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GATE 2017 EC – Question 57

Control Systems · Bode and Root-Locus Plots · 2 marks · Multiple choice

A linear time invariant (LTI) system with the transfer function

$$G(s) = \frac{K(s^2 + 2s + 2)}{(s^2 - 3s + 2)}$$

is connected in unity feedback configuration as shown in the figure.

For the closed loop system shown, the root locus for $0 < K < \infty$ intersects the imaginary axis for $K = 1.5$. The closed loop system is stable for

  1. $K > 1.5$
  2. $1 < K < 1.5$
  3. $0 < K < 1$
  4. no positive value of $K$

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Show answer and explanation

Correct answer: (A) $K > 1.5$

Explanation

The closed-loop characteristic equation is $(s^2 - 3s + 2) + K(s^2 + 2s + 2) = (1 + K)s^2 + (2K - 3)s + (2 + 2K) = 0$. For a second-order polynomial, all roots are in the left half plane only if all the coefficients have the same sign. That needs $2K - 3 > 0$, which is $K > 1.5$.