\subsection{Temporal Models}
For \bi{Cont-time n.-lin. system of ODE} $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.\\
Need linearised (around $\vec{f}_C(\vec{\overline{x}}, \vec{\overline{y}}) = 0$, at \bi{equilibrium}):\\
$\delta \vec{\dot{x}}(t) = \vec{f}_C(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F}_C \delta \vec{x}(t) + \mat{G}_C \delta \vec{u}(t) + \mat{L}_C \vec{w}(t)$\\
$\delta \vec{z}(t) = \mat{H} \delta \vec{x}(t) + \vec{v}(t)$.
Herein, $\mat{H}$ is measurements, $\mat{F}_C$ system, $\mat{G}$ input gain, $\vec{w}$ process noise, $\vec{v}$ measurement noise, both zero-mean \bi{Gaussian White Noise Process}.

For \bi{n-lin. cont-time system}:
$\vec{\dot{x}}(t) = \vec{f}_C(\vec{x}(t), \vec{u}(t), \vec{w}(t))$\\
$\vec{z}(t) = \vec{h}(\vec{x}(t)) = \vec{v})(t)$,
linearization is the same

To \bi{discretize}, integrate from $t_{k - 1}$ to $t_k$:\\
$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$
$\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
\bi{linearised}:\\
$\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$

\bi{Trapezoidal num. int} 
$\Delta \vec{x}_1 = \Delta t \vec{f}_C (\vec{x}_{k - 1}, \vec{u}_{k - 1}, t_{k - 1})$\\
$\Delta \vec{x}_2 = \Delta t \vec{f}_C (\vec{x}_{k - 1} + \Delta \vec{x}_1, \vec{u}_{k}, t_{k})$, then:\\
$\vec{x}_k = \vec{x}_{k - 1} + 0.5 \cdot (\Delta \vec{x}_1 + \Delta \vec{x}_2)$