control system 400 objective questions from gate & ies

Objective Questions

Chapter 2

1. For the network shown in Figure P2.43, ViðtÞ is the input and i(t) is the output. The transfer functionI(S)=V(S) of the network is

(a)Cs

LCs2 þ RCsþ 1(b)

C

LCs2 þ RCsþ 1

(c)Cs

RCs2 þ LCsþ 1(d)

C

RCs2 þ LCsþ 1 ½IES 1993�

2. For the field-controlled dc servomotor, as shown in Figure P2.44, the transfer function �ðsÞ=EðsÞcontains(a) Two times constants, no integration (b) Two times constants, one integration(c) One time constants, one integration (d) One time constants, one integration

½IES 1996�

3. Amechanical system consists of twomass-spring friction system, as shown in Figure P2.45. The orderof the transfer function X(s)=F(s) is(a) 1 (b) 2(c) 3 (d) 4

½IES 1996�

R L

j(t) CVi(t)

Figure P2.43 A network (Objective Question 1).

R Ig (Constant)

Singular relation

J1I

L

Figure P2.44 A filed-controlled servomotor.

Figure P2.45 Two mass-spring friction system.

4. Consider a multiple gear system, as shown in Figure P2.46. Which one of the following gives theequivalent inertia referred to shaft 1?

(a) J1 þ J2N1

N2

� �2þ Ja

N1N3

N2N4

� �2(b) J1 þ J2

N2

N1

� �2þ J3

N2N4

N1N3

� �2

(c) J1 þ J2N1

N2

� �2þ J3

N1N2

N3N4

� �2(d) J1 þ J2

N2

N1

� �2þ J3

N1N2

N3N4

� �2

½IES 2004�5. For the mechanical system, shown in Figure P2.47, the system is described as:

(a) Md2y1ðtÞdt2

þ Bdy1ðtÞdt

¼ K y2ðtÞ � y1ðtÞ½ �

(c) Md2y1ðtÞdt2

þ Bdy1ðtÞdt


½IES 2001�

B N

N

BBBN

N

J

J

J

Figure P2.46 A multiple gear system.

B

MK

y2(t) y1(t)

f(t)

Figure P2.47 A mechanical system.

(b) Md2y2ðtÞdt2

þ Bdy2ðtÞdt


(d) Md2y2ðtÞdt2

þ Bdy2ðtÞdt


2 . OBJECTIVE QUESTIONS

Chapter 3

1. Consider a signal flow graph, as in Figure P3.12.

Signal flow graphs, which have the same transfer function, would include:

(a) (i) and (ii) (b) (ii) and (iii)(c) (ii) and (iv) (d) (i) and (iv)

½IAS 1999�2. Consider a signal flow graph shown in Figure P3.13.

1s 2s 3s

1K 2K

3K

(i)1s 2s 3s

1K 2K

3K

(ii)1s 2s 3s1K

2K

3K

(iii)

Figure P3.12 Signal flow graphs for Objective Question 1.

1b

2b

3b

4b

5b 7b

8b 9b 10b

1Z 2Z 6Z


OBJECTIVE QUESTIONS . 3

Consider the following statements regarding the signal flow graph:

(i) There are three forward paths.

(ii) These are three individual loops.

(iii) These are two nontouching loops.

Of these statements:

(a) (i), (ii), and (iii) are correct. (b) (i) and (ii) are correct.(c) (ii) and (iii) are correct. (d) (i) and (iii) are correct.

½IES 1998�

3. Referring to Figure P3.14, match list I (signal flow graph) with list II (transfer function), and selectthe correct answer using the codes given in the following list:

List I List II

A. Figure (i) 1:P

1� Q

B. Figure (ii) 2:Q

1� PQ

C. Figure (iii) 3:PQ

1� PQ

D. Figure (iv) 4:PQ

1� P2



Codes: A B C D A B C D(a) 2 3 4 1 (b) 2 3 1 4(c) 3 2 1 4 (d) 3 2 4 1

½IAS 2000�4. A system block diagram is shown in Figure P3.15.

The overall transfer function of the system is

C

R¼ G1G2G3

1þ G1G2G3H1 þ G2H2 � G3G2H3

The value of X in the figure would be equal to(a) H3 (b) G3H3(c) G2H3 (d) G3H3

½IAS 2001�5. The signal flow graph of the system is shown in Figure P3.16.

The transfer function CðsÞ=DðsÞ of the system is

(a)G1ðsÞG2ðsÞ

1þ G1ðsÞHðsÞ (b)G1ðsÞG2ðsÞ

1� G1ðsÞG2ðsÞHðsÞ

(c)�G2ðsÞ

1þ G1ðsÞG2ðsÞHðsÞ (d)�G2ðsÞ

1� G1ðsÞG2ðsÞHðsÞ ½IAS 2001�

R(s) + + +

−1G 2G 3G

1H

2H

C

X




6. The closed loop system shown in Figure P3.17 is subjected to a disturbance NðsÞ.

The transfer function CðsÞ=NðsÞ is given

(a)G1ðsÞG2ðsÞ

1þ G1ðsÞG2ðsÞHðsÞ (b)G1ðsÞ

1þ G1ðsÞHðsÞ

(c)G2ðsÞ

1þ G2ðsÞHðsÞ (d)G2ðsÞ

1þ G1ðsÞG2ðsÞHðsÞ½IES 1997�

7. The transfer function of the system shown in Figure P3.18 is

(a)O

R¼ ABC

1þ ABC(b)

O

R¼ Aþ Bþ C

1þ ABþ AC

(c)O

R¼ ABþ AC

ABC(d)

O

R¼ ABþ AC

1þ ABþ AC

½IAS 1998�


R +

+−

−

A

B

O

C



8. Three blocks G1;G2 and G3 are connected in some fashion such that overall transfer function is

G1 þ G3ð1þ G1G2Þ1þ G1G2

The blocks are connected in the following manner:

(a) G1;G2 with negative feedback and combination in parallel with G3(b) G1;G3 with negative feedback and G2 in parallel(c) G1;G2 is cascade and combination in parallel with G3(d) G1;G3 in cascade and combination in parallel with G2

½IAS 2004�

9. In regeneration feedback, the transfer function is given by

(a)GðsÞRðsÞ ¼

GðsÞ1þ GðsÞHðsÞ (b)

GðsÞRðsÞ ¼

GðsÞHðsÞ1� GðsÞHðsÞ

(c)GðsÞRðsÞ ¼

GðsÞHðsÞ1þ GðsÞHðsÞ (d)

GðsÞRðsÞ ¼

GðsÞ1� GðsÞHðsÞ

½IAS 1992�10. The transfer function CðsÞ=RðsÞ of the system, whose block diagram is shown in Figure P3.19, is

(a)G1G2

1þ G1H1 þ G2H2 � G1G2H1H2(b)

G1G2

1þ G1H1 þ G2H2 þ G1G2H1H2

(c)G1G2

1þ G1H1 þ G2H2(d)

G1ð1þ G2H2ÞG2ð1þ G1H1Þ1þ G1H1 þ G2H2 þ G1G2H1H2

½IES 1993�

R(S) C(S)+ +

− −1G

1H2H

2G



11. The signal flow graph of a closed loop system is shown in Figure P3.20, wherein TD represents thedisturbance reduces by

(a) Increasing G2ðsÞ (b) Decreasing G2ðsÞ(c) Increasing G1ðsÞ (d) Decreasing G1ðsÞ

½IES 1997�

12. The response c(t) of a system to an input r(t) is given by the following different equation:

d2cðtÞdt2

þ 3dcðtÞdt

þ 5cðtÞ ¼ 5rðtÞ

The transfer function of the system is given by

(a) GðsÞ ¼ 5

s2 þ 3sþ 5(b) GðsÞ ¼ 1

s2 þ 3sþ 5

(c) GðsÞ ¼ 3s

s2 þ 3sþ 5(d) GðsÞ ¼ sþ 3

s2 þ 3sþ 5

½IES 1996�

13. The gain CðsÞ=RðsÞ of the signal flow graph, shown in Figure P3.21, is

(a)G1G2 þ G2G3

1þ G1G2H1 þ G2G3H1 þ G4(b)

G1G2 þ G2G3

1þ G1G2H1 þ G2G3H1 � G4

(c)G1G3 þ G2G3

1þ G1G3H1 þ G2G3H1 þ G4(d)

G1G3 þ G2G3

1þ G1G3H1 þ G2G3H1 � G4

½IES 2003�



14. The overall gain CðsÞ=RðsÞ of the block diagram, shown in Figure P3.22, is

(a)G1G2

1� G1G2H1H2(b)

G1G2

1� G2H2 � G1G2H1

(c)G1G2

1� G2H2 � G1G2H1H2(d)

G1G2

1� G1G2H1 � G1G2H2

½IES 2003�

15. From Figure P3.23, the transfer function of the signal flow graph is

(a)T12

1� T22(b)

T22

1� T12

(c)T12

1þ T12(d)

T22

1þ T12

½IES 1992�

R(S) + +

+

1G

1H2H

2G


1X 12T

22T

2X




Chapter 4

1. The unit-impulse response of a system is given by cðtÞ ¼ 0:5e�t=2. Its transfer function is

(a) 1=ðsþ 2Þ (b) 1= 1þ 2sð Þ(c) 2= 1þ 2sÞ (d) 2= sþ 2ð Þð

½IAS 1993�

2. If the unit-step response of a system is a unit impulse function, then the transfer function of such asystem will be

(a) 1 (b) 1=s

(c) s (d) 1=s2

½IAS 1994�

3. When a unit-step input is applied, a second-order underdamped system has a peak overshoot of OPoccurring at tmax: If another step input, equal in magnitude to the peak overshoot OP, is applied att ¼ tmax, then the system will settle down at

(a) 1þ OP (b) 1� OP

(c) OP (d) 1:0

½IAS 1994�

4. The system shown in Figure P4.58 is subjected to a unit ramp input on close the switch (s).

(a) Steady-state error will increase and damping coefficient j will decrease.

(b) Both-steady state error and damping coefficient j will increase

(c) Both steady-state error and damping coefficient j will decrease.

(d) Steady-state error will decrease and damping coefficient j will increase.

½IAS 1995�5. The impulse response of a system is given by

cðtÞ ¼ 1

2e�t=2

R(s) K

K1 s

s

s(s + a)

C(s)

–

+

–

+

Figure P4.58 Figure for Objective Question 4.


which of the following is its unit-step response?

(a) 1� e�t=2 (b) 1� e�t

(c) 2� e�t (d) 1� e�2t

½IAS 1998�

6. For the system, shown in Figure P4.59, the damping factor j and undamped natural frequencyvn arerespectively

(a)2ffiffiffiffiffiKJ

pf

and

ffiffiffiffiJ

K

r(b)

ffiffiffiffiK

J

rand

f

2ffiffiffiffiffiKJ

p

(c)f

2ffiffiffiffiffiKJ

p and

ffiffiffiffiK

J

r(d)

2FffiffiffiffiffiKJ

p andK

J

½IAS 1999�

7. Type of a system depends on the

(a) No. of its poles (b) Difference between the no. of poles and zeros(c) No. of its real poles (d) No. of poles it has at the origin

½IAS 2000�8. A unity feedback system has open loop transfer function as

GðsÞ ¼ 16

sðsþ 16ÞThe natural frequency of the system is

(a) 16 (b) 8(c) 2 (d) 4

½IAS 2002�9. The system

GðsÞ ¼ 0:8

s2 þ s� 2

is excited by a unit-step input. The steady-state output is

(a) 0.8 (b) 0.4(c) �0.4 (d) Unbonded

½IAS 2003�



10. The system shown in Figure P4.60 has a unit-step unit. In order that the steady-state error is 0.1, thevalue of K required is

(a) 0.1 (b) 0.9(c) 1.0 (d) 9.0

½IAS (EE) 1994�

11. The settling time of a feedback system with the closed-loop transfer function

C sð ÞR sð Þ ¼

v2s

s2 þ 2�vnsþ v2n

is

(a) ts ¼ 2

�vn(b) ts ¼ �vn

2

(c) ts ¼ 4

�vn(d) ts ¼ 4�vn

½IAS (EE) 1998�

12. The feedback control system shown in Figure P4.61 is

(a) Type 0 system (b) Type 1 system(c) Type 2 system (d) Type 3 system

½IES (EC) 1993�


+

−

2

2( 1)

ss +

2

( 1)

ss s

++

2 3

( 3)

ss s

++



13. A typical control system is shown in Figure P4.62. Assuming RðsÞ ¼ 1=s; the steady-state error is

(a)1

1þK(b) K

(c) Zero (d) 1

½IES (EC) 1995�14. The velocity-error constant Kv of a feedback system of a closed-loop transfer function

CðsÞRðsÞ ¼

GðsÞ1þ GðsÞHðsÞ

is

(a) Kv ¼ Lims!0

sG sð ÞH sð Þ (b) Kv ¼ Lims!0

s G sð Þ1þG sð ÞH sð Þ

(c) Kv ¼ Lims!0

sG sð Þ (d) Kv ¼ Lims!0

s 1þ G sð ÞH sð Þ½ �

½IES (EC) 1998�15. In the derivation of expression for peak percent overshoot

Mp ¼ exp��ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p !

� 100%

which one of the following condition is NOT required?

(a) The system is linear and time invariant.

