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I t h a n k m y a d v is o r s , P r o fe s s o r s S. M. C u k a n d R. D. M id d le b r o o k f o r p r o v id in g m e w it h a n o p p o r t u n i t y t o w o r k in th e f ie ld o f P o w e r E le c t r o n ic s . I a m g r a t e f u l f o r t h e i r g e n e r o u s s u p p o r t, g u id a n c e , a n d c o n s t r u c t iv e e n g a g e m e n t d u r in g th e c o u r s e o f m y s ta y a t C a lte c h .

I a m g r a t e f u l to P r o fe s s o r A. S a b a n o v ic * o f E n e r g o in v e s t o f Y u g o s la v ia , V is it in g P r o f e s s o r o f E l e c t r ic a l E n g in e e r in g a t C a lte c h 1 9 8 3 -8 5 . H e e x p o s e d t h e e x c it in g f ie ld o f s lid in g m o d e c o n t r o l t o m e , a n d o ffe r e d m e c o n s t a n t s u p p o r t a n d e n c o u r a g e m e n t.

I t h a n k C a lte c h f o r th e v a r io u s G r a d u a te F e llo w s h ip , T e a c h in g A s s is t a n ts h ip s , a n d R e s e a r c h A s s is ta n ts h ip s t h a t m a d e m y s t a y a t C a lte c h p o s s ib le .

I o w e a g r e a t d e a l to m y c o lle a g u e s a t th e P o w e r E le c t r o n ic s G r o u p a t C a lte c h f o r th e m a n y e n lig h t e n in g i n t e r a c t io n s , w h ic h m a d e m y w o r k p o s s ib le a n d p le a s a n t. M y p a r t i c u l a r t h a n k s a r e

iv

ABSTRACT

S w itc h e d m o d e p o w e r c o n v e r t e r s a r e b e in g u s e d e x te n s iv e ly f o r th e p u r p o s e o f e f f ic ie n t p o w e r c o n v e r s io n . S u c h c o n v e r t e r s a r e n o n lin e a r , tim e v a r i a n t s y s te m s . I n t h e p a s t s u c h c o n v e r t e r s w e r e b e in g m o d e lle d u s in g th e s ta te s p a c e a v e r a g in g m e th o d . T h e t h e o r y o f v a r ia b le s t r u c t u r e s y s te m s (VSS), a n d s lid in g m o d e c o n t r o l f o r m a m u t u a lly c o m p le m e n t a r y a n a ly s is a n d d e s ig n t o o ls f o r th e c o n t r o l o f s w it c h e d m o d e p o w e r c o n v e r t e r s . T h e a p p lic a t io n o f s lid in g m o d e c o n t r o l is p r e s e n te d f o r d c - t o - d c c o n v e r t e r s a n d e le c t r ic a l m o t o r d r iv e s in t h is th e s is . T h e c o n c e p t o f s lid in g m o d e c o n t r o l is b r o u g h t o u t t h r o u g h e x h a u s tiv e e x a m p le s o f s e c o n d o r d e r s y s te m s . T h e e q u iv a le n t c o n t r o l, a n a n a ly s is m e t h o d o f VSS, is a p p lie d to o b t a in t r a n s f e r f u n c t io n d e s c r ip t io n o f d c - t o - d c p o w e r c o n v e r t e r s . T h e s lid in g m o d e c o n t r o l is a p p lie d t o t h e c o n t r o l p r o b le m o f d c - t o - d c p o w e r c o n v e r t e r s a n d s p e e d c o n t r o lle d e le c t r ic a l d r iv e s to d e v e lo p p r a c t i c a l d e s ig n te c h n iq u e s . T h e p r a c t i c a l d e s ig n m e th o d s a r e c o n f ir m e d t h r o u g h e x p e r im e n t a l r e s u lt s .

V

T A B L E O F C O N T E N T S

itu m n UwLEDGEMEJNTS

A B S T R A C T i v

C H A P T E R 1 IN T R O D U C T IO N 1

C H A P T E R 2 V A R IA B L E S T R U C T U R E S Y S T E M S (V S S ) 5

2 .1 P o w e r P r o c e s s in g v s S ig n a l P r o c e s s in g 5

2 .2 V a r ia b le S t r u c t u r e S y s te m s 8

2 .3 S o m e E x a m p le s 9

C H A P T E R 3 S U D M G M O D E C O N T R O L 1 3

3 .1 P h a s e P la n e D e s c r i p t i o n o f S y s te m s 14

3 .2 S lid in g R e g im e s i n VSS 2 6

3 .3 G e n e r a l D y n a m ic S y s te m 4 4

3 .4 C o n d it io n s f o r E x is t e n c e o f S lid in g R e g im e 4 6

3 .5 E q u iv a le n t C o n t r o l 5 3

3 .6 S t a b i l i t y o f M o t io n in t h e S lid in g R e g im e 6 2

3 .7 D e s ig n M e th o d s 6 2

3 .8 S ta t e E s t i m a t i o n . 7 0

3 .9 S t a t ic O p t im iz a t io n 7 3

C H A P T E R 4 D C -T O -D C C O N V E R T E R S 7 7

4 .1 D c - t o - d c C o n v e r t e r T o p o lo g ie s 8 0

4 .2 D c - t o - d c C o n v e r t e r s a s VS S 8 7

4 .3 A n a ly s is o f D u t y R a t io C o n t r o lle d C o n v e r te r s 91 4 .4 S lid in g M o d e C o n t r o l o f B u c k C o n v e r t e r 101 4 .5 C u r r e n t P r o g r a m m e d D c - t o - d c C o n v e r t e r s 1 1 4 4 .6 V o lt a g e C o n t r o l o f D c - t o - d c C o n v e r t e r s 1 2 0

C H A P T E R 5 G E N E R A L T H E O R Y O F E L E C T R IC A L M A C H IN E S 1 4 1

5 .1 B a s ic T w o W in d in g M a c h in e 1 4 3

5 .2 F u n d a m e n t a l S lip r in g M a c h in e 1 4 7

5 .3 B a s ic C o m m u t a t o r M a c h in e 1 4 9

5 .4 P a s s iv e T r a n s f o r m a t i o n s 1 5 3

5 .5 D c M a c h in e 1 5 7

5 .6 S y n c h r o n o u s M a c h in e 161

vi

C H A P T E R 6 CO NTRO L OF E L E C T R IC A L D R IV E S 1 6 9

6 .1 D c M o t o r D r iv e 1 7 3

6 .2 E s t im a t io n o f S p e e d U s in g S lid in g M ode 1 8 5

6 .3 B r u s h le s s D c M o t o r D r iv e 1 9 0

6 .4 S lid in g M o d e S p e e d C o n t r o l l e r 2 1 3

6 .5 S o m e M o re P r a c t i c a l A s p e c ts 2 2 1

C H A P T E R 7 E X TE N S IO N TO S Y N C H R O N O U S MOTOR D R IV E 2 2 7

7 .1 S y n c h r o n o u s M o t o r D r iv e s 2 2 7

7 .2 L o s s e s in t h e M o t o r 2 3 0

C H A P T E R 0 C O N C LU SIO N 2 3 5

R E FE R E N C E S 2 3 7

A P P E N D IX I TR A N S FO R M A TIO N S IN E L E C T R IC A L M A C H IN E S 2 4 1

1

C H A P T E R 1

W T R O D U C T IO M

T h is th e s is d e a ls w it h t h e p r in c ip le s o f s lid in g m o d e c o n t r o l a n d t h e i r a p p lic a t io n t o p o w e r c o n v e r t e r s . T h e p o w e r c o n v e r t e r s t h a t a r e t a k e n u p f o r th e a p p lic a t io n a r e d c - t o - d c e l e c t r i c a l p o w e r c o n v e r t e r s , a n d e le c t r o m e c h a n ic a l p o w e r c o n v e r t e r s .

