Control Systems Test - 3 - PDF Flipbook
Control Systems Test - 3
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GATE
EEE
Control
Systems
Test-03Solutions
CONTROL SYSTEMS
1. Assertion (A): The continuous time differentiator is an unstable
system in the bounded-input, bounded-output (BI-BO) sense.
Reason (R): The bounded-input, x(t) = sint2 would result in an
unbounded-output.
a) Both A and R are true, and R is the correct explanation of A.
b) Both A and R are true, but R is not a correct explanation of
A.
c) A is true, but R is false.
d) A is false, but R is true.
Answer: (d)
Solution:
Continuous time differentiator may not necessarily lead to
unstable system. For input sint2, output would be either
unbounded or oscillatory.
2. In the formation of Routh-Hurwitz array for a polynomial, all
the elements of a row have zero values. This premature
termination of the array indicates the presence of
a) only one root at the origin
b) imaginary roots
c) only positive real roots
d) only negative roots
Answer: (b)
1
Solution:
If the entire row is zero, indicates the symmetrical roots of the
characteristic equation with respect to origin. Hence if the roots
are imaginary, they are symmetrical with respect to origin.
3. Assertion (A): If any one of the state variables is independent
of the control u(t), the process is said to be completely
uncontrollable.
Reason (R): There is no way of driving this particular state
variable to a desired state in finite time by means of a control
effort.
a) Both A and R are true and R is the correct explanation of A
b) Both A and R are true but R is NOT the correct explanation
of A
c) A is true but R is false
d) A is false but R is true
Answer: (a)
Solution:
Refer, the definition of state controllability.
4. Match List-I with List-II and select the correct answer using the
code given below the lists:
List-I List-II
2
Codes: D
ABC 4
1
a) 1 2 3 4
b) 4 3 2 1
c) 1 3 2
d) 4 2 3
Answer: (b)
Solution:
3
5. The open loop transfer function of a servo system with unity
negative feedback is given as:
( ) = 6
( +2)( +5)
What is the damping ratio and the natural frequency of
oscillation in rad/sec?
a) 7 and 4
8
b) 7 and 16
8
c) 4 and 4
7
d) 4 and 16
7
Answer: (a)
4
Solution:
6 6
s2+7s+16
T. F = (s+2)(s+5) =
1+(s+2)6(s+5)
Comparing with: s2 + 2ξωn + ωn2
ω2n = 16 ⇒ ωn = 4
2ξωn = 7 ⇒ ξ = 7
8
6. Match List-I (Properties) with List-II (Effect) and select the
correct answer using the code given below the lists:
List-I
A. Non-linear elements are sometimes intentionally introduced
B. Discrete data control system
C. Feedback can increase system gain
D. Sensitivity considerations are important
List-II
1. Are susceptible to noise
2. In one frequency range
3. Physical properties may change with environment and ageing
4. To impose system stability
Codes:
A B CD
a) 1 2 3 4
b) 4 1 2 3
c) 4 1 3 2
d) 1 2 4 3
Answer: (b)
5
Solution:
Discrete system is suspectable to noise. Sensitivity may change
with environment and ageing.
7. The transfer function is 1+1+0. 5 . It represents a
a) lead network
b) lag network
c) lag-lead network
d) proportional controller
Answer: (b)
Solution:
T. F = 1+0.5s
1+s
Its pole will be given by s = -1
and zero is = 1 = 2
0.5
Hence zero is greater than pole hence it’s a lag network.
8. Let, X = �01 21� x + �01� u, U = [b 0]x, where b is an
unknown constant.
This system is
a) observable for all values of b
b) unobservable for all values of b
c) observable for all non-zero values of b
d) unobservable for all non-zero values of b
Answer: (c)
6
Solution:
[CT: ATCT] = �b0 2bb� = 2b2 ≠ 0 for b ≠ 0
9. A unity feedback system has an open-loop transfer function of
the form ( ) = ( + ) ; b > a. Which of the loci shown in Fig.
2( + )
can be valid root-loci for the system?
