$s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0$

Mumbai University > Electronics Engineering > Sem 4 > Linear Control Systems

Topic: Stability Analysis in Time Domain

Difficulty : High

Marks : 10M

Given,

$s^6 + 2s^5 +8s^4 + 12s^3 + 20s^2 + 16s + 16$ = 0

$s^3$ row is zero

Auxillary equation using $s^4$ is given by,

A(s) = 2$s^4$ + 12 $s^2$ + 16

cliff A(s) wrt s,

$\frac{dA(s)}{ds}$ = 8$s^3$ + 24s

Since there is no sign change, system is stage,

by solving the Auxillary equation

$2s^4 + 12s^2 + 16$ = 0

put $s^2$ = y

= $2y^2 + 12y + 16$ = 0

y = $\frac{-12 \pm \sqrt{144-128}}{4} = -3 \pm 1 $

$\therefore$ y = - 2 , -4

$s^2$ = -2 , - 4

s= +i$(2)^{1/2}$,+i2

Since we have non repeated roods on jw axis, the system is marginally stage.