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Automatica. Vol. 23, No. 2, pp. 149 160. 1987Printed in G reat Britain.

0005 1098/87$3.00+0.00Pergamon Journals Ltd.

1987 nternationalFederationof AutomaticControl

G e n e r a l iz e d P r e d i c t iv e C o n t r o l P a r t II .

E x t e n s i o n s a n d I n t e r p r e t a t i o n s *D. W. CLARK Ft, C. M OHTAD It and P. S. TUFFS~

By relat ing a novel predict ive control ler to LQ designs s tabi l i ty theorems are deduced,and extension s to G eneral ized Predict ive Contro l give model fol lowing , s ta te-dead-beatand po le-placement control objectives.

Key W ords - -Se l f - tun in g con t ro l ; p red ic t ive con t ro l ; mode l - r efe rence con t ro l; LQ con t ro l ; dead-bea tcon t ro l ; nonmin imum -phase p lan t ; va r i ab le dead- t ime .

Abs t rac t - -The o r ig ina l GMV se l f - tune r was l a t e r ex tendedto provide a general f ramework which included feedforwardcom pensa t ion and use r-chosen po lynomia l s wi th de tuned mod -el- reference, opt im al Smith pre dictor and load-d is turban ce ta il -or ing object ives . This paper adds s imilar ref inements to theGP C a lgor i thm w hich a re i l lu s t r a ted by a se t o f simula t ions .The r e l a t ionsh ip be tween GP C and LQ des igns is inves t iga tedto show the compu ta t iona l advan tage o f the new approach . Thero les o f the ou tpu t and con t ro l ho r i zons a re exp lo red fo rp rocesses wi th non min im um-ph ase , uns tab le and va r i ab le dead-t ime mode ls . The robus tness o f the GPC approach to mode love r- and under-pa ram ete r i za t ion and to f a s t s ampl ing r at e s i sdemons t ra t ed by fu r the r s imula t ions . An append ix de r ivess tabi l i ty resul ts showing that cer ta in choices of control andou tp u t ho r i zons in GP C lead to cheap LQ, "m ean- l eve r ' , s t a t e -dead-bea t and po le -p lacemen t con tro l l er s .

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

THE BASIC G P C m etho d deve loped in Par t I is ana tura l successor to the GM V a lgor ithm of Cla rkeand Gawthrop (1975) in which a cos t - func t ion o fthe form:

JoMv = E{(y(t + k) - -w(t))2 + 2uZ(t)[t} (1)

was f i rs t def ined an d minimized. T he c ost of (1) iss ingle-stage and so i t is found that effective contro l

depends on know ledge o f the dead- t ime k o f thep lan t and fo r nonmin imum-phase p lan t s t ab i l i tyrequires a no nzero value of 2. The use of long-range p red ic t ion and a mul t i - s t age cos t in GPCovercom es the p rob lem of st ab il iz ing a nonm in i -mum-phase p lan t wi th unknown or va r iab le dead-time.

The re la t ive impor tance o fcont ro l le r perfor-mance cr i ter ia var ies with the appl icat ion area. For

* Received 2 M arch 1985; revised 7 July 1985; revised 3 Marc h1986; revised 22 September 1986. The or iginal vers ion of th is

pape r was no t p resen ted a t any IFAC mee t ing . Th i s pape r wasrecommended fo r pub l i ca t ion in r ev i sed fo rm by Assoc ia t eEd i to r M . Gever s under the d i r ec tion o f Ed i to r P. C . Pa rks .

" t Dep ar tm ent o f Engineer ing Science, P arks Road, OxfordOX1 3PJ, U.K.

:~ Alum inum Com pany o f Amer ica, A lcoa Techn ica l Cen te r,Pi t tsburgh, PA 15069, U.S.A.

14 9

example, in process control i t i s general ly foundthat energet ic control s ignals are undesirable , as lowly responding loop being prefered, and plantmode ls a re po or ly spec if ied in t e rms o f dead-t ime and order as wel l as their t ransfer funct ionparameters . Emphasis is therefore placed on robustand cons i s ten t pe r fo rmance desp i te va r ia t ions inquant i t ies such as dead-t ime and despi te sustainedload-d i stu rbances . H igh-per formance e lec t romech-an ica l sys tems tend to have wel l -unders tood m ode lsthough of ten with l ight ly-damped poles , and the

control requirement is for fas t response, accept ingthe fact that the actuat ion might saturate . I t i sdoubtful whether a s ingle cr i ter ion as in (1) candeal with such a wide range of problem s, so tocreate an effective gene ral-purpos e self- tuner i t i sessent ia l to be able to adapt the basic approach bythe incorpora t ion o f " tun ing-knobs" .

The GMV des ign was deve loped in to a use fu lself-tuning algori thm by the addi t ion of user-chosentransfer funct ionsP(q-1) , Q(q-1)and an observerpo lynomia l T(q-1) . These t ime-domain per for-mance-or ien ted "knob s" a l low the eng ineer to t ac -

kle different control problems within the sameovera ll scheme. For example , Gaw thro p (1977) andClarke and Gawthrop (1979) showed tha t mode l -fol lowing, detuned model-fol lowing, and opt imalSm ith predict ion w ere interpretat ions which couldbe invoked as well as the or iginal contro l weight ingcon cep t o f (1). Cla rke (1982) and Tufts (1984) givefur ther examples o f the use o f these po lynomia l s inpractice.

This paper in t roduces s imi la r po lynomia l s toGPC for spec i fy ing a des i red c losed- loop mode land for ta i lor ing the control led responses to loadd is tu rbances , and the der iva t ion o f G PC ise x p a n d e d t o i n c l u d e t h e m o r e g e n e r a l C A R I M Amod el . The prop ert ies of these extensions are ver-i fied by s imulat ions, which also sho w the robu stnessof the meth od to a range o f p rac t ica l p rob lem s such

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1 50 D . W . C LA R K Eet al .

a s m o d e l o v e r - a n d u n d e r - p a r a m e t e r i z a t i o n . A na p p e n d i x r e l a t e s G P C w i t h a s t a t e - s p a c e L Q c o n -t ro l l aw and de r ives s t ab i l i ty r e su l t s fo r two useso f G P C : l a rg e o u t p u t a n d c o n t r o l h o r i z o n s N 2 a n dN U l e a d i n g to a c h e a p L Q c o n t r o ll e r , a n dN U = 1

g iv ing a "mean- leve l " con t ro l l e r fo r s t ab le p lan t sw h i c h n e e d n o t b e m i n i m u m - p h a s e . M o r e o v e r,t h e r e l a t i o n s b e t w e e n s t a t e - d e a d - b e a t c o n t r o l a n dd e t e r m i n i s ti c p o le - a s s i g n m e n t w i t h G P C a r e e x a m -ined.

2. EX TENSIONS TO GEN ERA LIZED PREDICTIVECONTROL

2.1. M o d e l - f o l l o w i n g :P(q- x )Genera l i zed Pred ic i t ive Con t ro l a s desc r ibed in

Par t I is based on m in im iza t io n o f a se t o f p red ic teds y s t e m e r r o rs b a s e d o n a v a i l a b le i n p u t - o u t p u td a t a , w i t h s o m e c o n s t r a i n t s p l a c ed o n t h e p r o j e c t e dcontrol s ignals . I t i s poss ible to use ana u x i l i a r yf u n c t i o n o f t h e o u t p u t a s i n t h e G M V d e v e l o p m e n ta n d c o n s i d e r t h e p r e d i c t e d s y s t e m e r ro r s a s s o c i a t e dw i t h t h i s p s e u d o - o u t p u t a n d t h e p r o j e c t e d s e t -po in t s .

C o n s i d e r t h e a u x i l i a r y o u t p u t :

t~ ( t )= P ( q - 1 ) y ( t )w h e r ep(q- 1 )= P n ( q - 1 ) / P d ( q -1).

p (q -1 ) i s a t r ans fe r- func t ion g iven by po lynom ia l sP n a n d P d i n t h e b a c k w a r d s h if t o p e r a t o r q - 1 w i t hP(1) se t to un i ty to en sure offse t -f ree con trol . Th ecos t tha t the con t ro l l e r min im izes i s the expec ta t ionsub jec t to da ta ava i l ab le a t t ime t o f :

t J = N 1

}~ 2(j)[Au(t + j -- 1)] 2where :

O ( t )N 1N 2;~( j )

is P ( q - 1).~t);i s t h e m i n i m u m c o s t i n g h o r i z o n ;i s t h e m a x i m u m c o s t i n g h o r i z o n , a n di s a con t ro l -we igh t ing sequence .

T h e p r e d i c t i o n e q u a t i o n s g i v e n i n P a r t I m u s tthe re fo re be m odi f i ed to fo recas t @(t + j ) i n s t ead o f3~t +j) .

The loca l ly - l inea r i zed p lan t mode l to be cons id -e r e d is b a s e d o n a n A R I M A r e p r e s e n t a t i o n o f t h ed i s tu rbances :

A ( q - 1 ) y ( t ) = B ( q - 1 ) u ( t -1) + C ( q - 1 ) ~ ( t ) / A (2b)

and aga in fo r s impl i c i ty o f de r iva t ion i t is a s sume dtha t C(q-~) = 1 .

C o n s i d e r t h e D i o p h a n t i n e i d e n t i t y :

_~_ = iF:P n E s A A + q j = 1 , 2 , (3)P d Y d . . ..

