# Time Response Analysis

 Question 1
The block diagram of a closed-loop control system is shown in the figure. R(s),Y(s) and D(s) are the Laplace transforms of the time-domain signals $r(t),y(t),$ and $d(t)$, respectively. Let the error signal be defined as $e(t)=r(t)-y(t)$. Assuming the reference input $r(t)=0$ for all $t$, the steady-state error $e(\infty )$, due to a unit step disturbance $d(t)$, is _________ (rounded off to two decimal places).

 A -0.2 B -0.1 C 0 D -0.3
GATE EC 2022   Control Systems
Question 1 Explanation:
$Y(s)=\frac{R(s)\cdot \frac{10}{s(s+10)}}{1+\frac{10}{s(s+10)}} +\frac{D(s)\cdot \frac{10}{s(s+10)}}{1+\frac{10}{s(s+10)}}$
When, $r(t)=0\; and \; d(t)=U(t)\underleftrightarrow{L.T.}\frac{1}{s}$
$e(\infty )=-\lim_{s \to 0}\frac{s\cdot \frac{1}{s} \times 1}{s(s+10)+10} =-1/10=-0.1\;\;\;(\because e(t)=r(t)-y(t))$
 Question 2
Two linear time-invariant systems with transfer functions
$G_1(s)=\frac{10}{s^2+s+1}, G_2(s)=\frac{10}{s^2+s\sqrt{10}+10}$
have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?
 A $y_1(t)$ and $y_2(t)$ have the same percentage peak overshoot. B $y_1(t)$ and $y_2(t)$ have the same steady-state value. C $y_1(t)$ and $y_2(t)$ have the same damped frequency of oscillation. D $y_1(t)$ and $y_2(t)$ have the same 2% settling time.
GATE EC 2022   Control Systems
Question 2 Explanation:
\begin{aligned} G_1(s)&=\frac{10}{s^2+s+1}\\ \omega _{n1}^2&=1\\ 2\xi _1 \times 1&=1\\ \xi _1&=1/2\\ G_2(s)&=\frac{10}{s^2+s\sqrt{10}+10}\\ \omega _{n2}^2&=10\\ 2\xi _1 \times \sqrt{10}&=\sqrt{10}\\ \xi _2&=1/2\\ \end{aligned}
$\because M_P$ depends on $\xi$ only
$y_1(t) \; and \; y_2(t)$ have same percentage peak overshoot.
$\omega _{d_1}=\omega _{n_1}\sqrt{1-\xi ^2}$
$\Rightarrow \omega _{d_1}\neq \omega _{d_2} \;\; \because \omega _{n_1}\neq \omega _{n_2}$
damped frequency of oscillation is not same.
\begin{aligned} T_s &=\frac{4}{\xi \omega _n}(2%\; setting \; time) \\ T_{s_1} &\neq T_{s_2} \;\because \omega _{n_1}\neq \omega _{n_2}\\ C_1(s) &= \frac{10 \times 1/s}{s^2+s+1}\\ C_{ss} &=\lim_{s \to 0}s\cdot C_1(s) \\ &= 10\\ C_2(s) &= \frac{10 \times 1/s}{s^2+\sqrt{10}s+10}\\ C_{ss} &=\lim_{s \to 0}s\cdot C_2(s) \\ &= \frac{10}{10}1=1 \end{aligned}
Steady - state value is not same.
 Question 3
Consider the following closed loop control system

