The following system \(\frac{4 s+1}{4 s^{2}+1}\) is

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This question was previously asked in

ISRO Scientist Electronics 2024 Official Paper

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Answer 1

Concept:

A linear time-invariant (LTI) system is said to be:

Stable if all poles of the transfer function lie in the left half of the s-plane (have negative real parts).
Unstable if any pole lies in the right half of the s-plane (has positive real part).
Marginally stable if poles lie on the imaginary axis (pure imaginary), and none are repeated.

Given:

\( \text{Transfer function:} \; \frac{4s + 1}{4s^2 + 1} \)

Calculation:

The denominator polynomial determines the poles of the system.

\( 4s^2 + 1 = 0 \Rightarrow s^2 = -\frac{1}{4} \Rightarrow s = \pm j\frac{1}{2} \)

The poles are purely imaginary and non-repeated.

Conclusion:

Since the poles lie on the imaginary axis and are simple (not repeated), the system is marginally stable.