EES42042
Fundamental of Control Systems
Root Locus Root Locus
( ( Tempat Tempat Kedudukan Kedudukan Akar Akar =TKA) =TKA)
DR. Ir. Wahidin Wahab M.Sc.
Ir. Aries Subiantoro M.Sc.
Dept.E.E. University of Indonesia
2
Root Locus Plots
Means of tracking closed loop poles as some parameter e.g. gain varies
allows direct reading of damping ratio and
natural frequency
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.4
a. CameraMan®
Presenter Camera System
automatically follows a subject who wears infrared sensors on their front and back (the front sensor is also a microphone);
tracking
commands and audio are relayed to CameraMan via a radio frequency link from a unit worn by the subject.
b. block diagram.
c. closed-loop transfer function.
Courtesy of ParkerVision.
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
The System Transfer Function is:
G(s) = K . S 2 + 10 S + K
S 2 + 10 S + K = 0
(Quadratic Equation, Use ABC formula to find its roots)
0 )
( )
( 1
equation stic
characteri the
of
roots the
are system
this of
poles Closed
=
+ G s H s
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Table 8.1
Pole location as a function of gain for the system of
Figure 8.4
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.5
a. Pole plot from Table 8.1;
b. root locus
Example: Root Locus Plot 7
G(s) = K/[s (s + 0.5)(s 2 + 0.6 s + 10)]
Zeros : None
Poles : s = 0 s = -0.5
s = -3+j3.1480
s = -3-j3.1480
8
Root Locus Plots
+ - G(s)
R(s) C(s)
H(s)
) ( )
( 1
) ( )
( ) (
s H s
G
s G s
R s C
= +
9
Root Locus Plots
+ - G(s)
R(s) C(s)
H(s)
0 )
( )
( 1
equation stic
characteri the
of
roots the
are system
this of
poles Closed
=
+ G s H s
10
Root Locus Plots
1 )
( )
(
satisfying point
any for
condition angle
an and
magnitude a
to rise gives
equation This
function fer
open trans the
is ) ( )
( Now
1 )
( )
(
0 )
( )
( 1
−
=
−
=
→
= +
s H s
G
s s
H s
G
s H s
G
s H s
G
i
11
Root Locus Plots
{ } ( )
,....
2 , 1 , 0
180 1
2 )
( )
( arg
Criterion Angle
1 )
( )
(
Criterion Magnitude
1 )
( )
(
=
°
× +
=
=
−
=
k
k s
H s
G
s H s
G
s H s
G
i i
i i
i
i
12
Root Locus Plots
( )( ) ( )
( )( ) ( .... .... ) 0
1
0 )
( )
( 1
form in the
posed is
Problem
varies parameter
gain some
as poles
loop closed
of plot a
is plot locus
Root
2 1
2
1 =
+ +
+
+ +
+ +
→
= +
N M
p s
p s
p s
z s
z s
z K s
s H s
KG
K
13
Root Locus Plots
( )( ) ( )
( )( ) ( )
( )( is approximat ) ( .... ely ) 0
equation
stic charcateri
the of
values small
For
.... 0 1 ....
2 1
2 1
2 1
= +
+ +
+ = +
+
+ +
+ +
N
N M
p s
p s
p s
K
p s
p s
p s
z s
z s
z K s
For small values of gain closed loop poles are
equal to open loop poles
14
Root Locus Plots
( )( ) ( )
( )( ) ( )
( )( is approximat ) ( .... ely ) 0
equation
stic charcateri
the of
values large
For
.... 0 1 ....
2 1
2 1
2 1
= +
+ +
+ = +
+
+ +
+ +
N
N M
z s
z s
z s
K
K
p s
p s
p s
z s
z s
z K s
For large values of gain closed loop poles are
equal to open loop zeros
15
Root Locus Plots
Thus as gain K varies from zero to infinity, the closed loop poles moves from open loop poles to open loop zeros
If there are more open loop poles than open loop zeros then the deficiency is made up
by assuming an appropriate number of zeros
at infinity
16
Root Locus Plots
Rules for construction of root locus plots derived from magnitude and angle criteria
Rules are given in Ogata pp. 330-338
MATLAB program rl.m available for fast root locus plotting
¾ being able to do a fast root locus by hand is
important in understanding control system
behaviour
Root Locus Plotting Rules - 17
Example
Consider as an example
( 3 )( 10 )
) ( )
( = + +
s s
s s K
H s
G
Step 1 - Plot open loop poles and zeros
-10 -3 0
Root Locus Plotting Rules - 18
Example
( 3 )( 10 )
) ( )
( = + +
s s
s s K
H s
G
Step 2 -Real axis portions of locus lie to the left of an odd no. of poles/zeros
-10 -3 0
Root Locus Plotting Rules - 19
Example
Step 3 - Determine asymptotes of locus as K approaches infinity
( )
) ( )
( of
zeros no.
) ( )
( of
poles no.
...
