Control Systems

Classroom Workbook

ELECTRONICS AND COMM. ENGINEERING

ELECTRICAL ENGINEERING
INSTRUMENTATION ENGINEERING

GATE/ESE
Control Systems
Classroom Workbook

For ECE, EE and IN

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GATE Syllabus
ECE

Basic control system components; Feedback principle; Transfer function; Block diagram
representation; Signal flow graph; Transient and steady-state analysis of LTI systems;
Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots;
Lag, lead and lag-lead compensation; State variable model and solution of state equation of
LTI systems.

EE

Mathematical modelling and representation of systems, Feedback principle, transfer

function, Block diagrams and Signal flow graphs, Transient and Steadystate analysis of linear
time invariant systems, Stability analysis using RouthHurwitz and Nyquist criteria, Bode
plots, Root loci, Lag, Lead and Lead‐Lag compensators; P, PI and PID controllers; State space
model, Solution of state equations of LTI systems

IN

Feedback principles, signal flow graphs, transient response, steady-state-errors, Bode plot,
phase and gain margins, Routh and Nyquist criteria, root loci, design of lead, lag and lead-lag
compensators, state-space representation of systems; time-delay systems; mechanical,
hydraulic and pneumatic system components, synchro pair, servo and stepper motors, servo
valves; on-off, P, PI, PID, cascade, feedforward, and ratio controllers, tuning of PID
controllers and sizing of control valves.

ESE Syllabus
E&T

Basic control system components; Feedback principle; Transfer function; Block diagram
representation; Signal flow graph; Transient and steady-state analysis of LTI systems;
Frequency response; Routh-Hurwitz and Nyquist stability criteria; Bode and root-locus plots;
Lag, lead and lag-lead compensation; State variable model and solution of state equation of
LTI systems.

EE
Transfer Function, Principal of Feedback, Block Diagram, Signal Flow Paragraphs, Steady-
State Errors, Routh Hurwitz Creation, Nyquist Technology, Bode Plots, Root Loci, LED Leg
Compensation, Stability Analysis, State Page Model, State Translation Matrix, Controllability
and Observability, Linear State Variable Feedback, PID and Industrial Controller
 Contents

SChapters
r
.
11. Mathematical Modeling, Block Diagram & SFG
.
22. Time Response Analysis
.
33. Routh Hurwitz Stability
.
44. Polar Plot and Root Locus Diagram
.
55. Nyquist Stability Criterion
.
6.Bode Plot

7. State Space Analysis

8. Controllers and Compensators

6.
63.
6.


 1 Mathematical Modeling,
Block Diagram & SFG

MCQ, MSQ and NAT

Q.1 Determinant for the Signal Flow C s s  s  18 


Graph shown below is R s  s2  12s  28
(A)
C s s  28 

R s s  32s  48
2
(B)
C s s  s  18 

R s  s  12s  48
2
(C)
(A)Δ = 1 – d – g + f + ag – gf – ef – bef – C s s  s  28 

cef R s s2  32s  48
(D)
(B) Δ = 1 + d – g – f – ae + gf + dg +
bef – cef Q.3 A block diagram is shown in the
(C) Δ = 1 – d – g + f + ae – gf – dg + figure below:
bef + ced – cef
(D)Δ = 1 – d – g + f + ae – gf – dg – bef +
ced – cef
Q.2 The signal flow graph of a system
is shown below. The transfer
Find the expression of c(t) if r(t) =
C s
e–2t and n(t) = δ(t) is-
function R  s  of the system is-
c t  
3 2t 2 3t 5 6t
e  e  e
(A) 2 3 6

c t  
5 2t 2 3t 7 6t
e  e  e
(B) 2 3 6

c t  
1 9t 3 3t 5 4t
e  e  e
(C) 2 2 2
 c t  
1 2t 3 3t 5 6t
e  e  e
(D) 2 2 2

Q.4 The total number of loops in the

given signal flow graph is
G1G2G3  G1G4
_______. (A) 1 G1G2H1  G2G3H2  G1G2G3  G1G4  G4H2

(Rounded off to the nearest integer) G1G2G3  G1G4

1 G1G2H1  G2G3H2  G1G2G3  G1G4  G4H2
(B)
G1G2G3  G1G4
1 G1G2H2  G2G3H1  G1G2G3  G1G4  G4H2
(C)
G1G2G3  G1G4
1 G1G2H2  G2G3H1  G1G2G3  G1G4  G4H2
(D)
Q.7 Which of the rules and their
equivalent block diagrams given in the
options is/are correct for a control system?
Q.5 The transfer function of the given (A)
block diagram is-

(B)

20s  32
(A) 60s2  40s  50

27s  160 (C)

(B) 20s2  15s  12

10s  26
(C) 240s2  44s  15

40s  16
(D) 160s2  74s  5
(D)
C
Q.6 Determine the R for the given
block diagram of a control system.
 (B)
Q.8 A system is represented by the C(s) G1G2G3G4 G5H1

given equation, the transfer R2 (s) 1  G4  G1G2  G1G2G4  G1G4 G5H1H2

function of the system is-

(C)
x = x1 + B3u
C(s) G1G2G3G4 G5H1
x 1  A1x 1  x2  B2u 
R2 (s) 1  G4  G1G2  G1G2G4  G1G4 G5H1H2
x2  A 2x1  B1u
(D)
C(s) B3  B2 s  B1(s  A 1s  A2 )
2
(A) 
R(s) s2  A 1s  A2 C(s) G1G2G3  G4 G5

R2 (s) 1  G1G2G4  G1G2  G1G4H1H2
C(s) B2  B1s  B3 (s  A2s  A 1)
2
(B) 
R(s) s2  A 1s  A2
Q.10 For the given control system, the
C(s) B1  B2 s  B3 (s  A2s  A 1)
2
C(s)
(C)  transfer function is-
R(s) s2  A 2 s  A 1 R(s)

C(s) B1  B2 s  B3 (s  A 1s  A2 )
2
(D) 
R(s) s2  A 1s  A2

Q.9 The block diagram of the system

given below has two inputs R1(s)
and R2(s). If the input R1(s) = 0,
then the transfer function of the
system is-
1 1
(A) (B)
s2 s1
s1 1
(C) (D)
s  3s  2
2
s  3s  3
2

Q.11 The number of loops available in

the given signal flow graph is

C(s) G1G2G3  G4 G5
(A) 
R2 (s) 1  G1G2G4  G1G2  G1G4H1H2

(A) 4
(B) 5 d2 y  t dx  t
3 2
 4y  t   2x  t
(C) 3 dt dt

(D) 6 (where x(t) is input signal and y(t)

Q.12 The number of forward paths and is output signal)

the number of non-touching loop Consider the following statements.

pairs in the given signal flow graph 3s2  4

(1) Transfer function =
s 2
are respectively
(2) Transfer function = s2  2
s 2

(3) DC gain = 0.5

(4) DC gain = 2
Which of the following is/are
incorrect?

(A) 2 and 3 (A) 1 and 3 only

(B) 2 and 1 (B) 1 and 4 only

(C) 1 and 2 (C) 2, 3 and 4

(D) 3 and 2 (D) 1, 2 and 4 only

Q.13 Assertion (A) : Sensitivity Q.15 If the closed-loop transfer function

depends on the value of 1-GH. of a unity negative feedback

Reason (R) : For negative 3

system is , find the
s2  9s  17
feedback path, if the
corresponding open-loop transfer
gain is positive then
function.
sensitivity decreases.
3
(A) Both (A) and (R) are true, and (A)
s  9s  14
2

(R) is the correct explanation of

3
(B)
(A). s  9s  20
2

(B) Both (A) and (R) are true, but 3

(C)
(R) is not the correct s  9s
2

explanation of (A). 3
(D)
s  9s  17
2

(C) (A) is true, but (R) is false.

