# Control Systems

 Question 1
The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as $\zeta$ and $\omega _n$, respectively. The values of $\zeta$ and $\omega _n$ are

 A $\zeta =0.5 \text{ and }\omega _n=10 rad/s$ B $\zeta =0.1 \text{ and }\omega _n=10 rad/s$ C $\zeta =0.707 \text{ and }\omega _n=10 rad/s$ D $\zeta =0.707 \text{ and }\omega _n=100 rad/s$
GATE EE 2022      Time Response Analysis
Question 1 Explanation:
Reduced the block diagram:

Transfer function,
$\frac{C(s)}{R(s)}=\frac{100/(s(s+10))}{1+(100/s(s+10))}=\frac{100}{s^2+10s+100}$
Standard form,
$T.F.=\frac{\omega _n^2}{s^2+2\xi \omega _ns+\omega _n^2}$
On comparison : $\omega =\sqrt{100}=10rad/sec$ and $2\xi \omega _n=10$
$\Rightarrow \xi=\frac{10}{2 \times 10}=0.5$
 Question 2
The open loop transfer function of a unity gain negative feedback system is given as
$G(s)=\frac{1}{s(s+1)}$
The Nyquist contour in the s-plane encloses the entire right half plane and a small neighbourhood around the origin in the left half plane, as shown in the figure below. The number of encirclements of the point $(-1+j0)$ by the Nyquist plot of $G(s)$, corresponding to the Nyquist contour, is denoted as $N$. Then $N$ equals to

 A 0 B 1 C 2 D 3
GATE EE 2022      Frequency Response Analysis
Question 2 Explanation:
Given: P = 1 (Because, Nyquist contour encircle one pole i.e s = 0)
We have, N = P - Z
N = 1-Z
Characteristic equation
\begin{aligned} 1+G(s)H(s) &=0 \\ 1+\frac{k}{s(s+1)}&=0 \\ s^2+s+k&= 0 \end{aligned}
R-H criteria:
$\left.\begin{matrix} s^2\\ s^1\\ s^0 \end{matrix}\right| \begin{matrix} 1 & k\\ 1 & 0\\ k & \end{matrix}$
Hence, z = 0 (because no sign change in first column of R-H criteria)
(where, z = closed loop pole on RHS side of s-plane)
Therefore, N = 1
 Question 3
An LTI system is shown in the figure where
$G(s)=\frac{100}{s^2+0.1s+10}$
The steady state output of the system, to the input $r(t)$, is given as $y(t)=a+b\sin (10t+\theta )$. The values of $a$ and $b$ will be

 A a=1, b=10 B a=10, b=1 C a=1, b=100 D a=100, b=1
GATE EE 2022      Frequency Response Analysis
Question 3 Explanation:
We knaow, $y(t)=A|G(j\omega ) \sin (\omega t+\phi )$
Here, $G(j\omega )=\frac{100}{-\omega ^2+j0.1\omega +100}$
Put $\omega =0$
$|G(j\omega )|=1$
Now, put $\omega =10$ rad/sec
$|G(j\omega )|=\left | \frac{100}{-100+i1+100} \right |=100$
Therefore, $y(t)=1+0.1 \times 100 \sin(10t+\theta ) =1+10 \sin (10t+\theta )$
On comparision $: a=1 ,b=10$
 Question 4
The open loop transfer function of a unity gain negative feedback system is given by $G(s)=\frac{k}{s^2+4s-5}$.
The range of $k$ for which the system is stable, is
 A $k \gt 3$ B $k \lt 3$ C $k \gt 5$ D $k \lt 5$
GATE EE 2022      Root Locus Techniques
Question 4 Explanation:
Characteristic equation:
\begin{aligned} 1+G(s)H(s)&=0\\ 1+\frac{k}{s^2+4s-5}&=0\\ s^2+4s+k-5&=0 \end{aligned}
R-H criteria:
$\left.\begin{matrix} s^2\\ s^1\\ s^0 \end{matrix}\right| \begin{matrix} 1 & k-5\\ 4 & 0\\ k-5 & \end{matrix}$
Hence, for stable system,
$k-5 \gt 0 \;\; \Rightarrow \; k \gt 5$
 Question 5
The Bode magnitude plot of a first order stable system is constant with frequency. The asymptotic value of the high frequency phase, for the system, is $-180^{\circ}$. This system has