(b) The 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 condition.½GATE (EC) 2005�

16. For what values of a, does the system shown in Figure P4.63 have a zero steady-state error (timed)for a step input?(a) a ¼ 0 (b) a ¼ 1(c) a� 4 (d) For no value of a

1s + 20

s + 40s(s + 10)

+

–

R(s)K

C(s)



½GATE (EE) 1992�

17. A system has the following transfer function

G sð Þ ¼ 100 sþ 5ð Þ sþ 50ð Þs4 sþ 10ð Þ s2 þ 3sþ 10ð Þ

The type and order of the system are respectively

(a) 4 and 9 (b) 4 and 7(c) 5 and 7 (d) 7 and 5

½IES (EE) 1998�18. For the system shown in Figure P4.64, the state value of the output c(t) is

(a) 0 (b) 1(c) 1 (d) Dependent on the values of K and Kt

½IES (EE) 1999�

19. Consider the following statements regarding system shown in Figure P4.65, where m ¼ mass,B ¼ frictional coefficient and K ¼ spring constant:

1. It represents a conservative system.2. It has a natural frequency of undamped oscillation of

ffiffiffiffiffiffiffiffiffiK=m

p:

3. It has a time constant of m/K of these statements.

2

1

5

ss s a

++ +

1

4s +


K

s(s + 2s)–

+

C(s)

K

1 + 0.025

Input = Unit step



(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

½IAS 1997�

20. In Figure P4.66, spring constant is K, viscous friction coefficient is B, mass is M and the systemoutput motion is x(t) corresponding to input force F(t). Which of the following parameters relates tothe above system?

Here

1. The constant ¼ 1=M2. Damping coefficient ¼ B= 2

ffiffiffiffiffiffiffiKM

p� �3. Natural frequency of oscillation ¼ ffiffiffiffiffiffiffiffiffiffi

K=Mp

½IES (EE) 1995�21. The step response of a system is cðtÞ ¼ 1� 5e�t þ 10e�2t � 6e�3t. The impulse response of the

system is

(a) 5e�t � 20e�2t þ 18e�3t (b) 5e t � 20e2t þ 18e�3t

(c) 5e�t þ 20e�2t þ 18e�3t (d) 5e�t þ 20e�2t � 18e�3t

½IAS 2003�

22. Given a unity feedback with

G sð Þ ¼ K

s sþ 4ð Þ

B = 0

m


M

k

F(t)

x(t)



the value of K for damping ratio of 0.5 is

(a) 1 (b) 16(c) 4 (d) 2

½IAS 2003�23. A unity-feedback control system has a forward-path transfer function G(s) is given by

G sð Þ ¼ 10 1þ sð Þs2 sþ 1ð Þ sþ 5ð Þ

The steady-state error due to unit parabolic input

rðtÞ ¼ t2

2UðtÞ

is

(a) Zero (b) 0.5(c) 1.0 (d) Infinite

½IAS 2003�24. Damping factor and undamped natural frequency for a position control system is given by

(a) 2ffiffiffiffiffiKJ

p;

ffiffiffiffiffiKJ

prespectively (b)

K

2fJ;

ffiffiffiffiffiffiffiffiK=J

prespectively

(c)f

2ffiffiffiffiffiKJ

p ;ffiffiffiffiffiffiffiffiK=J

prespectively (d)

J

2ffiffiffiffiffiKf

p ;ffiffiffiffiffiKJ

prespectively

½IES (EE) 1992�25. Match List I (TimeDomain Specification) with List II (Equation for Finding Its Value), and select the

correct answer using the codes given below the lists.

List I (Time Domain Specification) List II (Equation for Finding Its Value)

A. Peak overshoot 1. �=ðvn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

pÞ

B. Peak time 2. 4= �vnð ÞC. Rise time 3. expð��=


pÞ%

D. Settling time (2%) 4. p� fcos�1½�=ðvn


pÞ�g

Codes: A B C D A B C D(a) 3 2 4 1 (c) 4 1 3 2(b) 3 1 4 2 (d) 4 2 3 1

½IAS 2004�26. In type I system, a constant output velocity at steady state will be possible, when

(a) There is no error.

(b) There is a constant steady-state error.


(c) There is a variable steady-state error.

(d) There is a fluctuating error.½IES (EE) 1992�

27. If the time response of a system is given by the following equation

y tð Þ ¼ 5þ 3 sin vt þ �1ð Þ þ e�3t sin vt þ �2ð Þ þ e�5t

then the steady-state part of the above response is given by

(a) 5þ 3 sin vt þ s1ð Þ (b) 5þ 3 sin vt þ �1ð Þ þ e�3t sin vt þ �2ð Þ(c) 5þ e5t (d) 5

½IES (EE) 1996�28. The impulse response of a system is 5e�10t; its step response is equal to

(a) 0:5e�10t (b) 5ð1� e�10tÞ(c) 0:5ð1� e�10tÞ (d) 10ð1� e�10tÞ

½IES (EE) 1996�29. The transfer function of a system is 10/(1þ s) when operated as a unity feedback system, the steady-

state error to a unit-step input will be

(a) Zero (b) 1/11(c) 10 (d) Infinity

½IES (EE) 1996�30. The unit-impulse response of a second-order system is 1=6e�0:8t sin 0:6tð Þ . Then the natural

frequency and damping ratio of the system are respectively

(a) 1 and 0.6 (b) 1 and 0.8(c) 2 and 0.4 (d) 2 and 0.3

½IES (EE) 2003�31. A second-order control system has

M jwð Þ ¼ 100

100� v2 þ 10ffiffiffi2

pjv

Its Mp (peak magnitude) is

(a) 0.5 (b) 1(c)

ffiffiffi2

p(d) 2

½IES (EE) 2003�32. Consider the following system, shown in Figure P4.67, where x(t) ¼ sin t.What will be the response

y(t) in the steady state?

x(t) y(t)

1

ss+



(a) sin(t � 458)=ffiffiffi2

p(b) sin(t þ 458)=

ffiffiffi2

p

(c)ffiffiffi2

p e�tsin t (d) sin t � cos t

½IES (EE) 2004�33. The damping ratio and natural frequency of a second-order system are 0.6 and 2 rad/s respectively.

Which of the following combinations gives the correct values of peak and settling time, respectively,for the unit-step response of the system?

(a) 3.33 s and 1.95 s (b) 1.95 s and 3.33 s(c) 1.95 s and 1.5 s (d) 1.5 s and 1.95 s

½IES (EE) 2004�34. Which of the following equations gives the steady-state error for a unity-feedback system excited by

us tð Þ þ tus tð Þ þ t2us2

(a)1

2þ Kpþ 1

Kvþ 1

Ka(b)

1

1þ Kpþ 1

Kvþ 2

Ka

(c)1

Kpþ 1

Kvþ 1

Ka(d)

1

1þ Kpþ 1

Kvþ 1

Ka

½IES (EE) 2004�35. The steady-state error, due to a ramp input for a type-2 system, is equal to

(a) Zero (b) Infinite(c) Non-zero number (d) Constant

½IES (EE) 2001�36. Which of the following is the steady-state error of a control system with step-error, ramp-error and

parabolic-error constants Kp; Ku Ka; respectively, for the input 1� t2ð Þ3� tð Þ?(a)

3

1þ Kp� 3

2Ka(b)

3

1þ Kpþ 6

Ka

(c)3

1þ Kp� 3

Ka(d)

3

1þ Kp� 6

Ka

½IES (EE) 2005�37. The steady-state error of the type-1 second-order system to unit-ramp input is

(a) 2�vn (b) 2�=vn

(c) 4�=vn (d) None of these

38. The unit-step response of a second-order linear system, with zero initial states, is given by

cðtÞ ¼ 1þ 1:25e�6t sinð8t � tan�11:333Þ; t � 0

The damping ratio and the undamped natural frequency of oscillation of the system are, respectively

(a) 0.6 and 10 rad/s (b) 0.6 and 12.5 rad/s(c) 0.8 and 10 rad/s (d) 0.8 and 12.5 rad/s

½IAS 2000�


39. If a second-order systemhas poles at ð�1� jÞ, then the step response of the systemwill exhibit a peakvalue at(a) 4.5 s (b) 3.5 s(c) 3.14 s (d) 1 s

½IAS 2001�40. In a continuous data system:

(a) Data may be a continuous function of time at all points in the system.(b) Data is necessarily a continuous function of time at all points in the system.(c) Data is continuous at the input and output parts of the system but not necessarily during

intermediate processing of the data.(d) Only the reference signal is a continuous function of time.

41. A control system, having a unit damping factor, will give

(a) A critically damped response (b) An oscillatory response(c) An undamped response (d) No response

½IES (EE) 1992�42. Principles of homogeneity and superposition are applied to

(a) Linear time-variant systems (b) Nonlinear time-variant systems(c) Linear time-invariant system (d) Nonlinear time-invariant systems

½IES (EE) 1993�43. The open-loop transfer function of a unity feedback control system is given by

G sð Þ ¼ K

s sþ 1ð ÞIf the gain K is increased to infinity, then the damping ratio will tend to become

(a) 1=ffiffiffi2

p(b) 1

(c) 0 (d) 1½IES (EE) 1993�

44. The transfer system of a control system is given as

T sð Þ ¼ K

x2 þ 4sþ K

whereK is the gain of the system in radians/amp. For this system to be critically damped, the value ofK should be

(a) 1 (b) 2(c) 3 (d) 4

½IES (EE) 1996�45. Consider the following statements with reference to a system with velocity-error constant,

Kc ¼ 1000:

1. The system is stable. 2. The system is of type 1.3. The test signal used is a step input.


Which of these statements are correct?(a) 1 and 2 (b) 1 and 3(c) 2 and 3 (d) 1, 2 and 3

½IES (EE) 2003�46. The response cðtÞ to a system is described by the differential equation

d2c tð Þdt2

þ 4dc tð Þdt

þ 5c tð Þ ¼ 0

The system response is:

(a) Undamped (b) Underdamped(c) Critically damped (d) Oscillatory

½IES (EC) 1999�47. Consider the following transfer functions:

1:1

s2 þ sþ 12:

4

s2 þ 2sþ 4

3:2

s2 þ 2sþ 24:

1

s2 þ 2sþ 1

5:3

s2 þ 6sþ 3

Which of the above transfer functions represents underdamped second-order systems?

(a) 4 and 5 (b) 1, 4 and 5(c) 1, 2 and 3 (d) 1, 3 and 5

½IES (EE) 2004�48. The open-loop transfer function of a unity-feedback control system is given by

G sð Þ ¼ K

s sþ 1ð ÞIf the gain K is increased to infinity, then the damping ratio will tend to become

(a) Zero (b) 0.707(c) Unity (d) Infinite

½IES (EE) 2005�49. Consider the following statements in connection with the differential equation

4d2y

dt2þ 36y ¼ 36x

1. The natural frequency of the response is 6 rad/sec.2. The response is always oscillatory.3. The percentage overshoot is 10%, and damping ratio of the system is 0.6.4. Both system time constant and settling time are infinite.


Which of the statements given above are correct?

(a) 1 and 3 (b) 2 and 4(c) 1, 2 and 3 (d) 2, 3 and 4

½IES (EE) 2005�50. A second-order system exhibits 100% overshoot. Its damping coefficient is:

(a) Equal to 0 (b) Equal to 1(c) Less than 1 (d) Greater than 1

½IES (EE) 1998�51. For a second-order system

2d2y

dt2þ 4

dy

dtþ 8y ¼ 8x

the damping ratio is

(a) 0.1 (b) 0.25(c) 0.333 (d) 0.5

½IES (EC) 1992�52. In the type-1 system, the velocity error is:

(a) Inversely proportional to the bandwidth of the system

(b) Directly proportional to gain constant

(c) Inversely proportional to gain constant

(d) Independent of gain constant½IES (EC) 1992�

53. A unity-feedback control system has a forward-path transfer function equal to

42:25

s sþ 6:5ð ÞThe unit-step response of this system, starting from rest, will have its maximum value at a timeequal to

(a) 0 sec (b) 0.56 sec(c) 5.6 sec (d) Infinity

½IES (EC) 1993�54. Match the system open-loop transfer functions given in List I with the steady-state errors produced

for a unit-ramp input. Select the correct answer using the codes given below the lists:

List I List II

A.30

s2 þ 6sþ 91. Zero

B.30

s2 þ 6s2. 0.2

C.30

s2 þ 9s2. 0.3

D.sþ 1

s24. Infinity



½IES (EC) 1993�55. A plant has the following transfer function

GðsÞ ¼ 1

s2 þ 0:2sþ 1

For a step input, it is required that the response settles to within 2% of its final value. The plantsettling time is

(a) 20 s (b) 40 s(c) 35 s (d) 45 s

½IES (EC) 2003�56. A second-order control system is defined by the following differential equation:

4d2c tð Þdt2

þ 8dc tð Þdt

þ 16c tð Þ ¼ 16u tð Þ

The damping ratio and natural frequency for this system are respectively

(a) 0.25 and 2 rad/s (b) 0.50 and 2 rad/s(c) 0.25 and 4 rad/s (d) 0.50 and 4 rad/s

½IES (EE) 2001�57. Assuming the transient response of a second-order system to be given by

c tð Þ ¼ 1� e�4tffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2

p sinðvn


pþ �Þ

the setting time for the 5% criterion will be

(a) 1/4 s (b) 3/4 s(c) 5/4 s (d) 4 s

½IES (EC) 1994�58. Consider the systems with the following open-loop transfer functions:

1:36

s sþ 3:6ð Þ 2:100

s sþ 5ð Þ3:

6:25

s sþ 4ð ÞThe correct sequence of these systems in increasing order of the time taken for the unit-stepresponse to settle is

(a) 1, 2, 3 (b) 3, 1, 2(c) 2, 3, 1 (d) 3, 2, 1

½IES (EC) 1994�


59. Match List I with List II and select the correct answer using the codes given below the lists:

List I (Characteristic equation) List II (Nature of damping)

A. s2 þ 15sþ 26:25 1. UndampedB. s2 þ 5sþ 6 2. Under-dampedC. s2 þ 20:25 3. Critically dampedD. s2 þ 4:55sþ 42:25 4. Overdamped


½IES (EC) 1994�60. For the control system in Figure P4.68 to be critically damped, the value of gain K required is:

(a) 1 (b) 5.125(c) 6.831 (d) 10

½IES (EC) 1995�61. A system has an open-loop transfer function

G sð Þ ¼ 10

s sþ 1ð Þ sþ 2ð ÞWhat is the steady-state error when it is subjected to the input

r tð Þ ¼ 1þ 2t þ 3

2t2 ?