M ul t ip ly in g (2b) byq J E j A and fo l lowing the samerou te a s in Pa r t I , we ob ta in :

~ ( t + j l t ) = G j A u ( t + j -1)+ F y ( t ) / P d ( q - x )

where Gj = E j B .(4)

T h e D i o p h a n t i n e r e c u r s i o n e q u a t i o n s d e v e l o p e dear l i e r a re iden t i ca l to those invo lved he re bu ts imply the s t a r t ing po in t i s d i ff e ren t :

Pn(O)E 1 - P d ( O ) ' F 1 = q ( P n - E 1 A ) , a n d A = A A P d .

The t r ans fe r func t ionP ( q - 1 )has tw o d i s t inc t in t e r-p r e t a t i o n s d e p e n d i n g o n t h e p a r t i c u l a r a p p l ic a t i o na n d o n t h e c o n t r o l s t r a t e g y e n vi s a ge d . F o r p r o ce s scon t ro l , when dea l ing wi th a " s imple" p lan t thep r i m a r y d e s i g n " k n o b s " a r e N 2 a n dN U . I f theprocess ou tpu t has a l a rge over shoo t to se t -po in ta n d l o a d c h a n g e s ,P ( q - 1 ) can be used to pena l i zet h i s o v e r s h o o t . I n h i g h - p e r f o r m a n c e a p p l i c a t i o n sN U i s c h o s e n l a rg e r t h a n u n i t y b y a n a m o u n td e p e n d i n g o n t h e c o m p l e x i t y o f t h e p l a n t .P ( q - 1 )c a n t h e n b e i n t e r p re t e d a s t h e " a p p r o x i m a t e i n v e rs ec l o s e d - lo o p m o d e l " a n d w h e n 2 = 0 a n d N 2= N U >~ kt he r e l a t ionsh ip i s exac t . For the no i se -f ree case th i s impl i e s tha t the c losed- loop responset o c h a n g e s i nw(t ) is given by:

1y( t ) ~ - -pW(t - k ) = M (q - 1 )w(t - k )

w h e r e M ( q - 1 ) i s the use r-chosen c losed- loop m ode l .

T h e m o d e l - f o l l o w i n g p r o p e r ti e s a r e d e t u n e d i ncases whereN U < N 2 ,bu t fo r l a rgeN U t h e c h a n g ein pe r fo rmance i s s l igh t wi thou t the a s soc ia t edp r o b l e m o f n o n m i n i m u m - p h a s e z e r o s in t h e e st i-m a t e d p l a n t w h i c h d e s t a b i l i z e t h e u s u a l m o d e l -reference control lers . I t i s poss ible to ach ieve mod el-f o l lo w i n g ( as i n m a n y M R A C a p p r o a c h e s ) b y u s i n ga pref i l terM ( q - 1),g iv ing an in te rmed ia te se t -po in tw ' ( t) = M ( q - 1 ) w ( t ) ,a n d w i t h a m i n i m u m - v a r i a n c econ t ro l r egu la t ing 3~ t) abo u t w ' (t ). Ho weve r, t hed i s tu rbance re j ec t ion p roper t i e s and the overa l lrobus tness a re no t then a ffec ted by the mode l a s i t

s imply opera tes on the se t -po in t wh ich i s independ-en t o f va r i a t ion s wi th in the loop . G PC in e ffect hasan inve r se se r ie s mode l w hose o u tp u t i s a ffec ted byb o t h w ( t ) a n d t h e d i s t u r b a n c e s a n d s o i s a m o r ep r a c t ic a l a p p r o a c h .

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G e n e r a l i z e d p r e d i ct i v e c o n t r o l - - P a r t I I 151

2.2. C o l o u r e d n o i s eC ( q - 1 ) a n d t h e d e s i g np o l y n o m i a lT(q - 1)

M o s t p r a c t i c a l p r o c e s s e s h a v e m o r e t h a n o n ed i s t u r b a n c e o r n o i s e s o u rc e a c t i n g o n t h e m t o g iv ean e ffect ive p lan t mod e l :

B C1 C.y( t ) = u ( t - k ) + ~ -A ~l ( t ) + .. . + ~ - -~ .( t ).

T h e n o i s e c o m p o n e n t s c a n b e c o m b i n e d i n t o aC

s ing le r an do m sequence-~-A( t ) where the no i se -

co lou r ing p o lyn om ia l C(q -1 ) has a l l o f i t s roo t swi th in the un i t c i r c l e p rov ided tha t a t l eas t onen o i s e e l e m e n t h a s n o n z e r o - m e a n a n d i s p e r s i st e n t lye x c it in g . N o t e t h a tC ( q - 1 )i s a t ime- invar i an t po ly -nom ia l on ly i f t he ind iv id ua l no i se va r i ances t r2r e m a i n c o n s t a n t . H o w e v e r, w i t h a t y p i c a l i n d u s t r i a lp rocess th i s w i ll r a re ly ho ld in p rac t i ce , so success fu liden t i f i ca t ion o fC ( q - 1 )i s un l ike ly. I f t he s t ruc tu reo f t h e v a r i a t i o n s c a n n o t b e e s t i m a t e d o n - l i ne , ad e s i g n p o l y n o m i a lT ( q - 1) can be used to represe ntp r i o r k n o w l e d g e a b o u t t h e p r o c e s s n o i s e .

O n e i n t e r p r e t a t i o n o fT ( q - 1) is as a f ixed obse rverfo r the p red ic t ion o f fu tu re (pseudo- ) ou t pu t s( A s t r 6 m a n d W i t t e n m a r k , 1 9 8 4) . I n p a r t i c u l a r, i fT ( q - 1 ) = C ( q - 1 )a n d t h e s t o c h a s t i c m o d e l o f (2 b)i s va l id , t hen the p red ic t ions a re a sympto t i ca l lyo p t i m a l ( m i n i m u m - v a r i a n c e ) a n d t h e c o n t r o l l e r w il lm i n i m i z e t h e v a r i a n c e o f th e o u t p u t s u b j e ct t o t h ep r e s p e c i f i e d c o n s t r a i n t s o n t h e i n p u t a n d o u t p u ts e q u en c e s . T h e f o l lo w i n g d e m o n s t r a t e s t h e i n c l u -s ion o fT ( q - 1) o r C ( q - i )i n t h e G P C c o n t r o l s c h e m ew i t h t h e fu ll C A R I M A m o d e l .

As be fo re , de f in ing a Diophan t ine iden t i ty :

T ( q - 1 ) = E j A A + q - J F j ( 5 )

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

r e c u r s io n e q u a t i o n o u t l i n e d i n A p p e n d i x A . C o m b i -ning (6) an d (7) g ives:

j ) ( t +j l t ) = G j A u ( t + j - 1) + F j A u Y ( t - 1) (8)+ F y f ( t ) .

T h e m i n i m i z a t i o n p r o c e d u r e t o p r o v i d e t h e o p t i m a lcon t ro l sequence i s then as g iven in Pa r t I .

T h e c h o i c e o f T ( q - 1 ) f o l l o w s t h e p r o c e d u r ea d o p t e d i n G M V d e s i g ns ( C l ar k e , 1 9 8 2 ; Tu f ts ,1 98 4). I f a c o n s t r a i n e d m i n i m u m - v a r i a n c e s o l u t i o nis required C ( q - 1 ) m u s t b e e s t i m a t e d a n d T p u tequa l to C , bu t fo r mos t p rac t i ca l app l i ca t ions Tcan be t ak en as a f ixed f i r s t -o rde r po lyn om ia l where1/Tis a low-pass f i l ter.

3. RELATIONOF GPC WITH STATE-SPACE LQDESIGN

A n y l i n e a r c o n t r o l l e r m a y b e i m p l e m e n t e d i n as t a t e - s p a c e f r a m e w o r k b y t h e a p p r o p r i a t e s t a t et r a n s f o r m a t i o n s a n d i f t h e c o n t r o l s c h e m e m i n i -m i z e s a q u a d r a t i c c o s t R i c c a t i i t e r a t i o n m a y b ee m p l o y e d . D e t a i l s o f s u c h a c o n t r o l l e r b a s e d o n aC A R I M A m o d e l r e p r e s e n t a t i o n a r e g i v en e l s ew h e r e(Cla rke et a l . , 1985). In o rde r to c ons ide r s t ab i l i tyand numer ica l p roper t i e s , however, i nc lus ion o fd i s tu rba nces i s no t necessa ry, bu t the ins igh t s basedo n w e l l - k n o w n s t a te - s p a c e c o n t r o l l e r d e s ig n c a n b e

a p p l i e d t o t h e G P C m e t h o d . T h i s i s e x p l o r e d i nde ta i l i n App end ix B , whe re in pa r t i cu la r i t is shownh o w v a r i o u s c h oi c e s o f c o n t r o l a n d o u t p u t h o r i z o n sl e a d t o c h e a p L Q , " m e a n - l e v e l ' , s t a t e - d e a d - b e a ta n d p o l e - p l a c e m e n t c o n t r o l le r s .

C o n s i d e r t h e p l a n t :

A Ay ( t ) = B A u ( t -1).