 A one LHP pole and one RHP zero at the same frequency B one LHP pole and one LHP zero at the same frequency C two LHP poles and one RHP zero D two RHP poles and one LHP zero.
GATE EE 2022      Frequency Response Analysis
Question 5 Explanation:
The given system is non-minimum phase system Therefore, transfer function, $T.F=\frac{s-1}{s+1}$
Hence, one LHP pole and one RHP zero at the same frequency.
 Question 6
The state space representation of a first-order system is given as
$\overset{\bullet }{x}=-x+u$
$y=x$
where,x is the state variable, u is the control input and y is the controlled output. Let $u=-Kx$ be the control law, where K is the controller gain. To place a closed-loop pole at -2, the value of K is ___________________.
 A 1 B 2 C 4 D 6
GATE EE 2021      State Variable Analysis
Question 6 Explanation:
$\dot{x}=-x-K x=x(-k I-I)$
Characteristic equation,
\begin{aligned} |S I+K I+I| &=0 \\ |(S+1+K) I| &=0 \\ \therefore \qquad\qquad S+1+K &=0 \\ S &=-1-K \\ -2 &=-1-K \\ K &=1 \end{aligned}
 Question 7
In the given figure, plant $G_{P}\left ( s \right )=\dfrac{2.2}{\left ( 1+0.1s \right )\left ( 1+0.4s \right )\left ( 1+1.2s \right )}$ and compensator $G_{C}\left ( s \right )=K\left [ \dfrac{1+T_{1}s}{1+T_{2}s} \right ]$. The external disturbance input is D(s). It is desired that when the disturbance is a unit step, the steady-state error should not exceed 0.1 unit. The minimum value of K is _____________.
(Round off to 2 decimal places.)

 A 12.25 B 14.12 C 9.54 D 6.22
GATE EE 2021      Time Response Analysis
Question 7 Explanation:
\begin{aligned} e_{s s} &=\lim _{s \rightarrow 0}\left[\frac{s R}{1+G_{C} G_{p}}-\frac{s D G_{p}}{1+G_{C} G_{P}}\right] \\ R(s) &=0 ; D(s)=\frac{1}{s} \\ \therefore \qquad\qquad e_{s s}&=\frac{2.2}{1+2.2 K}=0.1 \\ \therefore \qquad\qquad K_{\min }&=9.54 \end{aligned}
 Question 8
Consider a closed-loop system as shown. $G_{P}\left ( s \right )=\dfrac{14.4}{s\left ( 1+0.1s \right )}$ is the plant transfer function and $G_{c}(S)=1$ is the compensator. For a unit-step input, the output response has damped oscillations. The damped natural frequency is ___________________ $\text{rad/s}$. (Round off to 2 decimal places.)

 A 10.9 B 4.62 C 12.02 D 8.05
GATE EE 2021      Time Response Analysis
Question 8 Explanation:

\begin{aligned} q(s)&=s^{2}+10 s+144=0 \\ \omega_{n}&=12 ; \xi=\frac{5}{12} \\ \omega_{d}&=\omega_{n} \sqrt{1-\xi^{2}} \\ \quad&=12 \sqrt{\frac{119}{144}}=10.90 \end{aligned}
 Question 9
The Bode magnitude plot for the transfer function $\frac{V_{o}\left ( s \right )}{V_{i}\left ( s \right )}$ of the circuit is as shown. The value of R is _____________$\Omega$. (Round off to 2 decimal places.)

 A 0.1 B 0.2 C 0.25 D 0.05
GATE EE 2021      Frequency Response Analysis
Question 9 Explanation:
From response plot
\begin{aligned} M_{r} &=26 \mathrm{~dB}=20 \\ \therefore \qquad \qquad \frac{1}{2 \xi \sqrt{1-\xi^{2}}} &=20 \\ \therefore \qquad \qquad \xi &=0.025 \end{aligned}
From electrical network
\begin{aligned} \chi&=\frac{R}{2} \sqrt{\frac{C}{L}}=0.025 \\ \therefore \qquad \qquad R&=0.10 \Omega \end{aligned}
 Question 10
For the closed-loop system shown, the transfer function $\dfrac{E\left ( s \right )}{R\left ( s \right )}$ is

 A $\frac{G}{1+GH}$ B $\frac{GH}{1+GH}$ C $\frac{1}{1+GH}$ D $\frac{1}{1+G}$
GATE EE 2021      Mathematical Models of Physical Systems
Question 10 Explanation:
\begin{aligned} \frac{E(s)}{R(s)} &=\frac{R(s)-H \times C(s)}{R(s)}=1-H \times \frac{C(s)}{R(s)} \\ &=1-\frac{H G}{1+G H}=\frac{1+G H-G H}{1+G H} \\ &=\frac{1}{1+G H} \end{aligned}

There are 10 questions to complete.