Q.16 In the figure below the number of
(D) (A) is false, but (R) is true.
forward paths and loops are
Q.14 Transfer function of system
described by
 (A) 1.40
(B) 1.17
(C) 2.55

(A)1, 5 (B)1, 6 (D) 3.83

(C)2, 5 (D)2, 6 Q.19 The number of loops available in

Q.17 The block diagram of a control the given signal flow graph is

system is shown is figure. The ______. (Rounded off to nearest

Y(s)
integer)
transfer function G(s)  of
X(s)

the system is-

5 Q.20 Find the transfer function of the

(A)
s  20s  150
2
given block diagram –
5
(B)
s2  20s  50
5
(C)
s  20s  50
2

5
(D)
s2  20s  50
Q.18 Consider the feedback control (A)

system shown below – C(s) 1

 2
R(s) s R 1R 2C 1  R 1C 1  R 2C 2  R 1C 2  s  1

(B)
C(s) 1
 2
R(s) s R 1R 2C 1C 2  R 1C 1  R 2C 2  R 1C 2  s  1
Find the magnitude of the
sensitivity of transfer function to (C)
variation in parameter K at C(s) 1
 2
  6rad / sec . (Assume process R(s) s R 1R 2C 2  R 1C 1  R 2C 2  R 1C 2s  1

parameter K is 0.5).
(D) Q.23 The signal flow graph for a system
C(s) 1 is given below. The transfer
 2
R(s) s R 1R 2C 1  R 1C 1s  R 2C 2  R 1C 2s  1 Y s
function for this system is
U s
V2  s
Q.21 The transfer function of the
V1  s

circuit shown below is

s1
(A)
5s  6s  2
2

s1
(B)
s2  6s  2
s1
0.5s  1 (C) 3s 4s
62
(A) s
(B)
2

s 1 s 2
(D) 2 1
s 2 s 1
(C) (D)5s  6s  2
s 1 s 2
Q.24 For the following system
Q.22 The open-loop transfer function of
a dc motor is given as
  s 10
 . When connected
Va s 1 10s

in feedback as shown, the

When X 1  s  0 , the transfer
approximate value of K a that will
y s
reduce the time constant of the function is
x2  s
closed loop system by one hundred
s1 1
times as compared to that of the (A) (B)
s2 s1
open-loop system is s2 s1
(C) (D)
s  s  1 s  s  1

Q.25 Consider the following block

diagram in the figure.

(A)1 (B)5
(C)10 (D)100
 C(s)
The transfer function is
R(s)

G1G2
(A)
1 G1G2
(B) G_1 G_2+G_1+1

(C) G1G2  G2  1
s2  1
G1 (A)H(s) =
(D) 2s2  1
1 G1G2
s2  1
(B)H(s) =
Q.26 The steady state error of the system s3  2s2  s  1
shown in the figure for a unit step s 1
(C)H(s) =
input is _______. s  s 1
2

s2  1
(D)H(s) =
s3  s2  s  1
Q.29 The block diagram of a feedback
control system is shown in the
figure.
Q.27 Let Y(s) be the unit-step response
of a causal system having a transfer
function,
3s
G(s) =
(s  1)(s  3)

G(s)
That is, Y(s) = . The forced
s
response of the system is:
(A) u(t) –2e–t u(t) + e–3t u(t) Y(s)
The transfer function of the
(B) 2u(t) X(s)
(C) u(t) system is
–t –3t
(D) 2u(t) –2e u(t) + e u(t) G1  G2  G1G2H
(A)
Q.28 The block diagram of a system is 1 GH
1

illustrated in the figure shown, G1  G2

(B)
where X(s) is the input and Y(s) is 1  GH
1
 G2H

the output. The transfer function G1  G2

(C)
1 GH
Y(s) 1
H(s) = is:
X(s)
 G1  G2  G1G2H Reason (R): If the system has
(D)
1 GH
1
 G2H positive feedback, the stability
Q.30 A system is represented by a block increases.
diagram as shown below – (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
The unit impulse response of the above (C)A is true, but R is false
system for time t ≥ 0 is - (D)A is false, but R is true
(A)(1 – et) u(t) Q.34 In a closed loop system negative
–t
(B)(1 – e ) u(t) feedback increases which of the
–t
(C)(1 – e ) u(t – 1) parameter of the system.
(D)(1 + e-t) u(t) (A)Overall Gain (B)Noise
Q.31 A system is referred to as non- (C)Bandwidth (D)Distortion
minimum phase system is - Q.35 Analogous of spring constant (K)
(A)Poles and zeros do not lie on in force current analogy-
the right half of s-plane (A)C
(B)No zeros lie on the left half of (B)1/C
s-plane (C)1
(C)No poles lie on the left half of (D)1/L
s-plane
(D)Poles and zeros lie on the right
half of s-plane
Q.32 A function is given as
10
F  s  , when F(s) is
s  s2  s  4 

the Laplace transform of function

f(t), the initial value of f(t) is -
(A)0 (B)10
(C)2.5 (D)0.4
Q.33 Assertion (A): In control system,
the stability depends on the feedback.
Answer Key Q.21 [D]

Q.22 [C]

Q.1 [D] Q.23 [A]

Q.2 [D] Q.24 [D]

Q.3 [A] Q.25 [C]

Q.4 [8] Q.26 [0.5, Range 0.49 to 0.51]

Q.5 [D] Q.27 [D]

Q.6 [B] Q.28 [B]

Q.7 [A, D] Q.29 [C]

Q.8 [D] Q.30 [B]

Q.9 [B] Q.31 [D]

Q.10 [D] Q.32 [A]

Q.11 [B] Q.33 [C]

Q.12 [B] Q.34 [C]

Q.13 [D] Q.35 [D]

Q.14 [D]

Q.15 [A]

Q.16 [A]

Q.17 [A]

Q.18 [B]

Q.19 [5]

Q.20 [B]
 2 Time Response Analysis
c
MCQ, MSQ and NAT
Diode
Q.1 A unity feedback system is having value of |K| is ______. (Up to two
the closed-loop transfer function decimal places)
C s 6s  7 Q.4 A circuit is shown below. The

R s
 2s  3   52 s  6  value of resistance (in ohms), if
  . The 10% of overshoot occurs in the
steady-state error for the unit step voltage across the capacitor
input is _______. (Up to two is__________ (Up to two decimal
decimal places) places)
Q.2 The block diagram of a control
system is given below. The
minimum value of K for critical
damping is-

Q.5 An open-loop transfer function of a

unity feedback control system is
given below:
K
Gs 
s  1  sT 
(A)K = 0.0268
(B)K = 0.6295
If the damping ratio is increased
(C)K = –0.8567
from 0.3 to 0.9, then the amplifier
(D)K = –0.0182
gain changes from K1 to K2. Which
Q.3 A unity feedback control system
of the following relation(s) is/are
has an open-loop transfer
correct?
function
1
2s  5 K2 
OLTF  (A)If 2 , then K1 = 4.5
 s  6  3s  2K 
(B)If K2 = 9, then K1 = 81
If the peak time for the second
(C) If K1 = 9, then K2 = 4.5

(D)If K1 = 45, then K2 = 5
undershoot is 6 second, then the
Q.6 The open-loop transfer function of Q.9 The block diagram of a system is
a unity feedback control system given below.
is given by

Gs 
58
s 2
 s  6   2s2  3s  15 
. The steady-state error for The rise time of the system is
2
input r(t) = 5+6t+3t is ______ sec. (Up to three decimal
_______. (Up to two decimal places)
places) Q.10 The impulse response of an
Q.7 The block diagram of a control initially relaxed linear system is 6e–
system is given below. If the u(t). To produce a response of te–
3t

damping 3t
u(t),the input required is-
ratio is 0.5, then the value of K and
6e3t u(t )
rise time are respectively- (A) 5

e3t u(t )
(B) 6
e3t u(t )

(A) K = 0.42 and tr = 1.323 sec (C) 3

(B) K = 0.525 and tr = 1.33 sec (D) 6e3tu(t)

(C) K = 1.5 and tr = 0.925 sec Q.11 Calculate the DC gain of the

(D K = 0.116 and tr = 0.765 sec closed-loop negative feedback

Q.8 An open-loop transfer function of a system shown below.