I t s s t a t e - space r ep resen ta t ion in obse rvab le canon-ica l fo rm may be wr i t t en a s :

T ( q - 1 ) P ( t+ j l t ) = G j A u ( t + j - I) + F y ( t )

or:

~( t + j [ t) = G iA uf ( t + j -1) + F y Y ( t ) (6)

where "Y" deno tes a q ua n t i ty f i l t e red by1 / T ( q - 1 ) .T h e c o n s t r a i n t s a n d t h e c o s t a r e i n t e r m s o f

A u ( t + j )for j = 0 , 1 . . . ra th er th anA u ya n d t h e r e -f o r e t h e p r e d i c t i o n e q u a t i o n m u s t b e m o d i f i e d .C o n s i d e r t h e f o ll o w i ng :

G ~ (q - ~ ) = G ' j ~ q - ' ) T ( q - ' ) +q - ~ F ~ ( q - ' ) .(7)

The coe ff i c ien t s o f G ' a re those o f G where thein i t i a l i den t i ty o f (5) has T = 1 . These coe ff i c ien t st o g e t h e r w i t h t h o s e o f F j c a n b e f o u n d b y t h e

x(t + 1) = A x(t) + bA u(t) + if(t)y ( t ) = e T x ( t ) + W ( t) .

(9)

The ma t r ix A and the vec to r s b , e a re de f ined inAp pend ix B . The vec to r f f i s a s soc ia t ed wi th these t -po in t ( see C la rkeet al . , 1985). Fo r th i s m ode lt h e m u l t i - st a g e c o st o f G P C b e c o m e s :

J = x(t + N2)Tx(t + N2)N 2 + t - 1

+ ~ [x(i)rQx (i) +2( i )Au( i )2 ] .i= t

( l O )

T h e c o n t r o l g i v i n g m i n i m u m c o s t i s th e r ef o re :

Au(t) = kT~(t[t)k T = (2(0 + bTp(t)b) -~bTp(t)A

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152 [) . w . CLARKFe t a l .

where P ( t ) i s g iven by the backwards i t e r a t ion o fthe fo l lowing R icca t i equa t ion s t a r t ing f rom thet e r m i n a l c o v a r i a n c e Q :

P * ( i ) = P ( i + 1 ) - P ( i + 1 ) b ( ) .( i )

+ b T P ( i + 1 ) b ) l b T P ( i + 1 ) ( 1 1 )

P(i ) = ATP*(i )A + Q

Q = [1 ,0 ,0 . . . . . 0 ]T [1 ,0 ,0 . . . . . o ] . (12 )

F o r t h e c a s e s w h e r eN U < N 2 ( i .e . when someo f th e c o n t r o l i n c r e m e n t s i n t h e f u t u r e a r e a s s u m e dto be ze ro ) , t he va lue o f2(i) i s t ime-va ry ing , a sf ix ing the con t ro l s igna l i s equ iva len t t o employ inga v e r y la rg e p e n a l t y o n t h e p a r t i c u l a r c o n t r o li n c r e m e n t . I n t h is w a y a ll c o m b i n a t i o n s o f G P Cm a y b e i m p l e m e n t e d i n a s t a t e - s p a c e f r a m e w o r k .

Reca l l , however, t ha t because the s t a t e s a re no taccess ib l e a s t a t e -obse rve r o r s t a t e - r econs t ruc t ionscheme mus t be employed . La in (1980) uses at r a n s m i t t a n c e m a t r i x a n d h i s a p p r o a c h w a se m p l o y e d b y C l a r k ee t a l . (1985) in thei r LQ sel f -t u n e r. T h r e e p o i n t s a r e o f re l e v an c e .

( i) T h e s t a b il i ty p r o p e r t i e s o f G P C a n d t h ed e t e r m i n i s t i c L Q m e t h o d w i t h a f i n i t e h o r i z o n o fp red ic t ions a re iden t i ca l . Append ix B examines thes t ab i l i ty p ro pe r t i e s fo r spec ia l ca ses o f t he G P Cset t ings .

( i i ) I t i s known tha t numer ica l p rope r t i e s o f LQ

i n t h e s t a t e - s p a c e f o r m u l a t i o n a r e g o o d t h o u g h i t sd r a w b a c k is i n e x e c u t i o n t im e . T h e G P C a p p r o a c hrequ i re s the inv e r s ion o f (GTG + 21) wh ich can bed o n e u s i n g U D U f a c t o r i z a t i o n . Ve c t o r s f a n d wn e e d n e v e r b e f o r m e d s i n c e t h e m u l t i p l i c a t i o n o fGT(w - - f ) can a l so be im plem en ted r ecu r s ive ly a tt h e s a m e t i m e a s t h a t o f t h e G a n d F p a r a m e t e r s .I n t h e L Q c a se th e m e a s u r e m e n t - u p d a t e e q u a t i o n( 11 ) c a n b e d o n e u s i n g U D U a n d (1 2) c a n b e d o n ev i a a m o d i fi e d w e i g h te d G r a m - S c h m i d t a l g o ri th m .F o r l a rg eN U b o t h m e t h o d s r e q u i r edN U i t e r a t ionso f a U D U a l g o r i th m . A s s u m i n g t h e ti m e - u p d a t e so f (12 ) a re equ iv a len t t o ca l cu la t ing Gs and Fs int h e G P C a p p r o a c h , t h e b u r d e n o f c o m p u t i n g t h ef e e d b a c k g a i n s w i l l b e a p p r o x i m a t e l y e q u a l . H o w -eve r, w i th GP C the top row o f (GVG + 21) ~ iss imply mu l t ip l i ed by the vec to r GT(w - f l. In theLQ case typ ica l ly th ree ca l ls to the t r an smi t t a ncem a t r i x r o u t i n e s a r e r e q u i r e d t o o b t a i n s t a t e e s t i -m a t e s w h i c h a r e a v o i d e d b y G P C . F o rN U = 1i nve r s ion i s a s ca l a r ca l cu la t ion whi l s t fo r LQm a t r i x o p e r a t i o n s i n a d d i t i o n t o s t a t e e s t i m a t i o na r e r e q u i r e d . H e n c e t h e " o n e s h o t " a l g o r i t h m o fG P C is c o m p u t a t i o n a l l y l es s d e m a n d i n g .

( ii i) Th e R icca t i equ a t ion imp lem en ta t io n im pl i c-i t ly a s su m e s t h a t t h e r e a r e n o r a t e o r a m p l i t u d el imi ts on the con t ro l s igna l. Wi th G PC i t is poss ib l et o p e r f o r m a c o n s t r a i n e d o p t i m i z a t i o n w h i c hincludes these l imi ts .

4 . S IMULATION EXAMPLES

S i m u l a t i o n s w e r e p e r f o r m e d t o d e m o n s t r a t e t h ee ffec t o f t he des ign f ea tu res o f GP C us ing a sel f-t u n i n g c o n t r o l p a c k a g e FA U S T ( Tu f ts a n d C l a r k e ,1985). Tw o p r inc ipa l t ypes were unde r t aken : i n one

case the p l an t was cons ta n t a nd the exe rc ise wasi n t e n d e d t o s h o w t h e e f fe c t o f ch a n g i n g o n e o f t h edes ign "knobs" on the t r ans i en t r e sponse , wh i l s tthe second se t i nvo lved a t ime-va ry ing p lan t andt h e o b j e c t iv e w a s t o s h o w t h e r o b u s t n e s s o f t h ea d a p t i v e u s e o f G P C . S o m e s i m u l a t i o n s u s e d c o n -t inuou s - t ime mod e l s to i l l u s t r a t e the e ffec t o f s am-p l ing , t hough in a l l ca ses the e s t ima ted mode lused in the se l f- tuned ve r s ion o f G PC was in thee q u i v a l e n t d i s c r e te - t im e f o r m .

T h e p a r a m e t e r s o f t h e A a n d B p o l y n o m i a l s w e r ee s t i m a t e d b y a s t a n d a r d U D U v e r s io n o f R L S

(Bie rman , 1977) us ing the inc remen ta l mode l :

y : ( t ) = y Y(t - 1) + q(1 -A ( q - ~ ) ) Ay : ( t )+ B ( q - ~ ) A u f ( t ) + D ( q1)AvY(t)+ e,(t)

where " f " deno tes s igna l s f i l t e r ed by1 / T ( q - ~ ) ,ifu sed . Fo r s imp l i c ity o f exp os i t ion a f ixed fo rge t t ing -f a c t o r w a s a d o p t e d w h o s e v a l u e w a s n o r m a l l y o n e(no fo rge t t ing ) , un le s s o the rwise s t a t ed . The s igna lv ( t ) i s a measured d i s tu rbance s igna l ( f eed fo rward )w h i c h , a s w it h G M V, c a n r e a d i ly b e a d d e d t o G P C .The pa ramete r e s t ima tes were in i t i a l i zed in thes i m u l a t i o n s w i t h 6 o e q u a l t o o n e a n d t h e r e st e q u a lt o z e r o a n d w i t h t h e c o v a r i a n c e m a t r i x s e t t odiag{10}.

Th e f igu res cons i s t o f two se ts o f g raph s co ve r ingt h e b e h a v i o u r o v e r 40 0 o r 8 0 0 s a m p l es , o n e s h o w i n gthe se t -po in t w( t ) t o g e t h e r w i t h t h e p l a n t o u t p u t}~t) , and the o th e r show ing the co n t ro l s igna lu( t )and p oss ib ly a f eed fo rward s igna lv(t). The sca le sfo r each g raph a re show n o n the axes ; i n a llexamples the con t ro l was l imi t ed to l i e i n ther a n g e [ - 1 0 0 ,1 0 0 ]. L o a d - d i s t u r b a n c e s w e r e o f t w opr inc ipa l t ypes , one ( ca l led" d c u " )cons i s t ed o f st epsin d( t ) fo r t he mode l :

d ( q - l ). ~ (t )= B ( q - l ) u ( t -1) + d( t )

a n d t h e o t h e r( " d c y ' ) of s teps in d( t ) i n the mode l :

A ( q 1 ) y ( t )- - B ( q 1 ) u ( t -1) + A(q ~ )d ( t ) .

