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GATE 2021 IN – Question 39

Control Systems · Stability and frequency response · 2 marks · Multiple choice

A unity negative-feedback system has plant $G(s)=1/[(s+1)(s+3)]$ and PID controller $C(s)=K(s+3-j)(s+3+j)/s$, with scalar $K$. The closed loop is ______.

Diagram for GATE 2021 IN question 39
  1. Stable only for $K>0$
  2. Stable only for $-1<K<1$
  3. Stable only for $K<0$
  4. Stable for all $K$

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Show answer and explanation

Correct answer: (A) Stable only for $K>0$

Explanation

The controller numerator is K[(s+3)²+1]=K(s²+6s+10). The characteristic polynomial is s(s+1)(s+3)+K(s²+6s+10)=s³+(4+K)s²+(3+6K)s+10K. For K>0 all coefficients are positive and (4+K)(3+6K)−10K=6K²+17K+12>0. Routh therefore gives stability. K<0 gives a negative constant term and K=0 a pole at zero.

Thus **only K>0, A**.