(a) Zero (b) 0.4(c) 4 (d) infinity

½IES (EC) 1995�62. Consider a unit-feedback control system shown in Figure P4.69. The ratio of the time constants of

the open-loop response to the closed-loop response will be:

2s2 + 7s + 2

+

–

R(s) C(s)K


R(s) C(s)+

−

2

4s +



(a) 1:1 (b) 2:1(c) 3:2 (d) 2:3

½IES (EC) 1995�63. Consider the following overall transfer function for a unity feedback system

4

s2 þ 4sþ 4

Which of the following statements regarding this system are correct?

1. Position error constant Kp for the system is 4.2. The system is of type one.3. The velocity-error constant Kv for the system is finite.

Select the correct answer using the codes given below:

(a) 1, 2 and 3 (b) 1 and 2(c) 2 and 3 (d) 1 and 3

½IES (EC) 1996�64. A first-order system is shown in Figure P4.70. Its time response to a unit-step input is given by

(a) c tð Þ ¼ 1=Tð Þe�t=T (b) c tð Þ ¼ Tð1� e�t=TÞ(c) c tð Þ ¼ ð1� e�t=TÞ (d) c tð Þ ¼ Te�t=T

½IES (EC) 1996�65. For a unity-feedback system, the open-loop transfer function is

G sð Þ ¼ 16 sþ 2ð Þs2 sþ 1ð Þ sþ 4ð Þ

What is the steady-state error if the input is r tð Þ ¼ 2þ 3t þ 4t2ð Þu tð Þ?(a) 0 (b) 1(c) 2 (d) 3 ½IES (EC) 1996�

66. A system has a transfer function

C sð ÞR sð Þ ¼

4

s2 þ 1:6sþ 4

For the unit-step response, the settling time (in seconds) for 2% tolerance band is

(a) 1.6 (b) 2.5(c) 4 (d) 5

½IES (EC) 1996�

R(s) C(s)1

1 sT+



67. A second-order system has the damping ratio � and undamped natural frequency of oscillationvn. The settling time at 2% tolerance band of the system is

(a)2

�vn(b)

3

�vn

(c)4

�vn(c) �vn

½IES (EC) 2000�68. Which of the following is the steady-state error for a step input applied to a unity-feedback system

with the open-loop transfer function

G sð Þ ¼ 10

s2 þ 14sþ 50?

(a) ess ¼ 0 (b) ess ¼ 0:83(c) ess ¼ 1 (d) ess ¼ 1

½IES (EC) 2001�69. In the system shown in Figure P4.71, where

r tð Þ ¼ 1þ 2t t � 0ð Þthe steady-state value of the error eðtÞ is equal to(a) Zero (b) 2/10(c) 10/2 (d) Infinity

½IES (EE) 2001�70. Consider the unity-feedback system as shown in Figure P4.72. The sensitivity of the steady-state

error to change in parameter K and parameter a with ramp inputs are respectively(a) 1, �1 (b) �1, 1(c) 1, 0 (d) 0, 1

½IES (EC) 2002�71. When the time period of an observation is large, the type of error is:

(a) Transient error (b) Steady-state error(c) Half-power error (d) Position-error constant

½IES (EC) 2003�

r(t) e(t) C(t)+

− 2

10( 1)

( 2)

ss s

++


r(t) e(t) C(t)+

2

10( 1)

( 2)

ss s

++–



72. What is the unit-step response of a unity-feedback control system having forward-path transferfunction

G sð Þ ¼ 80

s sþ 18ð Þ ?

(a) Overdamped (b) Critically damped(c) Underdamped (d) Undamped oscillatory

½IES (EC) 2004�73. What is the steady-state error for a unity-feedback control system having

G sð Þ ¼ 1

s sþ 1ð Þdue to unit-ramp input?(a) 1 (b) 0.5

(c) 0.25 (d)ffiffiffiffiffiffiffi0:5

p½IES (EC) 2005�

74. Given a unity-feedback system with

G sð Þ ¼ K

s sþ 4ð Þwhat is the value of K for a damping ratio of 0.5?

(a) 1 (b) 16(c) 4 (d) 2

½IES (EC) 2005�75. Match List I (System G(s)) with List II (Nature of response), and select the correct answer using the

codes given.

List I (System G(s)) List II (Nature of response)

A.400

s2 þ 12sþ 4001. Undamped

B.900

s2 þ 90sþ 4002. Critically damped

C.225

s2 þ 30sþ 2253. Underdamped

D.625

s2 þ 0sþ 2254. Overdamped


½IES (EC) 2005�76. An underdamped second-order system with negative damping will have the two roots:

(a) On the negative real axis as real roots

(b) On the left-hand side of the complex plane as complex roots


(c) On the right-hand side of the complex plane as complex conjugates

(d) On the positive real axis as real roots½IES (EC) 2005�

77. Which of the following expresses the time at which second peak in step response occurs for a second-order system?

(a)�

vn


p (b)2�

vn


p(c)

3�

vn


p (d)�ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� �2p

½IES (EC) 2005�78. The steady-state error of a stable of type 0 unity-feedback system for a unit-step function is

(a) 0 (b) 1=ð1þ KpÞ(c) 1 (d) 1=Kp

½GATE (EC) 1990�79. A second-order system has a transfer function given by

G sð Þ ¼ 25

s2 þ 8sþ 25

If the system, initially at rest, is subjected to a unit-step input at t ¼ 0, the second peak in theresponse will occur at(a) � s (b) �=3 s(c) 2�=3 s (d) �=2 s

½GATE (EC) 1991�80. A unity-feedback control system has an open-loop transfer function

G sð Þ ¼ 4 1þ 2sð Þs2 sþ 2ð Þ

If the input to the system is a unit ramp, the steady-state error will be

(a) 0 (b) 0.5(c) 2 (d) Infinity

½GATE (EC) 1991�81. The step-error coefficient of a system

G sð Þ ¼ 1

sþ 6ð Þ sþ 1ð Þwith unity feedback is

(a) 1=6 (b) 1(c) 0 (d) 1

½GATE (EC) 1995�82. Consider a unity-feedback control system with open-loop transfer function

G sð Þ ¼ K

s sþ 1ð Þ


The steady-state error of the system due to a unit-step input is

(a) Zero (b) K(c) 1/K (d) Infinite

½GATE (EC) 1998�83. For a second-order system with the closed-loop transfer function

T sð Þ ¼ 9

s2 þ 4sþ 9

the settling time for 2% band, in seconds, is:

(a) 1.5 (b) 2.0(c) 3.0 (d) 4.0

½GATE (EC) 1999�84. If the characteristic equation of a closed-loop system is s2 þ 2sþ 2 ¼ 0, then the system is

(a) Overdamped (b) Critically damped(c) Underdamped (d) Undamped

½GATE (EC) 2001�85. Consider a system with a transfer function

G sð Þ ¼ sþ 6

Ks2 þ sþ 6

Its damping ratio will be 0.5 when the value of K is

(a) 2/6 (b) 3(c) 1/6 (d) 6

½GATE (EC) 2002�86. The transfer function of a system is

G sð Þ ¼ 100

sþ 1ð Þ sþ 100ð ÞFor a unit-step input to the system, the approximate setting time for 2% criterion is:

(a) 100 s (b) 4 s(c) 1 s (d) 0.1 s

½GATE (EE) 2002�87. For a feedback-control system of type 2, the steady-state error for a ramp input is:

(a) Infinite (b) Constant(c) Zero (d) Interminate

½GATE (EE) 1996�88. A unit-feedback system has an open-loop transfer function G(s). The steady-state error is zero for

(a) Step input and type-1 G(s) (b) Ramp input and type-1 G(s)(c) Step input and type-0 G(s) (d) Ramp input and type-0 G(s)

½GATE (EE) 2000�89. A unity-feedback system has an open-loop transfer function

GðsÞ ¼ 25

sðsþ 6Þ


The peak overshoot in the step-input response of the system is approximately equal to:

(a) 5% (b) 10%(c) 15% (d) 20%

½GATE (EE) 2000�90. If the ramp input is applied to a type-2 system, the steady-state error is:

(a) Positive constant (b) Negative constant(c) Zero (d) Positive infinity

½GATE (EE) 2000�91. Consider the unit-step response of a unity-feedback control system, whose open-loop transfer

functions is

GðsÞ ¼ 1

sðsþ 1ÞThe maximum overshoot is equal to

(a) 0.143 (b) 0.153(c) 0.163 (d) 0.173

½GATE (EE) 1996�92. An open-loop transfer function of a unity-feedback system is given by

K

s sþ 1ð ÞIf the value of gain K is such that the system is critically damped, the closed-loop poles of the systemwill lie at:

(a) �0.5 and �0.5 (b) � j0:5(c) 0 and �1 (d) 0.5 � j0:5

½GATE (EE) 2002�93. The block diagram shown in Figure P4.73 gives a unity-feedback closed-loop control system. The

steady-state error in the response of the above system to the unit-step input is:

(a) 25% (b) 0.75%(c) 6% (d) 33%

½IES (EE) 2001�94. A block diagram of a closed-loop control system is given in Figure P4.74. The values of K and P are

respectively (such that the system has a damping ratio of 0.7 and an undamped natural frequency,vn,of 5 rad/sec):

(a) 20 and 0.3 (b) 20 and 0.2(c) 25 and 0.3 (d) 25 and 0.2

u(t) + v(t)

−

3

15s +15

1s +



½IES (EC) 1997�95. The unit-impulse response of a second-order underdamped system, starting from rest, is given by

cðtÞ ¼ 12:5e�6t sin 8t; t � 0

The steady-state value of the unit-step response of the system is equal to:

(a) 0 (b) 0.25(c) 0.5 (d) 1.0

½GATE (EE) 2004�96. In the case of a second-order system described by a differential equation

Jd2�0dt2

þ Fd�0dt

þ k�0 ¼ k�i

where �i and �0 are the input and output shaft angles, the natural frequency is given by:

(a)ffiffiffiffiffiffiffiffiK=J

p(b)

ffiffiffiffiffiffiffiffiJ=K

p(c)

ffiffiffiffiffiKJ

p(d)

ffiffiffiffiffiffiffiffiffiffiffiK � J

p½IES (EC) 1997�

97. Assuming unit-ramp input match List I (System type) with List II (Steady-state error), and select thecorrect answer using the codes given below the lists:

List I (system type) List II (steady-state error)

A. 0 1. KB. 1 2. 1C. 2 3. 0D. 3 4. 1=4


½IES (EC) 2003�98. A unity-feedback second-order control system is characterized by

G sð Þ ¼ K

s Jsþ Bð Þ

s( s + 2)

K

–

+ C(s)R(s)

1 + sP



where J ¼moment of inertia, K¼ system gain and B¼ viscous damping coefficient. The transientresponse specification, which is not affected by variation of system gain, is the:

(a) Peak overshoot (b) Rise time(c) Settling time (d) Damped frequency of oscillation

½IES (EE) 1997�99. A linear second-order system with the transfer function

G sð Þ ¼ 49

s2 þ 16sþ 49

is initially at rest and is subjected to a step-input signal. The response of the system will exhibit apeak overshoot of:

(a) 16% (b) 9%(c) 2% (d) Zero

½IES (EE) 1998�100. The unit-impulse response of a linear time-invariant second-order system is

gðtÞ ¼ 10e�8t sin 6t ðt � 0ÞThe natural frequency and the damping factor of the system are respectively

(a) 10 rad/s and 0.6 (b) 10 rad/s and 0.8(c) 6 rad/s and 0.6 (d) 6 rad/s and 0.8

½IES (EE) 1999�101. If ð�a� jbÞ are the complex conjugate roots of a characteristic equation of a second-order system,

then its damping coefficient and natural frequency will be respectively:

(a)bffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p(b)

bffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p

(c)affiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2p and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p(d)

affiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2

p and a2 þ b2

½IES (EE) 2000�102. A unity-feedback control system has a forward-path transfer function

G sð Þ ¼ 10 1þ 4sð Þs2 1þ sð Þ

If the system is subjected to an input

rðtÞ ¼ 1þ t þ t2

2t � 0ð Þ

then the steady-state error of the system will be:

(a) Zero (b) 0.1(c) 10 (d) Infinity

½IES (EE) 2000�103. The effect of error-rate damping is:

(a) To reduce steady-state error (b) Delay the response(c) To provide larger settling time (d) None of the above


Chapter 5

1. If the characteristic equation of a system is s3 þ 14s2 þ 56sþ k ¼ 0 then it will be stable only if:

(a) 0 < K < 784 (c) 10 > K > 600(b) 1 < K < 64 (d) 4 < K < 784

½IAS 1994�2. The first two rows of Routh’s tabulation of a fourth-order system are:

s4 1 10 5s3 2 20

The number of roots of the system lying on the right half of the s-plane is:

(a) Zero (b) 2(c) 3 (d) 4

½IAS 1996�3. The first stability test showed the sign as follows:

Rows I II III IV V VI VIISigns + � + + + � +

The number of roots of the system lying the right half of the s-plane is:

(a) 2 (b) 3(c) 4 (d) 5

½IAS 1998�

4. For the block diagram shown in Figure P5.21, the limiting values of K for the stability of inner loopis found to be x < K < y, the overall system will be stable if and only if:

(a) 4x < K < 4y (b) 2x < K < 2y(c) x < K < y (d) x=2 < K < y=2

½IES (EE) 2000�



5. A system with the characteristic equation s4 þ 2s3 þ 11s2 þ 18sþ 18 ¼ 0 will have a closed-looppoles such that:

(a) All poles lie in the left half of the s-plane.(b) All poles lie in the right half of the s-plane.(c) Two poles lie symmetrically on the imaginary axis of the s-plane.(d) No pole lies on the imaginary axis of the s-plane.