T h e d c y d i s t u r b a n c e , e q u i v a l e n t t o a n a d d i t iv e s t e pon the p l an t ou tpu t , i s a pa r t i cu la r ly seve re t e s t o fa n a d a p t i v e a l g o r i t h m .

4 . 1 . T h e e f f e c t o fP ( q - 1)T h e s e s i m u l a t i o n s w e r e d e s i g n e d t o d e m o n s t r a t e

the uses o f P : pena l i z ing ove r sh oo t (p rocess con t ro l )a n d m o d e l - f o l l o w i n g ( h i g h - p e r f o r m a n c e r o le ).

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G e n e r a l i z ed p r e d ic t iv e c o n t r o l - - P a r t I I 1 53

IOO%

0 / *

S e t - p o i n t ( W )

L_A . . . .~ ~ n " v ' - V v - - -

I

0

O u t p u t ( Y )

L , , / ' L

P c h a n g i n g

! !

4 0 0

I 0 0 *A

05

O *A

Control s i g n a l a n dfeed-forward

F e e d - f o r w a r d s i g n o r

I I I I . , I I

4 0 0

FIG. 1. Th e effect of the P transfer-fun ctionon closed-loop performance.

T h e f i r s t p l a n t s i m u l a t e d w a s t h e t h i r d - o r d e rosc i l l a to r :

(1 + sX1 + s2)Xt) =u(t) + d(t) + v(t),

dwhe re s is the d i f f e ren t i a l op e ra to r ~ - , s amp led a t

1 s in tervals . Also , 3A ( q -1), 3B(q- 1)a n d 3D ( q -1)( f e e d -f o r w a r d ) p a r a m e t e r s w e r e e s t im a t e d , w i t h a na s s u m e d d e l a y o f u n i t y. T h e h o r i z o n N 2 w a s c h o s e nas 10 samples andN U = 1 . Th e con t ro l s igna l wasf ixed a t 20 un i t s fo r t he f i r s t 30 samples wh i l e thee s t i m a t o r w a s e n a b l e d . Tw o s e t - p o i n t c h a n g e s o f20 un i t s each w ere app l i ed a t i n t e rva l s o f 30 samples .U n m e a s u r a b l e s t e p l o a d - d i s t u r b a n c e s d c u o f + 1 0uni ts w ere add ed a t t = 150, 180 an d t = 210, 240.M e a s u r a b l e s t e p l o a d - d i s t u r b a n c e s v (t) o f t h e s a m esize wer e ad de d a t t = 90, 120 and t = 280, 310.P(q -~ ) was in i t ia l ly se t to uni ty. As seen in Fig . 1 ,t h e s e t - p o i n t r e s p o n s e a n d t h e l o a d - d i s t u r b a n c er e j e c t io n a l t h o u g h s t ab l e , h a d e x c e s s iv e o v e r s h o o t ,b u t o n c e t h e f e e d - f o r w a r d p a r a m e t e r s w e r e t u n e dt h e f e e d - f o r w a r d d i s t u r b a n c e r e j e c t io n w a s a l m o s te x a c t . N o t e t h a t i n o r d e r t o i n c l u d e f e e d - f o r w a r di n t h e p r e d i c t i o n e q u a t i o n s a m o d e l o f th e f o r m :

A ( q -1)Ay(t) = B ( q - l)Au (t - 1)+ D(q -1 )Av( t- I) + ~(t)

i s assumed where Av(t + j ) = 0 for N2 > j > 0 .F o r t h e s e c o n d h a l f o f th e s i m u l a t i o nP ( q - 1 )

w a s s e t t o ( 1 - 0.8q-1) /0 .2at t = 190. Both thed i s t u r b a n c e r e j e c t io n a n d t h e s e t - p o i n t r e s p o n s e sw e r e t h e r e b y d e t u n e d a n d t h e o v e r s h o o t r e m o v e da l t o g e t h e r. F e e d - f o r w a r d r e j e c t i o n , o n t h e o t h e r

AUT 2 3 /2 - S

h a n d , w a s u n a f f e c t e d b y t h e c h a n g e m a d e i n P.F o r t h e s e c o n d e x a m p l e s h o w n i n F ig . 2 , c o n s i d e r

t h e d o u b l e - o s c i l l a t o r p l a n t w h o s e t r a n s f e r f u n c t i o nis g iven by:

(1 + s2X1 + 1.5s2 )~(t)=u(t).

T h e m o d e l c h o s e n h a dPn(q -1)= ( 1 - 0 . 5 q - l ) 2,a n d Pd(q -1)s e t t o 0.2 77 8( 1 - 0 . 1 q - 1 ) . A c o n s t a n tcon t ro l s igna l o f one un i t was ap p l i ed fo r the f i rs t10 samples a f t e r wh ich i t was se t t o ze ro fo r7 0 s a m p l e s . T h e e s t i m a t o r w a s e n a b l e d f r o m t h eb e g i n i n g o f t h e r u n , w h e r e a s t h e c o n t r o l l e r w a ss w i t c h e d i n t h e l o o p a t t h e 8 0 t h s a m p l e a n d t h ese t -po in t was a l so changed to 20 un i t s a t t ha tin s t an t . N U was se t t o two a t t he s t a r t o f t hes i m u l a t i o n a n d a s s e e n th i s c h o i c e d i d n o t g i v eg o o d m o d e l - f o l l o w i n g c h a r a c t e r is t i c s a l t h o u g h t h ec losed- loop was s t ab le .N U was se t t o fou r a t t hed o w n w a r d s e t - p o i n t c h a n g e t o 2 0 u n it s a n d a t e a c hs u b s e q u e n t d o w n w a r d s e t - p o i n t c h a n g e t o 2 0 u n i t sN U w a s i n c r e m e n t e d b y t w o . C l e a r l y t h e c o n t r o lr ing ing inc reased wi th the inc rease inN U b u t t h ed e s i g n e d m o d e l - f o l l o w i n g c l o s e d - l o o p p e r f o r m a n c er e m a i n e d a l m o s t t h e s a m e f o r N U > 4 ( t h e n u m b e ro f p l a n t p o le s ). I f d e s ir e d , t h e c o n t r o l m o d e s c a nb e d a m p e d u s i n g a n o n z e r o v a l u e o f 2 to g i v e " f in e -t u n i n g " a s i n G M V.

4.2. The effec to f T ( q - 1 )A s e c o n d - o r d e r p l a n t w i t h ti m e - d e l a y w a s s im u -

la t ed where :

(1 + 15s + 50s2)3~t) = e -2S u(t) + lOd(t)+ (1 + 15s + 50s2)dc~t) +(1 + 15s + 5 0s2)~(t)

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154 D .W . CLARKEet al .

6 0 / *

- 1 0 %

S e t - p o i n t ( W )

~ A A ^ . , A A ^ . ~, O v " v , V V ' w . . . . l I

40 0

Con t ro t s i gno rio o /. ,.ill

0 * I

I

J/

_100/o] I 0 4 0 0

FIG. 2. (De tuned) m odel-fo llowing usingP(q J).

i n w h i c h ~ (t) w a s a n u n c o r r e l a t e d r a n d o m s e q u e n c ew i th z e r o m e a n a n d R M S v a l ue o f t w o u n i ts f o r210 < t < 240 . A s t ep -d i s tu rban ced c y of th ree un i t swas added to the ou tpu t a t 150 < t < 180. An o th e rs t e p - d i s t u r b a n c edcu , exc i t ing a ll o f t he m ode s o fthe p l an t , o f + 10 un i t s was add ed a t 120 < t < 150

and 270 < t < 300 . Tw o s t ep changes in se t -po in ta t t h e e n d o f a p e r i o d o f r e g u l a t i o n w e r e e m p l o y e dto see the e ffec t o f d i s tu rb ance s on the se rvop e r f o r m a n c e o f t h e c o n t r o l le r , a n d t h r e e A a n d f o u rB p a r a m e t e r s w e r e e s t i m a t e d w i t h a n a s s u m e d d e l a yo f u n i t y.

T h e i n i t i a l s e t - p o i n t r e s p o n s e a n d s u b s e q u e n tl o a d - d i s t u r b a n c e r e j e c t i o n w e r e g o o d , a s s e e n i nF ig . 3 . Th e r e j ec t ion o fdey, o n t h e o t h e r h a n d , w a sve ry ac t ive in i t i a l ly and incons i s t en t i n the secondc h a n g e o f lo a d . T h i s w a s d u e t o d y n a m i c p a r a m e t e rc h a n g e s c a u s e d b y n o t e s t i m a t i n g p a r a m e t e r s a s s o -c i a t e d w i t h t h e n o i se s t r u c t u r e . S u b s e q u e n t b e h a v i -o u r b a s e d o n t h e p o o r m o d e l w a s n o t v e r y g o o das the con t ro l was f a r t oo ac t ive .

