Automatic control by meiling CHEN1 Lesson 9 Root locus Automatic control 2. Analysis.

automatic control by meiling CHEN 1

Lesson 9

Root locus

Automatic control2. Analysis


Poles and zeros

)())((

)())(()(

21

21

n

m

pspsps

zszszsksF

n

m

ppp

zzz

,,

,,

21

21 zerospoles

axisRe

axisIm

pole

zero


2

2

2)(

2nn

n

sssT

Closed-loop transfer function :

axisRe

axisIm

21 nj

n

n

%100..

4

1

cos

21

2

eom

T

T

ns

n

p

j

)cos1sin(1

1)( 12

2

te

ty n

tn


j

1p

2p

3p

321

123

321

321

321

321

......

sss

nnn

ppp

TTT

sososo

TTT

n

21 nj


j

1p2p3p

321

123

321

321

321

321

......

sss

nnn

ppp

TTT

TTT

sososo

n

..)(

1)(

)(

2

soiii

Tii

Ti

pn

sn


j

0

01

1

01

0

1

10

10

11 1

1

1

1

10

0

0

Overdamped

Critically damped

Underdamped

Undamped

Negative damped


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 parameters to analyze the closed-loop system response

0k

Draw the s-plan root locus


Root locus properties:(i) The locus segments are symmetrical about the real axis.(ii)

(iii)

0,)()(

1k

sHsGk

zerossHsGk

polessHsGk

)()(,

)()(,0

j0s

1

2

4

3)()()( 432100 sHsG


Root locus construction

(i) Loci Branches each locus from poles to zeros 0k k

mn if for excess zeros or poles, locus segments extend from infinity.

(1) 0mn

(2) 0mn

branchesmn

zerosbranchesnm


(ii) Real axis segments

Poles + zeros = odd

Poles + zeros = even

0180

01801)()( sHskG


(iii) Asymptotic angles ,2,1,0,)12(

kmn

kk

0454

1802,6 mnif

045

045


(iv) Centroid of the asymptotes

mn

zerospoles

)186)(2(

3)()(

2

sss

ssHsG

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

13

0)33332(

jj

09013

180

example


(v) 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

577.1,423.0

0263

)23(

2

23

s

ssds

dk

sssk


(vi) Angle of departure and approach

)()(180

)()(1800

0

sHsG

sHsG

A

D

example)1)(1(

)2(

jsjs

skkGH

Angle of departure from the pole: js 1

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


(vii) The cross point of root locus and Im-axis

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

095.1

05

34 2

js

ks


)1.01)(5.01( sss

k

+ -

)(sC)(sR

)1.01)(5.01()(

sss

kskGH

,,

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

)0(

0

k)0(

2

k)0(

10

k

)12(5.4 kj

)12(5.4 kj

945.0s4

060


)1.01)(5.01()(

sss

kskGH

MATLAB method

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

ssss

skskGH

15

)93()(

234

gh=zpk([],[0 –2 -10],[1])rltool(gh)

Automatic control by meiling CHEN1 Lesson 9 Root locus Automatic control 2. Analysis.

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Transcript of Automatic control by meiling CHEN1 Lesson 9 Root locus Automatic control 2. Analysis.