# 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,
$0 \lt K \lt \frac{T+2}{T-2}$
 Question 6
Given the following polynomial equation $s^{3}+5.5s^{2}+8.5s+3=0$,
the number of roots of the polynomial, which have real parts strictly less than -1, is ________ .
 A 1 B 2 C 3 D 4
GATE EE 2016-SET-1   Control Systems
Question 6 Explanation:
$s^3+5.5s^2+8.5s+3=0$
Putting, $s=s_1-1$
$(s_1-1)^3+5.5(s_1-1)^2+(8.5)(s_1-1)+3=0$
$s_1^3+2.5s_1^2+0.5s_1-1=1$

As there is one sign change, hence, two roots of given polynomial will lie to the left of $s=-1$.
 Question 7
The transfer function of a second order real system with a perfectly flat magnitude response of unity has a pole at (2-j3). List all the poles and zeroes.
 A Poles at (2$\pm$j3), no zeroes. B Poles at ($\pm$2-j3), one zero at origin. C Poles at (2-j3),(-2+j3), zeroes at (-2-j3),(2+j3). D Poles at (2$\pm$j3), zeroes at (-2$\pm$j3).
GATE EE 2015-SET-1   Control Systems
Question 7 Explanation:
Response of transfer function is unit for all $\omega$.
$M=1; P_1=2-j3$
Second order system, hence number of poles =2
Therefore, second pole $P_2=2+j3$
Now for M=1, and due to x-axis symmetry of root locus of transfer function, position of zeroes must be
$Z_1=-2-j3$ and $Z_2=-2+j3$
 Question 8
A single-input single output feedback system has forward transfer function G(s) and feedback transfer function H(s). It is given that |G(s)H(s)| $\lt$1. Which of the following is true about the stability of the system ?
 A The system is always stable B The system is stable if all zeros of G(s)H(s) are in left half of the s-plane C The system is stable if all poles of G(s)H(s) are in left half of the s-plane D It is not possible to say whether or not the system is stable from the information given
GATE EE 2014-SET-3   Control Systems
 Question 9
A system with the open loop transfer function
$G(s)=\frac{K}{s(s+2)(s^{2}+2s+2)}$
is connected in a negative feedback configuration with a feedback gain of unity. For the closed loop system to be marginally stable, the value of K is ______
 A 4 B 5 C 6 D 7
GATE EE 2014-SET-2   Control Systems
Question 9 Explanation:
Given, $G(s)=\frac{K}{s(s+2)(s^2+2s+2)}$

The characteristic equation of given unity-negative feedback control system is given by
$1+G(s)H(s)=0$
or, $1+\frac{K}{s(s+2)(s^3+2s+2)}=0$
or, $s(s+2)(s^3+2s+2)+K=0$
or, $s^4+4s^3+6s^2+4s+K=0$
Forming Routh' array as shown below

For stability of the system, $K \gt 0$ and $\frac{20-4K}{5} \gt 0$or $K\gt 5$
$\therefore$ For stability, $0 \lt K \lt 5$
For given system to be marginally stable

K=5
 Question 10
For the given system, it is desired that the system be stable. The minimum value of $\alpha$ for this condition is ______.
 A 0.22 B 0.46 C 0.86 D 0.62
GATE EE 2014-SET-1   Control Systems
Question 10 Explanation:
Given, $G(s)=\frac{(s+\alpha )}{s^3+(1+\alpha )s^2+(\alpha -1)s+(1-\alpha )}$
and $H(s)=1$
Characteristic equation of given control system is given by
$1+G(s)H(s)=0$
or, $1+\left [ \frac{s+\alpha }{s^3+(1+\alpha )s^2+(\alpha -1)s+(1-\alpha )} \right ]=0$
or, $s^3+(1+\alpha )s^2+(\alpha -1)s+(1-\alpha )+s+\alpha =0$
or, $s^3+(1+\alpha )s^2+\alpha s+1=0$
Routh's array is

For the given system to be stable, there should not be any sign change in the first column of Routh's array.
Therefore, $1+\alpha \gt 0 \;or, \; \alpha \gt -1\;\; ...(i)$
Also, $\frac{\alpha ^2 +\alpha -1}{1+\alpha } \gt 0$
or, $\alpha ^2 +\alpha -1 \gt 0$
or, $(\alpha +1.618)(\alpha -0.618) \gt 0$
or, $\alpha \lt -1.618$
or, $\alpha \gt 0.618$
As $\alpha \gt -1$ (using equation (i))
therefore, $\alpha \gt 0.618 \;\;...(ii)$
Combining condition (i) and (ii),
$-1 \lt \alpha \lt 0.618$
Thus, for the system to be stable minimum value of $\alpha =0.618.$
There are 10 questions to complete.