For a standard second order polynomial given by G(s) = a2s2 + a1s + a0, according to the Routh test, the stability criterion is given by

Explanation:

Routh Stability Criterion:

The Routh stability criterion is a mathematical technique used in control systems to determine the stability of a linear time-invariant system.

For the system to be stable, all the roots of the characteristic equation must lie in the left half of the s-plane (stable region).

The characteristic equation of a second-order polynomial can be expressed as:

G(s) = a2s2 + a1s + a0

Where:

• a2, a1, and a0 are the coefficients of the polynomial.
• s is the complex Laplace variable.

According to the Routh stability criterion, for the system to be stable, all the coefficients of the characteristic equation must be positive, and none of the coefficients should be zero. Additionally, there should be no sign changes in the first column of the Routh array.

For a second-order polynomial, the stability criterion simplifies to:

a₂ > 0, a₁ > 0, a₀ > 0

This ensures that all roots of the characteristic equation lie in the left-half of the s-plane, and the system is stable.