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UBC Theses and Dissertations

Method of estimating the region of attraction for a system with many nonlinearities. Foster, William Robert 1971

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METHOD OF ESTIMATING THE REGION OF ATTRACTION FOR A SYSTEM WITH MANY NONLINEARITIES b y WILLIAM ROBERT FOSTER B.A.Sc., U n i v e r s i t y o f A l b e r t a , 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d Research S u p e r v i s o r Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA J a n u a r y , 1 9 7 1 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l . ,..fu1f i lment o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f fc l a C~t >r \ e g \ € The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date F e b I \ <\ 1 I ABSTRACT A method o f d e t e r m i n i n g r e g i o n s o f a t t r a c t i o n f o r a s y s t e m w i t h m u l t i p l e n o n l i n e a r i t i e s i s c o n s i d e r e d i n t h i s t h e s i s . . A p p l i c a t i o n o f t h e method i n v o l v e s f i n d i n g t h e g l o b a l minimum o f a n o n c o n v e x L y a p u n o v f u n c t i o n . T h i s i s d o n e by f i n d i n g a g r a p h i c a l s o l u t i o n u s i n g L a g r a n g e m u l t i p l i e r s and t h e n a p p l y i n g t h e p r o -j e c t e d g r a d i e n t method t o d e t e r m i n e t h e e x a c t s o l u t i o n . A t h r e e m a c h i n e power s y s t e m e x a m p l e i s i n c l u d e d t o i l l u s t r a t e t h e a p p l i c a t i o n . i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENT v i i NOMENCLATURE v i i i NOTATION i x 1. INTRODUCTION 1 1.1 Statement o f the problem . ." 3 2. THE THEORY ." 4 2.1 Method 1 - E x t e n s i o n o f the t h e o r y o f .. Walker and McClamroch( 3) 4 2.2 Method 2 - E x t e n s i o n o f the t h e o r y o f .. W i l l e m s H ) 5 3. APPLICATION OF THE THEORY 8 3.1 N o n l i n e a r i t i e s w i t h the same argument .. 8 3.2 f ( 0 i s o f d i a g o n a l type 9 3.2a A p p l i c a t i o n o f Method 1 10 3.2b A p p l i c a t i o n o f Method 2 21 3.3 f < • ) not of d i a g o n a l type 25 3.4 Case when P - l does not e x i s t 25 3.5 Another t e c h n i q u e 28 4. NUMERICAL METHODS 32 4.1 S u b r o u t i n e NR (Newton-Raphson) 32 4.2 S u b r o u t i n e PG ( P r o j e c t e d G r a d i e n t ) 34 4.3 S u b r o u t i n e PARF ( P a r a b o l i c F i t ) . 35 4.4 S u b r o u t i n e CONJ (Conjugate G r a d i e n t ) ... 35 4.5 Main Program 35 5. A PRACTICAL EXAMPLE - A t h r e e machine power . system w i t h u n i f o r m damping 37 5.1 System c o n s t a n t s 37 5.2 D e f i n i t i o n o f v a r i a b l e s 37 5.3 D i f f e r e n t i a l E q u a t i o n s 38 i i i Page 5.4 D e f i n i n g Region (1.8) 39 5.5 A p p l y i n g Method 1 40 5.6 A p p l y i n g Method 2 43 6. CONCLUSIONS 46 APPENDIX A GRAPHICAL SOLUTION BY METHOD 2 FOR f ( ' ) OF DIAGONAL TYPE ." 47 APPENDIX B GRAPHICAL SOLUTION WHEN, f ( * ) IS NOT OF DIAGONAL TYPE 50 REFERENCES ..• 53 . i v L I S T OF TABLES T a b l e Page 3.1 S o l u t i o n o f Example 1 ... 12 3.2 S o l u t i o n o f Example 3 24 v L I S T OF ILLUSTRATIONS F i g u r e Page 2.1 T h e o r y b e h i n d M ethod 1 - exa m p l e f o r m = 2.. 4 2.2 T h e o r y b e h i n d M ethod 2 - a 3 - d i m e n s i o n a l . . . . r e p r e s e n t a t i o n 6 3.1 S o l u t i o n o f Example 1 . 13 3.2 C o n s t r a i n t s and c o n t o u r s o f V ( x ) f o r Exam p l e 1 14 3.3 S o l u t i o n f o r t h e c a s e m = 3 - an e x a m p l e ... 16 3.4 O b t a i n i n g t h e g r a p h o f e q u a t i o n ( 3 . 1 3 ) ^ .... 18 3.5 O b t a i n i n g t h e g r a p h o f e q u a t i o n ( 3 . 1 3 ) 2 •••• 19 3.6 O b t a i n i n g t h e g r a p h o f e q u a t i o n ( 3 . 1 3 ) 2 •••• 19 3.7 G r a p h s o f e q u a t i o n ( 3 . 1 3 ) 1 and ( 3 . 1 3 ) 2 20 3.8 S o l u t i o n o f Exa m p l e 2 20 3.9 C o n t o u r s o f c o n s t a n t V ( x ) - ex a m p l e 4 30 3.10 C o n t o u r s o f c o n s t a n t V ( x ) - ex a m p l e 4 31 4.1 B l o c k D i a g r a m o f S u b r o u t i n e NR 33 4.2 B l o c k D i a g r a m o f S u b r o u t i n e PG 34 4.3 B l o c k D i a g r a m o f t h e M a i n P r o g r a m 35 5.1 The d e t e r m i n a t i o n o f u ^ ( i = l , . . . , 6 ) 40 5.2 C o n t o u r s o f V ( x ) f o r t h e t h r e e m a c h i n e power s y s t e m e x a m p l e 44 v i ACKNOWLEDGEMENT T h i s t h e s i s was p r o d u c e d u n d e r t h e i n s p i r a t i o n and g u i d a n c e o f my s u p e r v i s o r Dr.M.S. D a v i e s . I w o u l d l i k e t o a c k n o w l e d g e comments made b y Dr. E . V . Bohn and t h e d e t a i l e d s u g g e s t i o n s p r o v i d e d b y Dr.A.C. S o u d a c k . I am a l s o i n d e p t e d t o my w i f e f o r t y p i n g t h e t h e s i s and J o h n Wong f o r h i s c a r e f u l p r o o f r e a d i n g . The work was s u p p o r t e d f i n a n c i a l l y by t h e N a t i o n a l R e s e a r c h C o u n c i l . v i i y NOMENCLATURE n - d i m e n s i o n a l s t a t e v e c t o r n x n s y s t e m m a t r i x n - d i m e n s i o n a l c o n s t a n t v e c t o r s nxm t r a n s f o r m a t i o n m a t r i c e s n x n m a t r i c e s t h e t r a n s p o s e o f c m - d i m e n s i o n a l n o n l i n e a r f u n c t i o n v e c t o r t h e i t h component o f f ( « ) ( t h e a r g u m e n t may n o t be i n c l u d e d i f i t c a n be u n d e r -s t o o d f r o m t h e c o n t e x t ) t h e n t h o r d e r i d e n t i t y m a t r i x n t h o r d e r s q u a r e m a t r i x w i t h a l l z e r o e n t r i e s s u m m a t i o n o v e r k ( t h e l i m i t s o f t h e sum a r e t o be u n d e r s t o o d f r o m t h e c o n t e x t ) NOTATION Some of the n o t a t i o n used i n t h i s t h e s i s i s not s t a n d a r d and is. not e x p l a i n e d i n the main body o f the t e x t . The f o l l o w i n g n o t a t i o n i s used i n o r d e r to make some of the e q u a t i o n s i n the t h e s i s more compact. I D-1 i c. . = C . ' P c . l j 1 J C i A j = c i , A P " l c j C i A A j = V A P ^ A ' c . b. . = c. 'b . 13 1 j E q u a t i o n ( 5 . 3 ) 2 r e f e r s to the second e q u a t i o n of the s e t of e q u a t i o n s d e f i n e d by ( 5 . 3 ) . x^ = x^ ( CT"2 , x^, f 2 ( 0~2 ^  ) i m p l i e s t h a t x^ i s a f u n c t i o n o f o ^ j X ^ and f 2 ( o ~ 2 ) i x 1 1. INTRODUCTION Many n o n l i n e a r systems are encountered which a r e l o c a l l y a s y m p t o t i c a l l y s t a b l e b ut which are not asympto-t i c a l l y s t a b l e i n the l a r g e . Thus t h e r e i s a s t a b l e e q u i l i -brium p o i n t , and a r e g i o n around i t , such t h a t f o r i n i t i a l s t a t e s w i t h i n the r e g i o n the system s t a t e w i l l approach the e q u i l i b r i u m p o i n t as time approaches i n f i n i t y . F or i n i t i a l s t a t e s o u t s i d e the r e g i o n , the system s t a t e w i l l not approach the e q u i l i b r i u m p o i n t as time becomes i n f i n i t e . T h i s r e g i o n i s c a l l e d the r e g i o n o f a t t r a c t i o n o f the system and f o r most problems i t i s d e s i r a b l e to be a b l e to make an e s t i m a t e o f the r e g i o n w i t h o u t p e r f o r m i n g the time consuming procedure o f i n t e g r a t i n g the d i f f e r e n t i a l e q u a t i o n s . T h i s t h e s i s i s con-c e r n e d w i t h a method o f o b t a i n i n g an e s t i m a t e of the r e g i o n o f a t t r a c t i o n f o r a p a r t i c u l a r c l a s s o f problems. Much work i n the p a s t has been done on the system x = Ax + b f ( c r ) [ c r = c ' x j (1) where f ( . ) i s a n o n l i n e a r f u n c t i o n . Popov has o b t a i n e d a c r i t e r i o n p r o v i d i n g s u f f i c i e n t c o n d i t i o n s f o r the a b s o l u t e s t a b i l i t y o f the n u l l s o l u t i o n o f (1.1) f o r those n o n l i n e a r f u n c t i o n s f(«) s a t i s f y i n g the c o n d i t i o n 0<f ( c r ) / c r < k , c r ^ O f ( 0 ) = 0 ( 1 . 