I n t h e s e c o n d s i m u l a t i o n s h o w n i n F ig . 4,T(q - 1)w a s c h o s e n t o b e (1 - 0 . 8 q - 1 ) . N o t e t h a t a l t h o u g hT i m p r o v e d t h e d i s t u r b a n c e r e j e c ti o n o f t h e c l o se d -loo p i t had no e ff ec t on the se t -po in t r e sponse . Ina d d i t i o n , s i n c e t h e p a r a m e t e r e s t i m a t o r w a s b e t t e rc o n d i t i o n e d i n t h e s e c o n d c a s e , t h e f i n a l s e t - p o i n tr e sponses were a lmos t i den t i ca l t o the in i t i a l ones .

4.3. The e ffec t o fN 2 and the sampl ing pe r iodO n e o f th e m a j o r c r it ic i sm s o f d i g i ta l c o n t r o l l e r s

i s t h a t m o s t d e s i g n s o n l y w o r k w e l l i f t h e s a m p l i n gp e r i o d i s c h o s e n c a r e f u ll y ( a p p r o x i m a t e l y 1 /4 t o1 /10 o f the se t t l i ng - t ime o f the p l an t ) . A s low p lan twi th th ree r ea l po le s was chosen to inves t iga t e

w h e t h e r G P C s u ff e rs f r o m t h i s p r o b l e m :

(1 + 10s)3j~t) =u(t).

Five A and f ive B pa ram ete r s were e s t ima ted ; N1was chosen to be 1 andN 2 w a s in i t ia l ly se t to 10

b u t d o u b l e d a t e v e r y u p w a r d - g o i n g s t ep inw(t) to50 . A sam pl ing t ime o f 1 s was chosen ; no te tha tthe se t t li ng t ime o f the p l an t i s abou t 160s . F igu re5 shows tha t t he in i t i a l con t ro l was s t ab le bu t hada s m a l l r i n g i n g m o d e a n d a t t a i n e d t h e i m p o s e ds a t u r a t i o n l i m i ts . W h e n a n o u t p u t h o r i z o n o f 2 0s a m p l e s w a s c h o s e n t h i s m o d e w a s r e m o v e d . A tN 2 = 40 ( the r i s e - time o f the p l an t ) t he co n t ro l wasm u c h s m o o t h e r. I n c r e as i ng t h e h o r i z o n o f o u t p u tp red ic t ion to the se t t li ng t ime o f the p l an t(N2 = 160) caused the speed o f the c losed - loopu n d e r t h i s c o n d i t i o n t o b e a l m o s t t h e s a m e a s t h a to f t h e o p e n - l o o p , v e r i f y i n g t h e " m e a n - l e v e l " t h e o r yof Appe nd ix B . In a l l cases , t hen , t he r e sp onsesw e r e s m o o t h d e s p i t e t h e r a p i d s a m p l i n g .

4.4. Over-pa ramete r i za t ionO n e o f t h e p r o b le m s w i th m a n y a d a p t iv e c o n t r o l

s c h e m e s is t h a t a n e x a c t k n o w l e d g e o f t h e m o d e lo rd e r i s r equ i red ; o f pa r t i cu la r i n t e re s t i s t he ab i l i t yt o o v e r - p a r a m e t e r i z e t h e p l a n t p a r a m e t e r e s t im a t o ri n o r d e r t o m o d e l t h e p l a n t w e ll i n c a se o f d y n a m i cchanges .

A f i r s t -o rde r p l an t was s imu la t ed in d i sc re t e time :

( 1 - 0 . 9 q 1 ) 3 4 t ) =u ( t - 1 ) .

E s t i m a t i o n w a s d i s a b l e d a n d t h e p a r a m e t e r s w e r ef ixed a pr ior i t o the r equ i red va lues g iven be low.

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G e n e r a l i z e d p r e d i c ti v e c o n t r o l - - P a r t I I 1 5 5

i O 0 * A

!50/

0 O / o I0

S e t - p o i n t ( W I

O u t p u t ( Y )

i I i I

4 0 O

i o o ' A

_ , o oi1

Control s ignal

i , -- t

I P i

! I I I I I

4 0 0

FIG. 3. The con trol of a p lant with add itive disturbances (without the T polynom ial).

I 0 0 * /.

5 0 "/

0 / o

S e t - p o i n t ( W )

\ /k_W - ' v =

f , -!

O u t p u t ( Y )

I i I I I

4 0 0

Control s ignal

0

FIG, 4. The con trol of a p lant with additive disturbances (with the T polynom ial).

4 0 0

I n i ti a ll y a c o m m o n f a c t o r o f (1 + 2 q - 1 ) w a s s e tb e t w e e n t h e e s ti m a t e d A a n d B p o l y n o m i a l s , gi v in g :

. ~ ( q - 1 ) = 1 + 1 . 1q - 1 - - 1 . 8 q - 2

/~(q - 1) = 1 + 2q - 1.

N U w a s s e t t o o n e a t t h e s e t- p o i n t c h a n g e f r o m 0t o 2 0 , t o t w o f o r t h e c h a n g e 2 0 - 4 0 , f o u r fo r th ec h a n g e 4 0 - 2 0 a n d f in a ll yN U = 1 0 f o r t h e c h a n g e2 0 - 0 . T h e c o m m o n r o o t w a s t h en m o v e d t o - 0 . 5 ,0 . 5 a n d 2 i n s u c c e s s i o n a n d t h e t r a n s i e n t t e s t

r e p e a t e d . F i g u r e 6 s h o w s t h a t i n a l l c a s e s t h e c o n t r o lp e r f o r m a n c e o f G P C w a s u n a ff e ct e d b y th e c o m m o nf a c t o r.

4 .5 . Under-parameterizationM o s t i n d u s t r ia l p r o c e s s e s a r e n o n l i n e a r a n d

t h e r e fo r e m a y o n l y b e a p p r o x i m a t e d b y h i g h - o r d e rl in e a r m o d e l s . A g o o d c h o i c e o f s a m p l e - r a te , o n t h eo t h e r h a n d , e n a b l e s t h e d e s i g n e r t o u s e l o w - o r d e rm o d e l s f o r c o n t r o l : s l o w s a m p l i n g m a s k s t h e h i g h -o r d e r f a st d y n a m i c s .

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1 56 D . W . C L AR K Eet al ,

I00 ,~

50 */

0 */*

Set -po in t (W) O u t p u t ( Y }

| I I I I I I ,I |8 0 0

I0 0

- I O 0

Control signal

I[I ,II. , .

~ - ] r F L _ _ / - - - I _

I I I

8 0 0

FIG. 5. The effect of fast sampling and the predi ction hori zon.

5 0 *A

0 */, J L p u lJ

l I ,I

I

40 C

2 5 */*

0 */

- 2 5 * / .

Control signal

J,. ;i i 11 I I - I I

I t I I I !

0

I

4 0 0

FIG. 6. The effect of commo n factors in the esti mated parameters.

C o n s i d e r t h e f o u r t h - o r d e r p l a n t :

(1 + s)2(1 + 3S)2)~t) =u(t) + d(t) .

A s e c o n d - o r d e r m o d e l w a s a s s u m e d a n d t h e p l a n tw a s s a m p l e d a t 1 s i n t e rv a l s ; n o t e t h a t t h e s a m p l i n gp r o c e s s w a s n o t m a s k i n g t h e s l i g h t l y f a s t e r p o l e s .I n t h i s c a s e t w o A a n d t h r e e B p a r a m e t e r s w e r e

e s t i m a t e d a n d t h e a s s u m e d t i m e - d e l a y w a s u n i ty.N2 w as s e t t o 10 an dN U w a s s e t t o o n e . T h e s e t -p o i n t s e q u e n c e w a s a s q u a r e w a v e w i t h a p e r i o d o f4 0 s a m p l e s . L o a d - d i s t u r b a n c e s o f 1 0 a n d 2 0 u n i t sw e r e a d d e d a t t h e m a r k e d t i m e s a n d a s s h o w n i n

F ig . 7 , o f f s e t -f r ee con t ro l was ach i eve d . The ove ra l lp e r f o r m a n c e w a s g o o d d e s p i t e t h e w r o n g p a r a m e -t e r i z a t i o n ; t h e o v e r s h o o t c o u l d h a v e b e e n r e d u c e du s i n g P ( q - 1 ) a s s h o w n i n t h e p r e v i o u s s e c t i o n s .

4.6. U n k n o w n o r v a r i a b le t i m e - d e l a yT h e G M V d e s i g n i s s e n s it i v e t o c h o i c e o f d e a d -

t im e ; G P C is h o w e v e r r o b u s t p r o v i d e d t h a t6 B is

c h o s e n t o a b s o r b a n y c h a n g e i n t h e t i m e - d e l a y.C o n s i d e r t h e p l a n t :

( 1 - 1 . 1 q - 1 ) )~ t ) = - ( 0 . 1 + 0 . 2 q -l ) u ( t - k )

wh ere k = 1 , 2 , 3 , 4 , 5 a t d i fferent s tage s in the t r ia l .

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Genera l i zed p red ic tive co n t r o l - -P ar t I I 157

I O 0 %

5OO/

0 "/o

S e t - p o i n t ( W ) O u t p u t { Y )

O 4o0

Control signal

0 % " I L . _ . . _ _

I 0 0 / I I I I I I

FIG. 7. Th e effect of under-p aram eterization.

