t . The i n c r e a s i n g temperature a t t h e end of t h e r u n i s t y p i c a l and i s s e e n i n F i g . 8 t o o . I t i s caused by a d e c r e a s i n g g r a s s flow. When t h e r e i s l i t t l e g r a s s l e f t on t h e e l e v a t o r , t h e flow i s l e s s t h a n i n d i c a t e d by t h e f e e d r a t e s i g n a l , s i n c e t h e g r a s s i s not packed s o h a r d .
100"C
90.
80-
Y 1
hours
2
3
F i g . 10. S e t p o i n t and output from t h e drum d r i e r . Cascade c o n t r o l i s used. The LQk+l c o n t r o l l e r i n t h e i n n e r l o o p i s a d a p t i v e , and t h e P I - c o n t r o l l e r i n t h e outer loop i s time-invariant, c f . F i g . 7 . The p a r t shown s t a r t s when t h i s c o n t r o l l e r i s a c t i v a t e d and cont i n u e s t o t h e end of t h e r u n .
+ . . . . . . . . . . . . . . . .1
0 Fig.
20
40 minutes
60
80
9. S e t p o i n t and o u t p u t from t h e drum d r i e r . The c o n t r o l l e r i s a d a p t i v e and h a s no i n t e g r a l a c t i o n . I t i s t h e LQk+l w i t h r e f e r e n c e s i g n a l s ( 8 ) . The i n p u t r e f e r e n c e u r ( t ) was c o n s t a n t . The whole r u n i s shown except t h e s t a r t when t h e p l a n t was c o n t r o l l e d manually.
F i n a l l y i n Fig.10 and 1 1 a d a p t i v e c o n t r o l was used w i t h a P I - c o n t r o l l e r added i n an o u t e r l o o p , a s d e s c r i b e d by ( 1 3 ) and F i g . 7 . Now t h e e r r o r v a r i a n c e i s s m a l l , and t h e o u t p u t f o l l o w s t h e changes i n t h e s e t - p o i n t adequat e l y ( b e t t e r i n F i g . 11, where t h e g a i n i n t h e outer loop i s higher )
.
Obviously t h e c o n t r o l l e r used i n F i g . 10 and 1 1 g i v e s t h e b e s t performance. I t i s t h e o n l y one of t h e t e s t e d t h a t f u l f i l s t h e r e q u i r e ments
.
-
o Fig.
T
1
i hours
Set pointand output from the drum d r i e r . The same c o n t r o l l e r i s used a s i n F i g . 10. The f i r s t 3 hours and 20 minutes of t h e r u n a r e shown.
CONCLUSIONS D i f f e r e n t a d a p t i v e c o n t r o l l e r s have been t e s t e d on a p r o c e s s t h a t i s n o n l i n e a r and t i m e v a r i a b l e . One of t h o s e c o n t r o l l e r s was found t o g i v e a small o u t p u t v a r i a n c e , an a c c e p t a b l e i n p u t v a r i a n c e , and no o f f s e t from t h e r e f e r e n c e v a l u e . Conpared w i t h t h e conv e n t i o n a l t h r e e term c o n t r o l l e r t h e o u t p u t v a r i a n c e was c o n s i d e r a b l y reduced. The t e s t s a l s o i n d i c a t e d t h a t t h i s c o n t r o l l e r made it p o s s i b l e t o d r y f i e l d w i l t e d g r a s s and t h u s c o n t r i b u t e t o a c o n s i d e r a b l y reduced energy consuinption.
