I'm trying to understand concepts on Kalman Filters. Consider the overdetermined system $Ax=y$; $$\begin{bmatrix}1 \\ 1 \\ 1 \\1 \end{bmatrix} x = \begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$$

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 N\left(0, Q_{k}\right) \\ r_{k} & \sim N\left(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 the parameters of Kalman filter equations through this system. Since the model is static with trivial dynamics, I guess $q_{k-1} = 0$. I'm trying to identify $G_{k-1}$ and $u_{k-1}$ for this system. Can you please help me to identify other parameters?

How can we use the matrix $\begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$ to calculate $x_2^{+}, x_{2}^{-1}$ and other parameters?

There are similar questions in Chapter 2 of "Neural Control Engineering: The Emerging Intersection between Control Theory and Neuroscience". Thank you very much!

My working so far: Correct me if I'm wrong! Given that the model dynamics are trivial, we can assume that $$q_{k-1} = G_{k-1} =u_{k-1} = 0$$. Also, given that $F=A=I$. Then, Kalman filter equations become, $$x_k = x_{k-1}$$ $$y_k = x_k+r_k$$

Given the initial conditions are $y_1 = 3$ and $x_{1}^+ = y_1$. How can I find other parameters?

Since the initial condition $x_1^{+} = 3$, I can compute $$x_2^{-} = F_1x_1^{+}+G_1U_1 = I[3]+0 $$. I'm having a trouble finding $P_0^{+}$.

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    $\begingroup$ Numerous statements about the Kalman filter in this question don't make sense. $\begin{bmatrix}1 & 1 & 1 & 1\end{bmatrix}^T x$, for instance, is an invalid multiplication because $x$ is a 4-element column matrix. Also, the statement "Also, assume the uncertainty in both model, $F$, and observation, $A$, are both equal to $\sigma^2$" makes no sense -- the state transition matrix ($F$) and output matrix ($A$) in your nomenclature are assumed to be known, not uncertain. I could go on, but until the question makes sense, no answer can be formulated. $\endgroup$ – TimWescott Nov 22 '20 at 4:19
  • $\begingroup$ I suggest you hit the books again, or change the title of this question to "Why is this problem formulation not a valid Kalman filter formulation?" $\endgroup$ – TimWescott Nov 22 '20 at 4:19
  • $\begingroup$ @TimWescott Why do you think $x$ is a column vector? The problem in this way can be used to find $\hat{x}$ using the recursive Least square method. $\endgroup$ – ccc Nov 22 '20 at 4:30
  • $\begingroup$ Because in every Kalman filter formulation I've ever seen, $x$ is a column vector. Since you got other things considerably wrong, I have no reason to believe that you got that right. However, if you're working from a text where it isn't a column vector, I suggest you state what it is. And, of course, correct the other problems with your question. $\endgroup$ – TimWescott Nov 22 '20 at 4:38

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