2 ) The d e r i v a t i o n o f t h i s c r i t e r i o n d i d not i n v o l v e (2) Lyapunov f u n c t i o n methods, but i n a l a t e r paper, Kalman showed t h a t the c r i t e r i o n can be e s t a b l i s h e d u s i n g Lyapunov f u n c t i o n s o f the form c' x V(x) = x'Px + a/ f ( u ) d u (1.3) 0 ( 3) (4) Walker and McClamroch , and more r e c e n t l y Willems , made use o f these Lyapunov f u n c t i o n s to o b t a i n r e g i o n s o f a t t r a c t i o n f o r n o n l i n e a r i t i e s s a t i s f y i n g ( 1 .2 ) f o r o n l y a f i n i t e o r s e m i - i n f i n i t e range o f cr . L a t e r Anderson^ ^ extended Kalman 1s t h e o r y t o a system c o n t a i n i n g m u l t i p l e n o n l i n e a r i t i e s . C o n s i d e r the system ^ x = Ax + B f ( c r ) 1 . } (1.4) cr= C'x I where f ( * ) and 0~~ are m-vectors. Anderson gave s u f f i c i e n t c o n d i t i o n s f o r the s t a b i l i t y of ( 1 .4 ) s u b j e c t to the c o n d i -t i o n on f ( * ) t h a t f ' (cr )cr > f ' (cr )Kf (cr ) ( 1 .5 ) where K i s a n o n n e g a t i v e d e f i n i t e m a t r i x . T h i s was accomplished u s i n g a Lyapunov f u n c t i o n of the form •C'x V(x) = x'Px + 2]x / f ' ( c r ) d c r ~ (1 .6 ) 0 :-where P i s a p o s i t i v e d e f i n i t e symmetric m a t r i x and pi i s a c o n s t a n t s c a l a r such t h a t y>0. G l o b a l a s y m p t o t i c s t a b i l i t y can be checked i n p a r t i c u l a r c a s e s by examining the forms o f V(x) and V~(x). T h i s t h e s i s now w i l l attempt to extend the t h e o r y (3) (4) of Walker and McClamroch , and Willems , and o b t a i n an e s t i m a t e o f the r e g i o n of a t t r a c t i o n f o r a system w i t h m u l t i p l e n o n l i n e a r i t i e s . 3 1.1 Statement of the problem Suppose t h a t system (1.4) i s found to be g l o b a l l y a s y m p t o t i c a l l y s t a b l e f o r n o n l i n e a r i t y v e c t o r s f(«) s a t i s f y i n g some c o n d i t i o n (*) and assume t h a t t h i s was e s t a b l i s h e d by u s i n g a Lyapunov f u n c t i o n The problem i s to make an e s t i m a t e o f the r e g i o n o f a t t r a c t i o n when (*) ho l d s o n l y f o r f i n i t e o r s e m i - i n f i n i t e ranges o f the components o f C T . Thus, (*) i s t r u e o n l y f o r v a l u e s o f V ( x ) = x'Px + F(cr ) (1.7) where F ( c r ) i s a n o n l i n e a r f u n c t i o n o f the components of 0~" . 0~ = ( c r 1' , c r ) c o n t a i n e d i n a r e g i o n d e f i n e d by ( i = 1,2, m) (1 . 8 ) where e i t h e r u. or u. c o u l d be i n f i n i t e . i+m 2. THE THEORY 2.1 Method 1 - E x t e n s i o n o f the t h e o r y o f Walker and McClamroch The r e a s o n i n g behind t h i s method i s as f o l l o w s . S i n c e c r ^ = c^'x ( i = 1, ... ,m), where c^ i s the i t h column of C, (1.8) d e f i n e s a r e g i o n o f the s t a t e space w i t h i n a s e t of h y p e r p l a n e s . In t h i s r e g i o n , (1.7) must be a v a l i d Lyapunov f u n c t i o n s i n c e (*) i s s a t i s f i e d . Thus any s u r f a c e V(x) = a c o n s t a n t which i s c o m p l e t e l y c o n t a i n e d w i t h i n the r e g i o n (1.8) must e n c l o s e a su b s e t o f the r e g i o n o f a t t r a c t i o n and so i s an e s t i m a t e of the r e g i o n . The o b j e c t i v e then i s to f i n d the l a r g e s t s u r f a c e which i s w i t h i n r e g i o n ( 1 . 8 ) . r i l£i<m N o t a t i o n : L e t j = ) [ i-m m+l£i^2m and l e t =\x\ c^'x=u^} ( i = l , ... ,2m) Then f i n d = the g l o b a l minimum of V(x) such t h a t x£D^ f o r ( i = 1,2, ... ,2m) and take = minimum of £ , (i=l,...,2m)] Then the e s t i m a t e o f the r e g i o n o f a t t r a c t i o n i s R x = $ x l V ( x ) < V H ] A two d i m e n s i o n a l example w i t h m = 2 i s i l l u s t r a t e d i n F i g . 2.1 V(x) = v V(x) = V V ( x ) = V V(x) = V (D 1) The A " ( i = l , . . . , 4 ) are the p o i n t s where the g l o b a l minima o c c u r . Region (1.8) i s w i t h i n the p a r a l l e l o g r a m . In t h i s example V~M = and R 1 = { x l V(x)<V } . F i g u r e 2.1 Theory behind Method 1 - example f o r m= 2 D ( 4) 2.2 Method 2 - E x t e n s i o n o f the t h e o r y o f Willems S i n c e V\ i s the g l o b a l minimum o f V(x) s u b j e c t to the c o n s t r a i n t C j'x=u^, c^'x=u^ i s tangent t o the s u r f a c e V(x)=V\, and i n f a c t the tangency p o i n t i s the o n l y p o i n t o f i n t e r s e c t i o n o f c^'x=u^ w i t h the s u r f a c e V(x)=V^. L e t be t h i s p o i n t o f i n t e r s e c t i o n . I f A^ i s c o n t a i n e d i n the c l o s u r e o f ( 1 . 8 ) then a t V(x) = |^grad V(x) j 'x<0 S i n c e C j ' x i s tangent t o V(x)=V\, grad c.. 'x and grad V(x) are i n the same d i r e c t i o n a t A^. Thus |^grad V(x) J x < 0 i m p l i e s t h a t C j'x < 0 a t L e t L.j^  = ( x l c^'x = 0, xeD i) Then L^ s e p a r a t e s the hyperplane i n t o r e g i o n s i n which C j ' x has the same s i g n . Now take = the g l o b a l minimum of V(x) such t h a t x£L^ L e t n ± - $ x l V(x)<V i, xGD ±] S i n c e and L^ are d i s j o i n t s e t s , A ^ i s a r e g i o n i n which the s i g n o f c^'x i s the same throughout, and s i n c e c^'x < 0 a t A± and A^JT^ then c^'x < 0 over a l l j f ^ . (See F i g . 2 .2 ) Take V" M M = minimum of { V\ , ( i = l , . . . , 2m) 1 and c o n s i d e r the r e g i o n f x€ r e g i o n ( 1 . 8 ) R ? = / x l \ V ( X ) < V M M L e t wi. = $xl V ( x ) < V M M ? xeD i] Si n c e V M M < V i f o r a l l i€ J 1 2 m ^ T h e r e f o r e ( D . C O . x 1 S o l u t i o n p o i n t f o r g l o b a l minimum: X - Method 1 0 - Method 2 F i g u r e 2.2' Theory b e h i n d Method 2 - a 3-dimensional r e p r e s e n t a t i o n . (A) C r o s s - s e c t i o n through A^ p e r p e n d i c u l a r t o (B) C r o s s - s e c t i o n i n the plane o f D. Thus, i f i s not the n u l l s e t , C j ' x < 0 over and t r a j e c t o r i e s e n t e r a c r o s s the s e t ciK . A l s o V ( x)<0 on V ( x ) = w i t h i n (1.8) so t h a t a l l t r a j e c t o r i e s e n t e r 1*2, and i s t h e r e f o r e a s u b s e t o f the r e g i o n o f a t t r a c t i o n f o r the system. I t i s e v i d e n t t h a t s i n c e vjyjC V , R - ^ C F ^ J and a l a r g e r r e g i o n o f a t t r a c t i o n may be o b t a i n e d by Method 2 than by Method 1. These methods are e q u a l l y a p p l i c a b l e to systems f o r which the Lyapunov f u n c t i o n (1.7) o n l y proves the s t a b i l i t y , and not the a s y m p t o t i c s t a b i l i t y of the e q u i l i b r i u m p o i n t . In t h i s c ase t h e r e i s no " r e g i o n of a t t r a c t i o n " , but r a t h e r a r e g i o n of s t a b i l i t y . . 