system is given below as-

16
G(s)  ; H(s)  Ks
s  4s  16
2

If the damping ratio of the system

2
is 0.8, then - (A) 81 (B) 0
(A) Overshoot of the system is 1.52 3 1
%. (C) 127 (D) 11
(B) K= 2.12 Q.12 The output of a control system is
(C) Overshoot of the system is 3.22 given as-
%. c(t) = 1 + 0.5e–25t – 1.5e–10t
(D) K= 0.15
 then the damping ratio of the is excited by sin t .The steady-
system for unit step input is _____ state output of the system is zero at
. (Up to three decimal places) (A)   1 rad / s
Q.13 The differential equation of a
(B)   2 rad / s
closed-loop transfer function is
(C)   3 rad / s
given by-
(D)   4 rad / s
d2c 8dc
2
  64(r  c) Q.16 The open-loop transfer function of
dt dt
The peak overshoot of the system a dc motor is given as

is ______ %. (Up to one decimal   s 10

 . When connected
Va s 1 10s
place)
Q.14 The open loop transfer function of in feedback as shown, the

a plant is given as- approximate value of K a that will

reduce the time constant of the
Gs 
50
 s  0.2   s  5  s  50  closed loop system by one hundred
The approximate open loop times as compared to that of the
transfer function is obtained by the open-loop system is
retention of one pole closest to the
frequency response of the given
transfer function at low frequency.
The approximate OLTF is-
1
(A) 1 (B)5
(A) s
(C) 10 (D)10
0.2

(B)  s  0.2  Q.17 The forward path transfer function

of a unity negative feedback
50

(C)  s  0.2  system is given by

K
G(s) 
1 (s  2)(s  1)
(D)  s  0.2  s  5 
The value of Kwhich will place
Q.15 A system with transfer function both the poles of the closed-loop

G  s 
s 2
 9   s  2 system at the same location, is
s  1 s  3 s  4  ______.
Q.18 For the following feedback system 1
R(s) = as an input. Let C(s) be
1 s
G(s)  . The 2%
(s  1)(s  2) the corresponding output. The time
settling time of the step response is taken by the system output c(t) to
required to be less than 2 seconds. reach 94% of its steady-state value
lim c(t) , rounded off to two
t

decimal places, is:

(A)5.25 (B)2.81

Which one of the following (C)4.50 (D)3.89

compensators C(s) achieves this? Q.22 A system with transfer function

1
(A) 3 
1  G(s) = , a>0 is
 (s  1)(s  a)
 s  5
subjected to an input 5 cos3t. The
(B) 5 
0.03 
 1
 s  steady state output of the system is
(C)2(s + 4) 1
cos(3t  1.892). The value of
10
s 8
(D) 4  
 s 3 a is ______

Q.19 The natural frequency of an Q.23 Consider the following closed loop

undamped second-order system is control system.

40 rad/s. If the system is damped

with a damping ratio 0.3, the
damped natural frequency in rad/s
is _________
Q.20 The input -3e2t u(t), where (𝑡) is the
1
unit step function, is applied to a Where G(s) = and C(s) =
s(s  1)
s-2
system with transfer function s+3. If
s 1
K . If the steady state error for
the initial value of the output is −2, s 3
then the value of the output at a unit ramp input is 0.1, then the
steady state is _______. value of K is ___________.
Q.21 Consider a causal second-order Q.24 A control system has a closed loop
system with the transfer function transfer function as –
1
G(s) = with a unit-step
1 2s  s2
 The steady state error for unit ramp
input when K = 6 is -
(A) 0.16 (B) 1.6
(C) 6 (D)2
To make the response of the Q.27 The impulse response of an
system non-oscillatory at constant initially relaxed linear system is 3e–
frequency, the value of K is -
5t
u(t). To produce a response of te–
5t
(A) 1.12 (B) 0.62 u(t), the input should be-
(C) 0.16 (D) 0.06 0.2e5t u(t )
(A)
Q.25 Match List-I (Damping ratio) with 3

List-II (Nature of system) e5t u(t )

(B)
3

List-I List-II 2e5t u(t )

(C)
3
(P) < 1 (i) Overdamped
(D) 3e5tu(t)
(Q) > 1 (ii) Critically
Q.28 An LTI system is given as 2y"(t) +
damped
12y'(t) + 5y(t) = 6x'(t) + x(t), the
(R)  = 1 (iii) Underdamped
steady state value of the unit step
(S)  = 0 (iv) Undamped
response is-
(A) 3 (B) 0.2
Select the correct option using the
(C) 5 (D) 1/3
code given below.
Q.29 A closed loop control system pole
(A) (P) – (ii), (Q) – (i), (R) – (iv),
and zero is located at –1–3j and –
(S) – (iii)
1+3j respectively. The undamped
(B) (P) – (iv), (Q) – (i), (R) – (iii),
resonant frequency is-
(S) – (ii)
(A) 80 rad/ sec
(C) (P) – (iii), (Q) – (i), (R) – (ii),
(S) – (iv) (B) 10 rad/ sec

(D) (P) – (iv), (Q) – (i), (R) – (ii), (C) 2 2 rad/ sec
(S) – (iii)
(D) 2 rad/ sec
Q.26 In a unity negative feedback, the
Q.30 A system has velocity error
open loop transfer function is –
constant Kv = 200, then the system
K  s  1
G  s  is-
s  1 s  1 2s
 1
(1) Type = 0 and steady state error 
201
for unit step input.
1
(2) Type =1 and steady state error 
201 (A) 7/8 (B) 1/2
for ramp input. (C) 8/7 (D) 2
(3) Type = 1 and steady state error = 0 for
unit step input.
1
(4) Type = 1 and steady state error 
200
for ramp input.
Which of the option is correct-
(A) 1 and 2
(B) 2 and 3
(C) 3 and 4
(D) 1 and 4
Q.31 If we decrease the damping factor
() of the system, then peak
overshoot
1. Increases
2. Decreases
3. Remain constant
4. Makes the system more stable
Which of the statement(s) is/are
correct?
(A) Only 1
(B) Only 2
(C) Only 3
(D) 2 and 4
Q.32 Calculate the DC Gain of the
closed loop negative feedback
system shown below-
Answer Key Q.17 [2.25, Range 2.24 to 2.26]

Q.18 [C]

Q.1 [0.61, Range 0.58 to 0.63] Q.19 [38.157, Range 37.5 to 39]

Q.2 [C] Q.20 [0, Range -0.01 to 0.01]

Q.3 [25.90, Range 25 to 26.5] Q.21 [C]

Q.4 [30.52, Range 30.45 to Q.22 [4, Range 3.95 to 4.05]

30.60] Q.23 [30, Range 29.5 to 30.5]
Q.5 [A, B, D] Q.24 [D]
Q.6 [9.31, Range 9.28 to 9.35] Q.25 [C]
Q.7 [D] Q.26 [A]
Q.8 [A, D] Q.27 [B]
Q.9 [0.667, Range 0.63 to 0.69] Q.28 [B]
Q.10 [B] Q.29 [C]
Q.11 [D] Q.30 [C]
Q.12 [1.106, Range 0.98 to 1.12] Q.31 [A]
Q.13 [16.3, Range 16 to 16.5] Q.32 [D]
Q.14 [B]

Q.15 [C]

Q.16 [C]
 3 Routh-Hurwitz Stability

MCQ, MSQ and NAT

Q.1 The feedback system shown below

oscillates at 2 rad/s when

(A) K 2 and a  0.75

(A) the origin of the G(s) -plane once in
(B) K  3 and a  0.75
the counter-clockwise direction.
(C) K 4 and a  0.5
(B) the origin of the G(s) -plane once in
(D) K 2 and a  0.5
the clockwise direction.
Q.2 A polynomial
(C) the point −1+j0 of the G(s) -plane
f  x   a4x  a3x  a2x  a1x  a0
4 3 2
once in the counter-clockwise direction
with all coefficients positive has (D) the point −1+j0 of the G(s) -plane
(A) no real roots once in the clockwise direction.
(B) no negative real root Q.4 The loop transfer function of a negative
(C) odd number of real roots feedback system is
(D) at least one positive and one K(s  11)
G(s) H(s) = .
negative real root s(s  2)(s  8)
Q.3 The pole-zero map of a rational The value of K, for which the system is
function G(s) is shown below. When marginally stable, is __________.
the closed contour  is mapped into the Q.5 A unity feedback system that uses
G(s) -plane, then the mapping encircles: proportional-integral (PI) control is
shown in the figure.
 Q.8 Consider a unity negative feedback
The stability of the overall system is system, with open loop transfer
controlled by tuning the PI control
function as G(s) 

