Generalized Root Locus Outline Variable Parameter Example

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Outline
• Generalize the root locus to examine the
effect of varying other parameters.
• Rewrite characteristic equation.
• Example.
Generalized Root Locus
M. Sami Fadali
Professor EE
University of Nevada
1
Variable Parameter
2
Example: Variable Pole Location
R(s)

• Vary parameter other than the gain K.
• Rewrite characteristic equation in the
usual form with the parameter appearing
in K’s place.
• Closed-loop poles are still the roots of
the new equation.

10
s  2s  p 
C(s)
K L(s )
Loop gain
K L( s )
1  K L( s )
10
10

 2
s  2s  p   10 s  2s  10  p( s  2)
T (s) 
3
4
Root Locus: p Variable
Rewrite Characteristic Eq.
T (s) 
ଶ
ଶ
10
s 2  2 s  10  p ( s  2)
3
2
Imaginary Axis
Not the
loop gain
0
1
-1
-2
-3
-4
-10
-8
-6
5
Design: Tp about/2 s
-4
-2
0
2
Real Axis
6
Solve for Poles


Tp 

d
2
Root Locus
10
10
1
s2
1 p 2
0
s  2 s  10
4
Radius=
Root Locus
4
10 s 2  2 s  10 

1  p ( s  2) s 2  2 s  10 
Characteristic Eq.
ଶ
• Radius=
3
d  2 rad/s
2
Imaginary Axis
1
 n  2  10  4  2  6  4.45 rad/s
10
2
0
 n  ?
6
-1
n   n 2  d2  4.88 rad/s
  0 .9
-2
-3
-4
-10
-8
-6
-4
Real Axis
-2
0
2
7
8
Using MATLAB
Root Locus
System: g
Gain: 6.89
Pole: -4.44 + 2.01i
Damping: 0.911
Overshoot (%): 0.0953
Frequency (rad/sec): 4.87
4
3
2
• Calculate p using the magnitude condition
L p ( s) 
 n  4.44
n  4.87
1
Imaginary Axis
Determine p
1
s 2  2 s  10

 6.89
p
s2
L p ( s) ss
s  scl
p  6.89
0
-1
cl
-2
G (s) 
-3
-4
-10
s2
s 2  2s  10
-8
-6
-4
-2
0
2
9
Real Axis
Conclusion
• Can use root locus to determine the effect of
varying any parameter.
• Root contours: plot family of root loci with
each plot obtained for one value of the first
parameter (say K) as a second parameter
(say p) is varied.
11
10
s  2s  6.89
10
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