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Scholarship at UWindsor

Scholarship at UWindsor

Electronic Theses and Dissertations

Theses, Dissertations, and Major Papers

1-1-1966

Stability studies of automatic control systems.

Stability studies of automatic control systems.

Michael C. Wong

University of Windsor

Follow this and additional works at:

https://scholar.uwindsor.ca/etd

Recommended Citation

Recommended Citation

Wong, Michael C., "Stability studies of automatic control systems." (1966). Electronic Theses and

Dissertations. 6459.

https://scholar.uwindsor.ca/etd/6459

This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email

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S T A B I L I T Y STUD I E S O F A U T O M A T I C

C O N T R O L SYST E M S

by

M I C H A E L C. W O N G

A T h e s i s

S u b m i t t e d to the F a c u l t y of G r a d u a t e Stud i e s thro u g h the D e p a r t m e n t of E l e c t r i c a l E n g i n e e r i n g in P a r t i a l F u l f i l l m e n t

of the R e q u i r e m e n t s for the D e g r e e of M a s t e r o f A p p l i e d S c i e n c e at the

U n i v e r s i t y of W i n d s o r

*

W i n d s o r , O n t a r i o

(3)

IN F O R M A T IO N T O U S E R S

T h e quality of this reproduction is d e p e n d e n t upon the quality of the copy

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and th e re are m issing pages, th e s e will be noted. Also, if unauthorized

copyright m aterial had to be rem o ved , a note will indicate the deletion.

®

UMI

U M I M icroform E C 5 2 6 4 0

Copyright 2 0 0 8 by P ro Q u e s t LLC.

All rights reserved . This m icroform edition is protected against

unauthorized copying u n d er T itle 17, U nited S ta te s C o de.

P ro Q u e s t LLC 7 8 9 E. E ise n h o w e r P ark w ay

P O Box 1 34 6 A nn A rbor, M l 4 8 1 0 6 -1 3 4 6

(4)

.APP ROVED BY :

AX

S

A

' / f o t O Z f r

% P

(5)

This thesis p r e s e n t s a n e w and e f f e c t i v e m e t h o d for the a n a l y s i s

and syn t h e s i s of linear f e e d b a c k c o n t r o l systems. The p r o p o s e d

m e t h o d gives resu l t s c o m p a r a b l e to those o b t a i n e d b y the c o n v e n t i o n a l

N y q u i s t and R o o t L o c u s m e t h o d s and y i e l d s i n f o r m a t i o n s no t o b t a i n a b l e

f r o m the o t h e r two m e t hods.

B y m a p p i n g the s t a b i l i t y b o u n d a r y of the c o m p l e x s-pl a n e ont o

the i m a g i n a r y axis of the c o m p l e x z-plane, H u r w i t z c r i t e r i o n can be

appl i e d to y i e l d a n a l y t i c a l e x p r e s s i o n s , f r o m w h i c h s t a b i l i t y

b o u n d a r i e s c o r r e s p o n d i n g to d i f f e r e n t r e l a t i v e s t a b i l i t y c o n s t r a i n t s

can be d r a w n ( otand ^ d i a grams) for d i f f e r e n t v a l u e s of s y s t e m

parameters. The st a b l e and u n s t a b l e r e g i o n s are l o c a t e d and s y s t e m

a n a lysis is r e d u c e d to the si m p l e p r o c e s s of r e a d i n g the diagrams.

B y s u p e r p o s i t i o n of the «; and § d i agrams, the d o m i n a n t r oots can

b e located w h i c h can b e u s e d for d e s i g n purposes. T h e p r o p o s e d m e t h o d

#

is m o s t g e n e r a l and can be a p p l i e d to systems w i t h any d e g r e e of

c o m p l e x i t y and u n d e r an y p a r t i c u l a r r e l a t i v e s t a b i l i t y constraint.

It ha s d e f i n i t e a d v a n t a g e o v e r the p o p u l a r N y q u i s t and R o o t L o c u s

m e t h o d s in m a n y a p p l i c a t i o n s e s p e c i a l l y in the a n a l y s i s and s y n t h e s i s

of systems w i t h two or m o r e v a r i a b l e parameters.

iii

(6)

A C K N O W L E D G E M E N T

The a u t h o r w i s h e s to e x p r e s s his s i n c e r e a p p r e c i a t i o n and

g r a t i t u d e to Dr. H. H. Hw a n g , w h o s u p e r v i s e d this w ork, for his

c o n t i n u i n g e n c o u r a g e m e n t and h e l p f u l advice.

A c k n o w l e d g e m e n t is also due to 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

of C a n a d a for the a w a r d o f a s c h o l a r s h i p w h i c h f i n ances this project.

(7)

Pag e

A B S T R A C T ... iii

A C K N O W L E D G E M E N T S ... iv

C O N T E N T S ... v

C H A P T E R

I. I N T R O D U C T I O N ... 1

(a) I m p o r t a n c e of S t a b i l i t y Stud i e s

(b) C l a s s i f i c a t i o n of S t a b i l i t y

(c) C o n v e n t i o n a l M e t h o d s for R e l a t i v e S t a b i l i t y

Stud i e s

II. A N E W A N A L Y T I C A L A P P R O A C H T O R E L A T I V E S T A B I L I T Y

ST U D I E S ... 4

(a) F o r m u l a t i o n of T h e H u r w i t z C r i t e r i o n

(b) E x t e n s i o n of H u r w i t z C r i t e r i o n to R e l a t i v e

S t a b i l i t y Studies

III. A P P L I C A T I O N TO P H Y S I C A L S Y S T E M ... 12

(a) C h a r a c t e r i s t i c E q u a t i o n of G i v e n S y s t e m

(b) <*> - D i a g r a m s

(c) $ - D i a g r a m s

(d) L o c a t i o n of S t a b i l i t y R e g i o n s

(e) L o c a t i o n of D o m i n a n t R o o t s

(f) I n t e r p r e t a t i o n o f D i a g r a m s

(g) S y s t e m A n a l y s i s and D e s i g n

IV. D I S C U S S I O N A N D C O N C L U S I O N ... 26

v

(8)

A P P E N D I X ... 27

(1) T A B L E F O R C O M P U T I N G C O E F F I C I E N T S O F F(Z)

(2) ol - D I A G R A M S

(3) 3 “ D I A G R A M S

(4) S U P E R P O S I T I O N OF <*, A N D 1 D I A G R A M S

R E F E R E N C E S ... 38

V I T A A U C T Q R I S ... 39

(9)

I N T R O D U C T I O N

I m p o r t a n c e of S t a b i l i t y Studies

S t a b i l i t y stud i e s is one of the m o s t i m p o r t a n t topies in f e e d b a c k

c o n t r o l systems. F r o m a c l a s s i c a l p o i n t of view, a c e n t r a l p r o b l e m

of f e e d b a c k c o n t r o l the o r y can be i d e n t i f i e d as a s t a b i l i t y problem.

Th e purp o s e of a n y f e e d b a c k c o n t r o l s y s t e m is to h a v e the c o n t r o l l e d

o u t p u t of the s y s t e m b e a r a d e f i n i t e and k n o w n r e l a t i o n s h i p to the

r e f e r e n c e input. T h e r e f o r e it is i m p e r a t i v e that the t r a n s i e n t should

die d o w n after the c e s s a t i o n of the d i s t u rbance. Syst e m s in w h i c h

the t r a n s i e n t inc r e a s e s w i t h o u t b o u n d a f t e r c e s s a t i o n of the d i s t u r b a n c e

are said to b e unstable. I n s t a b i l i t y is u n d e s i r a b l e in that the

c o n t r o l l e d o u t p u t is not u n d e r c o n t r o l and it m a y cause h a r m and

fail u r e of the system. T h e r e f o r e it is m o s t i m p o r t a n t to h a v e an

e f f e c t i v e m e t h o d w h e r e b y the s t a b i l i t y p r o b l e m can be r e a d i l y studied.

