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: