FP5-3:OO

Richard Bellman

Departments of Mathematics, E l e c t r i c a l E n g i n e e r i n g , & Medicine Universi ty of Southern Cal i fornia

Los Angeles, CA 90007

ABSTRACT

A non l inea r sys t em wi th S - shape s t eady s t a t e cha r - a c t e r i s t i c i s r e f e r r e d t o a s a system with Arrhenius dynamics. The n e g a t i v e s l o p e p a r t of the S-shape curve r e p r e s e n t s a se t o f uns t ab le s t eady s t a t e s . Us ing two examples of Arrhenius systems (catalyt ic reactor and c o n t i n u o u s s t i r r e d t a n k r e a c t o r ) , i t i s shown t h a t i n - t r o d u c t i o n o f s u f f i c i e n t l y fas t o s c i l l a t i o n s i n s y s - tem's parameters generates a new Arrhenius system, the s t e a d y s t a t e c h a r a c t e r i s t i c o f which has a s m a l l e r nega t ive s lope pa r t . Resu l t s o f ana ly t i ca l i nves t iga - t i o n a s w e l l as numer ica l s imula t ion a re p resented . I t i s shown t h a t v i b r a t i o n a l s t a b i l i z a t i o n o f A r r h e n i u s systems gives an i n c r e a s e i n p r o d u c t i v i t y o f t h e p l a n t s

I . INTRODUCTIOK

De&i.nition 0 . A system with n s ta tes x = [XI, . a , xn] ' i s r e f e r r e d t o as a 4y4tm W i d l A?~?heni~cn dynamicn ( o r j u s t an A N L h e h nyb-tm) i f i t s behavior i s c h a r a c t e r i z e d by a t l e a s t two p o s i t i v e parameters , A and B , i n such a manner t h a t t h e norm , ixs(A,B) 1 ! of a s t e a d y s t a t e s o l u t i o n f = 0 has the form shown i n F i g . 1.

A simple example of a system with Arrhenius dynam- i c s can be given as :

i = - x + A(1-x)e , o 1. x 5 1. B X (1)

O b v i o u s l y , t h e s t e a d y s t a t e c h a r a c t e r i s t i c i s def ined here by x -Bx

. % = L e , 0 5 xs 2 1, 1-x

which f o r small B's has a form s i m i l a r t o c u r v e 1 i n Fig. 1, and fo r l a rge B's similar t o c u r v e 2 .

A l a r g e number of equa t ions o f d i f f e r e n t mathemat- i ca l na tu re can be r ecogn ized as systems with Arrhenius dynamics . These could be o rd inary and par t ia l d i f fe r - e n t i a l e q u a t i o n s o r other types of dynamic operators. Arrhenius dynamics appears natural ly in the systems k-here n o n l i n e a r i t i e s a r e of the form:

-bx. $ ( X . ) 1 1 3 ( x . )

2 1 e '-. ( 2 )

Here x i and X . a r e components o f the s ta te vec tor x ( p o s s i b l y i = j j , @,(x), s = 1, 2 , are po lynomia ls in x, and b i s a c o n s t a n t .

Obviously, due t o D e f i n i t i o n 0 and ( 2 ) , systems with Arrhenius dynamics can be recognized as a s p e c i f i c

VIBRATIONAL CONTROL OF SYSTEMS WIM ARRHENIUS DYNAMICS

1255 CH1788-9/820000-1255$00.75O1982 IEEE

* Supported by the U.S. Department o f Energy, Off ice of Basic Energy Sciences, under contract No. DE-ACOZ- 80ER10709.