C l a s s i f i c a t i o n o f S t a b i l i t y

In general, s t a b i l i t y can be stud i e d u n d e r two d i f f e r e n t h e a d i n g s

namely, (1) A b s o l u t e S t a b i l i t y and (2) R e l a t i v e Stab ility. A s y s t e m

is said to be a b s o l u t e l y stable if the t r a n s i e n t dies d o w n as time

a p p r o a c h e s i n f i n i t y , a n d a s y s t e m is s a i d to h e r e l a t i v e l y s t a b l e if

the t r a n s i e n t dies d o w n w i t h i n a c e r t a i n p e r i o d of time or a s p e c i f i e d

n u m b e r of cycles. F o r any f e e d b a c k c o n t r o l system, a b s o l u t e s t a b i l i t y

1

(10)

2

is a n e c e s s a r y condition. But it is not s u f f i c i e n t b e c a u s e a s y s t e m

m a y be a b s o l u t e l y stable, bu t if the t r a n s i e n t takes such a long time

to d e c a y that it is not p r a c t i c a l for use. There f o r e , r e l a t i v e

s t a b i l i t y is a m o r e i m p o r t a n t r e q u i r e m e n t for a s y s t e m t han a b s o l u t e

s t a b i l i t y b e c a u s e o n c e it is e s t a b l i s h e d that a s y s t e m is r e l a t i v e l y

stable, it is a u t o m a t i c a l l y a b s o l u t e l y stable.

The p r o b l e m of a b s o l u t e s t a b i l i t y is w e l l u n d e r s t o o d and m a n y

m e t h o d s h a v e b e e n d e v e l o p e d b y w h i c h the a b s o l u t e s t a b i l i t y of a

s y s t e m can be r e a d i l y determined. H o w e v e r , the m e a s u r e m e n t of r e l a t i v e

s t a b i l i t y ha s not b e e n d e v e l o p e d to the state w h e r e c o m p a r a b l e d e f i n i t e

t echniques are available. T h e q u a n t i t i e s are not w e l l d e f i n e d and the

i n t e r p r e t a t i o n of r e l a t i v e s t a b i l i t y d i f f e r s a mong individuals. In

general, there are two c r i t e r i a w h i c h are c o m m o n l y u s e d to s p e c i f y

the r e l a t i v e s t a b i l i t y r e q u i r e m e n t s of a system. O n e of these is the

r e q u i r e m e n t that all the t r a n s i e n t terms of the r e s p o n s e m u s t d e c a y

at least as r a p i d l y as the f u n c t i o n e -0<t. Thi s m e a n s that all the

r oots of the c h a r a c t e r i s t i c e q u a t i o n m u s t b e loca t e d o n the lefjt side

o f a line p a r a l l e l to and at a d i s t a n c e IT = -c* f r o m the i m a g i n a r y

as s h o w n in figure 1-1.

/

n

/

stable / u n s t able

*

cr

- ( 0

/

/

/

j

S - P L A N E

Fig. 1-1

table uns

s table

(11)

Th e other is the r e q u i r e m e n t that all oscillatory terms in the t r a n s i e n t

r e s p o n s e m u s t die d o w n in a s p e c i f i e d n u m b e r of cycles. T h i s is

e q u i v a l e n t to s p e c i f y i n g a m i n i m u m v a l u e of the d a m p i n g ratio J that

can be tolerated. This r e q u i r e s that all the r oots of the c h a r a c t e r i s t i c

eq u a t i o n m u s t be l o c a t e d w i t h i n the sector b o u n d e d b y the c o n s t a n t J

line as s hown in figure 1-2.

C o n v e n t i o n a l M e t h o d s for R e l a t i v e S t a b i l i t y Stud i e s

The m o s t p o p u l a r m e t h o d s w h i c h are b e i n g use d for r e l a t i v e s t a b i ­

lit y studies are the N y q u i s t and R o o t L o c u s m e t hods. T h e former is

a f r e q u e n c y d o m a i n a p p r o a c h w h i c h p e r m i t s the d e s i g n e r to m o d i f y the

o p e n loop s y s t e m in o r d e r to o b t a i n the a p p r o p r i a t e closed loop

f r e q u e n c y c h a r a c t e r i s t i c s . H o w e v e r , it can be a p p l i e d to systems

w h e r e the o p e n loop f u n c t i o n is k n o w n w i t h i n a g ain factor. E a c h

time a time c o n s t a n t or p a r a m e t e r is chan ged, a ne w N y q u i s t d i a g r a m

m u s t be drawn. A l s o if the t r a n s f e r f u n c t i o n is v e r y comp l e x and

no t in f a c t o r e d form, N y q u i s t plots m a y be v e r y difficult. T h e latter

is a pole and zero a p p r o a c h w h i c h r e a d i l y p r o v i d e s impormations, a b o u t

all the roots of ithe c h a r a c t e r i s t i c e q u a t i o n for a g i v e n v a l u e of the

o p e n loop gain. H o w e v e r , it a lso has m a n y s i g n i f i c a n t limitations.

T h e m o s t i m p o r t a n t of w h i c h is the fact that it is b a s i c a l l y a one

p a r a m e t e r m e t h o d w h i c h is v e r y i n c o n v e n i e n t for a n a l y s i s and s y n t h e s i s

of m u l t i - p a r a m e t e r systems. T h e r e f o r e b e t t e r and m o r e e f f e c t i v e

m e t h o d s are desirable. In the c h a p t e r s following, a ne w a n a l y t i c a l

m e t h o d is d e v e l o p e d w h i c h can be a p p l i e d to syslems of a n y d e g r e e of

complexity. Th e m e t h o d is e s p e c i a l l y s u i t a b l e for the a n a l y s i s and

syn t h e s i s of systems w i t h two or m o r e v a r i a b l e parameters.

(12)

CHAPTER I I

A N E W A N A L Y T I C A L A P P R O A C H T O R E L A T I V E S T A B I L I T Y STUD I E S

As m e n t i o n e d in the p r e v i o u s chapter, the p r o b l e m of r e l a t i v e

s t a b i l i t y is

a

p r o b l e m of d e t e c t i n g w h e t h e r or not any r oots of the

c h a r a c t e r i s t i c p o l y n o m i a l lie o u t s i d e a s p e c i f i e d s t a b i l i t y region.

T h u s s t a b i l i t y studies is r e d u c e d to the s tudy of the l o c a t i o n of

the roots of the c h a r a c t e r i s t i c e q u a t i o n of the system..

