Control Systems Test - 2 - PDF Flipbook
Control Systems Test - 2
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GATE
EEE
Control
Systems
Test-02Solutions
CONTROL SYSTEMS
1. Match List-I with List-II and select the correct answer using the
code given below the lists:
List-I List-II
A. Relative stability 1. State model
B. Eigenvalue 2. G.M
C. Difference equation 3. Bode plot
D. Corner frequency 4. Sampled-data system
Codes:
ABC D
a) 1 2 3 4
b) 1 2 4 3
c) 2 1 3 4
d) 2 1 4 3
Answer: (d)
Solution:
Gain margin is used for study of relative stability. Eigen values
roots of system matrix (A) hence in state space model. Corner
frequency is the frequency from where slope of Bode plot
changes.
2. Match List-I (Singular point) with List-II (Phase portrait)
and select the correct answer using the codes given below:
List-I
A. Unstable focus
B. Stable focus
1
C. Stable node
D. Saddle
List-II
1. A logarithmic spiral extending into the singular point.
2. Trajectories approach singular point adjacent to straight line
curve out and leave in vicinity of singular points.
3. A logarithmic spiral extending out of the singular point.
4. Trajectories are asymptotic to straight line.
Codes:
ABC D
a) 3 1 2 4
b) 1 2 3 4
c) 3 4 1 2
d) 1 4 3 2
Answer: (a)
3. Consider the following statements:
The roots of the characteristic equation of a control system lead
to its instability if they lie
1. on the imaginary axis of the S-plane
2. on the negative real axis of the S-plane
3. in the right-half of the S-plane
Which of the statements given above are correct?
a) 1, 2 and 3
b) 1 and 2
c) 2 and 3
2
d) 1 and 3
Answer: (d)
Solution:
If poles lie on R.H.S. or on imaginary axis, system becomes
unstable and output is unbounded.
4. The state-space representation of a system is given by
̇ = �−01 −02� + �10�
= �11�
Then the transfer function of the system is
a) 1
2+3 +2
b) 1
+2
c)
2+3 +2
d) 1
+1
Answer: (d)
Solution:
( ) = [ − ]−1
= [1 1] �( + 1) 0 −1 �10�
0 +
( 2)�
= [1 1] × 1 �( + 2) 0 1)� �01�
( +1)( +2) 0 +
(
= 1
( +1)
3
5. A unity feedback non-linear control system’s plot for -1/N and
G(jω) is shown in the diagram given below:
N is describing function of the non-linear device and G(s) is the
transfer function of the linear plant. Which one of the following
statements is correct?
The limit cycle is
a) stable
b) unstable
c) critically stable
d) None of the above
Answer: (b)
6. The forward voltage transfer function of a two-port network
s+ω . What will be the output voltage if the input voltage is
s2+ω2
(f)?
a) √2 sin(ωt − π/4)
b) cosωt – sinωt
c) cosδt
d) √2 sin(ωt + π/4)
Answer: (d)
4
Solution:
( ) = s+ω
( ) s2+ω2
( ) = ( )
Laplace transform ( ) = 1
( ) = s+ω
s2+ω2
= s + ω
s2+ω2 s2+ω2
Inverse Laplace transform,
( ) = cos ωt + sin ωt
( ) = √2 �cos ωt sin + sin ωt cos 4 �
4
= √2 �ωt + 4 �
[∵ sin( ± ) = sin cos ± cos sin ]
7. When a unit ramp input is applied to the unity feedback system
having closed loop transfer function ( ) = + , (a > 0, b >
( ) 2+ +
0, K > 0), the steady state error will be
a) 0
b) a/b
c) +
d) −
Answer: (d)
Solution:
= = +
1− 2+ +
1− 2 + + +
5
( ) = +
2+( − )
= l i→m0 . ( ) =
−
= 1 = −
8. The state-variable formulation of a system is
̇ = + ; = [1 0]
Where, = �−03 −12� , = �21�
The system transformation would be
a) +2
2+5 +6
b) 2 +5
2+5 +6
c) 2 −5
2+5 −6
d) +1
2+5 +6
Answer: (b)
Solution:
( ) = [ − ]−1. +
( )
∴ [ − ] = � + 3 −+12�
0
[ − ]−1 = 1 � + 2 −+13�
( +2)( +3) 0
( ) = [1 0] × 1 � + 2 1 3� �12�
( ) ( +2)( +3) 0 +
= [1 0] × 1 �2 ++35�
( +2)( +3)
= 2 +5
( +2)( +3)
6
9. Which one of the following methods is NOT used for the
analysis of nonlinear control systems?
a) Phase plane method
b) Describing function method
c) Lyapunov’s method
d) Piecewise linear method
Answer: (c)
Solution:
Lyapunov’s method is used for stability analysis of LTI control
system. Piecewise linear method is also used for general
investigation of non-linear system in addition to phase-plane and
describing function method.
