# Frequency Response Analysis

 Question 1
The pole-zero map of a rational function G(s) is shown below. When the closed counter $\Gamma$ is mapped into the G(s)-plane, then the mapping encircles.
 A the origin of the G(s)-plane once in the counter-clockwise direction. B the origin of the G(s)-plane once in the clockwise direction. C the point -1+j0 of the G(s)-plane once in the counter-clockwise direction. D the point -1+j0 of the G(s)-plane once in the clockwise direction.
GATE EC 2020   Control Systems
Question 1 Explanation:
s-plane contour is encircling 2-poles and 3-zeros in clockwise direction hence the corresponding G(s) plane contour encircles origin 2-times in anti-clockwise direction and 3-times in clockwise direction.
Therefore, Effectively once in clockwise direction.
 Question 2
For an LTI system, the Bode plot for its gain is as illustrated in the figure shown. The number of system poles $N_p$ and the number of system zeros $N_z$ in the frequency range $1Hz\leq f\leq 10^7Hz$ is
 A $N_p=5,N_z=2$ B $N_p=6,N_z=3$ C $N_p=7,N_z=4$ D $N_p=4,N_z=2$
GATE EC 2019   Control Systems
Question 2 Explanation:

Number of poles $(N_{P})$= 6
Number of zeros $(N_{Z})$ = 3
 Question 3
The figure below shows the Bode magnitude and phase plots of a stable transfer function $G(s)=\frac{n_{o}}{s^{2}+d_{2}s^{2}+d_{1}+d_{0}}$
Consider the negative unity feedback configuration with gain k in the feedforward path. The closed loop is stable for $k \lt k_{o}$. The maximum value of $k_{o}$ is ______.
 A 0.1 B 0.2 C 0.3 D 0.4
GATE EC 2018   Control Systems
Question 3 Explanation:
For G(s)
$M_{\mathrm{dB}}\left(\omega_{p c}\right)=20 \mathrm{dB}$
\begin{aligned} \mathrm{GM}_{\mathrm{dB}} &=-20 \mathrm{dB}-20 \log _{10}(k) \gt 0 \mathrm{dB} \\ 20+20 \log _{10}(k) & \lt 0 \\ 20 \log _{10}(k) & \lt -20 \\ k & \lt 10^{-1}=0.10\\ \text{So,}\quad k_{0}&=0.10 \end{aligned}
 Question 4
For a unity feedback control system with the forward path transfer function

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

The peak resonant magnitude $M_{r}$, of the closed-loop frequency response is 2. The corresponding value of the gain K (correct to two decimal places) is _________.
 A 5.38 B 14.92 C 20.48 D 25.84
GATE EC 2018   Control Systems
Question 4 Explanation:
Maximum resonant peak,
\begin{aligned} M_{r} &=\frac{1}{2 \xi \sqrt{1-\xi^{2}}}=2 \\ 2 \xi \sqrt{1-\xi^{2}} &=\frac{1}{2} \\ \xi^{2}\left(1-\xi^{2}\right) &=\frac{1}{16} \\ \xi^{4}-\xi^{2}+\frac{1}{16} &=0 \\ \xi^{2}&=\frac{1}{2} \pm \sqrt{\frac{1-\frac{1}{4}}{4}}=\frac{1}{2} \pm \frac{\sqrt{3}}{4} \\ \text { As } M_{r}=2 \gt 1, \xi& \lt \frac{1}{\sqrt{2}} \text { and } \xi^{2} \lt \frac{1}{2} \\ \text { So. } \quad\xi^{2}&=\frac{1}{2}-\frac{\sqrt{3}}{4}\\ Given, \quad G(s)&=\frac{K}{s(s+2)}=\frac{\omega_{n}^{2}}{s\left(s+2 \xi \omega_{n}\right)} \\ \text{So} \quad\omega_{n} &=\sqrt{K} \\ 2 \xi \sqrt{K} &=2\\ \sqrt{K} &=\frac{1}{\xi} \\ K &=\frac{1}{\xi^{2}}=\frac{1}{\left(\frac{1}{2}-\frac{\sqrt{3}}{4}\right)} \\ &=\frac{4}{2-\sqrt{3}}=14.928 \end{aligned}
 Question 5
The Nyquist stability criterion and the Routh criterion both are powerful analysis tools for determining the stability of feedback controllers. Identify which of the following statements is FALSE:
 A Both the criteria provide information relative to the stable gain range of the system. B The general shape of the Nyquist plot is readily obtained from the Bode magnitude plot for all minimum-phase systems. C The Routh criterion is not applicable in the condition of transport lag, which can be readily handled by the Nyquist criterion. D The closed-loop frequency response for a unity feedback system cannot be obtained from the Nyquist plot.
GATE EC 2018   Control Systems
 Question 6
A unity feedback control system is characterized by the open-loop transfer function
$G(s)=\frac{10K(s+2)}{s^{3}+3s^{2}+10}$
The Nyquist path and the corresponding Nyquist plot of G(s) are shown in the figures below.