8 3. APPLICATION OF THE THEORY In a p p l y i n g these methods the o u t s t a n d i n g problem, i s t h a t of o b t a i n i n g the g l o b a l minimum of the n o n l i n e a r f u n c t i o n V(x) s u b j e c t to e i t h e r a l i n e a r c o n s t r a i n t as i n Method 1, or s u b j e c t to both l i n e a r and n o n l i n e a r c o n s t r a i n t s , as i n Method 2. L o c a l minima may be o b t a i n e d by computer methods, but t h e r e a r e no such methods f o r o b t a i n i n g the g l o b a l minimum f o r a nonconvex programming problem. The above problem i s i n most cases o f the nonconvex v a r i e t y , and so i t i s n e c e s s a r y to r e s o r t t o g r a p h i c a l t e c h n i q u e s i n o r d e r to determine the l o c a t i o n o f a l l f e a s i b l e l o c a l minima. The p r i n c i p a l t e c h n i q u e used i n t h i s t h e s i s i n v o l v e s the use o f Lagrange m u l t i p l i e r s i n o r d e r to s o l v e the m i n i m i z a t i o n problem. The l i m i t a t i o n of t h i s t e c h n i q u e w i l l be shown as problems o f i n c r e a s i n g d i f f i c u l t y are attempted. L a t e r , a second t e c h n i q u e , i n v o l v i n g the p l o t t i n g o f c o n t o u r s o f c o n s t a n t V(x) i n s i d e the c o n s t r a i n t s u r f a c e , w i l l be d i s c u s s e d . Numerical methods of s o l u t i o n and a p r a c t i c a l example w i l l be c o n s i d e r e d . 3.1 N o n l i n e a r i t i e s w i t h the same argument In t h i s case the arguments of the n o n l i n e a r i t i e s are a l l the same and an exact s o l u t i o n i s o b t a i n e d by (4) a p p l y i n g the t e c h n i q u e s o f Willems L e t crV=c'x, ( i = l , . . . , m ) . The problem i s to f i n d the g l o b a l , minimum o f V(x) g i v e n i n (1.7), s u b j e c t to the c o n s t r a i n t s c'x=u^ and c'x=0. S u b s t i t u t e c'x=u^ i n t o the arguments of f ( . ) and F(«) i n e q u a t i o n s (1.4) and (1.7) 9 r e s p e c t i v e l y so t h a t the n o n l i n e a r p a r t s o f these e q u a t i o n s are c o n s t a n t s . The Lagrange m u l t i p l i e r t e c h n i q u e i s a p p l i e d and the g r a d i e n t c o n d i t i o n a t the minimum i s 2Px = X c + X 2A'c (3.1) The Lagrange m u l t i p l i e r s X^ and X 2 are found by m u l t i p l y i n g e q u a t i o n (3.1) by c'P ^ and c'AP ^ and s u b s t i t u t i n g the c o n s t r a i n t c o n d i t i o n s i n t o the r e s u l t i n g e q u a t i o n s . X^ and X 2 are then s u b s t i t u t e d i n t o the above g r a d i e n t c o n d i t i o n , which i s m u l t i p l i e d through by -^x' to o b t a i n x'Px a t the s o l u t i o n p o i n t . Thus V i = x'Px + F ( u i ) ( i = 1,2) ( a t the s o l u t i o n p o i n t ) u ? c ' A P _ 1 A ' c + 2 u i c ' B f ( u i ) c ' P ~ 1 A ' c + c ' P - 1 c ( c ' B f ( u ± ) ) 2 ( c ' P ~ 1 c ) ( c ' A P _ 1 A ' c ) - ( c ' A P ~ 1 c ) 2 VMM = m i n i m u m o f { V i » V 2 \ _ and the r e g i o n of s t a b i l i t y i s R 2 = [ x l V(x)<V F ] M, c'x£(u 1, u 2 ) \ 3.2 f ( . ) i s o f d i a g o n a l type In t h i s c a s e the i t h component of 0~ i s the argument f o r the i t h component o f f ( c r ). F ( c r ) i s taken to be o f the form F ( c r ) = a ' g ( c r ) where g ( c r ) = (g (cr ),...,g (cr ) ) ' 3 ^ 3 1 1 ' '^ m m ' a i s a c o n s t a n t m-vector w i t h i t h component a^ and g i ( o - . ) = J ^ i f . ( c r ) d c r , . which l e a v e s V(x) i n the form of (1.6). The assumption i s made t h a t f i ( u ± ) = 0 (i=l,...,2m) JLU T h i s i s t r u e of the problems c o n s i d e r e d h ere, a l t h o u g h i t i s not a n e c e s s a r y assumption f o r the f o l l o w i n g t e c h -n i q u e s to work. A f i n a l assumption t h a t d e t (P) / 0 i s made. T h i s i s u s u a l l y v a l i d s i n c e P i s r e q u i r e d to be ( 5) p o s i t i v e d e f i n i t e by Anderson . O c c a s i o n a l l y , however, P i s s i n g u l a r and so t h i s case w i l l be c o n s i d e r e d i n s e c t i o n 3.4. 3.2a A p p l i c a t i o n o f Method 1 C o n s i d e r the problem o f d e t e r m i n i n g V\. D e f i n e j as i n s e c t i o n 2.1. The g r a d i e n t c o n d i t i o n a t the minimum r e q u i r e s t h a t the g r a d i e n t of V(x) and t h a t of the l i n e a r c o n s t r a i n t c.'x=u. be i n the same d i r e c t i o n . m 2 P X + k ? ! ^ ^ = ( 3 # 2 > k / j - where X i s the Lagrange m u l t i p l i e r . M u l t i p l y (3.2) through by c . 1P to get -1 m 2 V X + ^ V j ^ i c = X c j j - ( 3 - 3 ) k ^ j S u b s t i t u t i n g u^=c..'x i n t o (3.3) and s o l v i n g f o r X 2u. m c., JJ k=l JJ . k ^ j M u l t i p l y (3.2) through by 1P 1 ( r = l , . . . , m , r ^ j ) to get m 2 cr ' x + E a k c r k f k = X c r j ( r = l , . . . , m , r ^ j ) (3.5) k ^ j S u b s t i t u t e (3.4) i n t o (3.5). m 2 ° ~ r c j j + Z . a k ( c r k c j j - c r j c j k ) f k = 2 u i c r j <r=l,...,m,i*j) JC — l • ( . i . b ; 1 1 ( 3 . 7 ) The problem now i s to f i n d a l l the s o l u t i o n s to (3.6) which are i n ( 1 . 8 ) . V(x) i s c a l c u l a t e d a t each s o l u t i o n p o i n t , and V\ i s the minimum o f thes e v a l u e s . To o b t a i n an e x p r e s s i o n f o r V ( x ) , m u l t i p l y (3.2) by -|-x' , m 2 1 kfel~2 k k T h e r e f o r e m V(x) = x'Px + a ^ d K . ) + E a k g k X m °~}c = 2 U i + X n ak (9k- ~ 2 f k ) + a j 9 j ( u i k = l J J where X i s o b t a i n e d from (3.4). C o n s i d e r the s o l u t i o n o f (3.6) f o r d i f f e r e n t v a l u e s o f m. Case when m = 2 Here (3.6) i s the s i n g l e e q u a t i o n 2°"r cH - 2 u i C r j = f ( c r r ) (3.8) A A , r r where A = a ( c . c . - c c . . ) r r j j r r r j j T h i s i s s o l v e d by p l o t t i n g y = LHS and y = RHS on the same graph paper w i t h Q - ^ as the x - a x i s f o r Q — . e ( u ^ , u r + m ) . -The p o i n t s where the two graphs i n t e r s e c t g i v e the v a l u e s of'Q-f o r the s o l u t i o n o f the e q u a t i o n . Example 1 c ^  ' x c 2 ' x V(x) = x'x + s i n ( c r ) d c r + s i n ( c r ) d c r Jo J 0 where u i = u 2 = - T t> u 3 = u 4 = n 12 and /1.00\ 0. 00 1.16 \0.00' f4. 68\ 0.00 0.00 i 0 . 0 0 ' To s o l v e ( 3 . 8 ) , y = s i n (q) i s p l o t t e d f o r q e ( - T C , T t ) and the f o u r l i n e s r e p r e s e n t i n g the l e f t hand s i d e o f t h a t e q u a t i o n are p l o t t e d on the same graph. (See F i g . 3.1) L i n e a r C o n s t r a i n t L i n e p l o t t e d q a t s o l u t i o n V(x) V. . l C-^  ' X = -TX y = -. 16q - 1. -1.00 8.88 V 8.88 -2. 50 9.14 'x = " f t y = -1.49q -: 1. -.41 2. 58 V 2 = 2. 58 1 X = TC y = -. 16q + 1. 1.00 8.88 8. 88 2. 50 9.14 C 2 ' X = TC y =-1.49q + 1. . 41 2. 58 V 4 = . 2. 58 T a b l e 3.1 S o l u t i o n o f Example 1 V M = minimum of {2.58, 8.88} = 2.58 The r e g i o n o f s t a b i l i t y would be R, {xl V(x)<2.58.} Note t h a t l i n e s y = -.16q+l. and y = - . 1 6 q - l . each i n t e r s e c t s i n (q) a t t h r e e o t h e r p o i n t s f o r q_gf (~TC j TC ) and r e s u l t i n t h r e e o t h e r s o l u t i o n s . A l l of the s o l u t i o n s , the c o n s t r a i n t s , and some c o n t o u r s o f V(x) are shown i n F i g . 3.2. X minimum O maximum V = 8.90 \ V = 9.16 S o l u t i o n s w i t h i n the c l o s u r e of r e g i o n (1.8) = 6.21 V = 9.16 o V = 8.90 \ / * c-^1 x=n; V(x) = 4. 76 "1 -V(x) = 6. 21 V(x) = 2 \ !. 58 J c o n t o u r s of: V(x) S o l u t i o n s w i t h i n JV = 9.14 the c l o s u r e of 1 V = 8.88 r e g i o n (1.8) ' x - it Region R^ Region N o t a t i o n : V denotes the v a l u e of V(x) a t a p o i n t . F i g u r e 3.2 C o n s t r a i n t s and c o n t o u r s of V(x) f o r Example 1 15 Case when m Here (3.6) c o n s i s t s o f two e q u a t i o n s cr. P JJ + a A r p r r + a, A k p k k = 2u. c i PJ (p = r,k) (3.9) where A = c c . . - c . c . = A pr pr j j pj j r r p and j+k+r = m(m+l)/2, j ^ r / k ^ j E l i m i n a t e f from the f i r s t e q u a t i o n and f, from the second e q u a t i o n . cr r = °~k = a k ( A k k A r r - ( A k r ) } 2c . .A. J J k r V A r r A k k " ( A k r } } rr, u. (c -A, - c, .A ) ^  r r ^ k l r j k r k j r r + A, + c . A , k r J J k r >(3.10) A k k cr, 2c..A , J J r k r u i ^ c k j A k r ° rj Akk^ A, + c . .A, k r j j k r The graphs o f these e q u a t i o n s are p l o t t e d t o g e t h e r w i t h cr, and c r as a x i s f o r cn €(u, , u, ) and c r £(u , u . ). An r k k ' k + m r r ' r + m example of t h i s i s shown i n F i g . 