Adaptive Control of Rotary Drum Driers ACKNOWLEDGMEBTS Part of this work was finacially supported by the Swedish Board for Technical Development under contract 77-3555, which is gratefully acknowledged. The authors also want to express their gratitude to GBran Carlsson, whose genuin knowledge of the process and assistance in the practical experiments were invaluable. REFERENCES AstrBm, K.J., U. Borisson, L. Ljung and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatica, 12,457-476. Berglund, L., P. Carlstrijm, K.-E. Svensson (1975). Studium av ett reglersystem f6r hetluftstorkar. Report no. 30. Dept. of Agric. Eng., Swedish University of Agricultural Sciences, Uppsala. 122 pp. (1n we dish) Bierman, G. J. (1975). Measurement updating using the U-D-factorization. Proc. 14th IEEE Conference on Decision and Control, 337-346. Carlsson, G. ( 1976). Nggra processtekniska faktorer vid hetluftstorkning. Report no. 31. Dept. of Agric. Eng., Swedish University of Agricultural Sciences, Uppsala.53 pp. (1n Swedish) Carlson, G., P. E. Mod&, A. Sandberg (1978). Adaptive optimal control of high-temperature drum driers. Proc. 2nd International Green Coop Drying Congress, 220-227. Clarke, D. W., S. N. Cope, and P. J. Gawthrop ( 1975). Feasibility Study of the Application of Microprocessors to Self-Tuning Controllers. Report no. 1137175. Dept. of Eng. Sc., University of Oxford, Oxford. 70 PP. Clarke, D. W., and P. J. Gawthrop (1979). Selftuning control. Proc. IEE, &, 633-640. Howarth, R. E. ( ~ d . )(1978). Proc 2nd International Green Coop Drying Congress. Extension Division, University of Saskatcheman, Saskatoon. 41 8 pp. Kurz, H., R. Isermann, and R. Schunann (1978). Development, comparison and application of various parameter - adaptive digital control algorithms. Preprints 7th IFAC World Congress, 443-452. Mod&, P. E., T. SBderstr6m (1979). On the achievable accuracy in stochastic control. Proc. 17th IEEE Conference on Decision and Control, 490-495. Nybrant , T. , P. E. ?ilod$n ( 19793 User' s manual for ZREG,a process control system. UPTEC Report no. 79 1OlR. Inst. of Tech., University of Uppsala, Uppsala.
.
Abstract. The process of drying green crops in a high-temperature rotary drum dehydrator is difficult to control. It is time-varying and there are stochastic disturbances. Adaptive control of the resulting moisture content is considered. The control laws discussed are based on linear quadratic gaussian optimal control with finite time horizon. They enable a suitable compromise between the minimal output variance wanted and the limited inputs available, and they are easy to apply adaotively. There will be a steady state error unless integral action is included. Different methods to add integral action are discussed. The resulting adaptive controllers were implemented on a microcomputer and applied to a drum drier. The practical results are presented and compared. Keywords. Adaptive control; stochastic control; steady state error; agriculture; drying process; temperature control; moisture control. INTRODUCTION
excessive comsumption of oil.
High-temperature drying of green crops is a way to conserve forage (~owarth,1978). In the process grass is dried and then pressed into cobs. It is very important to keep the moisture content of these cobs within certain limits for several reasons. The high-temperature drying process is known to be difficult to handle. This fact combined with high energy consum~tionof the process explains the need for a good controller. The conventional three-term controllers now in use give in many cases unsatisfactory results. (Carlsson, Mod& and Sandberg, 1978)
gross Input elevglor
--
press cooler A project "Adaptive Control of High-temperature Drum ~riers"has been performed in cooperation between the Institute of Technology at Uppsala Fig. 1. General view of a high-temperature drum drier for green crops. It is University and the Swedish University of Agrifurther described in the text. cultural Sciences. The goal has been to develop an adaptive controller in a microcomputer. In A direct measurement of the moisture in the this paper theoretical aspects as well as cobs seems to be the natural signal to use practical results are presented. for control. However, humidity sensors have not been reliable enough for dried fcrage and instead different temperatures in the proPROCESS DESCRIPTION cess could be used as output signal.
A view of a high-temperature drum drier is The chopped grass is fed in- These temperatures and other signals given in Fig. 1 . interest are: to the slowly rotating drum where it is dried by the hot gases from the oil burner. The grass u fuel valve (oil flow) M is separated in the cyclones from the hot gases v feed rate of grass and it is formed into cobs by the press. Before yl temperature in the oven they can be stored they have to pass a cooler. y2 temperature after the drum M y~ temperature of the dried w Too high a moisture content in the cobs would grass make them impossible to store. If too dry, they y ~ +temperature in the exhaust M would easily fall apart, their nutrition value pipe would be reduced and it would also cause an
-
of 0 , I L/s 1 ,60kg/s 8 0 0 ~ ~ 130-C 85°C 1 1 0 ~ ~
P.E. Mod6n an~d T. Nybrant E x i s t i n g systems use u o r v a s c o n t r o l i n p u t and y2, y3 o r YI, a s o u t p u t s i g n a l . y l i s u s e d t o check t h a t t h e t e m p e r a t u r e i n t h e oven i s n o t t o o h i g h . O t h e r r e l e v a n t d a t a of t h e p l a n t used i n t h e experiments a r e :
The i d e n t i f i c a t i o n e x p e r i m e n t s a l s o c l e a r l y indicated t h a t t h e coefficients i n t h e A, B, C and D polynomials were v a r y i n g w i t h t i m e and working l e v e l . I t i s concluded t h a t t h e system i s n o n - l i n e a r and timev a r i a b l e . F i g u r e 2 shows s t e p r e s p o n s e s o f models o b t a i n e d from d i f f e r e n t e x p e r i m e n t s .
