CONTROL-AUTOMATICO 13323_01; 2004

7/26/2019 CONTROL-AUTOMATICO 13323_01; 2004

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Chapter

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

i n t r o d u c t i o n

1 1 G e n e r a l s y s t e m s i d e a s

Th e ob j ec t s under s t udy i n con t ro l t heo ry a re sys tems . A sys t em i s any s e t o f e l emen t s

c o n n e c t e d t o g e t h e r b y i n f o r m a t i o n l in k s w i th i n s o m e d e l in e a t e d s y s t e m b o u n d a r i e s .

Refe r r i ng to F igu re 1 .1 no t e t ha t t he sys t em bound ary i s no t a phys i ca l bound ary

bu t r a t he r a conve n i en t f ic t i ona l dev i ce . No te a l so how in fo rm at ion li nks m ay pas s

th rough t he sys t em boundary .

element

Figu re 1 1 T he s t ruc ture o f a s y s t e m


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2 Control heory

S inc e c on t r o l t he o r y d e a l s w i th s t r uc tu ra l p r ope r t i e s , i t r e qu i r e s sy s t e m r e p r e se n -

t a t ions tha t ha v e be e n s t r ippe d o f a ll de ta i l, un t i l t he m a in p r op e r ty tha t r e ma ins i s

t h a t o f c o n n e c t e d n e ss . ( T h e m a s t e r l y m a p o f t h e L o n d o n U n d e r g r o u n d s y s t e m i s a n

e v e r y d a y e x a m p l e o f h o w u s e f u l a r e p re s e n t a ti o n c a n b e w h e n i t h as b e e n s t r i p p e d o f

a ll p r ope r t i e s e xc e p t t ha t o f c onne c te dne ss . )

C o nne c te d ne ss i s a c onc e p t f r om topo logy . To po logy , t he d i sc ip l ine tha t s tud ie s

the unde r ly ing s t r uc tu r e o f m a the m a t i c s , o f f e r s f a sc ina t ing r e a d ing to a sp i r ing sy s t e m s

the o r is t s . R e c o m m e n de d r e a d ing i s g ive n in the B ib l iog r a phy . C le a r ly , a sy s t e m

is a ve r y ge ne r a l c onc e p t ; c on t r o l t he o r y i s mos t i n t e r e s t e d in c e r t a in c l a s se s o f

sys t e m a nd to m a ke p r og r e s s we de l ine a t e the c l a s se s . F i r st i t i s i n t e r e s t e d in dyna m ic

s y s t e m s - t h e s e a r e s y s t e m s w h o s e b e h a v i o u r o v e r a t i m e p e r i o d i s o f in t er e st . T h u s

i f a s y s t e m w e r e c o n c e r n e d w i t h p o p u l a t io n a s p e c t s , a s im i l a r d y n a m i c s y s t e m w o u l d

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

S e c ond ly , i t i s m os t i n t e r e s t e d in a nd m os t pow e r f u l w he n de a l ing w i th l i ne a r sy s -

t e ms . A l ine a r sy s t e m i s c ha r a c t e r i s e d by the p r op e r ty show n in F igu r e 1 .2 . The uppe r

p a r t o f t h e f i g u re s h o w s a s y s t e m s r e s p o n s e t o s o m e a r b i t ra r y s ti m u l u s. T h e l o w e r

pa r t show s how, in the p r e se n c e o f l ine a r i ty , t he r e sponse to a s c a l e d - up ve r s ion o f t he

s t im u l u s i s s i m p l y a s c a l ed - u p v e r s i o n o f th e p r e v i o u s r e s p o n s e, w i t h p r o p o r t i o n a l it y

b e i n g p r e s e r v e d .

F ina l ly , it i s i n t e r e s t e d in f e e db a c k sys t e m s - t he se a r e sys t e m s wh e r e in f o r m a t ion

f l o w s i n o n e o r m o r e l o o p s , s o t h a t p a r t o f th e i n f o r m a t i o n e n t e ri n g a n e l e m e n t m a y

be in f o r m a t ion tha t p r e v iou s ly l e f t t ha t e l e me n t ( F igu r e 1 .3 ).

