The root locus is the path of the closed-loop poles of a system
as a parameter of the system is varied. Each point on the root
locus satisfies the angle condition, += 180)12()()( ksHsG . Using
this relationship, rules for sketching and finding points on the
root locus were developed and are now summarized:
Basic rule for sketching the root locus: 1. Number of branches
The number of branches of the root locus equals the number of
closed-loop poles.
2. Symmetry The root locus is symmetrical about the real
axis.
3. Real-axis segments On the real axis, for 0>K the root
locus exists to the left of an odd number of real-axis, finite
open-loop poles and/or finite open-loop zeros.
4. Starting and ending points The root locus begins at the
finite and infinite poles of )()( sHsG and ends at the finite and
infinite zeros of )()( sHsG .
5. Behavior at infinity The root locus approaches straight lines
as asymptotes as the locus approaches infinity. Further, the
equations of the asymptotes are given by the real-axis intercept
and angle in radians as follows:
zeros finite#poles finite#)12(
zeros finite#poles finite#zeros finitepoles finite
+=
=
pi
ka
a
6. Real axis breakaway and break-in points The root locus breaks
away for the real axis at a point where the gain is maximum and
breaks in to the real axis at a point where the gain is
minimum.
7. Calculation of j axis crossings The root locus crosses the j
- axis at the point where += 180)12()()( ksHsG . Routh-Hurwitz or a
search of the j - axis for + 180)12( k can be used to find the j -
axis crossing.
8. Angles of departure and arrival The root locus departs from
complex, open-loop poles and arrives at complex, open-loop zeros at
angles that can be calculated as follows. Assume a point close to
the complex pole or zero. Add all angles drawn from all open-loop
poles
and zeros to this point. The sum equals + 180)12( k . The only
unknown angle is that drawn from the close pole or zero, since the
vectors drawn from
all other poles and zeros can be considered drawn to the complex
pole or zero that is close to the point. Solving for the unknown
angle yields the angle of departure or arrival.
9. Plotting and calibrating the root locus All points on the
root locus satisfy the relationship += 180)12()()( ksHsG . The
gain, K , at any point on the root locus is given by
lengths zero finitelengths pole finite1
)()(1
===
MsHsGK
Sketching a root locus and finding critical points Problem 1:
page 397 example 8.7 Sketch the root locus for the system shown in
Figure 8.19(a)
)(sR
)4)(2()204( 2
++
+
ss
ssK)(sC
Closed-loop transfer function,
)208()46()1()204(
)204()4)(2()204(
)4)(2()204(1
)4)(2()204(
)(2
2
2
2
2
2
KsKsKssK
ssKssssK
ss
ssKss
ssK
sT
++++
+=
++++
+=
++
++
++
+
= .
Open-loop transfer function, )4)(2/()204()()( 2 +++= ssssKsHsKG
. The system has finite zeros; 42 j The poles are located on the
s-plane at 4,2 .A segment of the root locus exists on the real axis
between 42 and .
Number of asymptotes = 0.
The break-away point is estimated by evaluating )204()86(
)204()4)(2(
2
2
2 +
++=
+
++=
ss
ss
ss
ssK , between 4=s and 2=s .
Differentiating K with respect to s, and settling the derivative
equal to zero yields,
01522410
0)204()42)(86()62)(204(
42/)204(62/86
)204()86(
2
2
22
2
2
2
2
=
=
+
+=
=+=
==
+
++=
ss
ss
ssssss
dsdK
sdsdvssvsdsdussu
ss
ssK
Solving for s, we find [email protected])10(2)152)(10(42424
24
,
22
21 =
=
=
a
acbbss
The characteristic equation is 0)208()46()1( 2 =++++ KsKsK .
Therefore, the Routh table 2s K+1 K208 + 0 1s K46 0 0 0s ( )[ ] ( )
KKKK 20846/)208(46 +=+ 0 0
From row 1s , 5.1K-6,4K-,046 > K . Hence, the limiting value
of gain for stability is 5.1=K , and the roots of the auxiliary
equation are, (from row 2s ): 899.32.15,,0832.5s 212 jss ===+ .