Water e v a p o r a t i o n c a p a c i t y 4 500 kg/hour O i l consumption t y p 0,36 m3/hour or 0,3 L/kgofcobs Cob p r o d u c t i o n t y p 1 200 kg/hour Moisture content of t h e input g r a s s i s about 80 p e r c e n t and t h e c o r r e s p o n d i n g v a l u e o f t h e cobs i s a b o u t 10 p e r c e n t . Drying o f f i e l e w i l t e d g r a s s r e d u c e s t h e o i l c o s t s by 30-35 p e r c e n t ( C a r l s s o n , 1 9 7 6 ) . On t h e o t h e r hand i n t h i s c a s e t h e c o n v e n t i o n a l three-term c o n t r o l l e r o f t e n g i v e s t o o g r e a t a moisture content v a r i a t i o n i n t h e cobs. A good c o n t r o l l e r t h a t c o u l d h a n d l e t h i s c a s e b e t t e r c o u l d , even i f r a t h e r s o p h i s t i c a t e d , e a s i l y be m o t i v a t e d w i t h r e s p e c t t o t h e current o i l prices.
MODELLING Research done a t t h e Swedish U n i v e r s i t y o f A g r i c u l t u r a l S c i e n c e s showed t h a t ys ( g r a s s t e m p e r a t u r e i n t h e c y c l o n e ) was t h e temperat u r e t h a t had t h e s t r o n g e s t c o r r e l a t i o n w i t h t h e m o i s t u r e c o n t e n t ( B e r g l u n d , Carlstrijm and Svensson, 1 9 7 5 ) . It was t h e r e f o r e chosen a s t h e output signal. Off-line i d e n t i f i c a t i o n e x p e r i m e n t s were p e r formed on t h e a c t u a l p l a n t . They showed t h a t t h e p r o c e s s c o u l d b e w e l l des c r i b e d by t h e f o l l o w i n g l i n e a r model s t r u c ture:
Time (minutes) Fig. 2.
S t e p r e s p o n s e s o f l i n e a r models o b t a i n e d from d i f f e r e n t e x p e r i ment s
.
S i n c e u was found t o i n t r o d u c e l e s s t i m e d e l a y t h a n v , u was chosen a s t h e c o n t r o l i n p u t and v was r e g a r d e d a s a known d i s t u r b a n c e t h a t c o u l d b e manually o p e r a t e d .
ADAPTIVE CONTROL The p u r p o s e o f t h e c o n t r o l c a n b e d i v i d e d i n t o two t a s k s . The v a r i a n c e o f t h e c o n t r o l e r r o r s h o u l d b e s m a l l , and t h e mean v a l u e should be equal t o zero. A conventional t h r e e - t e r m c o n t r o l l e r c o u l d manage t h e mean value, but not t h e variance. Since t h e system i s t i m e - v a r y i n g , t h e r e q u i r e m e n t on t h e v a r i a n c e makes a d a p t i v e c o n t r o l n e c e s s a r y . I n p u t and Output V a r i a n c e s
k=0
t i n e d e l a y from
u
R=l
t i n e d e l a y from
v
g(t ! o u t p u t u( t ) o i l v a l v e
vl t d
:
feed r a t e d e s c r i b e s working l e v e l
e ( t ) w h i t e n o i s e w i t h z e r o mean q
-1
-1 backward s h i f t o p e r a t o r , q y ( t ) = y ( t - l )
A s t r a i g h t f o r w a r d method t o g e t a s m a l l o u t put variance i s t h e self-tuning regulator (Rstrdm and co-workers , 1 9 7 7 ) . I t combines r e c u r s i v e l e a s t s q u a r e s i d e n t i f i c a t i o n (RLS) w i t h minimum v a r i a n c e c o n t r o l (MV), a p p l y i n g t h e c e r t a i n t y equivalence p r i n c i p l e . A forg e t t i n g f a c t o r makes it p o s s i b l e t o compens a t e f o r time-varying system dynamics.
I t i s w e l l known t h a t i n o r d e r t o a c h i e v e t h e minimal o u t p u t v a r i a n c e , a h i g h i n p u t v a r i a n c e i s needed. The models f o r t h e drum d r i e r o b t a i n e d by o f f - l i n e i d e n t i f i c a t i o n i n d i c a t e t h a t t h e physical l i m i t s of t h e inp u t would b e r e a c h e d q u i t e o f t e n , i f MV c o n t r o l was a p p l i e d .
