
In (a), (b), (c), (d), and (e), the poles p
1
and p
2
are real and distinct. From Equation 4.262, the free
response is the linear combination of natural modes e
p
1
t
and e
p
2
t
. Since
lim
t!1
e
pt
¼
0, p < 0
1, p ¼ 0
1, p > 0
8
>
<
>
:
(4:264)
the two natural modes decay to zero in (a), and the limit L ¼0. Therefore, (a) corresponds to an
asymptotically stable system. In (b), one of the natural modes grows monotonically over time and
L fails to exist. Hence, (b) represents an unstable system. A similar analysis of the remaining cases
(c) through (k) leads to the results shown in Table 4.2.
In summary, the second-order system with transfer function in Equation 4.260 is