# Concepts of Stability

 Question 1
The characteristic equation of a linear time-invariant (LTI) system is given by
$\Delta (s)=s^4+3s^3+3s^2+s+k=0$
The system is BIBO stable if
 A $0 \lt k \lt \frac{12}{9}$ B $k \gt 3$ C $0 \lt k \lt \frac{8}{9}$ D $k \gt 6$
GATE EE 2019   Control Systems
Question 1 Explanation:
Routh array is For BIBO stability,
$\frac{\left ( \frac{8}{3}-3k \right )}{\left ( \frac{8}{3} \right )} \gt 0$
$\Rightarrow k \lt \frac{8}{9}$
and $k \gt 0$
$\therefore \;\; 0 \lt k \lt \frac{8}{9}$
 Question 2
The number of roots of the polynomial,
$s^{7}+s^{6}+7s^{5}+14s^{4}+31s^{3}+73s^{2}+25s+200$,
in the open left half of the complex plane is
 A 3 B 4 C 5 D 6
GATE EE 2018   Control Systems
Question 2 Explanation:
Characteristic equation, $s^7+s^6+7s^5+14s^4+31s^3+73s^2+25s+200=0$ Auxillary equation, $A(s)=8s^4+48s^2+200$
$\frac{d}{ds}A(s)=32s^3+96s$
Total number of poles =7
Two sign change above auxillary equation=2 poles in RHS.
Two sign changes below auxillary equation implies out of 4 symmetric roots about origin, two poles are in LHS and two poles are in RHS.
Therefore 3 poles in LHS and 4 poles in RHS.

 Question 3
The range of K for which all the roots of the equation $s^{3}+3s^{2}+2s+K=0$ are in the left half of the complex s-plane is
 A $0 \lt K \lt 6$ B $0 \lt K \lt 16$ C $6 \lt K \lt 36$ D $6 \lt K \lt 16$
GATE EE 2017-SET-2   Control Systems
Question 3 Explanation:
From the given equation, $s^3+3s^2+2s+K=0$
Using Routh's criterion, we get
$K \lt 6$ and $K \gt 0$ or $0 \lt K \lt 6$
 Question 4
A closed loop system has the characteristic equation given by $s^{3}+Ks^{2}+(K+2)s+3=0$. For this system to be stable, which one of the following conditions should be satisfied?
 A $0 \lt K \lt 0.5$ B $0\lt K \lt 1$ C $0 \lt K \lt 1$ D $K \gt 1$
GATE EE 2017-SET-1   Control Systems
Question 4 Explanation:
Characteristic equation is,
$s^3+Ks^2+(K+2)s+3=0$
For this system to be stable, using Routh's criterion, we can write,
$3 \lt K(K+2)$
or, $K^2+2K-3 \gt 0$
or, $(K+3)(K-1) \gt 0$
Here, the valid answer will be out of all the options given.
$i.e K \gt 1.$
 Question 5
The open loop transfer function of a unity feedback control system is given by
$G(s)=\frac{K(s+1)}{s(1+Ts)(1+2s)}, K\gt 0,T \gt 0$
The closed loop system will be stable if,
 A $0\lt T \lt\frac{4(K+1)}{K-1}$ B $0 \lt K \lt \frac{4(T+2)}{T-2}$ C $0 \lt K \lt \frac{T+2}{T-2}$ D $0 \lt T \lt \frac{8(K+1)}{K-1}$
GATE EE 2016-SET-2   Control Systems
Question 5 Explanation:
Open loop transfer function:
$G(s)=\frac{K(s+1)}{s(1+Ts)(1+2s)};$ $\; K \gt 0 \;$ and $\; t \gt 0$
For closed loop system stability, characteristic equation is,
$1+G(s)H(s)=0$
$1+\frac{K(s+1)}{s(1+Ts)(1+2s)}\cdot 1=0$
$s(1+Ts)(1+2s)+K(s+1)=0$
$2Ts^3+(2+T)s^2+(1+K)s+K=0$
Using Routh's criteria, For stability, $K \gt 0$
and $(2+T)(1+K)-2TK \gt 0$
$K(2+T-2T)+(2+T) \gt 0$
or
$-(T-2)K+2(2+T) \gt 0$
$-K \gt -\frac{(2+T)}{(T-2)}$ or $K \lt \frac{T+2}{(T-2)}$
Hence for stability,
$0 \lt K \lt \frac{T+2}{T-2}$

There are 5 questions to complete.