Closed-loop stability condition: for a feedback control system to be stable, which statement is correct?

Introduction / Context:
In linear time-invariant control, stability is fundamentally tied to the locations of the closed-loop poles. Whether you analyse from time domain (characteristic equation) or frequency domain (Nyquist, Bode), the conclusion is the same: stable systems have decaying natural responses, which correspond to poles with negative real parts for continuous-time systems.

Given Data / Assumptions:

• Continuous-time LTI system.
• Standard feedback configuration.
• No time discretisation effects.

Concept / Approach:
The closed-loop transfer function’s denominator roots (closed-loop poles) govern stability. Poles in the left half-plane (Re{s} < 0) yield exponentially decaying modes. Complex conjugate poles are permitted; they need not be real. Frequency-domain criteria (Nyquist, Bode gain/phase margins) are equivalent ways to ensure that all closed-loop poles reside in the left half-plane.

Step-by-Step Solution:

Verification / Alternative check:
Routh–Hurwitz gives algebraic tests ensuring left-half-plane poles; Nyquist ensures encirclement conditions around −1 that imply the same.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing necessary with sufficient conditions and misinterpreting frequency-response shapes as guarantees of stability.

Final Answer:
All poles of the closed-loop transfer function must lie in the left half of the complex plane.