½IAS 1993�

6. By properly choosing the value of K, the output c(t) of the system (as shown in Figure P5.22) can bemade to oscillate sinusoidally at a frequency (in rad/s) of:

(a) 1 (b) 2(c) 3 (d) 4

½IAS 1993�7. Which one of the following statements is true for the system shown in Figure P5.23?

(a) Open-loop system is unstable but closed-loop system is stable.(b) Open-loop system is stable but closed-loop system is unstable.(c) Both open-loop and closed-loop system are stable.(d) Both open-loop and closed-loop systems are unstable.

½IAS 1993�

K

s(s+1)(s+4)

c(t)





List I (Roots in the s-plane) List II (Corresponding impulse response)


Codes: A B C D A B C D

(a) 5 1 2 4 (b) 3 4 2 1(c) 5 2 1 4 (d) 4 3 1 5

½IAS 1995�

9. Consider the following statements regarding the stability analysis by Routh–Hurwitz criterion.

1. For a system to be stable, all the coefficients of the characteristic equation must be present and ofthe same sign.

2. If a system is to be stable, there should not be any sign change in the first column of the Routh’sarray.

3. The order of the auxillary equation obtained from the elements of the Routh’s table is alwaysodd.


(a) 1 and 2 are correct.(b) 2 and 3 are correct.(c) 1 and 3 are correct.(d) 1, 2 and 3 are correct.

½IAS 1999�

10. A closed-loop system is shown in Figure P5.24. The largest possible value of � for which the systemwould be stable is:

(a) 1 (b) 1.1(c) 1.2 (d) 2.3

½IES (EC) 1998�

11. The number of roots in the left-half of s-plane for the equation s3 � 4s2 þ sþ 6 ¼ 0 would be

(a) 1 (b) 2(c) 3 (d) 4

½IAS 2001�



12. The characteristic equation of a feedback-control system is s3 þ Ks2 þ 5sþ 10 ¼ 0 for the system tobe critically stable, the values of K should be

(a) 1 (b) 2(c) 3 (d) 4

½IES (EE) 1999�13. A closed-loop system is stable when all its poles in the s-plane lie

(a) On the positive real axis (b) On the imaginary axis(c) In the left half (d) In the right half ½IAS 2002�

14. Consider the equation 2s4 þ s3 þ 3s2 þ 5sþ 10 ¼ 0. The number of roots this equation has in theright half of the s-plane is:

(a) One (b) Two(c) Three (d) Four

½IAS 2003�15. The feedback system shown in Figure P5.25 is stable for all values of K is given by:

(a) K > 0 (b) K < 0(c) 0 < K < 42 (d) 0 < K < 60

½IAS 2003�16. For making an unstable system stable:

(a) Gain of the system should be increased.(b) Gain of the system should be decreased.(c) The number of zeros to the loop transfer function should be increased.(d) The number of poles to the loop-transfer function should be increased.

½IES (EE) 1992�17. While forming a Routh’s array, the situation of a row zeros indicates that the system:

(a) Has symmetrically located roots (b) Is not sensitive to variations in gain(c) Is stable (d) Is Unstable

½IES (EC) 1997�18. The characteristic equation of a closed-loop system is given by s4 þ 6s3 þ 11s2 þ 6sþ k ¼ 0: Stable

closed-loop behavior can be ensured when the gain K is such that:

(a) 0 < K < 10 (b) K > 10(c) �1 � K < 1 (d) 0 < K � 20

½IES (EE) 1993�



19. By a suitable choice of the scalar parameter K, the system shown in Figure P5.26 can be made tooscillate continuously at a frequency of:

(a) 1 rad/s (b) 2 rad/s(c) 4 rad/s (d) 8 rad/s

½IES (EE) 1993�

20. The open-loop transfer functions with unity feedback are given below for different systems:

1. G sð Þ ¼ 2

sþ 22. G sð Þ ¼ 2

s sþ 2ð Þ

3. GðsÞ ¼ 2

s2ðsþ 2Þ 4. G sð Þ ¼ 2 sþ 1ð Þs sþ 2ð Þ

Among these systems, the unstable system is

(a) 1 (b) 2(c) 3 (d) 4

½IES (EE) 1993�

21. The open-loop transfer function of a control system is given by

K sþ 10ð Þs sþ 2ð Þ sþ að Þ

The smallest possible value of a for which this system is stable in a closed loop for all positive valuesof K is:

(a) 0 (b) 8(c) 10 (d) 12

½IES (EE) 1994�

22. The open-loop transfer function of a unity-feedback control system is given by

G sð Þ ¼ K sþ 2ð Þsþ 1ð Þ s� 7ð Þ

For K > 6, the stability characteristic of the open-loop and closed-loop configuration of the systemare, respectively:

(a) Stable and stable (b) Unstable and stable(c) Stable and unstable (d) Unstable and unstable

½IES (EE) 1994�



23. The open-loop transfer function of a system is given by

G sð Þ ¼ K

s sþ 2ð Þ sþ 4ð ÞThe value of K which will cause sustained oscillations in the closed-loop unity feedback, is:

(a) 16 (b) 32(c) 48 (d) 64

½IES (EE) 1996�

24. The characteristic equation 1þ G(s)H(s) ¼ 0 of a system is given by s4 þ 6s3 þ 11s2 þ 6s þK ¼ 0. For the system to be stable, the value of the gain K should be:

(a) Zero (b) Greater than zero but less than 10(c) Greater than 10 but less than 20 (d) Greater than 20 but less than 30

½IES (EE) 1996]

25. The characteristic equation for a third-order is q(s)¼ a0s3 þ a1s2 þ a2sþ a3 ¼ 0: For the third-order system to be stable, besides that all the coefficients have to be positive, which one of thefollowing has to be satisfied as a necessary and sufficient condition?

(a) a0a1 � a2a3 (b) a1a2 � a0a3(c) a2a3 � a1a0 (d) a0a3 � a1a2 ½IES (EE) 2004�

26. For which of the following values of K, the feedback, shown in Figure P5.27, is stable?

(a) K > 0 (b) K < 0(c) 0 < K < 42 (d) 0 < K < 6 0

½IES (EE) 2005�

27. Consider the equation 2s4 þ s3 þ 3s2 þ 5sþ 10 ¼ 0: How many roots does this equation have inthe right half of the s-plane?

(a) One (b) Two(c) Three (d) Four

½IES (EE) 2005�



28. When all the roots of the characteristic equation are found in the left of an s-plane, the responsedue to the initial condition will:(a) Increase to infinity as time approaches infinity(b) Decrease to zero as time approaches infinity(c) Remain constant for all time(d) Be oscillating

½IES (EC) 1992�29. Match List I with List II and select the correct answer by using the codes given below the lists:

List I (characteristic root location) List II (system characteristic)

A. (�1 þj), (�1�j) 1. Marginally stableB. (�2 þj), (�2�j), (2j), (�2j) 2. UnstableC. �j, j, �1, 1 3. Stable

Codes: A B C A B C(a) 1 2 3 (b) 3 1 2(c) 2 3 1 (d) 1 3 2 ½IES (EC) 1992�

30. Match List I with List II and select the correct answer by using the codes given below the list:

List I (Roots in the s-plane) List II (Impulse response)

A. Two imaginary roots

B. Two complex roots in the right half plane

C. A single root on the negative real axis

D. A single root at the origin

Codes: A B C D

(a) 2 3 1 4(b) 1 2 3 4(c) 4 3 2 1(d) 3 2 4 1

½IES (EC) 1992�


31. Howmany roots of the characteristic equation s5 þ s4 þ 2s3 þ 2s2 þ 3sþ 15 ¼ 0 lie in the left halfof the s-plane?

(a) 1 (b) 2(c) 3 (d) 5

½IES (EC) 1993�32. Acontrol system is shown in Figure P5.28. Themaximumvalue of the gainK for which the system is

stable is:

(a)ffiffiffi3

p(b) 3

(c) 4 (d) 5 ½IES(EC) 1993�

33. Consider the following statements regarding the number of sign change in the first column of Routhin respect of the characteristic equation s2 þ 2asþ 4 :

1. If a ¼ þ", where " is near to zero, number of sign changes will be equal to zero.2. If a ¼ 0, the number of sign change will be equal to one.3. If a ¼ �"; where " ¼ near zero, the number of sign changes will be equal to two.


(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

½IES (EC) 1994�34. The value of K for which the unity-feedback system

G sð Þ ¼ K

s sþ 2ð Þ sþ 4ð Þcrosses the imaginary axis is

(a) 2 (b) 4(c) 6 (d) 48

½IES (EC) 1997�35. The first column of a Routh array is:

s5 1s4 2s3 3=2s2 �1=3s1 10s0 2



How many roots of the corresponding equation are there in the left of the s-plane?

(a) 2 (b) 3(c) 4 (d) 5

½IES (EC) 1996�36. The characteristic equation of a system is given by 3s4 þ 10s3 þ 5s2 þ 2 ¼ 0. This system is:

(a) Stable (b) Marginally stable(c) Unstable (d) None of (a), (b) or (c)

½IES (EE) 2002�37. The loop transfer function of Q closed-loop system is given by

G sð ÞH sð Þ ¼ k

s2 s2 þ 2sþ 2ð ÞThe angle of departure of the root locus at s ¼ �1þ j is

(a) Zero (b) 90�(c) �90� (d) �180�

½IES (EC) 1998�38. The Routh–Hurwitz criterion cannot be applied when the characteristic equation of the system

contains any coefficients which is:

(a) Negative real and exponential functions of s(b) Negative real, both exponential and sinusoidal functions of s(c) Both exponential and sinusoidal functions of s(d) Complex, both exponential and sinusoidal functions of s

½IES (EC) 2000�39. Which one of the following characteristics equations can result in a stable operation of the feedback

system?

(a) s3 þ 4s2 þ s� 6 ¼ 0 (b) s3 � s2 þ 5sþ 6 ¼ 0(c) s3 þ 4s2 þ 10sþ 11 ¼ 0 (d) s4 þ s3 þ 2s2 þ 4sþ 6 ¼ 0

½IES (EC) 2000�40. Consider the following statements: Routh–Hurwitz criterion gives

1. Absolute stability2. The number of roots lying on the right half of the s-plane.3. The gain margin and phase margin

Which of the statements are correct?


½IES (EC) 2000�41. The given characteristic polynomial s4 þ s3 þ 2sþ 3 ¼ 0 has:

(a) Zero root in the RHS of the s-plane (b) One root in the RHS of the s-plane(c) Two roots in the RHS of the s-plane (d) Three roots in the RHS of the s-plane

½IES (EC) 2001�


42. A system has a single pole at the origin. Its impulse response will be:

(a) Constant (b) Ramp(d) Decaying exponential (d) Oscillatory

½IES (EC) 2002�

43. Match List I (Pole–zero plot of linear control system) with List II (Response of the system), and selectthe correct answer using the codes given below the lists:

List I (Pole–zero plot of linear control system) List II (Response of the system)

1.

2.

4.

3.


Codes: A B C D

(a) 4 3 1 2(b) 4 3 2 1(c) 3 4 2 1(d) 3 4 1 2

½IES (EC) 2002�

44. The characteristic equation of a control system is given by

s6 þ 2s5 þ 8s4 þ 12s3 þ 20s2 þ 16sþ 16s ¼ 0

The number of the roots of the equation, which lie on the imaginary axis of the s-plane, is:

(a) Zero (b) 2(c) 4 (d) 6

½IES (EC) 2003�

45. The closed-loop system shown in Figure P5.29 becomes marginally stable, if the constant K ischosen to be:

(a) 10 (b) 20(c) 30 (d) 40

½IES (EE) 2002�

46. An open loop system has a transfer function

1

s3 þ 1:5s2 þ s� 1

It is converted into a closed-loop system by providing negative feedback having transfer function20(sþ 1). Which one of the following is correct?