* * Joseph Bentsman and Sernyon M. Meerkov

Department of E l e c t r i c a l E n g i n e e r i n g I l l i n o i s I n s t i t u t e o f T e c h n o l o g y

Chicago, IL 60616

subclass of nonl inear sys tems [ I ] .

namics. Indeed, if B B*, the sys tem has mul t ip le ( s t a b l e a n d / o r u n s t a b l e ) s t e a d y s t a t e s ; h y s t e r e s i s b e - h a v i o r i s observed i f parameter A i s s lowly va r i ed ; limit c y c l e s a r e known t o e x i s t i n t h e v i c i n i t y o f some s teady s ta tes . Therefore , Arrhenius sys tems a re o f a couidetLable bgbteR-ththeOhe~C intehiest.

t h e p r a c t i c a l s t a n d p o i n t as well . Indeed, every chemi- c a l o r biochemical process descr ibed by Arrhenius law i s , due t o ( 2 ) , a system with Arrhenius dynamics. There- fore , the major i ty o f chemica l reac tors a re Arrhenius systems. Here 1 I xS(A,B) ] I h a s i n t e r p r e t a t i o n as the p r o d u c t i v i t y o f t h e r e a c t o r i n a s t e a d y s t a t e xs(A,B) and maximizat ion of this product ivi ty in a s t a b l e s t e a d y s t a t e i s an impor t an t p rac t i ca l goa l . S ince i n most i n s t a n c e s B happens t o be l a rger than B* and s i n c e the upper branch of curve 2 (Fig. 1) i s n o t a c c e p t a b l e because of technological reasons, parameter A (which i s u s u a l l y i d e n t i f i e d w i t h t h e i n v e r s e o f t h e i n p u t f l o w r a t e o r t h e e x t e r n a l t e m p e r a t u r e ) i s chosen in such a manner t h a t t h e o p e r a t i n g p o i n t 1 / x s * / 1 is c l o s e t o m l (with a runaway margin R, - see F ig . 1). Because of t h a t ; f l u c t u a t i o n s i n i n p u t f l o w r a t e o r e x t e r n a l tem- pera ture could remove t h e s y s t e m i n t h e i n s t a b i l i t y a r e a and cause a "runaway". To p r e v e n t t h i s , a n d t o keep the sys tem in the des i rab le xs* , in many cases a feedback cooling system is u t i l i z e d . However, t h e dy- namics of this feedback i s s lower t han t ha t o f poss ib l e runaways. That i s why unstable behavior could be observed even i f the feedback cont ro l i s present , and t h a t i s why nw cOn.thak? ~echnLyuU doh chemicae he- a&5u me diesdub&.

Arrhenius systems are c h a r a c t e r i z e d by r i c h dy-

On t he o the r hand , t hese sys t ems a r e impor t an t from

In t he p re sen t pape r , v ib ra t iona l con t ro l app roach [ 2 ] , [ 3 ] i s a p p l i e d t o t h e p r o b l e m o f s t a b i l i z a t i o n o f systems with Arrhenius dynamics. I t i s s a i d [ 3 ] t h a t equ i l ib r ium pos i t i on xs i s v i b r a t i o n a l l y s t a b i l i z e d i f in t roduc t ion o f ze ro mean p e r i o d i c o s c i l l a t i o n s i n s y s - tem's parameters causes a b i fu rca t ion o f xs i n t o an a s y m p t o t i c a l l y s t a b l e limit cyc le xS( t ) hav ing t he averaged va lue c lose to xs . In case o f Arrhenius sys- tems, A and B a r e t he pa rame te r s where zero mean o s c i l l a - t i o n s a r e i n t r o d u c e d . The t h e o r e t i c a l aim o f t h i s p a p e r i s to g ive cond i t ions unde r which an uns t ab le s t eady s ta te o f an Arrhenius sys tem can be v ibra t iona l ly s tab i - l i z e d . The p r a c t i c a l g o a l s o f t h i s p a p e r a r e : 1 ) . t o show t h a t some o f t he uns t ab le s t eady s t a t e s , where p r o d u c t i v i t y i s l a r g e , a r e v i b r a t i o n a l l y s t a b i l i z a b l e ; 2 ) . t o g i v e recommendations on the choice o f ampl i tudes and f requencies of the vibrat ions which ensure the de- s i r e d p r o p e r t i e s o f a v i b r a t i o n a l l y s t a b i l i z e d s y s t e m .

