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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.

融合观测值 zn 和状态 xn,n-1 得到融合后的状态 xn,n: x_{n,n} = w1 zn + (1 - w1) xn,n-1. zn 和 xn,n-1 相互独立, 方差分别为 rn 和 pn,n-1. 根据 pn,n = w1^2 rn + (1 - w1)^2 pn,n-1 计算最优的 w1 值

Learn how to compute the optimal weight w1 for fusing an observation and a previous state, minimizing the variance p_{n,n}. This problem involves independent variances r_n and p_{n,n-1} and uses mathematical optimization techniques.

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