GATE 2022 CH – Question 58

Instrumentation and Process Control · Closed-loop stability and frequency response · 2 marks · Numerical answer

In the block diagram shown in the figure, the transfer function $G = \frac{K}{(\tau s + 1)}$ with $K > 0$ and $\tau > 0$. The maximum value of $K$ below which the system remains stable is __________ (rounded off to two decimal places).

a block diagram of six blocks G. The input $u$ goes through a G to the output summer, which gives $y$. The forward path of the main loop is a summer, then G, then the output summer. An inner loop is formed by two blocks G around the first summer. A feedback of the output $y$ through a G goes to the first summer.

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Correct answer: 0.70 to 0.72

Explanation

Let $w$ be the signal at the output of the first summer. The two blocks G in the inner path return $G^2w$ to this summer, and the feedback block returns $Gy$. With $y = Gw + Gu$ the summer equation is $w = G^2w + G(Gw + Gu)$, which gives $w(1 - 2G^2) = G^2u$. The characteristic equation is $1 - 2G^2 = 0$, that is $(\tau s + 1)^2 = 2K^2$ or $\tau s + 1 = \pm\sqrt{2}K$. The roots are $s = \frac{-1 \pm \sqrt{2}K}{\tau}$, which are in the left half plane only when $\sqrt{2}K < 1$, so $K < \frac{1}{\sqrt{2}} = 0.71$.