10.root locus techniques
TRANSCRIPT
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Root LocusTechniques
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IntroductionImportance of Pole Location
Performance is a function of pole location transient response
absolute stability (stable or not?)
relative stability (how stable?)
Poles migrate as control parameters vary function of controller gains, zeros, poles
what values produce good locations? design (place poles) using root locus
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Introduction- TransientResponse
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Introduction -Absolute Stability
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IntroductionRouth -Hurwitz Stability
Rouths criterion is a method for assessing stability
without finding roots!
"he method is tabular, finds the number of roots with
positive real parts, and is described in most controlste#tboo$s!
"he method was developed in the late %&''s when
finding roots was difficult!
Powerful calculation tools on the des$top havemade the method less useful!
Review it at a high level at this point!
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IntroductionRelative Stability
How stable is a system? compared to another system
distance to the border of instability
Measures of relative stability damping associated with each root
real parts of roots
gain and phase margins
(frequency response concept : study later)
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Close Loop System
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Close Loop System
)(
)()(
)()()(
sD
sNsH
sDsNsG
H
H
G
G
=
=Process "ransfer unction
eedbac$ "ransfer unction
lose *oop +ystem "ransfer unction
)()()()(
)()()(
)()(1
)()(
sDsKNsDsD
sDsKNsT
sHsKG
sKGsT
HGHG
HG
+=
+
=
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ReviewVector Representation of Comple
!umberVectorrepresentationof comple numbers!a" s " # j$
b" %s# a&$c"alternaterepresentationof %s# a&$
#"%s# '&(s) # j*
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ReviewVector Representation of Comple
!umber
he magnitude and angle of
!(s) at any point of s is :
=
+
+
==
=
=
anglespoleangleszero
zs
zs
lenghtspole
lenghtszeroM
m
j
j
m
i
i
__
_
_
1
1
=
+
+
=
=
=
factorscomlexsrdenumerato
factorscomlexsnumerator
ps
zs
sFm
j
j
m
i
i
__'
__'
)(
)(
)(
1
1
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+e,nin the Root Locus$utomatic Vi#eo Camera
System
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+e,nin the Root Locus$utomatic Vi#eo Camera System-Pole Location
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+e,nin the Root Locus$utomatic Vi#eo Camera System-Pole Location
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Root Locus
"efinition "he root locus is the path of the roots
of the characteristic e-uation plotted in the s.
plane as a system parameter is changed!
"esign hoose a parameter value for which thelocus lies in a good area of plane (where
dynamics meet specs)!
#teration /f no part of the root locus lies in a
good area of the s.plane, then change the
structure of the controller to modify the locus!
"hen choose parameter value!
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.roperties of the Root Locus
)()(1
)()(
sHsKG
sKGsT
+
=
oksHsKG 180)12(11)()( +
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S/etchin the Root Locus!umber of %ranch
"he number of
branches of the root
locus e-uals the
number of closed.looppoles!
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S/etchin the Root LocusSymmetry
"he root locus
symmetrical about the
real a#is
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S/etchin the Root LocusReal-ais se&ments
0n the real a#is, for 12', the root locus e#ists to
the left of an odd number of real a#is, finite
open.loop poles and3or finite open.loop zeros!
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S/etchin the Root LocusStartin& an# en#in& points
"he root locus begins at the finite and infinite
poles of 4(s)5(s) and ends at the finite and
infinite zeros of 4(s)5(s)!
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S/etchin the Root Locus%ehavior at in'nite
"he root locus approaches straight lines as asymptotes as
the locus approaches infinity! urther, the e-uation of the
asymptotes is given by the real.a#is intercept, 6a, and the
angle, 7a, as follows
8here , and the angle is given in radians
with respect to the positive e#tension of the real a#is!
zerosfinitenumberpolesfinitenumber
k
zerosfinitenumberpolesfinitenumber
zerosfinitepolesfinite
a
a
____
)12(
____
__
+=
=
3,2,1,0 =k
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0ample1S(etchin& a root locus with
asymptotes
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Re,nin the s/etchReal $is %rea(away an# %rea(-in
Points "he root locusbrea$s away fromthe real a#is at a
point where thegain is ma#imum,and brea$s intothe real a#is at a
point where thegain is minimum!
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Re,nin the s/etchReal $is %rea(away an# %rea(-in
Points
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Re,nin the s/etchReal $is %rea(away an# %rea(-in
Points
82.345.1
2
1
1
1
5
1
3
1
11
2
1
11
==
++
+=
+
+=
+
m
i
m
i pz
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Re,nin the s/etchThe )-$is Crossin&s
"o find the 9.a#is crossing, we can use the
Routh.5urwitz stability criterion!
he root locus crossed the *+ais at ,*-./0 and *-./0 at a gain of 0.1/.
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Re,nin the s/etch$n&le of *eparture an# $rrival
"he root locus departs from comple#, open loop
pole and arrive at comple#, open loop zeros!
"he angle of departure and arrival can be calculated
as follows :ssume a point close to the comple# pole or zero! :dd all
angles drawn from all open loop poles and zeros to this
point! "he sum e-ual (;$
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Re,nin the s/etch$n&le of *eparture
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Re,nin the s/etch$n&le of $rrival
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Re,nin the s/etchPlottin& an# calibratin& the root locus
:ll points on the root locus satisfy the relationship
=4(s)5(s)>(;$
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Re,nin the s/etch+ample"
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Re,nin the s/etch+ample"