2 , 1 , 0
1 2
asymptotes 180 of
Angles
s H s G m
s H s G n
k
m n
k
=
=
=
−
+
°
= ±
Root Locus Plotting Rules - 20
Example
Step 3 - Determine asymptotes of locus as K approaches infinity
( )
2 60
1 180
0 60
angles
0 3
Here
1 2
asymptotes 180 of
Angles
=
°
−
=
°
=
°
=
=
=
−
+
°
= ±
k k k , m n
m n
k
Root Locus Plotting Rules - 21
Example
Step 4 - Determine intersection of asymptotes with real axis
( ) ( )
m σ n
σ
a a
−
= −
=
zeros of
sum poles
of sum axis
real with
asymptotes of
on
intersecti
Root Locus Plotting Rules - 22
Example
Step 4 - Determine intersection of asymptotes with real axis
( )
3 13 3
10 - 3 - 0
case In this
= −
a =
σ
Root Locus Plotting Rules - 23
Example
-10 -3 0
3
− 13
+60 o
-60 o Steps 3&4 - determining
asymptotes
Asymptotes
Root Locus Plotting Rules - 24
Example
Step5 - Find breakaway/break-in points
These are points on the real axis where the locus leaves or “breaks into” as K varies
Break away point Break In point
Root Locus Plotting Rules - 25
Example
Step5 - Find breakaway/break-in points
These points occur where K is a maximum (breakaway) or a minimum (break in) for the locus portion on
the real axis
Break away point Break In point
Root Locus Plotting Rules - 26
Example
-10 -3 0
Step 5 - Here there is one breakaway point
and it lies midway between 0 and -3
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.13
Root locus example showing
real- axis
breakaway (-σ 1 ) and
break-in points (σ 2 )
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.14
Variation of gain along the real axis for the root locus of
Figure 8.13
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Table 8.2
Data for breakaway and break-in points for the root
locus of Figure 8.13
Root Locus Plotting Rules - 30
Example
Step 6 - Determine angle of departure or arrival at breakaway/break in point to complex pole/zero
⎟⎟ ⎠
⎜⎜ ⎞
⎝ + ⎛
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛
°
=
zeros
all from
pole complex
to vectors of
angles of
sum
poles other
all from
pole complex
to vectors of
angles of
- sum 180
pole complex
from departure
of
Angle
Root Locus Plotting Rules - 31
Example
Step 6 - Determine angle of departure or arrival at breakaway/break in point to complex pole/zero
⎟⎟ ⎠
⎜⎜ ⎞
⎝ + ⎛
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛
°
=
poles
all from
zero complex
to vectors of
angles of
sum
zeros other
all from
zero complex
to vectors of
angles of
- sum 180
zero complex
from arrival
of
Angle
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.16
Unity feedback system with complex poles
Open loop zero : s = -2
Open Loop poles : s = -3; s = -1 ± j1
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.17
Root locus for system of Figure 8.16
showing
angle of departure calculation
at pole -1+j1
Root Locus Plotting Rules - 34
Example
-10 -3 0
Step 6 does not apply to this case since there are
no complex poles or zeros in open loop t.f.
Root Locus Plotting Rules - 35
Example
Step 7 - Determine points where locus may cross imaginary axis
equation stic
characteri into
subsitute or
Routh test Apply
ω
j s =
This gives the critical gain for stability
36
Root Locus Example
-20 -15 -10 -5 0 5
-8 -6 -4 -2 0 2 4 6 8
Real Axis
Imaginary Axis
Root Locus plot
Typical Root Locus Plots
(Ogata Chapt.6)
38
Note :
being able to do a fast root locus by hand is important in understanding control system behaviour
Rules for construction of root locus plots derived from magnitude and angle criteria
rules given in Ogata pp. 330-338
MATLAB program crl.m available for fast root
locus plotting
Root Locus plotting with MATLAB 39
%MATLAB function to compute root locus for continuous time system
%
%syntax: [w]=rl(num, den, K)
% num: numerator polynomial e.g. [1 2 3]
= s^2+2*s+3
rl.m is available from Matlab Control System Toolbox
Root Locus Plotting with MATLAB 40
% den: denominator polynomial
% K: Gain values
% Denominator polynomial assumed
equal or greater in order than numerator
41
Root Locus Example
-3 -2 -1 0 1 2
-6 -4 -2 0 2 4 6
Real Axis
Imaginary Axis
°
=
→
=
= cos φ 0 . 6 φ 53 ς
65 . 1 2
.
1 j
s = − +
Finding value of gain k
Which gives a certain damping ratio ζ = 0.6
42
Root Locus Example
Applying magnitude criterion :
55 . 44
for solve
and
65 .
1 2
. 1 substitute
10 1
3
=
+
−
= + =
+ k
k
j s s
s s
k
43
Root Locus Example
{ }
s t
t
s
n n
s
5 . 2 2
. 1
3
1.2 s
Re now
3 ,
Time Settling
=
=
∴
=
=
= ςω
ςω
44
Root Locus example
Gain and phase margins from root locus
82 . 44.55 8
margin 393 gain
393 10 1
5 . 5 3
5 . 5 5
. 5
criterion magnitude
applying
5 . 5 at
axis imaginary
crosses locus
margin Gain
=
=
∴
=
→ + =
+
=
j k j
j
k
j
s
45
Root Locus example
Gain and phase margins
( )
{ } = °
=
→ + =
⋅ +
⋅
8 . 120 arg
3 . 1 0
. 1 10
3
55 . 44
1.0 is
gain where
frequency determine
First
margin Phase
2 2
2 2
g
g
g
g g
g
j GH ω
ω ω ω
ω
ω
46
Root Locus example
Gain and phase margins
( )
{ }
°
=
°
−
°
=
∴
°
=
2 . 59 8
. 120 180
margin phase
8 . 120 arg
margin Phase
j g
GH ω
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.11
System for
Example 8.2
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Table 8.3
Routh table for Eq. (8.40)
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.30
Root locus of pitch control loop
without
rate feedback,
UFSS vehicle
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.31 Computer simulation of step
response of pitch control loop
without rate feedback,
UFSS vehicle
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 8.32
Root locus of pitch
control loop with
rate feedback,
UFSS vehicle
Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