A d a p t i v e Control of R o t a r y Drum D r i e r s
Not o n l y t h e l i m i t s o f t h e i n p u t s i g n a l p u t a r e s t r i c t i o n o n ' t h e i n p u t v a r i a n c e . The f u e l v a l v e (which i s t h e c o n t r o l i n p u t ) i s a c t u a t e d by a r e l a y c o n t r o l l e d AC-motor w i t h c o n s t a n t r a t e o f change. T h i s means t h a t when t h e i n p u t i s changed t h e r e i s a time-delay which i s p r o p o r t i o n a l t o t h e i n c r e m e n t . The models used a r e v a l i d f o r r a t h e r s m a l l i n c r e m e n t s . With t o o l a r g e i n c r e m e n t s t h e d e l a y c o u l d make t h e system non-minimum-phase, and t h u s t h e c l o s e d system u n s t a b l e i n t h e c a s e o f MV c o n t r o l . A s m a l l e r i n p u t v a r i a n c e a l s o makes t h e i n p u t i n c r e m e n t s s m a l l enough. I n order t o decrease t h e input variance l i n e a r q u a d r a t i c o p t i m a l c o n t r o l c a n be a p p l i e d , choosing u ( t ) t o minimize
i s ~ o s i t i v er e a l ( ~ o d g nand b Q 8 ( p 1) / A ( < ' SoderstrGm, 1979)
-
4
.
aJ u
r3. m L
.r
0
>
C, 2
ab
C,
3
0
1.
.
Here p i s a p e n a l t y on t h e c o n t r o l i n p u t (~20)T . h i s c r i t e r i o n w i l l b e c a l l e d L%.
2
1
B
d
1
( i n p u t variance)/lO
1
&
Having a known t i m e - i n v a r i a n t system and a n i n f i n i t e t i m e h o r i z o n , L&m would g i v e a timei n v a r i a n t c o n t r o l l e r . Applying L&m adapt i v e l y would l e a d t o much more computation t h a n MV. However, v e r y simple computations a r e obtained with a f i n i t e time horizon j u s t k+l s t e p s a h e a d , LQk+l. I n t h i s c a s e u ( t ) minimizes ( 2 ) w i t h N=k+l a s w e l l a s
F i g . 3. The p o s s i b l e combinations o f i n p u t and o u t p u t v a r i a n c e s when LQk+l o r L&m c o n t r o l i s a p p l i e d t o a model o f t h e has drum d r i e r . The open system (p*) t h e o u t p u t v a r i a n c e 19. With MV cont r o l (p=O) t h e i n p u t v a r i a n c e i s 7 4 , and t h e o u t p u t v a r i a n c e i s 1 . (The variances a r e normalized.)
d e s c r i b e d by ( 1 ) , t h i s i s For a system e q u i v a l e n t t o a p p l y i n g MV t o a n augmented system which h a s t h e o u t p u t ( C l a r k and cow o r k e r s , 1975)
From F i g . 3 it i s e a s y t o chose a c o n t r o l l e r which g i v e s a n a c c e p t a b l e i n p u t v a r i a n c e and a n o u t p u t v a r i a n c e which i s c l o s e t o t h e minimal one. For example LQk+l w i t h p 0 . 0 1 w i l l r e d u c e t h e i n p u t v a r i a n c e by a f a c t o r o f 3 and i n c r e a s e t h e o u t p u t v a r i a n c e by o n l y 10 p e r c e n t compared t o MV. S i m i l a r r e s u l t s a r e obt a i n e d f o r a l l t h e o f f - l i n e i d e n t i f i e d drum d r i e r models.
