Linearize 2nd Order Diff Equations

[MstrGnrl]

Hello folks,

I am attempting to implement an LQR controller to a quadrotor and in order to do this I need to linearize the model's equations about a certain trim point, in this case hover, which makes all initial conditions equal to 0. However I am having a lot of trouble linearizing these equations because only one of each (theta, phi, and gamma) are in each equation which makes it very confusing... The inputs are U1, U2, U3 and the outputs are phi, theta, gamma. Can anyone guide me toward an example or explain where I can get started.

 

[hunt_mat]

How about writing:
$$\begin{array}{rcl}<br /> \theta & = & \theta_{0}+\varepsilon}\theta_{1} \\<br /> \phi & = & \phi_{0}+\varepsilon}\phi_{1} \\<br /> \gamma & = & \gamma_{0}+\varepsilon}\gamma_{1} <br /> \end{array}<br /> And look at the zeroth order equations and the first order equations, they should be linear...$$