engineers (chemical industries) by peter englezos [cap 6]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 6]

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Gauss-Newton Method for OrdinaryDifferential Equation (ODE) Models

In th i s chapter we are c onc e n t ra t i ng on t he G a u s s - N e w t o n m e t h o d for thee s t i m a t i o n o f u n k n o w n parameter s i n mode l s descr ibed by a se t of ordinary d i f -fe rent ia l e qua t ions ( O D E s ) .

6 .1 F O R M U L A T I O N O F T H E P R O B L E MAs i t was m e n t i o n e d in C hap t e r 2 , t h e m a t h e m a t i c a l m o d e l s are o f the

formdx( t ) = f(x(t ) , u, k) ; x( t 0) = x 0 (61d t

y(t) = Cx(t) (62o r m o r e g e n e r a l l y

y(t) = h(x( t ) ,k ) (63w h e r e

k = [ k h k2 , . . . , kp ] T is a p-dimensional vector o f parameters w h o s e n u m e r i c a lva l ues a re u n k n o w n ;84

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Gauss-Newton Method for ODE Models 85

x = [ X | , x 2 , . . . ,x n ] ' is an n-dimensional vec t o r o f s tate var iab les ;x 0 is an n-dimensional vec t o r o f i n i t i a l c o n d i t i o n s fo r state var iab les wh icha re a s s u m e d to be k n o w n p r e c i s e l y ;u = [ u b U 2 , . . . , u r ] T i s an r-dimensional ve c to r o f ma n i pu l a t e d v a r i a b l e s w h i c hare ei ther set by the e x p e r i m e n t a l i s t or t he y ha ve been measured and i t

is assumed tha t t he i r n u m e r i c a l va lues are prec i se ly k n o w n ;f = [ f i , f 2 , . . . , f n ] ' is a n-dimensional vector f unc t ion o f k n o w n form (the differ-e n t i a l e q u a t i o n s ) ;y = [ y , , y 2 , . . . ,y m ] T i s the m-dimensional o u t pu t vec t o r i.e., the se t of var iab lestha t are measu red e x p e r i m e n t a l l y ; an dC is the mxn obs e rva t i on ma t r i x , w h i c h i n d i c a t e s the s tate v a r i a b l e s (o r l i n -

ear c o m b i n a t i o n s of s tate var iab les) tha t a re m easured e xpe r ime n ta l l y .E x p e r i m e n t a l data are ava i l ab l e as m e a s u r e m e n t s of the output vec tor as

a f unc t ion of t ime , i.e., [ y j , t j ] , i = l , . . . , N w h e r e w i t h y j w e denote th e meas-ure me n t o f t he ou tpu t vector a t t ime t j . These are to be matched to the va l uesca lcu la ted by the m o d e l at the s a m e t i m e , y(t j) , in s o m e op t ima l fash ion . Basedon the sta t is t ical propert ies of the exper imen ta l error i nvo lve d in the measu re -ment o f the ou tput vec tor , we de termine the w e i g h t i n g m a t r ic e s Q j ( i = l , . . . , N )that shou ld be used i n the objec t ive fu nc t i o n to be m i n i m i z e d as m e n t i o n e d ear-l ier in Chapter 2. The ob jec t ive f u n c t i o n i s of the f o r m ,

S(k)= [ y , - y ( t f , k)] Q f [y ; - y ( t , , k)] (64M i n i m i z a t i o n o f S(k)c a n b e a c c o m p l i s h e d b y u s i n g a l m o s t a n y t e c h n i q u eava i lable f rom o p t i m i z a t i o n theory , however s ince each ob jec t ive fu nc t i o nevalua t ion r equ i r e s th e i n tegra t ion of the state equations, the use of quadra t ica l lyconvergen t a lgo r i thms is h i gh l y r e c o m m e n d e d . T h e G a u s s - N e w t o n m e t h o d isthe most appropria te one fo r ODE m o d e l s (Bard, 1970) and i t presen ted in de ta i lb e l o w .

6 . 2 T H E G A U S S - N E W T O N M E T H O DA g a i n , let us a s s u m e t h a t an es t i ma t e k1^ of the u n k n o w n p a r a m e t e r s isavai lab le at the j th i t e ra t ion . L inear i za t ion of the o u t p u t vector a round k 1 - '' an d

re t a i n i ng first order te rms y i e l d s



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86 Chapter 6

y ( t , k + l ) ) = y(t,k) O K A k 0 + l (6.5)

A s s u m i n g a l i near r e l a t i on s h i p be tween the ou t p u t vector and the s ta tevar iables (y = Cx),the above equa t i on b e c o m e s

= C x ( t , k G )) + C ox5k (6.6)

In th e case o f O D E m o d e l s , th e sen s i t i v i t y m a t r i x G(t) = (5xT/5k)T can-no t be ob ta i ne d by a s i m p l e d i f f e r e n t i a t ion . H o w e v e r , we can f i nd a di f feren t ia le qua t ion t ha t G(t) sat isf ies and hence , t he sens i t iv i ty mat r ix G(t) can be deter-m i n e d as a fu nc t i o n o f t i m e by s o l v i n g s i m u l t a n e o u s l y wi t h th e s ta te ODEs an -other set of di f feren t ia l equa t ion s . T h i s set of O D E s is ob ta i ne d by di f feren t ia t -in g both sides o f E q u a t i o n 6 .1 (the state e q u a t i o n s ) w i t h respect to k , n a m e l y

5k I dt 5k (6.7)

R e v e r s i n g the orde r o f d i f f e ren t i a t i o n on the l e f t -hand s ide of Equat ion6 .7 a nd p e r f o r m i n g th e imp l i c i t dif fe rent ia t ion of the r i gh t -hand s ide , w e obtain

dt

or better

5f5x 5k

of (6.8)

dG( t ) = o f 1dt 5x G(t) + 5k (6.9)

T h e i n i t i a l c o n d i t i o n G(t0) i s o b t a i n e d b y d i f f e r e n t i a t i ng th e i n i t i a l c o n d i -t i o n , x(to)=x 0 , w i t h respect t o k a n d s i n c e th e i n i t i a l state is i n d e p e n d e n t o f t h eparameters , w e ha ve :G(to) = 0. ( 6 .1 0)