I n p o w e r c o n v e r t e r s a h ig h p r e m iu m is p la c e d o n th e e f f ic ie n c y o f p o w e r c o n v e r s io n , b e s id e s t h e s te a d y s t a t e a n d d y n a m ic p e r f o r m a n c e r e q u ir e m e n t s . As a r e s u l t t h e to p o lo g ie s u s e d in p o w e r c o n v e r s io n a p p lic a t io n s r e ly o n h ig h s p e e d s w itc h e s f o r e f f ic ie n t o p e r a t io n . C h a p te r 2 c o n t r a s t s t h e p o w e r c o n v e r s io n to p o lo g ie s w it h s ig n a l p r o c e s s in g s y s te m s , a n d b r in g s o u t th e m o s t i m p o r t a n t f e a t u r e o f p o w e r c o n v e r t e r s , n a m e ly d y n a m ic s t r u c t u r a l c h a n g e s b r o u g h t a b o u t b e c a u s e o f t h e i r s w it c h in g n a t u r e . S u c h s y s te m s a r e n o n lin e a r a n d t im e v a r ia n t . T h e s w itc h in g p r o p e r t y m a k e s t h e p o w e r c o n v e r t e r s p r im e c a n d id a te s f o r th e a p p lic a t io n o f t h e t h e o r y o f V a r ia b le S t r u c t u r e S y s te m s (VSS).

2

t h o r o u g h l y t h e b a s ic p r in c ip le s in v o lv e d , C h a p te r 3 t r e a t s s im p le s e c o n d o r d e r s y s te m s e x h a u s t iv e ly , a n d b r in g s o u t th e r e le v a n t f e a t u r e s o f s lid in g m o d e c o n t r o l. T h e s e c o n c e p t s a r e t h e n e x te n d e d t o g e n e r a l h ig h e r o r d e r s y s te m s .

I n th e p a s t t h e s w it c h in g c o n v e r t e r s h a d b e e n a n a ly z e d u s in g th e t e c h n iq u e o f s t a t e s p a c e a v e r a g in g . S ta te s p a c e a v e r a g in g m e t h o d e s s e n t ia lly d e v e lo p e d l in e a r , s m a ll s ig n a l, f r e q u e n c y d o m a in m o d e ls o f t h e n o n lin e a r p o w e r c o n v e r t e r s . C h a p te r 4 b r ie f ly r e v ie w s th e s t a t e s p a c e a v e r a g in g m e th o d . T h e p o w e r c o n v e r t e r s a r e t h e n a n a ly z e d u s in g t h e t h e o r y o f YSS. T h e s lid in g m o d e c o n t r o l p r in c ip le s a r e t h e n a p p lie d t o d e v e lo p p r a c t i c a l d e s ig n c r i t e r i a f o r th e c o n t r o l o f d c - t o - d c c o n v e r t e r s . E x p e r im e n t a l r e s u lt s a r e t h e n p r e s e n te d .

A n o t h e r m a jo r a r e a o f p o w e r p r o c e s s in g a p p lic a t io n is t h e e le c t r o m e c h a n ic a l p o w e r c o n v e r s io n . S u c h s y s te m s c o n s is t o f s w it c h in g p o w e r c o n v e r t e r a n d e le c t r o m e c h a n ic a l a c t u a t o r s (d c o r a c m o t o r ) . I n t h e p a s t s u c h s y s te m s h a v e b e e n a n a ly z e d s e p a r a t e ly — a m o d e l f o r t h e p o w e r c o n v e r t e r a n d a m o d e l f o r t h e m o t o r . S lid in g m o d e c o n t r o l c a n b e u s e d as a n in t e g r a t e d c o n t r o l a p p r o a c h f o r s u c h c o m p o s it e s y s te m s . C h a p t e r 5 r e v ie w s th e g e n e r a l t h e o r y o f e l e c t r i c a l m a c h in e s . I t is s e e n t h a t a ll e l e c t r ic a l m a c h in e s c a n be d e s c r ib e d as q u a l i t a t i v e ly i d e n t i c a l s y s te m s . C h a p te r 6 g o e s o n t o a p p ly th e s lid in g m o d e c o n t r o l t o tw o r e p r e s e n t a t iv e m a c h in e s , n a m e ly t h e d c m o t o r a n d th e p e r m a n e n t m a g n e t s y n c h r o n o u s m o t o r . P r a c t i c a l d e s ig n c r i t e r i a f o r th e s p e e d c o n t r o l o f th e s e m a c h in e s a r e d e v e lo p e d . E x p e r im e n t a l r e s u lt s v e r if y in g t h e d e s ig n s t r a t e g y a r e p r e s e n t e d .

3

4

5

C H A P T E R 2

V A R IA B L E S T P U C T U R E SYSTEM S

I n t h is c h a p t e r s o m e o f t h e c h a r a c t e r i s t i c s d e f in in in g v a r ia b le s t r u c t u r e s y s te m s (VSS) a r e e x p la in e d . T h is is d o n e b y c o n t r a s t in g th e n a t u r e s o f p o w e r p r o c e s s in g s y s te m s a n d s ig n a l p r o c e s s in g s y s te m s . E x a m p le s o f a fe w o f th e VSS o f t e n e n c o u n t e r e d i n p o w e r p r o c e s s in g r e q u ir e m e n t s a r e h ig h lig h t e d .

2 .1 P o w e r P r o c e s s in g V s S ig n a l P r o c e s s in g

T h e e s s e n t ia l f e a t u r e s o f s ig n a l p r o c e s s in g s y s te m s a n d p o w e r p r o c e s s in g s y s te m s a r e s h o w n i n F ig . 2.1 [1 ],

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Fig. 2 .1 A s i g n a l p r o c e s s in g s y s t e m (a ), c o n t r a s t e d w i t h a p o w e r p r o c e s s in g s y s t e m (b).

Fig. 2 . 2 C o n te n ts o f a t y p i c a l p o w e r p r o c e s s i n g s y s te m . The c o m p o n e n ts a r e L im ite d to s w i t c h e s , a n d lossless r e a c t iv e e le m e n t s i n o r d e r to p r o c e s s p o w e r e f f ic ie n t ly .

c o n s id e r a t io n s o f e f f ic ie n c y r a r e l y b e c o m e c r i t i c a l . S iz e , c o s t a n d t h e c o n s e q u e n t d e m a n d t o i n t e g r a t e la r g e s ig n a l p r o c e s s in g f u n c t i o n s i n t o e v e r s m a lle r p a c k a g e s le a d t h e c h o ic e o f s y s t e m c o m p o n e n t s l i m i t e d t o r e s is t o r s , a c t iv e d e v ic e s , a n d o c c a s io n a lly c a p a c i t o r s . M a g n e tic e le m e n ts a r e , as a r u le , a v o id e d .

8

F ig u r e 2 .2 s h o w s t h e c o n t e n t s o f a t y p i c a l e le c t r ic a l p o w e r p r o c e s s in g s y s te m . I t c o n s is t s o f i n d u c t o r s , c a p a c it o r s a n d s w itc h e s . W ith t h e e le m e n ts o f t h e s y s t e m so l im i t e d , i t is n o t h a r d t o see t h a t t h e o n ly w a y o f a c h ie v in g t h e p o w e r p r o c e s s in g o b je c tiv e s is to d y n a m ic a lly v a r y t h e s t r u c t u r e o f t h e s y s t e m in a n i n t e l l i g e n t w a y . I n o r d e r t o d e v e lo p p r o p e r c o n t r o l s t r a t e g ie s f o r t h e p o w e r p r o c e s s in g s y s te m s , i t is n e c c e s a r y t o u n d e r s t a n d , a n a ly z e , a n d d e v e lo p m o d e ls f o r t h e p u r p o s e . S u c h d y n a m ic a lly v a r y in g s t r u c t u r e s r e s u l t in n o n lin e a r a n d t i m e v a r i a n t d e s c r ip t i o n . T h e t h e o r y o f VSS [ 2 ] p r o v id e s a m a t h e m a t ic a l f r a m e w o r k t o d e f in e s u c h s y s te m s a n d a r e w e ll s u it e d t o d e fin e , a n a ly z e a n d t o d e v e lo p c o n t r o l s t r a t e g ie s f o r p o w e r p r o c e s s in g s y s te m s . I t m a y b e m e n t io n e d h e r e t h a t t h e r e a r e a ls o o t h e r t o o ls a v a ila b le f o r t h i s p u r p o s e s u c h as s t a t e s p a c e a v e r a g in g [ 3 ] , a n d d e s c r ib in g e q u a t io n s [ 4 ] .

