As per least square estimation, the estimated value of X (Xest) is represented by (Where Z is measur

Question

As per least square estimation, the estimated value of X (Xest) is represented by (Where Z is measurement vector and H is the linear relation between Z and X)

Examoogle · Accepted Answer

$X_{est} = (H^T H)^{-1} H^T Z$

Explanation: In least square estimation for a linear system defined by $Z = HX + \epsilon$, the goal is to minimize the sum of squared residuals $J = \epsilon^T \epsilon = (Z - HX)^T (Z - HX)$. Setting the derivative of $J$ with respect to $X$ to zero yields the normal equations, which result in the estimator $X_{est} = (H^T H)^{-1} H^T Z$.

Working Principle: The principle relies on projecting the measurement vector $Z$ onto the column space of the matrix $H$. By minimizing the Euclidean norm of the residual vector, we obtain the Moore-Penrose pseudoinverse $(H^T H)^{-1} H^T$. This method assumes that $H^T H$ is non-singular and invertible, which corresponds to the system being observable.

Key Formulas: $J = (Z - HX)^T (Z - HX)$ — Objective function to minimize; $X_{est} = (H^T H)^{-1} H^T Z$ — Least square estimator solution

Key Points: The estimator is based on the minimization of the sum of the squares of the errors (residuals).; It assumes the errors are independent and identically distributed with zero mean.; If the matrix $H$ is not of full column rank, $H^T H$ is singular and the unique inverse does not exist.; This method is widely used in power system state estimation for processing SCADA data.

Advantages: Provides a mathematically optimal solution when errors are Gaussian.; Computational efficiency for linear systems.

Disadvantages: Sensitive to outliers (bad data) in the measurement vector $Z$.; Requires high computational overhead for extremely large sparse matrices.

Applications: Power system state estimation (monitoring bus voltages and angles).; Signal processing and sensor fusion.

Additional Info: The term $(H^T H)^{-1} H^T$ is known as the left Moore-Penrose pseudoinverse of $H$.. Option C is dimensionally inconsistent as $(H^T H)^{-1}$ is $n 	imes n$ and $H$ is $m 	imes n$, making their product undefined unless $m=n$.

Answer: B is correct — The least square estimator for a linear model $Z=HX$ is derived from the normal equations, resulting in $X_{est} = (H^T H)^{-1} H^T Z$.

Exam Tip: Always ensure dimensions match in matrix algebra; in $(H^T H)^{-1} H^T Z$, if $H$ is $m 	imes n$, then $(H^T H)$ is $n 	imes n$, and the total expression simplifies to an $n 	imes 1$ vector.

Related Questions

Discussion (0)