Question

If a system transfer function has some poles lying on the imaginary axis, it is _______.

• A. Unconditionally stable
• B. Conditionally stable
• C. Unstable
• D. Marginally Stable

Hint:

Poles on the imaginary axis (non-repeated) $\Rightarrow$ marginal stability due to sustained oscillations.

Answer 1

The Correct Option is D

Step 1: Define poles of transfer function
Poles of the transfer function are roots of the denominator of the Laplace-transformed system. They determine the time-domain behavior.
Step 2: Stability criteria
- Poles in left-half plane (LHP) $\Rightarrow$ system is stable.
- Poles in right-half plane (RHP) $\Rightarrow$ system is unstable.
- Poles on imaginary axis (excluding origin) $\Rightarrow$ marginally stable (oscillations persist, but do not grow or decay).
Step 3: Implication of imaginary axis poles
If the system has simple (non-repeated) poles at $s = \pm j\omega$, it results in sinusoidal oscillations that do not die out.
Step 4: Special case
If a pole is repeated on the imaginary axis (like a double pole at $j\omega$), the system becomes unstable. But for distinct simple poles on $j\omega$ axis, it is called marginally stable.