08. Linearized Model
Linearized Lateral-Directional Model
The full lateral-directional state vector contains 11 variables. When we linearize around a trim state we can ignore the x, z, and pitch variables.
We will also choose to include the sideslip angle \beta in the state, where:
\begin{aligned}
v &= V \sin \beta \\ \\
\bar{v} &= V^* \cos \beta^* \bar{\beta} \\ \\
\dot{\bar{\beta}} &= \frac{1}{V^* \cos \beta^*} \dot{\bar{v}}
\end{aligned}
The linearized model we use looks like this:
\begin{aligned}
\begin{bmatrix}
\dot{\bar{\beta}} \\
\dot{\bar{p}} \\
\dot{\bar{r}} \\
\dot{\bar{\phi}} \\
\dot{\bar{\psi}} \\
\end{bmatrix} &=
\begin{bmatrix}
Y_v & \frac{Y_p}{V^* \cos \beta^*} & \frac{Y_r}{V^* \cos \beta^*} & \frac{g \cos \theta^* \cos \phi^*}{V^* \cos \beta^*} & 0 \\
L_v V^* \cos \beta^* & L_p & L_r & 0 & 0 \\
N_v V^* \cos \beta^* & N_p & N_r & 0 & 0 \\
0 & 1 & \cos \phi^* \tan \theta^* & q^* \cos \phi^* \tan \theta^* \\
& & & -r^* \sin \phi^* \tan \theta^* & 0 \\
0 & 0 & \cos \phi^* \sec \theta^* & p^* \cos \phi^* \sec \theta^* \\
& & & -r^* \sin \phi^* \sec \theta^* & 0
\end{bmatrix}
\cdot
\begin{bmatrix}
\bar{\beta} \\
\bar{p} \\
\bar{r} \\
\bar{\phi} \\
\bar{\psi}
\end{bmatrix} \\
&+
\begin{bmatrix}
\frac{Y_{\delta a}}{V^* \cos \beta^*} & \frac{Y_{\delta r}}{V^* \cos \beta^*} \\
L_{\delta a} & L_{\delta r} & \\
N_{\delta a} & N_{\delta r} & \\
0 & 0 \\
0 & 0 \\
\end{bmatrix} \cdot
\begin{bmatrix}
\bar{\delta} a\\
\bar{\delta} r
\end{bmatrix}
\end{aligned}
Take a look at the fixed wing cheat sheet for a more detailed explanation of the various coefficients.