where $G(s)=\frac{1}{s(s+1)}$ and $C(s)=K\frac{s+1}{s+3}$. If the steady state error for a unit ramp input is 0.1, then the value of K is _______.
 A 10 B 20 C 30 D 40
GATE EC 2020   Control Systems
Question 3 Explanation:
Open loop transfer function for the system $=C(s)\times G(s)=\frac{K(s+1)}{ (s+3)}\times \frac{1}{s(s+1)}$
Since the system is type-1, so far a given unit ramp input steady state
$e_{ss}=\frac{1}{K_{V}}$
where, $K_{V}=\lim_{s\rightarrow 0}S\times \frac{K}{S(S+3)}=\frac{K}{3}$
so, $\; e_{ss}=\frac{1}{K/3}=\frac{3}{K}$
Given that, $\; e_{ss}=0.1$
So, $0.1=\frac{3}{K}\Rightarrow K=30$
 Question 4
A system with transfer function $G(s)=\frac{1}{(s+1)(s+a)}, a \gt 0$ is subjected an input $5\cos 3t$. The steady state output of the system is $\frac{1}{\sqrt{10}} \cos (3t-1.892)$. The value of $a$ is ____.
 A 3 B 4 C 5 D 6
GATE EC 2020   Control Systems
Question 4 Explanation:
Given that,
\begin{aligned}G(j\omega )&=\frac{1}{(1+j\omega )(\alpha +j\omega )} \\ \left |G(j\omega ) \right |&=\frac{1}{\sqrt{(\omega ^{2}+1)(\omega ^{2}+\alpha ^{2})}} \\ \text{According to }& \text{question,}\\ \left | G(j\omega ) \right |_{\omega =3}&=\frac{1}{5\sqrt{10}}\\ \Rightarrow \frac{1}{\sqrt{(\omega ^{2}+1)(\omega ^{2}+\alpha ^{2})}}&=\frac{1}{5\sqrt{10}} \\ \Rightarrow \frac{1}{\sqrt{10(a^{2}+9)}}&=\frac{1}{5\sqrt{10}} \alpha ^{2}+9=25 \\ \alpha ^{2}&=16 \\ \alpha &=4\end{aligned}
 Question 5
The loop transfer function of a negative feedback system is

$G(s)H(s)=\frac{K(s+11)}{s(s+2)(s+8)}$

The value of K, for which the system is marginally stable, is ________.
 A 160 B 120 C 180 D 200
GATE EC 2020   Control Systems
Question 5 Explanation:
Characteristic equation q(s) for the given open loop system will be
$q(s)=s^{3}+10s^{2}+16s+Ks+11K=0$
Using R-H criteria,

For System to be Marginally Stable
\begin{aligned}\frac{10(16+K)-11K}{10}&=0 \\ 160+10K-11K&=0 \\ K&=160\end{aligned}
 Question 6
Consider a causal second-order system with the transfer function

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

with a unit-step $R(s)=\frac{1}{s}$ as an input. Let C(s) be the corresponding output. The time taken by the system output c(t) to reach 94% of its steady-state value $\lim_{t \to \infty }c(t)$, rounded off to two decimal places, is
 A 5.25 B 4.5 C 3.89 D 2.81
GATE EC 2019   Control Systems
Question 6 Explanation:
\begin{aligned} G(s)&=\frac{1}{s^{2}+2 s+1}=\frac{1}{(s+1)^{2}} \\ R(s)&=\frac{1}{s} \\ \alpha (s)&=G(s) R(s)=\frac{1}{s(s+1)^{2}} \end{aligned}
Using partial fraction expansion, we get,
\begin{aligned} C(s) & =\frac{A}{s}+\frac{B}{(s+1)}+\frac{C}{(s+1)^{2}} \\ A\left(s^{2}+2 s+1\right)&+B\left(s^{2}+s\right)+C s=1 \\ A&=1 \\ A+B & =0 \Rightarrow B=-1 \\ 2 A+B+C & =0 \Rightarrow C=-1 \\ \therefore \quad C(s) & =\frac{1}{s}-\frac{1}{(s+1)}-\frac{1}{(s+1)^{2}} \\ \text{and}\quad c(t) & =\left(1-e^{-t}-t e^{-t}\right) u(t) \\ \lim _{t \rightarrow \infty} c(t) & =1 \end{aligned}
In order to reach 94 % of its steady-state value,
$\left(1-e^{-t}-t e^{-t}\right)=0.94$
By trial and error, we get,
$t \approx 4.50 \mathrm{sec}$
 Question 7
The open loop transfer function G(s)=$\frac{(s+1)}{s^{P}(s+2)(s+3)}$
Where p is an integer, is connected in unity feedback configuration as shown in figure.

