### Time Response & Transfer Function of a System - Topic wise GATE Questions on Control Systems (from 2003)

2003
1. The driving point impedance Z(s) of a network has the pole-zero locations as shown in figure. If Z(0) = 3, then Z(s) is

2. A second order system has the transfer function C(s)/R(s) = 4 / [s2 + 4s + 4] with r(t) as the unit step function, the response c(t) of the system is represented by

2004
3. The transfer function H(s) of an RLC circuit is given as 106/[s2+20s+106], then the Q – factor of this circuit is
a) 25
b) 50
c) 100
d) 5000

4. For the circuit shown in figure, the initial conditions are zero. Its transfer function Vo(t)/Vi(t) is

2005
5. In the derivation of expression for peak percent overshoot, which one of the following conditions is NOT required?
a) system is linear and time invariant
b) system transfer function has a pair of complex conjugate poles and no zeros
c) there is no transportation delay in the system
d) the system has zero initial conditions

6. A double integrator plant, G(s) = K/S2, H(s) = 1 is to be compensated to achieve the damping ratio ξ = 0.5, and an undamped natural frequency ωn = 5 rad/sec. Which one of the following compensator Gc(s) will be suitable?

2006
7. The unit step response of a system starting from rest is given by c(t) = 1 – e–2t for t ≥ 0. The transfer function of the system is
a) 1 / (1 + 2s)
b) 2 / (2 + s)
c) 1 / (2 + s)
d) 2s / (1 + 2s)

8. The unit impulse response of a system is h(t) = e–t, t ≥ 0. For this system, the steady state value of the output for unit step input is equal to
a) –1
b) 0
c) 1
d) ∞

2007
9. If the closed loop transfer function of a control system is given as T(s) = (s – 5)/[(s + 2)(s + 3)], then it is
a) an unstable system
b) an uncontrollable system
c) a minimum phase system
d) a non-minimum phase system

10. The transfer function of a plant is T(s) = 5/[(s + 5)(s2 + s + 1)]. The second order approximation of T(s) using dominant pole concept is

2008
11. The pole-zero plot given below corresponds to a
a) Low pass filter
b) High pass filter
c) band pass filter
d) Notch filter

12. Step response of a set of three second order under damped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems?

13. Group – I lists a set of four transfer functions. Group – II gives a list of possible step responses y(t). Match the step responses with the corresponding transfer functions.
a) P - 3, Q - 1, R - 4, S - 2
b) P - 3, Q - 2, R - 4, S - 1
c) P - 2, Q - 1, R - 4, S - 3
d) P - 3, Q - 4, R - 1, S – 2

14. A certain system has transfer function G(s) = (s + 8)/(s2 + αs – 4 ), where α is a parameter. Consider the standard negative unity feedback configuration shown below.
Which of the following statements is true?
a) The closed loop system is never stable for any value of α
b) For some positive values of α, the closed loop system is stable, but not for all positive values
c) For all positive values of α, the closed loop system is stable
d) The closed loop system is stable for all values of α, both positive and negative

15. The magnitude of frequency response of an under damped second order system is 5 at 0 rad/sec and peaks to 10/√3 at 5√2 rad/sec. The transfer function of the system is

2009
16. If the transfer function of the following circuit is Vo(s)/Vi(s) = 1/(2+sCR), then the value of the load resistor RL is
a) R/4
b) R/2
c) R
d) 2R

2010
17. The transfer function Y(s)/R(s) of the system shown is

18. A system with transfer function Y(s)/X(s) = s/(s + p) has an output y(t) = cos(2t – π/3) for the input signal x(t) = p cos(2t – π/2). Then, the system parameter 'p' is
a) √3
b) 2/√3
c) 1
d) √3/2

2011
19. 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

2012
20. A system with transfer function is excited by sin(ωt).
The steady state output of the system is zero at

2013
21. 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

22. A polynomial f(x) = a4x4+a3x3+a2x2+a1x–a0 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 :

23. 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)

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

25. 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

2014
26. 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
_________________

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

28. 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?

29. 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 ______________

30. 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

2015
31. 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 _____________

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

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