# Time Response Analysis

 Question 1
Consider a negative unity feedback system with the forward path transfer function $\frac{s^2+s+1}{s^3+2s^2+2s+K}$, where K is a positive real number. The value of K for which the system will have some of its poles on the imaginary axis is ________ .
 A 9 B 8 C 7 D 6
GATE EE 2020   Control Systems
Question 1 Explanation:
CE is
$1+G(s)H(s)=0$
$\Rightarrow \, 1+\frac{s^{2}+s+1}{s^{3}+2s^{2}+2s+k}=0$
$\Rightarrow \, s^{3}+3s^{2}+3s+(1+K)=0$
R.H. criteria:

$9 - (1 + K) = 0$
$\Rightarrow \, \, K=8$
 Question 2
Which of the following option is correct for the system shown below?
 A $4^{th}$ order and stable B $3^{rd}$ order and stable C $4^{th}$ order and unstable D $3^{rd}$ order and unstable
GATE EE 2020   Control Systems
Question 2 Explanation:
\begin{aligned} 1+\frac{20}{s^{2}(s+1)(s+20)}&=0\\ (s^{3}+s^{2})(s+20)+20&=0\\ s^{4}+20s^{3}+s^{3}+20s^{2}+20&=0\\ s^{4}+21s^{3}+20s^{2}+20&=0 \end{aligned}
Given system is fourth order system and unstable.