2 .2 V a r ia b le S t r u c t u r e s y s te m s (V S S )

VSS a r e s y s te m s w h o s e p h y s ic a l s t r u c t u r e o r n e t w o r k t o p o lo g y is c h a n g e d i n t e n t i o n a l l y d u r in g t h e t r a n s i e n t in a c c o r d a n c e w it h a p r e s e t s t r u c t u r e c o n t r o l la w [ 5 ] . T h e in s t a n t s o f t im e , a t w h ic h th e c o n t r o l a c t io n o f c h a n g in g t h e s t r u c t u r e o c c u r s , a r e n o t d e t e r m in e d b y a f ix e d p r o g r a m , b u t i n a c c o r d a n c e w it h t h e c u r r e n t s ta te o f t h e s y s te m . T h is p r o p e r t y d is t in g u is h e s VSS f r o m p r o g r a m m e d c o n t r o ll e r s . H o w e v e r , a s w i l l b e s e e n in s o m e l a t e r s e c tio n s , p r o g r a m m e d c o n t r o l l e r s m a y b e c o n s id e r e d as a s u b s e t o f VSS, b y s u it a b le m a t h e m a t ic a l m a n ip u la t io n s .

9

In g e n e r a l w e m a y s t a t e t h a t Y SS c o n s is t o f a n u m b e r o f w e ll d e fin e d e le m e n ts , l in e a r o r n o n lin e a r . U n d e r c o n t r o l a c t io n th e e le m e n ts m a y b e c o n f ig u r e d in t o a n u m b e r o f p o s s ib le w e ll d e fin e d s u b s t r u c t u r e s . O u t o f th e t o t a l n u m b e r o f a ll s u c h w e ll d e fin e d s u b s t r u c t u r e s , t h e r e w ill be a m in im u m s e t o f in d e p e n d e n t s u b s t r u c t u r e s . T h is m in im u m n u m b e r o f in d e p e n d e n t s u b s t r u c t u r e s a v a ila b le i n t h e s y s te m is c o n s id e r e d a s t h e n u m b e r o f c o n t r o l in p u t s t o t h e s y s te m . T h e t h e o r y o f VSS p r o v id e s a s y s t e m a t ic m e th o d o f d e fin in g s u c h s y s te m s u s in g d is c o n t in u o u s v a r ia b le s — k n o w n as s w it c h in g v a r ia b le s — a n d s e le c t in g a r a t i o n a l c o n t r o l la w t o p ic k o u t t h e s u b s t r u c t u r e u s e d a t a n y i n s t a n t in o r d e r t o a c h ie v e t h e c o n t r o l o b je c tiv e . B e fo r e w e g o o n t o se e th e s e a s p e c t s o f VSS in th e fo llo w in g c h a p t e r , tw o e x a m p le s — a s in g le i n p u t V S S a n d a m u lt ip le i n p u t VSS — o f VSS e n c o u n te r e d i n p o w e r c o n v e r s io n a p p lic a t io n a r e p r e s e n te d n o w .

2 .3 S o m e E x a m p le s o f VSS

10

Fig. 2 . 3 A n e x a m p le o f a s in g le c o n t r o l i n p u t v a r i a b l e s t r u c t u r e s y s t e m .

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f i g . 2 . 4 A n e x a m p le o f a m u l t i p l e c o n t r o l i n p u t v a r i a b l e s t r u c t u r e s y s te m .

T h is is a n e x a m p le o f a s in g le i n p u t VSS.

12

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C H A P T E R 3

S L ID IN G M O DE C O N TR O L

T h e t h e o r y o f V a r ia b le S t r u c t u r e S y s te m s (VSS) is th e a n a ly s is a n d s y n t h e s is o f s y s te m s , w h o s e s t r u c t u r e s a r e c h a n g e d i n t e n t io n a ll y d u r in g t h e t r a n s i e n t , a c c o r d in g t o a p r e s e t s t r u c t u r e - c o n t r o l la w , t o a c h ie v e th e c o n t r o l o b je c t iv e s . T h e u t i l i z a t i o n o f c o n t r o l h a r d w a r e is m o s t e f fe c t iv e i n s u c h s y s te m s . F u r t h e r m o r e , i t is p o s s ib le i n V SS t o o b t a in o v e r a l l s y s t e m p r o p e r t ie s t h a t a r e q u a l i t a t i v e ly d i f f e r e n t f r o m t h e c o n s t i t u e n t s u b s t r u c t u r e s . T h e s e a s p e c ts o f VSS a r e b e s t s e e n t h r o u g h t h e p h a s e p la n e d e s c r ip t io n o f t h e c o n s t i t u e n t s u b s t r u c t u r e s . S e c t io n 3 .1 e x p la in s t h e p h a s e p la n e d e s c r ip t io n o f s y s te m s a n d d e s c r ib e s t h e d if f e r e n t t y p e s o f p h a s e t r a j e c t o r i e s e n c o u n t e r e d in s e c o n d o r d e r s y s te m s .

O n t h e f o u n d a t io n o f t h e p h a s e t r a j e c t o r i e s d e s c r ib e d f o r s im p le s e c o n d o r d e r s y s te m s i n S e c t io n 3 .1 , t h e i m p o r t a n t f e a t u r e s o f s lid in g m o d e c o n t r o l a r e b u i l t u p i n S e c tio n 3 .2 . T h e c o n c e p t o f s lid in g r e g im e s is f o r m a l l y d e fin e d .

14

T h e a n a ly s is p r o b le m o f YSS c o n s is ts o f t e s t s f o r th e e x is t e n c e o f s lid in g r e g im e s , s y s te m d e s c r ip t io n u n d e r s lid in g m o d e c o n t r o l, a n d th e p e r f o r m a n c e o f t h e o v e r a ll s y s te m . T h e s e a s p e c ts a r e d e a lt w it h in d e t a il in S e c tio n s 3 .4 t h r o u g h 3 .6 . T h e s e c o n d a s p e c t o f th e t h e o r y o f VSS, n a m e ly th e d e s ig n m e th o d s , a re d e s c r ib e d in S e c tio n 3 .7 . A t t h e e n d in S e c tio n s 3 .8 a n d 3 .9 , o t h e r a p p lic a t io n s o f s lid in g m o d e c o n t r o l s u c h as s ta te e s t im a t io n a n d s t a t ic o p t im iz a t io n a r e in d ic a t e d .

3 .1 P h a s e P la n e D e s c r ip t io n off S y s te m s

T h e c o n t r o l a c t io n in VSS is t h e c h a n g e o f th e s y s te m s t r u c t u r e fo llo w in g a p r e s e t s t r u c t u r e - c o n t r o l la w . As a r e s u lt , VSS a r e t im e v a r y in g s y s te m s . I n o r d e r t o s t u d y s u c h s y s te m s , i t is a d v a n ta g e o u s t o s e le c t a s y s t e m d e s c r ip t io n w h e r e tim e i n f o r m a t io n is s u p p r e s s e d o r i m p l i c i t [ 6 ] . T h e p h a s e p la n e d e s c r ip t io n s a tis fie s t h is c o n d it io n a n d p r o v id e s v a lu a b le i n s ig h t in t o th e v a r io u s a s p e c ts p e r t a in in g t o VSS. T h e p h a s e p la n e d e s c r ip t io n is v e r y h a n d y f o r s e c o n d o r d e r s y s te m s a n d b r in g s o u t th e f e a t u r e s o f s lid in g m o d e c o n t r o l r e m a r k a b ly w e ll. F o r s y s te m s o f o r d e r h ig h e r t h a n tw o , th e p h a s e p la n e d e s c r ip t i o n is c lu m s y i f n o t im p o s s ib le . I n th o s e in s t a n c e s , t h e g r a p h i c a l id e a s o b t a in e d f o r s e c o n d o r d e r s y s te m s a re e x te n d e d m a t h e m a t ic a lly .

T h e p h a s e p la n e d e s c r ip t io n is u s e d w id e ly to c h a r a c t e r iz e s e c o n d o r d e r s y s te m s . T h e a x e s o f t h e p h a s e p la n e a r e th e s y s te m s ta te s . T h e in s t a n t a n e o u s s t a t e o f t h e s y s te m is r e p r e s e n t e d o n th e p h a s e p la n e b y a R e p r e s e n t a t iv e P o i n t (R P ) w h o s e c o o r d in a t e s o n th e

15

p h a s e p la n e a r e t h e p r e s e n t s t a t e s o f t h e s y s te m . T h e s t u d y o f th e s y s te m in v o lv e s t h e m o t io n o f t h e R P o n t h e p h a s e p la n e u n d e r d if f e r e n t i n p u t a n d i n i t i a l c o n d itio n s . T h e e v o lu t io n o f th e s y s te m s t a t e s w it h r e s p e c t t o t im e o n t h e p h a s e p la n e , r e f e r r e d t o as th e p h a s e t r a j e c t o r ie s o r th e s ta t e t r a j e c t o r i e s , r e p r e s e n t t h e d y n a m ic p r o p e r t ie s o f th e s y s te m . F o r a g iv e n s y s te m , th e p h a s e t r a j e c t o r ie s a r e a f a m ily o f c u r v e s s a t is fy in g t h e d y n a m ic p r o p e r t ie s o f th e s y s te m . T h e a x e s o f th e p h a s e p la n e a r e t h e s y s te m s ta te s a n d h e n c e , t h e tim e in f o r m a t io n is i m p l i c i t i n t h e s t a t e t r a je c t o r ie s .

