w(t) longrightarrow igg[frac{sqrt{2sigma ^2eta}}{s+eta}igg] longrightarrow igg[frac{1}{s}igg] longrightarrow y
$w(t) longrightarrow igg[frac{sqrt{2sigma ^2eta}}{s+eta}igg] longrightarrow igg[frac{1}{s}igg] longrightarrow y$
usepackage{amsmath} %可以使用oldsymbol加粗罗马字符；mathbf对罗马字符不起作用。
mathbf{x}_{k+1} = oldsymbol{phi}_k mathbf{x}_k + mathbf{w}_k
$mathbf{x}_{k+1} = oldsymbol{phi}_k mathbf{x}_k + mathbf{w}_k$
%注意{和}是特殊字符，使用{和}
mathbf{Q}_k=E[mathbf{w}_kmathbf{w}_k^T]
=Eig{ ig[ int_{t_k}^{t_{k+1}} oldsymbol{phi}(t_{k+1}, u) mathbf{G}(u) mathbf{w}(u)du ig] ig[ int_{t_k}^{t_{k+1}}oldsymbol{phi}(t_{k+1},v) mathbf{G}(v) mathbf{w}(v)dv ig]^T ig}
=int_{t_k}^{t_{k+1}} int_{t_k}^{t_{k+1}} oldsymbol{phi}(t_{k+1}, u)mathbf{G}(u)E[mathbf{w}(u)mathbf{w}^T(v)]mathbf{G}^T(v)oldsymbol{phi}^T(t_{k+1},v)dudv
$mathbf{Q}_k=E[mathbf{w}_kmathbf{w}_k^T]$
$=Eig{ ig[ int_{t_k}^{t_{k+1}} oldsymbol{phi}(t_{k+1}, u) mathbf{G}(u) mathbf{w}(u)du ig] ig[ int_{t_k}^{t_{k+1}}oldsymbol{phi}(t_{k+1},v) mathbf{G}(v) mathbf{w}(v)dv ig]^T ig}$
$=int_{t_k}^{t_{k+1}} int_{t_k}^{t_{k+1}} oldsymbol{phi}(t_{k+1}, u)mathbf{G}(u)E[mathbf{w}(u)mathbf{w}^T(v)]mathbf{G}^T(v)oldsymbol{phi}^T(t_{k+1},v)dudv$
left[egin{matrix}
dot{x_1}\dot{x_2}
end{matrix} ight]
= left[
egin{matrix}
0&1\0&-eta
end{matrix}
ight]
left[egin{matrix}
x_1\x_2
end{matrix} ight] +
left[egin{matrix}
0\sqrt{2sigma^2eta}
end{matrix} ight]w(t)
$left[egin{matrix}dot{x_1}\dot{x_2}end{matrix} ight] = left[egin{matrix}0&1\0&-etaend{matrix} ight] left[egin{matrix}x_1\x_2end{matrix} ight] + left[egin{matrix}0\sqrt{2sigma^2eta}end{matrix} ight]w(t)$
y=left[egin{matrix}
1&0
end{matrix} ight]
left[egin{matrix}
x_1\x_2
end{matrix} ight]
$y=left[egin{matrix}1&0end{matrix} ight]left[egin{matrix}x_1\x_2end{matrix} ight]$
三角形帽子表示估计
mathbf{hat{x}}_k^-=oldsymbol{Phi}_kmathbf{hat{x}}_{k-1}+mathbf{G}_kmathbf{u}_k
$mathbf{hat{x}}_k^-=oldsymbol{Phi}_kmathbf{hat{x}}_{k-1}+mathbf{G}_kmathbf{u}_k$