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GATE 2016 EE (EE2) – Question 49

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

A unity-feedback system has $G(s)=K(s+1)/[s(1+Ts)(1+2s)]$, K>0,T>0. The closed-loop system is stable if

  1. $0<T<4(K+1)/(K-1)$
  2. $0<K<4(T+2)/(T-2)$
  3. $0<K<(T+2)/(T-2)$
  4. $0<T<8(K+1)/(K-1)$

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Show answer and explanation

Correct answer: (C) $0<K<(T+2)/(T-2)$

Explanation

The characteristic polynomial is $2Ts^3+(T+2)s^2+(K+1)s+K$. For positive T,K, the remaining cubic Routh condition is $(T+2)(K+1)>2TK$.

It reduces to T+2>K(T−2). For T>2 this gives **$0<K<(T+2)/(T-2)$, C**, the keyed expression.

For 0<T≤2 the inequality already holds for every K>0. The source choice omits that separate regime; the full stability condition is the two-regime statement above, not the displayed bound applied blindly when its denominator is nonpositive.

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