I n th e c a s e o f VSS, th e p h a s e p la n e d e s c r ip t io n c o n s is ts o f a s e t o f a f a m ily o f p h a s e t r a j e c t o r i e s , o n e f o r e a c h o f th e s u b s t r u c t u r e s u s e d . T h e a n a ly s is p r o b le m o f VSS is to s tu d y t h e o v e r a l l s y s te m b e h a v io r u n d e r a g iv e n s t r u c t u r e - c o n t r o l la w . T h e d e s ig n p r o b le m in VSS is to d e v e lo p a r a t io n a le to s y n th e s iz e a s t r u c t u r e - c o n t r o l la w in o r d e r t o a c h iv e t h e p e r f o r m a n c e o b je c tiv e s o f t h e o v e r a ll s y s te m .

3 .1 .1 P r o p e r t ie s o f P h a s e T r a j e c t o r ie s

16

C o n s id e r a s e c o n d o r d e r s y s t e m d e s c r ib e d b y t h e fo llo w in g h o m o g e n e o u s d i f f e r e n t i a l e q u a t io n .

d 2x _ dx „

~ d f+ U ° z = 0 (3 .1 )

T h e a b o v e e q u a t io n m a y b e n o r m a liz e d t o o b t a in d 2x _ . dx

^ T + 2 t d 7 + x ~ 0 (3 .2 )

B y d e fin in g x x = x a n d x 2 = d x / d r , t h e a b o v e e q u a t io n m a y be w r i t t e n a s a s e t o f f i r s t o r d e r d if f e r e n t i a l e q u a t io n s .

dx. d r

d r

x 2

= - 2 { x 2- x l (3 .3 )

L e t g j a n d g2 b e t h e c h a r a c t e r i s t i c r o o t s o f t h e E q . ( 3 .2 ) . T h e n g i+ ? 2 =

9i92 = l

x 1 a n d x 2 a r e t h e d y n a m ic s t a t e s o f t h e s y s t e m a n d a r e t h e a x e s o f th e p h a s e p la n e . T h e d i f f e r e n t i a l e q u a t io n b e tw e e n z j a n d x 2 d ic t a t in g th e p r o p e r t i e s o f t h e p h a s e t r a j e c t o r i e s is o b t a in e d f r o m E q . ( 3 .3 ) .

dx g x j + 2 ^ 2

~ 771 = —

d x x x 2 (3 .4 )

m is t h e s lo p e o f t h e p h a s e t r a j e c t o r y p a s s in g t h r o u g h th e p o in t ( x j . x 2). z , a n d x 2 a r e r e l a t e d to t h e s lo p e b y

2:2 = ~ m + 2 f Z l (3 .5 )

T h e p h a s e t r a j e c t o r i e s a r e c o n s t r u c t e d s a t is f y in g E q . (3 .4 ) in th e p h a s e p la n e . T h e f o llo w in g p r o p e r t i e s o f t h e p h a s e t r a j e c t o r i e s a re

17

u s e d t o c o n s t r u c t th e m .

i) T h e s t r a i g h t lin e s x z = j a n d x z = qzxx , w h e n t h e y e x is t , a re p h a s e t r a j e c t o r ie s . T h is p r o p e r t y is s e e n b y s u b s t i t u t i n g m. - q Y a n d in. - qz in E q . (3 .5 ).

1 1

2 o = --- — 2 , = X , = q , 2 , 2 f f i + 2 f 1 g 2 1 y i 1

3:2 = = i ? * 1 = g2X l

T h e s t r a i g h t lin e s x z = g 1a:1 a n d x z = e x is t o n ly w h e n g x a n d g 2 a r e r e a l.

i i ) T h e p h a s e t r a j e c t o r i e s i n t e r s e c t t h e x z a x is w it h a s lo p e o f — 2£. dxr

dx.

i i i ) T h e p h a s e t r a j e c t o r i e s i n t e r s e c t t h e x t a x is a t r i g h t a n g le s .

d x x z z = 0 oo

iv ) T h e z e r o s lo p e o f t h e p h a s e t r a j e c t o r i e s o c c u r s a lo n g th e s t r a i g h t lin e x z - - ( 1 / 2 0 ^ 1 - T h is p r o p e r t y is f o u n d b y s o lv in g f o r m = 0 in E q . (3 .5 ).

v ) T h e t r a j e c t o r i e s c o n v e r g e f o r s ta b le s y s te m s ( e ig e n v a lu e s w it h n e g a t iv e r e a l p a r t s ) , a n d d iv e r g e f o r u n s t a b le s y s te m s ( e ig e n v a lu e s w it h p o s it iv e r e a l p a r t s ) . F o r m a r g i n a ll y s t a b le s y s te m s ( p u r e ly im a g in a r y e ig e n v a lu e s ) , t h e t r a j e c t o r i e s a r e c lo s e d c u r v e s .

18

A ll f r e e s e c o n d o r d e r s y s te m s e x h ib i t t h e a b o v e p r o p e r t ie s in t h e i r p h a s e t r a j e c t o r ie s . D e p e n d in g o n t h e n a t u r e o f th e e ig e n v a lu e s g t a n d g 2, th e p h a s e t r a j e c t o r i e s h a v e q u a lit a t iv e ly d if f e r e n t s h a p e s .

3 .1 .2 S ta b le S y s te m s m t h W e g a tiv e E e a l E ig e n v a lu e s

T h e e ig e n v a lu e s a n d g 2 b e in g r e a l, t h e s t r a i g h t lin e s w it h s lo p e s g i a n d g 2 a r e p h a s e t r a j e c t o r ie s . O th e r r e p r e s e n t a t iv e t r a j e c t o r ie s a r e c o n s t r u c t e d w it h th e p r o p e r t i e s g iv e n in t h e l a s t S e c tio n . T h e p h a s e t r a j e c t o r i e s in th e p a r a m e t r i c f o r m a r e g iv e n b y

Z i = —-1- [(gg^io-^ao)e?lT-( g ig io -^ 2 o )e ggT]

5 2 -9 1

* 2 = — — - [ { x 10- q 1x SQ) e qiT- { x l0- q zx 20) e qzT]

92 9 i

a:10 a n d x zo a r e t h e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y . q 1 a n d g 2 a r e n e g a t iv e . As t te n d s t o +°°, x 1 a n d x z t e n d t o z e r o . T h e r e f o r e t h e e q u i l ib r i u m p o in t f o r a ll t h e t r a j e c t o r i e s is th e o r ig in . F ig u r e 3.1 s h o w s t h e p h a s e t r a j e c t o r i e s f o r t h is c la s s o f s y s te m s w it h d is t in c t , n e g a t iv e , a n d r e a l e ig e n v a lu e s . I n t h e c a s e o f n e g a tiv e , r e a l, r e p e a t e d e ig e n v a lu e s t h e p h a s e t r a j e c t o r i e s i n t h e p a r a m e t r ic f o r m a r e g iv e n b y

x i = ® i o e9'r + ( : c 2 o + 2: i o b ‘ e ? T

x z - x Z0e^T- ( x Z0+ x 10)r e ^ T

Xio a n d x zo a r e t h e c o o r d in a t e s o f a n y p o in t o n a g iv e n t r a j e c t o r y , g is e q u a l t o —1. As r te n d s to +°°, x x a n d x z t e n d t o z e r o . T h e r e f o r e

19

•o

F ig . 3 .1 The p h a s e t r a je c t o r ie s o f a n o r m a liz e d \

(gr i = —0.5 ,qz = — = 1.25) s e c o n d o r d e r s y s te m w i t h d is tin c t, n e g a tiv e r e a l e ig e n v a lu e s . The s y s te m is s ta b le a n d a l l the t r a je c to r ie s c o n v e rg e to th e o r ig in .

