The question provided is a mathematical optimization problem related to fusing observations and previous states, involving weights and variances. The goal is to find the optimal weight $w_1$ by minimizing the expression for $p_{n,n}$, which is the fused variance.

Problem Breakdown:

• The fused state $x_{n,n}$ is given by a linear combination of the observation $z_n$ and the previous state $x_{n,n-1}$: $x_{n,n} = w_1 z_n + (1 - w_1) x_{n,n-1}$

• The variances of $z_n$ and $x_{n,n-1}$ are independent and denoted as $r_n$ and $p_{n,n-1}$, respectively.

• The fused variance $p_{n,n}$ is given by: $p_{n,n} = w_1^2 r_n + (1 - w_1)^2 p_{n,n-1}$

• The task is to find the optimal $w_1$ by minimizing $p_{n,n}$, with the hint being that the derivative of $p_{n,n}$ with respect to $w_1$ is set to zero: $\frac{dp_{n,n}}{dw_1} = 0$

Step-by-Step Solution:

1. Differentiate $p_{n,n}$ with respect to $w_1$: $\frac{dp_{n,n}}{dw_1} = 2 w_1 r_n - 2 (1 - w_1) p_{n,n-1}$

2. Set the derivative equal to zero to find the optimal $w_1$: $2 w_1 r_n - 2 (1 - w_1) p_{n,n-1} = 0$

3. Simplify the equation: $w_1 r_n = (1 - w_1) p_{n,n-1}$

4. Solve for $w_1$: $w_1 r_n = p_{n,n-1} - w_1 p_{n,n-1}$ $w_1 (r_n + p_{n,n-1}) = p_{n,n-1}$ $w_1 = \frac{p_{n,n-1}}{r_n + p_{n,n-1}}$

Thus, the correct answer is C: $w_1 = \frac{p_{n,n-1}}{p_{n,n-1} + r_n}$

Let me know if you would like further details or have any questions!

Additional Questions:

1. How does this formulation relate to Kalman filtering?
2. What are the roles of the variances $r_n$ and $p_{n,n-1}$ in the optimal weight calculation?
3. How would the result change if the variances were correlated?
4. What is the physical meaning of the fused state $x_{n,n}$?
5. Can this approach be applied to multi-dimensional states?

Tip:

When dealing with weighted combinations like this, the weights are often derived based on how reliable each source is, with the inverse variance playing a key role.