The open loop and closed loop system are respectively:

(a) Stable and stable (b) Stable and unstable(c) Unstable and stable (d) Unstable and unstable

½IES (EC) 2004�



47. An electromechanical closed-loop control system has the characteristic equation s3 þ 6Ks2 þK þ 2ð Þsþ 8 ¼ 0; where K is the forward gain of the system. The condition for the closed-loopstability is:

(a) K ¼ 0.528 (b) K ¼ 2(c) K ¼ 0 (d) K ¼ �2528

½GATE (EC) 1990�48. The characteristic equation of a feedback control system is given by s3 þ 5s2 þ K þ 6ð Þ sþ K ¼ 0;

where K > 0 is a scalar variable parameter. In the root–loci diagram of the system, the asymptotesof the root–locus for large values of K meet at a point in the s-plane, whose coordinates are:

(a) (�3, 0) (b) (�2, 0)(c) (�1, 0) (d) (2, 0)

½GATE (EC) 1991�49. For a second-order system, damping ratio, �; is 0 < � < 1, then the roots of the characteristic

polynomial are:

(a) Real but not equal (b) Real and equal(c) Complex conjugates (d) Imaginary

½GATE (EC) 1995�

50. The number of roots of s3 þ 5s2 þ 7sþ 3 ¼ 0 in the left half of the s-plane is:

(a) Zero (b) One(c) Two (d) Three

½GATE (EC) 1998�

51. The transfer function of a system is

2s2 þ 6sþ 5

sþ 1ð Þ2 sþ 2ð ÞThe characteristic equation of the system is:

(a) 2s2 þ 6sþ 5 ¼ 0 (b) sþ 1ð Þ2 sþ 2ð Þ ¼ 0(c) 2s2 þ 6sþ 5þ sþ 1ð Þ2 sþ 2ð Þ ¼ 0 (d) 2s2 þ 6sþ 5� sþ 1ð Þ2 sþ 2ð Þ ¼ 0

½GATE (EC) 1998�52. A system described by the transfer function

H sð Þ¼ 1

s3 þ �s2 þ Ksþ 3

is stable. The constraints on a and K are:

(a) � > 0; �K < 3 (b) � > 0; �K > 3(c) � < 0; �K > 0 (d) � > 0; �K < 0

½GATE (EC) 2000�


53. The characteristic polynomial of a system is q sð Þ ¼ 2s5 þ s4 þ 4s3 þ 2s2 þ 2sþ 1. The system is:

(a) Stable (b) Marginally stable(c) Unstable (d) Oscillatory

½GATE (EC) 2002�54. The open-loop transfer function of a unity feedback system is

G sð Þ ¼ K

s s2 þ sþ 2ð Þ sþ 3ð Þ

The range of K for which the system is stable is:

(a) 21/44 > K > 0 (b) 13 > K > 0(c) 21/4 < K < 1 (d) �6 < K < 1

½GATE (EC) 2004�55. For the polynomial P sð Þ ¼ s5 þ s4 þ 2s3 þ 2s2 þ 3sþ 15; the number of roots that lie in the right

half of the s-plane is:

(a) 4 (b) 2(c) 3 (d) 1

½GATE (EC) 2004�

56. A feedback system is shown in Figure P5.30. The system is stable for all positive values of K, if:

(a) T ¼ 0 (b) T < 0(c) T > 1 (d) 0 < T < 1

½IES (EE) 2000�57. The open-loop transfer function of a unity-feedback control system is

G sð Þ ¼ K sþ 10ð Þ sþ 20ð Þs2 sþ 2ð Þ

The closed-loop system will be stable, if the value of K is:

(a) 2 (b) 3(c) 4 (d) 5

½IES (EE) 1998�



58. The number of roots of the equation 2s4 þ s3 þ 3s2 þ 5sþ 7 ¼ 0 that lie in the right half of thes-plane is:


½GATE (EE) 1998�

59. The characteristic equation of a feedback control system is 2s4 þ s3 þ 3s2 þ 5sþ 10 ¼ 0. Thenumber of roots in the right half of the s-plane are:

(a) Zero (b) 1(c) 2 (d) 3

½GATE (EE) 2003�

60. First column elements of the Routh’s tabulation are 3, 5, �3/4, 1/2 and 2. It means that thereis/are:

(a) One root in the left half of the s-plane (b) Two roots in the half of the s-plane(c) Two root in the right half of the s-plane (d) One root in the right half of the s-plane

½IES (EC) 1999�

61. The loop gain GH of a closed-loop system is given by

K

s sþ 2ð Þ sþ 4ð Þ

The value of K for which the system just becomes unstable, is:

(a) K ¼ 6 (b) K ¼ 8(c) K ¼ 48 (d) K ¼ 96

½GATE (EE) 2003�

62. A unity-feedback system having an open-loop gain

G sð ÞH sð Þ ¼ K 1� sð Þ1þ s

becomes stable when:

(a) | K | > 1 (b) K > 1(c) | K | < 1 (d) K <�1

½GATE (EE) 2005�

63. The algebraic equation FðsÞ ¼ s5 � 3s4 þ 5s3 � 7s2 þ 4sþ 20 is given. F(s) ¼ 0 has:

(a) A single complex root with the remaining roots being real(b) One positive real root and four complex roots, all with positive real parts


(c) One negative real root, two imaginary roots and two roots with positive real parts(d) One positive real root, two imaginary roots and two roots with negative real parts

½GATE (EE) 2006�

64. The characteristic equation of a control system is given by sðsþ 4Þðs2 þ 2sþ 2Þ þ Kðsþ 1Þ ¼ 0.What are the angles of the asymptotes for the root loci for K � 0?

(a) 608, 1808, 3008 (b) 08, 1808, 3008(c) 1208, 1808, 2408 (d) 08, 1208, 2408

½IES (EE) 2005�

65. Figure P5.64 shows the Nyquist plot of the open-loop transfer function G(s)H(s) of a system. IfG(s)H(s) has one right-hand pole, the closed-loop system is:

(a) Always stable(b) Unstable with one closed-loop right-hand poles(c) Unstable with two closed-loop right-hand poles(d) Unstable with three closed-loop right-hand poles

½GATE (EC) 2003�

66. For the equation s3 � 4s2 þ sþ 6 ¼ 0; the number of roots in the left half of the s-plane will be:


½GATE (EE) 2004�

Figure P5.64


Chapter 6

1. Which one of the following application software is used to obtain an accurate root locus plot?

(a) LISP (b) MATLAB(c) dBASE (d) Oracle

½IES (EC) 2003�

2. Despite the presence of a negative feedback, control systems still have problems of instabilitybecause the:

(a) Components used have nonlinearities.(b) Dynamic equations of the subsystems are not known exactly.(c) Mathematical analysis involves approximations.(d) The system has a large negative phase angle at high frequencies.

½GATE (EC) 2005�

3. The open-loop transfer function of unity-feedback control system is

G sð Þ ¼ K

s sþ að Þ sþ bð Þ ; 0 < a � b

The system is stable if:

(a) 0 < K <aþ b

ab(b) 0 < K <

ab

aþ b

(c) 0 < K < ab aþ bð Þ (d) 0 < K <a

baþ bð Þ

½IES (EC) 2000�

4. The open-loop transfer function of a unity feedback control system is given by

G sð Þ ¼ K sþ 2ð Þs s2 þ 2sþ 2ð Þ

The centroid and angles of root locus asymptotes are respectively:

(a) Zero and þ90�; �90� (b) �2/3 and þ60�; �60�(c) Zero and þ120�; �120� (d) �2/3 and �90�; þ90�

½IAS 1993�

5. The open-loop transfer function is given by

G sð ÞH sð Þ ¼ K sþ 1ð Þ sþ 3ð Þs2 þ 4sþ 8


Its root locus diagram is

½IAS 1994�6. Which of the following are the features of the breakaway point in the root-locus of a closed-loop control

system with the characteristic equation 1þ KG1 sð ÞH1 sð Þ ¼ 0?

1. It need not always occur only on the real axis.2. At this point G1 sð ÞH1 sð Þ ¼ 0:

3. At this pointdk

ds¼ 0:



½IES (EC) 1997�7. The root-locus plot of an open-transfer function

GH sð Þ ¼ K

s sþ 2ð Þ sþ 4ð Þis


½IAS 1996�8. To improve the stability and time response of a control system, poles are often added to the system

transfer function. In this context, which one of the following pairs is correctly matched?

(a) Zero poles: makes the system stable and slow responding(b) One pole: makes the system less stable and slow responding(c) Two poles: makes the system less stable and fast responding(d) Three poles: makes the system conditionally stable and fast responding

½IAS 1997�9. Consider the following statements regarding root loci:

1. All root loci start from respective poles of G(s) H(s).2. All root loci end at the respective zeros of G(s) H(s) or go to infinity.3. The root loci are symmetrical about the imaginary axis of the s-plane.Of these statements:


½IAS 1997�10. If the open-loop transfer function of a feedback system is given by

G sð ÞH sð Þ ¼ K

s sþ 2ð Þ s2 þ 2sþ 5ð Þthen the centroid of the asymptotes will be:

(a) �1, 0 (b) 1, 0(c) 0, �1 (d) 0, 1

½IAS 1998�11. Consider the following statements regarding the root-locus technique for analyzing linear control

systems:1. Root-locus is the locus of the roots of the characteristic equation as the closed-loop gain K is

varied from zero to infinity.


2. The number of branches of the root-loci is equal to the number of poles of the open-loop transferfunction.

3. For a particular point in the s-plane to lie on the root-locus, the angle criteria to be satisfied is2I þ 1ð Þ�; where I ¼ 0;� 1; �2:


(a) 1 and 2 are correct. (b) 2 and 3 are correct.(c) 1 and 3 are correct. (d) 1, 2 and 3 are correct.

½IAS 1999�12. Figure P6.25 shows root-loci of the open-loop transfer function G(s)H(s) of a system. Consider the

following inferences drawn from the figure:

1. It has no zero. 2. It is a stable system.3. It is a second order system.

Which of these inferences are correct?


½IAS 2000�

13. The characteristic equation of a unity-feedback control system is given by s3 þ K1 s2 þ sþ K2 ¼ 0:Consider the following statements in this regard:1. For a given value of K1, all the root-locus branches will terminate at infinity for K2 in the positivedirection.

2. For a given value of K2, all the root-locus branches will terminate at infinity for the variable K1in the positive direction.

3. For a given value of K2, only one root-locus branch will terminate at infinity for the variable K1in the positive direction.


(a) 1 and 2 are correct (b) 3 alone is correct(c) 2 alone is correct (d) 1 and 3 are correct

½IES (EE) 1993�

σ

jω

s-plane



14. The transfer function for a system is

G sð Þ ¼ K sþ 2ð Þsþ 1ð Þ sþ 3ð Þ sþ 4ð Þ ; k > 0

Two branches (loci) of the plot directed along asymptotes are centered at a point:

(a) �3 (b) �4(c) �2 (d) �1

½IAS 2001�15. The root-locus plot of a system having open-loop transfer function

G sð ÞH sð Þ ¼ K sþ 10ð Þ sþ 70ð Þs3 sþ 100ð Þ sþ 200ð Þ

will have angle of asymptotes as:

(a) 608, 1008 (b) 608, 1808, 3008(c) 608, 1208, 1808 (d) 608, 908, 1208

½IAS – 2003�16. The intersection of root locus branches with the imaginary axis can be determined by use of:

(a) Nyquist criterion (b) Routh’s criterion(c) Polar plot (d) None of above

½IES (EE) 1992�17. Which of the following is not necessarily valid for root-locus pattern?

(a) The n finite zeros and m poles are plotted on the s-plane. Then (m – n) indicates the number ofnon-finite zeros.

(b) The number of poles gives the number of loci.(c) A value of s on the real axis is a point on the root locus, if the total number of poles and zeros on the

real axis to the right of the point is even.(d) There are as many asymptotes as non-finite zeros.

½IES (EE) 1992�18. The root locus of a unity feedback system is shown in Figure P6.26. The open-loop transfer function is

given by:



(a)K

s sþ 1ð Þ sþ 2ð Þ (b)K sþ 1ð Þs sþ 2ð Þ

(c)K sþ 2ð Þs sþ 1ð Þ (d)

Ks

sþ 1ð Þ sþ 2ð Þ ½IES (EE) 1993�19. Given the unity-feedback system with

G sð Þ ¼ K

s sþ 1ð Þ sþ 2ð ÞThe root-locus of the system is given by

½IES (EE) 2004�20. Figure P6.27 shows the root-locus of open-loop transfer function of a control system

* pole zero � origin

PQ ¼ 2:6 ¼ PQ0RP ¼ 1:4OR ¼ 2:0OQ ¼ 1:4 ¼ OQ0

jω

σ

jω

σ

jω

σ

jω

σ

(a) (b)

(c) (d)


The value of the forward-path gain K at the point P is:

(a) 0.2 (b) 1.4(c) 3.4 (d) 4.8

½IES (EE) 1994�21. For a unity-negative feedback control system, the open-loop transfer function is

G sð Þ ¼ K

s sþ 1ð Þ sþ 2ð ÞThe root-locus plot of the system is:

½IES (EC) 1999�

jω

RP

Q

Q

σ



22. The closed-loop transfer function of a feedback-control system is given by

C sð ÞR sð Þ ¼

K

s2 þ 3þ Kð Þsþ 2

Which one of the following diagrams represents a root locus diagram of the system for K > 0?

½IES (EE) 1996�23. The open-loop transfer function of a feedback-control system is given by

G sð ÞH sð Þ ¼ K sþ 2ð Þs sþ 4ð Þ s2 þ 4sþ 8ð Þ

In the root-locus diagram of the system, the asymptotes of the root-loci for large values ofK meet at apoint in the s-plane. Which one of the following is the set of coordinates of that point?

(a) (�1, 0) (b) (�2, 0)(c) (�10/3, 0) (d) (2, 0)

½IES (EE) 1996�


24. Consider the following statements with reference to the root loci of the characteristic equation ofunity-feedback control system with an open-loop transfer function of

G sð Þ ¼ K sþ 1ð Þ sþ 3ð Þ sþ 5ð Þs sþ 2ð Þ sþ 4ð Þ

1. Reach locus starts at an open-loop pole and ends either at an open-loop zero or infinity.2. Reach locus starts at an open-loop pole or infinity and ends at an open-loop zero.3. There are three separate root loci.4. There are five separate root loci.

Which of these statements are correct?


½IES (EE) 2003�

25. The loop transfer function of a system is given by

G sð ÞH sð Þ ¼ K sþ 10ð Þ sþ 100ð Þs sþ 25ð Þ2

The number of loci terminating at infinity is:

(a) 0 (b) 1(c) 2 (d) 3

½IES (EE) 2003�26. A control system has


s sþ 4ð Þ s2 þ 4sþ 20ð Þ ; 0 < K < �ð Þ

What is the number of breakaway points in the root locus diagram?

(a) One (b) Two(c) Three (d) Zero

½IES (EE) 2004�27. An open-loop transfer function of a feedback system has m poles and n zeros (m > n). Consider the

following statements:

1. The number of separate root loci is m.2. The number of separate root loci is n.3. The number of root loci approaching infinity is (m � n).4. The number of root loci approaching infinity is (m þ n).

Which of the statements given above are correct?


½IES (EE) 2005�28. Consider the root-locus diagram (Figure P6.28) of a system and the following statements:

1. The open-loop system is a second-order system.2. The system is overdamped for K > 1.3. The system is absolutely stable for all values of K.




