Decision Theoretical Remark on Sensitivity Analysis

DECISION THEORETICAL REMARK ON SENSITIVITY ANALYSIS

H . S t e h f e s t F e b r u a r y 1 9 7 5

R e s e a r c h R e p o r t s a r e p u b l i c a t i o n s r e p o r t i n g o n t h e w o r k o f t h e a u t h o r . Any views o r

c o n c l u s i o n s a r e t h o s e o f t h e a u t h o r , a n d d o n o t n e c e s s a r i l y r e f l e c t t h o s e o f IIASA.

D e c i s i o n T h e o r e t i c a l Remark o n S e n s i t i v i t y A n a l y s i s

H . ~ t e h f e s t * A b s t r a c t

The s e n s i t i v i t y o f dynamic s y s t e m s t o c h a n g e s o f t h e p a r a m e t e r v a l u e s c a n b e e s t i m a t e d by two f i r s t o r d e r methods:

e i t h e r by s o l v i n g t h e s e n s i t i v i t y s y s t e m , o r by s o l v i n g t h e o r i g i n a l s y s t e m t w i c e w i t h s l i g h t l y d i f f e r e n t p a r a m e t e r v a l - u e s . I n t h i s p a p e r t h e p r o b l e m i n v e s t i g a t e d i s which method s h o u l d b e p r e f e r r e d i n o r d e r t o m i n i m i z e l o s s e s d u e t o d e v i a - t i o n s o f t h o s e f i r s t o r d e r a p p r o x i m a t i o n s f r o m t h e a c t u a l s e n s i t i v i t y . I t i s shown t h a t t h e method u s i n g two d i f f e r - e n t p a r a m e t e r v a l u e s i s t o b e p r e f e r r e d if o n e i s t o f i n d o u t w h e t h e r t h e s o l u t i o n o f t h e dynamic s y s t e m c o u l d d e v i a t e

from t h e n o m i n a l s o l u t i o n by more t h a n a p r e s c r i b e d s t a n d - a r d . A p r o b l e m o f o p t i m a l c o n t r o l i s d e s c r i b e d f o r w h i c h t h e o t h e r method t u r n e d o u t t o b e p r e f e r a b l e . Some o t h e r p r o b l e m s a r e m e n t i o n e d f o r w h i c h t h e p r e f e r a b i l i t y o f o n e o f t h e two m e t h o d s c o u l d b e p r o v e d .

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

The c h a n g e s o f t h e s t a t e v a r i a b l e s o f a dynamic s y s t e m

- _dX d t - - f ( x , t , p ) ( x s t a t e v e c t o r , t t i m e , f v e c t o r - (1) v a l u e d f u n c t i o n , p p a r a m e t e r )

d u e t o d e v i a t i o n s o f t h e p a r a m e t e r v a l u e from t h e " n o m i n a l "

v a l u e

Po

c a n b e e s t i m a t e d by two f i r s t o r d e r methods [ I ] : A . One c a n compute t h e f u n c t i o n s

w h i c h a r e t h e s o l u t i o n o f t h e s o - c a l l e d s e n s i t i v i t y s y s t e m

*

The h e l p f u l d i s c u s s i o n s w i t h S . R i n a l d i r e g a r d i n g t h i s p r o b l e m a r e g r a t e f u l l y acknowledged.

where af i s t h e J a c o b i a n m a t r i x o f f .

System ( 3 ) i s a l i n e a r s y s t e m , whose s o l u t i o n r e q u i r e s t h e s o l u t i o n of t h e o r i g i n a l system ( 1 ) .

B . One c a n a l s o s o l v e t h e o r i g i n a l system t w i c e w i t h t h e p a r a m e t e r v a l u e s p and po

+

Ap and c o n s i d e r t h e f u n c t i o n

0

a s an e s t i m a t e f o r t h e s e n s i t i v i t y o f x w i t h r e s p e c t t o p. The v a l u e of Ap i s n o r m a l l y r e l a t e d i n some way t o t h e e x p e c t e d v a r i a t i o n s of p.