K s2  1  (a is
parameters KP and KI. The maximum
s  1s  a 
value of KI that can be chosen so as to
positive constant). If the system is
keep the overall system stable or, in the
stable for K >–1, then which of the
worst case, marginally stable (rounded
following is correct?
off to three decimal places) is _______.
(A) a < 2 (B) 0 < a < 1
Q.6 For a unity feedback control system, the
(C) a > 1 (D) No such value
open loop transfer function is given by
Q.9 The number of roots with the positive
e2s
G(s)  . If the system is stable, real parts does the characteristic
s(s p)
equation has –
then the possible value of
s3 + s2 + 2s + 1 = 0
p is
(A) One (B)Two
(A) 1.3
(C) Three (D)Zero
(B) 2.01
Q.10 The first column of Routh table consists
(C) 2
of the terms –2, –4, 6, –2.2, 1. Find the
(D) 1
number of unstable poles for the
Q.7 Consider the unity feedback system
respective system.
which employs feedback as shown in
(A) 1 (B) 2
the figure.
(C) 3 (D) 4
Q.11 The characteristics equation of a system
is given by s3 + s2 + 9s + 9 = 0. The
undamped natural frequency of the
system is _________ rad/sec. (Rounded
The frequency of oscillation of the off to nearest integer)
above system is Q.12 The characteristic equation of the
(A) ab  1 rad/sec system is given as:
q(s) = s4 + 6s3 + 14s2 + 16s + 8.
(B) ab rad/sec
The largest time constant of the
(C) ab  1 rad/sec characteristic equation that is greater
(D) ab + 1 rad/sec than 1 sec is ______sec. (Rounded off
to nearest integer)
Q.13 The number of poles lying on the left-
hand side of the jω axis for the
characteristic equation s6 + 2s5 + s4 – s2
– 2s – 1 = 0 is _______. (Rounded off
to nearest integer)
Q.14 The forward path transfer function for a
unity negative feedback system is:
The coefficients x, y, z and K are
Ke 0.2s
Gs  greater than zero. For critical stability,
1 s
The value of K for the system to be s1
stable is- K
the value of should be ________.
(A) K > 5 (B) –1 < K < 5 (Up to three decimal place)
(C) K < –1 (D)–1 < K < 0 Q. 17 For the equations s3 – 4s2 + 5 + 6 = 0
Q.15 Forward path transfer function of a the number of roots in the left half of s-
closed loop system is given as- plane will be
Gs  ; H s   1
K (A)Zero
s  s  1  s  9 
(B)One
When the gain of the system is set at the
(C)Two
finite value, the oscillation of the
(D) Three
system occurs at _________ rad/sec.
Q.18 For a feedback control system as shown
(Rounded off to nearest integer)
in figures. The system is stable for all
Q.16 The Routh array of a characteristic
positive values of k (i.e. for k > 0), if
equation is given as-
the value of T is greater than

The plot of the system is- Q.19 Consider the system shown in the figure
below. The value of system gain k for
which of the system becomes
oscillatory is .
 (D)0 < A < 12
Q.23 For a control system, the characteristic
equation is
s4 + 2s3 + 3s2 + 4s + 5 = 0
Then the nature of the system and the
number of poles at RHS respectively is
Q.20 A unity feedback system having open
……… .
loop gain, becomes stable when
(A) unstable, 3 (B) stable, 0
k(1 s)
G(s)H(s)  (C) unstable, 2 (D) unstable, 1
1 s
Q.24 A control system with the characteristic
(A) | k | > 1
equation given as,
(B) k > 1
(C)| k | < 1 s4 + 3s3 + 16s2 + 6s + K = 0
Then the value of K for which the
(D)k < -1
system is unstable is ……… .
Q.21 If the loop gain k of a negative
(A) 14 (B) 7
feedback system having a transfer
(C) 30 (D) 26
function k(s  3) , to be adjusted in
(s  8)
2
Q.25 For what positive value of k does the
induce a sustained oscillation then. polynomial,
(A)The frequency of this oscillation 2s4 + 8s3 + 12s2 + 24s + K = 0
must be 4 1 rad/sec have roots with zero real parts?
(B)The frequency of this oscillation Q.26 From the following statements given
must be 4 rad/sec below, which of these are true regarding
(C)The frequency of this oscillation the Routh-Hurwitz criterion?
must be 4 (or) 4 3 rad/sec (A) It gives the number of closed loop
(D)Such a k does not exist poles lying on the right half of the s-
Q.22 Consider the following equation given plane.
below, (B) It gives number of zeros lying on
s3 + 3s2 + 4s + A = 0. the right half of s-plane.
All the roots of this equation are in the (C) It gives the gain margin and the
left half of the s-plane provided that phase margin.
condition is___. (D) It provide absolute stability.
(A)5 < A < 12 Q.27 For the closed-loop system shown
(B) A > 12 below,
(C) –3 < A < 4
 (B) There is one root in the left half of
s-plane.
(C) There is one root in right half of the
s-plane.
Then the characteristic equation is - (D) there are three roots in the left half
(A) s2 + 2s + 120 = 0 of the s-plane.
(B)s2 + 2s + 124 = 0 Q.31 For the characteristic equation given
(C) s2 + 4s + 124 = 0 below,
2
(D)s + s + 124 = 0 s4 + Ks3 + s2 + s + 1 = 0
Q.28 For a unity feedback control system, the Then the range of k for stability is -
open-loop transfer function is given as (A) K > 1 (B) K < 1
– (C) K > 0 (D) none of these
K
G(s) = s(s+1)(s+2), Q.32 For a control system, the unity feedback
system is given as –
Then the system is stable if,
K
(A) 0 < K < 6 G(s) = (s+1)(s2+s+1)

(B) K > 6 Then the value of frequency of

(C) K > 0 oscillation (in rad/sec) for a marginally
(D) system is always unstable stable system is____.
Q.29 A system with the characteristic 1
(A) (B) 2
equation is given by – 2

s6 + s5 + 2s4 + 3s3 + 10s2 + 4s + 2 = 0 (C) 2 2 (D) 2

Then the number of the roots of the Q.33 An LTI system having a characteristic
equation that lies on the imaginary axis equation in negative unity feedback
of s-plane is - configuration is shown below –
(A) 4 (B) 3 s4 + 2s2 + 4 = 0
(C) 0 (D) 2 Then the number of poles located on j𝜔
Q.30 In Routh’s tabular method, the first axis is ______.
column elements are given as – Q.34 The characteristic equation of a control
1 2 system is s3 + 4s2 + 6s + 13 = 0
4,  2,3, ,
4 7
Then the number of roots of the
It means that equation which lie to the left of the line
(A) There are two roots in the right half
s + 1 = 0.
of s-plane.
Q.35 The characteristics equation of a system
is given by
3s4 + 10s3 + 6s2 + 11s + 2 = 0
The system is
(A) stable
(B) marginal stable
(C) unstable
(D) insufficient data
Q.36 The value of ‘K’ for which unity
feedback control system crosses the
imaginary axis is_____.
Q.37 The number of roots of the
characteristic equation that lies on the
right half of the s-plane.
3s4 + 2s3 + 3s2 + 5s + 10 = 0
Q.38 The stable unity feedback control
system whose open loop transfer
function is given below. The relation
between K1 and K2 is
K1
G  s 
s  s  s  5  K 2 

(A) K1< 10 K2
(B) 0 < K1< 5K2
(C) K1< 5K2
(D) 0 < K1< K2
Q.39 Which of the following characteristics
polynomial of a system shows stable
operation in the feedback system?
(A)s3 + 4s2 – 6s + 1 = 0
(B)s4 + 2s3 + s + 9 = 0
(C)s3 + 6s2 + 5s + 30 = 0
(D)s4 + 8s3 + 24s2 + 32s + 30 = 0
Answer Key Q.19 [6.67, Range 6.5 to 6.75]

