Page 16-15
• Laplace transform of a periodic function of period T
:
• Limit theorem for the initial value
: Let F(s) = L{f(t)}, then
• Limit theorem for the final value
: Let F(s) = L{f(t)}, then
Dirac’s delta function and Heaviside’s step function
In the analysis of control systems it is customary to utilize a type of functions
that represent certain physical occurrences such as the sudden activation of a
switch (Heaviside’s step function, H(t)) or a sudden, instantaneous, peak in an
input to the system (Dirac’s delta function, δ(t)). These belong to a class of
functions known as generalized or symbolic functions [e.g., see Friedman, B.,
1956, Principles and Techniques of Applied Mathematics, Dover Publications
Inc., New York (1990 reprint) ].
The formal definition of Dirac’s delta function
, δ(x), is δ(x) = 0, for x ≠0, and
Also, if f(x) is a continuous function, then
∫
∞
∞−
=− ).()()(
00
xfdxxxxf δ
∫
∞
=
s
duuF
t
tf
.)(
)(
L
∫
⋅⋅⋅
−
=
−
−
T
st
sT
dtetf
e
tf
0
.)(
1
1
)}({L
)].([lim)(lim
0
0
sFstff
st
⋅==
∞→→
)].([lim)(lim
0
sFstff
st
⋅==
→∞→
∞
∫
∞
∞−
= .0.1)( dxxδ
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