With b o t h methods p may be a l s o a n i n i t i a l v a l u e of an x-component. I n t h e f o l l o w i n g we r e f e r t o t h e f i r s t method a s " d i f f e r e n t i a l s e n s i t i v i t y a n a l y s i s " and t o t h e second one a s " f i n i t e s e n s i t i v i t y a n a l y s i s " . The c r i t e r i o n f o r c h o o s i n g between t h e two methods i s u s u a l l y t h e c o m p u t a t i o n a l e f f o r t . O f t e n d i f f e r e n t i a l s e n s i t i v i t y i s p r e f e r r e d b e c a u s e t h e

l i n e a r i t y of t h e s e n s i t i v i t y system a l l o w s g e n e r a l s t a t e m e n t s t o be d e r i v e d w i t h o u t n u m e r i c a l c a l c u l a t i o n . Another i m p o r t a n t a s p e c t , which seems n o t t o have been t a k e n i n t o a c c o u n t s o f a r , i s t h e l o s s e s c o n n e c t e d w i t h wrong p r e d i c t i o n s of t h e methods a b o u t system s e n s i t i v i t y . I n t h e f o l l o w i n g b o t h methods a r e c o n s i d e r e d o n l y w i t h r e g a r d t o t h i s a s p e c t .

I f , f o r i n s t a n c e , we e v a l u a t e t h e d i f f e r e n t i a l s e n s i t i v i t y ( 2 ) of t h e s y s t e m

w i t h r e s p e c t t o t h e p a r a m e t e r p we g e t t h e f u n c t i o n s which a r e shown i n F i g u r e 1 t o g e t h e r w i t h t h e nominal s o l u t i o n o f System

( 5 ) f o r po = 0.97. The i n i t i a l v a l u e s a r e

I f we a c t u a l l y i n c r e a s e t h e p a r a m e t e r p by 5X, however, w e g e t t h e s o l u t i o n shown i n F i g u r e 2. I t shows t h a t t h e d i f f e r e n t i a l s e n s i t i v i t y would l e a d t o a c o m p l e t e l y wrong p r e d i c t i o n a b o u t t h e model b e h a v i o r i n t h e c a s e of " s m a l l " p a r a m e t e r c h a n g e s . But examples c o u l d be g i v e n where f i n i t e s e n s i t i v i t y i s s i m i - l a r l y m i s l e a d i n g . The problem a r i s e s which method s h o u l d b e p r e f e r r e d i n o r d e r t o minimize l o s s e s due t o s u c h c a s e s . The r e s u l t depends on what c o n c l u s i o n s a r e drawn from t h e r e s u l t s of t h e s e n s i t i v i t y a n a l y s i s .

11. The Problem o f Maximum P e r m i s s i b l e D e v i a t i o n

A common problem f o r s e n s i t i v i t y a n a l y s i s i s t o f i n d o u t w h e t h e r t h e s o l u t i o n of a System ( 1 ) c a n d e v i a t e from t h e nom- i n a l s o l u t i o n by more t h a n a p r e s c r i b e d s t a n d a r d A i f t h e p a r a - m e t e r c a n v a r y o v e r a c e r t a i n r a n g e ( p o

- Ape,

po

+ Ape).

^{I f}

t h e s t a n d a r d i s e x c e e d e d c e r t a i n a c t i o n s a r e t a k e n : i f a s y s - l s y s t e m ( 5 ) c a n be looked upon a s t h e d e s c r i p t i o n of a t h r e e s p e c i e s e c o l o g i c a l s y s t e m w i t h x l = r a b b i t d e n s i t y , x2

= d e n s i t y o f g r a s s e a t e n by t h e r a b b i t s , x3 ⁼ d e n s i t y o f p l a n t s n o t e a t a b l e by r a b b i t s b u t i n t e r a c t i n g w i t h t h e g r a s s .

tem i s t o b e d e s i g n e d , t h e component which c a u s e s t h e e x c e s s h a s t o b e r e p l a c e d by a component w i t h h i g h e r s p e c i f i c a t i o n s . O r , i f t h e System ( 1 ) i s t o d e s c r i b e a n a t u r a l s i t u a t i o n , more a c c u r a t e measurements may b e n e c e s s a r y .

