# engineering mathematics – Matrix dynamics to Kalman equations

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 let’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?

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