Q.20 [C]

Q.1 [A] Q.21 [B]

Q.2 [D] Q.22 [D]

Q.3 [B] Q.23 [C]

Q.4 [160, Range 159 to 161] Q.24 [C]

Q.5 [3.125, Range 3 to 3.2] Q.25 [18, Range 17.5 to 18.5]

Q.6 [B] Q.26 [A, D]

Q.7 [C] Q.27 [C]

Q.8 [C] Q.28 [A]

Q.9 [B] Q.29 [C]

Q.10 [C] Q.30 [A]

Q.11 [3] Q.31 [D]

Q.12 [1] Q.32 [D]

Q.13 [3] Q.33 [0]

Q.14 [B] Q.34 [1]

Q.15 [3] Q.35 [A]

Q.16 [0.141, Range 0.1 to 0.2] Q.36 [120, Range 119 to 121}

Q.17 [B] Q.37 [2]

Q.18 [1] Q.38 [B]

Q.39 [D]

MCQ, MSQ and NAT

 4 Polar Plot and Root Locus
Diagram

MCQ, MSQ and NAT

Q.1 The polar plots of two systems,

G1(s) and G2(s) are shown below. (c)

(D)

The result of G1(s) xG2(s) is-

(A)

Q.2 A control system with unity

feedback has a forward path
transfer function

G s 
K
s  s  1
(B) . If the closed-
loop frequency response has a
peak resonant magnitude of 4,
then the 3dB bandwidth is
_______rad/sec. (Up to three
decimal places)
Q.3 The open loop transfer function for (D)
a system is given as

3  7s  5s  2

 s  2   s  1 s  5s  7  .
2
The
point where the asymptotes
intersect the real axis lies at
_________. (Up to two decimal Q.5 The open-loop transfer of a system
places) is-
Q.4 Determine the root locus of the
Gs H s 
s  1 1 0.2s 
system if closed loop transfer s  1 1 4s 
function is
The polar plot for the transfer
T s 
K
function is-
s2  K  8  s  1
. (A)
(A)

(B)
(B)

(C)

(C)

(D)
Q.6 The open-loop transfer function of
a unity feedback control system
is-
K
G(s) 
(s  6)(s  5) (B)
The value of K at the breakaway
point of the root locus plot of the
system is _____. (Up to two
decimal places)
Q.7 An open loop system is given
below. The centroid of the
asymptotes is-
k(s  2)
G(s) 
s(s  2s  2)(s2  2s  5)
2

(A) –0.5 (B) –1 (C)

(C) 1 (D) 2
Q.8 The open loop transfer function of
a system is given by-
K  s  2  s  4 
Gs H s 
s2  6s  13
The root locus diagram is-
(A)

(D)
 Q.10 In the root locus plot shown in the
figure, the pole/zero marks and the
arrows have been removed. Which one of
the following transfer functions has this
Q.9 The unity feedback function is root locus?

Gs 
1
s  s  2  s  4 
given as
. The plot of G(jω)H(jω) for 0
<ω<∞ is-

(A)

s1
(A)
(s  2)(s  4)(s  7)

s4
(B)
(B) (s  1)(s  2)(s  7)

s7
(C)
(s  1)(s  2)(s  4)

(s  1)(s  2)
(D)
(s  7)(s  4)

(C) Q.11 The characteristic equation of a

system is

s3+3s2+(K+2)s+3K=0 .
In the root locus plot for the given
system, as K varies from 0 to , the
(D) break-away or break-in point(s) lie
within
 (A) (–1, 0) (B) (–2, –1) asymptotic lines intersect at a point
(C) (–3, –2) (D) (–, –3) on Real axis.
Q.12 Consider the following systems: Which of the above is correct?
1 (A) S1, S2, S3 and S4
System 1: G(s) 
2s  1 (B) S3 and S4 only

1 (C) S2, S3 and S4 only

System 2: G(s) 
5s  1 (D) S1 and S2 only

The true statement regarding the Q.14 The closed loop transfer function

two given systems is of a system is –

K  s  8
(A) Bandwidth of system 1 is G  s H  s 
 s  2  s  3 
greater than the bandwidth of
system 2 Then the centroid of the root locus

(B) Bandwidth of system 1 is lower is located as -

than the bandwidth of system 2 (A) 4 (B)3

(C) Bandwidth of both the systems (C)–3 (D)2

is same Q.15 For the determination of the

(D) Bandwidth of both the systems transient response and the stability

is infinite of a system, the strongest method

Q.13 Consider the following statements used is -

about Root locus: (A)Routh Hurwitz

S1 : The root locus is symmetrical (B)Polar plot

with respect to both (real and (C)Bode

imaginary) axis. (D)Root locus

S2 : The root locus start from (K = Q.16 An open loop transfer function is

0) from the open loop poles/zeros given as-

K(s  1)(s  5)
and terminates (K = ) on either G(s)H(s) 
(s  8)(s  3)(s  2)
finite open loop zeros or infinity.
Find the centroid of the root locus.
S3 : If root locus intersect at
(A) +3 (B) –2
imaginary axis, the point of
(C) –1 (D) +2
intersection are conjugate,
Q.17 The open loop transfer function of
S4 : For higher values of K, root
a unity feedback system is-
locus can be approximated by
K2 (2s  3)
asymptotic lines and these G(s)H(s) 
s2 (s  5)
 In root locus plot if K varies from 0
to , find the value of K, when all
roots are real and equal-
125
(A) K  5 5 (B) K 
9 81

125
(C) K  (D) K  3 5
8 81
Q.18 The polar plot of an open loop
stable system is shown below. The
closed loop system is. The gain margin & the phase margin of the
system are respectively.
(A) 4, 60o
(B) 4, 140o
(C) 2, 60o
(D) 2, 140o
Q.20 The unity negative feedback
control system has an open loop
(A) Always stable transfer function is
(B) Marginally stable  4
k s  
3
(C) Unstable with one pole on the G(s)  2 , the values of ‘k’
s (s  12)
RHS plane
(D) Unstable with two poles on the such that all the closed loop poles

RHS plane are equal will be

Q.19 The polar plot of a negative Q.21 The root locus of a unity feedback

feedback, open loop control system control system is shown below,

as shown below

Then the open loop transfer

function is-
 K  s  5 K s  8
(A) (B)
 s  2 s  4
K s  8 K s  8
(C) (D)
s  4 s  4
Q.22 Which of the following properties
of the breakaway point in the root
locus of a closed loop control
system with characteristic equation
1 + KG(s) H(s) = 0 is/are correct?
dK
(A) At breakaway point, 0
ds

dK
(B) At breakaway point, 0
ds
(C) The breakaway point can lie
only on the real axis.
(D) At the breakaway point, K is
maximum for the root locus on the
real-axis.
Answer Key Q.21 [D]

Q.22 [A, D]
Q.1 [C]

Q.2 [6.096, Range 6.0 to 6.180]

Q.3 [-3.3, Range -3.5 to -3.01]

Q.4 [C]

Q.5 [D]

Q.6 [60.5, Range 59.5 to 61.5]

Q.7 [A]

Q.8 [C]

Q.9 [B]

Q.10 [B]

Q.11 [A]

Q.12 [A]

Q.13 [B]

Q.14 [B]

Q.15 [D]

Q.16 [A]

Q.17 [A]

Q.18 [D]

Q.19 [B]

Q.20 [48, Range 47.5 to 48.5]

MCQ, MSQ and NAT



5 Nyquist Stability Criterion

MCQ, MSQ and NAT

Q.1 The forward path transfer function

Gs 
K
is given as s that has

H s 
1
feedback of s s2.
2
The Nyquist plot can be given as-

(A)
(D)