½IES (EC) 1992�29. A transfer function G(s) has type pole-zero plot, as shown in Figure P6.29. Given that the steady-

state gain is 2, the transfer function G(s) will be given by:

(a)2 sþ 1ð Þ

s2 þ 4sþ 5(b)

5 sþ 1ð Þs2 þ 4sþ 4

(c)10 sþ 1ð Þs2 þ 4sþ 5

(d)10 sþ 1ð Þsþ 2ð Þ2

½IES (EC) 1993�30. In the root locus for open-loop transfer function

G sð ÞH sð Þ ¼ K sþ 6ð Þsþ 3ð Þ sþ 5ð Þ

the breakaway and break-in points are located respectively at:

(a) �2 and �1 (b) �2.47 and �3.77(c) �4.27 and �7.73 (d) �7.73 and �4.27

½IES (EC) 1994�31. If the open-loop transfer function of the system is

G sð ÞH sð Þ ¼ K sþ 10ð Þs sþ 8ð Þ sþ 16ð Þ sþ 72ð Þ

then a closed-loop pole will be located at s ¼ �12 when the value of K is:

(a) 4355 (b) 5760(c) 9600 (d) 9862

½IES (EC) 1994�




32. A unity-feedback system has

G sð Þ ¼ K

s sþ 1ð Þ sþ 2ð ÞIn the root locus, the breakaway point occurs between:

(a) s ¼ 0 and �1 (b) s ¼ �1 and �1(c) s ¼ �1 and �2 (d) s ¼ �2 and �1

½IES (EC) 1995�33. The loop-transfer function of a feedback control system is given by

GðsÞHðsÞ ¼ K

sðsþ 2Þðs2 þ 2sþ 2ÞThe number of asymptotes of its root loci is:

(a) 1 (b) 2(c) 3 (d) 4

½IES (EC) 1996�34. Which of the following effects are correct in respect of addition of a pole to the system loop transfer

function?

1. The root locus is pulled to the right.2. The system response becomes slower.3. The steady state error increases.


(a) 1 and 2 are correct. (b) 1, 2 and 3 are correct.(c) 2 and 3 are correct. (d) 1 and 3 are correct.

½IES (EC) 1998�35. The intersection of asymptotes of root-loci of a system with the open-loop transfer function


s sþ 1ð Þ sþ 3ð Þis

(a) 1.44 (b) 1.33(c) �1.44 (d) �1.33

½IES (EC) 2000�36. The root-locus plot of the system having the loop-transfer function


s sþ 4ð Þ s2 þ 4sþ 5ð Þhas(a) No breakaway point (b) Three real breakaway points(c) Only one breakaway point (d) One real and two complex breakaway points

½IES (EC) 2001�


37. An open-loop transfer function is given by

G sð ÞH sð Þ ¼ K sþ 1ð Þs sþ 2ð Þ s2 þ 2sþ 2ð Þ

It has:

(a) One zero at infinity (b) Two zeros at infinity(c) Three zeros at infinity (d) Four zeros at infinity

½IES (EC) 2001�38. Which of the following is the open-loop transfer function of the root loci shown in

Figure P6.30?

(a)K

s sþ T1ð Þ2 (b)K

sþ T1ð Þ sþ T2ð Þ2

(c)K

sþ Tð Þ3 (d)K

s2 sT1 þ 1ð Þ½IES (EC) 2002�

39. The instrument used for plotting the root locus is called:

(a) Slide rule (b) Spirule(c) Synchro (d) Selsyn

½IES (EC) 2002�40. A control system has

G sð ÞH sð Þ ¼ K sþ 1ð Þs sþ 3ð Þ sþ 4ð Þ

Root locus of the system can lie on the real axis:

(a) Between s ¼ �1 and s ¼ �3 (b) Between s ¼ 0 and s ¼ �4(c) Between s ¼ �3 and s ¼ �4 (d) Towards left of s ¼ �4

½IES (EC) 2002�

jω

r1

r2

r3σ



41. Figure P6.31 shows the root locus of a unity-feedback system. The open-loop transfer function of thesystem is:

(a)K

s sþ 1ð Þ sþ 2ð Þ (b)Ks

sþ 1ð Þ sþ 2ð Þ(c)

K sþ 1ð Þs sþ 2ð Þ (d)

K sþ 2ð Þs sþ 1ð Þ ½IES (EC) 2003�

42. The root loci of a feedback control system for large values of s are asymptotic to the straight lineswith angles u to the real axis given by the equation:

(a)p� zð Þ�2K þ 1

(b)2K þ 1ð Þ�p� z

(c) 2K(p� z) (d)2K

pz

½IES (EC) 2004�43. Consider the following statements: In root-locus plot, the breakaway points:

1. Need not always be on the real axis alone2. Must lie on the root loci3. Must lie between 0 and�1



½IES (EC) 1999�44. Which of the following is not the property of root loci?

(a) The root locus is symmetrical about jv axis.(b) They start from the open-loop poles and terminate at the open-loop zeros.(c) The breakaway points are determined from dK/ds = 0.(d) Segments of the real axis are part of the root locus, if and only if the total number of real poles

and zeros to their right is odd.

½IES (EC) 1995�



45. An open-loop transfer function is given by

K sþ 3ð Þsþ 5

Its root-loci will be as in

½IES (EC) 1997�46. Given a unity-feedback system with open-loop transfer function

GðsÞ ¼ K

s sþ 1ð Þ sþ 2ð ÞThe root locus plot of the system is of the form:

½GATE (EC) 1992�


47. If the open-loop transfer function is a ratio of a numerator polynomial of degreem and a denominatorpolynomial of degree n, then the integer (n� m) represents the number of:

(a) Break-away points (b) Unstable poles(c) Separate root loci (d) Asymptotes

½GATE (EC) 1994�48. Consider the loop-transfer function


In the root-locus diagram, the centroid will be located at:

(a) �4 (b) �1(c) �2 (d) �3

½IES (EC) 1999�49. Consider the points s1 ¼ �3þ j4 and s2 ¼ �3� j2 in the s-plane. Then, for a system with the

open-loop transfer function


sþ 1ð Þ4

(a) s1 is on the root locus, but not s2: (b) s2 is on the root locus, but not s1:(c) Both s1 and s2 are on the root locus. (d) Neither s1 nor s2 is on the root locus.

½GATE (EC) 1999�50. Which of the following points is NOT on the root locus of a system with the open-loop transfer

function


s sþ 1ð Þ sþ 3ð Þ(a) s ¼ �j

ffiffiffi3

p(b) s ¼ �1.5

(c) s ¼ �3 (d) s ¼ �1

½GATE (EC) 2002�51. The root locus of the system


s sþ 2ð Þ sþ 3ð Þhas the break-away point located at:

(a) (�0.5, 0) (b) (�2.548, 0)(c) (�4,0) (d) (�0.784, 0)

½GATE (EC) 2003�52. Given


s sþ 1ð Þ sþ 3ð Þthe 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

½GATE (EC) 2004�


53. A unity-feedback system has an open-loop transfer function, G sð Þ ¼ K=s2: The root loci plot is:

½GATE (EE) 2002�54. Figure P6.32 shows the root-locus plot (location of poles not given) of a third-order system whose

open-loop transfer function is:

(a)K

s3(b)

K

s2 sþ 1ð Þ(c)

K

s s2 þ 1ð Þ (d)K

s s2 � 1ð Þ

½GATE (EE) 2005�55. The root-locus plot for an uncompensated unstable system is shown in Figure P6.33. The system is to

be compensated by a compensated zero. The most desirable of the compensating zero would be thepoint marked:

(a) A (b) B(c) C (d) D



½IES (EE) 1997�56. The loop-transfer function GH of a control system is given by

GH ¼ K

s sþ 1ð Þ sþ 2ð Þ sþ 3ð ÞWhich of the following statements regarding the conditions of the system root loci diagram is/are correct?1. There will be four asymptotes.2. There will be three separate root loci.3. Asymptotes will intersect at real axis at A ¼ �2=3.

Which of the follwing is the correct answer?

(a) 1 alone (b) 2 alone(c) 3 alone (d) 1, 2 and 3

½IES (EE) 1998�57. Match List-I with List-II in respect of the open-loop transfer function

G sð ÞH sð Þ ¼ K sþ 10ð Þ s2 þ 20sþ 500ð Þs sþ 20ð Þ sþ 50ð Þ s2 þ 4sþ 5ð Þ

and select the correct answer using the codes given below the lists:

List I (Types of loci) List II (Numbers)

A. Separate loci 1. oneB. Loci on the real axis 2. twoC. Asymptotes 3. threeD. Break-away points 4. five


(a) 3 4 2 1 (b) 3 4 1 2(c) 4 3 1 2 (d) 4 3 2 1

½IES (EE) 1999�58. If the characteristic equation of a closed-loop system is

1þ K

s sþ 1ð Þ sþ 2ð Þ ¼ 0



the centroid of the asymptotes in root-locus will be:(a) Zero (b) 2(c) �1 (d) �2

½IES (EE) 1999�59. The characteristic equation of a linear control system is s2 þ 5Ksþ 10 ¼ 0: The root-loci of the

system for 0 < K < 1 is:

½IES (EE) 2000�60. The characteristic equation of a feedback control system is given by s3 þ 5s2 þ K þ 6ð Þsþ K ¼ 0:

In a root-loci diagram, the asymptotes of the root loci for large K meet at a point in the s-plane,whose coordinates are:

(a) (2, 0) (b) (1, 0)(c) (�2, 0) (d) (�3, 0)

½IES (EE) 2001�61. Which of the following are the characteristics of the root locus of


1. It has one asymptote. 2. It has intersection with jv-axis.3. It has two real axis intersection. 4. It has two zeros at infinity.

Select the correct answer using the codes given below:(a) 1 and 2 (b) 2 and 3(c) 3 and 4 (d) 1 and 3

½IES (EE) 2002�

(a)

jω

Re

K = 0

K = 0

K = ∞K = ∞

(b)

jω

ReK = 0K = 0

K = ∞

K = ∞

(c) (d)

jω

Re

K = 0

K = 0

K = ∞K = ∞

jω

Re

K = 0 K = 0

K = ∞

K = ∞


Chapter 7

1. A system is described by

X ¼ 0 1�2 �3

� �xþ 0

1

� �u and Y ¼ ½1 0�x

The transfer function is:

(a)1

s2 þ 2sþ 3(b)

1

2s2 þ 3sþ 1

(c)1

s2 þ 3sþ 2(d)

1

3s2 þ 2sþ 1[IAS 1995]

2. Consider the single-input system described by the vector-matrix state equation ¼ A ðtÞ þ BuðtÞ;and the output equation YðtÞ ¼ CðtÞ, where ðtÞ is the state vector, u(t) is the control input and

A ¼ 0 1�1 �2

� �; B ¼ 0

1

� �; C ¼ 1 1½ �

The system is:

(a) Controllable and observable (b) Controllable but unobservable(c) Uncontrollable but observable (d) Uncontrollable and unobservable

[IAS 1995]

3. Which of the following systems is completely state controllable?

(a)X1

X2

� �¼ �1 0

0 �2

� �x1x2

� �þ 2

1

� �u (b)

X1

X2

� �¼ 1 0

0 �2

� �x1x2

� �þ 2

0

� �u

(c)X1

X2

� �¼ �1 0

0 �1

� �x1x2

� �þ 1

0

� �u (d)

X1

X2

� �¼ 0 0

2 �2

� �x1x2

� �þ 2

2

� �u

[IAS 1996]

4. A control system is represented as x ¼ AX þ BU, y ¼ CX. Consider the following statements inthis regard:

1. The pair (AB) is controllable implies that the pair (ATBT) is observable.

2. The condition of controllability depends on the matrices A and B of the system.

3. The pair (AC) is observable implies that the pair (ATCT) is controllable.



[IAS 1997]

5. The state representation for a second-order system is given by X1 ¼ �x1 þ u, X2 ¼ x1 � 2x2 þ u.Consider the following statements regarding the above systems:

1. The system is completely state-controllable.

2. If x1 is the output, then the system is output controllable.

3. If x2 is the output, then the system is output controllable.


Of these statement:


[IAS 1997]

6. Given a system represented by equation

X ¼ 0 1�2 �3

� �xþ 0

1

� �u; Y ¼ 1 0½ �x

The equivalent transfer function representation G(s) of the system is:

(a) GðsÞ ¼ 1

s2 þ 5sþ 2(b) GðsÞ ¼ 1

s2 þ 3sþ 2

(c) GðsÞ ¼ 3

s2 þ 3sþ 2(d) GðsÞ ¼ 2

s2 þ 2sþ 2 [IES 1993]

7. A nonlinear system cannot be analyzed by Laplace transform because:

(a) It has no zero initial conditions.

(b) The superposition’s law cannot be applied.

(c) Nonlinearity is generally not well defined.

(d) All of the above. [IES 1992]

8. Which of the following statements regarding the state transition matrix is correct?

(a) fð0Þ ¼ 0 (b) fð0Þ ¼ 0 ¼ f 1=tð Þ(c) f t1 þ t2ð Þ ¼ fðt1Þ þ fðt2Þ (d) f t1 � t2ð Þfðt1 � t0Þ ¼ fðt2 � t0Þ

[IAS 2000]

9. A system G(s) is expressed in the state variable form as x¼ JXþ Bu, y¼ CXþDu. Consider thefollowing statements with regard to the properties of Jordan canonical matrix J.