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E q u a t i o n 6.9 i s a m a t r i x d i f f e r e n t i a l e q u a t i o n an d r ep res en t s a se t of nxpO D E s . O n c e th e s e n s i t i v i t y c o e f f i c i e n t s a re o b t a i n e d b y s o l v i n g n u m e r i c a l l y th ea b o v e O D E s , th e o u t p u t vec t o r , y ( t , k ^ l + 1 )) , c a n b e c o m p u t e d .Subs t i tu t ion of the latter i n to the objec t ive fu nc t i o n and use of the sta-

t i ona ry c ond i t i on 9 S (k t i+1)) /5k ( i+1) = 0, y i e l d s a l i n ea r equa t i on fo r A k t i + 1 )A A k M ) = b (6 .11)

w h e r eN

i = l

an dNb = ^GT(t,)CTQ, [ y , -Cx(t,kw)] (6 .13)

S o l u t i o n o f t h e a b o v e e q u a t i o n y i e l d s A k C r l > a n d h e n c e , k0''0 i s o b t a i n e dfromk G H ) = k(j)+ A k t , + i ) (6 .14)

w h e r e u i s a s tepping p a r a m e t e r (0 , . . . w h i c hof ten c o n v e r g e s to the opt imum, k* , i f the i n i t i a l guess , k (0 ), is suf f ic ien t ly close.T h e c o n v e r g e d p a r a m e t e r va l ues rep res en t t he L e a s t Squares (LS), W e i g h t e dLeast Squares (WLS) or G e ne ra l i z e d Leas t Squares (GLS)es t imates d e p e n d i n gon t he c ho ic e of the w e i g h t i n g mat r i ces Q j. Fur the rmore , if cer ta in a s s u m p t i o n srega rd ing th e s ta t i s t ica l d is t r ibut ion of the res idua ls hold , these parame te r va lue scould also be the M a x i m u m L i k e l i h o o d ( M L ) est imates .



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88 Chapter 6

6.2.1 Gauss-Newton Algorithm for ODE Models1 .2.3.

4.5 .6 .

I n p u t th e i n i t i a l g u e s s fo r t h e p aram e t e r s , k (0 ) a n d N S I G .F o r j = 0 , l , 2 , . . . , repea t .In tegra te state an d se ns i t iv i ty e q u a t i o n s to obtain x(t) an d G(t). A t eachs a m p l i n g p e r io d , t j , i = l , . . . , N c o m p u t e y( t j ,k^) , and G(t j ) to se t up ma-trix A and vec tor b .S o l v e th e l i n e a r e q u a t i o n A A k = b a n d ob ta in A k

G + i ) ( i ) ( j 1 " ' )D e t e r m i n e u . u s i n g the bisect ion r u l e an d ob ta in k =k + u A kC o n t i n u e u n t i l t h e m a x i m u m n u m b e r of i terat ions is reached or c o n v e r -

Ak(g e n c e is achieved (i .e., ox = [ g i . f c , . . . , g p ] ( 6 .1 5 )

I n t h i s case th e n-dimemional vector g , represents th e se ns i t iv i ty coeffi -cien ts of the state var iab les wi th respect to parame te r k[ an d sat isf ies the fol -l o w i n g ODE,Copyright 2001 by Taylor & Francis Group, LLC
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dg,(t) _ f of7dt ox ok i g i ( to )=0 (6 .16a)

Si mi l a r l y , the n-dimensional vector g 2 r e pre se n t s the sens i t iv i ty coeff i-c i en t s of the state var i ab l e s wi t h respect to parame te r k 2 an d sat isf ies th e fol-l o w i n g ODE,d g 2 ( t ) _ f o f T

dt oxd f I ; g 2 ( t o ) = ook, ( 6 . 1 6 b )

F i n a l l y for the last parame te r , k p , w e h a v e th e c o r r e s p o n d i n g se ns i t iv i tyvector g p

d g p ( t )dt

o f 1ox P(t)

fo f6k,, ; g p ( t 0 ) = o ( 6 . 1 6 c )

S i n c e m o s t of the n u m e r i c a l d i f f e r e n t i a l equa t i on solvers requi re thee qua t ions to be in tegrated to be of the formd zdt" = q > ( z ) ; z(to) = g iven (6 .17)

w e ge ne ra t e th e f o l l o w i n g nx.(p+l)-dimensional ve c to r z

x ( t )

oxox

oxok p

x(t)g , ( t )g 2 ( t ) (6 .18)



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90 Chapter 6

a n d t h e c o r r e s p o n d i n g nx(p+l)-dimensional vec t o r f u n c t i o n (p(z)

(p(z) =5x g l ( t )

x 82 (t)

-fOf g p ( t )

a (6 .19)

I f th e e q u a t i o n s o l v e r p e r m i t s i t , i n f o r m a t i o n ca n a lso b e p r o v i d e d a b o u tth e J acobean o f q>(z), pa r t i c u l a r l y w h e n w e a r e d e a l i n g w i t h st i f f d i f f e r e n t i a lequa t i on s . T he Ja c obe a n i s of the form

of

oz

oxo f 1

SfTox

.,.xT

(6 .20)

w h e r e the "*" i n t he first c o l u m n rep resen t s t e r m s tha t have second order d e-r iva t ives o f f w i th respect to x. In m o s t practical s i tua t i ons t he se t e r m s c a n bene g le c t e d and hence , t h i s Ja c obe a n can be c ons ide re d as a b lock d i a g o n a l mat r i x



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as fa r as the ODE s o l v e r is c o n c e r n e d . T h i s resu l t s i n s i g n i f i c a n t s a v i n g s int e r m s o f m e m o r y r e q u i r e m e n t s a n d r o b u s t n e s s o f t h e n u m e r i c a l i n t e g r a t i o n .I n a t y p ic a l i m p l e m e n t a t i o n , th e n u m e r i c a l i n tegra t ion r o u t i n e i s re-ques ted to prov ide z(t)at each s a m p l i n g poin t , t j , i = l , . . . , N an d h e n c e , x(t,) an dG ( t j ) b e c o m e ava i lable for the computa t ion of y(ti,k t l>) as w e l l as for a d d i n g th ea p p r o p r i a t e t e r m s in m a t r i x A a n d b .2. Implementation of the Bisection Rule

A s m e n t i o n e d in C h a p t e r 4, an a c c e p ta b le v a l u e fo r t h e s t ep p ing p a r a m e -ter u is o b t a i n e d b y s t a r t i ng w i t h u = l a n d h a l v i n g u u n t i l th e o b j e c t i v e f u n c t i o nb e c o m e s less t h a n t h a t o b t a i n e d i n t h e p r e v i o u s i t e r a t i o n , n a m e l y , th e f i rs t v a l u eo f |j. that sat isf ies th e f o l l o w i n g i n e q u a l i t y is accepted.

