### 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

### Steady State Error & Error Constants (Kp, Kv , Ka) - Topic wise GATE Questions on Control Systems (from 2003)

2005
1. A ramp input applied to a unity feedback system results in 5% steady state error. The Type number and zero frequency gain of the system are respectively
a) 1 and 20
b) 0 and 20
c) 0 and 1/20
d) 1 and 1/20

2009
2. The unit step response of an under damped second order system has steady state value of – 2. Which of the following transfer functions has these properties?

2014
3. The steady state error of the system shown in the figure for a unit step input is ___________

2015
4. 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 _______.

### State Space Analysis & State Transition Matrix (STM) - Topic wise GATE Questions on Control Systems (from 1987)

2003
1. The zero input response of a system given by the state space equation

2004
2. The state variable equations of a system are:
1 = -3x1 -x2 + u
2 = 2x1
y = x1 + u
Then the system is
a) controllable but not observable
b) observable but not controllable
c) neither controllable nor observable
d) controllable and observable

3. The state transition matrix eAt for the given matrix A, is

2005
4. A linear system is equivalently represented by two sets of state equations
Ẋ = AX + BU and Ẇ = CW + DU.
The eigen values of the representations are also computed as |λ| and |μ|. Which one of the following statements is TRUE?
Solution :

2006
5. A linear system is described by the following state equation, then the state transition matrix of the system is

2007
6. The state space representation of a separately excited DC servo motor dynamics is given as
where ω is the speed of the motor, ia is the armature current and u is the armature voltage. The transfer function of the motor is

7. The eigen value and eigen vector pairs (λi, vi) for the system are

Solution :
8. The system matrix A is

Solution :

2008
9. A signal flow graph of a system is given below.
The set of equations that correspond to this signal flow graph is

2009
10. consider the system,
where p and q are arbitrary real numbers, which one of the following statements about the controllability of the system is TRUE?
a) the system is completely state controllable for any non zero values of p and q
b) only p=0 and q=0 result in controllability
c) the system is uncontrollable for all values of p and q
d) we cannot conclude about controllability from the given data

2010
Common Data Questions:(11 and 12)
The signal flow graph of a system is shown below.
11. The state variable representation of the system can be

12. The transfer function of the system is

2011
13. 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 ẋ and the vector ẏ = [y1 y2]T is

2012
14. 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

2013
15. The signal flow graph for a system is given below.

The transfer function Y(s)/U(s) for this system is

16. 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

2014
17. 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

18. 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

19. 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

20. 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

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

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

2015
23. 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

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