2017年10月24日 1
The Root Locus Method
The Root Locus Concept The Root Locus Procedure The Root contour
The Root Locus Using Matlab
) (s G
) (s H ) (s E )
(s
R C(s)
) (
) ( )
( ) ) (
( q s
s p s
R s s Y
T
0 ) ( ) ( 1
)
( s G s H s q
Roots to find out the solution of the characteristic equation.
s K H s G
s H s KG s
q
1 1 ) ( ) (
0 ) ( ) ( 1
) (
1 1
1 1
...
2 , 1 , 0
0 for , 2 0
0 for , 2 180 )
( ) (
|
|
| 1 ) ( ) (
|
0 1
1 1 1
k where
K k
K k
s H s G
s K H s G
) 2 ( ) 4
( ) (
) 0 2 ( 1 4
0 ) 4 ( 2
0 4
2
0 ) ( ) ( 1 ) (
1 2 1
2 2 3
2 3
s s s s H s G
s s K s
s K s s
K Ks
s s
s H s G s
q
) 2 (
) 4 ) (
( )
(
2
s s
s s K
H
s
G
2017年10月24日 3
) 5 )(
4 (
) 1 ) (
( ) ( , ) 0 5 )(
4 (
) 1 1 (
) ( ) ( 1
)
(
1 1
s s s
s s H s s G
s s K s s
H s G s
q
D C B
A
j j
j s j
H s
G
s j2 2
2 2
2 2
2 2
2 1 1
1
2 4
2 3
2 1
2 0
5 2 1
)(
4 2 1
)(
2 1
(
1 2
| 1 ) ( ) (
|
) 5 )(
4 (
) 1 (
s s
s
s
＋ K
－
) (s
R Y (s )
2
1 1 j
s
-4
× × ×
-1 0 -5
S-plane A B
D C
2017年10月24日 5
3
~ 1 1
1 1
2 1 1
1
4 ) ( 2 tan 3 )
( 2 tan 1 )
( 2 tan 90
) 5 2 1
)(
4 2 1
)(
2 1
(
1 2 ) 1
( ) (
p p z
j
s
j j j
s j H s G
2
1
1 j
s
-4
× × ×
-1 0 -5
S-plane
z1
p 2
p 3
p
)
1 ( 2 tan 180
1 ) ( 2
tan
1 12017年10月24日 7
The Root Locus Procedure
Step Related equation or Rule
1. Write the characteristic equation so that the parameter of interest K appears as a multiplier.
0 ) ( ) ( 1
)
(s KG1 s H1 s F
2. Factor G1(s)H1(s) in terms of n poles and m zeros.
) (
) ( ) ( ) (
1 1 1
1
i n i
j m j
p s
z s s
H s G
3. Locate the open-loop poles and zeros of
) (s
F inthe s-planewithselected symbols.
: poles, : zeros,
or : roots of characteristic equation 4. Locate the segments of the real axis that
are root locus.
a). Locus begins at a pole and ends at zero.
b). Locus lies to left of an odd number of poles and zeros (
0 K
).
5. The number of branch on the root loci, . n, whennm;
n:numberoffinitepoles, m:numberof finite zeros
6. Therootlociaresymmetrical with respect to the horizontal real axis.
7. Intersect of the asymptotes (Centroid)
m n
z pi j
or
m n
z
pi j
Re( )
Re( ) 8. Angles of asymptotes of the root loci.
1 ,
, 2 , 1 , 0
; 0 2 ,
0 ) ,
1 2 (
k n m
m K n
k m K n
k
k
2017年10月24日 9 9. Breakaway points (saddle points) on the
root loci. Roots of G1(s)H1(s)0
ds
d or K 0
ds d
10. Intersection of root loci with imaginary axis.
Routh-Hurwitz criterion.
11.Anglesof departure and angles of arrival
of the root loci. 2 0, 0,1,2,
0 )
1 2 ) ( ( )
( 1
1
k
K k
K s k
H s
G
at s pi or szj . 12. Calculation of K at a specific root si.
i j m j
i n i
s s z s
p s
K
) (
) (
1 1
2017年10月24日 11
Step 4: The root locus on the real axis always lies in a section of the real axis to the left of an odd number of poles and zeros.
Odd segments
Step 5: Determine the number of separate loci,SL.The number of
separate loci is equal to the number of poles
Step 6: The root loci must be symmetrical with respect to the horizontal real axis.
