In the s^3 row the second element is (3(1)-1(3))/3 = 0, but the first element is 11/3. Is this a problem?

No, only the whole row being zero indicates symmetric roots; a zero in a later position is fine

Yes, it means roots on the jω axis

Yes, an epsilon must replace it

Yes, the system is marginally stable

What would a complete row of zeros in the array have indicated?

Roots symmetric about the origin, possibly on the jω axis

Asked 4/27
2066 Bhadra (old course) · 8 marks

R-H root distribution of s⁵+3s⁴+5s³+4s²+s+3

Short answer

Two roots lie in the right half-plane, three in the left half-plane, and none on the jω-axis. The system is unstable.

Using R-H criteria, tell how many roots of polynomial are in right half plane, in left half plane and on jω-axis. P(s) = s⁵ + 3s⁴ + 5s³ + 4s² + s + 3.

Result: RHP = 2, LHP = 3, $j\omega$ axis = 0. The system is unstable.

Routh–Hurwitz criterion: the number of sign changes in the first column of the Routh array equals the number of roots in the right half plane. A complete zero row would indicate roots on the $j\omega$ axis.

P(s) = s^5 + 3s^4 + 5s^3 + 4s^2 + s + 3

Routh array

Row	Col 1	Col 2	Col 3
$s^5$	1	5	1
$s^4$	3	4	3
$s^3$	$\frac{3(5)-1(4)}{3}=\frac{11}{3}$	$\frac{3(1)-1(3)}{3}=0$
$s^2$	$\frac{\frac{11}{3}(4)-3(0)}{\frac{11}{3}}=4$	$\frac{\frac{11}{3}(3)-0}{\frac{11}{3}}=3$
$s^1$	$\frac{4(0)-\frac{11}{3}(3)}{4}=-\frac{11}{4}$	0
$s^0$	3

First column: $1,\ 3,\ 3.67,\ 4,\ -2.75,\ 3$ .

$4 \to -2.75$ : one sign change
$-2.75 \to 3$ : second sign change

Root distribution

No row became entirely zero, so no root lies on the $j\omega$ axis.

Region	Number of roots
Right half plane	2
$j\omega$ axis	0
Left half plane	$5 - 2 = 3$

The roots ( $\approx 0.256\pm j0.720$ , $-0.941\pm j1.504$ , $-1.631$ ) confirm this.

Answer: 2 roots in the RHP, 3 in the LHP, none on the $j\omega$ axis. The system is unstable.

✓ Answer checked by Claude Opus 5.5 (high effort)

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