The va lue o f k was changed a t the dow nw ard-going s teps in set-point increasing ini t ia l ly f romone to f ive and then decreasing from f ive back toone aga in . The adap t ive con t ro l l e r es t imated twoA and s ix B parameters and a scalar forget t ing-fac to r o f 0 .9 was em ployed to enab le t rack ing o fvar ia t ions in the dead-t ime.N 2 w a s set to 10 an dN U t o o n e . T h e p e r f o r m a n c e o f G P C s h o w n i nFig. 8 is good; note that the plant was bothn o n m i n i m u m - p h a s e a n d o p e n - l o o p u n st a b le w i thvariable dead-t ime, yet s table control was achievedwith the defaul t set t ings o f this a lgori thm.

5 . C O N C L U S I O N ST h i s p a p e r h a s s h o w n t h a t G P C c a n b e e q u i p p e d

wi th the des ign featu res o f the wel l -known G M Vappro ach and g iven a wide range o f poss ib lecontrol object ives , which can be interpreted by i tsre la t ionsh ip wi th LQ a lgor i thms based on s ta te -space mode ls . These resu l t s a re summar ized inTable 1. I t migh t seem tha t the re a re ma ny poss ib lecho ices o f des ign param ete rs in GP C, bu t thetab le shows tha t many combina t ions l ead to wel l -understood control laws. In pract ice not a l l th isf l ex ib i l i ty would be requ i red and many processescan be effect ively contro l led using defaul t set tings.Closer inspect ion o f Tab le 1 show s that a " large"va lue o f N 2 i s genera l ly recommended and tha tN U and P can then be chosen accord ing to thecon t ro l ph i losophy appropr ia te fo r the p lan t andthe comput ing p owe r ava ilab le . Hence the "knobs"can be used to t a i lo r an adap t ive con t ro l l e r toprecise specif icat ions, which is of great value in thehigh-performance role . In par t icular, the methodno longer needs to em ploy con t ro l we igh t ing when

appl ied to a varying dead-t ime plant which is arequ i rement o f GM V designs .

The s imula t ions show tha t G PC can cope withthe control of com plex processes und er real is t icconditions. As it is relatively insensitive to basicassumptions (model order, e tc . ) about the process ,GPC can be eas i ly app l ied in p rac t i ce wi thou t aprolonged design phase. These features ensure thatthe metho d prov ides an e ffec tive appro ach to theadap t ive con trol of an industr ia l plant .

R E F E R E N C E S/~strrm, K. J. and B. Wittenmark, (1984).Computer Controlled

Sys t ems- -Theo ry and Des ign .Prent ice-Hal l , EnglewoodCliffs, NJ.

Bierm an, G. J. (1977).F actorization Meth ods for Discrete Sy stemEstimation. Academic Press, New York.

Clarke, D . W . (1982). The applica tion of self-tuning control.

Trans. Inst. M.C., 5,59-69.Clarke, D. W. an d P. J. Gaw throp, (1975). Self-tuningcontroller.Proc. IEE, 122, 929-934.

Clarke, D. W . and P. J. Gaw throp, (1979). Self-tuning control.Proc. IEE, 126, 633-640.

Clarke, D. W., P. P. K anjilal and C. M ohtad i, (1985). Ageneralised L QG appro ach to self-tuning control.Int. J.Control, 41, 1509-1544.

Gaw throp, P. J. (1977). Some inte rpreta tions of the self-tuningcontroller.Proc. lEE, 124, 889-894.

Kwa kernaak, H. and R. Sivan, (1972).Linear Optimal ControlSystems. Wiley, New York.

Lam, K. P. (1980). Implicit an d explicit self-tuning controllers.D. Phil Thesis, Oxford University.

Peterka, V. (1984). Predictor-ba sed self-tuning control.Automa-tica, 20, 39-50.

Tufts, P. S. 0984). Self-tuning control: algorith ms an d app lica-tions. D. Phil. Thesis, Oxford University.

Tufts, P. S. and D. W . Clarke, (1985). FA US T: a softwarepackage for self-tuning control.IEE Conf. " 'Control85",Cambridge.

Wellstead, P. E., D. Prager, a nd P. Zank er, (1979). Poleassignment self-tuning regulator.Proc. IEE, 126, 781-787.

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1 5 8 D . W . C L A R K Ee t a l .

ioo*/o 7 ~ -

5 0 *A

O */*0 800

too

-io

Control signoL

' I t I I I ,

8 0 0

F IG . 8 . T h e c o n t r o l o f a v a r i a b l e d e a d - t i m e p l a n t .

TABLE 1. SPECIAL CASES OF G P C SETTlNGS

N U N ~ N 2 P 2 P l a n t C o n t r o l l e r

1 1 10 1 0 s ,d "D efau l t "1 1 ---,o c 1 0 s , d " M e a n - l e v e l "N 2 1 / > k P 0 m p E x a c t m o d e l -

f o l l o w i n g P = I / M

< N 2 1 ~ >k P 0 ,2 " D e t u n e d " m o d e l -f o l l o w i n gN 2 1 - , o e 1 > 0 s , d L Q i n f i n i t e - s t a g eN 2 - - n + 1 1 - , o ~ 1 0 s , d C h e a p L Qn n >~2n - 1 1 0 o ,c S ta t e -de ad- bea tn n / > 2 n - 1 P 0 o ,c P o l e - a s s i g n m e n t

2 s ,d " D e t u n e d " p o l e -a s s i g n m e n t

s : s t a b i l i z a b l e; d : d e t e c t a b l e ; o : o b s e r v a b l e ; c : c o n t r o l l a b l e ; m p :m i n i m u m - p h a s e .

A P P E N D I X A . R E C U R S I O N O F T H E P O L Y N O M I A LG ' ( q - l )

C o n s i d e r t h e s u c c e s s iv e D i o p h a n t i n e i d e n ti t ie s :

G j = G j T + q - J F j (A.1)

G j + l = G ' j + I T +q - J - l F j r ( A .2 )

N o t e t h a t G j = E j B ,w h e r e :

E j( q l ) = e o + e l q l + . . . + e i _ l q - J + l .

S u b t r a c t i n g e q u a t i o n s ( A . I ) f r o m ( A . 2 ) w e o b t a i n :

q - J e jB = q - J g j + l T +q - J ( q - l F j + l - F j) . ( A .3 )

H e n c e t h e u p d a t e e q u a t i o n s b e c o m e :

gj t = 1 / to (e jbo+ ?O~o) (A.4 )

a n d , f o r i = 1 to m a x ( f B , f T J :

?o + 1~i = Yo i + e~bi +g j+ lt l ( A . 5 )

w h e r e ? u ~ d e n o t e s t h e i t h c o e f f ic i e n t o f th e p o l y n o m i a l F ~a s s o c i a t e d w i t h q - 1 N o t e t h a t t h e c o e f f i c ie n t s o f B o r T w i t hi n d i c e s g r e a t e r t h a n t h e i r r e s p e c t i v e d e g r e e s a r e z e r o .

A P P E N D I X B . S O M E S T A B I LI T Y R E S U LT S F O RL I M I T I N G C A S E S O F G P C

C o n s i d e r t h e p l a n t g i v e n i n s h i ft - o p e r a t o r f o r m b y :

A ( q - l )Ay(t) = B ( q -l )Au(t -- 1). (B.I)

A s t a t e - s p a c e m o d e l o f t h i s p l a n t c a n b e w r i t t e n a s :

x(t + 1) = Ax (t) + hA u(t) (B.2)

y(t) = erx( t) (B.3)

w h e r e A i s t h e s t a t e t r a n s i t i o n m a t r i x w h i c h w e t a k e t o b e i no b s e r v a b l e c a n o n i c a l f o r m , b i s t h e v e c t o r o f B p a r a m e t e r s a n dA A p o l y n o m i a l s ( C l a rk ee t a l . , 1 9 85 ). S i n c e d i s t u r b a n c e s d o n o ta f fe c t th e s t a b i l i t y p r o p e r t i e s o f t h e c o n t r o l l e r , o n l y d e t e r m i n i s t i ce l e m e n t s a r e c o n s i d e r e d i n t h e f o l l o w i n g s e c t i o n . D e f i n i n g t h ea u g m e n t e d p o l y n o m i a l ,~ t o b e :

~(q t) = A A = 1 + t] lq I + ~ 2q -2 q_ . .. .4_ t i ,q -"

t h e n t h e m a t r i c e s a n d v e c t o r s i n t h e s t a t e -s p a c e f o r m a r e:

A = - - a l 1 0

- a 2 0 1

- - t ~ n

b = [ b o , b l . . . . . b . 1 ] T

.. . 0

... 0

0

w h e r e a i = 0 fo r i > d e g ( ~ ) a n db ~ 0 for i > deg (B).

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T h e c o s t - f u n c t i o n i n t h e s t a t e - s p a c e f o r m u l a t i o n c a n b ew r i t t e n a s :

N2J = 3-'~ [x (t + i - 1)TQ x(t + i - 1)

+ 2(t + i -- l )Au (t + i - 1) 2 (B.4)

w h e r e Q = [ 1 , 0 , 0 . . . . . o ] T [ 1 ,O , O . . . . 0 ] . T h e s e t - p o i n t h a s b e e no m i t t e d f o r s i m p l ic i t y b e c a u s e t h e s t a b i l i t y p r o p e r t i e s a r e i n d e -p e n d e n t o f i n p u t s .

T h e s o l u t i o n i s o b t a i n e d b y i t e r a ti n g t h e e q u a t i o n s b e l o w.

