Lecture 10 example of root-locus

1

The forward-path transfer function of a negative unity feedback system is :

( )( 1)(0.5 1)

KG s

s S s

1. Sketch the root-loci of the system.

2. Analysis the stability of the system using the root-loci.

3. Determine the range of K value for which the response has no overshoot to

step input.

Solution:

Open loop poles 1 2 30, 1, 2P P P . No open loop zeros.

The asymptotes:

The intersection of the asymptotes is 1

(0 1 2) 13

a

The angles of the asymptotes are

( 0)3

(2 1) ( 1)

3

( 1)3

k

kk

k

Breakaway points on the root loci 3

13

ds

Intersection of the root loci with the imaginary axis.

3 2

3 2

1 ( 1)(0.5 1) 0.5 1.5 0

0.5( ) 1.5( ) 0

2, 3

GH s s s K s s s K

j j j K

K

So, when 0

Lecture 10 example of root-locus

2

Known : when 0.5 , 1,2 0.33 0.58s j

Another root: 1 2 3 1 2 3s s s p p p 3 2.34s

The magnitude of the real part of the pole 3s is about 7 times that of the poles

1,2s . so the pair of complex poles 1,2s are regarded as dominant poles.

The system is approximately regarded as the second-order system.

2

1 2

2 2 2

1

0.445( )

2 ( )( 2) 0.667 0.445

n

n n

s ss

s s s s s s s s

0.5 0.667n

% 16.3% 3.5

10.5 sn

t s