stablity status: since it has one missing term of 's' thus undoubtedly given transfer function is unstable.
 Question 3
Consider a linear time-invariant system whose input r(t) and output y(t) are related by the following differential equation.
$\frac{d^2y(t)}{dt^2}+4y(t)=6r(t)$
The poles of this system are at
 A +2j, -2j B +2, -2 C +4, -4 D +4j, -4j
GATE EE 2020   Control Systems
Question 3 Explanation:
\begin{aligned}\frac{d^2 y(t)}{dt^{2}}+4y(t) &=6 r(t)\\ [s^{2}+4]Y(s)&=6 R(s) \\ \frac{Y(s)}{R(s)}&=\frac{6}{s^{2}+4} \\ \text{Poles: } s^{2}+4&=0 \\ s&=\pm j2 \end{aligned}
 Question 4
The unit step response y(t) of a unity feedback system with open loop transfer function
$G(s)H(s)=\frac{K}{(s+1)^{2}(s+2)}$
is shown in the figure. The value of K is _______ (up to 2 decimal places).
 A 4 B 8 C 10 D 12
GATE EE 2018   Control Systems
Question 4 Explanation:
Closed loop transfer function,
$\frac{C(s)}{R(s)}=\frac{\frac{K}{(s+1)^2(s+2)}}{1+\frac{K}{(s+1)^2(s+2)}}$
$\frac{C(s)}{R(s)}=\frac{K}{(s+1)^2(s+2)+K}$
Given $R(s)=\frac{1}{s}$
$C(s)=\frac{K}{s((s+1)^2(s+2)+K)}$
$\lim_{s \to 0}sC(s)=0.8$
$\frac{K}{2+K}=0.8$
$\Rightarrow \; K=8$
 Question 5
C onsider a unity feedback system with forward transfer function given by
$G(s)=\frac{1}{(s+1)(s+2)}$
The steady-state error in the output of the system for a unit-step input is _________(up to 2 decimal places).
 A 0.25 B 0.45 C 0.66 D 0.85
GATE EE 2018   Control Systems
Question 5 Explanation:
Steady state error for type-0 and step input,
$e_{ss}=\frac{1}{1+k+p}$
$k_p=\lim_{s \to 0}\frac{1}{(s+1)(s+2)}=\frac{1}{2}$
$e_{ss}=\frac{1}{1+\frac{1}{2}}=\frac{2}{3}$
$\;\;=0.66$ uinits
 Question 6
Match the transfer functions of the second-order systems with the nature of the systems given below.
 A P-I, Q-II, R-III B P-II, Q-I, R-III C P-III, Q-II, R-I D P-III, Q-I, R-II
GATE EE 2018   Control Systems
Question 6 Explanation:
$P=\frac{15}{s^2+5s+15}$
$\omega _n=\sqrt{15}=3.872$ rad/sec
$2 \xi \times 3.872=5$
$\xi=\frac{5}{2 \times 3.872}=0.64 \;\;\;\;(Underdamped)$
$Q=\frac{25}{s^2+10s+25}$
$\omega _n=\sqrt{25}=5$ rad/sec
$2 \xi \times 5=10$
$\xi=1 \;\;\;\;\;(Critically \; damped)$
Observing all the options, option (C) is correct.
 Question 7
Which of the following systems has maximum peak overshoot due to a unit step input?
 A $\frac{100}{s^{2}+10s+100}$ B $\frac{100}{s^{2}+15s+100}$ C $\frac{100}{s^{2}+5s+100}$ D $\frac{100}{s^{2}+20s+100}$
GATE EE 2017-SET-2   Control Systems
Question 7 Explanation:
For maximum peak over shoot $M_P\propto \frac{1}{\xi}$
$\xi=0.25$ for option (C) which is least among all options. Therefore correct option is C.
 Question 8
When a unit ramp input is applied to the unity feedback system having closed loop transfer function
$\frac{C(s)}{R(s)}=\frac{Ks+b}{s^{2}+as+b}(a \gt 0,b \gt 0,K\gt 0)$,
the steady state error will be
 A 0 B $\frac{a}{b}$ C $\frac{a+K}{b}$ D $\frac{a-K}{b}$
GATE EE 2017-SET-2   Control Systems
Question 8 Explanation:
Closed loop transfer function $=\frac{Ks+b}{s^2+as+b}$
Open loop transfer function $= G(s)=\frac{Ks+b}{s^2+as+b-Ks-b}$
$G(s)=\frac{Ks+b}{s^2+as-Ks} =\frac{Ks+b}{s(s+a-K)}$
Steady state error for ramp input given to type-1 system $=1/K_V$
where, velocity error coefficient,
$K_V=\lim_{s \to 0}s\cdot \frac{Ks+b}{s(s+a-K)} =\frac{b}{a-K}$
$e_{ss}=\frac{a-K}{b}$
 Question 9
A second-order real system has the following properties:
a) the damping ratio $\xi$ = 0.5 and undamped natural frequency $\omega _{n}$=10 rad/s,
b) the steady state value of the output, to a unit step input, is 1.02.
The transfer function of the system is
 A $\frac{1.02}{s^{2}+5s+100}$ B $\frac{102}{s^{2}+10s+100}$ C $\frac{100}{s^{2}+10s+100}$ D $\frac{102}{s^{2}+5s+100}$
GATE EE 2016-SET-2   Control Systems
Question 9 Explanation:
Damping ratio $\xi=0.5$
Undamped natural frequency $\omega _n=10$ rad/sec
Steady state output toa unit step input $C_{ss}=1.02$
Hence steady state error $e_{ss}=1.02-1.00=0.02$
$\because$ Characteristic equation is,
$s^2+2 \xi \omega _ns+\omega _n^2=0$
$s^2+2 \times 0.5\times 10 s+100=0$
$s^2+10s+100=0$
From options, if we take option (B) then
$C_{ss}=\lim_{s \to 0}s\cdot C(s)$
$\;\; \; \; \;=\lim_{s \to 0 }s \times \frac{1}{s} \times \frac{102}{s^2+10s+100}$
$C_{ss}=1.02$
Hence, Option (B) is correct answer.
 Question 10
The unit step response of a system with the transfer function $G(s)=\frac{1-2s}{1+s}$ is given by which one of the following waveforms?
 A A B B C C D D
GATE EE 2015-SET-2   Control Systems
Question 10 Explanation:
$T.F.=\frac{1-2s}{1+s}$
$C(s)=\frac{1-2s}{1+s}\cdot \frac{1}{s}$
$\;\;=\frac{A}{1+s}+\frac{B}{s}$
For A,
$A=\lim_{s \to -1}(s+1)\cdot \frac{1-2s}{s(1+s)}$
$\;\;=\frac{(1-2(-1))}{(-1)}=-3$
For B,
$B=\lim_{s \to 0}s\cdot \frac{1-2s}{s(1+s)}=1$
$C(s)=\frac{1}{s}-\frac{3}{1+s}$
$C(t)=(1-3e^{-t})u(t)$
There are 10 questions to complete.