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Aims
Learn a specific techniquewhich shows how
changes in one of a systems parameter(usually the controller gain, K)
will modify thelocation of the closed-loop poles
in the s-domain.
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Definition
01 sHsKG
The closed-loop poles of the negative feedback control:
are the roots of the characteristic equation:
01 sHsKG
The root locus is the locus of the closed-loop poles
when a specific parameter (usually gain, K)
is varied from 0 to infinity.
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Root Locus Method The value of sin the s-plane that make the loop gain
KG(s)H(s) equal to -1 are the closed-loop poles
(i.e. )
KG(s)H(s) = -1 can be split into two equations byequating the magnitudes and anglesof both sides
of the equation.
101 sHsKGsHsKG
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Angle and Magnitude Conditions
Independent of K
,,,l 210
,,,l 210
12180
1
12180
1
1
0
lsHsG
KsHsG
lsHsKGsHsKG
sHsKG
o
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Learning Steps1) Sketch the root locus of the following system:
2) Determine the value of Ksuch that the dampingratio of a pair of dominant complex conjugateclosed-loop is 0.5.
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Rule #1Assuming npoles and mzeros for G(s)H(s):
The nbranchesof the root locus start at the n
poles. mof these nbranches end on the mzeros
The n-mother branches terminate at infinityalong asymptotes.
First step: Draw the npoles and mzeros of G(s)H(s)using x and o respectively
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Step #1Draw the npoles and m
zeros of G(s)H(s) using xand o respectively.
3 poles:
p1 = 0; p2 = -1; p3 = -2
No zeros
21
1
ssssHsG
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Step #1Draw the npoles and m
zeros of G(s)H(s) using xand o respectively.
3 poles:
p1 = 0; p2 = -1; p3 = -2
No zeros
21
1
ssssHsG
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Rule #2
The loci on the real axisare to the leftof an ODDnumberof REAL poles and REAL zerosof
G(s)H(s)
Second step: Determine the loci on the real axis.Choose a arbitrary test point. If the TOTAL number
of both real poles and zeros is to the RIGHT of thispoint is ODD, then this point is on the root locus
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Step #2Determine the loci on the
real axis:
Choose a arbitrary testpoint.
If the TOTAL number ofboth real poles and zeros
is to the RIGHT of thispoint is ODD, then thispoint is on the root locus
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Step #2Determine the loci on the
real axis:
Choose a arbitrary testpoint.
If the TOTAL number ofboth real poles and zeros
is to the RIGHT of thispoint is ODD, then thispoint is on the root locus
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Rule #3
Assuming npoles and mzeros for G(s)H(s): The root loci for very large values of s must be
asymptotic to straight lines originate on the real axisat point:
radiating out from this point at angles:
Third step: Determine the n - masymptotes of the root loci.Locate s = on the real axis. Compute and draw angles.Draw the asymptotes using dash lines.
mn
lo
l
12180
mn
zp
s mi
n
i
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Step #3
Determine the n - masymptotes: Locate s = on the real axis:
Compute and draw angles:
Draw the asymptotes usingdash lines.
13
210
03
321
ppps
mn
ll
12180
0
0
1
0
0
0
18003
112180
6003
102180
,,,l 210
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Step #3
Determine the n - masymptotes: Locate s = on the real axis:
Compute and draw angles:
Draw the asymptotes usingdash lines.
13
210
03
321
ppps
mn
ll
12180
0
0
1
0
0
0
18003
112180
6003
102180
,,,l 210
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Breakpoint Definition The breakpoints are the points in the s-domain
where multiplesroots of the characteristic
equation of the feedback control occur.
These points correspond to intersection points onthe root locus.
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Rule #4Given the characteristic equation is KG(s)H(s) = -1
The breakpoints are the closed-loop poles thatsatisfy:
Fourth step: Find the breakpoints. Express Ksuch as:
Set dK/ds = 0 and solve for the poles.
0dsdK
.
sHsGK
1
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Step #4Find the breakpoints.
Express Ksuch as:
Set dK/ds = 0 and solve for the
poles.
4226057741
0263
21
2
.s,.s
ss
sssK
sss
)s(H)s(G
K
23
211
23
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Step #4Find the breakpoints.
Express Ksuch as:
Set dK/ds = 0 and solve for the
poles.
4226057741
0263
21
2
.s,.s
ss
sssK
sss
)s(H)s(G
K
23
211
23
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Rule #1 AgainAssuming npoles and mzeros for G(s)H(s):
The nbranchesof the root locus start at the n
poles. mof these nbranches end on the mzeros
The n-mother branches terminate at infinityalong asymptotes.
Last step: Draw the n-mbranches that terminate atinfinity along asymptotes
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Last Step
Draw the n-mbranches thatterminate at infinity along
asymptotes
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Points on both root locus & imaginary axis?
Points on imaginary axissatisfy:
Points on root locus satisfy:
Substitute s=jinto thecharacteristic equation andsolve for.
jsj?
- j
01 sHsKG
20 or
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Learning Steps1) Sketch the root locus of the following system:
2) Determine the value of Ksuch that the dampingratio of a pair of dominant complex conjugateclosed-loop is 0.5.
See class notes
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