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zEe6ft0J$~T63_w$$aQ~VOF=6*NRET&mVxD9kQ@c(R)CeD4YXTmt5tIjS;!B_I#{fA zofzw)KF>z69dPi-Zjm~o&Znbs=12)q`>HI}t47eFNFySrk7|=TNsA-pNA;=~Yy;cD z4zLsSfnA^<>;`*yjZ&lfvLB;skt(B#(Tt8^n&1$o;T;Djz)62F3{HX5;0zc6XTdpM z(|<4u&Vvi!A`dk9OW-oNV&QkI+IiKixhNYm>m8F_GrQF^oFd|GI#IG@|Kydla*OI_+3_R z>66>C3QH&EWR+!>PX6Dw%F@@5gQagiIHn4Y_l=H^4UXwa$ETZ)&sC0@XO3@v?bF$Z zj-RuRxnGX?ddHt0M{&dp%-nO7j62F-Xb=5ix;?rsx*tqGGX2Q(Bh!yeKlWL+@YUaK ceii-!<_wD=m#QlT9t$-vH8Kh%B}Gq03X3Q9&j0`b diff --git a/electives/others/amr/parts/01_kinematics/03_temporal-models.tex b/electives/others/amr/parts/01_kinematics/03_temporal-models.tex index 754bbb3..40ad617 100644 --- a/electives/others/amr/parts/01_kinematics/03_temporal-models.tex +++ b/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:\\ +$\delta \vec{x}_k = \vec{f}(\vec{\overline{x}}, \vec{\overline{u}}) + \mat{F} \delta \vec{x}_{k - 1} + \mat{G}_k \delta \vec{u}_k + \mat{L}_k \vec{w}_k$; +$\delta \vec{z}_k = \mat{H}_k \delta \vec{x}_k$