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A Survey of Reduced Order Sensitivity Models
A SURVEY OF REDUCED ORDER SENSITIVIR MODELS
William R . Perkins Coordinated Science Laboratory and Department of E l e c t r i c a l Engineering U n i v e r s i t y of I l l i n o i s Urbana, I l l i n o i s 61801 USA
ABSTRACT Techniques f o r reducing t h e o r d e r of s e n s i t i v i t y models a r e surveyed. Work on s e n s i t i v i t y p o i n t s methods a r e d e s c r i b e d . Methods of o b t a i n i n g reduced o r d e r models u s i n g s t a t e r e p r e s e n t a t i o n s of systems a r e p r e s e n t e d . Some r e c e n t r e s u l t s on c o n t r o l l a b i l i t y and o b s e r v a b i l i t y of s e n s i t i v i t y models a r e p r e s e n t e d . INTRODUCTION Trajectory s e n s i t i v i t y functions, t h a t is, p a r t i a l d e r i v a t i v e s of s t a t e v a r i a b l e s w i t h r e s p e c t t o system parameters, a r e widely used i n c o n t r o l system a n a l y s i s and d e s i g n . Three comprehensive s u r v e y s , by Kokotovic and Rutman [ I ] , by Ngo [ 2 ] , and by S o b r a l [ 3 ] , a s w e l l a s t h e r e c e n t monograph e d i t e d by Cruz [ 4 ] , r e f e r e n c e many of t h e s e u s e s . Various schemes have been proposed f o r g e n e r a t i n g t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s . One important scheme i s t h e s e n s i t i v i t y model method. I n t h i s method the s e n s i t i v i t y f u n c t i o n s a r e d e f i n e d i m p l i c i t l y by t h e s e n s i t i v i t y e q u a t i o n s , o r p e r t u r b a t i o n e q u a t i o n s . These e q u a t i o n s may be obtained by d i f f e r e n t i a t i n g t h e o r i g i n a l system e q u a t i o n s with r e s p e c t t o t h e system parameters, provided a p p r o p r i a t e d i f f e r e n t i a b i l i t y c o n d i t i o n s a r e s a t i s f i e d [ 5 , 6 ] . A dynamic system d e s c r i b e d by t h e s e s e n s i t i v i t y e q u a t i o n s i s c a l l e d a s e n s i t i v i t y model. Use of t h e s e models avoids t h e need f o r forming f i n i t e d i f f e r e n c e s , a n u m e r i c a l l y hazardous endeavor. Moreover, t h e models s t r u c t u r a l l y resemble t h e o r i g i n a l system, and can be obtained o f t e n by simple manipulations of t h e o r i g i n a l system block diagram. However, i f t h i s approach i s used d i r e c t l y f o r s i m u l a t i o n by i d e n t i f y i n g each p a r t i a l d e r i v a t i v e a s a s t a t e v a r i a b l e o f t h e s e n s i t i v i t y model, p models, where p i s t h e number of parameters w i l l be needed, i n a d d i t i o n t o t h e o r i g i n a l system o r system model. Each of t h e s e models i s of o r d e r n. Thus a composite dynamic system of o r d e r (p+l)n w i l l be r e q u i r e d . This paper surveys techniques a v a i l a b l e f o r reducing t h i s o r d e r by c o n s i d e r i n g the v a r i a t i o n a l equations a s input-output rather than s t a t e e q u a t i o n s . Such r e d u c t i o n s r e s u l t i n important savings i n time and equipment needed f o r computer s i m u l a t i o n of t h e models. Formal
