# Stability Analysis

 Question 1
Consider an even polynomial $p(s)$ given by
$p(s)=s^4+5s^2+4+K$,
where $K$ is an unknown real parameter. The complete range of$K$ for which $p(s)$ has all its roots on the imaginary axis is ________.
 A $-4\leq K\leq \frac{9}{4}$ B $-3\leq K\leq \frac{9}{2}$ C $-6\leq K\leq \frac{5}{4}$ D $-5\leq K\leq 0$
GATE EC 2022   Control Systems
Question 1 Explanation:
$p(s)=s^4+5s^2+4+k$
$\begin{matrix} s^4 & 1& 5 &(4+k) \\ s^3& 0 & 0 & \\ s^2& & & \\ s& & & \\ s^0& & & \end{matrix}$
$s^4+5s^2+(4+k)=0$
$4s^3+10s=0$
$(4s^2+10)=0$
$s=\pm j\sqrt{5/2}$
$\begin{matrix} s^4 & 1& 5 &(4+k) \\ s^3& 4 & 10 & \\ s^2& 5/2 & (4+k) & \\ s^1&10-\frac{4(4+k)}{5/2} & & \\ s^0& (4+k) & & \end{matrix}$
\begin{aligned} (4+k) & \gt 0\\ k & \gt -4\\ 10 \times \frac{5}{2}& \gt 4(4+K)\\ (4+K)& \lt \frac{25}{4}\\ K& \lt 9/4 \end{aligned}
$\Rightarrow$ All roots be on imaginary axis $-4\leq K\leqslant 9/4$
 Question 2
Consider a unity feedback system, as in the figure shown, with an integral compensator $\frac{K}{s}$ and open-loop transfer function

$G(s)=\frac{1}{s^2+3s+2}$

where $k \gt 0$. The positive value of K for which there are exactly two poles of the unity feedback system on the $j\omega$ axis is equal to ______ (rounded off to two decimal places). A 2.45 B 4.28 C 6 D 6.25
GATE EC 2019   Control Systems
Question 2 Explanation:
$\frac{Y(s)}{X(s)}=\frac{K}{s^{3}+3 s^{2}+2 s+K}$
Two poles of this system lie the system is moro:
System is marginally stable.
$\qquad k_{\text{mar}}=3 \times 2=6$

 Question 3
A unity feedback control system is characterized by the open-loop transfer function
$G(s)=\frac{2(s+1)}{s^{3}+ks^{2}+2s+1}$
The value of k for which the system oscillates at 2 rad/s is ________.
 A 0.6 B 0.69 C 0.89 D 0.75
GATE EC 2017-SET-2   Control Systems
Question 3 Explanation:
The given open loop transfer function is,
$G(s)=\frac{2(s+1)}{s^{3}+K s^{2}+2 s+1}$
The closed loop transfer function is,
$T(s)=\frac{G(s)}{1+G(s)}=\frac{2(s+1)}{s^{3}+K s^{2}+4 s+3}$
When system oscillates, i.e., when system is marginally stable,
$(1)(3)=K(4)$
$\mathrm{So}, \quad K=0.75$
 Question 4
Which one of the following options correctly describes the locations of the roots of the equation $s^{4}+s^{2}+1=0$ on the complex plane?
 A Four left half plane (LHP) roots B One right half plane (RHP) root, one LHP root and two roots on the imaginary axis C Two RHP roots and two LHP roots D All four roots are on the imaginary axis
GATE EC 2017-SET-1   Control Systems
Question 4 Explanation:
$q(s)=s^{4}+s^{2}+1=0$
$\begin{array}{r|rrr} s^{4} & 1 & 1 & 1\\ s^{3} & 4 & 2 & 0\\ s^{2} & 0.5 & 1 & 0 \\ s^{1} & -6 & 0 & 0 \\ s^{0} & 1 & 0 & 0 \end{array} \frac{d A(s)}{d s}=4 s^{3}+2 s^{1}$
There are two sign Changes in the first column of the R-H table and the order of auxiliary equation is 4. So, four poles are symmetric about origin.
$\therefore$2 RHP roots and 2 LHP roots. Question 5
The first two rows in the Routh table for the characteristic equation of a certain closed-loop control system are given as The range of K for which the system is stable is
 A $-2.0 \lt K \lt 0.5$ B $0 \lt K \lt 0.5$ C $0\lt K\lt \infty$ D $0.5\lt K\lt \infty$
GATE EC 2016-SET-3   Control Systems
Question 5 Explanation:
$\begin{array}{c|cc} s^{3}& 1 & (2 k+3) \\ s^{2}&2 k & 4 \\ s&\frac{4 k^{2}+6 k-4}{2 k} & 0\\ s^{0}&4 \end{array}$
For stability, $K \gt 0$
\begin{aligned} 4 K^{2}+6 K-4& \gt 0 \\ 2 K^{2}+3 K-2& \gt 0 \\ (2 K-1)(K+2)& \gt 0 \\ \Rightarrow \quad K& \gt \frac{1}{2} or, k \lt -2 \\ \text{Hence, }\quad 0.5& \lt K \lt \infty \end{aligned}

There are 5 questions to complete.