Nonlinear differential equation to state space form

[casperl]

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 separate 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.

 

[KataKoniK]

Hello, thank you for your post. It looks like you have made some good progress so far. In order to continue, you need to separate the terms involving y and y* in the equation. You can do this by multiplying the equation by (y+1)^2. This will give you:

(y+1)^2*y**=9.81(y+1)^2-(0.981u(t))

Now, you can substitute x1=y and x2=y* back into this equation to get:

(x1+1)^2*x2*=9.81(x1+1)^2-(0.981u(t))

This is now in the form of x*(t)=f(x,u,t) that you were looking for. The state vector x(t)=[y(t),y*(t)] and the function f(x,u,t) is (x1+1)^2*x2*=9.81(x1+1)^2-(0.981u(t)).

I hope this helps. Good luck with the rest of your assignment!

 

[piercas]

I would approach this problem by first understanding the physical system being described - a metallic ball suspended under magnetic levitation. I would then use my knowledge of differential equations and state space representation to rewrite the given nonlinear equation in a form that is more suitable for analysis and control design.

To rewrite the equation in state space form, we need to define the state variables and their derivatives. In this case, we can choose the state vector x(t) = [y(t), y*(t)] and its derivatives as follows:

x1(t) = y(t)
x2(t) = y*(t)
x1*(t) = y*(t)
x2*(t) = y**(t) = 9.81 - (0.981u(t))/(y+1)2

Now, we can rewrite the given equation as:

x1*(t) = x2(t)
x2*(t) = 9.81 - (0.981u(t))/(x1+1)2

This can be further simplified by defining a new variable z = x1+1, which gives:

x1*(t) = x2(t)
x2*(t) = 9.81 - (0.981u(t))/z2

Now, we can write the equation in state space form as:

x1*(t) = x2(t)
x2*(t) = 9.81 - (0.981u(t))/z2
z*(t) = x1*(t) + 1 = x2(t) + 1

Note that the state variable z is not directly affected by the control input u(t), so we can treat it as a constant. This means that we can write the above equations in matrix form as:

(d/dt)[x1/x2/z] = [0 1 0; 0 0 -0.981/z2; 0 0 0] [x1/x2/z] + [0; 0; 0]u + [0; 0; 1]z

This is the desired state space representation of the given nonlinear differential equation. From here, we can proceed to linearize the system and design a control strategy to stabilize the metallic ball in its desired position.