(B) Q.2 The open loop transfer function of

a unity feedback control system G(s)
is unstable and has three poles on right
hand side of s-plane, but the
closed loop system is stable. The number
of encirclements made by the
Nyquist plot about (–1,0) point in the G(s)
plane is ________. (Rounded
(C)
off to nearest integer).
Q.3 Consider the feedback system
shown in the figure. The
Nyquist plot of G(s)is also
shown. Which one of the
 following conclusions is is 60. So, the gain margin (in dB)
correct? of the system is -
(A)16.01 (B)0.602
(C)3.82 (D)6.02
Q.6 For an unstable system, the phase
margin and the gain margin are -
(A)G(s) is an all-pass filter
(A)Positive (B)Zero
(B)G(s) is a strictly proper transfer
(C)Negative (D)Not
function
determined
(C)G(s) is a stable and minimum-
Q.7 The unity feedback has an open-
phase transfer function
loop transfer function given as
(D)The closed-loop system is
20
unstable for sufficiently large
s  4 
4

and positive k . The gain margin is

Q.4 The complete Nyquist plot of ________ dB.
the open-loop transfer function G(s) (A)25 (B)34
H(s) of a feedback control system is (C)38 (D)42
shown in the figure. Q.8 If the open-loop transfer
K
s  s  4  s  x 
function intersects at s =
±j20,then the phase crossover frequency
(in rad/sec) is_________
(Rounded off to the nearest integer)

Q.9 The open loop transfer function of

a feedback control system is-
If G(s) H(s) has one zero in the
(s  1)
right-half of the s-plane, the G(s)  H(s) 
s(s  3)(s  5)
number of poles that the closed-
The gain margin of the system is
loop system will have in the right-
_____(Up to two decimal places)
half of the s-plane is
(A)0 (B)1
Q.10 The open loop transfer function
(C)4 (D)3
with unity feedback is given as-
Q.5 The system is operating at the gain
of 30 and the critical value of gain
 (B) Unstable with two righthand
Gs 
K
s s  
2
pole
The gain margin is 21.94 dB and (C) Unstable with three right hand
the phase cross over frequency is 5 poles
rad/sec. The value of 'K' is (D) Unstable with four right hand
________. (up to two decimal poles
places)
Q.11 Consider a unity feedback system
with open loop transfer function
given as
1 10s
Gs 
s 2
 1 s 1 5s 
Which of the following
statement(s) is/are TRUE?
(A) The phase crossover frequency
is 2.5 rad/sec
(B) The phase crossover frequency
is 0.4 rad/sec
(C) The gain margin is 10.7
(D) The gain margin is 0.048
Q.12 The Nyquist plot of G(s)H(s) is
given below.

IfG(s)H(s) has one right hand pole,

then the corresponding closed loop
system is-
(A) Stable
Answer Key

Q.1 [D]

Q.2 [3]

Q.3 [D]

Q.4 [D]

Q.5 [D]

Q.6 [C]

Q.7 [B]

Q.8 [20, Range 19 TO 21]

Q.9 [13.33, Range 12.8 to 13.80]

Q.10 [19.99, Range 19.5 to 20.5]

Q.11 [B, D]

Q.12 [C]


 6 Bode Plot

MCQ, MSQ and NAT

Q.1 The Bode plot of a transfer

function G(s) is shown in the figure below.

If the system is connected in a

unity negative feedback
configuration, the steady state error
The gain (20 log |G(s)|) is 32 dB and
of the closed loop system, to a unit
8 dB at 1 rad/s and 10 rad/s ramp input, is ________.
respectively. The phase is negative for Q.4 For an LTI system, the Bode plot
all . Then G(s) is
for its gain is as illustrated in the
39.8 39.8
(A) (B) figure shown. The number of
s s2
system poles Np and the number of
32 32
(C) (D) 2 system zeros NZ in the frequency
s s
Q.2 The phase margin in degrees of range 1Hz f 107 Hz is:

10
G(s) 
(s  0.1)(s  1)(s  10)
calculated using the asymptotic
Bode plot is ______.
Q.3 The Bode asymptotic magnitude
plot of a minimum phase system is
shown in the figure.
 (A)NP = 4, Nz = 2
(B) NP = 7, Nz = 4
(C) NP = 6, Nz = 3
(D) NP = 5, Nz = 2
Q.5 The asymptotic bode plot of the
20
transfer function s  2s  3 is
2

shown in the figure below. The The transfer function of the system is-
magnitude of error (dB) at a s1
 s  1
2
frequency of   2 rad / sec is (A)
________. (up to 3 decimal places) 1 s
(B) 1 s

(C)   s  1
s  1

1
s  s  1
2
(D)
Q.6 The bode plot of a system is shown
Q.8 A Bode magnitude plot shown in
below.
the figure below, the transfer function
of the system is-

The value of ω1 + ω2 is _________

rad/sec. (Up to 3 decimal places)
Q.7 The bode plot of a system is shown
in the figure below. 10(s  1)(s  10)(s  10000)
H(s) 
(A) (s  100)2 (s  1000)2

10 10 (s  1)(s  10)(s  10000)

H(s) 
(B) (s  100)2 (s  1000)2

10 6 (s  1)(s  10)(s  1000)

H(s) 
(C) (s  100)2 (s  10000)2
 10 6 (s  1)(s  10)(s  10000) (A)
H(s) 
(D) (s  100) (s  1000)
2 2

Q.9 The asymptotic bode plot is shown

in the figure below.

(B)

The transfer function of the system G(s) is-

1000
 s  s 
 1   1 
(A)  10   200 
1000

(B) s  10 s  200  (C)

3
 s   s 
 1   1 
(C)  10   200 
3

(D)  s  10 s  250 

Q.10 If a transfer function consists of 1
zero and 4 poles, the final slope of the
bode magnitude plot is ________ (D)
dB/decade. (Rounded off to nearest
integer).
Q.11 The bode magnitude plot
104 (1 j)
H(j)  is
 10  j (100  j)

Q.12 The Asymptotic Bode Magnitude

plot of a minimum phase transfer function
is shown below. The difference between (P (A)
– Z) poles and zeroes is ________  s
2
 s 
 1   1 7 
 10   10 
2
 s  s  s 
 1   1   1  
 100   104   105 

(B)
2
 s  s 
10  1   1 5 
 10   10 
2
 s  s  s 
 1   1 4   1 7 
 100   10   10 

(C)
2
 s  s 
10  1   1 7 
 10   10 
(A) 0  s  s  s 
2

 1   1 4   1 5 
(B) 1  100   10   10 

(C) 2 (D)
2
(D) 3  s  s 
10  1   1 5 
Q.13 For the bode plot given, assume all  10   10 
2
 s  s  s 
poles and zeros are real valued.  1   1   1  
 100   104   107 

Then the value of f2 – f1 is ___________

kHz.(Upto 2 decimal places)

Q.14 The transfer function for minimum

phase system whose the bode
magnitude plot given below is -
Answer Key

Q.1 [B]

Q.2 [45, Range 42 to 48]

Q.3 [0.5, Range 0.45 to 0.55]

Q.4 [C]

Q.5 [3.010, Range 3 to 3.020]

Q.6 [78.528, Range 77 to 79]

Q.7 [B]

Q.8 [D]

Q.9 [A]

Q.10 [-60, Range -61 to -59]

Q.11 [A]

Q.12 [C]

Q.13 [49.99, Range 49.5 to 50.5]

Q.14 [C]


7 State Space Analysis

MCQ, MSQ and NAT

Q.1 The state variable description of an The state-variable equations of the

LTI system is given by system shown in the figure above
 x1   0 a1 0   x1   0  are
      
 x2    0 0 a2   x2    0  u  1 0   1
 x   a 0 0   x   1 (A) X    X   1 u
 3  3  3    1 1  

 x1  y = [1 –1] X + u
 
y   1 0 0   x2   1 0   1
x  (B) X    X   1 u
 3  1 1  
where y is the output and u are the y = [–1 –1] X + u
input? The system is controllable  1 0   1
(C) X    X   1 u
for  1 1  