3.3 where j = 1 and the e q u a t i o n s (3.10) a r e 0~2 = 2 s i n ( c r 3 ) + c r 3 + .5 .fj-3 = - s i n ( CT~2) - .5 cr, ~ -5 and u i = -TC ( i = l , 2 , 3 ) , u i = TC (i=4,5,6) Case when m = 4 Three e q u a t i o n s are o b t a i n e d . 2 r r c . . + a A f + a. A , f, + a A^ f u p J J s ps s k pk k r pr r 2u.c . (p=s,k,r) (3.11) l pj ^ ' ' where A i s d e f i n e d as i n (3.9) and j+k+r+s = m(m+l)/2, pr j V k ^ r / s ^ j . Take two e q u a t i o n s a t a time and e l i m i n a t e f from the f i r s t and second, f, from the f i r s t and t h i r d and f from the second and t h i r d to g i v e s F i g u r e 3.3 S o l u t i o n f o r t h e c a s e m= 3 - an examp le 17 2 c r k ^ s r c j j + a k ^ k k ^ s r ^ k r ^ s k ^ f k 2 c r s ^ k r c j j -A, A )f + 2u. (A c, .- A. c . ) ks s r s 1 s r k j k r s j a (A A ss kr 2 rr~ A, c . . + a (A A, - A A, ) f - ^ . . u r ks j j r r r ks r s k r r u k r s j j 2 c n A c.+ a, (A, , A - A . A. ) f. + 2u. (A. c .-A c, . ) k kk r s rk ks k I .ks r j r s k j S (3.12) <T- A . c . .+ a (A A , - A , A ) f u s r k j j s ss r k sk r s s = 2 o- A . c . . u r. sk j j + a ( A A - A ,A , ) f + 2u.(A , c .- A ,c .) r r r sk sk r k r l r k s j sk r j These t h r e e e q u a t i o n s can be s o l v e d g r a p h i c a l l y i n the f o l l o w i n g manner. For e q u a t i o n (3.12)^,y = L H S i s p l o t t e d w i t h CTV. as the x - a x i s , a n d on the same graph paper y = RHS i s a l s o p l o t t e d wither -, as an a l t e r n a t i v e x - a x i s . From t h i s a correspondence i s found between cn, and cr by com-K. S p a r i n g p o i n t s on each graph h a v i n g the same o r d i n a t e , and as a r e s u l t a graph ofcr, vs cr can be drawn. The K S same i s done f o r the o t h e r two e q u a t i o n s and graphs o f Cr vs cr and cr vs cr r e s u l t . I f cr vs cr and cr vs cr k r s r K S k r are p l o t t e d on the same graph paper then a graph o f °~s V S ®~r c a n ^ e ^ e ^ e r m i n e < ^ from t h i s . The two graphs o f <j~s v s cr^ . a r e n o w p l o t t e d t o g e t h e r and the p o i n t s o f i n t e r s e c t i o n r e p r e s e n t s o l u t i o n s to the e q u a t i o n s . T h i s method i s i l l u s t r a t e d by the f o l l o w i n g example: Example 2 C o n s i d e r the f o l l o w i n g e q u a t i o n s where j = 1 18 5 cr-3+ 2 s i n ( Q - ) = -. 5 o-2 - 2 s i n ( o-2 ) + .3 0~4+ . 5 s i n ( c r - 4 ) = C5~2+ 2sin(C7" 2) + .4 • 5 c r 4 + s i n ( c r ^ ) = . 5 c r 3 + s i n ( r j ~ 3 ) + 1 (3.13) F i g u r e s 3.4, 3.5 and 3.6 show the p l o t s o f y = LHS and y = RHS of e q u a t i o n s (3.13). F i g u r e 3.7 shows the graphs of C T " 2 vs <j- and cr" 2 vs cr^ o b t a i n e d from F i g u r e 3.4 and F i g u r e 3.5. F i g u r e 3.8 c o n t a i n s the two p l o t s o f cr~4 vs C T ^ J one o b t a i n e d from F i g u r e 3.7 and the ot h e r from F i g u r e 3.6. F i g u r e 3.4 O b t a i n i n g the graph o f e q u a t i o n (3.13)^ y -TC _L F i g u r e 3.6 O b t a i n i n g t h e graph of e q u a t i o n (3.13) Fig.ure 3.8 S o l u t i o n o f Example 2 21 The f i n a l s o l u t i o n i s c r 2 = -17, c r 3 = -.06, c r 4 = .63 To g i v e an i n d i c a t i o n o f the e r r o r i n v o l v e d the LHS-RHS i s t aken f o r each e q u a t i o n . LHS-RHS f o r ( 3 . 1 1 ^ = -.03 (3.11) (3.11) .02. -.01 m g r e a t e r than 4 In t h e s e c a s e s e q u a t i o n s such as (3.12) i n v o l v i n g two v a r i a b l e s cannot be found and as a r e s u l t the problem cannot be s o l v e d by simple g r a p h i c a l t e c h n i q u e s . 3.2b A p p l i c a t i o n o f Method 2 d f ± ( c r ± ) N o t a t i o n : L e t h. ( cr. ) = , ( i = l,...,m) There are now two c o n s t r a i n t s : c.'x = u. and c.'x = 0 .... . J 1 J (3.14) where m x = Ax + Bf(CT) = Ax + T b.f. k = l 1 The g r a d i e n t c o n d i t i o n becomes: m m 2Px + = ^1ci + A 2 [ A ' c . + ^ b . k c k h K J (3.15) (3.15) i s m u l t i p l i e d through by c^'P 1 and c^ *AP 1 m 2u.+ y a, c . f, = X.c. .+ X~ i . ,4*, k j k k 1 1 1 2 : k = l M l m L c J A i + ^ V ^ k j (3.16) k ^ j m ? , ( a k C j A k - 2 b j k ) f k = X l C i A i + X JAJ m L~jAAj + ] ^ 1 b j k C j A k h k k=l (3.16) and (3.17) are s o l v e d f o r X^ and X^ u s i n g Cramer'-s r u l e . (3.17) 22 M u l t i p l y (3.15) through by 'P 1 (r=1,...,m,r/j) m • m 2 ( T ~ r + k ? i a k C r i c f k = X l ° r J + X 2 l C j A r + jC b j k C r k h k J ( r = l , . . . , m , r / j ) k=l ( 3 . 1 8 ) k ^ j k ^ j The X^ and X^ o b t a i n e d from (3.16) and C3.17) are sub-s t i t u t e d i n t o (3.18) and e q u a t i o n s (3.18) are s o l v e d . V(x) i s c a l c u l a t e d f o r each o f the r e s u l t i n g s o l u t i o n p o i n t s and V\ i s taken to be the s m a l l e s t v a l u e o f V(x) o b t a i n e d . M u l t i p l y (3.15) through by ^-x' to get X, u . X 0 m m x'Px = + j - T b ( c r k h k - f k ) - T a k o - k f k k=l J k=l k ^ j k ^ j X, u . X 0 m m V ( x ) = + - J^bjfcCayv- f k ) + a.g.(u. ) + E ^ C g ^ c r ^ ) k ^ j k ^ j (3.19) Methods o f s o l u t i o n f o r . the cases m=2 and m=3 are c o n t a i n e d i n Appendix A. For l a r g e r v a l u e s o f m, t h r e e o r more e q u a t i o n s w i l l be o b t a i n e d h a v i n g t h r e e o r more v a r i a b l e s , and as a r e s u l t a two d i m e n s i o n a l g r a p h i c a l s o l u t i o n cannot be o b t a i n e d . P a r t i c u l a r Cases For systems o f low dimension i t may be e a s i e r t o a p p l y the g r a d i e n t c o n d i t i o n d i r e c t l y r a t h e r than use . the above e q u a t i o n s . Some systems may have c o r r e s p o n d i n g z e r o elements i n a l l c k ( k = l,...,m). For each c o r r e s -ponding zero the dimension o f the problem i s reduced by one. 23 C o n s i d e r the f o l l o w i n g example. Example 3 A = -1 0 0 0 -1 - 4 . 6 8 0 - 2 0 0 0 0 d 0 > - 3 0 b l = 0 0 - 2 - 2 - 2 V T - 3 0 0 = (1 0 1.16 0) c 1 = ( 4 . 68 0 0 0) The Lyapunov f u n c t i o n V(x) = x'x + ^ g v k=l K proves g l o b a l a s y m p t o t i c s t a b i l i t y o f the system p r o v i d e d the c o n d i t i o n o~fAo~)>0 ( i = l , 2 ) i s s a t i s f i e d . F or f ^ ( o ~ ) = s i n ( c r ) = fp^cr) a r e g i o n o f s t a b i l i t y can be found by Method 2. Si n c e t h e r e are c o r r e s p o n d i n g z e r o s i n c-^  and c 2 i n the second and f o u r t h p o s i t i o n s , the g r a d i e n t c o n d i t i o n i m p l i e s t h a t x 2=0=x 4 thus r e d u c i n g the problem to one of two dimensions. I n o r d e r to s o l v e f o r x^ and x^ t h e r e a re f o u r e q u a t i o n s . The two e q u a t i o n s r e m a i n i n g from the g r a d i e n t c o n d i t i o n a l s o c o n t a i n the unknowns X^ and X 2 and so are o f no use i n d e t e r m i n i n g and x^. T h i s l e a v e s the two c o n s t r a i n t e q u a t i o n s c.'x = 0 and c.'x = u.. J J i S i n c e crsin(cr)>0 f o r cr&(^Tc,7i;), take u ^ u 2= -n and u^= u^= TC. F o r i = 1 the two c o n s t r a i n t e q u a t i o n s are x^ + I . I 6 X 2 ~ ~ % x1 + 3.48x 3 + 4 . 6 8 : s i n ( 4 . 6 8 x 1 ) 0 x^ from the f i r s t e q u a t i o n i s s u b s t i t u t e d i n t o the second e q u a t i o n . 0.427x 1 + 2.01 = s i n ( 4 . 