The r e s u l t i n g
L%+l
Remembering t h a t LQk+l i s e q u i v a l e n t t o MV c o n t r o l of t h e introduced v a r i a b l e f ( t ) , t h e combination w i t h RLS t o form a n a d a p t i v e c o n t r o l l e r i s s t r a i g h t f o r w a r d ( ~ l a r k e ,Cope and Gawthrop, 1 9 7 5 ) -
controller is
S e t - p o i n t and S t e a d y S t a t e E r r o r where t h e p o l y n o m i a l s F ( q - l ) o f d e g r e e and ~ ( q - ') o f d e g r e e one l e s s t h a n t h e system a r e o b t a i n e d from t h e i d e n t i t y
k
The i n p u t p e n a l t y p i s t r e a t e d a s a d e s i g n v a r i a b l e which s h a l l b e chosen t o g i v e a c c e p t able variances. The p o s s i b l e combinations o f i n p u t and o u t p u t v a r i a n c e s , when L&k+, o r L& i s applied t o a t y p i c a l model o f t h e drum d r i e r , i s shown There i s a d i f f e r e n c e between i n Pig. 3 , LQk+l and L&m b u t it i s t o o s m a l l t o b e v i s i b l e i n t h i s f i g u r e . This i s not t h e case f o r a l l s y s t e m s , b u t good performance w i t h LQk+l i s g u a r a n t e e d i f t h e t r a n s f e r f u n c t i o n
The mean v a l u e o f t h e o u t p u t i s t o be k e p t a t t h e d e s i r e d l e v e l , t h e s e t - p o i n t ~ , ( t )T.h i s r e f e r e n c e v a l u e may be i n c l u d e d i n t h e LQk+l c o n t r o l l e r i n d i f f e r e n t ways. The s i m p l e s t method i s t o s u b s t i t u t e y ( t ) - y r ( t ) f o r y ( t ) i n t h e c o n t r o l law ( 5 ) . T h i s may, however, g i v e bad t r a n s i e n t r e s p o n s e s t o s e t - p o i n t changes. An a l t e r n a t i v e which i s b e t t e r i n t h i s r e s p e c t i s t o s u b s t i t u t e ~ ( i ) - ~ ,f(o tr ) y ( i ) i n t h e c r i t e r i o n ( 2 ) . Also a n i n p u t r e f e r e n c e u r ( t ) may b e i n c l u d e d by s u b s t i t u t i n g u ( i ) - u r ( t ) f o r u ( i ) i n t h e same c r i t e r i o n ( 2 ) . Then w i t h N=k+l ( f i n i t e t i m e h o r i z o n ) u ( t ) minimizes
P . E . Moden and T . Nybrant
I f t h e system S i s g i v e n by ( I ) , t h e c o r r e s ponding augmented system h a s t h e o u t p u t
It i s shown i n F i g . 4 . The r e s u l t i n g L Qk+1 controller with reference signals i s [ B ( ~ - ' I F ( ~ I- +'
$ c(q-'
)IU(~)=
0
The polynomials same a s i n ( 5 ) .
F(q-' )
and
~ ( q - )l
are the
T h i s means t h a t t h e r e i s a p e n a l t y on t h e i n c r e m e n t s ( M ( ~)=1 - ~) o r f i l t e r e d i n c r e m e n t s , and ~ ( l ) = l g u a r a n t e e s z e r o o f f s e t i f t h e c l o s e d system i s s t a b l e ( C l a r k e and Gawthrop, 1979). The above mentioned methods t o e l i m i n a t e t h e s t e a d y s t a t e e r r o r do n o t work u n l e s s t h e system p a r a m e t e r s a r e e x a c t l y known. When combining t h e c o n t r o l law w i t h a r e c u r s i v e p r e d i c t i o n e r r o r i d e n t i f i c a t i o n method ( e . g . RLS) t o an a d a p t i v e c o n t r o l l e r , t h e e s t i m a t e s o f t h e working l e v e l p a r a m e t e r d and t h e s t a t i c g a i n become r a t h e r poor and z e r o o f f s e t cannot be guaranteed. To a v o i d t h e o f f s e t , a f e e d b a c k from t h e integrated control e r r o r has t o be included i n t h e c o n t r o l l e r . The e x c e p t i o n i s when t h e c o n t r o l l e r ( 5 ) , o p e r a t i n g on t h e e r r o r y(t)-yr(t) instead of y ( t ) , i s applied t o a n i n t e g r a t i n g system. The system can b e made i n t e g r a t i n g , by cons i d e r i n g t h e input increments t o be t h e c o n t r o l i n p u t . Then t h e LQk+l c o n t r o l l e r w i l l have a z e r o which c a n c e l s t h e i n t e g r a t o r p o l e . T h i s z e r o c a n b e d e l i b e r a t e l y moved away from 1 , which l e a d s t o t h e c o n t r o l l e r i n F i g . 5 . (Kcrz , Isermann and Schumann, 1978)
.
F i g . 4 . Block scheme showing t h e c o n s t r u c t i o n o f t h e h e l p v a r i a b l e f ( t) Minimum v a r i a n c e c o n t r o l o f f ( t ) i s e q u i v a l e n t t o LQ control (with r e f e r e n c e s i g n a l s y A f t h e system S .
