$($a$)$ Hand Derivation of Kalman filter localization

Figure $1$ shows the time update and measurement update process of a Kalman filter localization

algorithm. Based on the provided equations, calculate the following signal model in two steps,

i$.$e$.k=1$ and $k=2.$

$x_k=A{x}_{k-1}+B{u}_{k-1}+w_{k-1},z_k=H{x}_k+v_k,k0$

where the state matrix $A$ is $1,$ the measurement matrix $H$ is $1,$ the process error covariance $Q$

$=0\mathrm{.}00001$ and the measurement noise covariance $R=0\mathrm{.}01.$ You need to calculate the Kalman

filter time update states, hat$(x{)}_k^{-}$and $P_k^{-}$as well as the measurement states $K_k,$hat$(x{)}_k,P_k$ for $k=0$ and

$1,$ respectively. Assume that $B=0,$ and widehat$(x{)}_0=0,P_0=1.$ The measurements are $z_1$ and $z_2,$

respectively.