The v a l u e s o f S1 = Max ( s l a p o ) o r S 2 = Max ( x ( p o + A p o )

t t

-

x ( p ) ) c a n b e t a k e n a s i n d i c a t o r s o f w h e t h e r t h e s u p p l e m e n t a r y 0

m o t i o n e x c e e d s t h e p r e s c r i b e d l i m i t s o r n o t . F o r t h e s a k e o f s i m p l i c i t y and w i t h o u t l o s s o f g e n e r a l i t y , w e c o n s i d e r a o n e v a r i a b l e s y s t e m and d i s r e g a r d t h e t i m e d e p e n d e n c e o f t h e s e n - s i t i v i t y . L e t SO b e t h e maximum d e v i a t i o n of x from t h e n o r m a l s o l u t i o n o v e r t h e i n t e r v a l ( p O-ApO,pO+ApO). Then t h e p a y o f f t a b l e f o r t h e c h o i c e between t h e two methods o f s e n s i t i v i t y a n a l y s i s h a s t h e f o l l o w i n g form:

L i s t h e l o s s s u s t a i n e d i f i t i s n o t r e c o g n i z e d t h a t t h e s t a n - 1

a r d i s v i o l a t e d , L2

i s

t h e l o s s i n t h e c a s e where s e n s i t i v i t y i s o v e r e s t i m a t e d . The a l t e r n a t i v e w i t h t h e minimum e x p e c t e d l o s s i s t o b e c h o s e n . From t h e p a y o f f t a b l e i t c a n b e s e e n

D i f f e r e n t i a l S e n s i t i v i t y

( i = 1 )

F i n i t e S e n s i t i v i t y ( i = 2 )

P r o b a b i l i t y

P1 1 P12 '1 3 '1 4

L o s s

0 1 0 L2

..-

So > H I S i > A 5 > A n S i < A

0

so

^{< A ~ < A s ~}

So

^{< A ~ s A ~ >}

I

i

P r o b a b i l i t y

P21 P22 '23

0 Loss

0

I

¹

0 L2

t h a t t h e d e c i s i v e e v e n t f o r t h i s c h o i c e i s t h e second o n e , s o t h a t we must compare P w i t h P 2 2 :

1 2 We have

h e c a u s e f o r any two e v e n t s a and b , we have

By d e f i n i t i o n o f S2 we have P ( S O > A1

s 2

^{> A ) = 1 .} F u r t h e r m o r e , we c a n u n d e r v e r y weak a s s u m p t i o n s p r o v e t h a t

P ( S 2 > A ) - > P ( S 1 > A ) : we c a n s e t

S1 ⁼

I B I

and S2 =

I B + R I ,

where B and R a r e random v a r i a b l e s which a r e r e a s o n a b l y assumed t o be i n d e p e n d e n t . (One c a n t h i n k of B and R a s t h e f i r s t o r d e r t e r m and t h e r e s t of a T a y l o r e x p a n s i o n . ) L e t v B ( b )

,

v R ( r )

,

and v

z

( z ) be t h e f r e q u e n c y f u n c t i o n s of B , R and B

+

^{R ,} r e s p e c t i v e l y , and l e t u s assume t h a t f o r b o t h v B and v R t h e f o l l o w i n g c o n d i - t i o n s a r e f u l f i l l e d :

With t h e s e weak a s s u m p t i o n s we have

v ( Z

-

r ) v (r) d r d z

= 2 1

1

^{B R}

A ^-03

4 - 1

+A ^{v B ( z}

^-

^{6 ) d z w i t h}

^-

^{< 6} - ^{< 0} -A

The l a s t i n t e g r a l i s maximal f o r

5

= 0. I n t h i s c a s e t h e i n t e - g r a l i s e q u a l t o P (

I

^B

I

^{< A )}

.

T h e r e f o r e w e have

This means that for the problem at hand the finite sensitivity analysis ought to be preferred to the differential sensitivity analysis.

The same arguments can be applied to the case where

So

^{= Max (x)}

-

^{Min (x)}

,

P P

111. The Problem of Owtimallv Sensitive Control

Another problem, for which the differential sensitivity turns out to be preferable, is the design of a system feed- back which corrects an optimal open loop control Go such that the performance index

remains as close as possible to the minimum if a system para- meter deviates from its nominal value. Let us again assume

a system with only one state variable, one control variable, and one parameter. Then a first order approximation to the solution of the problem is to add the feedback

to the nominal optimal control

GO

[2]. Here xo is the nominal motion of the state variable. Another possibility

would be to feedback

where Ap could be, for instance, the variance of p. By simple 0

geometrical reasoning, one can prove that using expression (8) with finite sensitivities gives a greater expected value of

the shortest distance between any point {GO

+

^Au2,

XI

and the curve G(x) if the parameter values are symmetrically distributed around the nominal value. Figure 3 illustrates for a certain time the relationship between the two approximations and the optimal control curve, for which a second order approximation was chosen. In general, greater distance from the optimal con- trol means a greater value of the performance index. Therefore feedback (7) should be preferred in order to minimize expected costs.

