# Control Systems

 Question 1
A stable real linear time-invariant system with single pole at p, has a transfer function $H(s)=\frac{s^2+100}{s-p}$ with a dc gain of 5. The smallest positive frequency, in rad/s at unity gain is closed to:
 A 8.84 B 11.08 C 78.13 D 122.87
GATE EE 2020      Frequency Response Analysis
Question 1 Explanation:
\begin{aligned} H(s)&=T.F.=[latex]\frac{s^{2}+100}{s-p} \\ \text{D.C. gain }&= 5 \\ \Rightarrow \; \; \frac{100}{-P}&=5=P=-20 \\ H(j\omega )&=\frac{-\omega ^{2}+100}{j\omega +20} \\ \left |H(j\omega ) \right |&=\frac{-\omega ^{2}+100}{\sqrt{\omega ^{2}+400}} \\ \frac{-\omega ^{2}+100}{\sqrt{\omega ^{2}+400}}&=1 \\ \Rightarrow \; \; \omega &=8.84 \text{rad/sec.} \end{aligned}
 Question 2
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      Time Response Analysis
Question 2 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 3
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      Time Response Analysis
Question 3 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 4
Consider a negative unity feedback system with forward path transfer function $G(s)=\frac{K}{(s+a)(s-b)(s+c)}$, where K, a, b, c are positive real numbers. For a Nyquist path enclosing the entire imaginary axis and right half of the s-plane in the clockwise direction, the Nyquist plot of $(1 + G(s))$, encircles the origin of $(1 + G(s))$-plane once in the clockwise direction and never passes through this origin for a certain value of K. Then, the number of poles of $\frac{G(s)}{1+G(s)}$ lying in the open right half of the s-plane is _________ .
 A 1 B 2 C 3 D 4
GATE EE 2020      Frequency Response Analysis
Question 4 Explanation:
\begin{aligned}O.L.T.F = G(s) &=\frac{K}{(s+a)(s-b)(s+c)} \\ N &= P - Z ; \; \;P = 1 \\ -1 &= 1 - Z ; \; \; N = -1 \\ Z &= 2\end{aligned}
 Question 5
Which of the options is an equivalent representation of the signal flow graph shown here?
 A A B B C C D D
GATE EE 2020      Mathematical Models of Physical Systems
Question 5 Explanation:
Simplifying given signal flow graph

 Question 6
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      Time Response Analysis
Question 6 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 7
Consider a state-variable model of a system
$\begin{bmatrix} \dot{x_1}\\ \dot{x_2} \end{bmatrix}=\begin{bmatrix} 0 &1 \\ -\alpha &-2\beta \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}+\begin{bmatrix} 0\\ \alpha \end{bmatrix}r$
$y=[1\;\;0]\begin{bmatrix} x_1\\ x_2 \end{bmatrix}$
where y is the output, and r is the input. The damping ratio $\xi$ and the undamped natural frequency $\omega _n$ (rad/sec) of the system are given by
 A $\xi =\frac{\beta }{\sqrt{\alpha }}; \; \omega _n=\sqrt{\alpha }$ B $\xi =\sqrt{\alpha }; \; \omega _n=\frac{\beta }{\sqrt{\alpha }}$ C $\xi =\frac{\sqrt{\alpha }}{\beta }; \; \omega _n=\sqrt{\beta }$ D $\xi =\sqrt{\beta }; \; \omega _n=\sqrt{\alpha }$
GATE EE 2019      State Variable Analysis
Question 7 Explanation:
Characteristic equation is,
$|sI-A|=0$
$|sI-A|=\begin{vmatrix} s &-1 \\ \alpha &s+2\beta \end{vmatrix}$
$\;\;=s^2+2s\beta +\alpha =0$
$\therefore \;\; \omega _n^2=\alpha$
$\;\;\; \omega _n=\sqrt{\alpha }$
$\;\;\;2\xi \omega _n=2\beta$
$\;\;\;\xi =\frac{\beta }{\sqrt{\alpha }}$
 Question 8
The transfer function of a phase lead compensator is given by
$D(s)=\frac{3\left ( s+\frac{1}{3T} \right )}{\left ( s+\frac{1}{T} \right )}$
The frequency (in rad/sec), at which $\angle D(j\omega )$ is maximum, is
 A $\sqrt{\frac{3}{T^2}}$ B $\sqrt{\frac{1}{3T^2}}$ C $\sqrt{3T}$ D $\sqrt{3T^3}$
GATE EE 2019      Design of Control Systems
Question 8 Explanation:
$T(s)=\frac{1+3Ts}{1+Ts}$
Frequency at which $\angle T(j\omega )$ is maximum,
$(\omega _m)=\frac{1}{T\sqrt{\alpha }}$
$\alpha =\frac{1}{1/3}=3$
$\omega _m=\frac{1}{T\sqrt{3}}=\sqrt{\frac{1}{3T^2}}$
 Question 9
The asymptotic Bode magnitude plot of a minimum phase transfer function G(s) is shown below.

Consider the following two statements.

Statement I: Transfer function G(s) has three poles and one zero.
Statement II: At very high frequency ($\omega \rightarrow \infty$), the phase angle $\angle G(j\omega)=-\frac{3\pi}{2}$.

Which one of the following options is correct?
 A Statement I is true and statement II is false. B Statement I is false and statement II is true. C Both the statements are true D Both the statements are false
GATE EE 2019      Frequency Response Analysis
Question 9 Explanation:
$G(s)=\frac{k}{s\left ( 1+\frac{s}{1} \right )\left ( 1+\frac{s}{20} \right )}$
Transfer function shows 2 poles and no zeros. So statement I is false.

$\angle G(j\omega )=-90-tan^{-1}\omega -tan^{-1}\frac{\omega }{20}$
$\angle G(j\omega )|_{\omega \rightarrow \infty }=-270^{\circ}=-\frac{3\pi}{2} \;rad$
So statement II is true.
 Question 10
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      Concepts of Stability
Question 10 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 $k \gt 0$
$\therefore \;\; 0 \lt k \lt \frac{8}{9}$

There are 10 questions to complete.