techniques f o r o b t a i n i n g minimal r e a l i z a t i o n s from i n p u t - o u t p u t d e s c r i p t i o n s , such a s [ 7 ] , could be used. However, t h e s e techniques r e q u i r e e x t e n s i v e computation, e s p e c i a l l y i n comparison t o t h e techniques t o be d i s c u s s e d h e r e . Repeated computation of minimal r e a l i z a t i o n s f o r d i f f e r e n t parameter v a l u e s , a s i n i t e r a t i v e o p t i m i z a t i o n schemes, f o r example, should be avoided. Thus t h e schemes t o be surveyed h e r e , while n o t of minimal o r d e r i n g e n e r a l , provide a simple method of g a i n i n g s u b s t a n t i a l r e d u c t i o n s from d i r e c t s i m u l a t i o n of t h e v a r i a t i o n s 1 e q u a t i o n s , b u t w i t h o u t t h e computational complexity of o b t a i n i n g a minimal r e a l i z a t i o n . Bihovski was t h e f i r s t t o s t r e s s t h e advantages of s e n s i t i v i t y models f o r g e n e r a t i n g s e n s i t i v i t y f u n c t i o n s f o r networks [ 8 ] . The pioneering a p p l i c a t i o n s of s e n s i t i v i t y models t o c o n t r o l problems i s due t o Kokotovic [ 9 ] . I n p a r t i c u l a r , Kokotovic [ 8 ] , Rutman [ l o ] , and o t h e r s [ l l ] have developed t h e s e n s i t i v i t y p o i n t s method. I n t h i s approach t h e o u t p u t s e n s i t i v i t y f u n c t i o n s f o r c e r t a i n s p e c i f i c system s t r u c t u r e s can be obtained a t c e r t a i n e a s i l y located " s e n s i t i v i t y p o i n t s " on a s i n g l e s e n s i t i v i t y model t h a t i s e s s e n t i a l l y a r e p l i c a of t h e system model. Thus o n l y one model i s needed, n o t p models a s with d i r e c t d i f f e r e n t i a t i o n of t h e system e q u a t i o n s . A p a r t i c u l a r l y impressive a p p l i c a t i o n of s e n s i t i v i t y p o i n t s i n system d e s i g n i s i n [ 1 2 ] . Recent developments i n t h e s e n s i t i v i t y p o i n t s approach i n c l u d e pseudosensi t i v i t y [13,14] and t h e " t h r e e - p o i n t s method" [ l o ]
.
Much c u r r e n t work on reduced o r d e r s e n s i t i v i t y models u t i l i z e s t h e s t a t e d e s c r i p t i o n of systems I n t h e following two s e c t i o n s , t h e s t a t u s of reduced o r d e r s e n s i t i v i t y model techniques f o r l i n e a r time i n v a r i a n t continuous time systems i s reviewed, with emphasis on s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models. REDUCED ORDER STATE MODELS I t i s r e a d i l y shown t h a t t h e phase canonic system representation has s p e c i a l s e n s i t i v i t y point p r o p e r t i e s [ I S ] . Moreover, i t i s w e l l known t h a t any s i n g l e i n p u t c o n t r o l l a b l e l i n e a r time i n v a r i a n t system can be put i n t o t h i s phase
canonic form. E x p l o i t i n g t h e s e n o t i o n s , Wilkie and Perkins [16] showed t h a t , f o r a s i n g l e i n p u t system, a l l s e n s i t i v i t y f u n c t i o n s of a l l s t a t e v a r i a b l e s w i t h r e s p e c t t o any number of system parameters could be g e n e r a t e d from a s i n g l e s e n s i t i v i t y model o f o r d e r n , t o g e t h e r w i t h a system model o r t h e a c t u a l o r i g i n a l system itself. A g e n e r a l i z a t i o n of t h i s approach was developed by Wilkie and P e r k i n s f o r m u l t i - i n p u t normal systems [ 1 7 ] , i n c l u d i n g systems having nonzero i n i t i a l c o n d i t i o n s . This approach r e q u i r e s (2m-1) models of o r d e r n f o r a system having m i n p u t s . More r e c e n t l y , Neuman and Sood, employing frequency domain (Laplace) system r e p r e s e n t a t i o n s , have s t u d i e d both s i n