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Stability classification: A system has two poles on the negative real axis and a conjugate pair of poles exactly on the jω axis. What is the stability status?

Difficulty: Easy

Correct Answer: limitedly stable

Explanation:


Introduction / Context:
System stability depends on the locations of closed-loop poles in the complex s-plane. Poles on the jω axis indicate sustained undamped oscillations, which are neither asymptotically stable nor divergent.


Given Data / Assumptions:

  • Two poles strictly in the left half plane (negative real axis).
  • One complex-conjugate pair exactly on the imaginary axis.
  • No right-half-plane poles.


Concept / Approach:
Definitions: Asymptotic stability requires all poles strictly in the LHP. Instability occurs if any pole is in the RHP or there are repeated poles on the jω axis. If there are simple (nonrepeated) poles on the jω axis and no RHP poles, the system is marginally (limitedly) stable with persistent oscillations.


Step-by-Step Solution:

Check pole locations: none in RHP → not unstable.Presence of jω poles → transients do not decay; sinusoidal component persists.Therefore classify as limitedly (marginally) stable.


Verification / Alternative check:

Impulse response contains undamped sinusoidal term e^{jωt}; magnitude neither grows nor decays.


Why Other Options Are Wrong:

Stable: incorrect—requires all poles strictly LHP.Unstable: would need RHP pole or repeated jω poles.“Either (a) or (c)” is ambiguous; the correct unique class is limitedly stable.


Common Pitfalls:

Confusing limited stability with asymptotic stability; overlooking the effect of imaginary-axis poles.


Final Answer:

limitedly stable
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