S ys t e m s a r e o f t e n v i sua l i s e d in the f o r m o f b lo c k d i a g r a m s , i l lu s t r a ting the ma in

f unc t ions , t he i r suppose d in t e r c onne c t ion a nd ( poss ib ly ) t he i r i n t e r c onne c t ion to the

e n v i r o n m e n t o f t h e s y s t e m . T h u s , a s i m p l e r ep r e s e n t at i o n o f th e h u m a n t e m p e r a t u r e

r e gu la t ion sy s t e m m igh t be a s shown in F igu r e 1 .4 .

magnitude

stimulus to

system

time

linear

system

magnitude

stimulus as above

but m agnified

linear

system

magnitude

systems

response

time

magnitude

response is as above

but magnified k times

Figure 1 2 Linear system characteristics


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Control concepts

system

bound ry

Figure 1 3

A simple feedback system

environment

Figure 1 4 simple representation of the human tempera ture regulation system

1 2

Wh at is control theory? an initial discussion

Many areas of study are fortunate in that their titles trigger an immediate image

of their scope and content. For instance, the names human anatomy , veterinary

medicine , aeronautical engineering and ancient history all conjure up coherent

visions of well-defined subjects. This is not so for control theory although almost

everyone is interested in control in the sense ofbeing able to achieve defined objectives

within some time frame. Rather specific examples occur in the named professions of

financial controller and production controller .


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Control theory

C o n t r o l t h e o r y a p p l i es t o e v e r y d a y s i t ua t io n s , a s i n t he e x a m p l e s g i v e n a b o v e , j u s t

a s we l l a s i t a pp l i e s to t he mor e e xo t i c ta sk o f m a n oe uv r in g spa c e ve h ic le s . I n f a c t, the

c o n c e p t s o f c o n tr o l t h e o r y a r e s i m p l e a n d a p p l i c at i o n -i n d e p e n d e n t . T h e u n i v e r s a li t y

o f c on t r o l t he o r y me a ns tha t i t i s be s t c ons ide r e d a s a pp l i e d to a n a bs t r a c t s i t ua t ion

tha t c on ta in s on ly the t opo log ic a l c o r e pos se s se d by a l l s i t ua t ions t ha t ne e d to be

c on t r o l l e d . S uc h a n a b s t r a c t s i tua t ion i s c a l le d a sy s t e m .

The a r gum e n t i s t ha t i f we kn ow h ow to c on t ro l a h igh ly ge ne r a l s i t ua tion c a l l e d

a sys t e m the n w e sha l l be a b l e t o c on t r o l a ny a nd e ve r y pa r t i c u l a r s i tua t ion . Th i s i s the

v ie w po in t o f c on t r o l t he o r y a n d i t is th i s v i e w po in t t ha t g ive s i t i ts e x t r a o r d ina r y po we r .

Thu s a n y s i t ua tion , de l ine a t e d f r om i ts e nv i r o nm e n t f o r s tudy , is c a l le d a sy s t e m .

W h e n c o n t ro l t h e o r y w i s h e s t o s t u d y t e m p e r a t u r e r e g u l a t io n i n t h e h u m a n b o d y , i t

c o n c e m s i t s e l f w i t h a s y s t e m i n v o l v i n g b l o o d c i r cu l a ti o n , h e a t g e n e r a t i o n a n d h e a t

l o s s m e c h a n i s m s a n d d e c i s i o n - m a k i n g b y t h e b r a in . S y s t e m s c a n u s e fu l l y b e d e f i n e d

in a lmos t a ny d i sc ip line - t he y a r e no t c o n f ine d to s c i e nc e o r e ng ine e r ing .