10. An closed-loop the s-plane lie system is stable when all its
poles
a) On the positive real axis
b) On the imaginary axis
c) In the left half
d) In the right half
Answer: (c)
Solution:
In LHS plane bounded input will result in bounded output.
7
11. It is given that
( ) =
2(1+ )
This system is operated in a closed-loop with unity feedback.
What is the order and type of the closed-loop system?
a) 3 and 2
b) 2 and 3
c) 3 and 1
d) 3 and 0
Answer: (a)
12. The state-variable description of a linear autonomous system is
̇ = AX where X is a two dimensional state vector and A is a
matrix given by = �20 20�.
The poles of the system are located at
a) -2 and +2
b) -2j and +2j
c) -2 and -2
d) +2 and +2
Answer: (a)
Solution:
By solving ( − ) = 0
�− 2 − 2 � = 0 ⇒ 2 ± 4
= ±2
8
13. A non-linear control system is described by the equation
̇ = sin = 0
The type of singular point at (0, 0) is
a) centre
b) focus
c) saddle point
d) none of the above
Answer: (a)
14. The system shown in the figure is
a) stable
b) Unstable
c) conditionally stable
d) stable for input u1, but unstable for input u2
Answer: (d)
Solution:
Let C1 and C2 are the outputs
9
= 1 � −+12�
1 �1+(( −+12))( −11)�
1 = ( −1) … … ( )
1 ( +3)
2 = 1 … … ( )
1 ( +3)
From (1) and (2) it is clear that the poles are left side.
∴Stable for any bounded input u1.
1 = −1 … … ( )
2 ( +3)
2 = ( +2) … … ( )
2 ( −1)( +3)
From (4) it is clear that the poles are right-side.
∴unstable for any bounded input u2.
15. The terms in the first column of Routh's array of characteristic
equation of certain systems are 5, 2, -4, 6, 3. The number of
roots of characteristic in the right half of the s-plane is equal to
a) 2
b) 1
c) none
d) 3
Answer: (a)
Solution:
Number of roots on the positive half of the s-phase is equal to
the number of sign changes. First sign change is from 2 to -4
and second sign change is from -4 to 6.
10
16. The transfer function of a phase-lead compensator is given by
G(s) = 1+3Ts where T > 0. What is the maximum phase-shift
1+Ts
provided by such a compensator?
a) 300
b) 450
c) 600
d) 900
Answer: (a)
Solution:
Phase lead compensator
( ) = 1+ ; < 1
1+
G(s) = 1+3Ts
1+Ts
=
= 3
⇒ = 1
3
Maximum phase,
sin = 1− = 1−31
1+ 1+31
sin = 1 ⇒ = 300
2
17. A unity feedback system has an open-loop transfer function
( ) =
( 2+14 +13)
The angles of asymptotes are given by
a) 450, 1350, 2250
11
b) 600, 1800, 3000
c) 900, 800, 2700
d) none of these
Answer: (b)
Solution:
Angle of asymptotes will be given by
= (2 +1)180 = (2 +1)180 = (2 +1)180
− 3−0 3
For q = 0, 1, 2 = 60, 1800 3000
18. A system is described by the characteristic equation given by
s3 + 4s2 + (K + 10) s + 5K = 0
What is the range of K for stable operation?
a) 0 < K < 20
b) 0 < K < 40
c) 0 < K < 60
d) 0 < K < 80
Answer: (b)
Solution:
Routh array
k > 0 and 40 – k > 0 ⇒ 0 < k < 40
12
19. Microprocessor based control systems can be classified as
a) continuous data based systems
b) learning control systems
c) sampled data control systems
d) stochastic control systems
Answer: (c)
20. The frequency at which the Nyquist diagram crosses the
negative real axis is known as
a) gain crossover frequency
b) phase crossover frequency
c) damping frequency
d) natural frequency
Answer: (b)
21. A second order system with no zeros has its located at – 3 + j4
and – 3 – j4 in the s-plane. The undamped natural frequency and
the damping factor of the system are respectively.
