Join 60,000+ competitive exam aspirants
or with email
Your Cart
Subtotal₹0
Total₹0
Examoogle • User • info@examoogle.com • EE-2024-8821
Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
Join 60,000+ competitive exam aspirants
The condition number for Gain Matrix ‘G’ is define as
Rank of Gain Matrix G
Ratio of largest to smallest eigenvalue
GG−1
Inverse of G
Ratio of largest to smallest eigenvalue
Quick Summary: The condition number of a matrix measures how sensitive a linear system is to small perturbations or numerical errors. For a gain matrix $G$, the condition number $\kappa(G)$ is defined as the ratio of the maximum singular value to the minimum singular value, which simplifies to the ratio of the largest eigenvalue to the smallest eigenvalue when the matrix is symmetric and positive definite.
The condition number of a matrix measures how sensitive a linear system is to small perturbations or numerical errors. For a gain matrix G, the condition number κ(G) is defined as the ratio of the maximum singular value to the minimum singular value, which simplifies to the ratio of the largest eigenvalue to the smallest eigenvalue when the matrix is symmetric and positive definite.
κ(G)=λminλmax — Definition of condition number for a symmetric positive definite matrix
G=HTR−1H — Construction of the Gain Matrix in state estimation
In power system state estimation, the gain matrix G=HTR−1H is used to solve the normal equations. If κ(G) is very large, the matrix is 'ill-conditioned,' meaning tiny errors in the input measurement vector result in large errors in the estimated state vector. A high condition number leads to numerical instability during matrix inversion.
The condition number is a quantitative measure of the stability of numerical matrix inversion.
A value of κ(G)=1 is ideal (perfectly conditioned).
Large condition numbers indicate ill-conditioning, often caused by high redundancy or poor measurement placement.
State estimation algorithms require robust matrix inversion techniques (like LU or QR decomposition) when G is ill-conditioned.
Predicts potential numerical instability before performing matrix inversion.
Helps in evaluating the observability of a power system network.
Computationally expensive for very large system matrices.
Requires full eigendecomposition or singular value decomposition.
Power System State Estimation
Numerical Analysis of Power Flow Jacobian
Optimal Sensor Placement
A matrix with a condition number approaching infinity is considered singular.
Option C (GG−1) is an identity matrix, not a condition number.
Option D is the matrix inverse itself, which is distinct from the scalar condition number.
B is correct — The condition number is the ratio of the largest to the smallest eigenvalue of the matrix.
In Power System State Estimation exams, remember that ill-conditioning is a primary cause of non-convergence in iterative solvers; monitoring the condition number of the gain matrix is standard practice.