Digital Control Systems [MEE 4003] - Kckong.info

2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 41
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Figure 2.11: Natural frequency and damping ratio of a transfer function
G(s) can be expressed in the following form:
G(s) =
3
s 2 +2ζω n s+ω 2 n
where ω n = 1, and ζ = 0.5. One can say that the transfer function has the natural
frequency of1and the damping ratio of0.5.
When a transfer function has more than two poles, it is not clear how the damping
ratio and the natural frequency are defined. Originally the transfer function in (2.13) is
obtained from a mass-spring-damper system, which follows a second-order linear differential
equation 8 . Therefore, in a strict sense the damping ratio and the natural frequency
can be defined only when the transfer function has two poles. In a general sense, ζ and
ω n are defined for each pair of complex poles.
Repeated poles
Suppose that a transfer function,G(s), hasmdistinct poles andn−m repeated poles. By
the partial fraction expansion, it can be expanded as
m∑
n−m
k i
∑ κ k
G(s) = +
s−p
i=1 i (s−p m ) k+1
k=1
where k i ’s and κ k ’s are scalars, p i ’s are the distinct poles and p m is the repeated pole. If
the system is under an impulse input, i.e.,U(s) = 1, the output is
8 mÿ +cẏ +ky = u
y(t) = L −1 {
=
m∑
i=1
m∑
k i e pit +
i=1
n−m
k i
∑ κ k
+
s−p i (s−p m ) k+1}
k=1
n−m
∑
k=1
α k t k e pmt
Digital Control Systems, Sogang University
Kyoungchul Kong