GATE EC 2024 Set 1 — Question 13
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Control Systems → Stability Analysis → Routh-Hurwitz Criterion
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Question
In the feedback control system shown in the figure below .
, and are the Laplace transforms of , and , respectively. If the input is a unit step function, then __________.

Correct answer
(D) _(t → ∞) e(t) does not exist, e(t) is oscillatory
Solution
The open-loop transfer function is given as . For a unity feedback system, the closed-loop characteristic equation is .To check the stability, we apply the Routh-Hurwitz criterion:
The presence of a row of zeros in the row indicates that the system is marginally stable. The auxiliary equation from the row is:The closed-loop poles are at and .The error signal for a unit step input is:The Final Value Theorem (FVT) states that , provided all poles of lie in the left half of the -plane. Here, has poles at and . Since there are poles on the imaginary axis, the steady-state error will be oscillatory and the limit does not exist. Thus, is oscillatory.
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