GATE 2023 - Important Concepts, Short cuts, Complete Solutions, Examples and Practice Problems

Root Locus Technique - Topic wise GATE Questions on Control Systems (from 1987 onwards)

2003
1. The root locus of the system G(s)H(s) = K / [s(s + 2)(s + 3)] has the break away point located at
a) (-0.5, 0)
b) (-2.548, 0)
c) (-4, 0)
d) (-0.748, 0)

Solution : https://www.youtube.com/watch?v=f_IljSih2eU

2004
2. Given G(s)H(s) = K /[s(s + 1)(s + 3)], the point of intersection of the asymptotes of the root loci with the real axis is
a) - 4
b) 1.33
c) - 1.33
d) 4

Solution : https://www.youtube.com/watch?v=1eQ1YNdEF_4

2005
3. A unity feedback system is given as G(s) = k(1 – s) / [s(s + 3)]. Indicate the correct root locus diagram.

Solution : https://www.youtube.com/watch?v=cCAfe9ydiU8

2007
4. A unity feedback control system has an open loop transfer function G(s) = – K/[s(s² + 7s + 12)]. The gain k for which s = –1+j1 will lie on the root locus of this system is
a) 4
b) 5.5
c) 6.5
d) 10

Solution : https://www.youtube.com/watch?v=ppJqQhOp_xk

2009
5. the feedback configuration and the pole-zero locations of G(s) = (s² – 2s + 2)/(s² + 2s +2), are shown below.

The root locus for negative values of k, i.e. for –∞ < k < 0, has break away/break in points and angle of departure at pole P (w.r.t. the positive real axis) equal to
a) ± √2 and 0^o
b) ± √2 and 45^o
c) ± √3 and 0^o
d) ± √3 and 45^o

Solution : https://www.youtube.com/watch?v=uagcsFAoEB0

2011
6. The root locus plot for a system is given below. The open loop transfer function corresponding to this plot is given by

Solution : https://www.youtube.com/watch?v=ARZuGJ_JqBo

2014
7. In the root locus plot shown in the figure, the pole/zero marks and the arrows have been removed. Which one of the following tranfer functions has this root locus?

Solution : https://www.youtube.com/watch?v=3X9gdDGti0Q
8. The characteristic equation of a unity negative feedback system is 1+KG(s) = 0. The open loop transfer function G(s) has one pole at zero and two poles at – 1. The root locus of the system for varying K is shown in the figure.

The constant damping ratio line, for ζ = 0.5, intersects the root locus at point A. The distance from the origin to point A is given as 0.5. The value of K at point A is ____________

Solution : https://www.youtube.com/watch?v=hEIZpN9XAPM

2015
9. A unity negative feedback system has the open loop transfer function

the value of the gain K (>0) at which the root locus crosses the imaginary axis is __________________

Solution : https://www.youtube.com/watch?v=taFVjNfxJ_Q

10. The open loop transfer function of a unity feedback configuration is given as

The value of a gain K( >0 ) for which -1+j2 lies on the root locus is _________.

Solution : https://www.youtube.com/watch?v=IhF_FZ9IJvI

11. A unity negative feedback system has an open loop transfer function G(s) = K / S(S+10), the gain K for the system to have a damping ratio of 0.25 is _____________

Solution : https://www.youtube.com/watch?v=LCcmOh0C1rY

12. For the system shown in figure, s = - 2.75 lies on the root locus if K is ____________

Solution : https://www.youtube.com/watch?v=mhCsqV9Gd2g

Polar Plot and Nyquist Plot - Topic wise GATE previous questions on Control Systems (from 1987 onwards)

2003
1. Figure shows the Nyquist plot of the open loop transfer function G(s)H(s) of a system. If G(s)H(s) has one right hand pole, the closed loop system is

a) always stable
b) unstable with one closed loop right hand pole
c) unstable with two closed loop right hand poles
d) unstable with three closed loop right hand poles

Solution : https://www.youtube.com/watch?v=DuipmIw_dtI

2. The gain margin and the phase margin of a feedback system with G(s)H(s) = s/(s + 100)³are
a) 0 dB, 0^o
b) ∞, ∞
c) ∞, 0^o
d) 88.5 dB, ∞

Solution : https://www.youtube.com/watch?v=n44i9T5koz4

2004
3. The gain margin for the system with open loop transfer function G(s)H(s) = 2(1 + s)/s², is
a) ∞
b) 0
c) 1
d) - ∞

Solution : https://www.youtube.com/watch?v=L2Y_rS_Zc9M

2005
4. Which one of the following polar diagrams corresponds to a lag network?