a) 4 rad/sec and 0.75
b) 3 rad/sec and 0.60
c) 5 rad/sec and 0.80
d) 5 rad/sec and 0.60
Answer: (d)
Solution:
[ − (−3 + 4)][ − (−3 − 4)]
= ( + 3 − 4)( + 3 + 4)
= 2 + 9 + 6 + 16 = 2 + 6 + 25
13
∴ 2 = 25
⇒ = 5
2 = 6
⇒ = 6 = 0.6
2×5
22. The unit - impulse response of a unity - feedback control
system is given by c(t) = -te-t + 2e-t, (t ≥ 0) the open loop
transfer function is equal to
a) +1
( +2)2
b) 2 +1
2
c) +1
( +1)2
d) +1
2
Answer: (b)
Solution:
= [− − + 2 − ]
= −1 + 2
( +1)2 ( +1)
= −1+2( +1)
( +1)2
= 2 +1
( +1)2
∴ = 1− � ( )=1
2 +1 = 2 +1
2
= ( +1)2
1−( 2 + +1)12
∴ = 2 +1
2
14
23. For stability poles should be in LH of S-plane. LH includes
a) real and imaginary axis
b) real axis but not j-axis
c) real axis but not – j-axis
d) negative real axis but not j-axis
Answer: (d)
24. The unit step response of a particular control system is given
by: c(t) = 1 – 5e-t.
What is the transfer function of this system?
a) 5
s+1
b) 5
s(s+1)
c) 1−4s
1+s
d) 1−4s
s(1+s)
Answer: (c)
Solution:
( ) = 1 − 5 −
Taking laplace transform
Output, ( ) = 1 − 5
+1
= −4 +1
( +1)
Since input, x(t) = u(t)
( ) = 1
Transfer function = ( ) = −4 +1 = 1−4
( ) ( +1) +1
15
25. If the gain margin of a certain feedback system is given as 20
db, the Nyquist plot will cross the negative real axis at the point
a) s = -0.05
b) s = -0.2
c) s = -0.1
d) none of these
Answer: (c)
Solution:
20 = 20 log10 � 1 �
⇒ � 1 � = (10)1
= 0.1
26. A second order differential equation is given by
2 + 5 + 7 = 7
2
The undamped natural frequency and damping ratio are
a) 1, 5
b) 5, 7
c) 1, √7
d) √7, 0.94
Answer: (d)
Solution:
2 + 5 + 7 = 7
2
Take Laplace transform
2 ( ) + 5. ( ) + 7 ( ) = 7 ( )
16
∴ ( ) = 7
( ) 2+5 +7
∴ = √7
2 = 5
⇒ = 0.94
27. A network has a pole at s = -1 and a zero at s = -2. If this
network is excited by sinusoidal input, the output
a) leads the input
b) lag the input
c) is in phase with input
d) decays exponentially to zero
Answer: (b)
Solution:
( ) = +2
+1
( ) = 2+
1+
This is the transfer function of lag network. Hence the output
will lag the input.
28. A control system is defined by the following mathematical
relationship
2 + 6 + 5 = 12(1 − −2 )
2
The response of the system as t → ∞ is
a) x = 6
b) x = 2
c) x = 2.4
17
d) x = -2
Answer: (c)
Solution:
2 + 6 + 5 = 12(1 − −2 )
2
Taking LT
( )[ 2 + 6 + 5] = 12 �1 − +12�
( ) = 12�1 − +12�
2+6 +5
= 12[ +2− ]
( 2+6 +5)( )( +2)
Final value = lim ( ) = lim ( )
→∞ → 0
= lim � ( +2()1(2 )2(+26) +5)�
→ 0
= (12)(2) = 2.4
(2)(5)
∴ Final value = 2.4
29. The root locus plot is shown below. The open-loop system has
a) two real poles
b) two complex poles and a zero
c) two complex zeros and a pole
d) two complex poles
Answer: (b)
18
30. Consider the following Nyquist plots of loop transfer functions
over ω = 0 to ω = ∞. Which of these plots represents a stable
closed loop system?
a) (1) only
b) all, except (1)
c) all, except (3)
d) (1) and (2) only
Answer: (d)
Solution:
For stability (-1, j0) should not be enclosed by the polar plot. In
figures (1) & (2) (-1, j0) s not enclosed.
Therefore systems represented by (1) & (2) are stable.
19
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