g l e and m u l t i - i n p u t systems [ 1 8 ] . Among o t h e r r e s u l t s , t h e y have o b t a i n e d a scheme r e q u i r i n g a t most m models of o r d e r n f o r a n mi n p u t n t h o r d e r l i n e a r time i n v a r i a n t c o n t r o l l a b l e ( n o t n e c e s s a r i l y normal) system. A s i m i l a r r e s u l t was developed a t about t h e same time by Denery [19]. Denery e x p l i c i t l y i n c l u d e s i n i t i a l c o n d i t i o n s . Moreover, t h e o r i g i n a l m u l t i - i n p u t system need o n l y be c y c l i c , n o t n e c e s s a r i l y c o n t r o l l a b l e . Denery's r e s u l t s f o l l o w from two i n t e r e s t i n g lemmas:
i) ii) iii)
(A,b) i s a c o n t r o l l a b l e p a i r bu, xo = 0 i = A x gu, zo = 0 ,
jc = Ax
+ +
t h e n x and z a r e r e l a t e d by a l i n e a r transformat i o n , z ( t ) = Mx(t). Lemma 2 : i) ii) iii)
If (A,b) i s a c o n t r o l l a b l e p a i r = Ax, xo = b i = Aw, wo = g ,
i
t h e n x and w a r e r e l a t e d by a l i n e a r transformation, w(t) = Fx(t). The t r a n s f o r m a t i o n s M and F a r e e a s i l y o b t a i n e d from t h e c o n t r o l l a b i l i t y m a t r i x f o r t h e p a i r (A,b). See Denery f o r d e t a i l s [ 1 9 ] . A p p l i c a t i o n of t h e s e lemmas t o t h e m u l t i - i n p u t v e r s i o n of ( 2 ) l e a d s immediately t o : Theorem [Deneryl. A l l t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s w i t h r e s p e c t t o a l l parameters of a m-input c y c l i c system of o r d e r n can be g e n e r a t e d by a t most (m+2) s e n s i t i v i t y models of o r d e r n . These r e s u l t s g i v e a n upper bound on t h e o r d e r of s e n s i t i v i t y models needed. However, t h e models so produced a r e n o t y e t of minimal o r d e r , i n g e n e r a l . Extension of t h e s e i d e a s t o o t h e r c l a s s e s of systems may be found i n [20,21]. SOME STRUCTURAL PROPERTIES OF SENSITIVITY MODELS I f a s e n s i t i v i t y model i s n o t of minimal o r d e r , i t w i l l n o t be both c o n t r o l l a b l e and observable. S e v e r a l i n v e s t i g a t o r s have s t u d i e d t h e
c o n t r o l l a b i l i t y and o b s e r v a b i l i t y of v a r i o u s s e n s i t i v i t y model s t r u c t u r e s . For example, i n [22] t h e s i n g l e - i n p u t reduced o r d e r model of [16] i s examined. I t i s shown t h a t t h e composite system c o n s i s t i n g of t h e o r i g i n a l system t o g e t h e r w i t h t h e s e n s i t i v i t y model i s always completely s t a t e c o n t r o l l a b l e , independent of t h e number of parameters a p p e a r i n g , provided t h e o r i g i n a l system i s completely s t a t e c o n t r o l l a b l e . The same c o n c l u s i o n was o b t a i n e d independently by Guardabassi , L o c a t e l l i , and R i n a l d i [23] . The o b s e r v a b i l i t y of t h e composite system depends on how t h e parameters e n t e r t h e system d e s c r i p t i o n . I f t h e parameters appear i n b o n l y , and n o t i n A , t h e n no dynamic s e n s i t i v i t y models a r e r e q u i r e d a t a l l . The t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s can be o b t a i n e d from t h e o r i g i n a l system model a l o n e . Thus t h e composite system i s n o t minimal, and t h u s cannot be o b s e r v a b l e . F i n a l l y , i t i s shown i n [22] t h a t f o r t h e important c a s e when t h e parameters a r e s t a t e feedback g