The H u r w i t z c r i t e r i o n ha s b e e n one of the m o s t p o p u l a r tools in

a b s o l u t e s t a b i l i t y analysis. W h e n a p p l i e d to a c h a r a c t e r i s t i c e q u a t i o n

w i t h real c o e f f i c i e n t s , the H u r w i t z c r i t e r i o n r e a d i l y i n d i c a t e s the

e x i s t e n c e of r o o t s w i t h p o s i t i v e real parts and thus i n d i c a t i n g

instability. H o w e v e r , it has the o b v i o u s l i m i t a t i o n of b e i n g o nly

a ble to be a p p l i e d to e q u a t i o n s w i t h rea l coefficients. A l s o it does

no t i n d i c a t e the d e g r e e of s t a b i l i t y of the s y s t e m and it gives little

i nsi g h t into s y s t e m design. Thus it has ha d v e r y little a p p l i c a t i o n

in r e l a t i v e s t a b i l i t y studies. H o w e v e r , w i t h some m a n i p u l a t i o n s ,

the H u r w i t z c r i t e r i o n can be d i r e c t l y a p p l i e d to y i e l d i n f o r m a t i o n

a b o u t r e l a t i v e stability. It a l s o y i e l d s a n a l y t i c a l e x p r e s s i o n s

f r o m w h i c h s t a b i l i t y b o u n d a r i e s can be drawn. T hus i n f o r m a t i o n a bout

the d y n a m i c a l b e h a v i o u r of the s y s t e m can be o b t a i n e d in a sim p l e and

s t r a i g h t f o r w a r d manner.

F o r m u l a t i o n of T h e H u r w i t z C r i t e r i o n

H u r w i t z f o r m u l a t e d .his-; c r i t e r i o n in a d e t e r m i n a n t a l f o r m and

(4)

stated it in the f o l l o w i n g way. F o r a p o l y n o m i a l

(13)

P(s) = an sn + a n - l811" 1 +

n

*=r- k

+ a, s + a 1 o

ai, s

(

2

-

1

)

k=0

w h e r e a^'s are real, all the roots of the c h a r a c t e r i s t i c e q u a t i o n

P(s) = 0 w i l l h a v e n e g a t i v e rea l p a r t s if the f o l l o w i n g c o n d i t i o n s

are satisfied.

C D ak > 0

(2) the H u r w i t z d e t e r m i n a n t s 0 for k = 1, 2,--- n-1

w h e r e

A

a^

aQ

0

0

----

0

a 3 a 2 a l a0 ... °

a^ a^ a^ ^*2. — — — — ——— — — — 0

0 a ,

n-1

(

2

-

2

)

If some of the H u r w i t z d e t e r m i n a n t s are n e g ative, this ind i c a t e s the

e x i s t e n c e of roots w i t h p o s i t i v e r eal p arts and the n u m b e r of such

r oots c o r r e s p o n d s to the n u m b e r of c h a n g e s of sign in the s e q u e n c e

A . ^ A2 ^ 3 " " " -^\i-l' ^ the ^ c h a r a c t e r i s t i c e q u a t i o n has roots

l y i n g o n the i m a g i n a r y axis, then the n - l t h H u r w i t z d e t e r m i n a n t

(3) w i l l v a n i s h

E x t e n s i o n of H u r w i t z C r i t e r i o n to R e l a t i v e S t a b i l i t y Studies

It h a s b e e n m e n t i o n e d that the H u r w i t z c riterion, w h e n appl i e d

to p o l y n o m i a l s w i t h rea l c o e f f i c i e n t s , r e a d i l y i n d i c a t e s the e x i s t e n c e

of roots w i t h p o s i t i v e rea l p arts and thus ind i c a t e s instability.

T h e r e f o r e the a p p l i c a t i o n of the H u r w i t z c r i t e r i o n to r e l a t i v e s t a b i l i t y

(14)

6

s tudies g e n e r a l l y c o n s i s t s of one or b o t h of the f o l l o w i n g processes;

(1) the m a p p i n g of the s t a b i l i t y b o u n d a r y o nto the i m a g i n a r y axis

of a n e w plane.

(2) the r e a l i z a t i o n of the c o e f f i c i e n t s of the t r a n s f o r m e d polynomial.

For systems u n d e r the f irst r e l a t i v e s t a b i l i t y c o n s t r a i n t

( m i n i m u m o< ), the r e q u i r e m e n t is that all the roots of the c h a r a c t e r ­

istic e q u a t i o n be located o n the left side of a line p a r a l l e l to and

at a d i s t a n c e 0 = - o< f r o m the i m a g i n a r y axis of the s - p l a n e as s hown

on figure 2-l(a). rr

H-S-PL A N E

a.

Fig. 2-l(a)

w

cr*

= 4 >

S *• Z -C X

Z - P L A N E

o

cr

o

e D

a.

i-i

S T A B I L I T Y B O U N D A R Y B E F O R E

v

T R A N S F O R M A T I O N

let the c h a r a c t e r i s t i c p o l y n o m i a l be

Fig. 2-l(b)

S T A B I L I T Y B O U N D A R Y A F T E ^ :

T R A N S F O R M A T I O N

P(s) = 2 . ak s k = 0

(

2

-

1

)

w h e r e a are real, k

B y u s i n g the t r a n s f o r m a t i o n s = Z - d , the £1= - « b o u n d a r y of the

complesc. s - p l a n e is m a p p e d o n t o the i m a g i n a r y axis of the C o m p l e x

(15)

P(Z-«) = ; >

ak (Z-rt)

k=0

.JL. k =2 ; ck z ;

k=0

(

2

-

2

)

w h e r e Ck 's are coeffs. of P(Z-<x)

S i n c e the ak 's a r e real and o< is real, t h e r e f o r e the c o e f f i c i e n t s

C ^'s of the t r a n s f o r m e d p o l y n o m i a l are real. Hence, the H u r w i t z

c r i t e r i o n can be a p p l i e d d i r e c t l y to P(Z-<tf) to yield i n f o r m a t i o n s

a b o u t the d y n a m i c a l b e h a v i o u r of the system.

Fo r systems u n d e r the second k i n d of r e l a t i v e s t a b i l i t y c o n s t r a i n t

( m i n i m u m | ), the r e q u i r e m e n t is that all the roots of the c h a r a c t e r i s t i c

e q u a t i o n be l o c a t e d w i t h i n the sector b o u n d e d b y the m i n i m u m $ lines

jY

S -PL A N E (Fig. 2-2).

Fig. 2-2(a)

7

s = Ze'

S T A B I L I T Y B O U N D A R Y C O R R E S P O N D I N G

TO TH E S E C O N D R E L A T I V E

S T A B I L I T Y C O N S T R A I N T

±j0

> /

/

/

Z - P L A N E

/

/ «

/

/

Fig. 2-2(b)

S T A B I L I T Y B O U N D A R Y OF

T R A N S F O R M E D -POLYNOMIAL

B y u s i n g s i m i l a r techniques, the s t a b i l i t y b o u n d a r y ca n be m a p p e d

o n t o the i m a g i n a r y axis of the complex: Z - p l a n e as s h o w n o n figure 2-2

(b).

Consider the characteristic polynomial.

(16)

8

(

2

-

1

)

(2-3)

(2-4)

j ^

Physic a l l y , the t r a n s f o r m a t i o n s = Ze c o r r e s p o n d s to a c l o c k w i s e

r o t a t i o n of the r o o t s of the c h a r a c t e r i s t i c p o l y n o m i a l P(s) t h r o u g h

an angle of 6 d e g r e e s as s hown in fi g u r e 2-3(a)(b).

Z - P L A N E

Fig. 2-3(b)

L O C A T I O N O F ZEROS O F P (Z) S - P L A N E

Fig. 2-3(a)

L O C A T I O N O F ZEROS OF P(s)

j 8

M a t h e m a t i c a l l y , the t r a n s f o r m a t i o n s = Ze g ives ris e to a

poly-(

6

)

n o m i a l P^(Z) w i t h complex; c o e f f i c i e n t s as shown in e q u a t i o n 2-4.