(A) a1  0,a2  0,a3  0 y = [–1 –1] X – u

(B) a1  0,a2  0,a3  0  1 1 1

(D) X    X   1 u
 0 1  
(C) a1  0,a2  0,a3  0
y = [1 –1] X – u
(D) a1  0,a2  0,a3  0
Q. 3 The state diagram of a system is
Q.2 The state diagram of a system is
shown below. A system is
shown below. A system is
described by the state-variable
described by the state-variable
equations
equations
X  AX  Bu; y  CX  Du
X  AX  Bu; y  CX  Du
 The state transition matrix eAt of (A) x1 t  1,x2 t  2

the system shown in the figure (B) x1 t  e t ,x2 t  2e t

above is
(C) x1 t  e t , x2 t  e 2t
 e t 0
(A)  t t  (D) x1 t  e t ,x2 t  2e t
 te e 
Q.6 Consider the state space system
 e t 0
(B)  t 
 te e t  expressed by the signal flow
diagram shown in the figure.
e t 0 
(C)  t 
e e t 

e t te t 
(D)  
0 e t 

Q.4 Consider the state space model of a

system, as given below The corresponding system is
(A)always controllable
 x1   1 1 0  x1  0  x1 
x    0 1 0  x   4 u;y  [1 1 1]x  (B)always observable
 2    2    2
x3   0 0 2 x3  0  x3  (C)always stable
. (D)always unstable

The system is
(A)controllable and observable Q.7 The state equation of a second-
(B)uncontrollable and observable order linear system is given by
(C)uncontrollable and x(t )  Ax(t ),x(0)  x0
unobservable 1
x0 = [ ] ,x(t)= [ e -t ] and for
-t
For
(D)controllable and unobservable -1 -e
&0
] ,x(t)= [ e-t -e -2t ]
-t -2t
x0 = [
&1 -e +2e
Q.5 &3
When x0 = [ ] ,x(t) is.
&5
An unforced linear time invariant (LTI) system is represented by
x 1   1 0  x 1   8e t  11e2t 
    (A)  t 2t 
x 2   0 2   x 2   8e  22e 

If the initial conditions are

 11e t  8e2t 
x1  0   1 and x2  0   1 , the (B)  t 2t 
 11e  16e 
solution of the state equation is
  3e t  5e2t  Q.9 For the given circuit, which one of
(C)  t 2t 
 3e  10e  the following is the correct state
equation?
 5e  3e 
t 2t

(D)  t 2t 
 5e  6e 

Q.8 Let the state-space representation


of an LTI system be x(t)= Ax(t) +
Bu(t),

y(t) = Cx(t) + du(t) where A, B, C

(A)
are matrices, d is a scalar, u(t) is
the input to the system, and y(t) is
its output. Let B = [0 0 1]T and d = d  v  4 4   v 0 4  i1 
     
dt i   2 4 i  4 0  i2 
0. Which one of the following
options for A and C will ensure
(B)
that the transfer function of this
LTI system is H(s) =
d  v  4 4  v 4 4 i1 
1      
? dt i   2 4  i  4 0  i2 
s  3s  2s  1
3 2

(C)
0 1 0
(A) 
A=  0 0 1  and C = [1 0 0]
 1 2 3 d  v  4 4  v 0 4  i1 
     
dt i   2 4 i  4 4 i2 
0 1 0
(B) 
A = 0 0 1  and C = [0 0 1]
 1 2 3 (D)

0 1 0
(C) A =  0 0 1  and C = [0 0 1]
 d  v  4 4  v 4 0  i1 
     
 3 2 1 dt i   2 4 i  0 4 i2 

0 1 0
(D) A =  0 0 1  and C = [1 0 0]
 Q.10 The electrical system shown in the
 3 2 1 figure converts input source current
is(t) to output voltage v0(t).
 Q.12 The state model of a system is
given as
  0 1  1
X  X   u
 2  2 2

Y = [1 -1]x
The above system is
(A) A non-minimum phase system

Current iL(t) in the inductor and (B) A minimum phase system

voltage vC(t) across the capacitor (C) An all-pass system

are taken as the state variables, (D) None of the above

both assumed to be initially equal Q.13 A linear time invariant system of

to zero, i.e., iL(0) = 0 and vC(0) = 0. second order has state space

The system is equations -

x1  4x1  x2  3u
g
(A)completely state controllable as
well as completely observable
x2  4x2  u
g

(B)completely state controllable

y = 3x1 – 2x2
but not observable
Then the output response of the
(C)completely observable but not
system is -
state controllable
(A)Critically damped
(D)neither state controllable nor
(B)Underdamped
observable
(C)Undamped
Q.11 The state matrix in a state space
(D)Overdamped
representation is given by
Q.14 The state space representation of a
1 0 0 
  system is given by-
A  0 2 0  . The closed loop
0 0 3   3 0  0 
  x    u
 1 2  1
poles of a system having state
y=[1 1]x
matrix [A]-1 will be
the system is-
(A) 1, 2, 3
(A) Controllable and observable
(B) -1, -2, -3
(B) Uncontrollable and
1 1
(C) 1, , Unobservable
2 3
(C) Uncontrollable and observable
(D) 1, 4, 9
(D) Controllable and unobservable
 The system represented above is
Q.15 The state space representation of a said to be-
system is given as- (A) Controllable and observable
0 1  1 (B) Observable but not controllable
x  x   u
 2 1 0  (C) Controllable but not observable
y = [ 0 1]x (D) Neither controllable nor
then the transfer function of the observable
Y(s) Q.18 A system is described by the state
system will be-
U(s)
model as shown below.
2(s  1)
(A)  1 1 1   x 1  0 
s s2
2      
x   1 0 1 x 2    1 u 
s1  1 1 0  x  0 
(B)    3  
s s2
2

y  0 1 1 x
2(s  1)
(C)
s2  s  2 Find the number of closed-loop
2 poles having positive real part.
(D)
s2  s  2 Q.19 A control system is represented by
Q.16 Determine the characteristic the state model as-
equation of the given matrix,  
 x 1    2 0   x 1   0  u
0 2 1     0 8  x  2 
  x 2     2   
A  2 1 0 
3 2 1
  The response of the system for the
(A) s3 – 9 = 0 unit step input is-
(B) s3 – s + 9 = 0 (A) Overdamped
(C) s3 – s – 9 = 0 (B) Underdamped
(D) s – 7s – 9 = 0
3
(C) Critically damped
Q.17 For a control system the state block (D) Undamped
diagram is given below as- Q.20 Consider a system G(s) represented
by the state model given below.
0 3  0 
A  B 
1 2 ,  1 and

C  2 2

If unity feedback is provided to

G(s), then which of the given
 statement(s) is/are true for the (A)–1, –2 and –3
system? (B)6, 11 and –5
(A)The steady-state error for the (C)–16, 5 and –11
unit ramp is infinite. (D)None of the above
(B)The steady-state error for the Q.23 A network is shown in the figure
unit ramp is zero. below:
(C)The steady-state error for the
unit step is 3.
(D)The steady-state error for the
unit step is 0.33.
Q.21 The forward path transfer function
 1 
 1 
Gs   s  1 
 1 1  The minimum number of states
is  s  2  and the
required to describe the network is
transfer function of the feedback
_____.
 1 0
H s      2 0 
x x
path is defined as 0 1 .  0 1
Q.24 A linear system, is
Then which of the following having an initial condition as
statement(s) is/are true?
 1
x(0)   
(A)The transfer function of the
2  . The system x(t) is-
s2  3s  3
T s  e2t 0
system is s2  5s  6  
(A)  0 et 
(B)The transfer function of the
et 0  2
s2  s  3
T s    
system is s2  5s  6 . (B)  0 e2t  1

(C)The damping ratio of the system e2t 0 

 
is 0.416.  0 et 
(C)
(D)The damping ratio of the
e2t 0  1
system is 1.02.   
(D)  0 et  2
Q.22 The eigenvalues of the matrix
Q.25 The representation of a system in
0 1 1
  
A   6 11 6 
 6 11 5 
state space is X  Ax  Bu , where
  are:
 2 4   1 Q.28 The transfer function of a system is
A  B 
a 8  and  1 . s  30
T(s) 
given by s  11s  30 . The
2
The value of ‘a’ for which the
state-space representation of the
system is uncontrollable is ______

.(Rounded off to the nearest transfer function is x  Ax  Bu
integer)  0 1 0 
A  B  
Q.26 The state transition matrix is  30 11 1 .
where and
 0 3  The sum of the elements of the
x(t )    x(t )
1 4  . The inverse of matrix C is-
the state transition matrix is- (A) 0 (B) 1
 e3t  3et 3e3t  3et  (C) 31 (D) 42
 3t 
3e3t  et 
(A)  e  e
t
Q.29 The matrices of a state space model