A l l t h e m a t e r i a l o f t h i s p a p e r i s based on two ex- amples of systems with Arrhenius dynamics, one being a Ca ta ly t i c Reac to r , which i s cons idered in Sec t ions I11 and IV, and the second - Cont inuous S t i r red Tank Re-

a c t o r d i s c u s s e d i n Sec t ions V and VI. *I Rem& I . Dynamics of chemical reactors with

Arrhenius k ine t ics was a subjec t o f ana lys i s o f numer- ous p u b l i c a t i o n s , s t a r t i n g from c l a s s i c a l p a p e r s [ 4 ] - [SI followed by [6]- [9] . We base our development on t h e models der ived in papers [ lo] , [ll] ( c a t a l y t i c r e - a c t o r ) and extremely complete and detailed papers [12], [13] ( c o n t i n u o u s s t i r r e d t a n k r e a c t o r ) .

Remath 2 . T h e r e e x i s t s a l s o v a s t l i t e r a t u r e on o s c i l l a t i o n s a s a too l o f cont ro l o f chemica l reac tors ( see a r ev iew [14 ] ) , Cond i t iona l ly , t h i s l i t e r a tu re can be devided into three groups: a ) . P e r i o d i c Optimi- z a t i o n ; B) . Push-pul l control ; and y ) . Asynchronous Quenching,

In group a ) , use i s made of t h e f a c t t h a t i n a nonl inear system with nonconvex admissible veloci ty set , i n t r o d u c t i o n o f p r o p e r o s c i l l a t i o n s g e n e r a t e s a new " e f f e c t i v e " v e l o c i t y s e t which is a convex h u l l o f t h e or ig ina l one . Therefore] the op t imized func t ion could assume "be t te r" va lues be ing def ined on the en la rged v e l o c i t y s e t . In o ther words , in the works of group a ) , a usual s ta tement of opt imal control problem i s comple- mented by a condi t ion :

x(0) = x(T), where T i s an unknown p e r i o d , and the problem i s t o f i n d - T

max - 1 i f o ( x , u ) d t . u E u ; ~ a

This approach was d i s c u s s e d , f o r i n s t a n c e , i n [ 1 5 ] - [ 1 9 ] . The main difference between per iodic opt imizat ion and v i b r a t i o n a l c o n t r o l i s tha t t he fo rmer one is a feed- back con t ro l whereas t he l a t t e r one i s not . Note that i n chemica l r eac to r s , some o f t h e s t a t e s , r e q u i r e d f o r feedback] are hardly measurable ( for example, the ra te of conversion) . The o t h e r d i f f e r e n c e i s t h a t p e r i o d i c opt imizat ion i s a s u b s t a n t i a l l y n o n l i n e a r phenomena whereas v ibra t iona l cont ro l is n o t ( s e e , f o r example [ 2 1 ) .

In group B) (see [20]-[21]) i t was assumed t h a t a s t a t e , which is t o b e s t a b i l i z e d , i s located between two u n s t a b l e s t e a d y s t a t e s . I n t h i s c a s e a feedback procedure can be specified under which the operating p o i n t i s being "push-pulled" from one unstable steady s t a t e t o a n o t h e r . As a r e s u l t , a system i s o s c i l l a t i n g a round t he des i r ed s t a t e . The difference between group B) and v i b r a t i o n a l c o n t r o l i s the same as between group a ) and v i b r a t i o n a l c o n t r o l .

Group y ) (see [22]-[25]) i s t h e c l o s e s t t o v i b r a - t iona l cont ro l . In [22] - [24] t ime dependent v ibra t ions were in t roduced in sys tem's parameters in o rder to modify the dynamics. However, the conditions under which v ib ra t iona l s t ab i l i za t ion can be ach ieved a s we l l a s t h e p r o p e r t i e s of necessa ry v ib ra t ions were not de- r ived . In [25] , a problem of inhibit ing runaway i n c a t - a l y t i c r e a c t o r s was considered under the assumption tha t i n t roduced o sc i l l a t ions a r e o f sma l l ampl i tudes and s u f f i c i e n t l y h i g h f r e q u e n c i e s . The r e s u l t s of the p r e s e n t p a p e r p e r t a i n i n g t o t h i s s p e c i f i c s i t u a t i o n a r e g iven in Sec t ion IV.

