[AMR] Almost finish temporal models

electives/others/amr/parts/01_kinematics/03_temporal-models.tex

@@ -1,6 +1,18 @@
 \subsection{Temporal Models}
-Often use cont. time n.-lin. system of ODE $\dot{\vec{x}} = \vec{f}_C(\vec{x}(t), \vec{u}(t))$, with measurements $\vec{z}(t) = \vec{h}(\vec{x}(t)) + \vec{v}(t)$.
+Often use 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
+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 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 discretize, integrate from $t_{k - 1}$ to $t_k$:
+$\vec{x}_k = \vec{f}(\vec{x}_{k - 1}, \vec{u}_k, \vec{w}_k)$
+$\vec{z}_k = \vec{h}(\vec{x}_k) + \vec{v}_k$,
+linearised:\\
+$\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$;
+$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$