modern control system ekt 308 root locus method (contd…)
DESCRIPTION
Step 2 (review): Locate the segments of the real axis that are root loci. The root locus on the real axis lies in a segment of the real axis to the left of an odd number of poles and zeros. Magnitude and Angle CriterionTRANSCRIPT
Modern Control SystemEKT 308
Root Locus Method (contd…)
Root Locus Procedure(contd…)
Root Locus ProcedureStep 1 (review):
KsPKsF
0 where0)(1 as, arrange-reThen 0)(1equation, sticcharacteri theWrite
(1) ---- 0)(
)(1
follows, as zeros and poles of form in the writeRe
1
1
n
jj
M
ii
ps
zsK
Locate poles and zeros in the s-plane (‘x’ for poles, ‘o’ for zeros)
Step 2 (review): Locate the segments of the real axis that are root loci. The root locus on the real axis lies in a segment of the real axis to the left of an odd number of poles and zeros.
Magnitude and Angle Criterion
0)(1 equation, sticcharacteri theSuppose,
sKP
(1) ---- 0)(
)(1
1
1
n
jj
M
ii
ps
zsK
1)(...)()()(....)()(
Magnitude,21
21
n
M
pspspszszszs
K
integer.an is where,
360.180)(-....)(-)(-
)(...)( )( Angle,
21
21
l
lpspsps
zszszsoo
n
M
Magnitude and Angle Criterion (contd…)
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21
4433
2211
r.rr.r
:refresh Background
rrrr
1. figurein shown are )1(s toingcorrespond Angles
11 2
4.1.K-42-,
areeqn sitccharacteri theof roots theSay,
)02 ,( 0)2(
1 equation, sticCharacteri
Example,
1
21
2
K
Kss
KssorssK
Figure 1: Angle for s = s1
Note: Because complex roots appear as complex conjugate pairs, root loci must be symmetrical with respect to horizontal real axis.
Step 3:The loci proceed to the zeros at infinity along asymptotes centered at
Mn
zp
MnsPofzerossPofPoles
n
j
M
iij
A
AA
1 1
)()(
)()( centroid, Asymptote
where, angle with and
)1,.....(2,1,0 ,180.12
,asymptotes of angle And
MnkMn
k oA
Where n, the order of numerator polynomial and M is the order of denominator Polynomial
Example for step 3.
2 figin shown are axis real on the lociroot and zeros and Poles
0)4)(2(
)1(11
equation, sticcharacteri heConsider t
2
ssssKGH(s)
Figure 2: Root loci on real axis
ly.respective 2 and 1, 0,for 300 ,180 ,60
60).12(180.12
are, asymptotes of angles The
.339
14)1()4(2)2(
),asymptotes ofon intersecti called (also centroid Asymptote
ooo
k
kMn
k ooA
A
Asymptotes are shown in Figure 3
Figure 3: Asymptotes
Step 4:Determine where the locus crosses the imaginary axis (if it does so), using Routh-Hurwitz criterion. Hint: When root locus crosses the imaginary axis from left to right, the system moves from stability to instability.
Example: Complete first four steps of sketching root locus of the characteristic equation
0)44)(44)(4(
1
jsjsss
K
Step 1: Poles and zeros are shown in figure 4.
Figure 4: Poles and zeros
Step 2: There is a segment of root locus on the real axis between s=0 to s=-4 as shown in figure 4 above.
34
444
centroid, Asymptotes.315 ,225 ,135 ,45
3. 2, 1, 0,k ,180.4
12
are, asymptotes of Angles :3 Step
A
oo0
oA
oA
k
The asymptotes are drawn in figure 5.
Figure 5: Asymptotes
Step 4.
Kc
Kb
K
ss
RouthKssss
11
12812641
sss
array, 01286412 equation, sticCharacteri
0
1
2
3
4
234
568.8912/)128(33.53)128(33.5312
012)128(33.53 ,33.53
12)128(33.531 ,33.531
KK
KSo
Kcb
above. 5 figurein shown as 266.3at axisimaginary thecrosses locusroot the568.89, when So,
)266.3)(266.3(33.53))266.3((33.53
)67.10(33.5389.56853.33s
by,given areequation aux theof Roots
222
22
jsK
jsjsjs
s