w(t) \longrightarrow \bigg[\frac{\sqrt{2\sigma ^2\beta}}{s+\beta}\bigg]  \longrightarrow \bigg[\frac{1}{s}\bigg] \longrightarrow y

$w(t) \longrightarrow \bigg[\frac{\sqrt{2\sigma ^2\beta}}{s+\beta}\bigg]  \longrightarrow \bigg[\frac{1}{s}\bigg] \longrightarrow y$

 

\usepackage{amsmath}  %可以使用\boldsymbol加粗罗马字符；\mathbf对罗马字符不起作用。

\mathbf{x}_{k+1} = \boldsymbol{\phi}_k \mathbf{x}_k + \mathbf{w}_k

$\mathbf{x}_{k+1} = \boldsymbol{\phi}_k \mathbf{x}_k + \mathbf{w}_k$

 

%注意{和}是特殊字符，使用\{和\}

\mathbf{Q}_k=E[\mathbf{w}_k\mathbf{w}_k^T]

=E\big\{   \big[ \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u) \mathbf{G}(u) \mathbf{w}(u)du \big]  \big[ \int_{t_k}^{t_{k+1}}\boldsymbol{\phi}(t_{k+1},v) \mathbf{G}(v) \mathbf{w}(v)dv \big]^T   \big\}

=\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u)\mathbf{G}(u)E[\mathbf{w}(u)\mathbf{w}^T(v)]\mathbf{G}^T(v)\boldsymbol{\phi}^T(t_{k+1},v)dudv

$\mathbf{Q}_k=E[\mathbf{w}_k\mathbf{w}_k^T]$

$=E\big\{   \big[ \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u) \mathbf{G}(u) \mathbf{w}(u)du \big]  \big[ \int_{t_k}^{t_{k+1}}\boldsymbol{\phi}(t_{k+1},v) \mathbf{G}(v) \mathbf{w}(v)dv \big]^T   \big\}$

$=\int_{t_k}^{t_{k+1}} \int_{t_k}^{t_{k+1}} \boldsymbol{\phi}(t_{k+1}, u)\mathbf{G}(u)E[\mathbf{w}(u)\mathbf{w}^T(v)]\mathbf{G}^T(v)\boldsymbol{\phi}^T(t_{k+1},v)dudv$

 

\left[\begin{matrix}

\dot{x_1}\\\dot{x_2}

\end{matrix}\right]

= \left[

\begin{matrix}

0&1\\0&-\beta

\end{matrix}

\right]

\left[\begin{matrix}

x_1\\x_2

\end{matrix}\right] +

\left[\begin{matrix}

0\\\sqrt{2\sigma^2\beta}

\end{matrix}\right]w(t)

$\left[\begin{matrix}\dot{x_1}\\\dot{x_2}\end{matrix}\right] = \left[\begin{matrix}0&1\\0&-\beta\end{matrix}\right] \left[\begin{matrix}x_1\\x_2\end{matrix}\right] + \left[\begin{matrix}0\\\sqrt{2\sigma^2\beta}\end{matrix}\right]w(t)$

y=\left[\begin{matrix}

1&0\

end{matrix}\right]

\left[\begin{matrix}

x_1\\x_2

\end{matrix}\right]

$y=\left[\begin{matrix}1&0\end{matrix}\right]\left[\begin{matrix}x_1\\x_2\end{matrix}\right]$

 

三角形帽子表示估计

\mathbf{\hat{x}}_k^-=\boldsymbol{\Phi}_k\mathbf{\hat{x}}_{k-1}+\mathbf{G}_k\mathbf{u}_k

$\mathbf{\hat{x}}_k^-=\boldsymbol{\Phi}_k\mathbf{\hat{x}}_{k-1}+\mathbf{G}_k\mathbf{u}_k$