### GATE 2011 ECE - Control Systems - Complete Video Solutions

2011

1. The differential equation 100d2y/dt2-20dy/dt+y = x(t) describes a system with an input x(t) and an output y(t).The system which is initially relaxed, is excited by a unit step input. The output y(t) can be represented by the waveform

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

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

4. The block diagram of a system with one input u and two outputs y1 and y2 is given below.
A state space model of the system in terms of the state vector and the vector = [y1 y2]T is

5. If F(s) = L[f(t)] = 2(s+1)/(s2+4s+7), then the initial and final values of f(t) are respectively
a) 0,2
b) 2,0
c) 0,2/7
d) 2/7,0

Common Data Questions 6 & 7:
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.

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

7. 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

### GATE 2012 Control Systems - Complete Video Solutions

2012

1. The unilateral Laplace transform of f(t) is 1/(s2+s+1). The unilateral transform of t.f(t) is

2. With initial condition x(1) = 0.5, the solution of the differential equation, t.(dx/dt)+x = t is
a) x = t – 1/2
b) x = t2 – 1/2
c) x = t2/2
d) x = t/2

3. A system with transfer function is excited by sin(ωt).
The steady state output of the system is zero at

4. The feedback system shown below oscillates at 2 rad/sec, when

a) K = 2 and a = 0.75
b) K = 3 and a = 0.75
c) K = 4 and a = 0.5
d) K = 2 and a = 0.5

5. The state variable description of an LTI system is by

where y is the output and u is the input. The system is controllable for
a) a1 0, a2 = 0, a3 = 0
b) a1 = 0, a2 0, a3 0
c) a1 = 0, a2 0, a3 = 0
d) a1 0, a2 0, a3 = 0

The transfer function of a compensator is given as Gc(s)= (s+a)/(s+b).

6. Gc(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

7. The phase of the above lead compensator is maximum at

### GATE 2013 Control Systems - Complete Video Solutions

2013

1. The Bode plot of a transfer function G(s) is shown in the figure below.

The gain (20log|G(s)|) is 32 dB and – 8 dB at 1 rad/sec and 10 rad/sec respectively. The phase is negative for all ω. Then G(s) is
a) 39.8/s
b) 39.8/s2
c) 32/s
d) 32/s2

2. Which one of the following is NOT TRUE for a continuous time causal and stable LTI system?
a) all the poles of the system must lie on the left side of the jω axis
b) zeros of the system can lie anywhere in the s-plane
c) all the poles must lie with in |s| = 1
d) all the roots of the characteristic equation must be located on the let side of the jω axis

3. A polynomial f(x) = a4x4+a3x3+a2x2+a1xa0 with all coefficients positive has
a) no real roots
b) no negative real root
c) odd number of real roots
d) at least one positive and one negative real root

Solution :

4. Assuming zero initial conditions, the response y(t) of the system given below for a unit step input, u(t) is __________________

a) u(t)
b) t.u(t)
c) (1/2)t2.u(t)
d) e-t.u(t)

5. The transfer function V2(s)/V1(s) of the circuit shown is

6. The open loop transfer function of a DC motor is given as ω(s)/Va(s) = 10/(1+10s). when connected in feedback as shown below.
The approximate value of Ka that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open loop system is
a) 1
b) 5
c) 10
d) 100

7. A system is described by the differential equation d2y/dt2+5dy/dt+6y(t) = x(t). Let x(t) be a rectangular pulse given as x(t) = 1 for 0<t<2 and zero otherwise. Assuming that y(0) = 0 and dy/dt = 0 at t=0, then the Laplace transform of y(t) is

8. The signal flow graph for a system is given below.
The transfer function Y(s)/U(s) for this system is

9. The state diagram of the system is shown below. A system is described by the state variable equations

a. State variable equations of the system shown in the above figure is

b. The state transition matrix eAt of the system shown in the figure above is

Solution (a & b) :     https://www.youtube.com/watch?v=wnfqGfnPFcs

### Control Systems - GATE Paper 2015 ECE Video Solutions

2015

1. The polar plot of the transfer function G(s) = 10(s+1)/(s+10) for 0 < ω < will be in the

2. 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 __________________

3. 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 ___________

4. A plant transfer function is given as
When the plant operates in a unity feedback configuration, the condition for the stability of the closed loop system is
a) KP > KI/2 > 0
b) 2KI > KP > 0
c) 2KI < KP
d) 2KI > KP

5. 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 _________.

