Response to
(non-repeated) square wave input (10)
020
T
and
where, again,
Summarizing, Equations 13 and 14 give the solution to the ODE in
Equation 9.
Laplace Transforms
Initial and Final Value Theorems
Initial and Final Value Theorems
Let be a continuous function with Laplace transform
)(tf
)(sF
Then, provided the limits exist,
Laplace Transforms
Response to a
periodic input
Response to a periodic input
We examine here the response of a stable, damped nth order
linear time-invariant system governed by
Consider the nth term:
The Laplace transform of such an input is
Response to a periodic input (2)
nTtt −=
Let . Then, considering the periodicity of ,
)(tg
where
Then, from Equations 16 through 19,
Response to a periodic input (3)
Now
so
and from Equations 1, 15 and 22 , the Laplace transform of the
system output is
)(ty
Response to a periodic input (4)
Theorem:
The response of a stable, damped nth order linear time-invariant
system to a bounded periodic input is eventually periodic.
Proof:
From Equation 22
and from our short table of Laplace transforms
Response to a periodic input (5)
+++
n
1
Hence the inverse Laplace transform of Equation 23 is
For this is
Tt
To show that the response to a periodic input eventually becomes
periodic, we must show that
Response to a periodic input (6)
The text discusses how we can be certain that the limit exists in
Response to a periodic input (7)
From the final value theorem
s
From Equation 19, and using the fact that is bounded
)(tg
From Equations 25 and 26
Response to a periodic input (8)
From Equations 24, 25 and 27
Homework Assignment 15
Read:
In text, Chapter 7, Section 7.6