Answer: (a)
Solution:
P=3&Z=1
No.of asymptotes = | − | = 2
7
Centroid(σ) = poles− zeros
P−Z
= 0− −(− ) is negative
−
Angle of asymptotes = = ±900
RLD is given below,
10. Statement (I): For radar tracking systems, signals are
available in the form of pulse trains.
Statement (II): The stability of a discrete-time system is
decreased as the sampling period is shortened.
Codes:
a) Both Statement (I) and Statement (II) are individually true
and Statement (II) is the correct explanation of Statement (I).
b) Both Statement (I) and Statement (II) are individually true
but Statement (II) is not the correct explanation of Statement
(I)
c) Statement (I) is true but Statement (II) is false.
d) Statement (I) is false but Statement (II) is true.
Answer: (b)
8
11. Match List-I (Evaluation of the Value of Function) with
List-II (Corresponding z-transform expression) and select the
correct answer using the code given below the lists:
List-I List-II
A. Final value 1. l i→m0(1 − −1) ( )
B. Initial value 2. l i→m1(1 − −1) ( )
3. l →im∞ ( )
Codes: 4. l →im∞ ( )
AB
a) 1 3
b) 1 4
c) 2 3
d) 2 4
Answer: (c)
Solution:
As per definition of initial value and final value theorem
12. When a transfer function model is converted into state space
model, the order of the system may be reduced during which
one of the following conditions?
a) Some of the variables are not considered
b) Some of the variables are hidden
c) Pole zero cancellation takes place
d) The order of the system will never get changed
9
Answer: (d)
13. What does introduction of negative feedback in a system result
in?
1. Increased accuracy
2. Reduced stability
3. Increased bandwidth
4. Increased sensitivity
Select the correct answer using the code given below:
a) 1, 2 and 3
b) 2, 3 and 4
c) 1, 3 and 4
d) 1, 2 and 4
Answer: (a)
Solution:
Negative feedback reduces sensitivity to parameter variation of
the system.
14. A discrete-time system is stable if all the poles of the Z-
transfer function of the system lie
a) outside the circle of unit radius on the Z-plane
b) within a circle of unit radius on the Z-plane
c) to the left of imaginary axis on the Z plane
d) to the right of imaginary axis on the Z-plane
Answer: (b)
10
15. A second order system is given as
( ) =
(1+ 1 )(1+ 2 )
The gain K is increased. Then the phase margin
a) gets increased
b) gets decreased
c) is unaffected
d) gets decreased and becomes stable
Answer: (b)
16. The figure shown is a phase-plane representation of
trajectories. The singular point shown is a
a) unstable node
b) saddle point
c) stable focus
d) stable node
Answer: (d)
17. If the gain of the open-loop system is doubled, the gain margin
a) is not affected
b) gets doubled
c) becomes half
d) becomes one-fourth
Answer: (c)
11
18. Consider a second order system whose state space
representation is of the form Ẋ = AX + BU. If x1(t) = x2(t), then
system is
a) Controllable
b) Uncontrollable
c) Observable
d) Unstable
Answer: (b)
Solution:
x1(t) = x2(t)
therefore dx1(t) = dx2(t)
dt dt
Let the state equation is
dx1(t) db� �xx12((tt))� + �ef� u(t)
�dxd2t(t)� = �ac
dt
Controllability matrix = [B AB]
AB = �ac db� �ef � = �acee + dbff�
+
Since x1(t) = x2(t)
a + b = c + d and e = f
|[B AB]| = �ef (ae + dbff))� = 0
(ce +
Hence uncontrollable
12
19. The state equations in the phase variable canonical from can be
obtained from the transfer function by
a) cascaded decomposition
b) direct decomposition
c) inverse decomposition
d) parallel decomposition
Answer: (b)
20. A control system is as shown in the given figure. The
maximum value of gain K, for which the system is stable, is
a) √3
b) 3
c) 4
d) 5
Answer: (d)
21. With reference to the signal flow graph shown below, which
one of the following will be the overall transfer function?