 1 3t 3 t 3 3t 3 t  0 4  0 
A , B   
 e  e  e  e  2 6 
 2 2 2 2  are given by   1
 1 e3t  1 et 3 3t 1 t 
e  e
(B)  2 
2 2 2 and
C = [1 0].
The location of poles of the system
 1 3t 3  t 3 3 
 e  e  e3t  e t 
is-
 2 2 2 2 
 1 e3t  1 e t 3 3t 1  t 
e  e (A) –4, –2 (B) 0, –4
(C)  2 2 2 2 
(C) –2, –6 (D) 4, 2
 3 3t 1 t 3 3t 3 t 
 e  e e  e  Q.30 The state-space model of the two
 2 2 2 2 
 1 e3t  1 e t e  et 
3 3t 1 systems is given below.
(D)  2 2 2 2 
SYSTEM 1:
Q.27 A difference equation is used to
 
x1   0 1   x 1  0 
    5 6  x    1 u 1  t 
represent a discrete system as-
x 2     2   
  1   5 
X(k  1)    X(k)
    x 
y 1   1 1  1 
The initial conditions X1(0) = X2(0) x 2 
= 0. If α = 2,then the position of SYSTEM 2:
pole that makes the system  
 x 3    4 1   x 3   0  u t
unstable is ______ . (Up to three     0 3  x  2  2  
x 4     4   
decimal places)
 x  Q.32 The state-space representation of a
y 2   1 2   1 
x 2  system is given by-

The two systems are cascaded as  5 0  0

x     u
shown below.  1 4 1
y = [1 –1]x
SYSTEM  1 y1t u2 t SYSTEM  2
Input   Output
T1  s T2 s The system is-
(A) Controllable

The transfer function of the resultant (B) Observable

system is- (C) Uncontrollable

(D) Unobservable
T s 
2
(A) s2  7s  11 Q.33 Determine the characteristic
4s  18 equation of the given matrix.
T s 
(B) s  12s  47s  60
3 2
0 3 1
 
A  3 2 0 
4  s  1
T s  5 1 2 
 
(C) s4  13s3  59s 2  107s  60
(A) s3– 31 = 0
2  2s  9  s  5 
T s  (B) s3 – 18s + 31 = 0
(D) s  13s3  59s 2  107s  60
4

(C) 2s3 – 6s – 9 = 0
Q.31 A linear time-invariant system has
(D) s3 – 12s – 18 = 0
the state and the output
Q.34 Find the state transition matrix for
equations
the given matrix, where X and Y
given as-
are the real numbers.
 
 x 1   2 1  x 1   0  X Y
   0 3  x   u A 
x 2     2   1 Y X 

x  (A)
y   1 1  1 
x 2  eXt cosYt eXt sin Yt
(t)   Xt Xt

If the initial conditions are given as  e sin Yt e cosYt
x1(0) = 3, x2(0) = –2 and u(0) = 0. (B)
dy  t 
 eXt sin Yt eXt cosYt
The value of dt at t = 0 is (t)   Xt  Xt

e cosYt e sin Yt
________ . (Rounded off to the
nearest integer)
 (C)

eXt cosYt eXt sin Yt 

(t)   Xt  Xt

 e sin Yt e cosYt
(D)

 eXt sin Yt eXt cosYt

(t)   Xt Xt

e cosYt e sin Yt 
Q.35 A system is represented by

X  AX  BU , where
0 1 0 
 
A  0 0 1
 
 5 K 11
. The range of K
for stability is-
5
0 K
(A) 11

5
K
(B) 11

11
K
(C) 5

5
K
(D) 11
Q.36 The signal flow diagram is shown
in the figure below.

The determinant for observability

is _______. (Rounded off to
nearest integer).
Answer Key Q.21 [A, D]

Q.22 [A]

Q.1 [D] Q.23 [3]

Q.2 [A] Q.24 [D]

Q.3 [A] Q.25 [-2, Range -2.1 to -1.9]

Q.4 [B] Q.26 [B]

Q.5 [C] Q.27 [5.274, Range 5.05 to 5.39]

Q.6 [A] Q.28 [C]

Q.7 [B] Q.29 [A]

Q.8 [A] Q.30 [B]

Q.9 [A] Q.31 [2, Range 1.9 To 2.1]

Q.10 [D] Q.32 [B, C]

Q.11 [C] Q.33 [A]

Q.12 [A] Q.34 [A]

Q.13 [A] Q.35 [B]

Q.14 [B] Q.36 [1]

Q.15 [D]

Q.16 [A]

Q.17 [B]

Q.18 [1]

Q.19 [A]

Q.20 [A, D]

 8 Controllers and Compensators

MCQ, MSQ and NAT

Q.1 G c  s is a lead compensator if

(A) a  1,b  2
(B) a  3,b  2
(C) a  3,b  1
Q.4 Consider the following statements:
(D) a  3,b  1
I. The steady state accuracy is
Q.2 The phase of the above lead
improved by lag networks.
compensator is maximum at
II. Phase lead compensators can be
(A) 2 rad / s
used to increase the bandwidth.
(B) 3 rad / s III. By using phase lag

(C) 6 rad / s compensators. The speed of the

response is reduced.
(D) 1/ 3 rad / s
IV. Phase lead compensation can
Q.3 Consider a unity feedback system,
be used to reduce settling time.
as in the figure shown, with an
Which of the above statements are
k
integral compensator and open- correct?
s
loop transfer function
(A) I, II and III
1
G(s) = 2 (B) III and IV
s  3s  2
(C) I, II and IV
Where K > 0. The positive value of
(D) All the statements
K for which there are exactly two
Q.5 The transfer function of a network
poles of the unity feedback system
is –
on the j axis is equal to ______
s
(rounded off to two decimal 1
G  s  16
places). s
1
49
 Then the frequency (in rad/sec) at (B) 2 and 3
which the network will provide (C) 1 and 3
maximum phase lead is - (D) 2 and 4
(A)28 (B)3.06 Q.9 The block diagram of a control
(C)1.75 (D)16.81 system is shown below:
Q.6 Effects of integral error
compensation on the various
parameters in a control system is -
(A) It does not have any effect on The damping ratio of the closed-
steady state response. loop system is 0.8 and the damped
(B) The steady state error is natural frequency is 3rad/sec. The
reduced parameters of PD controller are KP
(C) It decreases the type of the KP
system
and KD. The value of KD is
(D) It makes the system more stable
_______ (Up to three decimal
Q.7 The maximum phase shift of the
places)
5(3  0.9s)
compensator Gc (s)  Q.10 The open-loop transfer function of
4  0.4s
a unity feedback control system is
is-
Gs  
1
(A) 30° (B) 60°
s s  2
(C) 45° (D) 90° given as . If the

Q.8 Consider the following statement first-order compensator is used in

about the compensates: the system, then the poles of the

1. Pole is dominant in lag closed-loop system are –2 ± 4j and

compensator. –8. The transfer function of the

2. Pole is dominant in lead compensator is-

compensator.  s  10 
5 
3. Zero is dominant in lag (A)  s  5 

compensator.  s5 
32  
4. Zero is dominant in lead (B)  s  10 
compensator. s  9 s  5
Which of the above statement(s) (C) 5  s  10 
is/are correct?
(A) 1 and 4
 1

(D) 
s  50  s  10 

Q.11 The transfer function of the two

compensators are given as-
10  s  2  s  5
G1  , G2 
 s  5 10  s  1

Which one of the following

statements is correct?
(A) Both G1 and G2 are lag
compensator.
(B) Both G1 and G2 are lead
compensator.
(C) G1 is lead compensator and G2
is a lag compensator.
(D) G1 is lag compensator and G2 is
lead compensator.
Answer Key

Q.1 [A]

Q.2 [A]

Q.3 [6, Range 5.99 to 6.01]

Q.4 [D]

Q.5 [A]

Q.6 [B]

Q.7 [A]

Q.8 [A]

Q.9 [3.571, Range 3.45 to 3.62]

Q.10 [B]

Q.11 [C]