-
8/10/2019 Root Locus Examples
1/5
1
Root Locus Examples #4a and #4b
These two examples are nearly identical in terms of the
open-loop systems G(s)H(s).The only difference between thetwo is
the location of the imaginary parts of the complex-conjugate
open-loop poles. The real parts of those poles, theother poles, and
the zero are located at the same places in the two systems. Careful
note should be made of the effectsthat this change in location of
the imaginary part of the poles has on the root locus. Some
calculations will produce the
same results; some will produce different results. The root
locus plots will be different because of this, and you
shouldunderstand why the calculations produce their respective
plots.
A. Example #4a
G(s) = K(s + 3)
s (s + 2) (s2 + 4s + 5)=
K(s + 3)
s (s + 2) (s + 2 j) (s + 2 +j), H(s) = 1 (1)
n= 4, m= 1, n m= 3 (2)
Real axis part of root locus:
2 s 0, s 3 (3)Angle of asymptotes:
A =180(2l+ 1)
3 , l= 0, 1, 2 A = 60
, 180, 300 = 60, 180 (4)
Center of asymptotes:
A =0 + (2) + (2 +j) + (2 j) (3)
3 = 1 (5)
Break points:
1 + K(s + 3)
s (s + 2) (s2 + 4s + 5) = 0 at any closed-loop pole (6)
K = s (s + 2)
s2 + 4s + 5
(s + 3)
= s4 + 6s3 + 13s2 + 10s
s + 3 (7)
dK
ds =
(s + 3)
4s3 + 18s2 + 26s + 10
s4 + 6s3 + 13s2 + 10s
(1)
(s + 3)2
= 0 (8)
dK
ds = 0 = 3s4 + 24s3 + 67s2 + 78s + 30 (9)
dK
ds = 0 = (s + 0.7509) (s + 3.5424)(s + 1.8533 j0.5697) (10)
B = 0.7509, 3.5424 (11)
Angle of departure:
D = [180 + 90 + p3+ p4 z1] (12)
=
270 + tan1
1 0
2 0
+ tan1
1 0
2 (2)
tan1
1 0
2 (3)
(13)
= [270 + (26.5651 + 180) + 90 45] = [270 + 153.4349 + 90 45]
(14)
D = 468.4349
D = 108.4349 (15)
-
8/10/2019 Root Locus Examples
2/5
-
8/10/2019 Root Locus Examples
3/5
3
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
Real Axis
ImagAxis
Root Locus for G(s) = K(s+3)/[s(s+2)(s+2-j)(s+2+j)]
Fig. 1. Root locus plot for Example 4a.
B. Example #4b
G(s) = K(s + 3)
s (s + 2) (s2 + 4s + 13)=
K(s + 3)
s (s + 2) (s + 2 j3) (s + 2 +j3), H(s) = 1 (26)
n= 4, m= 1, n m= 3 (27)
Real axis part of root locus:
2 s 0, s 3 (28)
Angle of asymptotes:
A =180(2l+ 1)
3 , l= 0, 1, 2 A = 60
, 180, 300 = 60, 180 (29)
Center of asymptotes:
A =0 + (2) + (2 +j3) + (2 j3) (3)
3 = 1 (30)
-
8/10/2019 Root Locus Examples
4/5
4
Break points:
1 + K(s + 3)
s (s + 2) (s2 + 4s + 13) = 0 at any closed-loop pole (31)
K = s (s + 2)
s2 + 4s + 13
(s + 3)
= s4 + 6s3 + 21s2 + 26s
s + 3 (32)
dKds
=
(s + 3)
4s3
+ 18s2
+ 42s + 26
s4
+ 6s3
+ 21s2
+ 26s
(1)
(s + 3)2
= 0 (33)
dK
ds = 0 = 3s4 + 24s3 + 75s2 + 126s + 78 (34)
dK
ds = 0 = (s + 1.1845)(s + 3.9293) (s + 1.4431j1.8718) (35)
B = 1.1845, 3.9293 (36)
Angle of departure:
D = [180 + 90 + p3+ p4 z1] (37)
=
270 + tan1
3 02 0
+ tan1
3 0
2 (2)
tan1
3 0
2 (3)
(38)
= [270 + (56.3099 + 180) + 90 71.5651] = [270 + 123.6901 + 90
71.5651] (39)
D = 412.125
D = 52.125 (40)
Closed-loop transfer function:
TCL(s) = G(s)
1 + G(s)=
K(s + 3)
s4 + 6s3 + 21s2 + (26 + K) s + 3K (41)
Closed-loop characteristic equation:
CL(s) =s4 + 6s3 + 21s2 + (26 + K) s + 3K (42)
Routh Array:
s4 : 1 21 3Ks3 : 6 (26 + K)
s2 : 126 (26 + K)
6 3K
s1 : c1s0 : 3K
c1 =126 (26 + K)
6 (26 + K) 18K
126 (26 + K)
6
c1 = (100 K) (26 + K) 108K6
(43)
(K+ 70.7494) (K 36.7494) (44)
Limits on Kfor stability:
s0 : 3K >0 K >0 (45)
s2 : 100 K >0 K 0 70.7494< K
-
8/10/2019 Root Locus Examples
5/5
5
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
Real Axis
ImagAxis
Root Locus for G(s) = K(s+3)/[s(s+2)(s+2-j3)(s+2+j3)]
Fig. 2. Root locus plot for Example 4b.
0< K
