Suppose we have
We can have a linear approximation around the equilibrium point. Redefine
so
if 
, 
must also
be small.
More is know about system such as
Assume 
(or 
for sufficiently
large 
). Does there exist a transformation s.t.
Problem: find an integer , and a feedback of the
form 
(
is an external command
variable), such that when 
is substituted we have
reduces to
Suppose
If
then
So
For 
let
substitution gives
is equivalent to a first order system and we
can do a simple solution and establish feedback.
Example 
.
Let
so
then
This is called input-output linearization.
More generally: we have the Lie Derivative along 
, i.e.
a vector field , where
What if we have merely a linear system?
i.e.,
Then
The transfer function is given by
where 
's are the coefficients of the characteristic polynomial.
for 
, so 
,
, 
corresponds to 
.
Example 
.
Feedback linearization, suppose
if
is invertible. A nonlinear transformation of a nonlinear sytem that
is linear.
Substitute
Use control to cancel the nonlinearities and get a linear system. So
Let 
third row of the matrix above 
any linear combination 
such that
we get asymptotic stability 
.
