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

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

Solution : https://www.youtube.com/watch?v=cmSIdjxqW-s

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?

Solution : https://www.youtube.com/watch?v=FVflQ49wsF8

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

Solution : https://www.youtube.com/watch?v=FYTHpf7oG_w

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, R_a = 1 ohm, L_a = 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 T_d is _______.

Solution : https://www.youtube.com/watch?v=mbkr17M1Cng

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

Solution : https://www.youtube.com/watch?v=BzTXlRJ5M9k

2004
2. The state variable equations of a system are:
ẋ₁ = -3x₁ -x₂ + u
ẋ₂ = 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

Solution : https://www.youtube.com/watch?v=CVH0VsecVuQ

3. The state transition matrix e^At for the given matrix A, is

Solution : https://www.youtube.com/watch?v=w9zVGFsZ3Bw

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?

Answer :
Solution :

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

Solution : https://www.youtube.com/watch?v=SdKm17kCbFg

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, i_a is the armature current and u is the armature voltage. The transfer function of the motor is

Solution : https://www.youtube.com/watch?v=jTt-sWQZS0Y
Statement for Linked Answer Questions 7 & 8:

7. The eigen value and eigen vector pairs (λ_i, v_i) 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

Solution : https://www.youtube.com/watch?v=cxlX8Rpj9cs

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

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

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

Solution : https://www.youtube.com/watch?v=-4PqIhkiPD0
12. The transfer function of the system is

Solution : https://www.youtube.com/watch?v=kT5TsnSEl9g

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

Solution : https://www.youtube.com/watch?v=CsciojvPQF8

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

Solution : https://www.youtube.com/watch?v=dPCv96oD6Ow

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

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

Solution : https://www.youtube.com/watch?v=ndeVYkf7j6w

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 e^At 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

Solution : https://www.youtube.com/watch?v=6tsdXBAfClY

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

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

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

Solution : https://www.youtube.com/watch?v=PW4_N3VeGPs

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

Solution : https://www.youtube.com/watch?v=kx74sDKotOs

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

Solution : https://www.youtube.com/watch?v=2Yx8ux4wY4U

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

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

2015
23. The state variable representation of a system is given as

The response y(t) is
a) sin (t)
b) 1 – e^t
c) 1 – cos (t)
d) 0

Solution : https://www.youtube.com/watch?v=Ax6VxSUUJtA

24. A network is described by the state model as

then the transfer function H(s) = (Y(s)/U(s)) is

Solution : https://www.youtube.com/watch?v=z5t9oMUeA-U

Routh-Hurwitz Stability Criterion - Topic wise GATE Questions on Control Systems (from 1987)

2004
1. The open loop transfer function of a unity feedback system is G(s) = K/[s(s²+ s + 2)(s + 3)]. The range of K for which the system is stable is
a) 21/44 > K > 0
b) 13 > K > 0
c) 21/4 < K < ∞
d) - 6 < K < ∞

Solution : https://www.youtube.com/watch?v=KmUOC_i7USs

2. for the polynomial R(s) = s⁵ + s⁴ + 2s³ + 2s² + 3s + 15, the number of roots which lie in the Right Half of the S-plane is
a) 4
b) 2
c) 3
d) 1

Solution : https://www.youtube.com/watch?v=P_CSEbYpX0s

2006
3. The positive values of "K" and "a" so that the system shown in the figure below oscillates at a frequency of 2 rad/sec respectively are

a) 1, 0.75
b) 2, 0.75
c) 1, 1
d) 2, 2

Solution : https://www.youtube.com/watch?v=VYWtxlytlC8

2008
4. The number of open right half plane poles of G(s) = 10 / (s⁵ + 2s⁴ + 3s³ + 6s² + 5s + 3) is
a) 0
b) 1
c) 2
d) 3

Solution : https://www.youtube.com/watch?v=Seaj7dcM1yk

2012
5. 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

Solution : https://www.youtube.com/watch?v=IBPC5lyX_X4

2014
6. Consider a transfer function

with p a positive real parameter. The maximum value of p until which G_p remains stable is ______________

Solution : https://www.youtube.com/watch?v=2psgTIOta3M

2015
7. 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) K_P> K_I/2 > 0
b) 2K_I> K_P> 0
c) 2K_I< K_P
d) 2K_I> K_P

Solution : https://www.youtube.com/watch?v=1jJORFpuY5Y

8. 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 __________.

Solution : https://www.youtube.com/watch?v=oz77HAXHwQU

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

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

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

Routh-Hurwitz Stability Criterion - Topic wise GATE Questions on Control Systems (from 1987)

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

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