.
With t h i s c o n t r o l l e r ( 8 ) a p p l i e d t o a system ( 1 ) w i t h e x a c t l y known p a r a m e t e r s , t h e mean v a l u e o f t h e o u t p u t becomes
F i g . 5 . Block scheme showing a method t o i n c l u d e i n t e g r a l a c t i o n i n t h e cont r o l l e r . No feed-forward t e r m s have been i n c l u d e d .
if
y r , ur and v a r e c o n s t a n t . The o f f s e t i s z e r o i f e i t h e r p=O o r t h e p a r e n t h e s i s i s e q u a l t o z e r o . Thus w i t h p > 0 t h e i n p u t r e f e r e n c e u r ( t ) must b e chosen c o r r e c t l y . I t c a n a l s o be u p d a t e d by f i l t e r i n g o l d v a l u e s of t h e input. ur(t)=q-'hl(q-l
)u(t)
(10)
Here ~ ( q - )' i s a t r a n s f e r f u n c t i o n w i t h I n t h i s case t h e help variable i s M( 1 ) = l
.
A p e n a l t y on t h e i n t e g r a t e d e r r o r can be i n c l u d e d by t a k i n g t h e h e l p v a r i a b l e f ( t )t o be
-1 where P ( q ) i s i n t e g r a t i n g ( C l a r k e and Gawthrop, 1 9 7 9 ) . F i g 6 shows t h e r e s u l t i n g c o n t r o l l e r when ~ ( q - )l = p N ( q - l ) / (1-q-l )
.
E a r l i e r i n t h i s s e c t i o n one way o f u p d a t i n g t h e i n p u t r e f e r e n c e u r ( t ) was mentioned. It c a n a l s o b e u p d a t e d u s i n g t h e i n t e g r a t e d error. I n order t o r e t a i n t h e integration f o r t h e c a s e p=O, one c a n s u b s t i t u t e w ( t )
Adaptive Control of Rotary Drum D r i e r s PRACTICAL RESULTS During t h e autumn o f 1979 a d a p t i v e c o n t r o l was a p p l i e d t o t h e r o t a r y drum d r i e r . Some d i f f e r e n t s e l f - t u n i n g c o n t r o l l e r s based on LQk+l were t e s t e d . The d i f f e r e n c e was t h e method t o handle t h e s t e a d y s t a t e e r r o r , t h a t i s , how t h e i n t e g r a l a c t i o n was added. A l l of t h e c o n t r o l l e r s i n c l u d e d RLS i d e n t i f i c a t i o n of t h e c o n t r o l l e r parameters d i r e c t l y . I n t h e RLS t h e U-D-factorization a l g o r i t h m was used (Bierman, 1975). The c o n t r o l l e r s were implemented on a comp u t e r system f o r experiments i n automatic c o n t r o l . The system c o n s i s t s of a 2-80 microcomputer and a programpackage f o r o p e r a t o r s communication (Nybrant and ~ o d 6 n ,1979). The F i g . 6. Block scheme showing t h e c o n t r o l o f c o n t r o l l e r programs were w r i t t e n i n Z i l o g ' s t h e system S, when t h e LQk+l c o n t r o l h i g h - l e v e l language PLZ and t h e t y p i c a l s i z e l e r i s extended t o i n c l u d e a p e n a l t y of t h e c o n t r o l l e r programs was i n t h e o r d e r on t h e i n t e g r a t e d e r r o r . The h e l p v a r i of 8 kbyte. S i n c e f a s t a r i t h m e t i c o p e r a t i o n s a b l e f ( t ) i s i n t h i s case f ( t ) = p N ( q - l) I (1-q-I ) [ ~ ( t ) - ~ ~ ( t)I+- ~ - l were d e s i r e d , f l o a t i n g - p o i n t hardware was i n c l u d e d i n t h e system. This i n c r e a s e d t h e +p/bou(t-k-1 ) . The polynomials F and G computation speed about 60 times compared a r e given be t h e i d e n t i t y with arithmetic i n software.
for let
u ( t ) i n t h e c o n t r o l l e r ( 8 ) and
yr(t)+ bo
Here t h e c o n t r o l l e r s ' a b i l i t y t o reduce t h e o u t p u t v a r i a n c e and t o e l i m i n a t e t h e o f f s e t i s i l l u s t r a t e d i n diagrams showing t h e logged o u t p u t s and s e t - p o i n t s . The sequences shown a r e r e p r e s e n t a t i v e f o r t h e behaviour of t h e d r i e r , under normal o p e r a t i n g c o n d i t i o n s , when t h e d i f f e r e n t a d a p t i v e c o n t r o l l e r s a r e used.
