Control systems: for a stable second-order closed loop, which location of closed-loop poles is possible?

Introduction / Context:
Closed-loop stability in linear systems is determined by pole locations in the complex s-plane. For continuous-time systems, stability requires all poles to lie strictly in the left half-plane (negative real parts).

Given Data / Assumptions:

• Second-order system, continuous time.
• Standard stability notion: bounded-input bounded-output (BIBO) stability.
• No time delays or non-minimum phase complications considered.

Concept / Approach:
Poles with negative real parts correspond to decaying exponentials or damped oscillations. Real negative poles yield overdamped responses. Complex conjugate poles with positive real parts cause exponentially growing oscillations, which are unstable. Any pole in the right half-plane (positive real) causes instability.

Step-by-Step Solution:
Assess each option's pole locations.Both real and negative → left half-plane → stable (overdamped).Complex conjugate with positive real parts → unstable growth.Both real and positive → unstable.Mixed sign (one positive, one negative) → unstable due to RHP pole.

Verification / Alternative check:
Time-domain response y(t) for real negative poles is sum of decaying exponentials; integral of absolute response converges, meeting BIBO criteria.

Why Other Options Are Wrong:
Any case involving a pole with positive real part violates stability; mixed-sign or positive-real-part conjugates are unstable by definition.

Common Pitfalls:
Confusing “imaginary axis” marginal cases with stability; true stability requires strictly negative real parts (no repeated poles at the origin or on the jω-axis for strict BIBO stability).

Final Answer:
Both real and negative