GATE 2022 EC Q.21: solution and answer

GATE 2022 · ECQ.21 · 1 mark · MCQ

Consider a closed-loop control system with unity negative feedback and $K G (s)$ in the forward path, where the gain $K = 2$ . The complete Nyquist plot of the transfer function $G (s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G (s)$ has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is ________.

GATE 2022 EC question 21: Nyquist plot of G(s). — Figure from the GATE 2022 EC paper. It is part of the question.

(A)
0
(B)
1
(C)
2
(D)
3

Answer and solution

ANSWER

With K = 2 the critical point for the plot of G(s) is -1/K = -0.5. The clockwise outer curve and the clockwise left lobe of the figure-eight each enclose it once, so N = 2. With P = 0, Z = P + N = 2 closed-loop poles lie in the closed right-half plane. Answer: (C) 2.

Official answer: CSource: Official answer key, GATE 2022 EC (IIT Kharagpur)

Critical point. The closed-loop poles are the roots of $1 + K G (s) = 0$ , that is $G (s) = - \frac{1}{K} = - \frac{1}{2} = - 0.5$ . So we count how many times the plot of $G$ encircles $- 0.5$ . This is the same as counting encirclements of $- 1$ by the plot of $K G$ . The real-axis crossings of $K G$ are $2 (- 0.8) = - 1.6$ and $2 (- 0.4) = - 0.8$ , and these lie on either side of $- 1$ .
Nyquist criterion with a clockwise contour. $Z = P + N$ . Here $Z$ is the number of closed-loop poles inside the contour, $P$ is the number of open-loop poles of $K G (s)$ inside it, and $N$ is the net number of clockwise encirclements of the critical point.
Open-loop poles. The question says $G (s)$ has no poles in the closed right-half plane, so $P = 0$ and the contour needs no indentations.
Nothing on the imaginary axis. The plot meets the real axis only at $- 0.8$ , $- 0.4$ , $0$ and the right-hand crossing of the large curve. It never passes through $- 0.5$ , so $1 + K G (j ω) \neq = 0$ for every $ω$ . There are no closed-loop poles on the $j ω$ -axis, and $Z$ is the full count for the closed right-half plane.
Large outer curve. Its leftmost point is at $- 0.8$ and it goes around the origin, so it encloses $- 0.5$ . The arrows show it runs clockwise: the top arc moves right and the lower-right arc moves down and to the left. Contribution: $+ 1$ clockwise.
Left lobe of the figure-eight. It runs from $- 0.8$ to $- 0.4$ along the real axis, so $- 0.5$ is inside it. Its upper arc moves right and its lower arc moves left, so it runs clockwise. Contribution: $+ 1$ clockwise.
Right lobe. It runs from $- 0.4$ to $0$ , so it does not contain $- 0.5$ . It contributes $0$ whichever way it is traversed. (It runs counter-clockwise.)
Check with a ray. Draw a vertical ray upward from $- 0.5$ . It crosses exactly two branches: the upper arc of the left lobe and the upper-left part of the large curve. Both move left to right where they cross, so above the point that is two clockwise crossings. The right lobe never reaches $Re G = - 0.5$ .
Result. $N = 1 + 1 = 2, Z = P + N = 0 + 2 = 2$ The closed-loop system has 2 poles in the closed right-half plane, so it is unstable. Answer: (C).

KEY IDEA

When the plot shown is of $G (s)$ but the loop gain is $K G (s)$ , shift the critical point rather than rescaling the plot. Count clockwise encirclements of $- 1/ K$ (here $- 0.5$ ). Then take the direction of every loop from its own arrows and apply $Z = P + N$ for a clockwise contour. Only loops that actually contain the critical point contribute to $N$ .

WATCH OUT FOR

Counting encirclements of $- 1$ instead of $- 1/ K = - 0.5$ . Nothing in the figure encloses $- 1$ , so this gives $N = 0$ and the wrong answer (A). A second error is treating the figure-eight as one loop, or letting its two lobes cancel. The counter-clockwise right lobe does not contain $- 0.5$ , so it cancels nothing, and the clockwise left lobe adds one encirclement to the outer circle's one.

From the GATE 2022 EC paper. Every question of this paper · Download the full paper · All 11 Nyquist questions, with notes

GATE 2022 EC Q.21: solution and answer

Answer and solution

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