[SOLVED] CS Linearizing a non-linear system

Linearizing a non-linear system
Robert Platt Northeastern University

Pendulum EOM for pendulum:
How do we get this system in the standard form:
?

Pendulum EOM for pendulum:
How do we get this system in the standard form:

Pendulum EOM for pendulum:
How do we get this system in the standard form:
!!!!!!!

Linearizing a non-linear system Idea: use first-order Taylor series expansion
original non-linear system

Linearizing a non-linear system Idea: use first-order Taylor series expansion
original non-linear system
Linearize about
first order term

Linearizing a non-linear system
Idea: use first-ordeWr eTajuyslot rlinseraieriszedxpthaensiyosntem about x^*
original non-linear system
Linearize about
first order term

Linearizing a non-linear system
Suppose that x^* is a fixed point (or a steady state) of the system Then:

Linearizing a non-linear system
Suppose that x^* is a fixed point (or a steady state) of the system Then:
where
Change of coordinates

Example: inverted pendulum

Example: inverted pendulum
Linearize about:

Example: inverted pendulum
where

Example: inverted pendulum
Another way to think about this is: