$$ m \frac{dy^2}{dt^2} = u(t) - C_d \left( \frac{dy}{dt} \right)^2-mg $$
where $$\begin{align*} y(t)&=\text{missile altitude}\\ u(t)&= \text{force}\\ m&= \text{mass}\\ C_d&= \text{aerodynamic drag coefficient} \end{align*}$$ How do I linearize this beast? I want to obtain a transfer function so that I can create a PID controller for it..
I'm really stumped and could use some help.
If you want to design a PID controller for a nonlinear system based on linearization, the first step is to have to set an operation point for the altitude $x_1^*$, and another operation point for the velocity $x_2^*$.
Rewrite your system in state variable form
$$ \dot{x}_1 = x_2 \\ \dot{x}_2 = -\frac{C_d}{m}x_2^2 -g + \frac{u}{m} $$ , where $x_1$ is the altitude and $x_2$ is the velocity. In matrix form
$$ \begin{bmatrix}\dot{x}_1 \\ \dot{x}_2 \end{bmatrix}= \begin{bmatrix}0 & 1 \\ 0 & -\frac{C_d}{m}x_2\end{bmatrix} \begin{bmatrix}x_1 \\x_2\end{bmatrix} + \begin{bmatrix}0 \\ \frac{u}{m} - g\end{bmatrix} \\ y = Ix $$
I have assumed here that you can observe both, altitude and velocity.
Then, you have to compute the Jacobian of the system evaluated at your operation points (you will realize that the altitude does not matter), and you will have your constant matrices A, B, C and D.
To compute the desired transfer function, you can check the equivalences here State Space Representation
Note that your PID is only valid for a neighborhood about your desired velocity. I.E. you want to keep your desired velocity starting from this desired velocity, and not from another velocity.