S i n c e the H u r w i t z c r i t e r i o n is a p p l i c a b l e o n l y to p o l y n o m i a l s w i t h

r e a l c o e f f i c i e n t s , t h e r e f o r e furt h e r o p e r a t i o n s m u s t be p e r f o r m e d to

r e a l i z e the t r a n s f o r m e d polynomial. To do this, let

s = Z e _J& (2-5)

P ( s ) = 3 > aj,sk k=0

n n "l ,

= an s + an - l s + ' ...+ a l s + a0

w h e r e a^'s are real.

je let s = Ze

the n Pi (Z) a, (Ze'** )k k=0

o J h * 7n j. a jtn-i)* 7n-l

= a„e Z + a„ •. e Z +

---n n- 1

(17)

Ji-

- j fl k and f o r m P„(Z) = a, (Ze )

k=0

- j h0 n -j(w-') 0 „n-l

= a e Z + a , e Z +

n n-1

-je

+ a^e Z + Sq

(

2

-

6

)

Th i s c o r r e s p o n d s to a c o u n t e r c l o c k w i s e r o t a t i o n of the roots of P(s)

through an a ngle of Q d e g r e e s as s hown in fi g u r e 2-4(a)(b).

jod

JY

S - P L A N E

a

Fig. 2-4(a)

ZEROS OF P(s)

Z-PL A N E

Fig. 2-4(b)

ZEROS OF P2(Z)

By m u l t i p l y i n g P^(Z) and P ^ ( Z ) together, a n e w p o l y n o m i a l is obta ined.

F(Z) = P1( Z ) P2(Z) = : >

k=0

2n

a v ( Z e J D r > a. (Ze ) k=0

k=0

A k Z

(

2

-

6

)

w h e r e A ^ ' s are coeffs. o f F(Z)

Physic a l l y , e q u a t i o n (2-6) c o r r e s p o n d s to a s u p e r p o s i t i o n of the zeros

of P-^(Z) and P2(Z) as s h o w n in figure 2-5.

(18)

10

Z-PLANE

Fig. 2-5 ZEROS OF F(Z)

It can be seen that the zeros of F(Z) o c c u r in complex, c o n j u g a t e

pairs and t h e r e f o r e the c o e f f i c i e n t s of F(Z) are rea l c o e f ficients.

Ho w ever, it is o b s e r v e d that F(Z) has twice as m a n y r oots as the

o r i g i n a l p olynomial. T his is b e c a u s e o f the fact that in the proc e s s

of r e a l i z i n g the c o e f f i c i e n t s , w e i n t r o d u c e d c o n j u g a t e roots. Bu t

n e v e r t h e l e s s , this has no ef f e c t on our a n a l y s i s b e c a u s e in r e l a t i v e

s t a b i l i t y analysis, w e are o n l y i n t e r e s t e d in the l o c a t i o n of the

roots of the c h a r a c t e r i s t i c e q u a t i o n and no t the number. To i l l u s ­

trate the a bove process, c o n s i d e r a se c o n d o r d e r s y s t e m w h o s e

c h a r a c t e r i s t i c p o l y n o m i a l is g i v e n by,

2 P(s) = aQ + a^s + a 2 s

the n Pj. (Z) = a^ + a^eJ0 Z + a ^ e Z^

, v -je 9 2

and P2(Z) = a Q + a l & J Z + a 2 e Z

F(Z) = P1(Z ) P2(Z)

= (aQ + Z + a2e ^ e-Z^) (aQ + a^e Z + a 2 e J Z )

2 2 2

= a^ + (2aQa^ c o s 0 )Z + (a^ + 2 a Q a2 c o s 2 © )Z

+ ( 2 a ]_a2 c o s G > ) Z3 + a ^ Z4 (2-7)

4 ^

F(Z) = ; > A . Z (2-8)

k=0

(19)

p o l y n o m i a l c a n be o b t a i n e d b y the same procedure. The c o e f f i c i e n t s

A ^ ' s can be c o m p u t e d onc e and for all and put in a table. Thus for

an y g i v e n c h a r a c t e r i s t i c p o l y n o m i a l P(s), the c o r r e s p o n d i n g t r a n s ­

formed p o l y n o m i a l F(Z) ca n be o b t a i n e d r e a d i l y b y r e f e r r i n g to the

th

table. A table for c o m p u t i n g the c o e f f i c i e n t s of a 10 o r d e r

p o l y n o m i a l h a s b e e n w o r k e d o u t and is s h o w n in T a b l e I.

The above m e t h o d is m o s t g e n e r a l and can be a p p l i e d to the r e l a ­

tive s t a b i l i t y stud i e s o f an y s y s t e m u n d e r any p a r t i c u l a r k i n d of

r e l a t i v e s t a b i l i t y cons traint. B y m a p p i n g the s t a b i l i t y b o u n d a r y

o n t o the i m a g i n a r y axis of a ne w plane, and r e a l i z i n g the c o e f f i c i e n t s

of the t r a n s f o r m e d p o l y n o m i a l the H u r w i t z c r i t e r i o n can be a p p l i e d

ea s i l y to y i e l d i n f o r m a t i o n s a b o u t the d y n a m i c a l b e h a v i o u r of the

system.. F u r t h e r m o r e , by e q u a t i n g the n - l 1^ H u r w i t z d e t e r m i n a n t to

zero, a n a l y t i c a l e x p r e s s i o n s can be o b t a i n e d f rom w h i c h s t a b i l i t y

b o u n d a r i e s ca n be d r a w n for d i f f e r e n t v a l u e s o f s y s t e m parameters.

Thu s r e l a t i v e s t a b i l i t y studies is f u r t h e r s i m p l i f i e d to r e a d i n g

diagrams,..

(20)

CHAPTER I I I

A P P L I C A T I O N T O P H Y S I C A L S Y S T E M

T he v e r s a t i l i t y of the m e t h o d can. be b e s t i l l u s t r a t e d w h e n

a p p l i e d to systems w i t h two or three v a r i a b l e parameters. C o n s i d e r

a p h y s i c a l s y s t e m ^ ^ w h o s e d i a g r a m is g i v e n as follows.

L O A D

m o to r am plidyne

i eld

supply v o lta g e

Fig. 3-1 T Y P I C A L P O S I T I O N C O N T R O L S Y S T E M ^

T h e s y s t e m p a r a m e t e r s are g i v e n b y ^ P ;

K a = net c o n t r o l field a m p e r e s per v o l t error, signal

K g = n o - l o a d a m p l i d y n e t e r m i n a l v o l t a g e per ne t c o n t r o l f ield current,

L f

Tf = = time c o n s t a n t of a m p l i d y n e q u a d r a t u r e field in seconds

1^, = JRa/kj,- K e = time c o n s t a n t of m o t o r and l oad in seconds

(21)

Kt = torque f r o m m o t o r per a m p e r e of m o t o r a r m a t u r e c u r r e n t K & = v o l t a g e f r o m f e e d b a c k p o t e n t i o m e t e r per r a d i a n of m o t o r

R a = total a r m a t u r e r e s i s t a n c e of m o t o r a m p l i d y n e and leads.

F r o m the g i v e n system, the f o l l o w i n g b l o c k d i a g r a m is obtained.