F ig . 3 .2 The p h a s e tr a je c t o r ie s o f a n o r m a liz e d

(g 3 = g2 = q = —I f = l ) s e c o n d o r d e r s y s te m w i t h

20

th e e q u ilib r iu m p o i n t f o r a ll t h e t r a j e c t o r i e s is t h e o r ig in . F ig u r e 3 .2 s h o w s th e p h a s e t r a j e c t o r i e s w h e n t h e e ig e n v a lu e s a r e r e a l, n e g a t iv e , a n d r e p e a te d .

3.1.3 Stable Systems with Complex Eigenvalues

T h e e ig e n v a lu e s q x a n d q s a r e c o m p le x c o n ju g a t e s . T h e r e a r e n o s t r a i g h t lin e t r a j e c t o r i e s . A ll t h e o t h e r p r o p e r t i e s o f th e t r a j e c t o r ie s h o ld . T h e s y s t e m m a y b e s o lv e d t o o b t a in t h e p h a s e t r a j e c t o r ie s i n t h e f o llo w in g p a r a m e t r i c f o r m .

* -_ r € x 1 0 + x 2 0 . ^ -|

x x - e - vr|a:1 0c o s n T + ^ s in U T j

z2 = e~^T[2 2 0c o s Q r — ^ S- ^ — — sin Q T ]

0 = y / i - f

x 10 a n d z 2o a r e t h e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y . As r te n d s t o +<», x x a n d 2 2 t e n d t o z e r o . T h e t r a j e c t o r i e s a ll

t h e r e f o r e c o n v e r g e t o w a r d s t h e o r ig i n , w h ic h is t h e e q u i l i b r i u m p o in t . F ig u r e 3.3 s h o w s t y p i c a l t r a j e c t o r i e s f o r t h i s c la s s o f s y s t e m s . T h e y h a v e th e s h a p e o f c o n v e r g in g l o g a r i t h m i c s p ir a ls .

3.1.4 Marginally Stable Systems

M a r g in a lly s t a b le s y s te m s d o n o t h a v e a n y d a m p in g (£ = 0).

T h e e ig e n v a lu e s a r e p u r e l y im a g i n a r y . T h e s y s t e m is c o n s e r v a t iv e a n d th e p h a s e t r a j e c t o r i e s a r e c lo s e d c i r c le s o f r a d iu s r .

r - V 2 102 + 2 202

2 jo a n d 2 2 0 a r e t h e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y .

T h e s y s te m is a s y m p t o t i c a l l y s t a b le a n d h a s n o e q u i l i b r i u m p o in t .

21

* X2= - X a/ 2 £

F ig . 3 .3 The p h a s e t r a j e c t o r i e s o f a n o r m a liz e d

( 9 i9 s = 1.91 + 92 = - 2 ^ = - 0 .4 ) s e co T id o r d e r s y s te m w it h s ta b le , c o m p le x c o n ju g a te e ig e n v a lu e s . The s y s te m is s ta b le a n d th e t r a je c t o r ie s a r e c o n v e rg in g lo g a r it h m ic s p ir a ls .

A

X2=Xa/2 c t

F ig . 3 .4 The p h a s e t r a je c t o r ie s o f a n o r m a liz e d

22

3 .1 .5 U n s ta b le S y s te m s w it h C o m p le x E ig e n v a lu e s

T h e e ig e n v a lu e s a r e c o m p le x c o n ju g a t e s w it h p o s it iv e r e a l p a r t s . T h e p h a s e t r a j e c t o r y is g iv e n b y t h e fo llo w in g p a r a m e t r ic e q u a t io n s .

* ^ 2 0— 10 t

x x = e RT[3:1 0c o s O T + ^ ---s in O r]

CLjC o n Z i n

x z = e OT[2:3 0c o s O T + ^ --- s in O r]

Q = ; a =

a: i 0 a n d x zo a r e th e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y .

As r is t r a c e d b a c k i n t im e t o x 1 a n d x z a r e z e r o . T h e r e f o r e a ll th e t r a j e c t o r i e s e m e rg e f r o m t h e o r i g i n a n d d iv e r g e to w a r d s °° as t im e e v o lv e s . T h e p h a s e t r a j e c t o r ie s f o r t h i s c la s s o f s y s te m s a r e s h o w n i n F ig . 3 .4 . T h e y a r e d iv e r g in g l o g a r i t h m ic s p ir a ls .

3 .1 .6 U n s t a b le S y s te m s w it h P o s it iv e R e a l E ig e n v a lu e s

T h e e ig e n v a lu e s a r e r e a l. T h e s t r a i g h t lin e s w it h s lo p e s q x a n d qz a r e a ls o p h a s e t r a j e c t o r ie s . T h e p h a s e t r a j e c t o r i e s in p a r a m e t r i c f o r m a r e g iv e n b y

x i = „ [{<lzXl0 - x Z0)e q' r - ( q lx 1Xi- x Z0) e qzT] 92 ? 1

* 3 =

zr-^-z—

9 z ~ 9 i

2 j0 a n d x zo a r e th e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y . g x

a n d qz a r e p o s itiv e . As r is t r a c e d b a c k i n t im e to —«>, arj a n d x z t e n d to z e r o . T h e r e fo r e a ll t h e t r a j e c t o r i e s e m e r g e f r o m th e o r ig in a n d g r o w w it h t im e . F ig u r e 3 .5 s h o w s r e p r e s e n t a t iv e p h a s e t r a j e c t o r ie s .

23

‘2—

•o

F ig . 3 .5 The p h a s e t r a je c t o r ie s o f a n o rm a liz e d .

(g ! = 0 .5 ,g g = 2,£ = —1.25) s e co n d o r d e r s y s te m w it h d is tin c t, p o s itiv e r e a l e ig e n v a lu e s . The s y s te m i s u n s ta b le a n d a l l th e t r a je c t o r ie s d iv e r g e to <».

Pig. 3 .6 The p h a s e t r a je c t o r ie s o f a n o r m a liz e d

(g i = l , g2 = —l , f = 0) s e c o n d o r d e r s y s te m s w i t h d is tin c t

24

T h e y d iv e r g e in d ic a t in g u n s ta b le e ig e n v a lu e s a n d a r e t h e m i r r o r im a g e o f t h e t r a j e c t o r i e s o b t a in e d f o r s y s te m s w it h n e g a t iv e , r e a l e ig e n v a lu e s .

3 .1 .7 U n sta b le S y s t e m s vdth. E ig e n v a lu e s R e a l a n d U n lik e

S ig n

T h e e ig e n v a lu e s a r e r e a l. T h e s t r a i g h t lin e s o f s lo p e s q x a n d q2 a r e a ls o p h a s e t r a j e c t o r i e s . T h e p h a s e t r a j e c t o r i e s i n p a r a m e t r i c f o r m a r e g iv e n b y

* 1 = r - ^ ^ [ ( g 2* i o - *2o)egiT- ( g i * i o - *2o)e92T]

? 2—9 1

* 2 = -r -^ — [ ( x 10- q 1x Z0) e qiT- ( x 10- q 2x Z0) e qzT]

9 2 _ 9i

x10 a n d x 20 a r e th e c o o r d in a t e s o f a n y p o i n t o n a g iv e n t r a j e c t o r y . q 1

a n d q2 a re o f u n lik e p o l a r i t y . T h e r e f o r e t h e p h a s e t r a j e c t o r i e s a r e n o t u n if o r m l y d iv e r g e n t . T h e y e m e r g e f r o m » a n d e v o lv e t o w a r d s F ig u r e 3 .6 s h o w s t y p i c a l t r a j e c t o r i e s . T h e o v e r a l l b e h a v io r is d iv e r g e n t.

T h e n a t u r e o f t h e p h a s e t r a j e c t o r i e s f o r f r e e s e c o n d o r d e r s y s te m s g iv e n b y E q . ( 3 .1 ) is c o n s o lid a t e d i n F ig . 3 .7 . I t is p lo t t e d b e tw e e n d a m p in g f a c t o r £ a n d n a t u r a l f r e q u e n c y u02. T h e d if f e r e n t r e g io n s a r e n u m b e r e d a n d t y p i c a l t r a j e c t o r i e s h in t e d . S o m e o f th e s e t r a j e c t o r ie s a r e u s e d i n t h e f o llo w in g s e c t i o n t o h ig h l ig h t th e p r o p e r t ie s o f VSS a n d t h e c o n c e p t o f s lid in g r e g im e s .

