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One dimensional altitude control of a hovercraft. The model's dynamic equations that I came up with are, I treat the hover craft as a rigid particle and my goal is to have it hover at or reach a certain desired height.

$$F = ma$$

$$X(t) - mg = ma$$

$$ma + mg = X(t)$$

Taking Laplace Transform yields

$$m \cdot s^2 \cdot Y(s) - mg \cdot U(s) = X(s)$$

$$Y(s) = \frac{X(s)}{m \cdot s^2} + g \cdot U(s)$$

where X is the input thrust and Y is the output and U(s) is a constant unit step. However I can not seem to isolate for $$Y(s)/X(s)$$. I think it may have something to do with me including the gravity force in the dynamics equation when it may act as a disturbance but I am not sure.

When creating a dynamic model of a system How do we handle the constant force of $$mg$$ do we treat it as a disturbance, and if so how do we go about doing that?

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    $\begingroup$ Watch the sign of $mg$ in the third equation. $\endgroup$
    – Karlo
    Mar 4 '16 at 8:24
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It appears that you are trying to derive the transfer function of the system, that is, the input/output relationship in the frequency domain for zero initial conditions.

$ G(s) = \frac{Y(s)}{R(s)}$

Where $G(s)$ is the transfer function, $Y(s)$ is the system output and $R(s)$ is the input to the system and can be a summation of any number of inputs.

$R(s) = R_1(s) + R_2(s) + ... + R_n(s)$

(Note: normally in systems and controls we use $U$ for inputs, but you have already defined it as a unit step at time = 0 in your question so I will avoid that notation and use $R$ instead.)

How to treat the constant gravitational force

In a translational mass system, all external forces are inputs to the system. The gravity is an external force and therefore must be included in your input term when you derive the transfer function. Therefore you should have something like this:

$m\ddot{y} = x(t) + mg\dot{}u(t) = r_1(t) + r_2(t) = r(t)$

Taking the Laplace transform of both sides yields:

$ms^2Y(s) = X(s) + \frac{1}{s}mg$

$ms^2Y(s) = R_1(s) + R_2(s) = R(s)$

$G(s) = \frac{Y(s)}{R(s)} = \frac{1}{ms^2}$

How to derive Y(s)/X(s)?

I am guessing you still want know how to derive the transfer function for $Y(s)/X(s)$. As you stated, it is not possible to separate out the $X(s)$ term, but you can instead define your input thrust as having a variable (controlled) component $x_c$ and a constant (operating point) component $x_{o}$.

$x(t) = x_c(t) + x_{o}\dot{}u(t)$

Then all you have to do is set your operating point to be equal to the gravitational force:

$x_{o} = -mg$

This then cancels with the gravitational force, allowing you to derive a transfer function between the variable component of your thrust and the altitude of the hovercraft.

$m\ddot{y} = x_c(t) - mg\dot{}u(t) + mg\dot{}u(t)$

$\frac{Y(s)}{X_c(s)} = \frac{1}{ms^2}$

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