Estimate the two random variables by least square estimation method for a given measurement.
Z=0.50.

Question

Estimate the two random variables by least square estimation method for a given measurement.
Z=0.50.450.51
H=110101

Examoogle · Accepted Answer

[\begin{matrix} 0.29 \\ 0.35 \end{matrix}]

Explanation: Given: Measurement vector Z = [0.5, 0.45, 0.51]^T and Jacobian matrix H = [[1, 1], [1, 0], [0, 1]]^T (transposed form implied) Formula Used: $$\hat{X} = (H^T H)^{-1} H^T Z$$ Steps: Transpose H: Given the measurement model $Z = HX$, we compute the transpose $H^T$ of the matrix $H = [[1, 1], [1, 0], [0, 1]]$.. Compute product H^T H: Calculate the matrix product of $H^T$ and $H$.. Compute inverse of H^T H: The inverse of a $2 imes 2$ matrix $A = [[a, b], [c, d]]$ is $\frac{1}{ad-bc} [[d, -b], [-c, a]]$. Here $ad-bc = 2(2) - 1(1) = 3$.. Compute product H^T Z: Calculate the product of $H^T$ and the measurement vector $Z = [0.5, 0.45, 0.51]^T$.. Final Estimation: Multiply the inverse matrix by the vector $H^T Z$: $\hat{X} = \frac{1}{3} [[2, -1], [-1, 2]] [[0.95], [1.01]]$. Answer: B is correct because the least squares solution calculated using the formula $\hat{X} = (H^T H)^{-1} H^T Z$ yields approximately $[0.29, 0.35]^T$. Exam Tip: This concept is foundational in Power System State Estimation (PSSE), where weighted least squares (WLS) is used to process redundant meter readings to find the best estimate of bus voltages.

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