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.
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