In th e case of ODE mode l s , e va lua t ion of the objec t ive f u n c t i o n ,S (k (* + u A k (J ' :1 ') , for a par t icula r va l ue o f u . i m p l i e s the i n tegra t ion of the statee qua t ions . I t s hou l d b e e m p h a s i z e d here tha t it is unnecessa ry to in tegra te th esta te e q u a t i o n s f o r t h e e n t i r e data l e n g t h [t0, t N ] fo r each t r i a l v a l u e o f u .. Oncet h e o b j e c t i v e f u n c t i o n b e c o m e s greater t h a n S(k


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92 Chapter 6

6.3 THE G A U S S - N E W T O N M E T H O D - N O N L I N E A R O U T P U TR E L A T I O N S H I PW h e n th e o u tp u t vec tor (m e a sur e d var iab les ) are re la ted to the state vari-ables (and poss ib ly to the parameters) th rough a no n l i nea r r e la t ion s h i p of the form

y(t) = h(x(t) ,k) , w e need to m a k e s o m e addi t iona l m i n o r modi f ica t ions . T he sensi -t iv i ty of the ou tpu t vector to the parameters can be obta ined by p er fo rmi ng theimpl ic i t di f feren t ia t ion to yie ld :

oxSubs t i tu t ion i n to th e l i nea r i zed o u t p u t vector ( E q u a t io n 6.5) y ie ld s

O ' + i ) ( j ) (i +1 ) tc. T > - >y( t ,k ) = h(x ( t ,k )) + W ( t ) A k (6-2j)whe re

an d h e n c e the c or re spond ing norma l equa t i on s are ob t a i ned , i.e.,A A k " =b (6.25)

w h e r eN

A = wT(tj)QiW(tj) (6 .26)i=

an d N ,- . -,i = l

I f the n o n l i n e a r ou tpu t r e l a t ionsh ip is i n d e p e n d e n t of the parameters, i.e., itis of the form

y(t) = h(x( t)) (6.28)then W(tj) s impl i f ies to



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( T VW(t) = -- G(tj) (6 .29)lX J j

a nd the c or re spon d ing mat r i x A a n d vector b b e co m e

(6.30)' ' i o\ \ \ a\ \1 = 1 v )\ v y ;an d

Jt ^ (nhr^ r ,. i (6 .31)In other words , the observa t ion m a t r ix C from th e case o f a l i near ou tpu trelat ionship is subst i tuted w i th the Jacobean matr ix (9h T /9x)T in set t ing up matr ixA an d vector b.

6 .4 THE G A U S S - N EW T O N M E T H O D - SYSTEMS W ITH U N K N O W NI N I T I A L C O N D I T I O N S

L et us c o n s i d e r a system described by a se t of ODEs as in Sect ion 6.1.

= f ( x ( t ) , u , k ) ; x(t0) = x 0 (6.32)d ty(t) = C x ( t ) (6.33)

T he on ly dif fe rence here is that it is fur ther a s s u m e d that s o m e or all of thec ompone n t s o f t he i n i t ia l state vec tor x 0 are u n k n o w n . Let the q-dimensional v e c -tor p (0 < q < r i ) deno te the u n k n o w n c o m p o n e n t s of the vector x 0. In th is class ofparameter es t imat ion prob lems , the objec tive i s to de te rmin e no t on l y the parame-te r vector k bu t a l so the u n k n o w n vector p c o n t a i n i n g th e u n k n o w n e l e m e n t s o fth e in i t ia l state vec tor x(to).A g a i n , w e a s s u m e that exp er i men t a l data are avai lable as m e a s u r e m e n t s oft he ou tpu t vector at var ious p o i n t s in t ime , i . e . , [ y j , t j ] , i = l , . . . , N . T h e object ivefunc t ion tha t shou ld be m i n i m i z e d is the same as before. Tthe o n l y di f ference isthat the m i n i m i z a t i o n is carried ou t over k and p , n a m e l y the object ive func t ion isvie we d as



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94 Chapter 6

NS(k,p) = [y; - y ( t j , k , p ) F Q i | y j - y ( t , , k , p ) ] (6.34)i= l

Let us suppose tha t an es t imate k an d p of the u n k n o w n parameter an din i t ia l sta te v ectors is available a t the j" 1 i te ra t ion . L inea r i z a t i o n of the output v e c -to r a r ound k 1 ^ ' an d p y ie ld s ,

f .TWTVa x I I 5k'1

-*!]f-MA p C H ) (6-35)3x I ( d p I

A s s u m i n g a l i near o u t p u t r e l a t i o n sh ip (i.e., y(t) = Cx(t)) , th e a bove equa t ionbe c ome sy(t,,k c'" V)= Cx( t i ,k ( i ),p ( i) ) + CG(t) Ak(r" + CP(t) Ap^" (6-36)

w h e r e G(t)is the u su a l nxp parameter sens i t iv i ty matr ix (o x T /5k)T an d P(t)is thenxq i n i t i a l state sens i t iv i ty matr ix (9x T /9p)T .T he parameter sens i t iv i ty matr ix G(t)can be ob ta ine d as shown in the pre-v io u s sect ion b y so lv i n g th e m a t r ix dif ferent ia l e qua t ion ,

(6.37)d twith the i n i t ia l c o n d i t i o n ,

G(to) = 0. (6.38)S imi l a r to the param eter sens i t iv i ty m atr ix , the i n i t i a l state sens i t iv i ty matr ix ,P(t), c a n n o t b e o b ta i ned by a s imple d i f fe ren t ia t ion . P(t)is d e t e r m i n e d by s o l v i n ga mat r ix di f fe ren t ia l equat ion tha t i s ob ta ine d b y di f fe ren t i a t i ng both s ides o fEquat ion 6. 1 (state e q u a t i o n ) w i t h respect to p .Re v e r s i ng th e order o f di f fe ren t i a t ion an d p e r f o r m i n g imp l i c i t d i f f e ren t i a tion

on the r i g h t - h and s ide, w e arr ive atCopyright 2001 by Taylor & Francis Group, LLC