C o n t r o l t h e o r y c o n c e m s i t s e l f w i t h m e a n s b y w h i c h t o a l t e r t h e f u t u re b e h a v i o u r o f

sys t e m s . F o r c on t r o l t he o r y to be suc c e s s f u l ly a pp l i e d , t he r e ne e ds t o be a va i l a b l e :

i ) A pu r po se o r ob j e c t ive t ha t i s l i nke d w i th t he fu tu r e s t a te o f t he sys t e m. C le a r ly

the pa s t c a nno t be i n f lue nc e d no r , s i nc e no r e sponse c a n t a ke p l a c e i n a ny sys t e m

in z e r o t ime , c a n the p r e se n t . )

T h e o b j e c t i v e o f a n y c o n t r o l s y s t e m i n e v e r y c a s e i s c o n n e c t e d w i t h t h e

p e r f o r m a n c e o f t h e s y s t e m o v e r s o m e p e r io d o f t i m e - t h e a c c o u n t a n t a n d t he

i n du s tr ia l m a n a g e r w a n t t o s e e l o n g p e r i o d s o f s m o o t h a n d p r o f it a b l e o p e r a ti o n .

S ome t ime s th i s l e a ds t o c on f l i c t i ng r e qu i r e me n t s , i n t he s e nse t ha t sho r t t e r m

ob je c t ive s a r e f r e que n t ly i n d i r e c t oppo s i t i on to l ong t e r m ob je c t ive s . I n ge ne r a l

t e r m s t h i s o b j e c t i v e c an b e c o n s i d e r e d t o b e t h e d e s i r e d b e h a v i o u r o f th e s y s t e m .

i i) A se t o f pos s ib l e a c t ions t ha t o f f e r s a n e l e m e n t o f c ho ic e . I f no va r i a t ion o f

a c t ions i s pos s ib l e , c on t r o l c a nno t be e xe r c i s e d a nd the sy s t e m wi l l f o l l ow

a c ou r se t ha t c a nno t be m od i f i e d . )

i ii ) Un le s s a t ri a l a nd e r r o r s t r a t e gy i s t o be a dop te d ) som e m e a ns o f c hoo s ing the

c o r r e c t a c t ions i i) t ha t w i l l r e su l t i n t he de s i re d be h a v iou r i ) be ing p r oduc e d .

I n g e n e r a l te r m s , t h is r e q u i r e m e n t i s m e t b y a m o d e l c a p a b l e o f p re d i c t in g

the e f f e c t o f c on t r o l a c t ions on the sys t e m s t a t e. S uc h a mo de l m a y be im p l i c i t

a n d n o t e v e n r e c o g n i s e d a s a m o d e l o r it m a y c o n s i st o f a la r ge a n d c o m p l e x

se t o f e qua t ions .

F o r t he a c c oun ta n t , t he mode l i s a ba l a nc e she e t t oge the r w i th i nhe r i t e d

w i s d o m . F o r th e m i l i ta r y c o m m a n d e r , t h e m o d e l is a m a p o f l o c al t er r ai n a n d

a k n o w l e d g e o f t h e ty p e s a n d d e p l o y m e n t s o f m e n a n d e q u i p m e n t . F o r th e

c o n t ro l o f q u an t it ie s t h a t c a n b e m e a s u r e d b y s e n s o r s, m a t h e m a t i c a l m o d e l s i n

the f o r m o f s to r e d c u r ve s o r s e t s o f e qua t ions w i l l u sua l ly be u se d .

W e se e t he n tha t to a c h ie ve suc c e s s f u l c on t r o l w e m us t ha ve a de f ine d ob je c t ive

a nd be a b l e t o p r e d i c t a de qua te ly , ov e r som e su f f ic i e n t time sc a l e , a l l t he ou tc o m e s o f

a l l t he a c t ions t ha t a r e ope n to u s . F o r i n s t a nc e , a na t iona l powe r s t a t i on bu i ld -

i n g p r o g r a m m e c a n o n l y b e p l an n e d o n c e p r e d ic t i o n s o f t h e f u tu r e d e m a n d f o r

e l e c t r i c i t y a r e a va i l a b l e . F igu r e 1 .5 summa r i se s t he t h r e e r e qu i r e me n t s ne e de d f o r

suc c e s s f u l c on t r o l.