a i n s , t h e composite system i s completely o b s e r v a b l e , and thus i s a minimal o r d e r system f o r g e n e r a t i n g t h e s e n s i t i v i t y functions. Neuman and Sood have a l s o s t u d i e d s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models, and have o b t a i n e d s e v e r a l i n t e r e s t i n g r e s u l t s [24]. For example, they have o b t a i n e d upper and lower bounds on t h e minimum s e n s i t i v i t y model o r d e r . These bounds depend on t h e system o r d e r , t h e number o f i n p u t s , and t h e number of e s s e n t i a l parameters. I n a s e r i e s of papers Guardabassi, L o c a t e l l i , and R i n a l d i have explored v a r i o u s q u e s t i o n s concerning t h e s t r u c t u r e of s e n s i t i v i t y models. For example, s e e [25]. They have devoted s p e c i a l a t t e n t i o n t o t h e i n v e s t i g a t i o n of t h e s e n s i t i v i t y of t e r m i n a l c o n d i t i o n s t o parameter v a r i a t i o n s . A r e c e n t s t u d y of t h i s s o r t u s i n g s e n s i t i v i t y models i s [26]. The concept of p a r t i a l c o n t r o l l a b i l i t y i s i n t r o d u c e d t o formulate a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e t e r m i n a l s t a t e of a l i n e a r time i n v a r i a n t system t o be i n s e n s i t i v e , t o f i r s t o r d e r , w i t h r e s p e c t t o system parameter v a r i a t i o n s . A f u r t h e r development along t h e s e l i n e s , r e l a t i n g t o d e s i g n of s t a t e feedback s t r u c t u r e s t o o b t a i n terminal i n s e n s i t i v i t y , i s reported i n [27]. CONCLUSIONS Some r e s u l t s concerning reduced o r d e r s e n s i t i v i t y S e n s i t i v i t y p o i n t s methods models a r e reviewed. a r e d i s c u s s e d b r i e f l y . Recently developed t e c h n i q u e s f o r o b t a i n i n g reduced o r d e r s e n s i t i v i t y models from system s t a t e e q u a t i o n s a r e d e s c r i b e d . I n p a r t i c u l a r , a r e s u l t due t o Denery i s given. F i n a l l y , c u r r e n t work on s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models ( c o n t r o l l a b i l i t y , observab i l i t y ) i s d e s c r i b e d . These s t r u c t u r a l p r o p e r t i e s a r e r e l a t e d t o t h e o r d e r of t h e models, a s a nonminimal o r d e r model cannot be both c o n t r o l l a b l e and o b s e r v a b l e .
ACKNOWLEDGMENT This work was supported in part by the Joint Services Electronics Program (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-67-C-0199,and in part by the U. S. Air Force under Grant AFOSR-68-1579D REFERENCES 1.
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Automatic Control, Vol. AC-15, No. 3, pp. 382-384, June 1970. 24. C. P. Neuman and A. K. Sood, "Sensitivity Functions for Multi-input Linear TimeInvariant Systems - 11. Minimal-Order Models,'' Int. J. Control, Vol. 15, No. 3, 1972, pp. 451-463. 25. G. Guardabassi, A. Locatelli, and S. Rinaldi, "Structural Uncontrollability of Sensitivity Systems ,'I Automatica, Vol. 5, No. 3, May 1969, pp. 297-301. 26. U. Bertele and G. Guardabassi, "Can the Terminal Condition of Time-Invariant Linear Control Systems Be Made Immune Against Small Parameter Variations? ," IEEE Trans. on Automatic Control, Vol. AC-16, No. 5, pp. 460462, Oct. 1971. 27, G. Fronza and G. Guardabassi, "A Note on Terminal Insensitivity," IEEE Trans. on Automatic Control, Vol. AC-17, No. 4, pp. 559560, Aug. 1972.