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d (6.39)

or better

dP(t)dt 0\ P ( t ) (6.40)

T he ini t ia l c ond i t i on is obta ined by di f feren t ia t ing both sides of the in i t ia lco n d i t io n , x (t0)= x 0, w i t h respect to p , y i e l d i n g

P(to)= (n-q)xq(6 .41)

W i t h o u t an y loss o f ge ne r a l i ty , i t has been a s s u m e d tha t th e u n k n o w n i n i t ia lstates correspond to state variables that are placed as the first e le me n t s of the statevector x(t).Hen ce , the structure of the in i t ia l condi t ion in Equat ion 6.41.T hus , integrat ing the sta te and sensi t ivi ty e qua t ions (Equat ions 6 .1 , 6.9 and6.40), a total of nx(p+q+\) di f feren t ia l equa t ions , the output vector, y(t ,k t i + 1 ),p c 'T l) )is obta ined as a l i near func t ion of k t i + l ) an d p^1'. Nex t , subst i tut ion ofyfek^'^p^'1') i n to the objec t ive func t ion and use of the sta t ionary criteria

(6.42a)an d

= 0 (6.42b)

yie lds th e f o l low ing l i ne a r equat ion



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96 Chapter 6

^GT(t,)CTQCG(t,) ^GT(t,)CTQCP(t,)

2>T(t,)CTQCG(t,)A k ( J + 1 )A p < J + "

N^PT(t,)CTQ(y1 --Cx(t,,k( j ),p (j ))

(6.43)

So l u t i o n o f t h e above e q u a t i o n y i e l d s A k ^ r l ) a n d Ap (- i+1). T h e es t imates'and p*^1' are obta ined nex t as

k(j)P J A p ( j+ i ) (6.44)

where a s tepping parameter u (to be determined by the bisect ion ru le) is also used.I f th e in i t ia l guess k


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The model i s descr ibed by the f o l l o w i n g equa t ion

dt x(0) = 0 (6.45)

where a= 126.2, p=91.9 and x is the concent ra t ion of N O 2 . The concen t ra t ion ofN O 2 wa s me a sure d exp er i men t a l l y as a func t ion of t ime and the data are given inTable 6 .1T he m o d e l is of the form dx/dt=f(x,k 1 ,k 2) where f(x,k 1 ,k 2)=k | (a -x) (p-x)2-k? x 2. T he s ing le state var iab le x is also th e measured variable (i.e., y(t)=x(t)). T hesens i t iv i ty m a t r ix , G(t), is a (Ix2)-dimensiona/ matr ix w i t h e l e m e n t s :

G(t)= G2(t)] = (6.46)

Table 6.1 Data for the Homogeneous Gas PhaseReaction of NO with O 2.Time

012345679

1 11419242939

Concentration ofNO 20

1.46 .3

1 0 . 51 4 . 21 7 . 6

2 1 . 42 3 . 02 7 . 03 0 . 53 4 . 43 8 . 84 1 . 64 3 . 54 5 . 3

Source: B e l l m a n et al. (1967) .I n t h i s case, E q u a t i o n 6 . 16 s i m p l y be c ome s ,

d G ,dt

ofox 9 k , G,(0) = 0 (6.47a)



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98 Chapter 6

an d s i mi l a r l y fo r G 2(t),d G 2

dt5f

5k, G2(0) = 0 (6 .47b)

w h e r e

= -k,(p-x)2-2k,(a-xXp-x)-2k2x (6.48a)

df5 k , = (cc-x)(p-x)2 (6.48b)

of = -x " (6.48c)

E q u a t i o n s 6.47a an d 6.47b should be solved s im ul ta ne ous ly w i th the stateeq u a t io n ( E q u a t io n 6.45) . T h e three O D Es a r e pu t i n t o the s tandard form (dz/dt =(p(z)) used by dif ferent ia l e qua t ion solvers by set t ing

z(t) =x(t)

G,(t)G2(t)

(6.49a)

an d

q > ( z ) =k!(a-x)(p-x)2 -k 2x 2

)2 +2k1(a-x)(p-x) +2k2x]G1 +(a-x)(p-x)2-[k,(p-x)2 +2k ](a-x)(p-x) +2k2x]G2 -x2

(6.49b)

I n teg ra t ion of the above e q u a t i o n y i e l d s x(t) an d G(t) w h i c h a re used in set-t i ng up matr ix A a n d vector b at each iteration of the G a uss -N e wton m e t h o d .

6.5.2 Pyrolytic Dehydrogenation of Benzene to Diphenyl and TriphenylLe t u s now c o n s i d e r t he p y ro l y t i c dehydrogena t i on o f benzene t o d i p h e n y lan d t r iphenyl (Sein fe ld and Gavalas , 1970; Hougen and Watson, 1948):



16/31


2 C 2H 6 C ,2 H 1 0 + H 2C 6 H 6 + C i 2 H 1 0 ^ C I O H I 4 + H 2

T he f o l l o w i n g k i n e t i c m o d e l has been proposed

dt - = -r,-r2 (6.50a)

dt 2where(6.50b)

(6.5 la )

r2 = k 2 [x,x2 - ( l -x1 -2x2X2-2x1 -x2) /9K2 ] (6.5 I b )where X i denotes Ib-mole of b e n z e n e per Ib-mole of pure b e n z e n e feed an d x 2 de-no te s Ib-mole o f d i p h e n y l per Ib-mole of pure b e n z e n e feed. T h e parameters k j an dk 2 are u n k n o w n react ion rate cons tan t s whereas K] an d K2 are e qu i l ib r ium con-stants . T he data cons is t of m e a s u r e m e n t s of x , and x 2 in a f low reactor at eigh tvalues of the reciprocal space veloci ty t and are given be low: The feed to the re-actor w as pure be nz e n e .

Table 6.2. Data for the Pyrolytic Dehydrogenalion of BenzeneReciproca l SpaceVelocity (t) x 104

5 . 6 311.321 6 . 9 722.62

3 4 . 03 9 . 74 5 . 2

1 6 9 . 7

X i

0.8280.7040.6220 .5650.4990.4820.4700.443

X2

0.07370 . 1 1 30.13220.14000.14680.14770.14770.1476

Source: Seinfe ld and G ava la s (1970); H o u g e n and W atson (1948) .



