ODE -> Transfer Function Assistance

[BlueTempus]

Homework Statement:: ODE -> Transfer Function Assistance
Relevant Equations:: Newtonian physics, buoyancy, drag

[Mentor Note -- thread moved to DE from the schoolwork forums, since it is for work and not schoolwork]

Hello all,

I'm new here but I'm looking for a bit of guidance with a control engineering project I'm working on.

I am currently working on designing a buoyancy control module for a submersible. Using Newtonian physics equations, I have started with the following:

$$ma=mg−pgV+0.5pACv^2$$

where m = mass, a = acceleration, g = acceleration due to gravity, p = water density, V = Volume of displaced water (buoyancy), A = cross sectional area of craft, C = coefficient of drag and v = velocity

As I am trying to implement a linear controller, I decided to treat the drag as linear and remove the squared term. I'm not sure if this is appropriate:

$$ma=mg−pgV+0.5pACv$$

I then converted this to a differential equation in terms of displacement:

$$x′′(t)=mg−pgV(t)+0.5pACx′(t)$$

Finally, I carried out a Laplace tranform, assuming 0 initial conditions:

$$s^2X(S) = \frac{mg}{s} - pgV(S) + 0.5pACsX(S)$$

The input to my system is the V(S) term and the output is X(s). I need them as a ratio as X(S) / V(S) to derive the transfer function.

Due to the constant term mg/s, I am unable to separate the variables and obtain the transfer function.

I have looked into State Space Equations which may be a better alternative but I am not familiar with this.

Can anyone offer any advice or spot any errors with my workings?

 

[cnh1995]

I am no expert in fluid mechanics, but here is what I understood from your description:
You wrote the force-balance equation for a body sinking in a fluid. Its net velocity is downward, and the forces acting on it are buoyancy, drag and gravity. Is that right?

 

[BlueTempus]

Hi cnh1995,

Yes this is correct. I used the force due to gravity as the reference so 'negative is up'.

I have been thinking about this all day and the conclusion I have come to is there is no way to rearrange this into a transfer function form of output / input so I think the only way is to model it numerically, then try to approximate a linear transfer function for which I can design a controller.