chapter 5 - root locus - ho

8/4/2019 Chapter 5 - Root Locus - HO

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Chapter 5:Root Locus Design

Method

Part A: Introduction

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Goal

Learn a specific techniquewhich shows how

changes in one of a systemsparameter

(usually the controller gain, K)

w ill modify the location of the closed-

loop polesin the s-domain.

3

Definition

( ) ( ) 01 =+ sHsKG

The closed-loop poles of the negative feedbackcontrol:

are the roots of the characteristic equation:

( ) ( ) 0sHsKG1 =+

The root locus is the locus of the closed-loop poles

when a specific parameter (usually gain, K)

is varied from 0 to infinity.4

Root Locus Method Foundations

The value ofsin the s-plane that make theloop gain KG(s)H(s) equal to -1 are theclosed-loop poles

(i.e.)

KG(s)H(s) = -1 can be split into twoequations by equating the magnitudesand angles of both sides of the equation.

( ) ( ) ( ) ( ) 1sHsKG0sHsKG1 ==+


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Angle and MagnitudeConditions

Independent of K

K,2,1,0l =

K,2,1,0l =

( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

+=

=

+=

=

=

1l2180sHsG

K1sHsG

1l2180sHsKG

1sHsKG

1sHsKG

o

0

Chapter 5:Root Locus Design

Method

Part B: Drawing Root Locus directly

(= without solving for the closed-loop

poles)

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Learning by doing Example 1

1) Sketch the root locus of the followingsystem:

2) Determine the value ofKsuch that thedamping ratio of a pair of dominantcomplex conjugate closed-loop is 0.5.

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Rule #1

Assuming npoles and mzeros for G(s)H(s):

The nbranches of the root locus start at the

npoles. mof these nbranches end on the mzeros

The n-mother branches terminate atinfinity along asymptotes.

First step: Draw the npoles and mzeros ofG(s)H(s) using x and o respectively


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Applying Step #1

Draw the npoles and mzeros ofG(s)H(s) usingx and o respectively.

3 poles:

p1 = 0; p2 = -1; p3 = -2

No zeros

( ) ( )( )( )2s1ss

1sHsG

++=

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Applying Step #1

Draw the npoles and mzeros ofG(s)H(s) usingx and o respectively.

3 poles:

p1 = 0; p2 = -1; p3 = -2

No zeros

( ) ( )( )( )2s1ss

1sHsG

++=

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Rule #2

The loci on the real axis are to the l ef t of an ODD number of REAL poles and REAL

zeros ofG(s)H(s)

Second step: Determine the loci on the real axis.Choose a arbitrary test point. If the TOTALnumber of both real poles and zeros is to theRIGHT of this point is ODD, then this point is

on the root locus

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Applying Step #2

Determine the loci onthe real axis:

Choose a arbitrary

test point.

If the TOTAL numberof both real polesand zeros is to theRIGHT of this point isODD, then this pointis on the root locus


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Applying Step #2

Determine the loci onthe real axis:

Choose a arbitrarytest point.

If the TOTAL numberof both real polesand zeros is to theRIGHT of this point isODD, then this pointis on the root locus

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Rule #3Assuming npoles and mzeros for G(s)H(s):

The root loci for very large values of s must beasymptotic to straight lines originate on the realaxis at point:

radiating out from this point at angles:

Third step: Determine the n - masymptotes of the rootloci. Locate s = on the real axis. Compute and drawangles. Draw the asymptotes using dash lines.

( )

mn

1l2180o

l

+=

mn

zp

s mi

n

i

==

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Applying Step #3

Determine the n - masymptotes:

Locate s = on the real axis:

Compute and draw angles:

Draw the asymptotes using

dash lines.

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210

03

ppps 321 =

=

++==

( )mn

1l2180l

+=

( )

( )

=

+=

=

+=

0

0

1

00

0

18003

112180

6003

102180

K,2,1,0l =

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Applying Step #3Determine the n - m

asymptotes:

Locate s = on the real axis:

Compute and draw angles:

Draw the asymptotes usingdash lines.

13

210

03

ppps 321 =

=

++==

( )mn

1l2180l

+=

( )

( )

=

+=

=

+=

0

0

1

00

0

18003

112180

6003

102180

K,2,1,0l =


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Breakpoint Definition

The breakpoints are the points in the s-domain where multiples roots of thecharacteristic equation of the feedbackcontrol occur.

These points correspond to intersectionpoints on the root locus.

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Rule #4

Given 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.

0ds

dK=

( ) ( ).

sHsG

1K

=

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Applying Step #4Find the breakpoints.

Express Ksuch as:

Set dK/ds= 0 and solvefor the poles.

4226.0s,5774.1s

02s6s3

21

2

==

=

( )( )

s2s3sK

2s1ss

)s(H)s(G

1K

23=

++=

=

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Applying Step #4Find the breakpoints.

Express Ksuch as:

Set dK/ds= 0 and solvefor the poles.

4226.0s,5774.1s

02s6s3

21

2

==

=

( )( )

s2s3sK

2s1ss

)s(H)s(G

1K

23 =

++=

=


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Recall Rule #1

Assuming npoles and mzeros for G(s)H(s):

The nbranches of the root locus start at the npoles.

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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Applying Last Step

Draw the n-mbranchesthat terminate at infinityalong asymptotes

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Points on both root locus &imaginary axis?

Points on imaginary axissatisfy:

Points on root locus satisfy:

Substitute s=j into thecharacteristic equation andsolve for .

=s j?

- j

( ) ( ) 0sHsKG1 =+

2or0 ==

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Learning by doing Example 1

1) Sketch the root locus of the following system:

2) Determine the value ofKsuch that thedamping ratio of a pair of dominant complexconjugate closed-loop is 0.5.

chapter 5 - root locus - ho

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