R(s)

o

E(s),

G(s) C(s) = Q(s)

- ' B ( s )

H(s)

Fig. 3-2 B L O C K D I A G R A M

w h e r e

R(s) = r e f e r e n c e input

E(s) - e rror or a c t u a t i n g signal

B(s) = f e e d b a c k signal

G(s) = o p e n loop t r a n s f e r f u n c t i o n

H(s) = f e e d b a c k t r a n s f e r f u n c t i o n

C(s) = c o n t r o l l e d o u t p u t

In terms o f E(s), the t r a n s f o r m of the p o s i t i o n of the m o t o r

0(s

ca n be w r i t t e n as:

Cl)

Q(s)

G(s)

KaKgE(s')

K e s ( l + T f s ) (1+ Tm s)

K aK g / K e

s ( l + T f s)( l + T m s)

and H(s) = B(s) = Kg ©(s)

(22)

1 4

G(s) H(s) = K aK gK e / K P

s ( l + T f s ) ( 1 + T m s)

K

s ( l + T f s)(l+Tm s )

w h e r e K = K aK gK e / K e

= o v e r a l l s y s t e m gai n

T h e c h a r a c t e r i s t i c e q u a t i o n is l +G(s)H(s) = 0

1 + f = 0 or s ( l + T f s ) ( 1 + T m s)

f r o m w h i c h P(s) = K + s + (Tf + Tm ) s2 + T f T m s3

= K + s + a2s^ + ags^ (3-1)

w h e r e a2 = Tf + Tm

a3 = *f T *

o< - D i a g r a m s

The f i r s t c r i t e r i o n of r e l a t i v e s t a b i l i t y r e q u i r e s all the r oots

of the c h a r a c t e r i s t i c e q u a t i o n to h a v e n e g a t i v e r eal parts less

than a s p e c i f i e d v a l u e of (X = U s i n g the t r a n s f o r m a t i o n s = Z-ot,

the s t a b i l i t y b o u n d a r y ( (T = - c* line) is m a p p e d o nto the i m a g i n a r y

axis of the complex. Z-plane.

P(s) = K + s + a2s2 + a3s3 (341)

P(Z-ei) = K + (Z-ot) + a2(Z-«)2 + a3(Z-*)3

= agZ^ + (a2-3o<a3)Z^ + (1+3 o<.2a3-2o< a 2 )Z

+ (K+ cX2a2 - - cx ^ a 3 ) (3-2)

(23)

A , -k

(1+3 3- 2 oc a2 ) ( K + © <2a2-o< - c*3a 3 )

a3 (a2-3oc a 3 )

0 0

0

(l+3<*2a 3- 2oia2)

a 3

(3-3)

■f“V\

T h e n-1 H u r w i t z d e t e r m i n a n t w i l l v a n i s h if there are r o o t s of the

c h a r a c t e r i s t i c e q u a t i o n l o c a t e d on the i m a g i n a r y Z axis w h i c h is the

r e l a t i v e s t a b i l i t y boundary. He n c e , b y e q u a t i n g the n - l ^ H u r w i t z

d e t e r m i n a n t to zero, an a n a l y t i c a l e x p r e s s i o n can be o b t a i n e d f r o m

,(2,3)

w h i c h r e l a t i v e s t a b i l i t y b o u n d a r i e s can be d r a w n . F r o m (3-2).

A

n_x = A 2 =

1+3 o< a3~2cx a2 K + < x a2 - « - c x a.

a 3 a 3 - 3c* a3

E q u a t i n g A n - 1 = 0, get

(3-4)

(l +3cx2a3-2c*a2) ( a3-3c * a 3 ) - a3(K+ c*2a 2 - - e*3a 3 ) = 0

E x p a n d i n g and r e a r r a n g i n g , get

.2

K = 1 _ [ 8 <><. a3(a2- c * a 3 ) - 2cx(a2+ a 3 ) + a3 ] a 3

(3-5)

e q u a t i o n for p l o t t i n g r e l a t i v e s t a b i l i t y b o u n d a r i e s ( c* - diagrams).

E q u a t i o n (3-5) is p l o t t e d for v a r i o u s v a l u e s of cx and s y s t e m

p a r a m e t e r s as s hown on D i a g r a m s 1, 2 and 3 ( <x - diagram).

§ - D i a g r a m s

W h e n the r e l a t i v e s t a b i l i t y c o n s t r a i n t is the m i n i m u m v a l u e of

the d a m p i n g r a t i o | , the r e l a t i v e s t a b i l i t y r e g i o n b e c o m e s the sector

b o u n d e d b y the c o n s t a n t J line as s h o w n in Fig. 2-2(a). B y u s i n g + j0

the t r a n s f o r m a t i o n s = Ze and T a b l e I, the t r a n s f o r m e d p o l y n o m i a l

F(Z) can b e o b t a i n e d fro m w h i c h the H u r w i t z d e t e r m i n a n t c a n be

(24)

16

formed^- B y e q u a t i n g the n - 1 ^ H u r w i t z d e t e r m i n a n t to zero, an a n a l y ­

tical e x p r e s s i o n is o b t a i n e d f r o m w h i c h r e l a t i v e s t a b i l i t y b o u n d a r i e s

c o r r e s p o n d i n g to a p a r t i c u l a r v a l u e o f S can be d r a w n (]§ - diagrams).

Th e m i n i m u m v a l u e of the d a m p i n g r a t i o d i f f e r s f o r different' syst e m s

a c c o r d i n g to the p a r t i c u l a r r e q u irement. . But. for o r d i n a r y systems,

a d a m p i n g r a t i o of 0.5 is u s u a l l y adequate,

let

j

= 0.5

then 'f' = cos = 60°

0 = 90° - 60° = 30°

T h e c h a r a c t e r i s t i c e q u a t i o n of the s y s t e m u n d e r s tudy is

2 3

P(s) = K + s + a2S + a^s (3-1)

T h e t r a n s f o r m e d p o l y n o m i a l F(Z) is w r i t t e n in g e n e r a l f o r m as:

F(Z) = A q + A j Z + A2Z2 + A3Z3 + A4Z4 + + A & Z6 (3-6)

w h e r e the c o e f f i c i e n t s are c o m p u t e d b y r e f e r r i n g to T a b l e I,

o

<

=

4

- k

2

A l

= 2 a ^ a Q C o s G — b K w h e r e : b *= 1 . 7 3 2

A2 =

2

+

2 a 2 & Q C O s 2

0

=

1

+

Ka2

a2 =

T f

+

A 3 =

ba£

a 4

=

a 2

2 3 a 3

= T£Tm

A 5 =

ba£a^

(25)

From, equation 3-6:

A 1 A o 0 0 0

A3 A2

A 1 A 0 0

^ n-1 = ^ 5 = a 5 A 4 A 3 A 2 A 1

0 a6

A 5 A 4 A 3

0 0 0 A 6 A 5

bK K2 0 0 0

b a2 1+Ka2 bK K2 0

= ba£ a3 2 a 2+ a

3 b a 2 1+ K a2 bK (3-7)

0 a32 b a2a2 3.22"' 33 b^2

0 0 0 a 32 b a 2 a 3

E q u a t i n g Z \ n _q to zero and simplifying, w e obtain:

3 2 3 3 2 2 4 s 2 5 3 2.