M e a s u r e m e n t u p d a t e :P * ( i ) = P ( i + 1 ) -P(i + l)b(2(i)

+ hrp ( i + 1 )h) -1hT p( i + 1 ). (B.5)

T i m e u p d a t e :P( i ) = Q + ArP *( i )A (B.6)

Au( t ) = - ( 2 (0 + bTp( t )b ) - thTP( t )Ax( t ) . (B .7)

P i s c a l le d t h e " c o v a r i a n c e m a t r i x " ( f r o m d u a l i t y w i t h t h ee s t i m a t i o n e q u a t i o n s ) . N 2 i t e r a ti o n s a r e p e r f o r m e d b a c k w a r d ss t a r t i n g f r o m Q , t h e t e r m i n a l c o v a r i a n c e m a t r i x . N o t e t h a t s i n c eb o t h t h e " o n e - s h o t " ( G P C ) m e t h o d o f co s t m i n i m i z a ti o n a n dt h e d y n a m i c p r o g r a m m i n g a p p r o a c h o f st a te - s p ac e ( L Q ) s u c ce e di n m i n i m i z i n g t h e s a m e c o s t u n d e r c e r t a i n c o n d i t i o n s (i.e . li n e a rp l a n t a n d n o c o n s t r a i n t s o n t h e c o n t r o l s i g n a l) , t h e r e s u l t in gc o n t r o l l a w m u s t b e t h e s a m e b e c a u s e t h e r e is o n l y o n e m i n i m u ma n d s o t h e i r s t a b i l i t y c h a r a c t e r i s t i c s m u s t b e i d e n t i c a l .

N o t e t h a t f i x i n g t h e p r o j e c t e d c o n t r o l s i g n a l in t h e f u t u r e i se q u i v a l e n t t o e m p l o y i n g a l ar g e p e n a l t y o n t h e a p p r o p r i a t ei n c r e m e n t ( i. e. 2 ( 0- -- , o o ). T h i s m e a n s t h a t t h e m e a s u r e m e n tu p d a t e n e e d n o t b e p e r f o r m e d f o r t h e p a r t i c u l a r v a l u e o f i. T h r e es p e c ia l c a s e s o f G P C a r e c o n s i d e r e d b e l o w.

T h e o r e m 1 . T h e c l o s e d - l o o p s y s t e m i si s s t a b i l i z a b l e a n d d e t e c t a b l e a n d i f :

(i) N 2 --* or3,N U = N 2 a n d 2 ( 0

s t a b l e i f t h e s y s t e m ( A , b , )

> 0 o r

(ii) N 2 --* ~ , N U - -* o o w h e r e N U ~< N 2 - n + 1 an d2(0 = O.

P r o o f . P a r t ( i) i s e a s i l y p r o v e n f r o m t h e s t a b i l i t y c o n d i t i o n s o ft h e s t a t e - s p a c e L Q c o n t r o ll e r . T h e c o s t o f (B . 4) t e n d s t o t h ei n f in i t e s ta g e c o s t a n d f o r c o n v e r g e n c e t o t h e a l g e b r a i c R i c c a t ie q u a t i o n ( A R E ) s o l u t i o n . Q c a n b e p o s i t i v e s e m i - d e f i n i t e i f 2 isp o s i t i v e d e f i n it e f o r al l t e r m i n a l c o v a r i a n c e s ( K w a k e r n a a k a n dS i v a n , 1 9 7 2) . F o r p a r t O i L n o t e t h a t f o r t h e f i r st n - 1 i t e r a t i o n so f t h e R i c c a t i e q u a t i o n o n l y t h e t i m e - u p d a t e s ( B . 6) a re n e c e s s a r y( 2( 0 - ~ ~ ) . N o t e , m o r e o v e r , t h a t t h e t e r m i n a l c o v a r i a n c e Q i so f r a n k o n e . E a c h o f t h e t i m e - u p d a t e s i n c r e a se s th e r a n k o ft h e m a t r i x P b y o n e a n d a f t e r n - 1 i t e r a ti o n s t h e m a t r i xP(N2 - n + 1) i s o f fu l l r ank . I t i s seen tha t P(N2 - n + 1) i sZ A ~ r e e r A ~ -- t h e o b s e rv a b i li t y G r a m m i a n w h i c h i s g u a r a n t e e dp o s i t iv e d e f i ni t e f or t h e s t r u c t u r e a s s u m e d . T h e n a sN U ~ o ot h e i t e r a t i o n s ( B . 5, B . 6 ) c o n v e rg e t o t h e A R E s o l u t i o n f o r a llv a l u e s 2 ( 0 = 2 / > 0 .

R e m a r k 1 . Pa r t ( i i) i s a spec ia l case o f the s tab i l i z ing con t ro l le rof Pe t e rk a (1984) , us in g 2(0 = tos wh ere 0 < ~os < oo .

R e m a r k 2 . F o r 2 --* 0 t h e G P C l a w s a b o v e a r e e q u i v a l e n t t o t h ec o n s t r a i n e d m i n i m u m - v a r i a n c e r e g u l a t o r d e r i v e d b y sp e c t r a lf a c t o r i za t i o n f o r n o i s e m o d e l s o f r e g r e s s io n t y p e.

T h e o r e m 2 . F o r o p e n - l o o p s t a b l e p r o c e s s es t h e c l o s e d -l o o p i ss t a b l e a n d t h e c o n t r o l t e n d s t o a m e a n - l e v e l l a w f o rN U = 1an d 2( i) = 0 as N~ - , oo .

P r o o f . F o r s i m p l i c it y a s s u m e t h a t t h e m a t r i x A h a s d i s t i n c te i g e n v a l u e s a n d c a n t h e r e fo r e b e w r i t t e n a s :

A = ~ .2 iq i r ~ ( B . 8 )

w h e r e 2 a r e t h e e ig e n v a l u e s a n dI A i l< 1 for al l i # ! and 2~ = 1a n d q l a n d r i a r e r i g h t a n d l e f t e i g e n v e c t o r s a s s o c i a t e d w i t h t h ep a r t i c u l a r e i g e n v a l u e o f A .

T h e r i g h t a n d l ef t e i g e n v e c t o r s a s s o c i a te d w i t h t h e e i g e n v a l u ea t 1 a r e g i v e n b y :

q l r = [ l , 1 + f i t , I + a I + ' ~ 2 , . . . ]

r~ = I-l, 1, 1, 1 .... 1].

H e n c e , t h e m a t r i x

A = - - , q t r ~ a sm - ~ o o .

N o t e , h o w e v e r , t h a t t h e c h o i c eN U = 1 i m p l i e s t h a t t h e r e w i l lb e N 2 - 1 t i m e - u p d a t e s f o l l o w e d b y a s in g l e f u ll u p d a t e a t t h el a s t i t e r a t i o n , g i v i n g :

P ( t + 1 ) = Q + A T Q A + A 2 T Q A 2 + A a T Q A 3 + . . . .

In the l imi t as N2 ~ ~ the ma t r ix P ( t + 1 ) wi l l sa t i s fy :

P( t + l ) / m ~ q l r r Q q , r X a s m - ~ o,

i.e. P(t + 1 ) / m - - *r l q t r [ l , 0 . . . . . 0 ] T E l , 0 , . . . , 0 ] q l r tT

o r:

P( t + 1 ) / m - - *r l r ~ . ( B . 9 )

T h i s i m p l i e s t h a t:

hXP(t + 1)h/m ~ (Zbl) 2

a n d f i na l ly s u b s t i t u t i n g t h e a s y m p t o t i c v a l u e o f P ( t + 1 ) f r o m(B.9) int o (B.7) giv es for 2 = 0:

Au( t ) = - (Eb b) -1 11 , 1 , 1 . . . . 1 Ix( t ). (B .10)

R e c a ll t h e f o r m u l a t i o n i s t h a t o f t h e o b s e r v a b l e c a n o n i c a l f o r ms o t h a t i n a d e t e r m i n i s t i c e n v i r o n m e n t t h e s t a t e s m a y b e w r i t te nas fo l lows :

x . ( t ) = - ~ . y ( t - 1) + b . _ l A u ( t - 1)

x. _ 1 = - t in _ ly(t - 1) - t i .y{t - 2)

+ b ._ tAu( t - 2 ) +b . _ 2 A u ( t - 1)

x d t ) = - 8 l y ( t - 1) - ... + b o A u ( t - 1) + ... (B . I1)

a n d

~ t ) = x d t ) .

C o m b i n i n g ( B .11 ) w i t h t h e e x p r e s s i o n f o r t h e c o n t r o l s i g n a l( B .1 0 ) y i e l d s t h e p a r a m e t r i c r e p r e s e n t a t i o n o f t h e c o n t r o l le r :

( Y, b 3 A u ( t ) = - y ( t ) -(1 + f i t ) y ( t - 1) ... ( E b , - b o )

Au (t - 1) - (Xbl - bo - b t)A u(t - 2) - . . . (B.12)

o r:

G ( q - l )Au(t) = - A ( q - 1 ) / B ( I ) y ( t )

w h e r e G i s th e s o l u t i o n o f t h e D i o p h a n t i n e i d e n t i ty

G ( q - 1 ) A ( q - ' ) A + q - l A ( q - l ) B ( q - t ) / B ( l ) = A ( q - l ) . (B.13)

T h i s c a n b e v e r i fi e d s i m p l y b y c o m p a r i n g t h e c o e f fi c ie n t s o f Gw i t h t h o s e d e r i v e d i n ( B .1 2 ). T h e s e a r e t h e e q u a t i o n s o f p o l e -a s s i g n m e n t p l a c i n g t h e c l o s e d - l o o p p o l e s a t t h e o p e n - l o o pp o s i t i o n s . H e n c e f o r o p e n - l o o p s t a b l e p r o c e s s e s t h e c l o s e d - l o o pi s s t a b l e a s N 2 ~ o o , a n d b e c a u s e t h e c l o s e d - l o o p p o l e s a r ep l a c e d i n e x a c tl y t h e s a m e l o c a t i o n s a s t h o s e o f t h e o p e n - l o o pp o l e s , t h i s c o n t r o l l e r i s a m e a n - l e v e l c o n t r o l l e r.

