automation 8

8/8/2019 Automation 8

1/16

Ch6

The Root Locus Method


2/16

Main content The Root Locus Concept

The Root Locus Procedure Generalized root locus or Parameter RL

Parameter design by root locus method

PID controllers and RL method Examples and simulation by MATLAB

Summary


3/16

IntroductionIn the preceding chapters we discussed the

relationship between the performance and the

characteristic roots of feedback system.

The root locus is a powerful tool for designing

and analyzing feedback control system, it is a

graphical method by determining the locus ofroots in the s-plane as one system parameter is

changed.


4/16

6.1 The root locus concept Definition: The root locus is the path of the

roots of the characteristic equation traced

out in the s-plane as a system parameter is

varied.

Root locus and system performance

Stability

Dynamic performance

Steady-state error


5/16

Root locus equation Relationship between the open-loop and

closed-loop poles and zeros

Root locus equation:

!

ss!!

4

4

!

!

!!

j

m

j

i

n

i

n

i

i

m

j

j

zs

ps

K

kkpszs

1

1

11

)2,1,0()12()()( .T


6/16

Basic task of root locus How to determine the closed-loop poles

from the known open-loop poles and zeros

and gain by root locus equation.

Angle requirement for root locus

Magnitude requirement for root locusNecessary and sufficient condition for root locus plot

Gain evaluation for specific point of root locus


7/16

6.2 The Root Locus Procedure

Step 1:Write the characteristic equation as

Step 2: Rewrite preceding equation into the

form of poles and zeros as follows:

0)(1 ! sF

0

)(

)(

1

1

1 !

!

!

n

i

i

j

j

ps

zs


8/16

6.2 Root locus procedure

Step 3: Locate the poles and zeros with

specific symbols, the root locus begins at

the open-loop poles and ends at the open-

loop zeros asKincreases from 0 to infinity.

If open-loop system has n- zeros at infinity,

there will be n- branches of the root locus

approaching the n- zeros at infinity.


9/16

Step 4: The root locus on the real axis lies in

a section of the real axis to the left of an odd

number of real poles and zeros.

Step 5: The number of separate loci is equal

to the number of open-loop poles.

Step 6: The root loci must be continuous andsymmetrical with respect to the horizontal

real axis.



10/16

Step 7: The loci proceed to zeros at infinity

along asymptotes centered at and with

angles :


aW

aJ

mn

zpn

i

m

j

ji

a

!

! !1 1W

)1,2,1,0()12(

!

! mnk

mn

ka

.T

J


11/16

Step 8: The actual point at which the rootlocus crosses the imaginary axis is readily

evaluated by using Routh criterion.

Step 9: Determine thebreakaway point d(usually on the real axis):


! !

!

m

j

n

i ij pdzd1 111


12/16

Step 10: Determine the angle of departureof locus from a pole and the angle of

arrival of the locus at a zero by using

phase angle criterion.


ip

iz

)(180,11

0

{!!

!n

ijj

pp

m

j

pzp ijijiUNU

)(1801,1

0 !{!

!n

j

zp

m

ijj

zzz ijijiUNN


13/16

Step 11: Plot the root locus that satisfy thephase criterion.

Step 12: Determine the parameter valueK1

at a specific root using the magnitude

criterion.


.,2,1)12()( !! kksP T

1s

11

11

)(

)(

ss

m

j

j

n

i

i

zs

psK

!!

!

!


14/16

An example Fourth-order system

Refer to Table7.2

Illustration of complete procedure

Page347-349

Summary of root locus procedure


15/16

Typical root locus diagrams Refer to Table 7.7

(P381-383)

An summary of 15 typical root locus diagrams is

shown in Table 7.7


16/16

Assignment E7.6

E7.18

automation 8

Documents