Solution:
We are given:
\begin{eqnarray*}
A &=& 1 \\
B &=& 2 \\
G &=& 3 \\
C &=& 1 \\
H &=& 4 \\
\mu_k &=& 5 \\
\Sigma_k &=& 0.3 \\
\Sigma_\theta &=& 0.5 \\
\Sigma_\psi &=& 1.5 \\
u_k &=& 0.5 \\
y_k &=& 7
\end{eqnarray*}
From there, apply the steps of the filter:
- Compute $\Sigma^\prime_{k+1}$: \begin{eqnarray*} \Sigma^\prime_{k+1} &=& A \Sigma_k A^\top + G \Sigma_\theta G^\top \\ &=& 1 \cdot 0.3 \cdot 1 + 3 \cdot 0.5 \cdot 3 \\ &=& 0.3 + 4.5 = 4.8 \end{eqnarray*}
- Compute $L_{k+1}$: \begin{eqnarray*} L_{k+1} &=& \Sigma^\prime_{k+1}C^\top ( C \Sigma^\prime_{k+1} C^\top + H \Sigma_\psi H )^{-1} \\ &=& 4.8 \cdot 1 \cdot ( 1 \cdot 4.8 \cdot 1 + 4 \cdot 1.5 \cdot 4 )^{-1} \\ &=& 4.8 \cdot (28.8)^{-1} = 1/6 \end{eqnarray*}
- Compute $\mu_{k+1}$: \begin{eqnarray*} \mu_{k+1} &=& A \mu_k + B u_k + L_{k+1}( y_k - C(A \mu_k + B u_k) ) \\ &=& 1 \cdot 5 + 2 \cdot 0.5 + (1/6)( 7 - 1 \cdot (1 \cdot 5 + 2 \cdot 0.5)) \\ &=& 6 + (1/6) \\ &=& 37/6 \end{eqnarray*}
- Compute $\Sigma_{k+1}$: \begin{eqnarray*} \Sigma_{k+1} &=& (I - L_{k+1}C)\Sigma^\prime_{k+1} \\ &=& (1 - (1/6)\cdot 1)\cdot 4.8 \\ &=& 4 \end{eqnarray*}