Question

The closed loop poles of a stable second order system could be

• A. Real and positive
• B. Complex conjugate with positive real part
• C. Real and negative
• D. One real positive and the other real negative

Hint:

Stability Mapping Rule: - Real part is Positive \(\rightarrow\) Unstable system (Right Half Plane) - Real part is Negative \(\rightarrow\) Stable system (Left Half Plane)

Answer 1

The Correct Option is C

Concept: The stability of any linear control system is determined by the locations of its closed-loop transfer function poles (the roots of the characteristic equation) on the complex \(s\)-plane. - If any pole lies in the Right-Half Plane (RHP) where the real part is positive, the system response contains a term \(e^{+at}\), causing it to grow unboundedly and become unstable. - For a system to be stable, all its poles must strictly reside in the Left-Half Plane (LHP), meaning their real parts must be negative.

Step 1: Checking the stability condition.
We are given that the system is stable. This automatically eliminates any options containing positive real components, since positive real components yield unbounded, unstable outputs.

• Option (A) is unstable (positive real parts).

• Option (B) is unstable (positive real parts).

• Option (D) is unstable (one pole is positive, causing divergence).

Step 2: Assessing a stable second-order system.
A stable second-order system can possess either:

• Complex conjugate poles with a negative real part (if under-damped).

• Real and negative distinct or repeated poles (if overdamped or critically damped).
Among the given choices, option (C) correctly matches the necessary conditions for a stable system.