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## Homework Statement

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

## Homework 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)]

## 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 y

^{2}, 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.