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a nd ~ < 8 < 0 . . ~ence, these segments are t he p art of root locus. I n o ther
'
t he e ntIre real aXIS m t he lefthalf plane, except t he segment between  2 a nd ~:d~,
a p art o f t he r oot locus.
' IS 2. A real axis segment is a p art of t he root locus if t he sum of t he real axis poles a nd zeros o f G (s)H(s) t hat lie to t he right of the segment is odd. Moreover,
t he r oot loci are symmetric about real axis.
We c an readily verify in Fig. 6.41 t hat t he real axis segment t o t he right of  8
has only o ne pole (and no zeros). Hence, this segment is a p art of the root
locus.
3. T he n  m r oot loci terminate a t 0 0 a t angles k7r / (n  m) for k = 1, 3, 5, . ...
Note t hat, according t o rule 1, m loci terminate on the open loop zeros, and
the remaining n  m loci terminate a t 0 0 according to this rule. In Fig. 6.41,
we verify t hat n  m = 2 loci terminate a t 0 0 a t angles k7r / 2 for k = 1 and 3.
Now we s hall make an interesting observation. I f a transfer function G(s) has
m (finite) zeros and n poles, t hen lims+ooG(s) = sm/sn = l /snm. Hence,
G(s) h as n  m zeros a t 0 0. This fact shows t hat although G(s) h as only m
finite zeros, there are additional n  m zeros a t 0 0. According t o rule 1, m loci
terminate o n m finite zeros, and according to this rule t he remaining n  m
loci t erminate lit 0 0, which are also zeros of G(s). This result means all loci
begin o n o pen loop poles a nd t erminate on open loop zeros.
4. T he c entroid of the asymptotes (point where t he asymptotes converge) of the
(n  m) loci t hat t erminate a t 0 0 is a= (PI + 112 + ... + Pn)  (Zl + Z2 + . " + Z m) ( nm) where P I, 1>"2, . .. , Pn are t he poles and Z l, Z 2, . .• , Z m are t he zeros, respectively,
of t he openloop transfer function.
Figure 6.41 verifies t hat t he centroid of the loci is [ (8 + 0)  0]/2 =  4.
5. T here are additional rules, which allow us t o compute t he points where the
loci intersect and where they cross the j w axis to enter in the righthalf plane.
These rules allow us t o draw a quick and rough sketch of the root loci. But 441 Rul~ 3: n  m = 3. Hence, (all) t he t hree loci terminate a t 0 0 along asymptotes a t
a ng es k7r / 3 for k = 1, 3 a nd 5. T hus, t he a symptote angles are 60 0 1200 a d 180 0
4. Rule 4: T he c entroid (where all t he t hree asymptotes converge) is (O~ 2 _4)/n =  2'
3
We d raw three asymptotes s tarting a t  2 a t angles 60 0 120 0 a nd 1800
h
..
F ig 6 43 T his'o£
t'
£Ii
"
a s S Own m
. I' .'
1 o rma IOn s u ces t o give a n i dea a bout t he r oot locus. T he a ctual
are
'g
root OCI . also shown in F1 . 6. 43 . T wo 0 f t he a symptotes cross over t o t he R HP
'
3. behaVIOr which shows t hat for some range of K , t he s ystem becomes unstable. o .' C omputer E xample C 6.5
Solve E xample 6.21 using MATLAB.
T he MATLAB commands to find t he r oot locus for this case are:
n um=[O 0 0 1];
d en=conv(conv([1 0 ],[1 2 ]),[14]);
r !ocus(num,den),grid 0 6 .74 SteadySt...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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