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1 6 0 D . W . C L A R K Ee t a l .

R e m a r k 1 . A m e a n - l e v e l c o n t r o l l e r p r o v i d e s a s t e p i n c o n t r o lf o l l o w i n g a s t e p i n t h e s e t - p o i n t w h i c h w i ll d r i v e t h e p l a n to u t p u t e x a c t l y t o t h e s e t - p o i n t a n d h e n c e p r o v i d e t h e s a m ec l o s e d - l o o p d y n a m i c s a s o f t h e o p e n - l o o p . N o t e t h a t s t e p s i nl o a d - d i s t u r b a n c e a r e , h o w e v e r, r e j e c t e d s i n c e t h e c o n t r o l l e ri n c l u d e s a n i n t e g r a t o r.

R e m a r k 2 . T h e s p e c tr a l d e c o m p o s i t i o n o f th e m a t r i x A s h o w st h a t a v a l u e o fN 2 e q u a l t o t h e s e t t l i n g t i m e o f t h e p l a n t i se q u i v a l e n t f o r c o n t r o l p u r p o s e s t o N 2 - , o o.

R e m a r k 3 . F o r c a s e s w h e r e t h e r e a r e m u l t i p l e e i g e n v a l u e s t h es p e c t r a l f a c t o r i z a t i o n (B . 8) is n o t v a l i d a n d t h e J o r d a n c a n o n i c a lf o r m m u s t b e e m p l o y e d . H o w e v e r , b e c a u s e a ll p l a n t e i g e n v a l u e sa r e a s s u m e d t o l ie i n s i d e t h e u n i t c i r c l e t h e v a l u e o f A " a s m - +o o r e m a i n s t h e s a m e a s f o r t h e d i s t i n c t e i g e n v a l u e c a s e a n d t h er e s t o f th e a rg u m e n t f o l l o w s .

T h e o r e m 3 . T h e c l o s e d - l o o p s y s t e m i s e q u i v a l e n t t o a s t a b l es t a t e - d e a d - b e a t c o n t r o l l e r i f

( 1) t h e s y s t e m ( A , h , e ) i s o b s e r v a b l e a n d c o n t r o l l a b l e a n d

(2) N 1 = n, N 2 >i 2n - 1,N U = n a n d 2 = 0 , w h e r e n i s t h en u m b e r o f s ta t e s o f t h e p l a n t .

Proof . I n i t ia l l y t h e c a s e o f N 2 = 2 n - I i s c o n s i d e r e d . T h e c o s tm i n i m i z e d b y t h e- c o n t r o l l e r p r o p o s e d a b o v e i s

N2 +t

d = j~=t x( i ) rQ( i)x( i ) (B.14)

w h e re Q ( i ) = 0 f or i < t + N 1 a n d Q ( i ) = e c T f or i > / t + N vT h e K a l m a n c o n t r o l g a i n s c a n t h e r e f o re b e c a l c u la t e d u s i n g t h ei te ra t ion s (B.5 , B .6, B .7). Fo r t he f i r s t n - 1 i t e ra t ion s of theR i c c a t i e q u a t i o n o n l y (B . 6) is u s e d w i t h Q = c C , h e n c e o b t a i n i n gt h e o b s e r v a b i li t y G r a m m i a n i n T h e o r e m ( 1 ). F o r t h e n e x t ni t e r a t i o n s ( B .5 , B . 6) a r e u s e d e m p l o y i n g Q = 0 . N o t e t h a t t h e s ei t e r a t io n s y i e ld a u n i q u e K a l m a n c o n t r o l g a i n w i t hP( t + N2) = c c T .

F r o m t h e p r e d i c t i v e p o i n t o f v ie w, c a l c u l a t io n o f t h e c o n t r o l

s i g n a l Au( t ) i s t a n t a m o u n t t o s o l v i n g th e s e t o f s i m u l t a n e o u se q u a t i o n s :

g, g , - 1 . . . g 1

g 2 . 2 g 2 . - 1 . .- g . 1

A u ( t + l ) = w - J ( t + n + l ) .

Au( t + n - I ) [ w - J ~ t + 2 n - I ) ]

Clear ly i f the mat r ix G i s o f fu l l rank then Au( t) i s un ique .

B y a s s u m p t i o n 1 t h e c o n t r o l l e r m i n i m i z i n g t h e c o s t a b o v e i su n i q u e b e c a u s e G i s f u l l r a n k .

A s t h e s y s t e m i s a s s u m e d c o n t r o l l a b l e t h e r e e x i s t s a u n i q u ef e e d b a c k ga i n k s u c h t h a t ( A - b k r ) " x ( t ) = 0 , i n d e p e n d e n t o fx (t ). S u c h a c o n t r o l l e r i s t h e s t a t e - d e a d - b e a t c o n t r o l l e r a n d i ss t a b le , i n d e p e n d e n t o f t h e p o l e - z e r o l o c a t i o n s o f t h e s y s t e m .N o t e t h a t t h e c o s t i n c u r r e d b y c h o o s i n g k a s t h e d e a d -b e a t c o n t r o l l e r i s e x a c t l y z e r o ( i n t h e d e t e r m i n i s t i c c a s e ) - - t h em i n i m u m a c h i e v a b l e c os t . R e c al l h o w e v e r , th a t t h e c o n t r o l l e rm i n i m i z i n g t h e c o s t w a s s h o w n t o b e u n i q u e . T h e r e f o r e , t h ec o n t r o l l e r d e r i ve d f r o m i t e r a ti o n s o f t h e R i c c a t i e q u a t i o n o rs o l v in g t h e s e t o f s i m u l t a n e o u s e q u a t i o n s i s th e d e a d - b e a tc o n t r o l l e r a n d h e n c e s t ab l e . N o t e t h a t s i n c e t h e c o s t i n c u r r e d i sz e r o , in c r e a s i n g t h e h o r i z o n N 2 d o e s n o t a f fe c t t h e m i n i m u m

n o r t h e s o l u t i o n o f t h e o p t i m i z a t i o n p r o b l e m .

R e m a r k 1 . N o t e t h a t t h e s t a t e - d e a d - b e a t c o n t r o l i s e q u i v a l e n tt o p l a c i n g a l l o f t h e c l o s e d - l o o p p o l e s a t t h e o r i g i n .

R e m a r k 2 . A s s e e n i n t h e a s s u m p t i o n ( 1 ) , t h e c o n d i t i o n o fm i n i m a l r e a l i z a t i o n i s n e c e s s a r y f o r t h e c o n t r o l c a l c u l a t i o n a n dt h e r e f o re t h e c h o i c e o f h o r i z o n s a b o v e m u s t i n p r a c t ic e b e u s e dw i t h c a u t i o n . T h i s i s t h e s a m e c o n d i t i o n r e q u i r e d w h e n s o l v i n gt h e D i o p h a n t i n e i d e n t i ty f o r a p o l e - a s s i g n m e n t c o n t ro l l e r . N o t et h a t 2 m a y b e u s e d t o i m p r o v e t h e c o n d i t i o n o f t h e G m a t r i x .

R e m a r k 3 . I f i n s t e a d o f (B .1 ) t h e a u g m e n t e d p l a n t :

A ( q - ) A P ( q - 1)y(t) = B(q ' ) A P ( q - l ) u (t - I )

i s c o n s i d e r e d a n d d e a d - b e a t c o n t r o l i s p e r f o r m e d o n i p( t) =P~(t)i n s t e a d o f ~ (t ), t h e c l o s e d - l o o p p o l e s w i ll b e l o c a t e d a t t h e z e r o so f P(q- 1 ) .T h i s i s b e c a u s e t h e c l o s e d - l o o p p o l e s o f t h e a u g m e n t e dp l a n t ( w - -, ~ ( t) ) a r e a l l a t t h e o r i g i n a n d t h e r e f o r e t h e c l o s e d -l o o p p o l e s o f t h e a c t u a l p l a n t a r e a t t h e z e r o s o fP ( q - t ) . Thep r e d i c t i o n s f i t + j ) c a n b e o b t a i n e d f r o m ( 6 ) r e p l a c i n g / (t ) f o r3 ~t) a n d t h e G - p a r a m e t e r s f r o m t h e a l g o r i t h m g i v e n in A p p e n d i xA . T h e s i g n a lPA u ( t ) a n d s u b s e q u e n t l yu(t) a r e c a l c u l a t e d f r o mt h e e q u a t i o n s g i v e n a b o v e . T h i s i s t h e n e q u i v a l e n t t o t h es t a n d a r d p o l e - p l a c e m e n t c o n t r o l le r .