6 8 x 1 ) 24 T h i s e q u a t i o n i s s o l v e d the same way as (3.8) and no s o l u t i o n i s found f o r X ^ £ ( - T C , T C ) . Thus t h e r e i s no s o l u t i o n f o r . The same approach i s f o l l o w e d f o r i = 2,3 and 4. i V (x) f o r v a l i d minima V. 1 none no s o l u t i o n 2 10. 20 r 10. 20 4. 76 3 11. 48 4. 76 12.09 4. 76 4 11.48 4. 76 12.09 T a b l e 3.2 S o l u t i o n o f Example 3 VMM = m i n i m u m of i 1 0 . 2 , 4.76, 4.76] T h e r e f o r e V M M = 4.76 and the r e g i o n o f s t a b i l i t y i s _ \ V(x)<4.76 - ) K 2 " ( x 2 € ( - . 67,. 67) ) A comparison of Methods 1 and 2 i s o b t a i n e d here s i n c e the same problem was examined i n example 1 u s i n g Method 1. Region o b t a i n e d here i s c o n s i d e r a b l y l a r g e r than the r e g i o n ^ x l V(x)<2.58} found by Method 1. A s k e t c h o f both o f thes e r e g i o n s i s g i v e n i n F i g u r e 3.2. 25 3.3 f ( ' ) not o f d i a g o n a l type The approach f o l l o w e d i n o r d e r to o b t a i n a g r a p h i c a l s o l u t i o n i n t h i s case i s s i m i l a r to t h a t used i n S e c t i o n 3.2 and i s i n c l u d e d i n Appendix B. G r a p h i c a l s o l u t i o n s are o b t a i n a b l e by Method 1 and Method 2 f o r m = 2 and m = 3. F o r m>3, as b e f o r e , t h e r e i s no g r a p h i c a l s o l u t i o n . 3.4 Case when P ^ does not e x i s t In some c a s e s a Lyapunov f u n c t i o n o f the form r (1.7) may be o b t a i n e d i n which P i s o n l y p o s i t i v e semi-d e f i n i t e . In these c a s e s P ^ does not e x i s t and the e q u a t i o n s which have been developed here are no l o n g e r v a l i d a l t h o u g h the same g e n e r a l procedures s t i l l a p p ly. To s i m p l i f y the e q u a t i o n s somewhat c o n s i d e r the f o l l o w i n g . Suppose the m a t r i x P i s of rank p<n. Then by i n t e r c h a n g i n g the rows and columns of P i t w i l l be p o s s i b l e to o b t a i n a m a t r i x which has a pxp submatrix P^ i n the upper l e f t hand c o r n e r such t h a t d e t (P-j_) A 0. Then by doi n g a s e r i e s o f elementary row t r a n s f o r m a t i o n s , the l a s t n-p rows can be s e t to zer o w h i l e P^ remains i n i t s o r i g i n a l p o s i t i o n . The new m a t r i x P w i l l then be o f the form: 26 A l l the elementary row t r a n s f o r m a t i o n s i n c l u d i n g row i n t e r c h a n g e s which were done to P are a l s o done to an nxn i d e n t i t y m a t r i x t o o b t a i n a m a t r i x H. In t e r c h a n g e s are made i n the elements of x and c^ (k=l,...,m) c o r r e s -ponding to i n t e r c h a n g e s made i n the columns o f P and the r e s u l t i n g g r a d i e n t c o n d i t i o n f o r the system i s m u l t i p l i e d through by H. H,x and c k are p a r t i t i o n e d as f o l l o w s : H H H, x = x l x 2 \ / k (k=l,..•,m) where i s a pxn m a t r i x and x^ and d^ are p v e c t o r s . As an example, t h i s p rocedure i s a p p l i e d to. e q u a t i o n (3.2) to g i v e : m 2P x •+ 2 P 2 x 2 + E a k H c f = XH c k = l J m y a, H~c, f, = XH„c . k 2 k k 2 i k = l J The r e m a i n i n g e q u a t i o n s f o r t h i s problem are: d r ' x l + e r ' X 2 = c r r ( r = l , . . . , m , r ^ j ) (3.20) d j ' x l + e j ' X 2 u. (3.21) M u l t i p l y i n g ( 3 . 2 0 ) 1 by d lP ] L ( r = l , ,m) and sub-s t i t u t i n g i n (3.21) 2 r r + 2(d *P, 1 P „ - e • ) x 0 + Y a. d 'P. ±H, c, f. u r r 1 2 r 2 £-> k r 1 1 k k m -1, ^dr'Pl~ H xc. ( r = l , . . . ,m,r^j ) (3.22) 27 -1 m - l 2u. + 2(d.'P 1 P „ - e.')x„ + i d . ' P . H, c, f. i j 1 2 j 2 k j 1 1 k k k ^ j = . X d j • P 1 " 1 H 1 c j ( 3 . 2 2 ) 2 The m e q u a t i o n s (3.22) c o n t a i n m+n-p unknowns, namely, x 2 » ^ » C T r ( r = l , . . . , m , r ^ j ) . n-p o f these unknowns may be e l i m i n a t e d w i t h the use o f ( 3 . 2 0 ) 2 thus l e a v i n g m e q u a t i o n s and m unknowns. Example C o n s i d e r a system s i m i l a r to (3.2) w i t h m = 2, j = 1, and 1 0 - 1 0 . f 1 0 1 1 - 2 0 P = c = - 1 1 2 - 2 x 2 0 - 2 - 2 5 1 a 2 = a 3 = 1 The t h i r d and f o u r t h rows and columns of P are i n t e r c h a n g e d to l e a v e a 3x3 m a t r i x i n the upper l e f t hand c o r n e r w i t h a non-zero d e t e r m i n a n t . The f i r s t row i s added to the f o u r t h row and the second row s u b t r a c t e d from the f o u r t h row to g i v e P. P = 1 0 0 -1 0 1 -2 1 0 -2 5 -2 0 0 0 0 P e r f o r m i n g the same row t r a n s f o r m a t i o n s to an i d e n t i t y m a t r i x , H i s o b t a i n e d . 28 H = 1 0 0 1 0 1 0 -1 0 0 0 1 0 0 1 0 H 2 J The t h i r d and f o u r t h elements of and are i n t e r c h a n g e d c o r r e s p o n d i n g to the column i n t e r c h a n g e s i n P. 1 0 1 2 C 2 = -1 2 0 0 E q u a t i o n s (3.22) become 2o~2 + ^ x2 + 21f^ = 3X 2u.- 2x~ + 7 f 0 = 3X 1 . 2 2 and e q u a t i o n ( 3 . 2 0 ) 2 - f 2 = X (3.23) (3.24) X from (3.24) i s s u b s t i t u t e d i n t o (3.23), ( 3 . 2 3 ) 2 i s m u l t i p l i e d by 3, and (3.23)^ and (3.23)2 are added to e l i m i n a t e X 2 « The r e s u l t i s 6u1 + 2cr2 + 5 4 f 2 = 0 (3.25) (3.25) can be s o l v e d g r a p h i c a l l y f o r cr^* 3.5 Another t e c h n i q u e Another t e c h n i q u e c o n s i d e r e d i n t r y i n g to s o l v e the m i n i m i z a t i o n problem was t h a t of drawing c o n t o u r s o f c o n s t a n t V ( x ) . F o r example, c o n s i d e r f i n d i n g by Method f o r a case of f o u r dimensions and two n o n l i n e a r i t i e s . From c^'x = x, ( <j~~ , x_ , x . ) , u^ and c 2 ' x = cr^ determine (3.26) 23 and s u b s t i t u t i n g (3.26) i n t o c-^'x = 0 o b t a i n x 3 = x 3 ( c r - 2 > x 4 » f 2 ( 0~2^ ) which i s s u b s t i t u t e d i n t o (3.26) t o get = x n ( o-„,x /,,f„(cr„)), x~( cr0,x , f ~ ( cr0 )) (3.27) (3.28) I f (3.27) and (3.28) are s u b s t i t u t e d i n t o V(x) - a c o n s t a n t and cr^ i s s e t to some v a l u e w i t h i n the i n t e r v a l ( U p , u^) then t h i s e q u a t i o n becomes a q u a d r a t i c i n x^ which can be e a s i l y s o l v e d to f i n d p o i n t s which are on the c o n t o u r . Example 4 C o n s i d e r a system w i t h 1. 5 -1.0 1. 0 1.0 0.0 A ' c 1 = 1. 0 c l = 1.0 c 2 = 1.0 • 5 -2.0 1.0 """ • 5 b l l = 1. = b 12' P = I ' a l = 1. = a2 , u x = —TC U 3 = TC = u 4 » f l (cr ) = f 2 ( o - ) = s i n ( 0 ~ ) u. The r e s u l t i n g c o n t o u r s are shown i n F i g u r e 3.9 and 3.10 i n d i c a t i n g t h a t t h e r e i s o n l y one s o l u t i o n f o r O -^£(-T C , T C ) a l t h o u g h t h e r e are o t h e r s f o r c T p o u t s i d e t h i s i n t e r v a l . T h i s t e c h n i q u e i s o b v i o u s l y of v e r y l i m i t e d v a l u e as i t can o n l y be used w i t h a problem up to f o u r dimensions and h a v i n g as many as t h r e e n o n l i n e a r i t i e s . For problems of h i g h e r dimension o r g r e a t e r number of n o n l i n e a r i t i e s the c o n t o u r s cannot be r e p r e s e n t e d on a two d i m e n s i o n a l p l o t . F i g u r e 3.9 C o n t o u r s o f c o n s t a n t V ( x ) - e xamp le 4 F i g u r e 3.10 Contours o f c o n s t a n t V(x) - example 4 to I—1 32 4. NUMERICAL METHODS The t e c h n i q u e s d e s c r i b e d above are u s e f u l f o r f i n d i n g the number o f s o l u t i o n s , approximate l o c a t i o n o f the p o i n t s a s s o c i a t e d w i t h the s o l u t i o n s and approximate v a l u e s of V.