Root locus asymptotes for sum of two polynomials of same degre

Krall, A. M.

English

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Root Locus Asymptotes 
For The Sun O f  Two Polynomials 
Of The Same Degree 
1 Al1a.n M. Kra.11 
In t roduct ion .  There seems t o  be a considerable  amount o f  
confusion concerning the asymptotes of the  root- locus diagram 2311 
t h e  s u m  of  two polynomials having the same degree,  
i s  e i t h e r  ignored [8], an improper impression i s  given because t h e  
parameter involved i s  r e s t r i c t e d  t o  non-negative values [4 j ,  or 
e l s e  i n c o r r e c t  s ta tements  a r e  made concerning it  [7;  page 135, l a s t  
l i n e ] .  
r e a l  valued and a r e  important i n  the cons t ruc t ion  of the  r o o t - l x v s  
diagram. The confusion a r i s e s  because the root- locus does not 
The siabject 
The f a c t  i s  t h a t  asymptotes do e x i s t  when the  parane ter  i s  
a.pproach the asymptotes a s  the parameter approaches + 03, but i n s t ead  
a s  i t  a.pproaches -1. 
- 
The reason f o r  t h i s  i s  qu i t e  simple. A s  t he  parameter appraaches 
-1, t h e  c o e f f i c i e n t  of t he  h ighes t  power of the polynomia.1 approaches 
zero,  and the  polynomial i s  reduced i n  degree 
A Theorem. Let 
GPO PRICE s 
n-1 
0 
Hard copy (HC) 
Microfiche (MF) ,A, 
ff 653 July 65 
1. McAll is ter  Bui lding,  The Pennsylvania S t a t e  Uni?ersity,  Unix7ersit.y 
P a r k .  PennsvlvgniR 16;80?_ 
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https://ntrs.nasa.gov/search.jsp?R=19670002747 2020-03-16T16:59:35+00:00Z
n-1 
zi 
h ( z )  = zn + b n-1 9 
i) 
be re1a.t ively prime, and l e t  
i s  the  coef- a re  complex constants,and a.n - b  n-m - where a,i, bi 
f i c i e n t  of the h ighes t  power of g ( z )  - h ( z )  which i s  non-zera.  
Clear ly  m < n. We wish t o  consider t h e  root- locus of 
F ( z )  = g ( z )  - Ke i 0  h ( z )  = 0 
a s  K va.ries from --co t o  +w. 0 i s  a. f i xed ,  r e a l  consta.nt..  We 
prove the  following. 
Theorem. Let 8 # 0 ,  8 # T .  - L e t  
The roo t  locus of F ( z )  ( a s  K v a r i e s  between --oo and f ~ )  l i e s  
w i t h i n  a c i r c l e  centered a t  the  o r i g i n  wi th  rad ius  p = M + 1. 
Let 8 = T .  ( 0  = 0 i s  t h e  same except K changes s i g n . )  
A s  K a.pproaches -1 from above and below, the  root  iocus of  F ( z )  
becomes asymptotic t o  t h e  2(n-m) l i n e s  passing through 
-
- 
z = [ b l - [  a n -m+l 
n-m an-m - b  
ma,king angles  
- b ) - a.rg(K+l) + (2r+l)n]/(n-m.) , [ a.rg ( a,n -m n-m 
where r = 0, 1, ..., n-m-1, a_--- Fur ther  the  2(n-m) q u a n t i t i e s  (for 
K >-1 and  K <--1) 
-
3 .  
become a r b i t r a z i l y  good estima.tes for n-m roo t s  of F ( z )  = 0 
a s  K a.pproa.ches -1. 
Proof.  Let 8 # 0, 8 # T. Consider 
$ ( z )  = F(z)/[l - Ke 18 
J 
n-1 - Ke i e  5n 
- l z e  = z n + I  I: n - i  - Ke" 
0 
The roots  of $ ( z )  = 0 and F ( z )  = 0 a.re t h e  sa.me. It i s  wel l  
known tha . t  i f  M i s  a.n upper bound f o r  t he  c o e f f i c i e n t s  of z 0 , 
n-1 ..., z , then  a c i r c l e  centered a.t  t he  o r i g i n  wi th  rad ius  p = M + 1 
conta.ins a.11 of the r o o t s  # ( z ) .  Thus we must f i n d  an upper bound 
f o r  t h e  q u a n t i t i e s  
of K. 
(a.i - Ke i o  b i ) / ( l  - Kebe) which i s  independent 
Let a, p be any of the p a . i r s  a,i, bi, and consider  the  
ma.pp i n g  
f = ( a  - wp) / (1  - w ) .  