5/9

Control concepts 5

~

ty t o ~

te and )

actions /

ii2

~ t a c t ~

[ or to make

I mo difications hat when applied )

\ to the system will result in the /

desired behaviour i) being obtained /

Figure 1 5 The three elements need ed fo r successful control design

A m ajo r p rob l em in con t ro l u sing a l ong t e rm ho r i zon i s unce r t a in ty o f the l ong

t e r m a c c u r a c y o f m o d e l s , c o m p o u n d e d b y t h e l i k e l ih o o d o f u n f o r e s e e n e v e n ts . T h a t

i s t o s ay, t he pos s ib i l i ty mu s t be f aced t ha t , once uncer t a in ty r i s es a bov e a p a r t i cu l a r

l eve l , no m ean ing fu l con t ro l can be imp lem en ted and t ha t po l i c i e s t ha t l ook ahead t o

an t i c ipa t e fu tu re con t i ngenc i es m ay ca ll fo r imm ed ia t e s ac r i f i ces t ha t w i ll nev er be

r e p a i d b y t h e c r e a t io n o f m o r e f a v o u r a b l e f u tu r e e n v i r o n m e n t s .

F eed back con t ro l , in wh ich an e r ro r i n it ia t e s co r rec t i ve ac t i on , can be used on ly

wh ere co r rec t i ve ac t i ons t ake e f fec t r e l a t i ve ly qu ick ly . I t i s c l ea r l y unsa t i s f ac to ry

t o w a i t u n t il e l e c t r ic i ty d e m a n d e x c e e d s t h e m a x i m u m p o s s ib l e s u p p l y l e v el b e f o r e

s t a rt ing t o bu i l d a new po w er s t a t ion . On t he o the r hand , i t is u sua l l y pe r fec t l y f eas ib l e

t o c o n t r o l th e s p e e d o f a m o t o r b y a n e r r o r -d r i v e n fe e d b a c k c o r r e c t io n .

N o n e o f th e p r o c e s s e s t h at w e a r e c a ll e d u p o n t o c o n t r o l c a n b e m a d e t o c h a n g e

i ts s ta te ins tan taneous ly . Th is is becau se a l l p ro ces ses have the equ ivalen t of iner t i a .

S upp ose tha t we hav e t he t a sk o f m ov in g a l a rge spher ica l bou lde r f rom A to B by

brute forc e F igure 1 .6) .

C l ea r l y , cons ide rab l e i n i t i a l e f fo r t mus t be expended t o ge t t he bou lde r ro l l i ng

and a s imi lar ef fo r t m us t be ex pe nd ed to br ing i t to res t. In the case i l lus t ra ted , i t

w i l l be a l l t oo easy t o over s hoo t t he t a rge t o r to spend t oo l on g a r r i v ing the re i f any

misca l cu l a t i on i s mad e . The d i f f i cu l t y o f ach i ev ing con t ro l i n th i s s i tua t i on i s en t i r e l y

t y p ic a l a n d o c c u r s b e c a u s e o f th e e n e r g y th a t n e e d s t o b e s t o re d i n a n d t h e n r e m o v e d

f r o m t h e b o u l d e r to a l l o w t h e t a sk t o b e a c h i e v e d . O n l y w h e n w e p o s s e s s a p r i o r


6/9

6 Control heory

initial position of bou lder

A

required final position of boulder

B

Figure 1 6 The problem of moving the boulder

q u a n t it a ti v e k n o w l e d g e o f t h e e n e r g y s to r a g e m e c h a n i s m c a n w e h o p e t o a c h i e v e f a st

a nd a c c u r a t e c on t r o l .

A sys t e m wi th in te r na l e ne r g y s to ra ge i s c a l l e d a dyn a m ic sys t e m. Thus , we c a n

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

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

1 3 W h a t is a u t o m a t i c c o n t r o l ?

C o n t r o l t h e o r y w a s d e v e l o p e d t o s u p p o r t t h e e m e r g e n t a c t i v i t y o f a u t o m a t i c c o n t r o l. I t

i s t he r e f o r e a u se f u l m o t iva t ion to tu r n o u r a t t e n t ion to a u tom a t i c c on t r o l . H i s to ri c a l ly ,

t h e d i sc i p li n e o f a u t o m a t i c c o n t ro l w a s c o n c e r n e d w i t h t h e r e p l a c e m e n t o f t h e h u m a n

w or ke r o f F igu r e 1 .7 by the a u to m a t i c c on t r o l l e r o f F igu r e 1 .8 .