1. The diagonal elements of J are poles of G(s).

2. All the elements below the principal diagonal are zeros.

3. All the elements above the principal diagonal are zeros.

Among these statements:

(a) 1 and 3 are correct (b) 1 and 2 are correct(c) 2 alone is correct (d) 3 alone is correct

[IAS 1999]

10. For

fðsÞ ¼sþ 6

s2 þ 6sþ 5

1

s2 þ 6sþ 5

�5

s2 þ 6sþ 5

s

s2 þ 6sþ 5

2664

3775


the coefficient matrix A is:

(a)6 �5

�6 0

� �(b)

5 �50 �6

� �

(c)6 0

�5 �6

� �(d)

0 1�5 �6

� �[IAS 1998]

11. A linear second-order single-input continuous time system is described by the following set ofdifferential equations:

X1 ¼ �2x1 þ 4x2, X2 ¼ 2x1 � x2 � u(t)

where x1 and x2 are state variables and u(t) is the input. The system is:

(a) Controllable and stable (b) Controllable but unstable(b) Uncontrollable and unstable (d) Uncontrollable but stable

[IAS 1998]

12. The second-order system X ¼ AX has

A ¼ �1 �11 0

� �

The values of its damping factor x and natural frequency on are respectively:

(a) 1 and 1 (b) 0.5 and 1(c) 0.707 and 2 (d) 1 and 2

[IES 1996]

13. Which of the following properties are associated with the state transition matrix (t)?

1. fð�tÞ ¼ f�1ðtÞ2. fðt1=t2Þ ¼ fðt1Þ � f�1ðt2Þ3. f t1 þ t2ð Þ ¼ fð�t2Þ � fðt1ÞSelect the correct answer using the codes given below:


[IES 1996]

14. The state variable description of a single-input single-output linear system is given by: XðtÞ ¼AXðtÞ þ buðtÞ and yðtÞ ¼ CXðtÞ; where

A ¼ 1 12 0

� �; B ¼ 0

1

� �; C ¼ 1 �1½ �

The system is:

(a) Controllable and observable (b) Controllable but on observable(c) Uncontrollable but observable (d) Uncontrollable and unobservable

[IES 1996]


15. A system is described by state equation

X1

X2

� �¼ 2 0

0 2

� �x1x2

� �þ 1

1

� �u

The state transition matrix of the system is:

(a)e2t 10 e2t

� �(b)

e�2t 00 e�t

� �

(c)e2t 10 e2t

� �(d)

e�2t 10 e�2t

� �[IES 1994]

16. The state equation of a system is

X ¼ 0 1�20 �9

� �X þ 0

1

� �u

The poles of this system are located at:

(a) �1, �9 (b) �1, �20(c) �4, �5 (d) �9, �20

[IES 1995]

17. The system matrix of a continuous-time system, described in the state variable form, is

A ¼x 0 00 y �10 1 �2

24

35

The system is stable for all values of x and y satisfying:

(a) x <1

2; y >

1

2(b) x >

1

2; y > 0

(c) x < 0; y < 2 (d) x < 0; y <1

2 [IES 1993]18. The state representation of a second-order system is x1 ¼�x1þ u, x2¼ x1� 2x2þ u. Consider the

following statements regarding the above system:

1. The system is completely state controllable.

2. If x1 is the output, then the system is completely output controllable.

3. If x2 is the output, then the system is completely output controllable.

Of these statements


[IES 1993]19. The state space representation of a system is given by:

X ¼ �1 00 �2

� �X þ 1

0

� �u and Y ¼ 1

1

� �X


Then the transfer function of the system is:

(a)1

s2 þ 3sþ 2(b)

1

sþ 2

(c)s

s2 þ 3sþ 2(d)

1

sþ 1 [IES 2003]

20. Consider the following statements with respect to a system represented by its state space modelX ¼AX þ BU and Y ¼ CX:

1. The static vector X of the system is unique.

2. The eigen values of A are the poles of the system transfer function.

3. The minimum number of state variables required is equal to the number of independent energystorage elements in the system.


(a) 1 and 2 (b) 2 and 3(c) 1 and 3 (d) 1, 2, and 3

[IES 2003]

21. A certain linear time invariant system has the state and the output equations given below:

X1

X2

� �¼ 1 �1

0 1

� �x1x2

� �þ 0

1

� �u; Y ¼ 1 1½ � x1

x2

� �

If x1ð0Þ ¼ 1; x2ð0Þ ¼ �1; uð0Þ ¼ 0 then dydt

��t¼0 is:

(a) 1 (b) �1(c) 0 (d) None of above

[GATE 1997]

22. A linear discrete-time system has the characteristic equation z3 � 0.81z ¼ 0. The system:(a) Is stable(b) Is marginally stable(c) Is unstable(d) Stability cannot be accessed from the given information

[GATE 1992]

23. A linear time-invariant system is described by the state variable model

X1

X2

� �¼ �1 0

0 �2

� �x1x2

� �þ 0

1

� �u 1 2½ � x1

x2

� �

(a) The system is completely controllable.

(b) The system is not completely controllable.

(c) The system is not completely observable.

(d) The system is not completely observation.

[GATE 1992]


24. A linear second-order single-input continuous time system is described by the following set ofdifferential equations:

X1ðtÞ ¼ �2x1ðtÞ þ 4x2ðtÞ; X2ðtÞ ¼ 2x1ðtÞ � x2ðtÞ þ 4ðtÞwhere x1ðtÞ and x2ðtÞ are the state variables and u(t) is the control variable. The system is:

(a) Controlled and stable (b) Controlled but unstable(c) Uncontrolled and unstable (d) Uncontrolled but stable

[GATE 1992]

25. A system described by

X1

X2

� �¼ 0 1

�1 �1

� �x1x2

� �

To test its stability by Lyapunov’s method, the following V-functions are considered:

V1 ¼ 2x12 þ x22; V2 ¼ x12 þ x22

Which of these V-functions are suitable in this case?

(a) Only V1 (b) Only V2

(c) Both V1 and V2 (d) Neither V1 nor V2

[IES 1993]

26. The transfer function of a zero-order-hold system is:

(a) ð1=sÞ ð1þ e�sTÞ (b) ð1=sÞ ð1� e�sTÞ(c) 1� ð1=sÞe�sT (d) 1þ ð1=sÞe�sT

[GATE 1998]

27. Given

A ¼ 1 00 1

� �

the state transition matrix eAt is given by:

(a)0 e�t

e�t 0

� �(b)

et 00 et

� �

(c)e�t 00 e�t

� �(d)

0 et

et 0

� �[GATE 2004]

28. The state variable equations of a system are X1 ¼ �3x1 � x2 þ u; X2 ¼ 2x1; Y ¼ x1 þ u: Thesystem is:

(a) Controllable but not observable (b) Observable but not controllable(c) Neither controllable nor observable (d) Controllable and observable

[GATE 2004]


29. The zero-input response of a system given by the state-space equation

x1x2

� �¼ 1 0

1 1

� �x1x2

� �and

x1 ð0Þx2 ð0Þ

� �¼ 1

0

� �

is

(a)tet

t

� �(b)

et

t

� �

(c)et

tet

� �(d)

ttet

� �[GATE 2003]

30. The transfer function y(s)/u(s) of a system described by the state equations xðtÞ ¼ �2xðtÞ þ 2uðtÞand yðtÞ ¼ 0:5xðtÞ is:

(a)0:5

s� 2(b)

1

s� 2

(c)0:5

sþ 2(d)

1

sþ 2 [GATE 2002]

31. For the system described by the state equation

x ¼0 1 00 0 10:5 1 2

24

35xþ 0

01

24

35u

If the control signal u is given by u ¼ ½�0:5 � 3 � 5�xþ v; the eigen values of the closed-loopsystem will be:

(a) 0, �1, �2 (b) 0, �1, �3(c) �1, �1, �2 (d) 0, �1, �1

[GATE 1999]

32. The system model described by the state equations is

X ¼ 0 12 �3

� �xþ 0

1

� �u; Y ¼ 1 1½ �x

(a) Controllable and observable (b) Controllable but not observable(c) Observable but not controllable (d) Neither controllable nor observable

[GATE 1999]

33. A system is described by a state equationX ¼ AXþ BU. The output is given by Y¼ CX, where

A ¼ �4 �13 �1

� �; B ¼ 1

1

� �; C ¼ 1 0½ �

Transfer function G(s) of the system is:

(a)s

s2 þ 5sþ 7(b)

1

s2 þ 5sþ 7


(c)s

s2 þ 3sþ 2(d)

1

s2 þ 3sþ 2 [GATE 1995]

34. The eigen values of the matrix

a 1a 1

� �

are:(a) ðaþ 1Þ; 0 (b) a; 0(c) ða� 1Þ; 0 (d) 0; 0

[GATE 1994]

35. The matrix of any state-space equations for the transfer function CðsÞ=RðsÞ of the system, shown inFigure P7.17, is

(a)�1 00 �1

� �(b)

0 10 �1

� �

(c) ½�1� (d) ½3�[GATE 1994]

36. The transfer function for the state variable representation X ¼ AX þ Bu, Y ¼ CX þ Du isgiven by:

(a) Dþ CðsI� AÞ�1B (b) BðsI� AÞ�1CþD(c) DðsI� AÞ�1Bþ C (d) CðsI� AÞ�1Dþ B

[GATE 1993]

37. Consider a seconds order system whose state-space representation is of the form X ¼ AX þ Bu. Ifx1ðtÞ ¼ x2ðtÞ, the system is:

(a) Controllable (b) Uncontrollable(c) Observable (d) Unstable

[GATE 1993]

38. A linear system is described by the following state equation XðtÞ ¼ AXðtÞ þ BUðtÞ; where

A ¼ 0 1�1 0

� �



The state-transition matrix of the system is:

(a)cos t sin t� sin t cos t

� �(b)

� cos t sin t� sin t � cos t

� �

(c)� cos t � sin t� sin t cos t

� �(d)

cos t � sin tcos t sin t

� �[GATE 2006]

39. A system with input x[n] and output y[n] is given as

y½n� ¼ sin5

6pn

� �x½n�

The system is:(a) Linear, stable and invertible (b) Nonlinear, stable and non-invertible(c) Linear, stable and non-invertible (d) Linear, unstable and invertible

[GATE 2006]

40. The eigen value of the system represented by

X ¼0 1 0 00 0 1 00 0 0 10 0 0 1

2664

3775X

are:

(a) 0, 0, 0, 0 (b) 1, 1, 1, 1(c) 0, 0, 0, �1 (d) 1, 0, 0, 0

[GATE 2002]

41. For the system

X ¼ 2 30 5

� �Xþ 1

0

� �u

which of the following statements is true?

(a) The system is controllable but unstable (b) The system is uncontrollable and unstable(c) The system is controllable and stable (d) The system is uncontrollable and stable

[GATE 2002]

42. The equation in Objective Question 41 may be organized in the state-space form as follows

d2odt2

dodt

2664

3775 ¼ P

dodto

" #þ QVa

where the P matrix is given by:

(a)

�B

J

�K2

LJ

1 0

264

375 (b)

�K2

LJ

�B

J

0 1

264

375


(c)

0 1

�K2

LJ

�B

J

24

35 (d)

1 0

�B

J

�K2

LJ

24

35

[GATE 2003]

43. The state transition matrix for the system X ¼ AX with initial state X(0) is:

(a) ðsI� AÞ�1 (b) eAt � ð0Þ(c) Laplace inverse of ðsI� AÞ�1� �

(d) Laplace inverse of ðsI� AÞ�1 � ð0Þ� �[GATE 2002]

44. The state variable description of a linear autonomous system is X ¼ AX, where X is thetwo-dimensional state vector, and A is the system matrix given by

A ¼ 0 22 0

� �

The roots of the characteristic equation are:

(a) �2 and þ2 (b) �j2 and þj2(c) �2 and �2 (d) þ2 and þ2

[GATE 2004]

45. For the matrix

P ¼3 �2 20 �2 10 0 1

24

35

one of the eigen values is equal to �2. Which of the following is an eigen vector?

(a)3

�21

24

35 (b)

�32

�1

24

35

(c)1

�23

24

35 (d)

250

24

35

[GATE 2005]

46. If

R ¼1 0 �12 1 �12 3 2

24

35

then the top view of R�1 is:

(a) 5 6 4½ � (b) 5 �3 1½ �(c) 2 0 �1½ � (d) 2 �1 1=2½ �

[GATE 2005]


47. For a system with the transfer function

HðsÞ ¼ 3ðs� 2Þ4s2 � 2sþ 1

the matrix A in the state-space form X ¼ AX þ Bu is equal to:

(a)1 0 00 1 0

�1 2 �4

24

35 (b)

0 1 00 0 1

�1 2 �4

24

35

(c)0 1 03 �2 11 �2 4

24

35 (d)

1 0 00 0 1

�1 2 �4

24

35

[GATE 2006]

48. A linear system is described by the state equations

X1

X2

� �¼ 1 0

1 1

� �x1x2

� �þ 0

1

� �r ; C ¼ x2

where r and C are the input and output respectively. The transfer function is:

(a)1

sþ 1(b)

1

ðsþ 1Þ2

(c)1

s� 1(d)

1

ðs� 1Þ2[IES 1997]

49. Consider the following state equations for a discrete system:

x1 K þ 1x2 K þ 1

� �¼ �1=2 0

�1=4 �1=4

� �x1 Kð Þx2 Kð Þ

� �þ 1

1

� �u Kð Þ;

y Kð Þ ¼ 1 �1½ � x1 Kð Þx2 Kð Þ

� �� 4u Kð Þ

The system given above is:

(a) Controllable and observable (b) uncontrollable and unobservable(c) Uncontrollable and observable (d) Controllable and unobservable

[IES 1997]

50. The state and output equations of a system are as under:

State equation :x1 tð Þx2 tð Þ

� �¼ 0 1

�1 �2

� �x1 tð Þx2 tð Þ

� �þ 0

1

� �uðtÞ

Output equation : CðtÞ ¼ 1 1½ � x1 tð Þx2 tð Þ

� �


The system is:

a. Neither state controllable nor output controllableb. State controllable but not output controllablec. Output controllable but not state controllabled. Both state controllable and output controllable

[IES 1998]

51. A linear time-invariant system is described by the following dynamic equation:

dxðtÞdt

¼ AxðtÞ þ BuðtÞ; yðtÞ ¼ CsðtÞ

where

A ¼ 0 1�2 �3

� �; B ¼ 0

1

� �; C ¼ 1 1½ �

The system is:

(a) Both controllable and observable (b) Controllable, but not observable(c) Observable, but not controllable (d) None of the above

[IES 2002]

52. The state-space representation in a phase-variable form for the transfer function

GðsÞ ¼ 2sþ 1

s2 þ 7sþ 9

is

(a) x ¼ 0 1�9 �7

� �xþ 0

1

� �u; y ¼ 1 2½ �x (b) x ¼ 1 0

�9 �7

� �xþ 0

1

� �u; y ¼ 0 1½ �x

(c) x ¼ �9 00 �7

� �xþ 0

1

� �u; y ¼ 2 0½ �x (d) x ¼ 9 �7

1 0

� �xþ 0

1

� �u; y ¼ 1 2½ �x

[IES 2002]

53. The asymptotic approximation of the log magnitude versus frequency plot of a minimum phasesystem with real poles and one zero is shown in Figure P7.18.