Solution : https://www.youtube.com/watch?v=essJrzYsfPg

5. The polar diagram of a conditionally stable system for open loop gain K = 1 is shown in figure. The open loop transfer function of the system is known to be stable. The closed loop system is stable for

Solution : https://www.youtube.com/watch?v=1mBpd1vOX78

2006
6. The Nyquist plot of G(jω)H(jω) for a closed loop control system, passes through ( –1, j0) point on the GH – plane. The gain margin of the system in dB is equal to
a) infinite
b) greater than zero
c) less than zero
d) zero

Solution : https://www.youtube.com/watch?v=skWe48mXRqg

2009
Common Data Questions: (7 & 8)
The Nyquist plot of a stable transfer function G(s) is shown in the figure. We are interested in the stability of the closed loop system in the feedback configuration shown.

7. Which of the following statements is TRUE?
a) G(s) is an all pass filter
b) G(s) has a zero in the right half plane
c) G(s) is the impedance of a passive network
d) G(s) is marginally stable

Solution : https://www.youtube.com/watch?v=2t0EyW64kfI

8. The gain and phase margin of G(s) for closed loop stability are
a) 6 dB and 180^o
b) 3 dB and 180^o
c) 6 dB and 90^o
d) 3 dB and 90^o

Solution : https://www.youtube.com/watch?v=RlFKoEWXgw8

2011
9. for the transfer function G(jω) = 5 + jω, the corresponding Nyquist plot for positive frequency has the form

Solution : https://www.youtube.com/watch?v=F0oR9sCCy80

2014
10. Consider the feedback system shown in the figure. The Nyquist plot of G(s) is also shown. Which of the following conclusions is correct?

a) G(s) is an all pass filter
b) G(s) is a strictly proper transfer function
c) G(s) is a stable and minimum phase transfer function
d) The closed loop system is unstable for sufficiently large and positive k

Solution : https://www.youtube.com/watch?v=ujZuye5co_M

2015
11. The polar plot of the transfer function G(s) = 10(s+1)/(s+10) for 0 < ω < ∞ will be in the
a) first quadrant
b) second quadrant
c) third quadrant
d) fourth quadrant

Solution : https://www.youtube.com/watch?v=-2VWgQ6uM1I

Gain Margin and Phase Margin - Topic wise Questions in Control Systems (from 1987 onwards)

2003
1. The gain margin and the phase margin of a feedback system with G(s)H(s) = s/(s + 100)³are
a) 0 dB, 0^o
b) ∞, ∞
c) ∞, 0^o
d) 88.5 dB, ∞

Solution : https://www.youtube.com/watch?v=n44i9T5koz4

2004
2. The gain margin for the system with open loop transfer function G(s)H(s) = 2(1 + s)/s², is
a) ∞
b) 0
c) 1
d) - ∞

Solution : https://www.youtube.com/watch?v=L2Y_rS_Zc9M

2005
Statement for Linked Answer Questions 3 & 4:
The open loop transfer function of a unity feedback is given by G(s) = 3e^–2s/ [s(s + 2)].
3. The Gain and Phase cross over frequencies in rad/sec are, respectively
a) 0.932 and 1.26
b) 0.632 and 0.485
c) 0.485 and 0.632
d) 1.26 and 0.632

Solution : https://www.youtube.com/watch?v=eKZo_zPxjYI

4. based on the above results, the Gain and Phase Margins of the system will be
a) –7.09 dB and 87.5^o
b) 7.09 dB and 87.5^o
c) 7.09 dB and –87.5^o
d) –7.09 dB and –87.5^o

Solution : https://www.youtube.com/watch?v=aHVPuRMM0q8

2006
5. The open loop transfer function of a unity gain feedback control system is given by G(s) = K/[(s + 1)(s + 2)]. The gain margin of the system in dB is given by
a) 0
b) 1
c) 20
d) ∞

Solution : https://www.youtube.com/watch?v=_GEVQWlLBOs

Statement for Linked Answer questions: (6 and 7)
Consider a unity gain feedback control system whose open loop transfer function is G(s) = (as + 1)/s².
6. The value of "a" so that the system has a phase margin equal to π/4 is approximately equal to
a) 2.40
b) 1.40
c) 0.84
d) 0.74

7. With the value of "a" set for phase margin of π/4, the value of unit impulse response of the open loop system at t = 1 second is equal to
a) 3.40
b) 2.40
c) 1.84
d) 1.74

Solution (10 & 11) : https://www.youtube.com/watch?v=DlzZDVlirog

2011
Common Data Questions 8 & 9:
The input – output transfer function of a plant H(s) = 100/s(s+10)². The plant is placed in a unity negative feedback configuration as shown in figure below.

8. The signal flow graph that DOES NOT model the plant transfer function H(s) is

Solution : https://www.youtube.com/watch?v=gSg5jPv06fQ

9. The gain margin of the system under closed loop unity negative feedback is
a) 0 dB
b) 20 dB
c) 26 dB
d) 46 dB

Solution : https://www.youtube.com/watch?v=HreyReFPqag

2012
10. A system with transfer function is excited by sin(ωt).