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100 Chapter 6

A s both sta te var iab les are meas u red , th e o u t p u t vector is the s a m e w i t h th estate vector, i.e., y i = X i an d y2=x 2. T he feed to the reactor w as pure benzene . T heequi l ib r ium constants K, a nd K 2 were de termined from the run a t the lowest spacevelocity to be 0.242 an d 0.428, respectively.U s i n g our s tandard notat ion, the above problem is w ri t ten as fo l lows:d x , _dt = f , ( x ] , x 2 ; k 1 , k 2 ) (6.52a)

d x 2~dT = f2 (x ],x 2 ,k,,k2 ) (6.52b)

where f ]=( - r ] - r2) an d f2=r]/2-r2 .T h e sensi t iv i ty matr ix , G( t) , is a (2x2)-dimensional m a t r ix w i th e l e m e n t s :

= [g , ( t ) , g 2 ( t ) ] =

G , , ( t ) G 1 2 ( t ) "G 2 1 ( t ) G 2 2 ( t )

f tqo k j5 x 2

(6.53)

Equat ions 6.16 then become,d g i ( t ) =dt

of5 k , g i ( to )=0 (6 .54a)

an dd g 2(t ) _ 8 f ]

dt oxof

g2(to)=0 (6 .54b)

Taking in to accoun t Equat ion 6 .53, the above equat ions can also be wri t tenas fo l lows :



18/31


an d

f d r "dtd G 2 1dt

\ _ i v j j ^dtd G 2 2d t

~ 5 f i 5fi5 x , ax25 f 2 5 f 2a x i a x 2

o f i o f i5 x j 3x25 f 2 5 f25 x ] 3x2

1 1_G21 .

G12

L . 22}

pl]5 k ,5 f25 k ]

" a f 1ia k 2S f28'k 2

; G , , ( t o ) = 0 , G 2 , ( to )=0 (6 . 55a )

; G 2, (to)=0, G 22(to)=0 (6.55b)

Fina l ly , w e obtain the f o l l o w i n g e qua t ionsdG,, af,

d t a x 5 k , Gn(0) =0 (6.56a)

( f , \ ~ ~5f2 5f2^ x ~ r 2 i + ^ 7 G 2 1 (0 ) = 0 (6 .56b)d G 1 2 _ of,

dt ax, (6.56c)

dG 22d tSf2 r . 5 f2 U ^ 5 f 2-^ G12+ - G22+^SxJ l a x 2 ) 5k2 ( 6 .56d)

w h e r e

a x -Kx2- ( 4 x ,'Jk' 2 ^ Of V 'J K, y R 9 (6 .57a)ax, =-k 3 K , x, -2)|-k2[ x, - (5x, -5 +4x 2 ) | (6.57b)vis..

af, k ,a x = 2 x , + -

2X23 K , 9K - (4 X l -4 + 5x 2 ) (6.57c)



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102 Chapter 6

^T~ =y ( 2 x 2 +2x, -2) -kJ x, - ( 5 x , +4x2 -5)| (6 .57d )5 f,5 k , x j " + ( x 2 + 2 x , x 7 -2x7 (6 .57e )

! 3 K ( x2 + 2 x , x 2 - 2 x 2 ) (6 .57f )

S k 7 9 K 2X | X 7 - px2 -4x, + 5 x , x 2 -5x2 + 2 x 2 (6.57g)5 f7 = -I x,x, -5k, ' 2 9 K 7 (2 x

2 -4x, + 5 x , x 2 -5x2 +2x2 + 2 J ( 6 . 57 h)

T h e fou r sens i t iv i ty e q u a t i o n s ( E q u a t i o n s 6 .56a -d) sh o u ld be so lved s imu l -t a ne ous ly w i t h the two state e qua t ions (Equa t ion 6.52). In tegra t ion o f these s ix[ = n x ( p + l ) = 2 x ( 2 + l ) ] e qua t ions y ie lds x(t)an d G(t)w h i c h are used in set t ing upmat r ix A . and vector b at each iteration o f t he Gaus s -Newt on me t hod .

T h e o rd i na ry di f fe ren t ia l e q u a t i o n tha t a par t icu lar e l e m e n t , G y, of the (nxp)-dimensional sens i t iv i ty mat r ix sat is f ies , c a n b e wri t ten direct ly u s i n g t h e f o l l o w i n ge xpr e ss ion ,

d t (6.58)

6.5.3 Catalytic Hydrogenation of 3-Hydroxypropanal (HPA) to1,3-Propanediol (PD)T h e h y d r o g e n a t i o n o f 3 - h y d r o x y p r o p a n a l (HPA) to 1,3 -propanedio l (PD)over N i / S i C V A K O j catalyst p o w d e r w as s tudied by Professor H o f f m a n ' s group a tt h e F r i ed r i ch - Alex and er U n ive r s i t y i n Er la ge n , G e r m a n y (Zhu et al., 1997). PD isa po ten t ia l ly attractive m o n o m e r fo r p o l y m e r s l ike p o l yp rop y l ene terephthala te .

T h e y used a batch s t i r red autoclave. The e xpe r im e n ta l da ta were k i n d l y providedby Professor H o f f m a n and cons i s t o f meas u remen t s o f t he concen t r a t i on o f HPAa nd P D (C Hp,\, C P D ) ver su s t i m e at var ious opera t ing tempera tures an d pressures.



