How can I linearize the drag force without any constants terms?

Question

How can I linearize the drag force at $V=V_{medium}$:

$F_d = \frac{1}{2}C_d.A.\rho .V(t)^2$ without any constant term? in the form:

$F_d=K_1V(t)+K_2$

K2 is the constant term and should be 0.

EDIT: If I expand by Taylor I can get the linearization, but K2 will be not 0. I need K2=0 to put the function in a space state for control theory. Do you have any ideas? I solved this kind of problems for other terms doing a linear transformation at the state variable (in the drag force case, this variable is V)

I'm kind stuck

Answer 1

Let $k=C_d A \varrho$, which means that $$F(V)=\frac{1}{2}kV^2$$ Using Taylor's theorem, we have that $$F(V)\approx F(V_{medium})+F'(V_{medium})[V-V_{medium}]$$ I.e. $$F(V) \approx \frac{1}{2}kV_{medium}^2+kV_{medium}[V-V_{medium}]$$

But the constant term won't be $0$. It happens because we can linearize it when the difference $V-V_{medium}$ is small, but in that case the constant is too big to neglect.