-1 where t h e t r a n s f e r f u n c t i o n H(q ) i s i n t e g r a t i n g and y r o ( t ) i s t h e s e t - p o i n t . This l e a d s t o cascade c o n t r o l . I t i s d e s c r i b e d i n F i g . 7. I n t h e i n n e r loop t h e r e i s t h e LQk+l c o n t r o l l e r w i t h r e f e r e n c e s i n a l s , and i n t h e o u t e r l o o p t h e r e i s t h e ~ ( q -) , which t y p i c a l l y i s a PIc o n t r o l l e r . This can be compared t o t h e cascade c o n t r o l suggested by Clarke and Gawthrop ( 1979). Note however, t h a t t h e i n t e r p r e t a t i o n of ( 13) labove may g i v e some h e l p i n t h e choice of ~ ( q) .
7
90
F i g . 7. Block scheme showing cascade c o n t r o l , with a PI-controller i n t h e outer l o o p . The s i g n a l w( t ) r e p l a c e s y r ( )t+ ~ / b o u , ( t) i n t h e LQk+l c o n t r o l l e r (8). Figures 5, 6 and 7 r e p r e s e n t t h r e e d i f f e r e n t methods t o i n c l u d e i n t e g r a l a c t i o n by some ext e n s i o n of t h e LQk+l c o n t r o l l e r . There a r e of course o t h e r methods t o o .
100 110 minutes
120
Fig. 8 . S e t p o i n t and o u t p u t from t h e drum d r i e r . The c o n t r o l l e r i s a d a p t i v e and i n c l u d e s a p e n a l t y on t h e i n t e g r a t e d e r r o r , c f . F i g . 6 . The sequence shown i s t h e l a s t 40 minutes of t h e r u n . I n Fig. 8 t h e controller includes a penalty on t h e i n t e g r a t e d e r r o r , a s d e s c r i b e d by ( 1 2 ) and F i g . 6 , w i t h ~ ( q - lkpN(q-l ) / ( 1-q-l ) The performance i s r a t h e r poor. The v a r i a n c e i s t o o l a r g e , and t h e r e a l s o seems t o be an o f f s e t . One p o s s i b l e e x p l a n a t i o n i s t h a t t h e
.
P.E.
Mod6n and T. Nybrant
model s t r u c t u r e used f o r t h e RLS i d e n t i f i c a t i o n i s not s u i t a b l e f o r t h i s c o n t r o l method. With t h i s s t r u c t u r e t h e i d e n t i f i c a t i o n does not t a k e advantage of t h e f a c t t h a t t h e t r a n s f e r f u n c t i o n ~ ( ~ - i ln ) ( 1 2 ) i s known. Adaptive c o n t r o l without i n t e g r a l a c t i o n was a l s o t e s t e d . This was t h e c a s e i n F i g . 9. The s h o r t term v a r i a n c e i s a c c e p t a b l e from 40 minutes and on, but t h e r e i s an o f f s e t . The i n c r e a s i n g temperature a t t h e end of t h e r u n i s t y p i c a l and i s s e e n i n F i g . 8 t o o . I t i s caused by a d e c r e a s i n g g r a s s flow. When t h e r e i s l i t t l e g r a s s l e f t on t h e e l e v a t o r , t h e flow i s l e s s t h a n i n d i c a t e d by t h e f e e d r a t e s i g n a l , s i n c e t h e g r a s s i s not packed s o h a r d .
100"C
90.
80-
Y 1
hours
2
3
F i g . 10. S e t p o i n t and output from t h e drum d r i e r . Cascade c o n t r o l i s used. The LQk+l c o n t r o l l e r i n t h e i n n e r l o o p i s a d a p t i v e , and t h e P I - c o n t r o l l e r i n t h e outer loop i s time-invariant, c f . F i g . 7 . The p a r t shown s t a r t s when t h i s c o n t r o l l e r i s a c t i v a t e d and cont i n u e s t o t h e end of t h e r u n .
+ . . . . . . . . . . . . . . . .1
0 Fig.
20
40 minutes
60
80
9. S e t p o i n t and o u t p u t from t h e drum d r i e r . The c o n t r o l l e r i s a d a p t i v e and h a s no i n t e g r a l a c t i o n . I t i s t h e LQk+l w i t h r e f e r e n c e s i g n a l s ( 8 ) . The i n p u t r e f e r e n c e u r ( t ) was c o n s t a n t . The whole r u n i s shown except t h e s t a r t when t h e p l a n t was c o n t r o l l e d manually.
