# A Study of Emergency Aircraft Control During Landing
## 5 Aircraft's Model Identification
### Mathematical Model Structure  
The aircraft's mathematical model can be expressed in a form of the linear state-space equations, separately for longitudinal and lateral motion. A general form of such a model is presented in Equations (1) and (2). \[\dot{\mathrm{X}}=\mathrm{A}\cdot\mathrm{X}+\mathrm{B}\cdot\mathrm{U}+\mathrm{E} \cdot\mathrm{Z} \tag{1}\]  
\[\mathrm{Y}=\mathrm{C}\cdot\mathrm{X} \tag{2}\]  
A full state linear representation of the longitudinal channel according to [15] is given in (3). \[ \begin{bmatrix}\dot{\mathrm{u}}\\ \dot{\mathrm{w}}\\ \dot{\mathrm{q}}\\ \dot{\mathrm{o}}\end{bmatrix} = \begin{bmatrix}\mathrm{X}_{\mathrm{u}}&\mathrm{ X}_{\mathrm{w}}&0&-\mathrm{gcos}\,\Theta_{0}\\ \mathrm{Z}_{\mathrm{u}}&\mathrm{Z}_{\mathrm{w}}&\mathrm{U}_{\mathrm{0}}&- \mathrm{gsin}\,\Theta_{0}\\ \mathrm{M}_{\mathrm{u}}&\mathrm{M}_{\mathrm{w}}&\mathrm{M}_{\mathrm{q}}&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}\mathrm{u}\\ \mathrm{w}\\ \mathrm{q}\\ \mathrm{q}\\ \mathrm{o}\end{bmatrix} + \begin{bmatrix}\mathrm{X}_{\mathrm{g}}&\mathrm{X}_{ \mathrm{g}}\\ \mathrm{Z}_{\mathrm{g}}&\mathrm{Z}_{\mathrm{g}}\\ \mathrm{M}_{\mathrm{g}}&\mathrm{M}_{\mathrm{g}}\\ 0&0\end{bmatrix}\begin{bmatrix}\delta\mathrm{E}\\ \mathrm{M}_{\mathrm{g}}\\ \mathrm{M}_{\mathrm{g}}\\ 0\end{bmatrix} + \begin{bmatrix}\mathrm{X}_{\mathrm{g}}&\mathrm{X}_{\mathrm{g} }\\ \mathrm{L}_{\mathrm{g}}&\mathrm{L}_{\mathrm{g}}\\ \mathrm{N}_{\mathrm{g}}&\mathrm{N}_{\mathrm{g}}&\mathrm{N}_{\mathrm{g}}\\ 0&0\end{bmatrix}\begin{bmatrix}\delta\mathrm{A}\\ \delta\mathrm{R}\\ \mathrm{N}_{\mathrm{g}}\\ 0\end{bmatrix}  \tag{3}\]  
A steady state is considered to be the aerodynamic equilibrium state of the aircraft in straight-line flight. Since the variable \(\mathrm{w}\) is not actually represented by any on-board measurement, through the use of relation (4), it is possible to transform Equation (3) into a form that is useful from a control system design perspective. \[\alpha(\mathrm{t})=\frac{\mathrm{w}(\mathrm{t})}{\mathrm{U}_{0}} \tag{4}\]  
By omitting the disturbance part, which is not considered in this paper, the final form of the longitudinal state equation takes the form (5) [15]. \[ \begin{bmatrix}\dot{\mathrm{u}}\\ \dot{\alpha}\\ \dot{\mathrm{q}}\\ \dot{\mathrm{o}}\end{bmatrix} = \begin{bmatrix}\mathrm{X}_{\mathrm{u}}&\mathrm{ X}_{\mathrm{\alpha}}&0&-\mathrm{gcos}\,\Theta_{0}\\ \mathrm{Z}_{\mathrm{u}}&\mathrm{Z}_{\mathrm{\alpha}}&1&\frac{-\mathrm{gsin} \,\Theta_{0}}{\mathrm{U}_{0}}\\ \mathrm{M}_{\mathrm{u}}&\mathrm{M}_{\mathrm{\alpha}}&\mathrm{M}_{\mathrm{q}}&0 \\ 0&0&1&0\end{bmatrix}\begin{bmatrix}\mathrm{u}\\ \alpha\\ \mathrm{q}\\ \mathrm{o}\end{bmatrix} + \begin{bmatrix}\mathrm{X}_{\mathrm{g}}&\mathrm{X}_{ \mathrm{g}}\\ \mathrm{L}_{\mathrm{g}}&\mathrm{L}_{\mathrm{g}}\\ \mathrm{M}_{\mathrm{g}}&\mathrm{N}_{\mathrm{g}}\\ 0&0\end{bmatrix}\begin{bmatrix}\delta\mathrm{E}\\ \delta\mathrm{T}\end{bmatrix}  \tag{5}\]  
To represent the aircraft's lateral motion, the full linearised state equation is given in (6) [15]. \[ \begin{bmatrix}\dot{\mathrm{v}}\\ \dot{\mathrm{p}}\\ \dot{\mathrm{r}}\\ \dot{\mathrm{o}}\end{bmatrix} = \begin{bmatrix}\mathrm{Y}_{\mathrm{v}}&\mathrm{ Y}_{\mathrm{p}}&\mathrm{Y}_{\mathrm{r}}-\mathrm{U}_{0}&\mathrm{gcos}\,\Theta_{0}\\ \mathrm{L}_{\mathrm{v}}&\mathrm{L}_{\mathrm{p}}&\mathrm{L}_{\mathrm{r}}&0\\ \mathrm{N}_{\mathrm{v}}&\mathrm{N}_{\mathrm{p}}&\mathrm{N}_{\mathrm{r}}&0\\ 0&1&\mathrm{tg}\Theta_{0}&0\end{bmatrix}\begin{bmatrix}\mathrm{v}\\ \mathrm{P}\\ \mathrm{r}\\ \dot{\mathrm{o}}\end{bmatrix} + \begin{bmatrix}\mathrm{Y}_{\mathrm{g}}&\mathrm{ X}_{\mathrm{g}}\\ \mathrm{L}_{\mathrm{g}}&\mathrm{L}_{\mathrm{g}}\\ \mathrm{N}_{\mathrm{g}}&\mathrm{N}_{\mathrm{g}}\\ 0&0\end{bmatrix}\begin{bmatrix}\delta\mathrm{A}\\ \delta\mathrm{R}\\ \mathrm{0}\end{bmatrix} + \begin{bmatrix}\mathrm{Y}_{\mathrm{v}}_{\mathrm{g}}\\ \mathrm{L}_{\mathrm{v}}\\ \mathrm{N}_{\mathrm{v}}_{\mathrm{g}}\\ 0\end{bmatrix}