A l t h o u g h a u t o m a t i c c o n t r o l is n o w a d a y s a c o m p l e x d i s c ip l in e , n o l o n g e r p r i m a r -

i ly c on c e r ne d w i th the r e p la c e m e n t o f hum a n o pe r a to r s , i t i s a u se f u l s t a r t ing po in t

t o c o n s i d e r w h a t s o r t o f s k i ll s a re n e c e s s a r y t o m o v e f r o m a n e x i st in g , m a n u a l l y

c on t r o l l e d s i t ua t ion to a ne w a u tom a t i c a l ly c on t r o l l e d s i tua t ion , a s i n F igu r e 1 .8 .

1 ) A c e n t r a l ide a o f c on t r o l t he o r y i s t he c on tr o l l oop . A l l c on t r o l l oops ha ve the

sa m e ba s i c f o r m, r e g a r d le s s o f t he pa r t i c u l a r a pp l i c a t ion a re a . Th us , c on t r o l

the o r y u se s a n a pp l i c a t ion - inde pe nd e n t no ta t ion to c onv e r t a l l c on t r o l p r ob le m s

in to the sa m e s t a nda r d p r ob le m . W e c a n c ons ide r t ha t c on t ro l t he o r y c onc e n t r a t e s

on s tudy ing the u n ive r sa l s i t ua t ions tha t unde r l ie a l l a pp l i c a t ions o f qua n t i t a t ive

cont ro l .

I n b r oa d e s t f o r m a c on t r o l l oop a ppe a r s a s in F igu r e 1 .9 . Th e de c i s ions

gov e r n a c t ions tha t a re t a ke n . The e f f e c t o f t he a c t ions i s r e po r t e d ba c k by the

in f o r m a t ion c ha nne l . F u r the r de c i s ions a r e t a ke n a nd the loop op e r a t e s c on t inu -

ous ly a s de sc r ibe d . A c on t r o l l oop p r ov ide s a n e x t r a o r d ina r i ly powe r f u l me a ns

o f c on t r o l bu t , a t t he s a m e t im e , t he e x i s t e nc e o f t he loop a lw a ys b r ings the

p o s s i b i li t y o f t h e p o t e n t i a ll y v e r y d e s t ru c t i v e p h e n o m e n o n o f i n s ta b il it y .

2 ) A l l c on t r o l loops a r e e r r o r -d r ive n , w he r e e r r o r i s de f ine d a s t he d i ff e r e nc e

b e t w e e n t h e b e h a v i o u r t h at i s d e s ir e d a n d t h e b e h a v i o u r t h a t is m e a s u r e d .

3 ) A n i m p o r t a n t p e r f o r m a n c e m e a s u r e f o r a c o n tr o l s y s t e m re l a te s t o ra t e o f e r r o r

r e duc t ion . Of t e n , pe r f o r m a nc e i s quo te d in t e r ms o f t he h ighe s t f r e que nc y tha t

the c on t r o l sy s t e m c a n f o l low , wh e n r e qu i r e d to do so .


7/9

Control concepts 7

Figure 1 7

system o be controlled

factory process

aircraft

ship

military hardware

~ - - consum er evice)

levers of

power

measured

behaviour

process

operator

A manually controlledprocess

Figure 1 8

process

supervisor

automatic system o be

controller controller

measured

behaviour

The process of Figure 1 7 now under automatic control

decision

actions

i n f o r m t i o n o n

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

Figure 1 9 A control loop in its broadest form


8/9

8

4 )

Control theory

A l l c o n t r o l l o o p s t e n d to b e c o m e u n s ta b l e a s h i g h e r a n d h i g h e r p e r f o r m a n c e is

s o u g h t . A g o o d u n d e r s t a n d i n g o f t h e to p i c o f s t a b il it y is c e n t ra l t o u n d e r s t a n d i n g

c o n t r o l t h e o r y .

