what is root locus ?

What is Root Locus ?

The characteristic equation of the closed-loop system is

1 + K G(s) = 0

The root locus is essentially the trajectories of roots of the characteristic equation as the parameter K is varied from 0 to infinity.

A simple example

A camera control system:

How the dynamics of the camera changes as K is varied ?

A simple example (cont.) : pole locations

A simple example (cont.) : Root Locus

(a) Pole plots from the table. (b) Root locus.

The Root Locus Method (cont.)• Consider the second-order system

• The characteristic equation is:

,...540,180 and 12

sG

:requiringby found is variedisK gain theas roots theof locus The022

02

11

222

sGssK

ssKsssssKsKGs

nn

Introduction

)2s(sK

+

-

x y

Ks2sK

)s(G1)s(G

)s(X)s(Y

2

Characteristic equation 0Ks2s2

1Kj1Roots

K1K11Roots

1K0

S plane

xxK=0 K=0

-2

K=1

K>1

K>1

j

1-

221

cos that know we1For

1,

nnss

The Root Locus Method (cont.)• Example:

– As shown below, at a root s1, the angles are:

11

11

11

11

s to2- from vector theof magnitude the:2s origin to thefrom vector theof magnitude the:

2

122

1

ss

ssK

ssK

ssK

ss

The Root Locus Method (cont.)• The magnitude and angle requirements for the root locus are:

• The magnitude requirement enables us to determine the value of K for a given root location s1.

• All angles are measured in a counterclockwise direction from a horizontal line.

integeran :

360180......

1......

2121

21

21

qqpspszszssF

pspszszsK

sF

G(s)

H(s)

R +-

C

GH1G

RCF

01nn

01m

1mm

b.......bs)a......sas(K

)s(D)s(KNGH

nm

KNDGD

D/KN1GF

The poles of the closed loop are the roots ofThe characteristic equation

0)s(KN)s(D

Root locus

Closed loop transfer function

Open loop transfer function

Root locus (Evans)

0K 0)s(KN)s(D)s(P

oK=

x x

oK=0 K=0

K= j

s plane

• Root locus in the s plane are dependent on K

• If K=0 then the roots of P(s) are those of D(s): Poles of GH(s)

• If K= then the roots of P(s) are those of N(s): Zeros of GH(s)

Open looppoles

Open loopzeros

K=0 K=

Root locus example

)2s(s)1s(KGH

H=14/K1

2K2s 2

1

4/K12

K2s 22

K)2K(ss

)1s(K)s(F 2

K=1

,5

K=1

,5

j

x-1-2

x

K= 0

K=0K=

oo

K=

1s2sPoleZero

K>0

Any information from Rooth ?

Root locus example

K)2K(ss)1s(K)s(F 2

K)2K(ss2

0

1

2

s

s

s

K2K

1

00K

As K>0

Rules for plotting root loci/loca•Rule 1: Number of loci: number of poles of the open loop transfer Function (the order of the characteristic equation)

)4s(s)2s(KGH 2

Three loci (branches)

XXX

-4

O-2

Poles

Zero

•Rule 2:Each locus starts at an open-loop pole when K=0 and finishesEither at an open-loop zero or infinity when k= infinity

Problem: three poles and one zero ?

Rules for plotting root loci/loca

• Rule 3Loci either move along the real axis or occur as complex Conjugate pairs of loci

S plane

xxK=0 K=0

-2

K=1

K>1

K>1

j


• Rule 4A point on the real axis is part of the locus if the number ofPoles and zeros to the right of the point concerned is odd for K>0

K=1

,5

K=1

,5

j

x-1-2

xK

= 0

K=0K=

oo

K=

1s2sPoleZero

K>0


)4s(s)2s(KGH 2

Three loci (branches)

XXX

-4

O-2

Poles

Zero

Locus on the real axis No locus

Example

Rules for plotting root loci/loca•Rule 5: When the locus is far enough from the open-loop poles and zeros,It becomes asymptotic to lines making angles to the real axisGiven by: (n poles, m zeros of open-loop)

mn180)1L2(

There are n-m asymptotes

90)0L(mn

180

270)1L(mn

180*3

L=0,1,2,3…..,(n-m-1)

Example

2n,0m)2s(s

K)s(G


S plane

xxK=0 K=0

-2

K=1

K>1

K>1

j

Rules for plotting root loci/loca• Rule 6 :Intersection of asymptotes with the real axisThe asymptote intersect the real axis at a point given by

j

4mn

mn

zpn

1i

m

1iii

polepi

zerozi


)4s(s)2s(KGH 2

XXX

-4

O-2

12

24

-1

• Example

Rules for plotting root loci/loca• Rule 7The break-away point between two poles, or break-in pointBetween two zero is given by:

X X XO O

m

1i i

n

1i i z1

p1

XX

O OBreak-away Break-in

First method

Rules for plotting root loci/locaExample

)2s)(1s(sK)s(G

3 asymptotes: 60 °,180 ° and 300°

13

21

02

11

11

Break-away point

0263 2

577.1,423.0

Part of real axis excluded

xxx-1-2

-0.423


Second method

The break-away point is found by differentiating V(s) withRespect to s and equate to zero

)s(G1)s(v

)2s)(1s(sK)s(G

Example

Ks2s3s

K)2s)(1s(s)s(V

23

0K

2s6s3dsdV 2

423.0,577.1s


• Rule 8: Intersection of root locus with the imaginary axisThe limiting value of K for instability may be found using the Routh criterion and hence the value of the loci at theIntersection with the imaginary axis is determined

0)j(D)j(NK1

1

1

Characteristic equation

)2s)(1s(sK)s(G

Example

Characteristic equation 0Ks2s3s 23


0Ks2s3s 23

0

1

2

3

s

s

s

s

What do we get with Routh ?

KK6

31

00K2

If K=6 then we have an pure imaginary solution


0Ks2s3s 23

js

0)2(j)3K(Kj23j 2223

63K

22

2

Example

xxx-1-2

-0.423

2j

2j

K=6K=0

Rules for plotting root loci/loca• Rule 9Tangents to complex starting pole is given by

)'GHarg(180S

GH’ is the GH(starting p) when removing starting p

Example )j1s)(j1s()2s(KGH

)j2()j1(K)j1s('GH

459045'ArgGH

13545180S

j

-1-j

X

X

Rules for plotting root loci/loca• Rule 9`Tangents to complex terminal zero is given by

)''GHarg(180T

GH’ is the GH(terminal zero) when removing terminal zero

Example)1s(s

)js)(js(KGH

)j1(j)j2(K)js('GH

45'ArgGH

225)45(180S

j

-j

O

O

Rules for plotting root loci/locaExample

2)1s()2s(K)s(GH

Two poles at –1One zero at –2One asymptote at 180°Break-in point at -3

XX-1

O-2

=-3

K=2

K=2

K=4 K=0

21

12

142


Example Why a circle ?

Characteristic equation 01K2)K2(ss2

For K<4

2)K4(Kj)K2(

s 2,1

For K>4

2)4K(K)K2(

s 2,1

2)K4(Kj)K2(

2s 2,1

Change of origin

222 KK44K4K)K4(K)2K(m4

1m


XX-1

O-2

=-3

K=2

K=2

K=4 K=0

what is root locus ?

Documents

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