GATE 2023 IN Q.24: solution and answer

GATE 2023 · INQ.24 · 1 mark · NAT

The number of zeros of the polynomial $P (s) = s^{3} + 2 s^{2} + 5 s + 80$ in the right-half plane is ______.

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Answer and solution

ANSWER

$P (s) = s^{3} + 2 s^{2} + 5 s + 80$ has 2 zeros in the right-half plane.

Official answer: 2Source: Official answer key, GATE 2023 IN (IIT Kanpur)

Every coefficient is positive. For a cubic that is necessary for stability but not sufficient, so we build the Routh array to count the right-half-plane (RHP) roots.
The first two rows come straight from the coefficients: $s^{3} s^{2} : 15 : 280$
Row $s^{1}$ : $\frac{( 2 ) ( 5 ) - ( 1 ) ( 80 )}{2} = \frac{10 - 80}{2} = - 35$
Row $s^{0}$ : $80$ , carried down from the $s^{2}$ row.
First column: $1, 2, - 35, 80$ . The sign changes twice, once from $2$ to $- 35$ and once from $- 35$ to $80$ .
By the Routh criterion, the number of sign changes in the first column equals the number of roots in the right-half plane. So $P (s)$ has 2 RHP zeros.
Cross-check with the cubic rule. For $s^{3} + a_{2} s^{2} + a_{1} s + a_{0}$ with all coefficients positive, the roots all lie in the left-half plane only if $a_{2} a_{1} > a_{0}$ . Here $a_{2} a_{1} = 2 \times 5 = 10 < a_{0} = 80$ , so the polynomial is unstable. Also $P (- 5) = - 20$ and $P (- 4) = 28$ , so there is one real root between them, near $s \approx - 4.64$ . The roots sum to $- a_{2} = - 2$ , so the complex pair has real part $\approx \frac{- 2 + 4.64}{2} \approx + 1.32$ . Both of those roots are in the RHP, which confirms the count of 2.

KEY IDEA

In the Routh array, the number of sign changes down the first column equals the number of roots in the right-half plane. For a cubic $s^{3} + a_{2} s^{2} + a_{1} s + a_{0}$ with positive coefficients, the one test that matters is whether $a_{2} a_{1} > a_{0}$ . When it fails, as it does here, the $s^{1}$ entry turns negative and two roots lie in the RHP.

WATCH OUT FOR

Seeing that every coefficient is positive and answering 0. Positive coefficients are necessary but not sufficient for stability once the order is 3 or more. A second mistake is to spot that the system is unstable and answer 1. The number of sign changes is the number of RHP roots, and here there are two, $2 \to - 35$ and $- 35 \to 80$ .

From the GATE 2023 IN paper. Every question of this paper · Download the full paper · All 15 Routh–Hurwitz questions, with notes

GATE 2023 IN Q.24: solution and answer

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