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T h e comp l e t e data se t w i l l b e g iven i n t h e case s tud ies sec t ion . I n th i s chapte r , w ew i l l discuss how we se t up the e q u a t i o n s for the regress ion of an i so thermal da tase t g iven in Tables 6 .3 or 6 .4 .T h e s a m e g r o u p also pr opose d a r e a ct io n s c h e m e a n d a m a t h e m a t i c a l m o d e ltha t descr ibe th e ra tes o f HPA c o n s u m p t i o n , PD f o r ma t i o n as w e l l as the f o r ma -t ion of acrole in (Ac) . T he m o d e l is as f o l low s

dt 5.59a)

d C P Dd t (6 .59b)

dt - = r3 - r4 - r_3 (6.59c)where C k i s the c on cen t ra t ion of the ca ta lys t (10 g/L). T he reaction rates are g ivenbe low

r, =-H KLPH

0 .5(6.60a)

1 +k 2 C P D C H P A

K1PH(6.60b)

+ K 2 C H P A

r, = k,Cr-=

H PA

Ac

(6 .60c)( 6 .60d)

r4 = k 4 C A c C H P A (6.60e)I n the above equat ions , k j ( j -1 , 2, 3, -3, 4) are ra te con s tan ts (U(mol min g),

K] an d K2 are the adsorpt ion e qu i l ib r ium cons tan t s (L/mol) for H 2 and HPA re -spect ively. P is the hydrogen p ressu re (MPa) in the reactor and H is the H e n r y ' sla w cons tan t wi t h a va lue equa l to 1379 (L bar/mol) at 298 K . T he seven parame-ters ( k j , k 2, k 3, k.3, k 4, K, and K 2) are to be d e t e r m i n e d from the m e a s u r e d c o n c e n -t ra t ions of H PA and PD.



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104 Chapter 6

Table 6.3 Data for the Catalytic Hydrogenation of 3-Hydroxypropanal(HPA) to 1,3-Propanediol (PD) at 5.15 M Pa and 45 < C

t ( m i r i )0.01 0203040506080100120140160

CHPA ( m o l / L )1.349531.363241.258821 . 1 7 9 1 80.9721020.8252030.6971090.4214510.2322960.1280950.02898170.00962368

Cp D ( m o l / L )0.00.002628120.07003940.1843630.3540080.4697770.6073590.8524311.035351 . 1 6 4 1 31.300531 . 3 1 9 7 1

Source: Zhu e t al. (1997) .

Table 6.4 Da ta or the Catalytic Hydrogenation of 3-Hydroxypropanal(HPA) to 1,3-Propanediol (PD) at 5.15 M pa and80 C

t ( m i r i )0.051 01 5202530

CHPA ( m o l / L )1.349530.8735130.447270.1409250.03500760.01308590.00581597

C P D ( m o l / L )0.00.3885680.8160320.9670171 . 0 5 1 2 51.082391.12024

Source: Zhuetal. (1997) .

In order to use our s tandard nota t ion we in t roduce the f o l l o w i n g vectors:x = [x , , x 2 , x 3]T = [C H PA , C PD , C A c] Tk = [ k , , k2, k 3, Ic,, k 5 , k ^ k y f = [kh k 2 , k3, k.3, kt, K,, K2]Ty = [ y i , y 2f = [ C H P A , C P D ] T

Hen ce , t h e di f fe ren t ia l e q u a t i o n m o d e l t a ke s th e f o r m ,



22/31


dx-z-= f , ( x 1 , x 2 , x 3 ; k i , k 2 , . . . , k 7 ; u i , u 2 )= f 2 ( x 1 , x 2 , x , ; k ] , k 2 , . . . , k 7 ; u , , u 2 )

dtdx,

dtdx.-= f - ) ( x 1 , x - , , x 3 ; k 1 , k 2 , . . . , k 7 ; U ] , u 9dt

and the observa t ion mat r ix i s s imply

(6.6 la)

(6.6 I b )

(6.6 I c )

C = 0 0 (6 .62)

In E qua t i on s 6 .61 , U [ deno tes the concent ra t ion of catalyst present in the re-actor (Ck) an d u 2 the hydrogen pressure (P). As far as the est imation problem isc onc e rne d , both these variables are assumed to be known precisely. Actual ly, as itw i l l be discussed later on exp er i men t a l design (Chapter 12), the value of suchvariables i s chosen by the exp er i men t a l i s t and can have a p a r a m o u n t effect on thequal i ty of the parameter es t imates . Equa t ions 6 . 6 1 are rewri t ten as f o l l o w i n gd x ,"dT

d x 2 = U i ( r rr 2 )

(6.63a)

(6 .63b)

whered t

r , = - k , u 2 x2 ,H

p, =

0 .5

k6"2H k7Xl

(6.63c)

(6.64a)

(6.64b)



23/31

106 Chapter 6

r_3 = k4x.r3 =k 3 x, (6.64c)

(6.64d)r4 = k 5 x 3 x , (6.64e)

T he sens i t iv i ty m a t r ix , G(t), is a (3x7)-ditnensional matr ix with e lements :

G ( t ) = [g,(t),g2(t),...,g7(t)]= dx dx

G(t) =. . G 1 7 ( t ) '

G 2 ] ( t) . . . G 27 G37(t)

E q u a t io n s 6 . 16 then b e c o m e ,

=(-lT,,,+(-*dtd g 2 ( t ) _ 3 f '

dt d\ g 2 ( t )

dkj

g i ( to )=0

g2(to)=0

(6.65a)

(6.65b)

(6 .66a)

(6 .66b)

dt ox g7(to)=0 (6 .66c)

w h e r eCopyright 2001 by Taylor & Francis Group, LLC


24/31


( T VU J

an d

N =5k.ijT a k i n g i n t o a c c o u n t th e

dGu ( d f i ] ( d f , ]

r r ^ o po r s f . ^ ii^oxj 2J ^ o x 3 J

3f2 5 f2 6 f 2I5xlj UX2J 13J/ _,, \ f ^- \ / ^,. \5 f3 o f3 o f3

_ ^ 5 x , J ^ 0 X 2 J l , S x 3 J _

I*1 1 1lS kJ(^) :J-U....7M

above e q u a t io n s w e ob ta in

, ,f^l V. , 5fl . r, -d t -UJ'' U2J "'' / ok, ' ^1IW UdG2, fSf, V ,f3f,V , f V , af2 . r^ /- m A- G11+U G21+K G31+a, ' G2l() d t v o x i y v^7/ 1^ *37 ^ i

dG'7 f 5 f i V /5f'V , fa f i V , af' r rm n,t -kG'7+U G27+k G37+ 3, J G17() d t ^ox,j ^ 0 x 3 J l ^ o x 3 J 5k7, _ f -v? \ f ~-c \ f ~-c \ ^cd G 2 7 [ or 2 ol^ '2 *2-\ pi? + -, G 2 7 + G37 + ; G 2 7 ( 0 ) 0Qt I (7X i j 1 (7X 7 J I t/X ^ y C'K y

d G 3 7 fS f3V , f S fO r , f 5 fO r , 5fl r rm n (._L ( T ^ ^ _ L (T,4- ( I -,-, 1 U 1 (I

(6 .67a)

(6 .67b)

(6.68)

dt 6x 1 v ~JI ^ '' Io k 7 J



25/31

108 Chapter 6

T h e par t ia l der iva t ives w i t h respect to the state var iab les in E q u a t i o n 6.67a tha t a rene e de d i n the above O D E s are g iven nex t

(6 .69a)

(6.6%)

(6.69c)

( 6 - 6 9 d )OX(6 .69e )

(6 .69f)

(6.69g)

(6.69h)

(6.69i)

The part ial der iva t ives wi th respect to the parameters i n Equat ion 6.67b tha tare needed in the above O D E s are g iven nex t

ox.