11. NOTATIONS & DEFINITION

Assume tha t v ib ra t ions o f t he fo rm

A = A. + Df(wt) 1 (3)

are in t roduced in parameter A of an Arrhenius system.

-'Due to space l imi t a t ion , Sec t ions V and VI a r e

t h e JowinaL 06 Math&& Andy& and Appfic(Lti0~6, omi t ted . The f u l l t e x t o f t h e p a p e r will appear in

1982.

Here and D a r e p o s i t i v e c o n s t a n t s and f(wt) is a per - i o d i c z e r o mean sca l a r func t ion w i th f r equency w .

I t is e v i d e n t t h a t i f w is smal l enough, s tab i l iza- t i o n o f an uns tab le s teady s ta te cannot be ach ieved s i n c e i n t h e limit w -+ 0 t h e s y s t e m j u s t s t a y s i n a (poss ib ly , uns t ab le ) s t eady s t a t e , x , def ined by A. + cons t ( see Sec t ion IV) . Therefore , 3t is equal ly ev i - d e n t t h a t t o a c h i e v e t h e d e s i r e d e f f e c t , it is necessary t o assume t h a t

w > > - , 1

Tr where Tr is the r i s e t ime o f t he sys t em. The r i s e time is def ined as the t ime necessary to reach a s u f f i c i e n t l y small ne ighborhood of the equi l ibr ium pos i t ion , s ta r t ing from an i n i t i a l d e v i a t i o n o f o r d e r 1.

I n t r o d u c i n g r e l a t i v e t i m e

from ( 3 ) we f i n d t h a t

A1 = A. + Df($) , where, as i t follows from (4),

Because of formal reasons, we assume t h a t

D=: , ~ < a < l . ( 6 )

This does no t necessar i ly imply tha t the ampl i tude o f o s c i l l a t i o n s is l a r g e , s i n c e a can be s u f f i c i e n t l y s m a l l . However, ( 6 ) d e s c r i b e s a l s o l a r g e a m p l i t u d e s , f o r i n - s t ance , t he ca se where a / E f ( t / c ) i s a sequence of large impulses added t o a cons t an t pa r t o f t he form

A. - 1 /T I06( t -kT)dt , k = 0, 1, * * * ; here & ( a ) i s a 6 -

func t ion and T i s a per iod of f ( t /E) ( see 121 where it i s i n d i c a t e d t h a t 6 - f u n c t i o n i s t h e most e f f i c i e n t s h a p e of v i b r a t i o n s ) .

T

Thus , t he o sc i l l a t ing pa rame te r i n an Arrhenius system we represent in the form:

A = A. + 5 f(;) , t 1

where t i s t ime normalized to T r #

system i s s a i d t o b e v i b r a t i o n a l l y s t a b i l i z a b l e i f f o r every 6 t h e r e e x i s t s 0 < cO << 1 such tha t the Arrhenius system with vibrations of the form (S), ( 7 ) and 0 < E <

z g ( t / E ) , -m < t < m, c h a r a c t e r i z e d by

De.bin&on 1 . A s t e a d y s t a t e xs of an Arrhenius

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

where {(T) i s the averaged value of u ( T ) .

In t he fo l lowing Sec t ions , su f f i c i en t cond i t ions fo r v i b r a t i o n a l s t a b i l i z a b i l i t y f o r two examples of Arrhe- nius systems are given. I t is shown t h a t t h e b e h a v i o r of an Arrhenius system with vibrations (S), ( 7 ) i s , i n a s ense , equ iva len t t o t he behav io r o f a nw Arrhenius sys t em (wi th cons t an t A) , t he s t eady s t a t e cha rac t e r i s - t i c of which, 1 lys(A,B) I I ! has a s m a l l e r i n s t a b i l i t y a r e a . The per iodic so lu t lon , xS( t /E) , ment ioned in Def- i n i t i o n 1 i s centered around the corresponding s teady s t a t e , y s , o f t h e new Arrhenius system, and the asymp- t o t i c s t a b i l i t y o f x S ( t / E ) f o l l o w s from the asymptot ic s t a b i l i t y of ys.