Given that the steady state error is zero for unit step input and is 6 for unit ramp input, the value of the parameter p is _________.
 A 1 B 2 C 3 D 4
GATE EC 2017-SET-1   Control Systems
Question 7 Explanation:
Steady state error of type 1 system, for step input is zero, for ramp input is $\frac{1}{K_{v}}$ and for parabolic input is infinity.
So, the given system must be of type- 1 and p=1
 Question 8
For the unity feedback control system shown in the figure, the open-loop transfer function G(s) is given as

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

. The steady state error $e_{ss}$ due to a unit step input is
 A 0 B 0.5 C 1 D $\infty$
GATE EC 2016-SET-3   Control Systems
Question 8 Explanation:
\begin{aligned} E(s) &=\frac{R(s)}{1+G(s) H(s)} \\ R(s) &=\frac{1}{s} ; G(s)=\frac{2}{s(s+1)}; \\ H(s) &=1 \\ E(s) &=\frac{\frac{1}{s}}{1+\frac{s}{s(s+1)}}=\frac{s+1}{s^{2}+s+2} \\ e_{s s} &=\lim _{s \rightarrow 0} s E(s) \\ &=\lim _{s \rightarrow 0} \frac{s(s+1)}{s^{2}+s+2} \\ e_{s s} &=0 \end{aligned}
 Question 9
In the feedback system shown below $G(s)=\frac{1}{(s^{2}+2s)}$. The step response of the closed-loop system should have minimum settling time and have no overshoot.

The required value of gain k to achieve this is ________
 A 0.5 B 1 C 2 D 3
GATE EC 2016-SET-2   Control Systems
Question 9 Explanation:
\begin{aligned} G(s)&=\frac{1}{s^{2}+2 s} \\ \frac{Y(s)}{R(s)}&=\frac{\frac{K}{s^{2}+2 s}}{1+\frac{K}{s^{2}+2 s}}=\frac{K}{s^{2}+2 s+K} \end{aligned}
Minimum settling time and no overshoot implies critical damping i.e.
\begin{aligned} \xi &=1 \\ \omega_{n} &=\sqrt{K} \\ \therefore \quad 2 \times \xi \cdot \omega_{n} &=2 \\ \omega_{n} &=1 \\ \Rightarrow \quad \sqrt{K} &=1 \text { or } K=1 \end{aligned}
 Question 10
The response of the system $G(s)=\frac{s-2}{(s+1)(s+3)}$ to the unit step input u(t) is y(t). The value of $\frac{dy}{dt}$ at t=$0^{+}$ is _________
 A 0 B 1 C 2 D 3
GATE EC 2016-SET-2   Control Systems
Question 10 Explanation:
\begin{aligned} L\left(\frac{d y}{d t}\right) &=(s Y(s)-y(0)) \\ Y(s) &=G(s) \times \frac{1}{s}=\frac{s-2}{s(s+1)(s+3)} \\ y(0)&=\lim _{s \rightarrow \infty} s Y(s) \\ \text{(Applying } & \text{initial value theorem) } \\ &=\lim _{s \rightarrow \infty} \frac{s-2}{(s+1)(s+3)} \\ &=\frac{\left(1-\frac{2}{s}\right)}{s\left(1+\frac{1}{s}\right)\left(1+\frac{3}{s}\right)} \\ y(0)&=0 \\ L\left(\frac{d y}{d t}\right) &=s Y(s)=\frac{s \times(s-2)}{s(s+1)(s+3)} \\ &=\frac{s-2}{(s+1)(s+3)} \\ \left.\frac{d y}{d t}\right|_{t=0} &=\lim _{s \rightarrow \infty} s L\left(\frac{d y}{d t}\right) \\ &=\lim _{s \rightarrow \infty} \frac{s \times(s-2)}{(s+1)(s+3)}\\ &=\frac{\left(1-\frac{2}{s}\right)}{\left(1+\frac{1}{s}\right)\left(1+\frac{3}{s}\right)}=1 \end{aligned}
There are 10 questions to complete.