The Routh-Hurwitz criterion is a fundamental tool in control systems engineering used to determine the stability of a linear time-invariant (LTI) system without actually solving the characteristic equation. It's particularly crucial for analyzing the stability of closed-loop systems in applications like power systems and machines, especially in the context of competitive exams like GATE.

The stability of an LTI system is directly related to the location of the roots of its characteristic equation, which is typically a polynomial in the complex variable 's'. For a system to be stable, all roots of the characteristic equation must lie in the left half of the s-plane (i.e., have negative real parts). The Routh-Hurwitz criterion provides a systematic way to check this condition.

The Routh array is constructed using the coefficients of the characteristic equation, $a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0 = 0$. The first two rows are populated directly from the coefficients, alternating between them.

Subsequent rows are calculated using a specific formula involving elements from the preceding two rows. The general formula for an element $b_k$ in the third row, based on elements $a_1, a_2, a_3$ and $b_1, b_2$ from the first two rows, is: $b_k = \frac{(a_2 b_1 - a_1 b_2)}{b_1}$.

The stability of the system is determined by the signs of the elements in the first column of the Routh array. The Routh-Hurwitz criterion states:

The number of sign changes in the first column of the Routh array is equal to the number of roots of the characteristic equation that lie in the right-half of the s-plane.

There are a few special cases that require careful handling during Routh array construction:

1. Zero in the First Column: If a zero appears in the first column, it can lead to division by zero. This is handled by replacing the zero with a small positive number $\epsilon$ and analyzing the limit as $\epsilon \to 0$. Alternatively, a common practice is to multiply the characteristic equation by $(s+a)$ where 'a' is a positive constant, effectively shifting the roots.

2. Entire Row of Zeros: If an entire row of the Routh array becomes zero, it indicates the presence of roots that are symmetrically located with respect to the origin. These roots can be on the imaginary axis (marginal stability) or in conjugate pairs in the right-half plane. The auxiliary polynomial, formed from the coefficients of the row just above the row of zeros, helps in finding these roots.

For GATE Electrical Engineering, understanding the Routh-Hurwitz criterion is essential for solving problems related to system stability, transient response, and controller design. Be prepared to construct the Routh array for given characteristic equations and interpret the results to determine stability, the number of unstable roots, and conditions for marginal stability (roots on the imaginary axis).