(x) a s s o c i a t e d w i t h t h e s e p o i n t s . To o b t a i n more e x a c t v a l u e s , n u m e r i c a l methods must be used. The method used i n t h i s t h e s i s i s the p r o j e c t e d g r a d i e n t method. N o t a t i o n : g = ( g l t . . . , g )' = 0 are the p c o n s t r a i n t e q u a t i o n s , g. = g r a d i e n t of the i t h c o n s t r a i n t , 9 X = "gg t 'lx t \ Jpx 6x = ' 6x n \ 6x„ where 6x. i s the I g g ' P = I - g 'I ~g ^x^x ' g x gg ^x \ n/ ,change i n x^ -1 The program i s d i v i d e d i n t o a main program f o l l o w e d by s e v e r a l s u b r o u t i n e s . The s u b r o u t i n e s are as f o l l o w s : 4.1 S u b r o u t i n e NR (Newton-Raphson) T h i s s u b r o u t i n e makes x move towards the c o n s t r a i n t s u r f a c e . (See F i g u r e 4.1) I f x i s c l o s e to the c o n s t r a i n t s u r f a c e (gg<.5) then y,=l. i s s e t so t h a t q u a d r a t i c convergence i s a l l o w e d . For x f a r from the c o n s t r a i n t s u r f a c e ]i<l. i s s e t i n o r d e r to reduce the s t e p s i z e i n case the c o n s t r a i n t i s h i g h l y n o n l i n e a r and caused d i v e r g e n c e i n s t e a d . At f i r s t ]x was s e l e c t e d so t h a t ||6x|| = oc ||x|| ( t h a t i s \i =obJx1 x/g • l ^ ) but s m a l l v a l u e s of or were r e s u l t i n g i n long convergence times because of too small a step s i z e , and too l a r g e an oc r e s u l t e d i n o s c i l l a t i o n s about the c o n s t r a i n t . Because of the d i f f i c u l t y i n s e l e c t i n g « , u was set a r b i t r a r i l y as .3 and t h i s was found to g i v e reasonable convergence times. ST was i n i t i a l l y s e l e c t e d to be .01 but V(x) was found to vary s u b s t a n t i a l l y f o r changes of x w i t h i n gg< .01 . The f i n a l c h o i c e was ST = . 0 0 0 1 . 1 ^ CALL NR ^ COMPUTE g ' g x ' ^ g " 1 ' P g g = T i g ± i 1=1 SET ]l = 1. YES YES C^gg< 5 ^ Vi = . 3 3. x = x - ug 'I x gg ^x RETURN Fi g u r e 4.1 Block Diagram of Subroutine NR 4.2 S u b r o u t i n e PG ( P r o j e c t e d G r a d i e n t ) Increments o f x are made i n the n e g a t i v e g r a d i e n t d i r e c t i o n o f V(x) w i t h i n the tangent plane t o the c o n s t r a i n t s u r f a c e . (See F i g u r e 4 .2 ) S u b r o u t i n e NR i s c a l l e d each s t e p to r e t u r n the p o i n t t o the con-s t r a i n t s u r f a c e . S e l e c t i o n of K was made so t h a t ||&x|| = 61 = a c o n s t a n t , and 51 = .1 gave r e a s o n a b l e convergence times f o r the problems c o n s i d e r e d . V = 1000. V = V CALL NR COMPUTE V COMPUTE K = 61 /V V 'P V ' v x g x F i g u r e 4 .2 B l o c k Diagram o f S u b r o u t i n e PG 35 4 .3 S u b r o u t i n e PARF ( P a r a b o l i c F i t ) T h i s f o l l o w s the c a l l i n g o f s u b r o u t i n e PG and does a p a r a b o l i c f i t along the p r e v i o u s p r o j e c t e d g r a d i e n t d i r e c t i o n . I t must be f o l l o w e d by S u b r o u t i n e NR so as to r e t u r n the p o i n t to the c o n s t r a i n t s u r f a c e . 4.4 S u b r o u t i n e CONJ (Conjugate G r a d i e n t ) s e a r c h i s made i n the tangent p l a n e to the c o n s t r a i n t s u r f a c e . T h i s was found to work v e r y w e l l u n t i l the Newton-Raphson s u b r o u t i n e was used to r e t u r n x to the c o n s t r a i n t s u r f a c e , when i t was found t h a t x d i v e r g e d from the minimum. T h i s s u b r o u t i n e was not used i n t h i s t h e s i s but c o u l d g i v e v e r y f a s t convergence i f coupled w i t h a Newton-Raphson s u b r o u t i n e which would move towards the c o n s t r a i n t s u r f a c e so as to minimize V(x) as w e l l . 4 .5 Main Program When x i s near the minimum a c o n j u g a t e g r a d i e n t CALL PG F i g u r e 4 .3 B l o c k Diagram o f the Main Program 36 Example C o n s i d e r the problem from example 4. With M = 3 (See S e c t i o n 4 . 5 ) the minimum v a l u e o f V(x) was - 5 found to w i t h i n an e r r o r of l e s s than l . x l O , t h a t i s the minimum v a l u e of V(x) was = 4 .45300 and the p o i n t a s s o c i a t e d w i t h t h i s was w i t h c r 2 = - 1 . 15102 The l o c a t i o n o f t h i s p o i n t i s shown i n F i g u r e 3 .9 . 0 .26490 - 0 . 1 7 8 1 7 0.38339 - 1 . 26951 3 7 5. A PRACTICAL EXAMPLE - A t h r e e machine power system w i t h u n i f o r m damping 5.1 System constants-C o n s i d e r the example o f the t h r e e machine power (6.) system used by Willems v a l u e s . The c o n s t a n t s are i n per u n i t M = M„= M = 1 where M.= the i n e r t i a c o n s t a n t o f the i t h 1 2 3 i machine. E l ~ E 2 ~ E 3 ~ 1 E.= the i n t e r n a l v o l t a g e , a l ~ a 2 ~ a 3 " a a l l m achines). P a^= the damping c o n s t a n t (equal f o r ml P ~= 3, P 0= 0, P .= the mechanical power i n p u t . m3 ' m2 ' mi ^ G 1 = G 2 = G 3= 0 G.= the s h o r t c i r c u i t ' admittance. YL2= ^' ^ 1 3 = ^23~ 2 ' • ^ i j = ^ r a n s f e r i n d u c t a n c e between the i t h and j t h machines. 5.2 D e f i n i t i o n o f v a r i a b l e s The s t a t e v e c t o r f o r the system i s g i v e n by "2 -X = c cO 6 2 - b°2 where 6^= the a n g l e i n e l e c t r i c a l degrees between the r o t o r s h a f t and the s h a f t r u n n i n g a t synchronous speed, 5? i s the e q u i l i b r i u m v a l u e o f 5., and co.= the s l i p v e l o c i t y = d ^ i . 1 i . d t 38 D e f i n e : 5 i - 6 2 c r 2 - V 5 3 5 3 . Note t h a t CT^- ^ 3 (5 .1 ) L e t g -^ r e p r e s e n t the e q u i l i b r i u m v a l u e for cr^* For t h i s p a r t i c u l a r problem the e q u i l i b r i u m p o i n t was found to be .o 18 .8 c r 2 = 5 8 . 9 ° 4 i . r A l s o d e f i n e : e = (e-^, e 2 , e^) where e^= 0-^-CT"^ ' (4-=l,2,3) 5.3 D i f f e r e n t i a l E q u a t i o n s _ The d i f f e r e n t i a l e q u a t i o n s f o r the system are g i v e n by x = Ax - B f ( e ) e = Cx (5 .2 ) - a ] L / M 1 0 0 0 - a 2 / M 2 0 ° 3 - a l 3 ° 3 A = 0 . 0 - a 3 / M 3 — -X3 ° 3 _ X3 ° 3 B = I / M 2 • l / M , 0 0 - 1 / M , 0 - 1 / M 3 _ 1 / M 3 1 ' 1 0 -1 .0 1 0 -1 -1 3 39 1 -1 0 1 0 0 -1 1 -1 c 1 c ' c 2 f ( ' ) i s o f d i a g o n a l type w i t h f l ( E l > = E 1 E 2 ^ 1 2 L f 2 ( £ 2 ) = E l E 3 y 1 3 s i n ( e 1 + cr~2) - s i n s i n ( e 2 + cr~2) ~ s i n ( c H (5.3) f 3 ( e 3 ) = E 2 E 3 y 2 3 s i n ^ e 3 + C T " 3 ) - s i n t e r ^ ) 5.4 D e f i n i n g Region (1.8) The Lyapunov f u n c t i o n used i n t h i s problem was ( 6 ) d e r i v e d by Willems V ( x ) = x'Px + 2 3 f C x y M . f • A l XJ 0 (e)de where M 1 M 2 + M l M 3 - M ] LM 2 -M M 1 3 -M M M l 2 M 1M 2+ M2M 3 " M 2 M 3 i °3 P = ' - M lM 3 - « 2 M 3 °3 M l M 3 + M 2M 3 i °3 (5.4) ( 6 ) Willems has shown t h a t the Lyapunov f u n c t i o n (5.4) proves s t a b i l i t y (not a s y m p t o t i c ) f o r c o n d i t i o n s on the n o n l i n e a r i t i e s e .f.(e.)>0 (1=1,2,3) (5.5) T h i s c o n d i t i o n s t a t e s t h a t f ^ ( e ^ ) must be c o n t a i n e d w i t h i n the f i r s t and t h i r d quadrants, and f o r the n o n l i n e a r i t i e s c o n s i d e r e d here i t i s met o n l y over a f i n i t e range of e.-.: ( t h a t i s e.e(u.,u. ,,)). l I i ' i+3 40 C o n s i d e r the graph of y y s i n ( e j _ + c r ^ ) s i n ( 0~ ^ ) •180°- 20-° i E.= 180°- 2o-° i x y = - s i n ( crv ) F i g u r e 5.1 The d e t e r m i n a t i o n of u^ ( i = l , . . . , 6 ) I t i s e v i d e n t from F i g u r e 5.1 t h a t ( i = l , 2 , 3 ) u. = - 180° - 2cr? I . i u. = x 180°- 2 o - ° (5.6) (i=4,5,6) 5.5 A p p l y i n g Method 1 Note t h a t P = 2 -1 -1 j -1 2 -1 | °3 -1 -1 °3 2 | °3 i s o n l y p o s i t i v e s e m i d e f i n i t e w i t h rank equal to two and when the f i r s t and second rows are added to the t h i r d row, the t h i r d row becomes a l l z e r o s . -Hence a p p l y i n g S e c t i o n 3. H l = 2 -1 -1 2 1 0 0 0 0 0 0 1 0 0 0 0 - 1 0 0 0 - 1 0 0 0 H 2 = 1 1 1 1 0 0 °3 1 0. 