(a2 a 3 )K + (a3

a 2 a 3 - 2 a 2 a ^ ) K + (a2 + 2a2a 3 - 232^3)^

+ (a| a3 - af) = 0 (3-8)

F r o m e q u a t i o n (3-8), s t a b i l i t y b o u n d a r i e s are d r a w n for d i f f e r e n t

v a l u e s of s y s t e m p a r a m e t e r s as s h o w n on d i a g r a m s 4, 5, and: 6

( J - diagrams).

L o c a t i o n of S t a b i l i t y R e g i o n s o n oi and 5 D i a g r a m s

(A) c< - D i a g r a m s

A f t e r the s t a b i l i t y b o u n d a r i e s h a v e b e e n plotted, the nex t thing

is to locate the stable and the u n s t a b l e regions. T his can be done

s o m e t i m e s b y inspection. F o r c o m p l i c a t e d cases w h e r e the s t a b i l i t y

r e g i o n s c a n n o t be d e t e r m i n e d b y i nspection, the f o l l o w i n g m e t h o d can

(26)

18

be used.

(1) S e l e c t two points, one on each side of the s t a b i l i t y boun dary.

(2) A p p l y the H u r w i t z c r i t e r i o n to test the two points.

(3) Th e p o i n t w h i c h sat i s f i e s the H u r w i t z c r i t e r i o n lies on the stable

r e g i o n w h i l e the one, w h i c h does not, lies on t h e _ u n s t a b l e region.

For example, r e f e r r i n g to d i a g r a m 3, let the r e l a t i v e s t a b i l i t y

c o n s t r a i n t be = 1.0. T h e n the s t a b i l i t y b o u n d a r y c o r r e s p o n d i n g to

this p a r t i c u l a r r e q u i r e m e n t is the = 1.0 curve. To locate the s t a ­

b i l i t y region, select two points one on e ach side of the = 1.0

boundary. L e t the po i n t s be:

(1) P oint A w i t h T^ = 0. 4 (2) P o i n t B w i t h T^ = 0. 4

Tm = 0.0 8 Tm = 0.08 a2 = 0.48

K = 1 a3 = 0 .032 K = 3

A p p l y i n g H u r w i t z c r i t e r i o n to p o i n t A, and f r o m (3-3)

A k

J

1 + 3 ( . 0 3 2 ) - 2 ( . 48) 1 + . 4 8 - 1 - . 0 3 2 0

.032 .4 8 - 3 ( . 032) .136

0 0 .032

F r o m w h i c h

A

^ =.136,

A

2 = 0.0379, and

A

3

= 0.001 2. S ince

A

and

A 3 are all positive, the r e f o r e p o i n t A lies on the stable region.

Fo r p o i n t B.

.136 2.448 0

0 11 .032 .384 .136

0 0 .032

(27)

A x = 0.136

A 2 = - 0 . 0 2 6 2

A , = - 0 . 0 0 0 8 4

S i n c e A ] _ A2 a n d A g h a v e d i f f e r e n t signs, t h e r e f o r e p o i n t B lies o n the

u n s t a b l e region. F r o m the a b o v e two tests, w e can c o n c l u d e that the

r e g i o n on the left of the s t a b i l i t y b o u n d a r y c o r r e s p o n d s to the stable

r e g i o n w h i l e the r e g i o n on the r i g h t c o r r e s p o n d s to the u n s t a b l e

region.

(B) ^ - D i a g r a m s

T h e stable and u n s t a b l e r e g i o n s of the t - d i a g r a m s can be l o c a t e d

b y similar p rocedures. It is f ound that for the J - di a g r a m s , the

st a b l e r e g i o n also lies on the left of the r e l a t i v e s t a b i l i t y

b o u n d a r y w h i l e the u n s t a b l e r e g i o n lies on the right. W i t h the

and ^ d i a g r a m s a v a i l a b l e and the s t a b i l i t y regi o n s located, the

d y n a m i c a l b e h a v i o u r of the s y s t e m can be e a s i l y v i s u a l i z e d , and r e l a ­

tive s t a b i l i t y a n a l y s i s b e c o m e s a simple m a t t e r of r e a d i n g the d i a g ­

r ams .

L o c a t i o n of -Dominant P o les

job /

J

/

a - t < 0

/

/ /

Fig. 3-3(a)

S T A B I L I T Y B O U N D A R Y OF

<x - D I A G R A M S

Fig. 3-3(b) Fig. 3-3(c)

S T A B I L I T Y B O U N D A R Y OF

j - D I A G R A M S

S U P E R P O S I T I O N OF

ot A N D I D I A G R A M S

(28)

2 0

Th e b e h a v i o u r of a s y s t e m is l a r g e l y d e t e r m i n e d by the pair of roots

l o c a t e d c l o s e s t to the i m a g i n a r y axis. Fo r this reason, such roots

are called d o m i n a n t r oots b e c a u s e they d o m i n a t e the b e h a v i o u r of the

system. F o r a g i v e n r e l a t i v e s t a b i l i t y r e q u i r e m e n t , the d o m i n a n t

roots can be loca t e d b y a simple s u p e r p o s i t i o n of the and J

diagrams. Eac h U d i a g r a m c o r r e s p o n d s to a s t a b i l i t y b o u n d a r y for

a g i v e n v a l u e of (Fig. 3-3(a)), a n d - e a c h 5 d i a g r a m c o r r e s p o n d s

to a s t a b i l i t y b o u n d a r y for a g i v e n v a l u e of J (fig. 3-3(b)). T h e r e ­

fore by s u p e r p o s i n g the and f dia g r a m s , the point of i n t e r s e c t i o n

b e t w e e n the two curves gives the l o c a t i o n of the d o m i n a n t p o l e s w h i c h

w i l l s a t i s f y the r e q u i r e d r e l a t i v e s t a b i l i t y c o n s t r a i n t s and the r e f o r e

w i l l give the d e s i r e d s y s t e m performance.

I n t e r p r e t a t i o n of D i a g r a m s

Th e o<. d i a g r a m s are r e l a t i v e s t a b i l i t y b o u n d a r i e s for v a r i o u s

v a l u e s of and s y s t e m p a r a m e t e r s w h e n the m a g n i t u d e of the real

part of the r o o t s is the r e l a t i v e s t a b i l i t y constraint. F o r example,

any point on the = 1.0 c u r v e c o r r e s p o n d s to a root of c h a r a c t e r ­

istic e q u a t i o n w i t h n e g a t i v e real part = 1.0. Po i n t s lying on

the left of. the b o u n d a r y c o r r e s p o n d to r oots w i t h n e g a t i v e real part

less than 1.0 w h i l e po i n t s o n the r i g h t c o r r e s p o n d to r o o t s w i t h

n e g a t i v e r eal par t grea t e r than 1.0. T hus the r e g i o n on the left

r e p r e s e n t s the st a b l e r e g i o n and the one on the r i g h t r e p r e s e n t s

the u n s t a b l e region. T h e ’I d i a g r a m s are s t a b i l i t y b o u n d a r i e s w h e n

the m i n i m u m v a l u e of the d a m p i n g r a t i o is the s t a b i l i t y constraint.

A n y point on the J d i a g r a m c o r r e s p o n d s to a r o o t of the c h a r a c t e r ­

(29)

b o u n d a r y c o r r e s p o n d to r oots w i t h d a m p i n g r a t i o grea t e r than the

m i n i m u m r e q u i r e m e n t w h i l e points on the r i g h t c o r r e s p o n d to r oots

w i t h d a m p i n g r a t i o less than the r e q u i r e d minimum. Thus the stable

r e g i o n is loca t e d on the left w h i l e the u n s t a b l e r e g i o n is located

on the r i g h t of the s t a b i l i t y boundary.

