A complete mathematical model of my quadrotor is presented through Euler's equation of motion, as follows.
$$ \begin{aligned} \ddot x &= \frac1m \left(\cos\phi \sin\theta \cos\psi - \sin\phi \sin\psi \right) U_{1} \\ \ddot y &= \frac1m \left (\cos\phi \sin\theta \sin\psi + \sin\phi \cos\psi \right) U_{1} \\ \ddot z &= −g + \frac1m \left (\cos\phi \cos\theta \right) U_{1} \\ \ddot \phi &= \frac{1}{I_{}} \left (\dot\theta \dot\psi (I_{z} - I_{y}) - J_{r} \dot\theta \Omega + lU_{2} \right) \\ \ddot \theta &= \frac{1}{I_{y}} \left( \dot\phi \dot\psi (I_{x} - I_{z}) + J_{r} \dot\phi \Omega + lU_{3} \right) \\ \ddot \psi &= \frac{1}{I_{z}} \left( \dot\phi \dot\theta (I_{y} - I_{x}) + U_{4} \right) \end{aligned} $$
where
$U_{1}$ is the total thrust generated by the four rotors; $U_{2}$,$U_{3}$ and $U_{4}$ are the respective roll, pitch and yaw thrusts;
$m$ denotes the mass of the quadrotor;
$g$ denotes the acceleration due to gravity;
$l$ represents the length from the motor to the center of mass;
$J_{}$ signifies the moment of inertia;
$I_{}$,$I_{}$ and $I_{}$ are the moments of inertia in the $x$, $y$ and $z$ axes for the whole body;
$Ω$ is the overall residual propeller angular speed which is considered in the gyroscopic torque as the quadrotor rolls or pitches.
I am seeking guidance on transforming this system model to general normal form; specifically, I am thinking third-degree. Any guidance is appreciated, feel free to request further information. My ultimate goal is to achieve a form which meets the below discrete-time nonlinear uncertain system, which is in general normal form:
$$\left\{\begin{matrix} x_{1} (k+1) = x_{2}(k) \\\vdots \\ x_{n-1} (k+1) = x_{n} (k) \\ x_{n} (k+1) = f(x(k)) + g(x(k))u(k) \\ y(k) = x_{1}(k) \end{matrix}\right.$$