111. VIBRATIONAL CONTROL OF CATALITIC REACTOR: THEORY

I n [ l o ] , [ l l ] and [25] , a s e t o f n o n l i n e a r p a r t i a l --

1256

di f fe ren t ia l equa t ions , which descr ibe hea t and mass transfer i n a s p h e r i c a l l y s h a p e d c a t a l y t i c p e l l e t , was reduced t o a dimensionless f i rs t o r d e r o r d i n a r y d i f f e r - e n t i a l e q u a t i o n which we r e p r e s e n t i n t h e form:

where x i s d imens ion le s s t empera tu re i n t he r eac to r , t i s time e x p r e s s e d i n u n i t s o f T,, A i s dimensionless ex- te rna l t empera ture , B i s parameter def ined by the react- ing substances, and g(x,B) , B '> B*, i s o f t h e form shown i n F i g . 2 .

S i n c e i n t h i s a p p l i c a t i o n , p a r a m e t e r B remains con- s t a n t and i s no t be ing v ib ra t ed , we omit B , denot ing g(x,B) = g ( x ) .

s i n c e xs(A) has a form as in F ig . 1.

t r i v i a l : t h e s t e a d y s t a t e s on t h e n e g a t i v e s l o p e p a r t o f t he cu rve i n F ig , 3 a re uns t ab le ( i nc lud ing po in t s K O and Ro), whereas a l l o t h e r p o i n t s xs a re a sympto t i - c a l l y s t a b l e . C o n s e q u e n t l y , x s ~ [ m l , mz] a r e n o t a c c e s - s i b l e i n a s t e a d y s t a t e o p e r a t i o n . S i n c e t h e p r o d u c - t i v i t y o f t h e r e a c t o r i s a monotonica l ly increas ing funct ion of x and s ince the upper branch of g(x) i s not accep tab le due t o t echno log ica l r ea sons , s t ab i l i za t ion of xSE[ml, m 2 ] is an impor tan t p rac t ica l p roblem. To address th i s p roblem we assume, following [25], that parameter A i s o s c i l l a t i n g a c c o r d i n g t o t h e law given i n ( 7 ) . As a r e s u l t , t h e f o l l o w i n g p e r i o d i c e q u a t i o n i s obtained:

Obviously, (8) i s a system with Arrhenius dynamics

S t a b i l i t y a n a l y s i s i n t h e f irst order sys tem (8) i s

Theohem I . Assume t h a t g(x) i s an a n a l y t i c f u n c - t i o n on XE[O, 1 ) . I n t h i s c a s e xs i s v i b r a t i o n a l l y s t a b i l i z a b l e i f t h e r e e x i s t s A s u c h t h a t t h e e q u a t i o n

T 1

= f f(T)dT - [ ! f(T)dT] 0 0

has a s t e a d y s ta te ys = xs which i s asymptot ica l ly sta- b l e .

Rem& 3. Equation (10) i s a new system with Ar rhen ius dynamics , t he s t eady s t a t e cha rac t e r i s t i c o f which i s def ined by

Since d g/dx2 i s n e g a t i v e i n t h e v i c i n i t y o f t h e maximum o f g ( x ) a n d p o s i t i v e i n t h e v i c i n i t y o f t h e minimum of g ( x ) , t h e r e l a t i v e p o s i t i o n o f g ( x ) a n d g l ( x ) a r e a s i t i s shown i n F i g . 4.

T h e r e f o r e , i n t r o d u c t i o n o f f a s t o s c i l l a t i o n s r e - s u l t s i n m o d i f i e d s t e a d y s t a t e c h a r a c t e r i s t i c g l ( x ) , lower branch of which yields larger xs ( p r o d u c t l v l t y ) then the lower branch of g(x) .

RematLiz 4 . I f a spec i f i c ana ly t i c fo rm o f g (x ) i s assumed, gl(x) can be analyzed in more d e t a i l . F i g . 5 shows g(x) and gl(x,y2) for

2

Note t h a t t h e a r e a o f i n s t a b i l i t y ( n e g a t i v e s l o p e p a r t o f t h e c u r v e s ) i s decreas ing as y 2 i s i n c r e a s i n g .

