Tutorial 6Solutions to Problem 1

Problem 1Investigate each of the transfer functions below for

BIBO stability and asymptotic stability.

))(1()( 22 ++= ssK

sg

)34()( 2 ++= ssK

sg

)34()1()( 2 ++

+=

sss

sKsg

Problem 1Solution:We shall find the roots and of the pole polynomial and judge stability accordingly.

))(1()(

22 ++=

ss

Ksg

The roots of the pole polynomial are:

jss

==

3,2

1 /1

The pure imaginary roots indicate that the system is not asymptotically stable. Unit step-response will oscillate continuously.

Problem 1 Now consider the second TF

)34()(

2 ++=

ss

Ksg

The roots are located at:

31

2

1

=

=

s

s

Both are real and negative, implying Stable System

Problem 1 Consider now,

)34()1()(

2 ++

+=

sss

sKsg

The roots are located at:

031

3

2

1

=

=

=

s

s

s

There are two poles real and negative and one at the origin. First note that the pole at 1 will cancel with the zero in the numerator, but this does not create a problem since it is a stable pole. The pole at the origin indicates critical stability stability,