10.Root Locus Techniques

7/26/2019 10.Root Locus Techniques

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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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Points


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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"

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10.Root Locus Techniques