Rem&k 5. Assume t h a t s y s t e m (8) has A = Ak which

r e s u l t s i n an uns t ab le s t eady s t a t e x s* . As i t fo l lows from Fig. 5, with y2 = 0 . 0 4 4 , xs* i s asymptot ica l ly st:- ble , and therefore system (9) will o s c i l l a t e a r o u n d xs in asymptot ica l ly s tab le manner , i f A = A1 < A K . This means tha t i n o rde r t o keep t he sys t em in x s* , pa rame te r A shou ld be dec reased t o t he va lue o f A 1 a long wi th in - t roduct ion of v ibra t ions . Otherwise the in t roduct ion of v i b r a t i o n s will cause the system to jump bnto the upper branch of g,(x).

s t i t u t i o n Ptiood 06 Theahem 7 . In t roducing T = t / E and a sub-

X(T) Z(T) + a$'(T) I ( 14 ) where $(T) i s def ined by (12) , we r e w r i t e (9) a s

- = dz dT E [ A - g(Z+a$(T)) I* (15)

This i s an equat ion in s o cal led s tandard form. S ince g (x) is ana ly t i c , t he ave rag ing p r inc ip l e [26 ] i s a p p l i c a b l e t o (15) . A p p l i c a t i o n o f t h i s p r i n c i p l e y i e l d s (wi th accu racy t o O ( a 4 ) ) :

where y i s d e f i n e d i n (11).

i s asymptot ica l ly s tab le , the secona theorem of the averaging pr inc ip le [26 , p . 4971 g u a r a n t e e s t h a t f o r every 6 > 0 t h e r e e x i s t s E O such t ha t equa t ion (15 ) with 0 < E < EO has an a sympto t i ca l ly s t ab le pe r iod ic so lu - t i o n , z ~ ( T ) , -- < T < m , c h a r a c t e r i z e d by

L

I f t h e e q u i l i b r i u m p o s i t i o n , y , of equat ion (16)

/Zs(T)-Ys/ < 6 , - < T < , (17)

I f t h e r e e x i s t s A such t ha t f16'1 has an asvmotot i - c a l l y s t a b l e e q u i l i b r i u m p o s i t i o n ys = xs, from (14 ) and (17) fo l lows:

. I , L ~

This , acco rd ing t o Def in i t i on 1, means tha t equ i l ib r ium p o s i t i o n xs is v i b r a t i o n a l l y s t a b i l i z a b l e . Theorem i s proven.

IV. VIBRATIONAL CONTROL OF CATALYTIC REACTOR: -- NUMERICAL SIMULATIONS

Kumer ica l s imula t ions were car r ied ou t for a model of the form:

& = A _- d t 0 1-X

X -Bx e ,

The r eac to r w i th i n t roduced v ib ra t ions was desc r ibed a s :

dxl X - B x - - - AO(l + C s i n t ) - - ' e 1 d t l - X 1 (19)

Note t h a t i n (19) v ib ra t ions a r e i n t roduced i n a form showing the re la t ive ampl i tude C .

The averaged equation (10) i n t h i s c a s e i s of the form:

where 2

y E T .

40

In t he cour se o f s imu la t ions , t he p rope r t i e s o f (18) , (19) and (20) were comparatively analyzed. Solu- t i o n s o f (18)- (20) were obtained for various values of parameters Ao, B , C and 0 u n d e r d i f f e r e n t i n i t i a l con- d i t i ons u s ing t he Gea r ' s a lgo r i thm. The r e s u l t s a r e described below.

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In Sec t ion :I, small parameter was introduced as

1 E = - ,

'rW For vibrat ions of (19)

where To is the per iod of s inwt.

xs(A) and ys(A) , of equations (18) and (20) (denoted as 1 and 2 , r e s p e c t i v e l y ) and t h e s o l u t i o n s , x ( t ) , x l ( t ) and y( t ] , of (18)-(20) (denoted as 3 , 4 and 5, respec- t i v e l y * ) f o r v a r i o u s T , C and A . The va lue o f E i n the exper iment p resente2 in F ig . 6A i s

Fig. 6 r e p r e s e n t s t h e s t e a d y s t a t e c h a r a c t e r i s t i c s ,

1 E = - 0.32 ,

E v i d e n t l y , t h i s E does not ensure a? accep tab le p rec i - s ion of averaged descr ip t ion s ince xs does not approxi- mate ys.

