GATE 2020 EC Q.23: solution and answer

GATE 2020 · ECQ.23 · 1 mark · NAT

The loop transfer function of a negative feedback system is

G (s) H (s) = \frac{K ( s + 11 )}{s ( s + 2 ) ( s + 8 )} .

The value of $K$ , for which the system is marginally stable, is ________.

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Answer and solution

ANSWER

The system is marginally stable when $K = 160$ . It then oscillates at $ω = 176 \approx 13.27 rad/s$ .

Official answer: 160Source: Official answer key, GATE 2020 EC (IIT Delhi)

Characteristic equation. For negative feedback, $1 + G (s) H (s) = 0$ : $s (s + 2) (s + 8) + K (s + 11) = 0.$ Since $s (s + 2) (s + 8) = s^{3} + 10 s^{2} + 16 s$ , this becomes $s^{3} + 10 s^{2} + (16 + K) s + 11 K = 0.$
Routh array: $s^{3} s^{2} s^{1} s^{0} : 116 + K : 1011 K : \frac{10 ( 16 + K ) - 11 K}{10} = \frac{160 - K}{10} : 11 K$
Stability conditions (no sign change in the first column): the $s^{0}$ row needs $K > 0$ , and the $s^{1}$ row needs $160 - K > 0$ . So the system is stable for $0 < K < 160$ .
Marginal stability: at $K = 160$ the $s^{1}$ row becomes entirely zero, which signals a pair of roots on the $j ω$ -axis.
Auxiliary equation from the $s^{2}$ row: $10 s^{2} + 11 (160) = 0 \Rightarrow s^{2} = - 176 \Rightarrow s = \pm j 176 \approx \pm j 13.27 rad/s$ .
Check: at $K = 160$ the characteristic polynomial is $s^{3} + 10 s^{2} + 176 s + 1760 = (s^{2} + 176) (s + 10)$ . The roots are $s = - 10$ and $s = \pm j 13.27$ . The imaginary pair is simple (not repeated) and the third root is in the left half-plane, so the system oscillates steadily: it is marginally stable.
Therefore $K = 160$ .

KEY IDEA

Build the Routh array of $s^{3} + 10 s^{2} + (16 + K) s + 11 K$ . Marginal stability happens at the value of $K$ that turns the $s^{1}$ row entirely to zero, provided the other first-column entries stay positive. The auxiliary equation from the $s^{2}$ row then gives the $j ω$ -axis roots and the oscillation frequency.

WATCH OUT FOR

Forgetting the zero term $K (s + 11)$ when forming the characteristic equation. This drops the $K s$ term or the $11 K$ constant and gives a wrong $s^{1}$ entry. Another mistake is using the third-order shortcut $a_{2} a_{1} = a_{3} a_{0}$ with the coefficients in the wrong order: the correct form is $10 (16 + K) = 1 \cdot 11 K$ , which gives $K = 160$ . Some students also answer $K = 0$ . That makes the $s^{0}$ row zero, which only puts a root at the origin (the open-loop pole at $s = 0$ ), so it does not give the sustained oscillation the question asks about.

From the GATE 2020 EC paper. Every question of this paper · Download the full paper · All 15 Routh–Hurwitz questions, with notes

GATE 2020 EC Q.23: solution and answer

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