William R . Perkins Coordinated Science Laboratory and Department of E l e c t r i c a l Engineering U n i v e r s i t y of I l l i n o i s Urbana, I l l i n o i s 61801 USA
ABSTRACT Techniques f o r reducing t h e o r d e r of s e n s i t i v i t y models a r e surveyed. Work on s e n s i t i v i t y p o i n t s methods a r e d e s c r i b e d . Methods of o b t a i n i n g reduced o r d e r models u s i n g s t a t e r e p r e s e n t a t i o n s of systems a r e p r e s e n t e d . Some r e c e n t r e s u l t s on c o n t r o l l a b i l i t y and o b s e r v a b i l i t y of s e n s i t i v i t y models a r e p r e s e n t e d . INTRODUCTION Trajectory s e n s i t i v i t y functions, t h a t is, p a r t i a l d e r i v a t i v e s of s t a t e v a r i a b l e s w i t h r e s p e c t t o system parameters, a r e widely used i n c o n t r o l system a n a l y s i s and d e s i g n . Three comprehensive s u r v e y s , by Kokotovic and Rutman [ I ] , by Ngo [ 2 ] , and by S o b r a l [ 3 ] , a s w e l l a s t h e r e c e n t monograph e d i t e d by Cruz [ 4 ] , r e f e r e n c e many of t h e s e u s e s . Various schemes have been proposed f o r g e n e r a t i n g t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s . One important scheme i s t h e s e n s i t i v i t y model method. I n t h i s method the s e n s i t i v i t y f u n c t i o n s a r e d e f i n e d i m p l i c i t l y by t h e s e n s i t i v i t y e q u a t i o n s , o r p e r t u r b a t i o n e q u a t i o n s . These e q u a t i o n s may be obtained by d i f f e r e n t i a t i n g t h e o r i g i n a l system e q u a t i o n s with r e s p e c t t o t h e system parameters, provided a p p r o p r i a t e d i f f e r e n t i a b i l i t y c o n d i t i o n s a r e s a t i s f i e d [ 5 , 6 ] . A dynamic system d e s c r i b e d by t h e s e s e n s i t i v i t y e q u a t i o n s i s c a l l e d a s e n s i t i v i t y model. Use of t h e s e models avoids t h e need f o r forming f i n i t e d i f f e r e n c e s , a n u m e r i c a l l y hazardous endeavor. Moreover, t h e models s t r u c t u r a l l y resemble t h e o r i g i n a l system, and can be obtained o f t e n by simple manipulations of t h e o r i g i n a l system block diagram. However, i f t h i s approach i s used d i r e c t l y f o r s i m u l a t i o n by i d e n t i f y i n g each p a r t i a l d e r i v a t i v e a s a s t a t e v a r i a b l e o f t h e s e n s i t i v i t y model, p models, where p i s t h e number of parameters w i l l be needed, i n a d d i t i o n t o t h e o r i g i n a l system o r system model. Each of t h e s e models i s of o r d e r n. Thus a composite dynamic system of o r d e r (p+l)n w i l l be r e q u i r e d . This paper surveys techniques a v a i l a b l e f o r reducing t h i s o r d e r by c o n s i d e r i n g the v a r i a t i o n a l equations a s input-output rather than s t a t e e q u a t i o n s . Such r e d u c t i o n s r e s u l t i n important savings i n time and equipment needed f o r computer s i m u l a t i o n of t h e models. Formal
techniques f o r o b t a i n i n g minimal r e a l i z a t i o n s from i n p u t - o u t p u t d e s c r i p t i o n s , such a s [ 7 ] , could be used. However, t h e s e techniques r e q u i r e e x t e n s i v e computation, e s p e c i a l l y i n comparison t o t h e techniques t o be d i s c u s s e d h e r e . Repeated computation of minimal r e a l i z a t i o n s f o r d i f f e r e n t parameter v a l u e s , a s i n i t e r a t i v e o p t i m i z a t i o n schemes, f o r example, should be avoided. Thus t h e schemes t o be surveyed h e r e , while n o t of minimal o r d e r i n g e n e r a l , provide a simple