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GATE 2022 EC – Question 44

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

Consider an even polynomial $p(s)$ given by

$$p(s)=s^4+5s^2+4+K,$$

where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is ________.

  1. $-4\le K\le\frac94$
  2. $-3\le K\le\frac92$
  3. $-6\le K\le\frac54$
  4. $-5\le K\le0$

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Show answer and explanation

Correct answer: (A) $-4\le K\le\frac94$

Explanation

Let $u=s^2$, so $u^2+5u+(4+K)=0$. All $s$ are imaginary exactly when both roots $u$ are real and $\le0$. Real roots need $25-4(4+K)\ge0$, i.e. $K\le\frac94$. Both roots non-positive needs the product $4+K\ge0$, i.e. $K\ge-4$ (the sum $-5$ is already negative). So $-4\le K\le\frac94$.