If $0 \lt K \lt 1$, then the number of poles of the closed-loop transfer function that lie in the right half of the s-plane is
 A 0 B 1 C 2 D 3
GATE EC 2017-SET-2   Control Systems
Question 6 Explanation:
Given that, the open loop transfer function of a unity feedback system is,
$G(s) =\frac{10 k(s+2)}{s^{3}+3 s^{2}+10}$
From the given Nyquist plot, for $0 \lt k \lt 1$ the encirclements about the point (-1+j 0) is,
$\begin{array}{l} N=0 \\ N=P-Z \end{array}$
P= number of open loop poles in the right half of s-plane
Z= number of closed loop poles in the right half of s-plane
To determine the value of P :
Applying R-H criteria to G(s)
$G(s)=\frac{10 k(s+2)}{s^{3}+3 s^{2}+10}$
$\begin{array}{c|cc} s^{3} & 1 & 0 \\ s^{2} & 3 & 10 \\ s^{1} & -10 / 3 & 0 \\ s^{0} & 10 & 0 \end{array}$
Two sign changes are there in the first column of the RH table. So, two open loop poles are there in right half of s-plane.
So,
$P=2$
$\begin{array}{l} N=P-Z \\ 0=2-Z \\ Z=2 \end{array}$
Z=2 indicates, there are two closed loop poles in right half of s-plane.
 Question 7
The Nyquist plot of the transfer function

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

does not encircle the point (1+j0) for K=10 but does encircle the point (-1+j0) for K=100. Then the closed loop system (having unity gain feedback) is
 A stable for K = 10 and stable for K = 100 B stable for K = 10 and unstable for K = 100 C unstable for K = 10 and stable for K = 100 D unstable for K = 10 and unstable for K = 100
GATE EC 2017-SET-1   Control Systems
Question 7 Explanation:
Given that,
$G(s)=\frac{K}{\left(s^{2}+2 s+2\right)(s+2)}$
P= open loop poles in RHS of s-plane
Z= closed loop poles in RHS of s-plane
N= number of encirclements about the
$\begin{array}{l} \text { point }(-1+j 0) \\ \text { For } K=10 ; \\ N=P-Z=0 \Rightarrow Z=0: \text { stable } \\ \text { For } K=100 ; \\ N=P-Z=1 \Rightarrow Z \neq 0: \text { unstable } \end{array}$
 Question 8
Consider a stable system with transfer function
$G(s)=\frac{s^{p}+b_{1}S^{p-1}+....+b_p}{s^{q}+a_{1}S^{q-1}+....+a_{q}}$

where $b_{1},...b_{p}$ and $a_{1},...a_{q}$ are real valued constants. The slope of the Bode log magnitude curve of G(s) converges to -60 dB/decade as $\omega \rightarrow \infty$ . A possible pair of values for p and q is
 A p=0 and q=3 B p=1 and q=7 C p=2 and q=3 D p=3 and q=5
GATE EC 2017-SET-1   Control Systems
Question 8 Explanation:
Final slope = -60 dB/decade, which indicates that, q-p=3
Among the given options, option (A) satisfies this condition
 Question 9
The asymptotic Bode phase plot of $G(s)=\frac{1}{(s+0.1)(s+10)(s+p_{1})}$, with k and $p_{1}$ both positive, is shown below. The value of $p_{1}$ is ________
 A 0.5 B 1 C 1.5 D 2.5
GATE EC 2016-SET-2   Control Systems
Question 9 Explanation:
\begin{aligned} G(s)=\frac{K}{(s+0.1)(s+10)\left(s+p_{1}\right)} \\ \angle G(s)=-\tan ^{-1} \frac{\omega}{0.1}-\tan ^{-1} \frac{\omega}{10}-\tan ^{-1} \frac{\omega}{p_{1}} \\ \left.\angle G(s)\right|_{\omega=1}=-\tan ^{-1} \frac{1}{0.1}-\tan ^{-1} \frac{1}{10}-\tan ^{-1} \frac{1}{p_{1}} \\ =-135^{\circ} \\ -\tan ^{-1} 10-\tan ^{-1} 0.1-\tan ^{-1} \frac{1}{p_{1}}=-135^{\circ} \\ -84.28^{\circ}-5.71^{\circ}-\tan ^{-1} \frac{1}{p_{1}}=-135^{\circ} \\ -\tan ^{-1} \frac{1}{p_{1}}=-135^{\circ}+90^{\circ}\\ \tan ^{-1} \frac{1}{p_{1}} =45^{\circ} \\ \frac{1}{p_{1}} =1 \Rightarrow p_{1}=1 \end{aligned}
 Question 10
In the feedback system shown below $G(s)=\frac{1}{(s+1)(s+2)(s+3)}$

The positive value of k for which the gain margin of the loop is exactly 0 dB and the phase margin of the loop is exactly zero degree is ________
 A 40 B 50 C 60 D 70
GATE EC 2016-SET-2   Control Systems
Question 10 Explanation:
$1+G(s) H(s)=1+\frac{K}{(s+1)(s+2)(s+3)}=0$
$(s+1)\left(s^{2}+5 s+6\right)+K=0$
$s^{3}+5 s^{2}+6 s+s^{2}+5 s+6+k=0$
$\Rightarrow s^{3}+6 s^{2}+11 s+6+K=$
Gain margin =0dB and phase margin $=0^{\circ}$
It implies marginal stable system
By Routh Array
$\begin{array}{r|rr} s^{3} & 1 & 11 \\ s^{2} & 6 &(6+k) \\ s & \frac{66-6-K}{6} & 0 \\ s^{\circ} & 6+K & \end{array}$
For marginal stable system,
$60-K=0$
$K=60$
There are 10 questions to complete.