Modern Control System Theory and Design, 2nd Edition

2.5.  USEFUL LAPLACE TRANSFORMS

The Laplace transforms for various time functions will now be considered. These are readily obtainable through a direct application of Eq. (2.54).

A.  Laplace Transform of a Unit Step

For the unit step function defined by

the Laplace transform is

Therefore,

From here on we assume that f(t) = 0 for t < 0.

B.  Laplace Transform of an Exponential Decay

For the function

we have the Laplace transform

Therefore,

C.  Laplace Transform of a Unit Ramp

For the function

the Laplace transform is

Integrating by parts,

with u = t, dv = e^−st dt, the following is obtained: ...