F r o m b o t h the <A and 5 d i agrams, it is o b s e r v e d that for c o n s ­

tant v a l u e s of T , the s t a b i l i t y b o u n d a r i e s b e n d towards the left as

m ’

J

increases. Thi s m e a n s that an i n c r e a s e in T^ m u s t be a c c o m p a n i e d

b y a d e c r e a s e in the o v e r a l l g a i n to m a i n t a i n s y s t e m stability. W h e n

T is also a l l o w e d to increase, the s t a b i l i t y b o u n d a r i e s b e n d m o r e

m

towards the left. T his m e a n s that the s y s t e m gai n m u s t be furt h e r

reduced. Thus not o n l y can w e d e t e r m i n e the d y n a m i c a l b e h a v i o u r of

the s y s t e m u n d e r any g i v e n co n d i t i o n s , bu t w e can a l s o v i s u a l i z e

the e f f e c t of s i m u l t a n e o u s v a r i a t i o n of s y s t e m p a r a m e t e r s on the

p e r f o r m a n c e of the system. Thi s i n f o r m a t i o n is p a r t i c u l a r l y i m p o r t ­

ant e s p e c i a l l y in g u i d e d m i s s i l e systems and o ther p o s i t i o n c o n t r o l

systems w h e r e the time c o n s t a n t s are f u n c t i o n s of t e m p e r a t u r e and

o t h e r e n v i r o m e n t a l conditions.

F r o m d i a g r a m 8, the s u p e r p o s i t i o n of oL and 5 dia g r a m s , it is

o b s e r v e d that the 5 = 0 . 5 b o u n d a r y does not cut the oi d u r v e s for

v a l u e s of ot less than 0.6. Thi s m e a n s that the d o m i n a n t r oots c a n n o t

h a v e real parts w i t h m a g n i t u d e less than 0.6 for the g i v e n r a n g e of

va l u e s of s y s t e m parameters. Thus this m e t h o d pl a c e s a l i m i t a t i o n on

the m a g n i t u d e of the d o m i n a n t roots for a g i v e n r e l a t i v e s t a b i l i t y

requirement.

(30)

2 2

S y s t e m A n a l y s i s and D e s i g n

The c* and 5 d i a g r a m s can be u s e d m o s t e f f e c t i v e l y for s y s t e m

a n a l y s i s as w e l l as design. Th e d i a g r a m s r e a d i l y give i n f o r m a t i o n

w i t h regard to the r e l a t i v e s t a b i l i t y of the s y s t e m for a n y g i v e n

v a l u e s of s y s t e m parameters. F o r example, for a s y s t e m w i t h p a r a m e ­

ters g i v e n b y T^ = 0 . 6 4 sec., Tm = 0.06 sec. and K = 1, i n f o r m a t i o n

a b o u t the d y n a m i c a l b e h a v i o u r of the s y s t e m can be o b t a i n e d f r o m the

- d i a g r a m or the J d i a g r a m or the c o m b i n a t i o n of b o t h d e p e n d i n g

o n the p a r t i c u l a r r e l a t i v e s t a b i l i t y constraint. L e t it be r e q u i r e d

that the c h a r a c t e r i s t i c e q u a t i o n m u s t h a v e r oots w i t h n e g a t i v e real

parts less than -0.8 and d a m p i n g r a t i o not less than 0.5. R e f e r r i n g

to d i a g r a m 8, it is o b s e r v e d that the p o i n t T^ = 0 . 6 4 sec., Tm = 0.06

sec. and K = 1 (point c) falls on the left of the J = 0 . 5 b o u n d a r y

b u t on the r i g h t of the 'o< = 0 . 8 boundary. T his m e a n s that the roots

o f the c h a r a c t e r i s t i c e q u a t i o n w i l l h a v e d a m p i n g r atio g r e a t e r than

0.5 but w i t h n e g a t i v e real parts g r e a t e r than -0.8. T h u s the r e l a t i v e

st a b i l i t y r e q u i r e m e n t s are not s a t i s f i e d and the system, u n d e r -the

a b o v e g i v e n co n d i t i o n s , is r e l a t i v e l y unstable. S i milarly, for o t h e r

g i v e n s y s t e m c o n d i t i o n s and r e l a t i v e s t a b i l i t y r e q u i r e m e n t s , the

d y n a m i c a l b e h a v i o u r of the s y s t e m can be d e t e r m i n e d b y sim p l y r e f e r r ­

ing to the diagrams.

The p e r f o r m a n c e of a s y s t e m can be a p p r o x i m a t e d in terms of a

sec o n d o r d e r s y s t e m w h e n the d o m i n a n t poles are located. R e f e r r i n g

to d i a g r a m 8, the J = 0.5 and <x = 1.2 curves i n t e r s e c t at the p o i n t

w h e r e Tf = 0 .36 sec., Tm = 0.06 sec. and K = 2.05 (Point D) f r o m

(31)

Fig. 3-4 Fig. 3-5 U NIT S T E P R E S P O N S E

I = 0.5 0 = 60° OF 2nd O R D E R S Y S T E M

cR = 1.2

p = c* tan 0 = 1.2 tan60° = 2.08

the d o m i n a n t r oots o c c u r at

s = -1:2+ j 2 ; 08

the n a t u r a l f r e q u e n c y of the s y s t e m is

for a u n i t step i nput the b a n d w i d t h is g i v e n by

the time for the c o n t r o l l e d o u t p u t to r e a c h its m a x i m u m o v e r s h o o t is: W n = J?L = q* ^ = 2; 4 radions/sec.

(1)

= 2. 4 x 1.27

= 3.05

II

H 3.14

(32)

2 4

T h e s e t t l i n g time is g i v e n by:

3.0 3.0

2.5 secs. C s w n 0 . 5 x 2 . 4

The m a x i m u m o v e r s h o o t is:

= 0 . 1 6 6 % e

and the n u m b e r of o c i l l a t i o n s N for the s y s t e m to settle is:

N = 1.5 J l - $ 2 = 0.82

IT S

T h u s c o m p l e t e i n f o r m a t i o n a bout s y s t e m p e r f o r m a n c e is o b t a i n e d w h e n

the d o m i n a n t r oots are located. F u r t h e r m o r e the <* and .? d i a g r a m s

can be u s e d to o b t a i n the v a l u e s of s y s t e m p a r a m e t e r s that w i l l give

a r e q u i r e d response. F o r example, if it be d e s i r e d that the s y s t e m

F r o m d i a g r a m 8, the d o m i n a n t r oots are o b t a i n e d w h i c h is the i n t e r ­

s e c t i o n of the J = 0.5 and o< = 1.1 curves. The s y s t e m p a r a m e t e r s

that w i l l y i e l d the s p e c i f i e d r e s p o n s e are g i v e n b y T f = 0 . 4 sec.,

T m = 0 .06 sec., and K = 1.9. S i m i l a r l y for o t h e r s y s t e m r e q u i r e m e n t s ,

the r e q u i r e d s y s t e m p a r a m e t e r s can be e a s i l y o b t a i n e d b y the same should h a v e a r e s p o n s e w i t h d a m p i n g r a t i o 0.5 and r e s o n a n t

f r e q u e n c y w n = 2.2 radians/sec. T h e n

(33)

p r o c e d u r e .

