Question

If system has multiple poles on \( j\omega \)-axis, the system is________.

• A. stable
• B. unstable
• C. conditionally stable
• D. marginally stable

Hint:

Always analyze the pole positions to determine stability. \( j\omega \)-axis poles = marginal stability, but repetition can lead to instability.

Answer 1

The Correct Option is D

In control systems, the location of poles in the \( s \)-domain (Laplace domain) determines the system stability. Here's a brief summary:
- Stable: All poles lie in the left half of the s-plane.
- Unstable: At least one pole lies in the right half of the s-plane.
- Marginally stable: Poles lie on the imaginary axis \( j\omega \) but none are repeated.
- Unstable (again): If multiple or repeated poles lie on the \( j\omega \)-axis, then the system becomes unstable due to unbounded response over time.
But the exception is simple non-repeated poles on the \( j\omega \)-axis: these systems are marginally stable.
These typically include systems like undamped oscillators or LC circuits where the output oscillates forever but doesn't diverge.
Hence, the system with multiple non-repeated poles on \( j\omega \)-axis is marginally stable.