a) C = G
R
b) C = G
R 1+H2
c) C = G
R (1+H1)(1+H2)
d) C = G
R 1+H1+H2
13
Answer: (c)
Solution:
T = 1 K PKΔK
Δ
Forward path, P1 = G → only one forward path
Loop 1 = -H1
Loop 2 = -H2
Both are non-touching loop,
Δ = 1 − (−H1 − H2) + H1H2
= 1 + (H1 + H2) + H1H2
Δ1 = 1
C = G
R 1+H1+H2+H1H2
= G
(1+H1)(1+H2)
22. Consider the following properties attributed to state model of a
system:
1. State model is unique.
2. Transfer function for the system is unique.
3. State model can be derived from transfer function of the
system.
Which of the above statements are correct?
a) 1, 2 and 3
b) 1 and 2 only
c) 2 and 3 only
d) 1 and 3 only
Answer: (c)
14
Solution:
The state model of a system is not unique. But whereas transfer
function for the system is unique and state model can be derived
from transfer function of the system.
23. A linear time-invariant system initially at rest, when subjected
to a unit-step input, gives a response
y(t) = te−t, t > 0
The transfer function of the system is.
a) 1
( +1)2
b) 1
( +1)2
c)
( +1)2
d) 1
( +1)
Answer: (a)
24. If stability error for step input and speed of response be the
criteria for design, what controller would you recommend?
a) P controller
b) PD controller
c) PI controller
d) PID controller
Answer: (d)
15
25. The open-loop transfer function of a unity feedback control
system is G(s) = (s+12)2. The closed-loop transfer function will
have poles at
a) – 2, – 2
b) – 2, – 1
c) – 2, ±j
d) – 2, 2
Answer: (c)
Solution:
Closed loop transfer function have poles which are given by the
roots of its characteristics equation
1 + 1 = 0
(s+2)2
⇒ (s + 2)2 + 1 = 0
⇒ s2 + 4 + 4s + 1 = 0
⇒ s2 + 4s + 5 = 0
⇒ s = −2 ± j
26. A system is described by the state equation ̇ = + . The
output is given by = .
Where A = �−34 −−11� , = �11� , = [1 0]. Transfer
function G(s) of the system is
a)
2+5 +7
b) 1
2+5 +7
c)
2+3 +2
16
d) 1
2+3 +2
Answer: (a)
Solution:
[SI − A] = s �10 10� − �−34 −−11�
= �(s−+34) s 1 1�
+
[SI − A]−1 = �s+31 s−+14�
(s+1)(s+4)+3
TF = Y(s) = C[SI − A]−1B + D
U(s)
= [1 0]�s+31 s−+14��11�
(s+1)(s+4)+3
= (s+1)−1
s2+5s+7
∴ TF = Y(s) = s
U(s) s2+5s+7
27. The purpose of the series quadrature windings in an amplidyne
is to
a) neutralize the effect of armature reaction
b) reduce commutation difficulties
c) increase the gain
d) increases the response lime
Answer: (b)
17
28. Given the homogeneous state-space equation ̇ =
�−03 −12� , the steady state value = l →im∞ ( ), given the
initial state value of (0) = [10 −10] , is
a) = �00�
b) = �−−32�
c) = �−1100�
d) = �∞∞�
Answer: (a)
29. The state representation of a second order system is
x1̇ = x1 + u, x2̇ = x1 − 2x2 + u
Consider the following statements regarding the above system;
1. The system is completely state controllable
2. If 1is the output, then the system is completely output
controllable.
3. If 2is the output, then the system is completely output
controllable.
Of these statements
a) 1, 2 and 3 are correct
b) 1 and 2 are correct
c) 2 and 3 are correct
d) 1 and 3 are correct
Answer: (c)
18
30. The error detectors elements in a control system gives
a) the sum of the reference signal and feedback signal
b) the sum of the reference signal and error signal
c) the difference of the reference signal and feedback signal
d) the difference of the reference signal and output signal
Answer: (c)
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