Ch6

The Root Locus Method

Main content

The Root Locus ConceptThe Root Locus ProcedureGeneralized root locus or Parameter RLParameter design by root locus methodPID controllers and RL methodExamples and simulation by MATLABSummary

Introduction

In 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 of roots in the s-plane as one system parameter is changed.

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

Root locus equationRelationship between the open-loop and

closed-loop poles and zeros

Root locus equation:

j

m

j

i

n

i

n

ii

m

jj

zs

psK

kkpszs

1

1

11

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

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 locus

Necessary and sufficient condition for root locus plot

Gain evaluation for specific point of root locus

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

ii

m

jj

ps

zs

K

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 as K increases from 0 to infinity.

If open-loop system has n-m zeros at infinity, there will be n-m branches of the root locus approaching the n-m zeros at infinity.

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 and

symmetrical with respect to the horizontal

real axis.

6.2 Root locus procedure

Step 7: The loci proceed to zeros at infinity along asymptotes centered at and with angles :

6.2 Root locus procedure

aa

mn

zpn

i

m

jji

a

1 1

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

mnkmn

ka

Step 8: The actual point at which the root locus crosses the imaginary axis is readily evaluated by using Routh criterion.

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

6.2 Root locus procedure

m

j

n

i ij pdzd1 1

11

Step 10: Determine the angle of departure of locus from a pole and the angle of arrival of the locus at a zero by using phase angle criterion.

6.2 Root locus procedure

ip

iz

)(180,11

0

n

ijjpp

m

jpzp ijiji

)(1801,1

0

n

jzp

m

ijjzzz ijiji

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

Step 12: Determine the parameter value K1

at a specific root using the magnitude criterion.

6.2 Root locus procedure

,2,1)12()( kksP

1s

11

11

)(

)(

ss

m

jj

n

ii

zs

psK

An example

Fourth-order system

Refer to Table7.2

Illustration of complete procedure

Page347-349

Summary of root locus procedure

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

Assignment

E7.6

E7.18