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GATE 2025 EE – Question 41

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Multiple choice

Let $G(s)=\dfrac1{(s+1)(s+2)}$. Then the closed-loop system shown in the figure below, is

Diagram for GATE 2025 EE question 41
  1. stable for all $K>2$.
  2. unstable for all $K>2$.
  3. unstable for all $K>1$.
  4. stable for all $K>1$.

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Show answer and explanation

Correct answer: (B) unstable for all $K>2$.

Explanation

The open loop is $\dfrac{K(s-1)}{(s+1)(s+2)}$, so the characteristic equation is $(s+1)(s+2)+K(s-1)=s^2+(3+K)s+(2-K)=0$. For a second-order polynomial all coefficients must be positive: $3+K>0$ and $2-K>0$, i.e. $-3<K<2$. For $K>2$ the constant term is negative, so a root lies in the right half-plane and the system is unstable.