^X^



26/31


o f9kT

- u

(6.70a)

dtdk,

- u

- u (6.70b)

d f5 kT (6.70c)

of5k 4

df, (6.70d)



27/31

n o Chapter 6

ofok 5

ok 53k, 0x 3 x .

(6.70e)

d fok,***6

-u , or, dr.5k6 dktor. or-.

ok 6 3k t (6.70f)

5k 7

^L3k 7

f Sr,U 11*7f Sr,u , --1*70

5r2 ^ |f okjor2 " j*7J (6.70g)

T he 21 equa t i on s (given as Equat ion 6.68)should be solved s im ul ta ne ous lyw i t h th e t h ree state eq u a t io n s ( E q u a t io n 6.64) . I n tegra t ion of these 24 e q u a t i o n sy ie lds x(t)an d G(t)w h i c h are used in se t t ing up matrix A and vector b at eachiteration of the Gauss -Newton m e t h o d . Gi ven the complexi ty of the O D E s w h e nth e d i m e n s i o n a l i t y of the problem i ncreases , it is q u i t e h e l p f u l to ha v e a ge ne r a lpurpose c ompute r program tha t sets up the sensi t ivi ty e qua t ions automat ica l ly .F u r th e rmo re , s i nce ana ly t ica l der iva t ives a re s u b j e c t to user i n p u t error, n u -mer ica l eva lua t ion of the derivat ives can also be used in a typica l compu te r im -p lementa t ion of t he G a uss -N e wton me thod . Detai ls for a success fu l imple me n ta -t ion of the method are given in Chapter 8.



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Gauss-Newton Method for ODE Models 1 1 1

6 .6 E Q U I V A L E N C E O F G A U S S -N E W T O N W I T H T H E Q U A S I -L I N E A R I Z A T I O N M E T H O DT h e q u as i l i n ea r i z a t i o n m e t h o d (QM) is ano t he r me t hod fo r s o l v i n g of f - l i neparameter es t ima t io n p r o b l e m s descr ibed by E q u a t i o n s 6.1,6.2 and 6 .3 ( B e l l m a n

and Kalaba , 1965) . Qua s i l i ne a r i z a t ion c onve rge s quadra t ica l ly to the o p t i m u m bu thas a smal l r egion o f c o n v e r g e n c e ( S e i n fe ld an d Gavaias , 1970). Kalogerakis andL u u s (1983b) presen ted an a l t e rna t ive development o f the QM that e na b le s a m o r eeff ic ient i m p l e m e n t a t i o n of the a l go r i t hm.Fur the rmore , they s h o w e d tha t th is s impl i f i ed Q M i s very s imi l a r to theG a u s s - N e w t o n m e t h o d . N e x t the q u as i l i n ea r i z a t i o n m e t h o d as w e l l as the s i m p l i -fied q u a s i l i n e a r i za t i o n m e t h o d a r e descr ibed a nd the e q u i v a l e n c e of QM to theG a u s s - N e w t o n m e t h o d is demons t r a t ed .

6.6.1 The Quasilinearization Method and its SimplificationA n es t imate k ^ * o f t h e u n k n o w n parameter vector is avai lab le at the j th it-era t ion . Equa t ion 6 .1 t he n b e c o m e s

- = f ( x ( - i ) ( t ) , k ( i ) ) (6 .71)dt x 'U s i n g the p arame t e r e s t im a te k 1-'" '" 1 ' from th e n e x t i t e ra t ion we obtain fromEqua t ion 6 .1

dtB y u s i n g a T a y l o r ser ies e x p a n s i o n on t he r i g h t h and s ide o f E q u a t io n 6.72

an d k e e p i n g o n l y th e l i n ea r t e rms w e o b tain the f o l l o w i n g e qua t ion

' (6.73)

w h e r e th e par t ia l der ivatives are evaluated at x (t).T he above e qua t ion is l i n e a r in x^1' an d k1-' . I n tegra t ion o f E q u a t io n 6.72w i l l resu l t i n the f o l l o w i n g e qua t ion



29/31

112 Chapter 6

xti+1)(t) = g(t)+G(t)ku+" (6.74)whe re g(t) is an n - d i m e n s i o n a l vec tor an d G(t ) is an nxp m a t r ix .Equat ion 6.74 is different iated and the RHS of the r e su l t an t equat ion isequated wi th t he RHS o f Equa t ion 6.73 to yield

(6.75)dtan d

dtT he i n i t i a l c o n d i t i o n s fo r E q u a t i o n s 6 .75 an d 6.76 are as fo l lows

g ( t 0 ) = x0 (6.77a)G(t0) = 0. (6 .77b)

Equa t ions 6 .7 1 , 6 .75 an d 6.76 can be so lved s imul taneous ly to y ie ld g(t) an dG ( t ) w h e n the in i t ia l state vector x 0 and the parameter est imate vector k^ areg i v e n . I n order t o de t e rmi ne k ^+ 1 ' the ou tpu t vector (given by Equat ion 6.2)i s in-serted i n to th e objec t ive f u nc t i o n (E q u a t i o n 6.4) and t h e s tat ionary co nd i t i o nyie lds ,

(6.78)

T he case of a n o n l i n e a r observat iona l re la t ionsh ip (Equat ion 6.3) w i l l bee x a m i n e d later. Equa t ion 6.78 y ie lds the f o l l o w i n g l i ne a r equa t ion w h i c h i s solvedby L U de c om pos i t ion (or any other t echn ique) to ob ta in k 0 + l )iN

^GT(t,)CTQ1CG(t,) (6 .79)

A s matr ix Q s is posi t ive def in i te , th e a b o v e e qua t ion g ive s th e m i n i m u m o fthe ob jec t ive f u n c t i o n .



