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
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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
IN F O R M A T IO N T O U S E R S
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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
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.
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
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
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
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
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.
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 thec 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
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
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 Da.
i-iS 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
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.
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
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.
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
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,..
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
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)
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)
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
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
= 0.5then '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 2A2 =
2
+
2 a 2 & Q C O s 20
=1
+Ka2
a2 =
T f
+
A 3 =
ba£
a 4
=a 2
2 3 a 3= T£Tm
A 5 =
ba£a^
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
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, andA
3
= 0.001 2. S inceA
andA 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
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
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
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.
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
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
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
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
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.
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
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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.
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.