Root locus - Department Of Electrical...

Root locus

Control system by: Nafees Ahmed 1

Root locus

What is Root Locus ?

H(s)

The characteristic equation of the closed-loop system is

1 + K G(s)H(s) = 0

The root locus is essentially the trajectories of roots of thecharacteristic equation as the parameter K is varied from 0 toinfinity.

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.

Root locus

k G(s)

H(s)

＋－

)(ty)(tr


)()(1

)(

)(

)()(

sHskG

skG

sR

sysT

0)()(1 sHskG poles

)12()()(

1)()(

1)()(

0)()(1

nsHskG

sHskG

sHskG

sHskGOpen loop transfer function

Using open loop transfer function + system parametersto analyze the closed-loop system response


Using open loop transfer function + system parametersto analyze the closed-loop system response

0k

Draw the s-plan root locus

zerossHsGk

polessHsGk

)()(,

)()(,0

)1.01)(5.01()()(

sss

ksHskG

Example: Draw Root locus

Sol:In Matlab>>n=[1];>>d=[0.05 0.6 1 0];>>rlocus(n,d)


Sol:In Matlab>>n=[1];>>d=[0.05 0.6 1 0];>>rlocus(n,d)

Root locus construction

(ii) Loci Branches: Each locus starts from an open pole (K=0) andterminates at either on an open loop zero oron infinity (K=∞)

P=No of finite polesZ=No of finite zeros

(i) Root Locus is symmetrical about real axis

No of branches in root loci =P if P>Z=Z if Z>P

Let


if ZP

if togoesbranchesZP

zerostofromcomesbranchesPZ

=>Poles are more

=>Zeros are more

No of branches in root loci =P if P>Z=Z if Z>P

PZ

(iii) Existence of Root loci on real axis

Poles + zeros = odd

0180

Note: Only real axis Poles+ Zeros not complex


Poles + zeros = even

01801)()( sHskG

If Poles + Zeros on RHS= Odd=> That portion will be in root locusIf Poles + Zeros on RHS= Even=> That portion will not be in root locus

(iv) Asymptotic angles,2,1,0,

)12(

kZP

kk

0454

1802,6 ZPif

Reference for angle


045

045

Total no of asymptotes =P-Z

Reference for angle

(v) Centroid of the asymptotes

ZP

zerospoles

)186)(2(

3)()(

2

sss

ssHsG

Zero : 0Poles: -2, -3+j3, -3-j3 4

13

0)33332(

jj

example


Zero : 0Poles: -2, -3+j3, -3-j3 4

13

0)33332(

jj

09013

180

(vi) Breakaway and entry points 0ds

dk

example)2)(1(

sss

kkGH

01 kGH The characteristic function of closed loop system

0)2)(1(

231

23

sss

kssskGH


0)2)(1(

231

23

sss

kssskGH

577.1,423.0

0263

)23(

2

23

s

ssds

dk

sssk

(vii) Angle of departure and arrival

example)1)(1(

)2(

jsjs

skkGH

Angle of departure from the pole: js 1

)()(1800 sHsGA

)()(1800 sHsGD Angle of departure from a complex pole

Angle of arrival at a complex zero


0

0

0

0

225

)11()21(180

180)11()21(

180)1()2(

180)1()1()2(

D

D

D

D

jjj

jjj

jss

jsjss

Angle of approach to the zero:

example )1(

))((

ss

jsjskkGH

js

0

0

0

0

0

135

)1()(180

180)1()(

180)1()(

180)1()()(

A

A

A

A

jjjj

jjjj

ssjs

ssjsjs


0

0

0

0

0

135

)1()(180

180)1()(

180)1()(

180)1()()(

A

A

A

A

jjjj

jjjj

ssjs

ssjsjs

(viii) Intersection with Img axis: By Routh Hurwitz criterion

example)22)(3( 2

ssss

kkGH

The characteristic function of closed loop system:

0685

0)22)(3(234

2

kssss

kssss

ks

ks

ks

s

ks

0

1

2

3

4

34

252045

3465

81

16.8

034

25204

k

k

rowzerogetTo


ks

ks

ks

s

ks

0

1

2

3

4

34

252045

3465

81

16.8

034

25204

k

k

rowzerogetTo

095.1

05

34 2

js

equauxilaryks

)1.01)(5.01( sss

k

+ -

)(sC)(sR

)1.01)(5.01()()(

sss

ksHskG

,,

10,2,0

zeros

poles

05.7,945.0

0)6.005.0(

06.005.0

0)1.01)(5.01(

21

23

23

ss

sssds

d

ds

dk

ksss

ksss(i)

Example


,,

10,2,0

zeros

poles

6003

180

403

0)10()2(0

k

05.7,945.0

0)6.005.0(

06.005.0

0)1.01)(5.01(

21

23

23

ss

sssds

d

ds

dk

ksss

ksss(i)

(ii)

(iii)

ks

ks

ks

s

ksss

6.0

05.06.06.0

105.0

06.005.0

1

2

3

23

5.4

0126.0

122

js

s

k

)12(5.4 kj


)0(

0

k)0(

2

k)0(

10

k

)12(5.4 kj

945.0s4

060

Determination of K on Root loci

point the toG(s)H(s)ofzeros thefromdrawnlengths vectorallofProduct

point the toG(s)H(s)ofpoles thefromdrawnlengths vectorallofProductK

A

BxCK


AB

C

Root Loci

)1.01)(5.01()(

sss

kskGH

MATLAB method

ssss

skskGH

15

)93()(

234

gh=zpk([],[0 –2 -10],[1])rltool(gh)

Example 1

Example 2


n=[-3 -9]m=[1 –1 –1 –15 0]gh=tf(n,m)rltool(gh)

ssss

skskGH

15

)93()(

234

Example 3:

1)(,)84(

)(2

sHsss

KsG

>> n=[1];>> d=[1 4 8 0];>> rlocus(n,d)

Ord may be written as

d=conv([1 0],[1 4 8]);


>> n=[1];>> d=[1 4 8 0];>> rlocus(n,d)

Ord may be written as

d=conv([1 0],[1 4 8]);

Thanks?


Thanks?

Root locus - Department Of Electrical...

Documents

Transcript of Root locus - Department Of Electrical...