41 E q u a t i o n (3.20)^ i m p l i e s t h a t co^= a)3* With j = '1 e q u a t i o n (3.20)2 becomes 6f 2= -X -6f 2 -6f 3 = 0 (5.7) which i m p l i e s t h a t f 2= - f 3 (5.8) E q u a t i o n s ( 5 . 3 ) 2 a n d ( 5 . 3 ) 3 are s u b s t i t u t e d i n t o e q u a t i o n (5.8) to g i v e s i n ( c r 2 ) + s i n ( o ~ 3 ) = s i n t e r ^ ) + s i n ( o ~ 3 ) (5.9) A second e q u a t i o n i s o b t a i n e d from the l i n e a r c o n s t r a i n t e^= c r ^ - c r = c^'x = u^ ( i = l , 4 ) (5.10) E q u a t i o n (5.1) i s s u b s t i t u t e d i n t o e q u a t i o n (5.10) to g i v e cr - o~_= cr + u. 2 3 1 1 ( i = l,4.) (5.11) and l e t q.= u.+ c r • ( i = l , . . . , 6 ) so t h a t e q u a t i o n (5.11) s i m p l i f i e s to C r i = o " 2 - c r 3 = q. ( i = l , 4 ) (5.12) S u b s t i t u t e e q u a t i o n (5.12) i n t o e q u a t i o n ( 5 . 9 ) . s i n ( c r + q ± ) + s i n ( c r 3 ) = s i n ( c r ° ) + s i n ( c r ° ) (5.13) Expand e q u a t i o n (5.13). s i n ( c r 3 ) ( l + c o s ( q i ) ) + cos ( 0~ 3 ) s i n ( q i ) = s i n t e r ^ ) •+ s i n t e r " ) ( i = l , 4 ) L e t Rcos(oc) = XI + c o s ( q i ) ) and Rsin(cx) = s i n ( q ^ ) (5.14) 42 Thus e q u a t i o n ( 5 . 14 ) becomes c r 3 = a r c s x n — ( s i n ( c r 2 ) + s i n ( o ~ 3 ) ) — oc ( 5.15) where R =V2(1 + c o s ( a . ) ) of = a r c t a n / s i n ( q ^ ) ( i = l , 4 ) 1 + co s ( q ^ ) The q^ are determined from e q u a t i o n ( 5 . 6 ) : q 1= - 1 9 8 . 8 ° , q 2= - 2 3 8 . 9 ° , q 3 = - 2 2 1 . 1 ° , q 4 = 1 6 1 . 2 ° , o 0 q<-= 121.1 , and q^= 138.9 . When q^ and q^ are s u b s t i t u t e d i n t o e q u a t i o n ( 5 . 1 5 ) , jj^ l s i n ( o ~ 2)+ sin ( c r ^ ) 1>1 i s found so t h a t the a r c s i n e does not e x i s t . Thus t h e r e i s no s o l u t i o n to e q u a t i o n ( 5 . 1 5 ) . S i m i l a r l y f o r j = 2 e q u a t i o n ( 3 . 2 0 ) 2 y i e l d s f 1 = f 3 from which an e q u a t i o n f o r c r ^ i s o b t a i n e d . cr~-^ = a r c s i n ^(2sin(cr-°) - s i n t e r ^ ) ) - OC ( 5 . 16 ) where R =~\J 5 + 4cos ( q ± ) ' / _ cx = a r c t a n s i n ( q ^ ) \ (i = 2 , 5 ) 2 + co s ( q ^ ) j When q 2 i s s u b s t i t u t e d i n t o e q u a t i o n ( 5 . 16 ) the s o l u t i o n i s 30.0 - 2 38.9° ( 5 . 17 ) - 2 6 8 . 9 The r e g i o n ( 1 . 8 ) i n t h i s example i s d e f i n e d by e i e ( u i , u i + 3 ) '(1 = 1 , 2 ,3 ) orcr.£ ( q i , q i + 3 ) (1 = 1 , 2 , 3 ) . 43 S i n c e c r Q = -268.9 j£(q-,,q A), t h e s o l u t i o n p o i n t (5.17) i s not v a l i d . When q,. i s s u b s t i t u t e d i n t o e q u a t i o n (5.16) the s o l u t i o n i s c r = / 3Cr.0°\ 121. l c (5.18) \ 91.1°/ S o l u t i o n (5.18) i s w i t h i n r e g i o n (1.8) and i s t h e r e f o r e v a l i d . 3 = 3 S o l v i n g e q u a t i o n ( 3 . 2 0 ) 2 f o r j = 3, f^= • ~ f 2 i s o b t a i n e d and' 0~^= a r c s m | ( 2 s i n ( c r j ) + s i n ( 0-°)) - « (5.19) where R =~\J5 + 4 c o s ( q i ) ' ' s i n t q ^ ) oc = a r c t a n 2 + c o s ( q i ) (i = 3 , 6 ) For both q^ and q^ e q u a t i o n (5.19) has no s o l u t i o n s i n c e i-l2sin(cr°) + s i n t e r ^ ) I > 1 i n b o t h c a s e s . Hence (5.18) i s the o n l y v a l i d s o l u t i o n and v"M= V(x) i s c a l c u l a t e d a t t h i s s o l u t i o n p o i n t . In t h i s case V^- 4.19 so t h a t the r e g i o n o f s t a b i l i t y (not the r e g i o n o f a t t r a c t i o n ) i s . R 1 = 1x1 V(x)<4.19} Contours o f V(x) are shown i n F i g u r e 5.2 f o r V(x) = 4.19 and V(x) = 10. A l s o i n c l u d e d on the diagram are the s i x h y p e r p l a n e s c..'x = u^ ( i = l , . . . , 6 ) . 5.6 A p p l y i n g Method 2 I t i s o n l y n e c e s s a r y t o c o n s i d e r the case i = 5 s i n c e t h i s i s the o n l y case where a v a l i d s o l u t i o n was 44 F i g u r e 5.2 C o n t o u r s o f V ( x ) f o r t h e t h r e e mach ine power s y s t em examp le 45 o b t a i n e d by Method 1. C o n s i d e r the two c o n s t r a i n t s c 2 ' x = 6 - 6 2- (6° - 6°) = u 5 (5.20) and c 2 ' x = 0 (5.21) In e v a l u a t i n g c 2 ' x, note t h a t c 2 ' B = 0 and c 2 * A = (10 -1 0 0 0 ). Thus the second c o n s t r a i n t i s c 2 ' x = a> - co3= 0 (5.22) In o r d e r f o r t h e r e t o be a s o l u t i o n o t h e r than t h a t o b t a i n e d by Method 1 i t i s n e c e s s a r y t h a t t h e s e two c o n s t r a i n t s (5.20) and (5.22) i n t e r s e c t . I t i s e v i d e n t t h a t they do not i n t e r s e c t i n t h i s case so the r e g i o n o f s t a b i l i t y o b t a i n e d by Method 2 w i l l be the same as f o r Method 1. 4b 6. CONCLUSIONS Methods 1 and 2 have extended the t h e o r y of (3) (4) Walker and McClamroch and Willems t o the case o f more than one n o n l i n e a r i t y . The i d e a s behind these methods have a l s o been i n c l u d e d i n a paper by J.L. Willems (7) and J.C. Willems b ut development o f t h i s t h e s i s was done i n d e p e n d e n t l y . G r a p h i c a l t e c h n i q u e s of s o l u t i o n and e q u a t i o n s were p r e s e n t e d f o r a system w i t h up to t h r e e n o n l i n e a r i t i e s . F or the s p e c i a l case i n which the v e c t o r f u n c t i o n f O ) was of d i a g o n a l t y p e , Method 1 was used to g i v e a s o l u t i o n f o r a system w i t h f o u r n o n l i n e a r i t i e s . I t was found t h a t f o r a system w i t h a g r e a t e r number o f n o n l i n e a r i t i e s a..graphical s o l u t i o n c o u l d not be o b t a i n e d and hence t h a t t h e r e was no way of knowing the number of f e a s i b l e s o l u t i o n s and t h e i r l o c a t i o n s i n the s t a t e space. A n u m e r i c a l t e c h n i q u e was a l s o d i s c u s s e d f o r o b t a i n i n g more a c c u r a t e s o l u t i o n s once the approximate s o l u t i o n s had been o b t a i n e d by g r a p h i c a l t e c h n i q u e s . F i n a l l y , the t e c h n i q u e s were a p p l i e d to a t h r e e machine power system example. For t h i s example a s l i g h t l y l a r g e r s t a b i l i t y (6) r e g i o n (V(x)<5.13) was o b t a i n e d by Willems u s i n g a d i f f e r e n t method. However t h i s i s not a v a l i d comparison of the methods s i n c e Willems' m e t h o d r e q u i r e s t h a t V(x)^0 everywhere which i s a sev e r e l i m i t a t i o n on the c h o i c e o f Lyapunov f u n c t i o n s t h a t can be made. 47 APPENDIX A GRAPHICAL SOLUTION BY METHOD 2 FOR f ( • ) OF DIAGONAL TYPE Case when m = 2 L e t k = 3 - j . Then from (3.16) and (3.17) AX 1 = 2 u . c j A A j + ( a k c . k c j A A . + 2 b . k c . A j - a k c . A j c . A k ) f ] c + 2 u i b j k C j A k h k + 2 ( b j k ) c j k h k f k ( A . l ) AA2= - 2 u i C j A . + ( a k c . . c j A k - 2 b j k c . . - a k c j k c . A j ) f k (A.2) where A = c. .c . .. . • J J J A A J (c.„.) + b ., (c . . c . „, - c ., c . . . ) h, J A J ] k J J j A k j k J A J k Thus X 1 and X^ a r e o b t a i n e d from ( A . l ) and ( A . 