F i n a l l y i n Fig.10 and 1 1 a d a p t i v e c o n t r o l was used w i t h a P I - c o n t r o l l e r added i n an o u t e r l o o p , a s d e s c r i b e d by ( 1 3 ) and F i g . 7 . Now t h e e r r o r v a r i a n c e i s s m a l l , and t h e o u t p u t f o l l o w s t h e changes i n t h e s e t - p o i n t adequat e l y ( b e t t e r i n F i g . 11, where t h e g a i n i n t h e outer loop i s higher )
.
Obviously t h e c o n t r o l l e r used i n F i g . 10 and 1 1 g i v e s t h e b e s t performance. I t i s t h e o n l y one of t h e t e s t e d t h a t f u l f i l s t h e r e q u i r e ments
.
-
o Fig.
T
1
i hours
Set pointand output from the drum d r i e r . The same c o n t r o l l e r i s used a s i n F i g . 10. The f i r s t 3 hours and 20 minutes of t h e r u n a r e shown.
CONCLUSIONS D i f f e r e n t a d a p t i v e c o n t r o l l e r s have been t e s t e d on a p r o c e s s t h a t i s n o n l i n e a r and t i m e v a r i a b l e . One of t h o s e c o n t r o l l e r s was found t o g i v e a small o u t p u t v a r i a n c e , an a c c e p t a b l e i n p u t v a r i a n c e , and no o f f s e t from t h e r e f e r e n c e v a l u e . Conpared w i t h t h e conv e n t i o n a l t h r e e term c o n t r o l l e r t h e o u t p u t v a r i a n c e was c o n s i d e r a b l y reduced. The t e s t s a l s o i n d i c a t e d t h a t t h i s c o n t r o l l e r made it p o s s i b l e t o d r y f i e l d w i l t e d g r a s s and t h u s c o n t r i b u t e t o a c o n s i d e r a b l y reduced energy consuinption.
Adaptive Control of Rotary Drum Driers ACKNOWLEDGMEBTS Part of this work was finacially supported by the Swedish Board for Technical Development under contract 77-3555, which is gratefully acknowledged. The authors also want to express their gratitude to GBran Carlsson, whose genuin knowledge of the process and assistance in the practical experiments were invaluable. REFERENCES AstrBm, K.J., U. Borisson, L. Ljung and B. Wittenmark (1977). Theory and applications of self-tuning regulators. Automatica, 12,457-476. Berglund, L., P. Carlstrijm, K.-E. Svensson (1975). Studium av ett reglersystem f6r hetluftstorkar. Report no. 30. Dept. of Agric. Eng., Swedish University of Agricultural Sciences, Uppsala. 122 pp. (1n we dish) Bierman, G. J. (1975). Measurement updating using the U-D-factorization. Proc. 14th IEEE Conference on Decision and Control, 337-346. Carlsson, G. ( 1976). Nggra processtekniska faktorer vid hetluftstorkning. Report no. 31. Dept. of Agric. Eng., Swedish University of Agricultural Sciences, Uppsala.53 pp. (1n Swedish) Carlson, G., P. E. Mod&, A. Sandberg (1978). Adaptive optimal control of high-temperature drum driers. Proc. 2nd International Green Coop Drying Congress, 220-227. Clarke, D. W., S. N. Cope, and P. J. Gawthrop ( 1975). Feasibility Study of the Application of Microprocessors to Self-Tuning Controllers. Report no. 1137175. Dept. of Eng. Sc., University of Oxford, Oxford. 70 PP. Clarke, D. W., and P. J. Gawthrop (1979). Selftuning control. Proc. IEE, &, 633-640. Howarth, R. E. ( ~ d . )(1978). Proc 2nd International Green Coop Drying Congress. Extension Division, University of Saskatcheman, Saskatoon. 41 8 pp. Kurz, H., R. Isermann, and R. Schunann (1978). Development, comparison and application of various parameter - adaptive digital control algorithms. Preprints 7th IFAC World Congress, 443-452. Mod&, P. E., T. SBderstr6m (1979). On the achievable accuracy in stochastic control. Proc. 17th IEEE Conference on Decision and Control, 490-495. Nybrant , T. , P. E. ?ilod$n ( 19793 User' s manual for ZREG,a process control system. UPTEC Report no. 79 1OlR. Inst. of Tech., University of Uppsala, Uppsala.
.