EDIT: Background on my application
I am making a few measurement models for an extended Kalman filter (EKF). The EKF is meshing an INS and a GPS currently, but the GPS update rate does not update fast enough to keep the filter converged. So I am tasked with adding some other measurement systems so that I can increase it's performance. Due to it being a Kalman filter I need the models to be linearized. So currently I want to linearize the Pitot tube model for that purpose. I am hoping if I find a good solution, I can also use it for other models with the same issues. So far, all of the work will be done purely as a simulation.
I'm trying to get a linearized model for a pitot tube, but I've got an asymptote in the linearized equation that I don't know how to deal with. I have Bernoulli's equation
$P_o = P + 1/2\rho v^2$
solved for velocity.
$v = \sqrt{\frac{2(P_o-P)}{\rho}}$
I would like a linearized equation so that I can use it as a linear model.
I've taken the derivative with respect to $\Delta P$
$\Delta P = P_o - P$
so that
$\frac{dv}{dt} = \frac{1}{\sqrt{2\rho \Delta P}}\delta \Delta P$
as $\Delta P$ approaches 0 I get an asymptote that I wouldn't get in the nonlinear equation.
My questions are:
1) Is this the proper approach to linearizing this equation?
2) How do I deal with an asymptote, in what should be the physical realm with no disconnect?