2 ) . E q u a t i o n (3.18) becomes: d j l + d j 2 a " k + d j 3 f k + W d j 5 ° " k h k + d j 6 f k h k = ° ( A * 3 ) where d ^ , . . . , d ^ g are c o n s t a n t s d e f i n e d by: d . = 2u . (c . . . c . ., - c . . . . c ., ) J l i J A j j A k j A A j j k d. 2= 2 C J J C J A A j ( C j A j } d j 3 ~ a k C j A A j " C k k ( C j A j ) 2 c . . c, , - (c .. ) ' L J J kk j k + a, 2 c j k C j A j C j A k C j j ( G j A k ) + 2b., ( c . . c . . - c . , c . A . ) j k J J j A k j k j A j d j 4 = 2 u i b j k ( c k k C j A j - C k j C j A k ) d j 5 = 2 b j k ( c j j c j A k - • c j k c j A j > : d . r= 2 (b ., ) 2 c . .c, , j 6 j k j j kk The s o l u t i o n to e q u a t i o n (A.3) can be found by computing the l e f t hand s i d e a t increments o f CJ~k between u k and u k + m and d e t e r m i n i n g where the s i g n changes are 48 l o c a t e d . A f t e r t h i s i t i s a simple matter t o keep r e -d u c i n g the s t e p s i z e u n t i l the d e s i r e d a c c u r a c y f o r o~ k i s o b t a i n e d . Case when m = 3 L e t j+k+r = 6, j / k ^ r / j . From (3.16) and (3.17) A V 2 u i C j A A j + 2 u i b j k C j A k h k + 2 u i b j r C J A r h r + a k C j k C j A A j f k + a r C j r C j A A j f r + 2 ( b j k } SkVk+ 2 ( b j r ) 2 c j r f r h r + b j r ( c j k b j A r a k - a.c. c..,+ 2c. b ., ) f, h + b., (a c. c . ., -c . . c .. a k j r jAk . jr j k k r jk r j r j A k j A r jk r + 2c ., b . ) f h, jk j r r k ( A . 4 ) AX ~ = -2u. c . . . + (a. c . . c ... - 2c . . b .. - c .. c .. . a. ) f. 2 i j A j k J J jAk j j j k j k j A j k k + (a c . . c . . - 2c..b. - a c. b . . . ) f r j j j A r j j j r r j r j A j r ( A . 5) where A = c . . c . , , . - (c.„.) + b ., (c . . c .' - c ., c .. . ) h. J J J A A J J A J j k J J jAk j k J A J k + b . ( c . . c . . - c . c . A . ) h j r J J j A r j r J A J r which are used to o b t a i n X^ and X^' There are now two e q u a t i o n s f o r (3.18) d n f r + d , r r h, + d ~n- h + d _ f . + d .f + d ,-h, f. pO u p p l u p k p 2 u p r p3 k p4 r p5 k k + d ,h f + d „h. f + d 0 h f , + d o = 0 (p=k,r) p6 r r p7 k r p8 r k p9 r i . (A.6) where d „= 2 c . . c . . . . pO L JJ JAAj j A j - (c d , = 2b ., ( c . . c . A 1 - c ., c . n . ) p l j k J J jAk j k J A J d „ = 2b. (c . . c .. - c . c .. . ) p2 j r J J j A r j r J A J d _ = a. p3 k C p k ( G j j C j A A j ~ ( C J A J } > + C j A P ( c j k C j A j " C j j ^ A k ^ + 2c . .c . . b ., JJ jAp Jk 'JAj 'jAp j r j A j j j j A r ' d „ = a [c ( c . c . . . . - (c.„.)2) + c . B J c . . c ; , . - c..c.,.) p4 r L pr J J jAAj -+ 2c..c . . b . 33 JAp j r d c= 2(b., ) (c , c . . - c .c., ) p5 j k pk J J pj j k d 2(b. ) 2 ( c c. .- c .c. ) p6 j r pr J J pj j r d „= a b., p7 r j k c ( c . . c . „ n - c . , c . . . ) + c , ( c . C . . . - ' C . . C . A ) . pr j j jAk j k J A J pk j r J A J J J j A r + C P J ( C J k C j A r " c j r c j A k > + 2b ., b . (c , c . . - c .c., ) j k j r pk J J pj j k d Q= a. b . p8 k j r c . (c . .c . . - c. c . '. ) + c ( c . , c . . . - c..c.., ) L pk J J j A r j r J A J pr j k J A J J J jAk + c . ( c . c.., - c.,c.„ ) + 2b. b . , ( c c..- c .c. ) PJ j r jAk j k j A r J j r j k pr J J pj j r d 0 = 2 u . ( c . „ c . . . - c . c . „ n . ) p9 l JAp J A J p j JAAj A graph of each of the e q u a t i o n s (A.6) can be drawn by s e t t i n g c r e q u a l to some c o n s t a n t and s o l v i n g f o r cn, by r K the method used f o r s o l v i n g (A. 3 ) . In t h i s way a s e t of p o i n t s f o r d i f f e r e n t v a l u e s o f cr £(u ,u ) are o b t a i n e d . ^ r r r+m The i n t e r s e c t i o n o f the two graphs g i v e s the s o l u t i o n p o i n t . 50 and l e t APPENDIX B GRAPHICAL SOLUTION WHEN f ( • ) IS NOT OF DIAGONAL TYPE In t h i s case assume V(x) t o be of the form (1.7) dF , . 0 f r Method 1 der.. F k ' a n d der,, f r k The g r a d i e n t c o n d i t i o n i s m 2Px + V c, F, = Xc . (B. 1) kfel k k J F o l l o w i n g the same procedure as b e f o r e m 2c. .cr + y A , F. = 2u.c . ( r = l . . . . , m , r ^ j ) (B.2) j j r k 4^i rk k l r j ' ' ' r j Xu, m cr- k and V(x) = x'Px + F ( c r ) = ~^ =- + F ( c r ) - £ ~o^F]c (B.3) k=l 2u. m c where X = — — +. Y —^—F, c . . , ' i c . . k J J k=l J J F o r m = 2 eq u a t i o n s (B.2) become 2 c r c . . + A . F . + A F = 2 u . c . (B.4) r. J J r j j r r r I r j where r = 3 - j . T h i s e q u a t i o n can be s o l v e d the same way as d e s c r i b e d f o r s o l v i n g (A.3). For m = 3 e q u a t i o n s (B.2) become 2c . . c r + A . F . + A , F. + A F = 2u. c . J J u r r j j r k k r r r l r j 2 c j j c r k + A k j F j + A k k F k + A k r F r = 2 u i c k j j (B.5) where r+k+j = 6, r/^k^j/^r. These e q u a t i o n s can be s o l v e d the same as (A.6) For m>3 t h e r e i s no g r a p h i c a l s o l u t i o n . 51 Method 2 The g r a d i e n t c o n d i t i o n i s : m m m 2Px + rc,F,= Lc.+ L(A'c.+ T T b. c e f ) (B.6) ^ k k 1 j . 2 j ^ ^ j r s r s The f o l l o w i n g e q u a t i o n s are used to s o l v e f o r X^ and X,>. 2u.+ y c.,F, = X, c . . + X„ (c .. . + y Vb. c. f ^ ^ ) i V Jk k 1 J J 2 J A J £-? ^  j r j s r s • r s ( B > 7 ) F (c . „, F, - 2 b f, ) = X, C . . . + X „ ( c . A A . + y Th . c . . f ) V; jAk k j k k 1 J A J 2 jAAj L , j r jAs r s where A V 2 u i C j A A j + 5 ( C J k C j A A j - C j A j C j A l c ) F k + 2 ^ b j k C j A j f k + 2 u i £ X > j r c j A s f r s + 5 ^ ^ b J r C j A s ( c j k " c j A k ) F k f r s + 2 r b. c . A f f • l^L-'L-' j k j r j A s r s k K JT S AX 2 = - 2 u ± c j A j + E ( c jj cjAk" c j k c j A j ) F k + £ 2 b j k c j j f k where A = C i J CJAAj- ( c j A j ) 2 + E D > j r c . A s ( c . r c j A j ) f r s and X^ and X 2 are s u b s t i t u t e d i n t o 2 c r + 7c , F, = X, c .+ X~ (c ..+ YVh. c f ) (p=l,...,m,p/j) P V Pk k 1 PJ 2 pAj LJL, j r ps r s ^ ' ' K r S (B.8) to g i v e • 0 r p r s r s V k k V k k v L ^ ^sk k r s ^ r s k . k J c r s + E F E e r s k f r s f k + F E ers frs= 0 <P=1, • • • p ^ j ) k r s r s where d„= 2 u . (c . A .c . .-< c ., . .c .) 0 l J A J pAj J A A J pj d n= 2 f " c . . c . A . . - ( c . A . ) 2 1 L J J JAAj j A j d = 2 b . c .. (c . .-c . A . ) r s j r jAs J J J A J d, = c , k pk C j j C j A A j ^ j A j 5 C p j ( G j k C j A A j C j A j C j A k ) " C p A j ( C j j C j A k C j k C j A j - ) e,= - 2 b . , ( c . c . , . + c A . c . . ) k j k pj J A J pAj J J d , = c , b . c . , (c . .- c . . . ) - c .b. c . A (c ., - c . ) r s k pk j r jAs J J J A J pj j r jAs j k jAk - b . c (c . .c .., - c ., c .. . ) j r ps J J jAk j k J A J e ,= -2b.,b. ( c . c - . + c c..) r s k j k j r pj jAs ps J J e = 2 u.b. ( c . . . c - c -c.. ) r s 1 j r J A J ps pj jAs I f a l l f e a s i b l e s o l u t i o n s of (B . 9 ) can be o b t a i n e d then V\ can be c a l c u l a t e d from v + ^ ( £ £ b ( o - a f „ - fr>> + F(O-) r s J (B.10) For m = 2 e q u a t i o n (B . 9 ) can be s o l v e d u s i n g the approach taken i n (A.3) and f o r m = 3 the method c o r r e s p o n d s t o t h a t o f (A. 6 ) . 53 REFERENCES 1. M.A. Aizerman and F.R. Gantmacher, A b s o l u t e S t a b i l i t y  o f R e g u l a t o r Systems, (book), Holden Day (San F r a n c i s c o ) , 1964. 2. R.E. Kalman, Lyapunov F u n c t i o n s f o r the Problem o f  L u r ' e i n Automatic C o n t r o l , P r o c e e d i n g s N a t i o n a l Academy of S c i e n c e s , Volume 49, 201, 1963. 3. J.A. Walker and N.H. McClamroch, F i n i t e Regions of  A t t r a c t i o n f o r the Problem of L u r ' e , I n t e r n a t i o n a l J o u r n a l o f C o n t r o l , Volume 6, 331, 1967. 4. J . L . W i l l e m s , The Computation of F i n i t e S t a b i l i t y  Regions by means of Open" Lyapunov S u r f a c e s , I n t e r -n a t i o n a l J o u r n a l of C o n t r o l , Volume 10, 537, 1968. 5. B.D.O. Anderson, S t a b i l i t y o f C o n t r o l Systems w i t h  M u l t i p l e N o n l i n e a r i t i e s , J . F r a n k l i n I n s t i t u t e , Volume 282, 155, 1966. 6 . J . L . W i l l e m s , Optimum Lyapunov F u n c t i o n s and S t a b i l i t y  Regions f o r M u l t i m a c h i n e Power Systems, P r o c e e d i n g s IEEE, Volume 117, 573, 1970. 7. J . L . Willems and J.C. W i l l e m s , The A p p l i c a t i o n of  Lyapunov methods to the Computation of T r a n s i e n t  S t a b i l i t y Regions f o r M u l t i m a c h i n e Power Systems, IEEE T r a n s a c t i o n s , Volume PAS-89, 795, 1970. 

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