method of g a i n i n g s u b s t a n t i a l r e d u c t i o n s from d i r e c t s i m u l a t i o n of t h e v a r i a t i o n s 1 e q u a t i o n s , b u t w i t h o u t t h e computational complexity of o b t a i n i n g a minimal r e a l i z a t i o n . Bihovski was t h e f i r s t t o s t r e s s t h e advantages of s e n s i t i v i t y models f o r g e n e r a t i n g s e n s i t i v i t y f u n c t i o n s f o r networks [ 8 ] . The pioneering a p p l i c a t i o n s of s e n s i t i v i t y models t o c o n t r o l problems i s due t o Kokotovic [ 9 ] . I n p a r t i c u l a r , Kokotovic [ 8 ] , Rutman [ l o ] , and o t h e r s [ l l ] have developed t h e s e n s i t i v i t y p o i n t s method. I n t h i s approach t h e o u t p u t s e n s i t i v i t y f u n c t i o n s f o r c e r t a i n s p e c i f i c system s t r u c t u r e s can be obtained a t c e r t a i n e a s i l y located " s e n s i t i v i t y p o i n t s " on a s i n g l e s e n s i t i v i t y model t h a t i s e s s e n t i a l l y a r e p l i c a of t h e system model. Thus o n l y one model i s needed, n o t p models a s with d i r e c t d i f f e r e n t i a t i o n of t h e system e q u a t i o n s . A p a r t i c u l a r l y impressive a p p l i c a t i o n of s e n s i t i v i t y p o i n t s i n system d e s i g n i s i n [ 1 2 ] . Recent developments i n t h e s e n s i t i v i t y p o i n t s approach i n c l u d e pseudosensi t i v i t y [13,14] and t h e " t h r e e - p o i n t s method" [ l o ]
.
Much c u r r e n t work on reduced o r d e r s e n s i t i v i t y models u t i l i z e s t h e s t a t e d e s c r i p t i o n of systems I n t h e following two s e c t i o n s , t h e s t a t u s of reduced o r d e r s e n s i t i v i t y model techniques f o r l i n e a r time i n v a r i a n t continuous time systems i s reviewed, with emphasis on s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models. REDUCED ORDER STATE MODELS I t i s r e a d i l y shown t h a t t h e phase canonic system representation has s p e c i a l s e n s i t i v i t y point p r o p e r t i e s [ I S ] . Moreover, i t i s w e l l known t h a t any s i n g l e i n p u t c o n t r o l l a b l e l i n e a r time i n v a r i a n t system can be put i n t o t h i s phase
canonic form. E x p l o i t i n g t h e s e n o t i o n s , Wilkie and Perkins [16] showed t h a t , f o r a s i n g l e i n p u t system, a l l s e n s i t i v i t y f u n c t i o n s of a l l s t a t e v a r i a b l e s w i t h r e s p e c t t o any number of system parameters could be g e n e r a t e d from a s i n g l e s e n s i t i v i t y model o f o r d e r n , t o g e t h e r w i t h a system model o r t h e a c t u a l o r i g i n a l system itself. A g e n e r a l i z a t i o n of t h i s approach was developed by Wilkie and P e r k i n s f o r m u l t i - i n p u t normal systems [ 1 7 ] , i n c l u d i n g systems having nonzero i n i t i a l c o n d i t i o n s . This approach r e q u i r e s (2m-1) models of o r d e r n f o r a system having m i n p u t s . More r e c e n t l y , Neuman and Sood, employing frequency domain (Laplace) system r e p r e s e n t a t i o n s , have s t u d i e d both s i n g l e and m u l t i - i n p u t systems [ 1 8 ] . Among o t h e r r e s u l t s , t h e y have o b t a i n e d a scheme r e q u i r i n g a t most m models of o r d e r n f o r a n mi n p u t n t h o r d e r l i n e a r time i n v a r i a n t c o n t r o l l a b l e ( n o t n e c e s s a r i l y normal) system. A s i m i l a r r e s u l t was developed a t about t h e same time by Denery [19]. Denery e x p l i c i t l y i n c l u d e s i n i t i a l c o n d i t i o n s . Moreover, t h e o r i g i n a l m u l t i - i n p u t system need o n l y be c y c l i c , n o t n e c e s s a r i l y c o n t r o l l a b l e . Denery's r e s u l t s f o l l o w from two i n t e r e s t i n g lemmas:
