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
∫
∞
∞
−
=
−
).
(
)
(
)
(
0
0
x
f
dx
x
x
x
f
δ
∫
∞
=
s
du
u
F
t
t
f
.
)
(
)
(
L
∫
⋅
⋅
⋅
−
=
−
−
T
st
sT
dt
e
t
f
e
t
f
0
.
)
(
1
1
)}
(
{
L
)].
(
[
lim
)
(
lim
0
0
s
F
s
t
f
f
s
t
⋅
=
=
∞
→
→
)].
(
[
lim
)
(
lim
0
s
F
s
t
f
f
s
t
⋅
=
=
→
∞
→
∞
∫
∞
∞
−
=
.
0
.
1
)
(
dx
x
δ