A s  w v a r i e s  a.long the  s t r a . i gh t  l i n e  Keie, which pasees through 
t h e  o r i g i n  wi th  inc l ina . t i on  8 ,  b descr ibes  a. c i r c l e  wi th  cen2er 
cen te r  = ( a  - - e2ie) 
and  ra.dius 
2 i.8 r ad ius  = 1p - a\/ll - e 1 .  
Therefore  a. ma.ximwn .for f on the  c i r c l e  i s  \ c e n t e r \  + ra,dius. 
4 
Using the  t r i a n g l e  inequa. l i ty ,  we note tha.t  
l f l  5 [ lal+lBll/lsin e l  e 
Thus i f  
M = l c sc  61 sup[ l a , i \+ \b i l ]9  
i 
the  result fol lows.  
Y )/(K + L), e l imina t ion  'n-m Let 8 = K .  If W = ( a  n-m 
of K i n  F ( z )  = 0 yi.elds 
h j z )  . ,  + Wk(z) = 0, 
The r e s u l t  then  fcllows from thsorem 2 cf [5]. 
An Example. L e t  us consider  
It i s  apparent  from the  relative l c c a t i o n s  of t h e  F o l n t s  
Z = 0 ,  -1 and -2 and t h e  poin ts  z -- -3, ifi and -ic, t h a t  
t he  p o s i t i v e  root - locus  (rC > 9) c o r l s i s t s  o f  t w o  b r m c h e s  
beginning a t  z = 0 and z = -1, which meet a t  z = -1 + 9 , s n d  
roo t - locus  (K < C> i s  T o t  c-t 211 c l e s r ,  
The p o s i t i v e  root- locus ( W  > 0 )  c o n s i s t s  o f  a branch beginning 
a t  z = is, becoming asymptotic t o  the l i n e  through the cen te r  
of g r a v i t y  a t  an angle o f  60'; a branch beginning a t  z = -is, 
becoming asymptotic to  the l i n e  through the cen te r  of g r a v i t y  a t  
an angle of 300'; and a t h i r d  branch beginning a t  z = -3, which 
approaches z = -03 along the r e a l  a x i s .  
The negat ive root - locus  ( W  < 0) has two branches which begin 
a t  z = f is and meet a t  z = -1 + -5. h/5 One branch then 
approaches + 03 along the r e a l  a x i s .  The o t h e r  moves lef tward 
along the r e a l  a x i s  and meets the t h i r d  branch, which began a t  
z = -3, a t  z = -1 
t o  the l i n e s  through the cen te r  o f  g r a v i t y  a t  angles  o f  120' 
and 240'. 
J3 These then s p l i t  t o  become asymptotic - T o  
It i s  now apparent what the  o r i g i n a l  negat ive r o o t  l o c u s  
c o n s i s t s  o f .  
If z = x + i y ,  the  root - locus  s a t i s f i e s  the equation 
That i s ,  i t  c o n s i s t s  o f  the l i n e  y = 0 and the hyperbola 
3(x + 1)2 - y 2 = 1, (See [ 6 ] . )  
T h i s  a r t i c l e  was supported i n  p a r t  by NASA Grant NGR 39-009-041. 
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C #  
, REFERENCES 
1. Gerald Berman and R. G. Stanton, "The Asymptotes o f  the  Root Locus" ,  
S.I.A.M. Review, 5, (1963), pp. 209-218. 
2. W ,  R e  Evans, "Graphical Analysis of Control Systems", 
Trans. A . I .E .E . ,  67, (1948), pp. 547-551. 
3 .  , "Control System Synthesis  by the  Root-Locus Method", 
b a n s .  A . l . E . E . ,  69, (1950), pp. 66-69. 
4. Wilfred Kaplan, "Operational Methods for Linear Systems", 
Addison-Wesley Publishing Co., Reading, Mass., 1962. 
5. Allan M. W a l l ,  "An Extension and Proof  o f  the Root-Locus Method", 
J. S.I.A.M., 9, (1.961)~ pp. 644-653. 
6. , ''A CUsed Expression f o r  the  Root-Locus Method", 
J. S.I.A.M., 11, (1.963)~ pp. 700-704. 
7. S i eg f r i ed  H. Lehnigk, " S t a b i l i t y  Theorems f o r  Linear Motions", 
P ren t i ce  Hall, Inc.  , Englewood C l i f f s ,  N. J. , 1966. 
8. C .  J. Savant, Jr., "Basic Feedback Control System Design", 
McGraw H i l l  Book Co, Inc . ,  New York, 1958. 


Root Locus Asymptotes for the Sum of Two Polynomials of the Same Degree

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