Consider the system

$$begin{bmatrix}1 \ 1 \ 1 \1 end{bmatrix} begin{bmatrix}x_1 \ x_2 \ x_3 \ x_4 end{bmatrix} = begin{bmatrix} 3 \ 5 \ 4 \ 8 end{bmatrix}$$

Let $R$ be the errors in measurements $y$ and $P$

be the errors in the estimate of $hat{x}$. Assume that the uncertainty in measurements, $sigma^{2}$. Then, $R=begin{bmatrix}sigma^{2} & 0 & 0 & 0\ 0 & sigma^{2}& 0 & 0 \ 0 & 0 &sigma^{2}& 0 \ 0 & 0 & 0 & sigma^{2}end{bmatrix}$.

Now consider Linear Kalman filter equations

begin{aligned}

x_{k} &=F_{k-1} x_{k-1}+G_{k-1} u_{k-1}+q_{k-1} \

y_{k} &=A_{k} x_{k}+r_{k} \

q_{k} & sim Nleft(0, Q_{k}right) \

r_{k} & sim Nleft(0, R_{k}right) \

end{aligned}

Assume a static model, with **trivial dynamics** $F = I$, so that $F_{k+1} = F_k$

and letâ€™s assume again that the variable $x$ is directly observable, so that $A = I$, and $A_{k+1} = A_k$. Also, assume the uncertainty in both model, $F$, and observation, $A$, are both equal to $sigma^{2}$.

I’m trying to identify parameters of Kalman filter equations through this system. Since the model is static with trivial dynamics, I think $q_{k-1} = 0$. Can you please help me to identify other parameters?