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Gauss-Newton Method for ODE Models 1 1 3

S i n c e l i near iza t ion of the d i f fe ren t ia l E q u a t i o n 6 . 1 a r o u n d th e t r a jec toryx(t) , r esul t ing from the choice of k has been used, the above m e t h o d givesk ^ + 1 ) w h i c h i s a n a p p r o x i m a t i o n t o t h e be s t p a r a m e t e r ve c to r . U s i n g t h is v a lue a sk a new k ^ + l ' can be ob ta ined and t h u s a se que nc e of vectors k (0 ), k ( l ) , k < 2 ) . . . isobta ined. This s e q u e n c e c o n v e r g e s r a p id ly to the op t im um pr ov ide d that thei n i t ia l guess is su f f i c i en t l y g o o d . T h e above desc r ibed me thodo logy cons t i tu testh e Quasi l inear iza t ion M e t h o d ( Q M ) . The to ta l n u m b e r of dif fe rent ia l equa t ionswhi ch mus t be in tegra ted at each i teration step is nx(p+2).Kalogerakis an d L u u s (1983b) not iced tha t Equat ion 6.75 is r e d u n d a n t .S i n c e E q u a t i o n 6.74 is o b t a i n e d b y l i n e a r i z a t i o n a r o u n d th e n o m i n a l t r a j ec to ryx(t) r esul t in g from k, i f we le t k1-'*1' be k t he n Equa t ion 6 .74 b e c o m e s

v ( j ) (t \ _ f r / t \ _ i _ r ^ ^ t \ L - 0 ) (t\ Q(\\X ( ^ I) g^l) + lj(I J K ^ O . o U JEquat ion 6.80 i s exac t ra ther t h a n a first order a p p r o x i m a t i o n as E q u a t io n 6.74 is .Th i s is s i m p l y because Equat ion 6.80 is Equat ion 6.74 evaluated at the po in t ofl inear iza t ion , k . T h u s Equat ion 6.80 can be used to compu te g(t) as

fyA\__v(J)(+ f""1 ftM?0) I01\&V^/ VV vJ^l^H ^U.O 1)I t is obvious tha t the use of Equation 6.81 leads to a s impl i f ica t ion because

th e n u m b e r of di f feren t ia l equat ions tha t now need to be integrated is nx(p+l).Kalogerakis an d L u u s (1983b) then proposed th e f o l l o w i n g a lgor i thm fo r t he Q M .Step 1 . Select an i n i t i a l guess k (0 >. H e nc e j = 0 .Step 2. In tegra te Equa t ions 6.71 an d 6.76 s i m u l t a n e o u s l y to obtain x(t) an d

G(t) .Step 3 . Use eq u a t io n 6.81 to ob ta in g(tj), i = l , 2 , . . . , N and se t u p m a t r i x A a n dvector b in E q u a t i o n 6.79.Step 4. Solve equation 6.79 to obtain k"' ' .Step 5 . C o n t i n u e u n t i l

k ( J + D _ k ( j ) < T O L (6.82)w h e r e TOL is a preset sma l l n u m b e r to ensure te rmina t ion of the it-erations. I f the above inequa l i ty i s not satisfied then we set k = k (J+ l ),increase j by on e and go to Step 2 to repeat the calculat ions.



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114 Chapter 6

6.6.2 Equivalence to Gauss-Newton MethodI f w e c o m p a r e E q u a t i o n s 6.79 an d 6 . 1 1 w e no t i c e that the o n l y di f fe rencebetween th e quas i l i near i za t ion method a nd the G a uss -N e w ton m e thod i s t he na tu reof the equat ion tha t yie lds th e parameter es t imate vec tor k(rl>. I f o n e subst i tutes

Equat ion 6.81 i n to Equat ion 6.79ob t a i n s the f o l l o w i n g equat iont C i + i ) =

(6.83)= lN

B y t a k i n g th e last term on the r ight h a n d side o f Equat ion 6.83 to the l e f th a n d side on e obta ins Equa t ion 6.11 that is used for the G a u s s - N e w t o n m e t h o d .He nc e , w h e n the ou t p u t vector is l i near ly related to the state vector (Equat ion 6.2)then the s impl i f ied qua s i l i ne a r i z a t ion m e t h o d is computa t iona l ly identical to theGauss -Newton me thod .Kalogerakis a n d L u u s (1983b) compared the comp u t a t i ona l effort requiredb y G a u s s - N e w t o n , s i mp l i f i e d q u a s i l i n e a r i z a t i o n a n d s tandard q u a s i l i n e a r i za t i o nmethods . They f ound tha t a l l methods produced the s a m e new es t imates a t eachi teration as expected. Fur thermore , the requi red com puta t ion a l t ime for the G auss-N e w t o n and the s impl i f ied quasi l inear iza t ion was the same and about 90% of thatrequired by the standard quas i l i near i za t ion m e t h o d .

6 . 6 . 3 Nonlinear Outpu t RelationshipW h e n the ou tpu t vector is n o n l i n e a r l y related to the state vector (Equa t ion6.3) then su b s t i t u t i o n of x ^ 1 + 1 > from E q u a t i o n 6.74 i n to the Equa t ion 6 .3 f o l l o w e d

by su b s t i t u t i o n of the r e su l t i ng eq u a t io n i n to the ob jec t ive f u n c t i o n ( E q u a t io n 6.4)yie lds the f o l l o w i n g equat ion after appl ica t ion of the s ta t ionary condi t ion (Equa-t ion 6.78)

T he above e qua t ion represents a se t of p non l i ne a r e qua t ions w h i c h can beso lved to ob ta in k ^ 4 " 1 ' . T h e s o lu t i o n o f t h i s se t o f eq u a t io n s can b e acco mp l i sh ed b ytw o methods . First, by e m p l o y i n g N e w t o n ' s me thod o r a l te rna t ive ly by l i near i z ingthe output vector a round th e trajectory x^(t). Kalogerakis an d L u u s (1983b)

engineers (chemical industries) by peter englezos [cap 6]

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