In F ig . 68 , cor responding to w = 0.25,

E = - 0.16, 1 2 7

and xs approximates y( t ) wi th 10% p r e c i s i o n . We con- c lude tha t for the sys tem under cons idera t ion

E 0.16 . max - F i g s . 6C. D i l l u s t r a t e t h a t xs -f v- when E + 0 .

Consequently, *given 6 we can f i nd c O i 8.16 ( o r w 2 0.25) such tha t

The discrepancy between ;s and ys f o r E = 0.32 (F ig . 6A) is due t o t h e f a c t t h a t h e r e s o l u t i o n x , ( t ) r e a c h e s t h e s t e a d y s t a t e c o r r e s p o n d i n g t o AO(1-C) = A m i n . This means t h a t f o r small f r equenc ie s , t he dynamics is "par t ia l ly e l imina ted" and the sys tem i s p a r t i a l l y d r i f t i n g a l o n g t h e s t e a d y s t a t e c h a r a c t e r i s - t i c . Under t h e s e c o n d i t i o n s v i b r a t i o n a l c o n t r o l e f f e c t does no t t ake p lace . For E = 0.16 (Fig. 68) Amin i s not reached and the averaged descr ipt ion is acceptab le .

I n a m e i n PhoductivLty Consider the or iginal system (18) with A = A

(Fig. 7A). Denote the runaway margin, correspondlng t o the r e su l t i ng ope ra t ing po in t , x s , as Rm. Along with (18), consider (19) and (20) with C = C 1 and o = 0.25 (which ensures E = cmaX). The s t e a d y s t a t e c h a r a c t e r - i s t i c o f ( 2 0 ) i s shown i n F i g . 7A f o r C - 0 . 5 ; 1; and 1.5. Choose operat ing points on each 01 ihese charac- t e r i s t i c s i n s u c h a manner t h a t t h e runaway margin, Rm, remains unchanged. Denote these operating points as ys(C1). The r a t i o

I

0

d e s c r i b e s t h e i n c r e a s e i n p r o d u c t i v i t y . F i g . 7A shows t h a t e ( 0 . 5 ) = 1.25; e(1) = 1 . 7 5 ; e ( l . 5 ) = 2 . 3 . F i g . 7B shows the t ime so lu t ion of (19) for C 1 = 1.25.

t a n g u l a r waveform with relative amplitude 1 (see Sec- t ion VII) .

Note t h a t C 1 = 1.25 i s achieved i f f ( o t ) i s a rec-

The r e s u l t s p r e s e n t e d i n F i g s . 7A, B a r e ob ta ined

* 'The same n o t a t i o n is used in Fig. 7 . T i n F i g s . 6 - 7

denotes the s imula t ion t ime.

f o r a system with B = 5.5 . For o ther B's the conclus ion m i g h t b e d i f f e r e n t . F o r i n s t a n c e , i f B = 4 . 5 , o s c i l l a - t i o n s w i t h C1 = 0.9 and o = 0 . 3 ( F i g . 7C) r e s u l t i n an Ar rhen ius sys t em wi thou t i n s t ab i l i t y a r ea a t a l l . There- f o r e , t h e i n c r e a s e i n p r o d u c t i v i t y i s by a f a c t o r o f 213. Note t h a t i n F i g . 7C s t a b i l i z a t i o n was ach ieved fo r a new s t e a d y s t a t e d e f i n e d b y t h e same A as in system (18) wi thou t v ib ra t ions . F ig . 7D shows t h e t i m e s o l u t i o n f o r t h i s s i t u a t i o n .

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a * 5.5

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Fig. 4

Fig !

Fig. 2

1259

Fig. 6 ! b )

'Q '0

Fig. ?(a)

j B = 5 . 5 C - 1.25 A,z 0.06 w = 0.25

x , * O T . 1 5 0

I I

3 . 5.5 c - 0.03 An:0.1056 w.0.3

I

Flg. ?(d)

1260