ANS:B - poles of the closed loop transfer function should lie in the left half of the complex plane.

For a feedback control system to be stable, the correct statement is: Poles of the closed-loop transfer function should lie in the left half of the complex plane. Here’s an explanation of each option:

1. Roots of the characteristic equation should be real: This is related to the location of the poles of the closed-loop transfer function. Real roots (or poles) do not guarantee stability by themselves; stability depends on their location relative to the imaginary axis.
2. Poles of the closed-loop transfer function should lie in the left half of the complex plane: This is a fundamental stability criterion known as the Routh-Hurwitz criterion. Poles in the left half-plane (where the real part of the pole is negative) indicate stable behavior because they contribute to exponential decay of any perturbations.
3. Bode plots of the corresponding open-loop transfer function should monotonically decrease: Bode plots are used to analyze frequency response and gain/phase margins but do not directly determine stability. However, certain characteristics in Bode plots can provide insights into stability margins rather than absolute stability.
4. Poles of the closed-loop transfer function should lie in the right half of the complex plane: Poles in the right half-plane (where the real part of the pole is positive) typically indicate instability because they lead to unbounded or oscillatory responses.

Therefore, the key criterion for stability in a feedback control system is that the poles of the closed-loop transfer function must reside in the left half of the complex plane. This ensures that the system response decays over time rather than grows or oscillates uncontrollably.