I V . Possible Extensions

A possible generalization of the problem in section 2 is to ask whether expression (2) or (4) is more appropriate for estimating the probability that a parameter value is drawn for which the state variable exceeds a prescribed limit. Though it has not yet been proved, some numerical experiments indicate that, for this problem also, finite sensitivity is to be pre- ferred. Figure 4 shows such an experimental result: for a great number of functions f('p), with p normally distributed around the nominal value, those probabilities are computed and compared with' the estimates according to (2) (Fig. 4a) and (4) (Fig. 4b). (The estimates have been computed with s l 0 ( p

-

^po)

and S2* (p

-

po) as approximations to f(p) . ) Ap in (4)was chosen to be equal to the variance of the parameter. The functions f were 5th order polynomials with coefficients randomly selected from normal distributions with zero mean and the variance de-

c r e a s i n g a s t h e o r d e r o f t h e p o l y n o m i a l t e r m i n c r e a s e s . The c r i t i c a l c a s e s a r e a g a i n t h o s e i n which t h e p r o b a b i l i t y o f ex- c e e d i n g t h e s t a n d a r d i s g r e a t e r t h a n a c e r t a i n d e c i s i o n t h r e s h - o l d , s a y 5 % , w h i l e t h e s e n s i t i v i t y e s t i m a t e r e m a i n s below

t h i s t h r e s h o l d . The f i g u r e c l e a r l y shows t h a t t h e r e a r e more c a s e s i n t h e c r i t i c a l ( s h a d e d ) a r e a w i t h t h e d i f f e r e n t i a l s e n - s i t i v i t y ( F i g . 4 a ) t h a n w i t h t h e f i n i t e s e n s i t i v i t y ( F i g . 4 b ) .

There a r e c e r t a i n l y s t i l l more problems s o l v e d by means of s e n - s i t i v i t y f u n c t i o n s f o r which one c a n p r o v e t h a t o n e o f t h e two

methods i s i n g e n e r a l " b e t t e r " ( i n t h e s e n s e d e s c r i b e d i n t h e i n t r o d u c t i o n ) . I t would a l s o be of i n t e r e s t t o compare b o t h methods f o r h i g h e r o r d e r s e n s i t i v i t i e s . ÿ his means, f o r i n - s t a n c e , f i n d i n g o u t w h e t h e r it i s b e t t e r i n a c e r t a i n s i t u a t i o n t o know t h e v a l u e s o f a f u n c t i o n a t t h r e e p o i n t s o r t o have a t one p o i n t t h e f i r s t t h r e e t e r m s o f t h e T a y l o r e x p a n s i o n o f t h a t f u n c t i o n .

Figure 1. Nominal solutions xi (- ) and sensitivity functions

( - - - ) ^f

'li or System ( 5 ) .

Figure 2. Solution of System ( 5 ) with p = 1.05 po.

Figure 3. Illustration of the relationship between optimal feedback ; and t h e approximations (7) and (8) of

;.

(It is Dist (P~,;) - < 5 Dist (P2,;) if x < x0.)

F i g u r e 4 a . P r o b a b i l i t y o f e x c e e d i n g a maximum p e r m i s s i b l e d e v i a t i o n f r o m t h e f u n c t i o n v a l u e £ ( p o l f o r a l a r g e number o f f u n c t i o n s f ( p ) , p n o r m a l l y d i s t r i b u t e d a r o u n d po. A c t u a l p r o b a b i l i t y Po c o m p a r e d w i t h t h e d i f f e r e n t i a l s e n s i t i v i t y e s t i m a t e P1. The numbers i n d i c a t e how many p o i n t s (PO,P1) f e l l i n t o t h e same i n t e r v a l .

F i g u r e 4b. A s i n F i g u r e 4 a , b u t Po compared w i t h t h e f i n i t e s e n s i t i v i t y e s t i m a t e P2.

R e f e r e n c e s

[l] C r u z , J . J .

,

^Ed. S y s t e m S e n s i t i v i t y A n a l y s i s . S t r o u d s b u r g , P a . , Dowden, H u t c h i n s o n & R o s s , I n c . , 1 9 7 3 .

[2] K o k o t v i c , P . , J . B . C r u z , J . H e l l e r , a n d P . S a n n u t i .

" S y n t h e s i s o f O p t i m a l l y S e n s i t i v e S y s t e m s . " P r o c . IEEE, 5 6 , 8 ( 1 9 6 8 ) t 1 3 1 8 - 1 3 2 4 .

-

-