1 4 Som e exam ples of control sys tems

F o u r c o n t r o l s y s t e m s a r e i l l u s tr a t e d in F i g u r e 1 .1 0 . A l l c a n b e s e e n t o h a v e t h e f o r m

o f F i g u r e 1 .1 1 . A u s e r , u n i n t e r e s t e d i n t h e m e c h a n i c s o f a ll th i s, w i ll s e e t h e s i m p l e r

v i e w o f F i g u r e 1 .1 2 . W e r e f e r to t h is s i n g l e b l o c k ( th a t h a s t h e c o n t r o l l o o p h i d d e n

i n s id e ) a s t h e c o n t r o l s y s t e m .

T h e f o l l o w i n g f u r t h e r p o i n t s a r e i m p o r t a n t :

( 5 ) C o n t r o l s y s te m p e r f o r m a n c e c a n o n l y b e m e a n i n g f u l l y s p e c i f ie d in r e la t io n to

t h e ( t o ta l ) c o n t r o l s y s t e m o f F i g u r e 1 .1 2 .

desired interest rate measured

inflation / ] inf latio n

rat e ? e c o n o m y [ r ate

cen tral ban k -~

steam flow

measured

des ired ~ ] [ s t eam ] sp ee d

speed ~ Governor ] en gine ]

desired

temperature inside

building ~

heating

controller

hot w ater f low

radiators

and

building

measured

tempe rature inside

building

desired

aircraft

heading

automatic

pilot

movem ent of

control surfaces

aircraft

measured

aircraft

heading

Figure 1 10 Some examples o f particular control applications


9/9

Control concepts 9

comparison

\ error

decision

c t i o n 1 s y s t e m I

r- to be

I controlled [

information

measured

behaviour

Figure 1.11

The general form o al l the control systems in Figure 1.10

desired

behaviour

measured

behaviour

Figure 1.12 A user s view of the control system of Figure 1.11

6) The con t ro l sys t em des igner a lmos t a lways has to inco rpo ra t e i n to t he con t ro l

l oop an e l em en t whose i n tr i n si c behav iou r is l a rge ly ou ts ide h is ow n in f luence .

F or i n s t ance , t he con t ro l sys t ems des igner may have l i t t l e i n f l uence on t he

des ign o f a bu i ld ing a l t hough l a t e r he w i ll be ca l l ed upon t o des ign t em pera tu re

con t ro l sys t em s fo r i t. )

7 ) To qu it e a la rge ex t en t, t he con t ro l l e r mu s t neu t ra li s e adverse charac t e r i st i c s i n

the p roces s , compensa t i ng fo r non- idea l p roces s con f igu ra t i ons and fo r sho r t

and l ong t e rm per tu rba t i ons an d va r i ab i li ti e s .

8 ) F o r 7 ) t o be possib l e , t he p roce s s cha rac t e r i s ti c s mu s t be know n to some deg re e

o f a c c u r a c y a n d b e r e a s o n a b l y c o n s t an t .

9 ) Idea l l y [ s ee 6 ) ] t he con t ro l sys t em des igne r wi l l ensu re t ha t t he p roce s s has the

bes t pos s ib l e i nheren t behav iou r , even wi th no con t ro l . Th e con t ro l des ign cy c l e

i s t he re fo re rough ly :

a ) De c ide on a nece s sa ry pe r fo rm ance spec i f ica t i on .

b ) Q u a n t i fy t h e p e r f o r m a n c e o f a n y s y s t e m - t o - b e - c o n t r o l l e d e l e m e n t t h a t is t o

be i nc luded i n t he con t ro l l oop .

c ) Des ign , b y one o r o the r con t ro l des ign t echn iques , a con t ro l l e r so t ha t t he

con t ro l sys t em m ee t s the spec i f ica t i on o f a ) .

d ) Cons t ruc t , comm iss ion and t e s t t he con t ro l sys t em.

In the nex t chapter , w e take these ideas fur ther .

Documents

CONTROL-AUTOMATICO 13323_01; 2004