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GATE 2019 EE – Question 25

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 1 mark · Multiple choice

The characteristic equation of a linear time-invariant (LTI) system is given by $\Delta(s)=s^4+3s^3+3s^2+s+k=0$.

The system is BIBO stable if

  1. $0<k<\frac{12}{9}$
  2. $k>3$
  3. $0<k<\frac89$
  4. $k>6$

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Correct answer: (C) $0<k<\frac89$

Explanation

The Routh array is: $s^4$: $1,\ 3,\ k$; $s^3$: $3,\ 1$; $s^2$: $\frac83,\ k$; $s^1$: $\frac{\frac83-3k}{\frac83}$; $s^0$: $k$. All first-column entries are positive when $\frac83-3k>0$, i.e. $k<\frac89$, and $k>0$. So $0<k<\frac89$.