25

I STA B LE REAL ROOTS

STABLE

COMPLEX

ROOTS

V

III

UNSTABLE COMPLEX

ROOTS

IV. V

UNSTABLE REAL

ROOTS

m * > X i

F ig . 3 . 7 The q u a l it a t iv e d if fe r e n c e i n th e p h a s e t r a je c t o r ie s o f a s e c o n d o r d e r s y s te m as a f u n c t i o n o f th e n a t u r a l f r e q u e n c y cj0 a n d d a m p in g S y s te m s lo c a te d i n r e g io n I

2 6

3 .2 Sliding R eg im es in 'VSS

I t w a s a s s e r te d t h a t , in t h e c a s e o f VSS, p h a s e t r a j e c t o r ie s t h a t a r e q u a l i t a t i v e ly d i f f e r e n t f r o m th o s e o f th e s u b s t r u c t u r e s c a n b e o b t a in e d . T h is p o i n t a n d o t h e r f e a t u r e s o f VSS a r e n o w i l l u s t r a t e d . C o n s id e r t h e fo llo w in g tw o s u b s t r u c t u r e s ( f r o m S e c tio n 3 .1 .5 a n d 3 .1 .7 r e s p e c t iv e ly ) .

S u b s t r u c t u r e I is g iv e n b y x x = x z

- 1 < K < 0

x z - - 2^ 2- X y

T h e e ig e n v a lu e s o f s u b s t r u c t u r e I a r e c o m p le x w it h p o s it iv e r e a l p a r t s . T h e p h a s e t r a j e c t o r i e s o f t h is s u b s t r u c t u r e a r e s h o w n in F ig . 3 .8 a a n d is th e s a m e as s h o w n i n F ig . 3 .4 . T h e t r a j e c t o r ie s a r e d iv e r g in g l o g a r it h m ic s p ir a ls a n d t h e s u b s t r u c t u r e I is u n s ta b le .

S u b s t r u c t u r e I I is g iv e n b y

Z

=

9- =

: 93 =

T h e e ig e n v a lu e s o f s u b s t r u c t u r e I I a r e r e a l a n d o f u n lik e p o la r i t y . T h e p h a s e t r a j e c t o r ie s o f s u b s t r u c t u r e I I a r e g iv e n i n F ig . 3 .8 b . T h e y d iv e r g e to °° in d ic a t in g i n s t a b i l i t y a n d a r e t h e s a m e as s h o w n e a r li e r i n F ig . 3 .6 .

3.2.1 Simple VSS

I t m a y b e o b s e r v e d t h a t th e t r a j e c t o r i e s o f s u b s t r u c t u r e I a r e b o u n d e d in e v e r y q u a d r a n t . T h e t r a j e c t o r i e s o f s u b s t r u c t u r e II a r e b o u n d e d in q u a d r a n t s 2 a n d 4 . A V S S m a y be s y n th e s iz e d

27

X2

-o

fa]

SUBSTRUCTURE I

XS

(h) SUBSTRUCTURE I I

28

A

Xg

SUBSTRUCTURE I

SUBSTRUCTURE I

(a)

(b)

M g . 3 .9 A s im p le VSS m a d e u p o f th e tw o s u b s tr u c tu r e s g iv e n e a r l ie r i n F ig . 3 .8 . The s u b s tr u c tu r e s a c tiv e i n th e d if f e r e n t re g io n s o f th e p h a s e p la n e a re s h o w n . The r e s u lt a n t o v e r a ll t r a je c t o r y o f the o v e r a ll VSS is s h o w n i n (a ). T y p ic a l t r a je c t o r ie s s t a r t in g f r o m a r b i t r a r y i n i t i a l c o n d itio n s A (x 11,x 21) a n d B ( x 12.x22) a re s h o w n i n (b). The VSS m a d e u p o f u n s ta b le s u b s tr u c tu r e s is s e e n to be s ta b le .

29

c o m p o s in g o f th e s e d e s ir a b le p o r t i o n s o f t h e t r a j e c t o r i e s o f t h e s u b s t r u c t u r e s I a n d II. T h e s t r u c t u r e - c o n t r o l la w m a y b e s t a t e d as

S u b s t r u c t u r e I h o ld s f o r s 1( z2- g 1a;1) > 0

S u b s t r u c t u r e I I h o ld s f o r x ^ x z — g ^ i ) < 0

A c c o r d in g t o t h e s t r u c t u r e - c o n t r o l la w t h e p h a s e p la n e is d iv id e d in t o s e v e r a l r e g io n s , a s s h o w n in F ig . 3 .9 a . T h e s u b s t r u c t u r e s a c t iv e in e a c h o f th e s e r e g io n s a r e la b e le d . T h e o v e r a l l p h a s e t r a j e c t o r y is c o m p o s e d o f t h e t r a j e c t o r i e s o f e a c h o f t h e s u b s t r u c t u r e s i n t h e a p p r o p r ia t e r e g io n s . T h e p h a s e t r a j e c t o r i e s f o r t h e c o m p o s it e VSS a r e s h o w n in F ig . 3 .9 a . T h e t r a j e c t o r i e s o f t h e s y s t e m s t a r t i n g f r o m a r b i t r a r y i n i t i a l c o n d it io n s A (a: n .2:2 1) a n d B{x Xz<x zz) a r e s h o w n in F ig .

3 .9 b . S t a r t i n g f r o m a n y i n i t i a l c o n d i t i o n t h e s t a t e s o f t h e VSS c o n v e r g e to z e r o a n d s o th e o v e r a l l s y s t e m is s t a b le .

I n t h e a b o v e e x a m p le i t w a s a s s u m e d t h a t t h e s t r u c t u r e - c o n t r o l la w w a s r e a li z e d p e r f e c t l y . I n o t h e r w o r d s t h e c h a n g e o f s t r u c t u r e f r o m o n e s u b s t r u c t u r e t o t h e o t h e r a c t e d w h e n e v e r t h e s y s t e m R P c r o s s e d t h e b o u n d a r ie s s e t u p b y th e s t r u c t u r e - c o n t r o l la w . I n p r a c t i c e s u c h p e r f e c t s e n s in g o f t h e l o c a t i o n o f t h e RP w it h r e s p e c t t o t h e v a r io u s b o u n d a r ie s is im p o s s ib le . T h e r e w i l l a lw a y s be n o n id e a lit ie s s u c h as d e la y , h y s t e r e s is e tc , i n h e r e n t in p h y s ic a l h a r d w a r e u s e d f o r c h a n g in g t h e s t r u c t u r e . We n o w see t h e e f f e c t o f s u c h n o n id e a l r e a l i z a t i o n o f t h e s t r u c t u r e - c o n t r o l la w . L e t t h e r e a l s t r u c t u r e - c o n t r o l la w b e

30

SUBSTRUCTURE I

SUBSTRUCTURE I

(a)

H> X ,

l b )

F ig . 3 .1 0 S w itc h in g b o u n d a r ie s i n a r e a l VSS (a ). The b o u n d a rie s e x te n d o v e r a s m a l l A n e ig h b o u rh o o d o f th e s w itc h in g lin e s . The e ffe c t o f th e n o n id e a l s w itc h in g b o u n d a r y o n th e o v e r a ll t r a j e c t o r y (b ) is s e e n to be s e n s itiv e to the h y s te r e s is i n th e s w it c h in g b o u n d a r y .

31

o i t h e VSS h a r d w a r e a n d g o e s t o z e r o as th e s e n o n id e a litie s v a n is h . F ig u r e 3 .1 0 a s h o w s th e p a r t i t i o n i n g o f t h e p h a s e p la n e in t o v a r io u s r e g io n s a n d th e s u b s t r u c t u r e s t h a t a r e a c tiv e in e a c h o f th e s e r e g io n s a c c o r d in g t o t h e s t r u c t u r e - c o n t r o l la w . T h e s w itc h in g b o u n d a r ie s a r e s h o w n e x te n d e d t o in c lu d e th e h y s te r e s is z o n e . S y s te m t r a j e c t o r ie s s t a r t i n g f r o m a n a r b i t r a r y i n i t i a l c o n d it io n A ( x x itx2i) a r e s h o w n i n F ig . 3 .1 0 b . T h e e f f e c t o f t h e s w itc h in g n o n id e a lit ie s r e s u lt s in t r a j e c t o r ie s t h a t a r e d if f e r e n t f r o m th e id e a l c a s e . H o w e v e r th e o v e r a ll s y s te m c o n t in u e s to b e s ta b le . I t m a y a ls o b e o b s e r v e d t h a t as A a p p r o a c h e s z e r o , th e r e a l t r a j e c t o r y a p p r o a c h e s th e id e a l t r a j e c t o r y .