The steady state output of the system is zero at
a) ω = 1 rad/sec
b) ω = 2 rad/sec
c) ω = 3 rad/sec
d) ω = 4 rad/sec

Solution : https://www.youtube.com/watch?v=UYEEuh71lxc

2014
11. The phase margin in degrees of

calculated using the asymptotic Bode plot is ____________

Solution : https://www.youtube.com/watch?v=ZwdvClI0_mU

2015
12. The phase margin (in degrees ) of the system G(s) = 10 /s(s+10) is _____________

Solution : https://www.youtube.com/watch?v=WXHKVhdLeOc

P, PD and PID controllers & Lead and Lag Compensators - Topic wise Questions in Control Systems ( from 1987)

2003
1. A PD controller is used to compensate a system. Compared to the uncompensated system, the compensated system has
a) a higher type number
b) reduced damping
c) higher noise amplification
d) larger transient overshoot

Solution : https://www.youtube.com/watch?v=flVQh4IItX4

2005
2. Which one of the following polar diagrams corresponds to a lag network?

Solution : https://www.youtube.com/watch?v=essJrzYsfPg

2006
3. The transfer function of a phase lead compensator is given by G_c(s) = (1 + 3Ts)/(1 + Ts) where T ≥ 0. The maximum phase shift provided by such a compensator is
a) π/2
b) π/3
c) π/4
d) π/6

Solution : https://www.youtube.com/watch?v=B6_LDoAZ3-I

2007
4. A control system with a PD controller is shown in the figure.

If the velocity error constant K_v = 1000 and the damping ratio ξ = 0.5, then the values of K_p and K_D are
a) K_p = 100, K_D = 0.09
b) K_p = 100, K_D = 0.9
c) K_p = 10, K_D = 0.09
d) K_p = 10, K_D = 0.9

Solution : https://www.youtube.com/watch?v=vxHx4zTs5vg

5. The open loop transfer function of a plant is given as G(s) = 1/(s² – 1). If the plant is operated in a unity feedback configuration, then the lead compensator that can stabilize this control system is

Solution : https://www.youtube.com/watch?v=HJ09FtxcO-I

2008
6. Group – I gives two possible choices for the impedance Z in the diagram. The circuit elements in Z satisfy the condition R₂C₂>R₁C₁. The transfer function V_o/V₁ represents a kind of controller. Match the impedance's in Group – I with the types of controllers in Group – II.

a) Q – 1, R – 2
b) Q – 1, R – 3
c) Q – 2, R – 3
d) Q – 3, R – 2

Solution : https://www.youtube.com/watch?v=Q3O1G2CU5Xw

2009
7. The magnitude plot of a rational transfer function G(s) with real coefficients is shown below. Which of the following compensator has such a magnitude plot?

a) Lead compensator
b) Lag compensator
c) PID compensator
d) Lead – Lag compensator

Solution : https://www.youtube.com/watch?v=JVBUSrYEb7Q

8. If the transfer function of the following circuit is V_o(s)/V_i(s) = 1/(2+sCR), then the value of the load resistor R_L is

a) R/4
b) R/2
c) R
d) 2R

Solution : https://www.youtube.com/watch?v=0bJX_BCy6bc

2010
9. A unity negative feedback closed loop system has a plant with the transfer function G(s) = 1/(s² + 2s + 2) and a controller G_c(s) in the feed forward path. For a unit step input, the tranfer function of the controller that gives minimum steady state error is

Solution : https://www.youtube.com/watch?v=zdwjtNkiBxg

2012
Statement for Linked Answer Questions : 10 and 11
The transfer function of a compensator is given as G_c(s)= (s+a)/(s+b).
10. G_c(s) is a lead compensator if
a) a = 1, b =2
b) a = 3, b =2
c) a = –3, b = –1
d) a = 3, b =1

11. The phase of the above lead compensator is maximum at
a) √2 rad/sec
b) √3 rad/sec
c) √6 rad/sec
d) 1/√3 rad/sec

Solution (6 & 7) : https://www.youtube.com/watch?v=CSWEFDPoSsA

2015
12. A lead compensator network includes a parallel combination fo R and C in the feed-forward path. If the transfer function of the compensator is GC(s) = (s+2)/(s+4) , the value of RC is ___________

Solution : https://www.youtube.com/watch?v=DeKH-DPUdMo

13. The transfer function of a first order controller is given as Gc(s) = K(s+a) / (s+b), where K, a and b are positive real numbers. The condition for this controller to act as a phase lead compensator is
a) a < b
b) a > b
c) K < ab
d) K > ab

Solution : https://www.youtube.com/watch?v=uKr3VkkMkow

Root Locus Technique - Topic wise GATE Questions on Control Systems (from 1987 onwards)

Polar Plot and Nyquist Plot - Topic wise GATE previous questions on Control Systems (from 1987 onwards)

Gain Margin and Phase Margin - Topic wise Questions in Control Systems (from 1987 onwards)

P, PD and PID controllers & Lead and Lag Compensators - Topic wise Questions in Control Systems ( from 1987)

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