i) ii) iii)
(A,b) i s a c o n t r o l l a b l e p a i r bu, xo = 0 i = A x gu, zo = 0 ,
jc = Ax
+ +
t h e n x and z a r e r e l a t e d by a l i n e a r transformat i o n , z ( t ) = Mx(t). Lemma 2 : i) ii) iii)
If (A,b) i s a c o n t r o l l a b l e p a i r = Ax, xo = b i = Aw, wo = g ,
i
t h e n x and w a r e r e l a t e d by a l i n e a r transformation, w(t) = Fx(t). The t r a n s f o r m a t i o n s M and F a r e e a s i l y o b t a i n e d from t h e c o n t r o l l a b i l i t y m a t r i x f o r t h e p a i r (A,b). See Denery f o r d e t a i l s [ 1 9 ] . A p p l i c a t i o n of t h e s e lemmas t o t h e m u l t i - i n p u t v e r s i o n of ( 2 ) l e a d s immediately t o : Theorem [Deneryl. A l l t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s w i t h r e s p e c t t o a l l parameters of a m-input c y c l i c system of o r d e r n can be g e n e r a t e d by a t most (m+2) s e n s i t i v i t y models of o r d e r n . These r e s u l t s g i v e a n upper bound on t h e o r d e r of s e n s i t i v i t y models needed. However, t h e models so produced a r e n o t y e t of minimal o r d e r , i n g e n e r a l . Extension of t h e s e i d e a s t o o t h e r c l a s s e s of systems may be found i n [20,21]. SOME STRUCTURAL PROPERTIES OF SENSITIVITY MODELS I f a s e n s i t i v i t y model i s n o t of minimal o r d e r , i t w i l l n o t be both c o n t r o l l a b l e and observable. S e v e r a l i n v e s t i g a t o r s have s t u d i e d t h e
c o n t r o l l a b i l i t y and o b s e r v a b i l i t y of v a r i o u s s e n s i t i v i t y model s t r u c t u r e s . For example, i n [22] t h e s i n g l e - i n p u t reduced o r d e r model of [16] i s examined. I t i s shown t h a t t h e composite system c o n s i s t i n g of t h e o r i g i n a l system t o g e t h e r w i t h t h e s e n s i t i v i t y model i s always completely s t a t e c o n t r o l l a b l e , independent of t h e number of parameters a p p e a r i n g , provided t h e o r i g i n a l system i s completely s t a t e c o n t r o l l a b l e . The same c o n c l u s i o n was o b t a i n e d independently by Guardabassi , L o c a t e l l i , and R i n a l d i [23] . The o b s e r v a b i l i t y of t h e composite system depends on how t h e parameters e n t e r t h e system d e s c r i p t i o n . I f t h e parameters appear i n b o n l y , and n o t i n A , t h e n no dynamic s e n s i t i v i t y models a r e r e q u i r e d a t a l l . The t r a j e c t o r y s e n s i t i v i t y f u n c t i o n s can be o b t a i n e d from t h e o r i g i n a l system model a l o n e . Thus t h e composite system i s n o t minimal, and t h u s cannot be o b s e r v a b l e . F i n a l l y , i t i s shown i n [22] t h a t f o r t h e important c a s e when t h e parameters a r e s t a t e feedback g a i n s , t h e composite system i s completely o b s e r v a b l e , and thus i s a minimal o r d e r system f o r g e n e r a t i n g t h e s e n s i t i v i t y functions. Neuman and Sood have a l s o s t u d i e d s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models, and have o b t a i n e d s e v e r a l i n t e r e s t i n g r e s u l t s [24]. For example, they have o b t a i n e d upper and lower bounds on t h e minimum s e n s i t i v i t y model o r d e r . These bounds depend on t h e system o r d e r , t h e number o f i n p u t s , and t h e number of e s s e n t i a l parameters. I n a s e r i e s of papers Guardabassi, L o c a t e l l i , and R i n a l d i have