Consider a unity negative feedback control system with forward path gain G(s)=K(s+1)(s+2)(s+3) as shown. The impulse response of the closed-loop system decays faster than e-t if .

GATE 2024 EC – Question 37

Control Systems · Routh-Hurwitz and Nyquist Stability Criteria · 2 marks · Multiple choice

Consider a unity negative feedback control system with forward path gain $G(s)=\dfrac{K}{(s+1)(s+2)(s+3)}$ as shown.

The impulse response of the closed-loop system decays faster than $e^{-t}$ if ________.

$1\le K\le5$
$7\le K\le21$
$-4\le K\le-1$
$-24\le K\le-6$

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Correct answer: (A) $1\le K\le5$

Explanation

The response decays faster than $e^{-t}$ when all closed-loop poles have real part less than $-1$. Substitute $s=z-1$ in $(s+1)(s+2)(s+3)+K=0$ to get $z^3+3z^2+2z+K=0$, and require all its roots to be in the left half-plane. Routh's criterion gives $K>0$ and $3\times2>K$, i.e. $0<K<6$. Of the options, only $1\le K\le5$ lies inside this range. For negative $K$ the constant term is negative and there is a root in the right half of the $z$-plane.