Step 7: The linear asymptotes are centered at a point on the real axis given by The angle of the asymptotes with respect to the real axis is
m n
zeros poles
A
1800
) 1 2
(
m n
q
A 180 , 0,1,2, 1
1 4
) 1 2 , ( 1 ,
4 1 0
q q n m
m
n
2017年10月24日 13
Step 8:The actual point at which the root locus crosses the imaginary axis is readily evaluated by utilizing the Routh-Hurwitz criterion.
Step 9: Determine the breakaway point
the tangents to the loci at the breakaway point are equally over
0 ds dK
360
0) 0 4 )(
2 ( 1 1
) (
1
1
KG s K s s
3 ,
0 ) 6 2 (
) 4 )(
2 (
s ds s
dK
s s
K
ds dK
s 2
-2
2017年10月24日 15
ds y dK
s
b a 0
2 / ) (
0
2 / ) (
0
2 / ) (
0 ,
0
c b d
y
c a d
if y c s
b a c
y b
s y
a s
c c
b a
c
數值解法(1):半區間法
y
ay
by
c, ds y
dk
s y 1
m
s 1
s 2
m y s ms
s s
m y
y
2 1 12 1
1
0
S y y
m y s
2 ms
1
1S
1y
1m S
2|
| 1
y
數值解法(2):牛頓法
2017年10月24日 17
Step 10:Determine the angle of departure of the locus from a pole and the
angle of arrival of the locus at a zero,using the phase angle criterion.
Examples
) 2 (
1 1 )
(
s K s s
F ( 2 )( 3 )
1 1 )
(
s s
K s s
F
-4 -3 -2 -1 0 1 2
-6 -4 -2 0 2 4 6
Real Axis
Imag Axis
-3 -2 -1 0 1 2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Real Axis
Imag Axis
Effects adding poles to G1(s)H1(s)
2017年10月24日 19
) 4 )(
3 )(
2 (
1 1 )
(
s s
s K s s
F
-6 -5 -4 -3 -2 -1 0 1 2
-4 -3 -2 -1 0 1 2 3 4
Real Axis
Imag Axis
-5 -4 -3 -2 -1 0 1 2
-5 -4 -3 -2 -1 0 1 2 3 4 5
Real Axis
Imag Axis
) 3 )(
2 (
1 4 )
(
s s s
K s s
F
Effects adding zeros to G1(s)H1(s) Effects adding poles to G1(s)H1(s)
s s
s K s
s
F 12 64 32
1 1 )
(
4 3 2
-6 -5 -4 -3 -2 -1 0 1 2
-6 -4 -2 0 2 4 6
Real Axis
Imag Axis
-6 -5 -4 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3 4
Real Axis
Imag Axis
) 2 2
)(
3 (
1 1 )
(
2
K s s s s s
F
Effects adding poles to G1(s)H1(s) -<K<
2017年10月24日 21
0 )
( s s
3 s
2 s
q 1
3 2 0
s s
s
) 0 1 1 (
0
2 2 3
s s
s s
The Root contour
) 2 (
1
s s
k
s k 2
Specifications:
1. Steady-state error for a ramp input
2. Damping ratio of dominant roots sec.
3. Settling time to within 2 % of the final value sec.
%
35
707 .
0
3
1 1
2
2
,
0 2
) ( 1
k k
k
s s s
s GH
) (s R
)
(s
Y
2017年10月24日 23
2017年10月24日 25 ) 0
( 1 1
) ( ) (
1 1 1 2
s s a
K s s
H s G
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-10 -8 -6 -4 -2 0 2 4 6 8 10
Root Locus
Real Axis
Imaginary Axis
10 a
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
-8 -6 -4 -2 0 2 4 6 8
Root Locus
Real Axis
Imaginary Axis
9 a
-8 -7 -6 -5 -4 -3 -2 -1 0
-8 -6 -4 -2 0 2 4 6 8
Root Locus
Real Axis
Imaginary Axis
8 a
-7 -6 -5 -4 -3 -2 -1 0
-8 -6 -4 -2 0 2 4 6 8
Root Locus
Real Axis
Imaginary Axis
7 a
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Root Locus
Real Axis
Imaginary Axis
2 a
-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Root Locus
Real Axis
Imaginary Axis
1 a
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2017年10月24日 29