6. 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 _____________

7. For the signal flow graph shown in the figure, the value of C(s)/R(s) is

8. By performing cascading and/or summing/differencing operations using transfer function blocks G1(s) and G2(s), one CANNOT realize a transfer function of the form

9. The transfer function of a mass-spring-damper system is given by
the frequency response data for the system are given in the following table.
The unit step response of the system approaches a steady state value of _____________

10. The state variable representation of a system is given as
The response y(t) is
a) sin (t)
b) 1 et
c) 1 – cos (t)
d) 0

11. The output of a standard second order system for a unit step input is given as
The transfer function of the system is

12. Consider the differential equation dx/dt = 10 – 0.2x with initial condition x(0) = 1. The response x(t) for t > 0 is
a) 2 – e-0.2t
b) 2 – e0.2t
c) 50 – 49e-0.2t
d) 50 – 49e0.2t

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

14. Consider the Bode plot shown in figure.
Assume that all the poles and zeros are real valued. The value of fHfL (in Hertz) is ____________________

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

16. The characteristic equation of an LTI system is given by F(s) = s5 + 2s4 + 3s3 + 6s2 – 4s – 8 = 0. The number of roots that lie strictly in the left half S – plane is __________.

17. A network is described by the state model as
then the transfer function H(s) = (Y(s)/U(s)) is

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

19. The position control of a DC servo-motor is given in the figure. The values of the parameters are KT = 1 N-mA, Ra = 1 ohm, La = 0.1 H, J = 5 kg-m2, B = 1 N-m/(rad/sec) and Kb = 1 volt/(rad/sec). The steady state position response(in radians) due to unit impulse disturbance torque Td is _______.

### Control Systems - GATE 2014 ECE Video Solutions

2014

1. The forward path transfer function of a unity feedback system is given by
The value of K which will place both the poles of the closed loop system at the same location is
_________________

2. 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

3. In the circuit shown, the value of capacitor C (in mF) needed to have critically damped response i(t) is ______________________

4. A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system.
When y(0) = 1 and u(t) is a unit step function, y(t) is

5. consider the state space model of a system, as given below.
The given system is
a) controllable and observable
b) uncontrollable and observable
c) uncontrollable and unobservable
d) controllable and unobservable

6. The phase margin in degrees of

calculated using the asymptotic Bode plot is ____________

7. For the following feedback system
The 2% settling time of the step response is required to be less than 2 seconds.
Which one of the following compensators C(s) achieves this?

8. The natural frequency of an undamped second order system is 40 rad/sec. If the system is damped with a damping ratio 0.3, the damped natural frequency in rad/sec is ______________

9. For the system shown,
when the X1(s) = 0, the transfer function Y(s)/X2(s) is

10. An unforced linear time invariant (LTI) system is represented by
If the initial conditions are x1(0) = 1 and x2(0) = - 1, then the solution of the state equation is

11. The Bode asymptotic magnitude plot of a minimum phase system is shown in the figure.
If the system is connected in a unity negative feedback configuration, the steady state error of the closed loop system, to a unit ramp input is _____________

12. Consider the state space system expressed by the signal flow diagram shown in the figure.
The corresponding system is
a) always controllable
b) always observable
c) always stable
d) always unstable

13. Consider the following block diagram in the figure.
The transfer function C(s)/R(s) is

14. The input -3e2tu(t), where u(t) is the unit step function, is applied to a system with transfer function (S-2)/(S+3). If the initial value of the output is -2, then the value of the output at steady state is _____________

15. The steady state error of the system shown in the figure for a unit step input is ___________

16. The state equation of a second order linear system is given by

17. 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?

18. In a Bode magnitude plot, which one of the following slopes would be exhibited at high frequencies by a 4th order all pole system?

19. For the second order closed loop system shown in the figure,
the natural frequency (in rad/sec) is
a) 16
b) 4
c) 2
d) 1

20. The state transition matrix Φ(t) of a system,

21. Consider a transfer function
with p a positive real parameter. The maximum value of p until which Gp remains stable is ______________

22. 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 ____________

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