Lecture Root Locus - KMUTT
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Transcript of Lecture Root Locus - KMUTT
DC motor
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TF of DC motor
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mbfafaaa
m
faa
m
a
KKKRsKLJRJsLK
KJsRsLK
sVs
2
1
610J610aLApproximate to the first order:
mbfafaaa
m
a KKKRsKLJRJsLK
sVs
2
)( 1pss
KsVs
a
(No load)
Stability and step response
• What do you think about stability and step response from the system?
• How can we improve step response?
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)( 1pss
KsVs
a
Camera man
Object Trackingusing infrared
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Varying gain (K)
Varying K, closed‐loop poles are moving!!!
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Transient:• K<25 overdamped• K=25 critically damped• K>25 underdamped
• Settling time remains the same under underdamped responses.
Stability:• Root locus never crosses over
into the RHP, system is always stable.
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What is root locus and why is it needed?
• Fact I: poles of closed‐loop system are an important key to describe a performance of the system (transient response, i.e. peak time, %overshoot, rise time), and stability of the system.
• Fact II: closed‐loop poles are changed when varying gain.
• Implication: Root locus = paths of closed‐loop poles as gain is varied.
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Concept of Root Locus
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Sketching Root Locus
1. Number of branches2. Symmetry3. Real‐axis segment4. Starting and ending points5. Behavior at infinity
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1. Number of branchesNumber of branches = number of closed‐loop poles
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2. SymmetryRoot locus is symmetrical about the real axis
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3. Real‐axis segment
On the real axis, the root locus exists to the left of an odd number of real‐axis
180)12()()( ksHsKG
• Sum of angles on the real axis is either 0 or 180 (complex poles and zeroes give a zero contribution).
• Left hand side of odd number of poles/zeros on the real axis give 180 (path of root locus)
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Example
root locus on the real axis
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4. Starting and ending points
Root locus starts at finite/infinite poles of G(s)H(s) and ends at finite/infinite zeros of G(s)H(s)
)()(1)()(
sHsKGsKGsT
closed‐loop transfer function
)()()(
sDsNsG
G
G)()()(
sDsNsH
H
H
)()()()()()()(
sNsKNsDsDsDsKNsT
HGHG
HG
K=0 (beginning) poles of T(s) are
K=∞ (ending) poles of T(s) are
)()( sDsD HG
)()( sNsKN HG
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Example
5. Behavior at infinity
Root locus approaches asymptote as the locus approaches∞, the asymptotes is given by
zeros finite#poles finite#
zeros finitepoles finite
a
...,2,1,0zeros finite#poles finite#
)12(
k
ka
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# of poles = # of zeroes
has 3 finite poles at 0 ‐1 ‐2, and 3 infinite zeroes at infinity
)2)(1()()(
sssKsHsKG
Rule of thumb
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Example Sketch the root locus of the system
Example
Sketch root locus
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14)3()421(
0
2kfor , 3/51kfor , 0kfor , 3/
zeros finite#poles finite#)12(
k
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