Discussion

Match List I with List II and select the correct answer using codes given below :

List I (Characteristic Equation of the System)	List II (Stability of the System)
(a) \(s^{5} + s^{4} + 2s^{3} + 2s^{2} + 3s + 15 = 0\)	(i) marginally stable
(b) \(s^{6} + 2s^{5} + 8s^{4} + 12s^{3} + 20s^{2} + 16s + 16 = 0\)	(ii) unstable
(c) G(s) H(s) = 10/s (s + 2) (s + 5)	(iii) stable
(d) \(G(s) = \dfrac{s + 5}{s^{2} + 4s + 9}\)	(iv) phase angle of the system varies between 0° and -90°.

Answer: A

The correct matching is a-ii, b-i, c-iii, d-iv.

(a) In the Routh array of s⁵ + s⁴ + 2s³ + 2s² + 3s + 15, the s³ row starts with zero (2 - 2 = 0); replacing it by a small ε gives two sign changes in the first column, so two roots lie in the right half plane: unstable (ii).
(b) The Routh array of the sixth-order polynomial produces a row of zeros whose auxiliary polynomial has roots on the imaginary axis, with no sign change in the first column: marginally stable (i).
(c) With GH = 10/s(s + 2)(s + 5), the characteristic equation is s³ + 7s² + 10s + 10 = 0; since \(7 imes 10 = 70 \gt 10\), all roots are in the left half plane: stable (iii).
(d) G(s) = (s + 5)/(s² + 4s + 9) has phase 0° at zero frequency and approaches -90° at high frequency, since it has one more pole than zero (iv).

The other codes call the polynomial with a right-half-plane pair marginally stable or the third-order loop unstable, which the Routh test rules out.

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