Linearizing a system of equations?

[Jayalk97]

Homework Statement

So these are the equations of motion for a quad-copter. I am supposed to create a MATLAB model for the z-axis. In order to do this I have to linearize the equations around these points, and arrange them in state space representation.

Homework Equations

As above

The Attempt at a Solution

So there are supposed to be two equations for each state. The first equation, linearized, would just be
x' = v_x
x'' = v_x'
As would the next two, but with y and z correct?

As for the rest I would just take the partial derivative with respect to each point of linearization, evaluate each by all of the given points, multplied by the value, like this (pardon my lack of formatting):

x'' = v_x'
v_x' = (df/dx)x+(df/dy)y+df(dphi)phi+...etc?

I hope that all made sense, am I going about this correctly?

Thanks in advanced.

 

[scottdave]

I'm not sure if this is allowed, but you may be able to use small angle approximations for sine and cosine of the two tilt angles. As far as the voltage hint, what do you know about how these operate?

 

[FactChecker]

I'm not sure if this is allowed, but you may be able to use small angle approximations for sine and cosine of the two tilt angles. As far as the voltage hint, what do you know about how these operate?

I agree. I think that the constant and x term of the power series of the trig functions at the point would be enough for linearization. That is, the value and slope at the point should be good enough.

 

[Jayalk97]

I'm not sure if this is allowed, but you may be able to use small angle approximations for sine and cosine of the two tilt angles. As far as the voltage hint, what do you know about how these operate?

The voltage hint is that since we are only operating in the z-axis they will always have the same value. I think it has to do with converting it to state space representation, not

I agree. I think that the constant and x term of the power series of the trig functions at the point would be enough for linearization. That is, the value and slope at the point should be good enough.

I'm certain we use small angle approximations, he said we would prior to getting the assignment. What confuses me is how to o this for both x and u. Do I use linearization with the voltages as well? Or would I just take the square root of the function to get the voltages.

 

[FactChecker]

Do I use linearization with the voltages as well? Or would I just take the square root of the function to get the voltages.

If the voltages change in response to the dynamics of the system, then you need to linearize them. If they remain constant in the condition that you are studying, then their value at that point should be used.

 

[Jayalk97]

If the voltages change in response to the dynamics of the system, then you need to linearize them. If they remain constant in the condition that you are studying, then their value at that point should be used.

I see, a followup question on linearization I have is how would I treat the velocities? An example would be w_z'. In this equation, when I take the partial derivative with respect to phi, how would I treat w_x, since it is the first derivative of phi?