Laplace transform - Continuous and Discrete Time Signals and Systems - page 301

Property 9 A causal LTIC system with n poles { p l } ,1 ≤ l ≤ n , will be abso-
lutely BIBO stable if and only if the real part of all poles are
non-zero negative numbers, i.e. if

Re p l < 0 for all l . (6.44)

Equation (6.44) states that a causal LTIC system will be absolutely BIBO stable
if and only if all of its poles lie in the left half of the s-plane, (i.e. to the left of the
j ω -axis). In other words, a causal LTIC system will be absolutely BIBO stable
and causal if the ROC occupies the entire right half of the s-plane including the
j ω -axis.
We illustrate the application of the stability condition in Eq. (6.44) in Example
6.20.

Example 6.20
In Example 6.18, we considered the following LTIC systems:

( s + 4)( s + 5)
s 2 ( s + 2)( s − 2) ;

(i) H 1 ( s ) =

( s + 4)
s 3 + 5 s 2 + 17 s + 13 ;

(ii) H 2 ( s ) =

H 3 ( s ) = 1 (iii)

e s + 10 .
Assuming that the systems are causal, determine if the systems are BIBO stable.

Solution
(i) The LTIC system with transfer function H 1 ( s ) has four poles located at
s =− 2, 0, 0, 2. Since all the poles do not lie in the left half of the s-plane,
the transfer function does not represent an absolutely BIBO stable and causal
system. The impulse response of the causal implementation of the LTIC system
was calculated in Example 6.19. It can be easily verified that the time-domain
stability condition, Eq. (6.39), is not satisfied because of the rising exponential
function 21 / 8 exp(2 t ) u ( t ) and the ramp function 5 t , which have infinite areas.
(ii) The LTIC system with transfer function H 2 ( s ) has three poles located
at s =− 1, − 2 j3. Since all the poles lie in the left-half s-plane, the transfer
function represents an absolutely BIBO stable and causal system. The impulse
response of the causal implementation of the LTIC system was calculated in
Example 6.19. It can be easily verified that the time-domain stability condition,
Eq. (6.39), is satisfied as all terms are decaying exponential functions with finite
areas.
(iii) The LTIC system with transfer function H 3 ( s ) has multiple poles located
at s =− 2 . 3 + j(2 m + 1) π , for m = 0 , 1 , 2, . . . Since all the poles lie in the
left-half s-plane, the transfer function represents an absolutely BIBO stable and
causal system.