explored v a r i o u s q u e s t i o n s concerning t h e s t r u c t u r e of s e n s i t i v i t y models. For example, s e e [25]. They have devoted s p e c i a l a t t e n t i o n t o t h e i n v e s t i g a t i o n of t h e s e n s i t i v i t y of t e r m i n a l c o n d i t i o n s t o parameter v a r i a t i o n s . A r e c e n t s t u d y of t h i s s o r t u s i n g s e n s i t i v i t y models i s [26]. The concept of p a r t i a l c o n t r o l l a b i l i t y i s i n t r o d u c e d t o formulate a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e t e r m i n a l s t a t e of a l i n e a r time i n v a r i a n t system t o be i n s e n s i t i v e , t o f i r s t o r d e r , w i t h r e s p e c t t o system parameter v a r i a t i o n s . A f u r t h e r development along t h e s e l i n e s , r e l a t i n g t o d e s i g n of s t a t e feedback s t r u c t u r e s t o o b t a i n terminal i n s e n s i t i v i t y , i s reported i n [27]. CONCLUSIONS Some r e s u l t s concerning reduced o r d e r s e n s i t i v i t y S e n s i t i v i t y p o i n t s methods models a r e reviewed. a r e d i s c u s s e d b r i e f l y . Recently developed t e c h n i q u e s f o r o b t a i n i n g reduced o r d e r s e n s i t i v i t y models from system s t a t e e q u a t i o n s a r e d e s c r i b e d . I n p a r t i c u l a r , a r e s u l t due t o Denery i s given. F i n a l l y , c u r r e n t work on s t r u c t u r a l p r o p e r t i e s of s e n s i t i v i t y models ( c o n t r o l l a b i l i t y , observab i l i t y ) i s d e s c r i b e d . These s t r u c t u r a l p r o p e r t i e s a r e r e l a t e d t o t h e o r d e r of t h e models, a s a nonminimal o r d e r model cannot be both c o n t r o l l a b l e and o b s e r v a b l e .
ACKNOWLEDGMENT This work was supported in part by the Joint Services Electronics Program (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DAAB07-67-C-0199,and in part by the U. S. Air Force under Grant AFOSR-68-1579D REFERENCES 1.
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19. D. G. Denery, "Simplification in the Computation of the Sensitivity Functions for Constant Coefficient Linear Systems,It IEEE Trans. on Automatic Control, Vol. AC-16; No. 4, ~ugust 1971, 348-350. 20. C. P. Neuman and A. K. Sood, "Sensitivity Models of Canonical Form Delay-Differential Systems," IEEE Trans. on Automatic Control, Vol. AC-16, No. 4, pp. 365-366, Aug. 1971. 21. R. K. Varshney, "Reduced-Order Sensitivity Models of Linear Systems with Applications," Report R-545, Coordinated Science Lab., Univ. of Illinois, Urbana, UILU-ENG 72-2204. 22. W. R. Perkins and D. F. Wilkie, "Controllability and Observability of Sensitivity ~odels Proc. 4th ~nnualPrinceton conference on Information Sciences and Systems, March 1970. 23. G. Guardabassi, A. Locatelli, and S Rinaldi, "On the Optimality of the Wilkie-Perkins LowOrder Sensitivity Model," IEEE Trans. on
Automatic Control, Vol. AC-15, No. 3, pp. 382-384, June 1970. 24. C. P. Neuman and A. K. Sood, "Sensitivity Functions for Multi-input Linear TimeInvariant Systems - 11. Minimal-Order Models,'' Int. J. Control, Vol. 15, No. 3, 1972, pp. 451-463. 25. G. Guardabassi, A. Locatelli, and S. Rinaldi, "Structural Uncontrollability of Sensitivity Systems ,'I Automatica, Vol. 5, No. 3, May 1969, pp. 297-301. 26. U. Bertele and G. Guardabassi, "Can the Terminal Condition of Time-Invariant Linear Control Systems Be Made Immune Against Small Parameter Variations? ," IEEE Trans. on Automatic Control, Vol. AC-16, No. 5, pp. 460462, Oct. 1971. 27, G. Fronza and G. Guardabassi, "A Note on Terminal Insensitivity," IEEE Trans. on Automatic Control, Vol. AC-17, No. 4, pp. 559560, Aug. 1972.