# Nonlinear differential equation to state space form

1. Sep 27, 2008

### casperl

1. The problem statement, all variables and given/known data
The problem gives a nonlinear differential equation for a metallic ball suspended under maggnetic levitation:

y**(t)=9.81- (0.981u(t))/(y+1)2
(note: y**= second derivative of y)\
where y(t) is the position of the ball and u(t) is the voltage applied ti the magnet

2. Relevant equations
Rewrite the above equation in the form of x*(t)=f(x,u,t) where the state vector x(t)=[y(t)/y*(t)]

3. The attempt at a solution
First I try to bring the equation down to ist order by assuming x1=y, x2=y*
then
(d/dt)x1=x1*=y*=x2 => x2=x*1
and the given equation can be rewriten as
(d/dt)x2=x2*=y**=9.81-(0.981u(t))/(y+1)2
Then I believe I have to write the equation in the form of matrix
which I got
(d/dt)[ x1/x2 ]= [ 0 1/X X ] [x1/x2] + [X/X]u+ [0/9.81]
(note: square bracket means matrix, where "/" is used to divide the rows of the matrix)
X is where I got stuck and don't know what to put in there, I am thinking I have to somewhat seperate the u(t) and y2, but can't figure out a way to do it...and this is only the first part of the question later I have to linearize...

Any hint or point in direction will be great! This is a homework question from my first year Master in control theory, and my major in undergrad was not control so I am pretty clueless right now...

Thank you for looking at my question.

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