3.2.2 Sliding Modes in VSS

I n t h e a b o v e e x a m p le s , d e s ir a b le s e c t io n s o f th e p h a s e t r a j e c t o r ie s o f u n s a t is f a c t o r y s y s te m s a r e p ie c e d t o g e t h e r t o o b t a in a VSS w it h a d e s ir a b le p r o p e r t y ( s t a b i li t y in th e s e e x a m p le s ) . A m o r e fu n d a m e n t a l a s p e c t o f VSS is t h e p o s s ib ilit y t o o b t a in r e s u l t a n t p h a s e t r a j e c t o r ie s t h a t a r e n o t i n h e r e n t i n a n y o f t h e s u b s t r u c t u r e s u s e d . T h is a s p e c t is b r o u g h t o u t in t h e e x t e n s io n o f t h e s a m e e x a m p le .

C o n s id e r t h e VSS t h a t c o n s is t s o f t h e s a m e s u b s t r u c t u r e s I a n d I I as b e fo r e . T h e s t r u c t u r e - c o n t r o l la w is n o w m o d ifie d .

S u b s t r u c t u r e I h o ld s f o r i . ( s 2f c z j ) > A

S u b s t r u c t u r e I I h o ld s f o r z , ( z2 + c z ] ) < A

c > 0 ; | c | < | g ! |

32

SUBSTRUCTURE I

SUBSTRUCTURE

o X ,

Xg-cflaXft

(a)

(b )

_(c)

F ig . 3 .1 1 S lid in g r e g im e i n VSS

(

).

33

t h e r e g io n s o f t h e p h a s e p la n e s a r e a l l s h o w n i n F ig . 3 .1 1 a . T h e s w it c h in g b o u n d a r ie s a r e t h e z z a x is a n d t h e s t r a i g h t lin e z 2-hcz1 = o. T h e lin e x 2 + c z 1 = 0 is t h e s w it c h in g lin e . T h e s t r u c t u r e c h a n g e s

w h e n e v e r t h e R e p r e s e n t a t iv e P o i n t (R P ) e n t e r s a r e g io n d e fin e d b y th e s w it c h in g b o u n d a r ie s . T h e i m p o r t a n t p r o p e r t y o f th e p h a s e t r a j e c t o r i e s o f t h e s u b s t r u c t u r e s is t h a t , in t h e v i c i n i t y o f th e s w it c h in g lin e , t h e p h a s e t r a j e c t o r i e s c o n v e r g e t o t h e s w it c h in g lin e . T h e im m e d ia t e c o n s e q u e n c e o f t h i s p r o p e r t y is t h a t , o n c e t h e RP h it s t h e s w it c h in g lin e t h e s t r u c t u r e - c o n t r o l la w e n s u r e s t h a t t h e R P d o e s n o t m o v e a w a y f r o m t h e s w it c h in g lin e . F ig u r e 3 .1 1 b s h o w s a t y p i c a l t r a j e c t o r y s t a r t i n g f r o m a n a r b i t r a r y i n i t i a l c o n d i t i o n i4(arllPa:21). T h e r e s u l t a n t t r a j e c t o r y is s e e n t o b e c o n f in e d a lo n g t h e s w it c h in g b o u n d a r y z z + czj^ = A. F ig u r e 3 .1 1 c s h o w s t h e s a m e t r a j e c t o r y as in F ig . 3 .1 1 b , w h e n t h e n o n i d e a l i t i e s o f t h e s w it c h in g b o u n d a r ie s a p p r o a c h z e r o (A = 0). I n t h e c a s e o f id e a l s w it c h in g i t m a y b e s e e n

t h a t o n c e t h e R P m o v e s o n t o t h e s w it c h in g lin e z z + c x 1 = 0, t h e

s y s te m m o t i o n is t h e n a lo n g t h e s w it c h in g lin e . T h e s w it c h in g lin e x z + c z 1 = 0 is d e fin e d b y t h e s t r u c t u r e - c o n t r o l la w a n d is n o t p a r t o f

th e t r a j e c t o r i e s o f a n y o f t h e s u b s t r u c t u r e s o f t h e VSS. This m o t io n o f th e s y s te m R P a lo n g a t r a j e c t o r y, o n w h ic h th e s t r u c t u r e o f th e s y s te m c h a n g e s , a n d t h a t is n o t p a r t o f a n y o f th e s u b s tr u c tu r e

t r a je c t o r ie s , is c a lle d th e s li d i n g m o d e . T h is p r o p e r t y is o n e o f th e

34

zg + c z j = 0 o n w h ic h t h e o v e r a ll s y s te m t r a j e c t o r i e s a r e c o n fin e d .)

3 .2 .3 S t a b i l i t y o f S lid io g M o d e s

A n o t h e r e x a m p le o f VSS is n o w s y n t h e s iz e d to i l l u s t r a t e th e a s p e c t o f s t a b i l i t y o f t h e s y s te m o p e r a t in g in a s lid in g m o d e . T h e VSS c o n s id e r e d h e r e a ls o is c o m p o s e d o f t h e s a m e s u b s t r u c t u r e s I a n d I I as i n t h e p r e v io u s e x a m p le s . T h e s t r u c t u r e - c o n t r o l la w is n o w g iv e n as

S u b s t r u c t u r e I h o ld s f o r z , ( z s + c :c j) > A S u b s t r u c t u r e I I h o ld s f o r z ^ z j + c z j ) < A c < 0 ; | c | < |q 2 \

T h e id e a liz e d s w it c h in g b o u n d a r ie s , t h e s u b s t r u c t u r e s a c t iv e in th e v a r io u s r e g io n s o f t h e p h a s e p la n e , a n d t h e t r a j e c t o r i e s in e a c h o f t h e r e g io n s o f t h e p h a s e p la n e s a r e a l l s h o w n in F ig . 3 .1 2 a . T h e s w it c h in g b o u n d a r ie s a r e t h e x z a x is a n d t h e s t r a i g h t lin e x z + c x t - 0.

T h e s t r u c t u r e c h a n g e s w h e n e v e r t h e R e p r e s e n t a t iv e P o in t (R P) e n te r s a r e g io n d e fin e d b y t h e s w it c h in g b o u n d a r ie s . As in t h e p r e v io u s e x a m p le t h e p h a s e t r a j e c t o r i e s o f t h e s u b s t r u c t u r e s in t h e v i c i n i t y o f t h e b o u n d a r y i 2+ c z1 = 0, c o n v e r g e t o t h e b o u n d a r y z2 + c z , = 0.

F ig u r e 3 .1 2 b s h o w s a t y p i c a l t r a j e c t o r y s t a r t i n g f r o m a n a r b i t r a r y i n i t i a l c o n d it io n A { x l l t x zi ). T h e r e s u l t a n t t r a j e c t o r y is s e e n c o n fin e d a lo n g t h e s w it c h in g b o u n d a r y x z + c x 1 = A. F ig u r e 3 .1 2 c s h o w s th e s a m e t r a j e c t o r y a s i n F ig . 3 .1 2 b , w h e n th e n o n id e a lit ie s o f th e s w it c h in g b o u n d a r ie s a p p r o a c h z e r o (A = 0). I n t h e id e a l c a s e th e

s y s te m m o t io n , as i n t h e p r e v io u s e x a m p le , is a lo n g t h e s w it c h in g lin e x z + c xj = 0. H o w e v e r u n lik e t h e p r e v io u s e x a m p le , th e s y s te m

35

SUBSTRUCTURE

I I

SUBSTRUCTURE

(a)

(b)

_(c)

36

m o tio n is fo r e v e r a w a y f r o m th e o r ig in a n d is u n s ta b le . T h e

i m p o r t a n t p o in t m a d e t h r o u g h th is e x a m p le is t h a t , w h e n s lid in g

m o d e s e x is t, a n a ly s is o f th e s t a b ilit y o f t h e o v e r a ll s y s te m is d o n e

s im p ly b y e x a m in in g if th e t r a j e c t o r i e s o n th e s w itc h in g b o u n d a r ie s

le a d to a s ta b le o p e r a tin g p o in t o r n o t.

