GATE 2025 IN – Question 50
The feedback-system plant is $P(s)=a/(s^2-b^2)$, with $a>0$ and $b>0$. Which controller types $C(s)$ cannot stabilize this plant?
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Correct answer: (A) proportional (P) controller; (B) integral (I) controller; (C) proportional-integral (PI) controller
Explanation
With P control the characteristic quadratic has no damping term.
I control gives s³−b²s+aKi and PI gives s³+(aKp−b²)s+aKi; both lack an s² term and cannot be Hurwitz.
PD adds a positive s coefficient and can make aKp−b² positive.
Thus the controllers that cannot stabilize are **A,B,C**.