C o m p a r i s o n of R e s u l t s w i t h R o o t L o c u s M e t h o d

The v a l i d i t y of the p r o p o s e d m e t h o d can be v e r i f i e d b y c o m p a r i n g

the resu l t s w i t h those o b t a i n e d b y the R o o t L o c u s m e t h o d } F o r T £ = 0 . 2 5

sec., Tm = 0 .625 sec. and ^ = 0.5, the R o o t L o c u s meth o d ^ - g i v e s the

f o l l o w i n g results:

o v e r a l l g a i n K = 2.6

d o m i n a n t P oles s^ 2 ~ " i j^ ~ - 1 . 6 + j 2 . 8

R e f e r r i n g to d i a g r a m 10, it is o b s e r v e d that for = 0.2 5 sec.,

Tm = 0.0625 sec., § = 0.5, the f o l l o w i n g r e s u l t s are obtained:

o v e r a l l g a i n K = 2.56

* = 1.6

= 1.6 tan 60°

= 2.78

d o m i n a n t Poles s l,2 = - + j |6 = -1.6 + j2. 78

Thu s the ne w m e t h o d not o n l y g ives r e s u l t s c o m p a r a b l e to the R o o t

L o c u s and N y q u i s t m e t h o d s b u t it als o y i e l d s i n f o r m a t i o n n o t o b t a i n ­

abl e f r o m the o t h e r two methods.

/ 3 &

UNIVERSITY

OF WINDSOR

LIBRARY

(34)

CHAPTER I V

D I S C U S S I O N A N D C O N C L U S I O N

A n e w m e t h o d ha s b e e n d e v e l o p e d w h i c h can be e f f e c t i v e l y a p p l i e d

to the a n a l y s i s and s y n t h e s i s of linear f e e d b a c k c o n t r o l Systems. .

B y m a p p i n g the s t a b i l i t y b o u n d a r y o n t o the i m a g i n a r y axis of a new

c o m p l e x p l a n e and r e a l i z i n g the c o e f f i c i e n t s of the t r a n s f o r m e d

p o l y n o m i a l , the H u r w i t z c r i t e r i o n can be a p p l i e d to y i e l d a n a l y t i c a l

e x p r e s s i o n s f r o m w h i c h s t a b i l i t y b o u n d a r i e s can be d r a w n for v a r i o u s

v a l u e s of s y s t e m parameters. B y u s i n g the s y s t e m p a r a m e t e r s as

c oo r d i n a t e s , the e f f e c t of s i m u l t a n e o u s v a r i a t i o n of s y s t e m p a r a m e ­

ters on the d y n a m i c a l b e h a v i o u r of the s y s t e m can be r e a d i l y v i s u a l ­

ized. By l o c a t i n g the st a b l e and u n s t a b l e regions, r e l a t i v e s t a b i l i t y

a n a l y s i s is r e d u c e d to the si m p l e p r o c e s s of r e a d i n g the diagrams.

By s u p e r p o s i n g the and 3 d i a grams, the d o m i n a n t poles can be

located an d s y s t e m p a r a m e t e r s a r e d e t e r m i n e d to give a d e s i r e d

response. T hus the ne w m e t h o d p r o v i d e s a p o w e r f u l tool for s y s t e m

*

analy s i s as w e l l as design.

The p r o p o s e d m e t h o d is m o s t g e n e r a l and can be a p p l i e d to systems

w i t h a n y d e g r e e of c o m p l e x i t y and u n d e r any s p e c i f i c k i n d of r e l a t i v e

s t a b i l i t y constraint. It is e s p e c i a l l y a d v a n t a g e o u s w h e n a p p l i e d to

systems w i t h two or m o r e v a r i a b l e parameters.

(35)

T A B L E I

T A B L E F O R C O M P U T I N G TH E C O E F F I C I E N T S OF F(Z)

C O E F F I C I E N T IN T E R M S OF C O E F F I C I E N T S OF P(s)

A 10

a5

A n 2a,_a,cos9

9 5 4

2

Ag

a^ + 2a^agCos29

Ay

2a^agCos9 + 2aga2Cos39

2

Ag

a^ + 2a^a2Cos29 + 2aga^cos49

Ag 2a g a2C o s9 + 2a^a-^cos39 + 2 a g a g C o s 5 9

2

A/ a_ + 2 a 0a.,cos29 + 2 a , a . c o s 4 9

4 2 3 1 4 0

Ag

2a2a^cos9 + 2agaQCos39

2

A2

a-^ + 2a2aQCos29

A^ 2 a ^ a Q C o s 9

2

A 0 a0

w h e r e P(s) = aQ + a^s 4- a

2

S^ + ags^ + a ^s^ + a^s"*

F(Z) = Aq + A ^ Z + + A g Z ^ + A ^ Z ^ + A ^ Z + A ^ Z ^ + A y Z

8

9

.

10

+ A g Z + A g Z + A g g Z

27

(36)

table

(37)

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D

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3 2

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\—

in

CO o

pi

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0 < M P 1

v n

o <3

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(43)
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o

F.eP?0' idvice1

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pvoP't W A * *

(46)

0°-R E F E 0°-R E N C E S

(1) " S e r v o m e c h a n i s m and R e g u l a t i n g S y s t e m D e s i g n " , Vol. I, 2n d E d i t i o n

H. C h estnut, R.W. Mayer. J o h n W i l e y & Sons Inc. N e w Y ork, N.Y.

1951, P p 2 3 5 - 2 3 6 , Pp386-388.

(2) "On the Zeros o f P o l y n o m i a l s and the D e g r e e of S t a b i l i t y of

L i n e a r S y s tems", J.F. K o l n i g , Journ. of Appl. P h y s . , Vol. 24,

P p . 476, 1953.

(3) "On the R e p r e s e n t a t i o n of the S t a b i l i t y R e g i o n in O s c i l l a t i o n

P r o b l e m s w i t h the A i d of H u r w i t z D e t e r m i n a n t s " , E. Spon der,

N A C A Tech. Mem. 1348, Aug., 1952.

(4) " The M a t h e m a t i c s of C i r c u i t A n a l y s i s " , E.A. G u i l l e m i n , J o h n W i l e y

& Sons, N.Y. 1949, P p . 395-409.

(5) " N y q u i s t D i a g r a m s and the R o u t h H u r w i t z S t a b i l i t y Criter i o n " ,

IRE Proc. 38, 1 3 4 5 - 1 3 4 8 (1950).

(6) " A N e w A p p l i c a t i o n of the H u r w i t z - R o u t h S t a b i l i t y C r i t e r i o n " ,

T h e r o n U s h e a Jr. A I E E S u m m e r G e n e r a l M e e t i n g J u n e 24-28, 1957.

P p . 530-533.

(47)

1939 B o r n o n 4 ^ July, in Canton, China.

1939 C o m p l e t e d H i g h Sc h o o l at W a h Y a n C o l l e g e H o n g Kong.

1965 G r a d u a t e d f r o m Th e U n i v e r s i t y of W i n d s o r , W i n d s o r , O n t a r i o , w i t h T h e D e g r e e of B.A.Sc. in E l e c t r i c a l Engineering.

1966 C a n d i d a t e for The D e g r e e of M.A.Sc. in E l e c t r i c a l E n g i n e e r i n g at T h e U n i v e r s i t y of W i n d s o r , W i n d s o r , Ontario.

Figure

figure1-1.
Fig. 2-l(a)
Fig. 2-2(a)
Fig. 2-3(a)
+7

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