3 .2 .4

Practical Example

The e xam ples c o n s id e re d in th e p re v io u s s e c tio n s w e re a ll fre e s e c o n d o rd e r syste m s . T h e y b r o u g h t o u t th e v a r io u s fe a tu r e s o f th e s lid in g m odes in VSS. To r e in f o r c e th e c o n c e p ts illu s t r a t e d b y th e p re v io u s e xa m ple s, a p r a c tic a l e x a m p le o f a d c - to - d c e le c t r ic a l p o w e r c o n v e r te r is p re s e n te d , b e fo re we go o n to g e n e ra liz e th e s e c o n c e p ts f o r g e n e ra l d y n a m ic syste m s.

C o n s id e r th e d c -to -d c c o n v e r t e r c i r c u i t s h o w n in F ig . 3 .1 3 a .

D e p e n d in g o n th e p o s itio n o f th e s in g le p o le d o u b le t h r o w (S P D T )

s w itc h , th e s y s te m c o n s is ts o f tw o s u b s t r u c t u r e s . T h e s w itc h in g

v a r ia b le u is r e la t e d to th e S PD T p o s itio n , u = 1 w h e n th e i n d u c t o r is

c o n n e c te d t o Vg. u = 0 w h e n th e in d u c t o r is c o n n e c t e d to g r o u n d .

T h e s u b s t r u c t u r e s 0 a n d 1 a r e th e r e s u l t a n t c ir c u it s f o r u - 0 a n d

u = 1 r e s p e c tiv e ly , a n d a r e s h o w n in F ig . 3 .1 3 b , a n d F ig . 3 .1 3 c . T h e

d y n a m ic e q u a tio n s o f th e s y s te m a r e ,

d u 0 v 0

C d t R

di.

L dF = V9u ~v °

T h e s a m e e q u a tio n s r e p r e s e n t b o th s u b s t r u c t u r e s 0 a n d 1 d e p e n d in g

o n t h e v a lu e ta k e n b y th e s w itc h in g v a r ia b le it . T h e s y s t e m e q u a tio n s

37

1 = 1 -s> 2

(a)

38

m a y b e p u t in t h e fo llo w in g f o r m .

d zv 0 _ i _ i d vQ V g

d t z ~ TCrV° EC d t + I c “

W ith th e d e fin it io n o f v 0 = x 1: d v c/ d t = x 2 ; 1 / L C = co02 ; 2<fo>0 = 1 / E C ,

th e fo llo w in g r e p r e s e n t a t i o n is o b ta in e d .

x i ~ x 2

x 2 ~ —u 0z x 1—2 ^ a 0x z + u 0z Vgu

I n n o r m a l c o n v e r t e r s t h e d a m p in g f a c t o r ^ is le s s t h a n 1, r e s u lt in g in

c o m p le x c o n ju g a t e e ig e n v a lu e s w ith n e g a t iv e r e a l p a r t s .

F o r s u b s t r u c t u r e 0, u = 0 a n d t h e s y s t e m e q u a t io n s a r e

i d e n t i c a l to th o s e g iv e n in S e c tio n 3 .1 .3 . T h e e q u i li b r i u m p o in t is

g iv e n b y v 0 = d v 0/ d t = 0. T h e p h a s e t r a j e c t o r i e s f o r th e s u b s t r u c t u r e

0 a r e s h o w n in F ig . 3 .1 4 . F o r s u b s t r u c t u r e 1, u = 1 a n d th e s y s te m

e q u a t io n s h a v e a n e x t r a fo r c in g t e r m u 02 Vg . T h e e q u i li b r i u m p o in t is

t h e n u a = Vg . T h e e ig e n v a lu e s b e in g c o m p le x w ith n e g a t iv e r e a l p a r ts ,

t h e p h a s e t r a j e c t o r i e s a r e s im ila r to th o s e g iv e n in S e c tio n 3 .1 .3 , a n d

a r e a ls o s h o w n in F ig . 3 .1 4 .

We n o w s y n th e s iz e a c o n t r o l la w g iv e n b y

d v 0

S u b s tr u c tu r e 0 h o ld s fo r (v c— F0 ) + - > 0

d v 0 S u b s tr u c tu r e 1 h o ld s fo r (^ 0 —F0*) + t ^ < 0

T h e s w itc h in g b o u n d a r y e s ta b lis h e d b y t h e a b o v e c o n t r o l la w , a n d

p a r t o f th e t r a j e c t o r i e s v a lid in t h e tw o r e g io n s d e fin e d b y th e

s w itc h in g b o u n d a r ie s a r e s h o w n in F ig . 3 .1 5 . T h e c o n s e q u e n c e s of

t h e a b o v e s t r u c t u r e - c o n t r o l la w a r e a s fo llo w s .

i) T h e p h a s e t r a j e c t o r i e s e v e r y w h e r e in t h e p h a s e p la n e a r e

39

U=

U= 1

=

1

■Vo

u —

F ig . 3 . 1 4 The t r a j e c t o r i e s o f th e s u b s t r u c t u r e s i n v o l v e d i n th e b u c k c o n v e r t e r . T h e t r a j e c t o r i e s m a r k e d u = 0, a n d u — 1 c o r r e s p o n d to t h e a c t i v e a n d n o n a c t iv e s u b s t r u c t u r e s o f th e b u c k c o n v e r t e r r e p e c t i v e l y .

d t

=0

(Vo-V o) J -rd V o /d t^ O

=0

0

=0

Vo

u = l

u=

F ig . 3 . 1 5 Th e d e s i r e d r e s p o n s e o f th e b u c k c o n v e r t e r

( dvo

v c ) + T ~jj±~ ^ s e e n to be a s l i d i n g r e g i m e . The

40

d ir e c te d to w a rd s th e s w itc h in g b o u n d a ry .

ii) W hen th e s w itc h in g b o u n d a ry is c o n s id e re d as a t r a je c to r y , th e r e e x is ts a s ta b le o p e ra tin g p o in t g iv e n b y v 0 = F0°.

The s te a d y s ta te o p e ra tin g p o in t d e fin e d b y th e s w itc h in g b o u n d a ry is d iffe r e n t f r o m th o s e o i e ith e r o f th e s u b s tru c tu re s . S ta r tin g f r o m z e ro in it ia l c o n d itio n s , th e s y s te m m o tio n is show n in Fig. 3.16. The o v e r a ll s y s te m m o tio n c o n s is ts o f tw o p a r ts . The f i r s t p a r t is th e m o tio n f r o m a n y a r b it r a r y in i t i a l c o n d itio n on th e p h ase p la n e t i l l th e RP re a c h e s th e s w itc h in g b o u n d a ry o r th e s lid in g lin e . The tim e ta k e n f o r th is p a r t o f th e m o tio n depends o n th e s y s te m p a ra m e te rs a n d w ill be a s m a ll f r a c t io n o f th e t o t a l re s p o n s e tim e in a w e ll d e s ig n e d s y s te m . The s e c o n d p a r t o f th e m o tio n is fr o m w h e re th e RP h its th e s lid in g lin e , to th e s te a d y s ta te o p e ra tin g p o in t o n tn e s lid in g lin e . In th is e x a m p le th is m o tio n is w ith a. tim e c o n s ta n t o f r , a n d is in d e p e n d e n t o f th e s y s te m p a ra m e te rs .

3 .2 .5 E q u a tio n s o f M o tio n o n th e S w itc h in g B o u n d a ry

The sam e e x a m p le sh o w n in th e p re v io u s s e c tio n is now s tu d ie d f o r th e e q u a tio n s o f m o tio n o f th e s y s te m RP a lo n g th e s w itc h in g b o u n d a ry . I t was seen t h a t th e r e s u lt a n t m o tio n o f th e RP, in th e case o f id e a l s w itc h in g , is d ir e c te d a lo n g th e s w itc h in g b o u n d a ry . One m a y th e n ta k e th e e q u a tio n o f th e s lid in g b o u n d a ry to r e p r e s e n t th e s y s te m m o tio n . A lte r n a tiv e ly , th e a c tu a l s y s te m m o tio n ta k in g in to a c c o u n t th e n o n id e a litie s in s w itc h in g m a y be c o m p u te d a n d th e r e s u lt a n t m o tio n a r r iv e d a t b y th e p ro c e s s o f ta k in g th e n o n id e a litie s to th e lim it o f z e ro . T h is m e th o d o f a r r iv in g a t th e

41

(Vo-Wo) + T d V o /d t = <

0

F ig . 3 . 1 6 T y p ic a l s t a r t i n g t r a n s i e n t o f th e b u c k c o n v e r t e r s e e n o n th e p h a s e p l a n e . The s t e a d y s t a t e o p e r a t i n g p o i n t is v 0 = V0 *.

t = A t

- A t

c=0

(V o-V o )