–60 dB/dec

–60 dB/dec

–40 dB/dec54

0.1 2 5 25 rad/sec

dB

–40 dB/dec

Figure P7.18 Closed-loop poles and response: a. stable system; b. unstable system.


Its transfer function is:

(a)20ðsþ 5Þ

sðsþ 2Þ ðsþ 25Þ (b)10ðsþ 5Þ

ðsþ 2Þ2 ðsþ 25Þ

(c)20ðsþ 5Þ

s2ðsþ 2Þ ðsþ 25Þ (d)50ðsþ 5Þ

s2ðsþ 2Þ ðsþ 25Þ[GATE (EE) 2001]

54. The Bode plot shown in Figure P7.19 has G joð Þ as:

(a)100

joð1þ j0:5oÞð1þ j0:1oÞ (b)100

joð2þ joÞð10þ joÞ

(c)10

jo 1þ 2joð Þ 1þ 10joð Þ (d)10

joð1þ 0:5joÞð1þ0:1joÞ

[IES (EE) 1999]

55. The system, with the Bode magnitude plot, shown in Figure P7.20, has the transfer function:


0

20

40

60

0.01 0.05 0.1

1s + 1x (t) y (t)

ω



(a)60 sþ 0:01ð Þ sþ 0:1ð Þ

s2 sþ 0:05ð Þ2 (b)10 1þ 10sð Þs 1þ 20sð Þ

(c)3 sþ 0:05ð Þ

s sþ 0:1ð Þ sþ 1ð Þ (d)5 sþ 0:1ð Þs sþ 0:05ð Þ

[GATE (EE) 1991]

56. A system has 14 poles and 2 zeros. The slope of its highest frequency asymptote in its magnitudeplot is:

(a) �40 dB=decade (b) �240 dB=decade

(c) �280 dB=decade (d) �320 dB=decade

[IES (EE) 2000]

57. Consider the Bode-magnitude plot shown in Figure P7.21.

The transfer function HðsÞ is:

(a)sþ10

ðsþ1Þðsþ 100Þ (b)10ðsþ1Þ

ðsþ1Þðsþ 100Þ

(c)102ðsþ1Þ

ðsþ10Þðsþ 100Þ (d)103ðsþ100Þðsþ1Þðsþ 10Þ

[GATE (EC) 2004]

58. In the Bode plot of a unity-feedback control system, the value of phase ofG(jo) at the gain crossoverfrequency is �125�, the phase margin of the system is:

(a) �125� (b) �55�(c) 55� (d) 125�

[GATE (EC) 1998]



59. Non-minimum phase-transfer function is defined as the transfer function, which has:

(a) Zeros in the right-hand s-plane (b) Zeros only in the right-half s-plane(c) Poles in the right-half s-plane (d) Poles in the left-half s-plane

[GATE (EC) 1995]

60. The 3-dB bandwidth of a typical second-order system given by with the transfer function

CðsÞRðsÞ ¼

o2n

s2tþ2xonsþo2n

is

(a) on ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2x2

q(b) on ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 2x2Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix4� x2þ1

qr

(c) on ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 2x2Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4x4� 4x2þ2

qr(d) on ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 2x2Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4x4� 4x2þ 2

qr

[GATE (EC) 1994]

61. The log-magnitude Bode plot of non-minimum system is shown in Figure P7.22.

Its transfer function is given by:

(a) G sð Þ ¼ s� 10

sþ 100(b) G sð Þ ¼ sþ 10

s� 100

(c) G sð Þ ¼ s� 10

1� 100(d) G sð Þ ¼ sþ 10

sþ 100

[IES (EC) 1997]



62. For the minimum phase system to be stable:

(a) Phase margin should be negative and gain margin should be positive.

(b) Phase margin should be positive and gain margin should be negative.

(c) Both phase margin and gain margin should be positive.

(d) Both phase margin and gain margin should be negative.[IES (EC) 2005]

63. With negative feedback in a closed-loop control system, the system sensitivity to parametervariations:

(a) Increases (b) Decreases(c) Becomes zero (d) Becomes infinite

[IES (EC) 2005]

64. Consider the following statements regarding the frequency response of a system as shown in FigureP7.231. The type of the system is one.2. o3 ¼ static error coefficient (numerically).3. o2 ¼ ðo1 þ o2Þ=2:



[IES (EC) 2003]

65. A system with zero initial conditions has the closed-loop transfer function

T sð Þ ¼ s2 þ 4

sþ1ð Þ sþ4ð ÞThe system output is zero at frequency:

(a) 0.5 rad/sec (b) 1 rad/sec(c) 2 rad/sec (d) 4 rad/sec

[GATE (EE) 2001]

20 db/dec

40 db/dec

log0

Mag

nitu

de (

db)

1ω 2ω 3ω



66. The forward-path transfer function of a unity-feedback system is given by

G sð Þ ¼ 100

s2þ10sþ100

The frequency response of the system will exhibit the resonance peak at:

(a) 10 rad/s (b) 8.66 rad/s(c) 7.07 rad/s (d) 5 rad/s

[IES (EC) 2004]

67. A second-order overall transfer function is given by

4

s2þ2sþ4

Its resonant frequency is:

(a) 2 (b)ffiffiffi2

p(c)

ffiffiffi3

p(d) 3

[IES (EC) 1997]

68. Consider the following statements associated with phase and gain margins:

1. They are a measure of closeness of the polar plot to the �1þ j0 point.2. For a non-minimum phase to be stable, it must have positive phase and gain margins.3. For a minimum phase system to be stable, both the margin must be positive.

Which of the above statements are correct?

(a) 2 and 3 (b) 1 and 3(c) 1 and 2 (d) 1 alone

[IES (EC) 2002]

69. An open-loop transfer function of a unity-feedback control system has two finite zeros, two poles atorigin and two pairs of complex conjugate poles. The slope of high-frequency asymptote in a Bodemagnitude plot will be

(a) þ40 dB/decade (b) 0 dB/decade(c) �40 dB/decade (d) �80 dB/decade

[IES (EC) 2001]

70. The phase angle for the transfer function

G sð Þ ¼ 1

1þsTð Þ3

at the corner frequency is

(a) �458 (b) �908(c) �1358 (d) �2708

[IES (EC) 1998]



List I (Transfer functions) List II (Description)

A.1� s

1þs1. Non-minimum phase system

B.1þs

ð1þsÞð1þ 2sÞð1þ 3sÞ 2. Minimum phase system

C.1� 3s

ð1þ 4sÞð1þ 2sÞð1þsÞ 3. All phase system

Codes A B C A B C(a) 1 3 2 (b) 3 2 1(c) 3 1 2 (d) 2 1 3

[IES (EC) 1995]

72. What is the slope change ato ¼ 10 of themagnitude vs. frequency characteristic of a unity-feedbacksystem with the following open-loop transfer function?

G joð Þ ¼ 5 1þj0:1oð Þjo 1þj0:5oð Þ 1þj0:6 o=50ð Þþ jo=50ð Þ2� �

(a) �40 dB/dec to �20 dB/dec (b) 40 dB/dec to 20 dB/dec(c) �20 dB/dec to �40 dB/dec (d) 40 dB/dec to �20 dB/dec

[IES (EC) 1995]

73. The magnitude plot for a minimum phase function is shown in the given Figure P7.24.

The phase plot for this function:

(a) Cannot be uniquely determined(b) Will be monotonically increasing from 08 to 1808



(c) Will be monotonically decreasing from 1808 to 08(d) Will be monotonically decreasing from 1808 to �908

[IES (EC) 1995]

74. For the second-order transfer function

T sð Þ ¼ 4

s2 þ 2sþ 4

the maximum resonance peak will be:

(a) 4 (b) 4=ffiffiffi3

p(c) 2 (d) 2=

ffiffiffi3

p[IES (EC) 1994]

75. A linear state time-invariant system is forced with an input xðtÞ ¼ A sinot under steady-stateconditions (see Figure P7.25),

the output y(t) of the system will be:

(a) A sin ðot þ xÞ; where x ¼ tan�1 jGðjoÞj (b) jGðjoÞjA sin ot þ ffG joð Þ½ �(c) jGðjoÞjA sin 2ot þ ffG joð Þ½ � (d) AGðjoÞ sin ot þ ffG joð Þ½ �

[IES (EC) 1993]

76. Consider the following statements with regard to the bandwidth of a closed-loop system:

1. In a system, where the low-frequency magnitude is 0 dB on the Bode diagram, the bandwidth ismeasured at the �3-dB frequency.

2. The bandwidth of the closed-loop control system is a measurement of the range of fidelity ofresponse of the systems.

3. The speed of response to a step input is proportional to the bandwidth.4. The system with the larger bandwidth provides a slower step response and lower fidelity ramp

response.

Which of the statements give above are correct?

(a) 1, 2 and 3 (b) 1, 2 and 4(c) 1, 3 and 4 (d) 2, 3 and 4

[IES (EE) 2005]

x(t) y(t)G(s)



77. The Bode phase angle plot of a system is shown in Figure P7.26.

The type of the system is:

(a) 0 (b) 1(c) 2 (d) 3

[IES (EE) 2003]78. Which one of the following transfer functions represents the Bode plot, as shown in Figure P7.27?

(a) G ¼ 1� s

1þ s(b) G ¼ 1

ð1þ sÞ2

(c) G ¼ 1

s2(d) G ¼ 1

sð1þ sÞ

[IES (EE) 1993]

79. The Bode plot of the transfer function GðsÞ ¼ s is:

(a) Zero magnitude and phase shift(b) Constant magnitude and constant phase-shift angle(c) 20 dB/decade and phase shift of p=2(d) �20 dB/decade and constant phase-shift angle

[IES (EE) 1992]


90°

0 db

–180°

= 1

= 1ω

ω



80. Aminimumphase unity-feedback system has a Bode plot with a constant slope of�20 dB/decade forall frequencies. What is the value of the maximum phase margin for the system?

(a) 08 (b) 908(c) �908 (d) 1808

[IES (EE) 2004]

81. A decade frequency range is specified by:

(a) o1=o2 ¼ 2 (b) o1=o2 ¼ 10(c) o1=o2 ¼ 8 (d) None of the above

[IES (EE) 1992]

82. Which one of the following is the minimum phase transfer function corresponds to Bode magnitudeplot, as shown in the Figure P7.28?

(a)1

2sþ1(b) 2sþ 1

(c)1

ðs=2Þ þ 1(d)

1

2sþ1

[IAS 2004]

83. The open-loop transfer function of a unity feedback system is given as

G sð Þ ¼ 1

sþ 1

The bandwidth for this system, under open-loop and closed-loop operations, are respectively:

(a) 0.5 and 1.0 rad/s (b) 1.0 and 0.5 rad/s(c)1.0 and

ffiffiffiffiffiffiffi2:0

p(d) 2.0 and 1.0 rad/s

[IAS 2004]

84. The magnitude plot for a transfer function is shown in Figure P7.29. What is the steady-state errorcorresponding to a unit-step input?

20 db/dec

Frequency

Mag

nitu

de

= 2ω



(a) 1=101 (b) 1=100(c) 1=41 (d) 1=40

[IES (EC) 1995]85. Loop gain versus phase plot is known as:

(a) Nyquist plot (b) Bode plot(c) Nichol’s chart (d) Inverse Nyquist plot

[IAS 2002]86. The transfer function of a system is given by

G joð Þ k

joð Þ joTþ1ð Þ ; k <1

T

Which one of the following (Figure P7.30) is the Bode plot of this function?

[IES (EE) 1998]




87. The frequency response for a network function H(s) is given in Figure P7.31.

H(s) is given by:

(a)s

sþ1(b)

10s

sþ1

(c)10

sþ1(d)

1

s sþ1ð Þ [IAS 2003]

88. Consider the following statements regarding the transfer function G(s) plotted in Figure P7.32:

1. G(s) has corner frequencies at o ¼ 0:1, 1 and 10.

2. G sð Þ ¼ 100 sþ1ð Þs sþ10ð Þ :

3. The magnitude of 20 log10 G joð Þj j at o ¼ 1000 is �20 dB.



[IAS 2000]

|H( j )| db

(rad/s)

20–20 db/dec

ω

ω


100 (rad)10.1 10

40

20 lo

g 10 |G

(j

)|

20

ω

ω



89. The magnitude of a transfer function is shown in Figure P7.33.

The transfer function in question is:

(a)4 1þ s

2

�s 1þ s

10

� (b)4s 1þ s

2

�1þ s

10

(c)4 1þ 2sð Þs 1þ 10sð Þ (d)

4s 1þ 2sð Þ1þ 10s

[IES (EC) 1998]

90. Straight line asymptotic Bode-magnitude plot for a certain system is shown in Figure P7.34.What will be its transfer function?

(a)4s

1þ s

4

�1þ s

10

�2 (b)4 1þ s

4

�1þ s

10

�2(c)

0:25

1þ s

4

�1þ s

10

� (d)0:25s

1þ s

4

�1þ s

10

�2[IAS 1997]

10 2 (log scale)20.1

6 db

–6 db/octane

–6 db/octane

db

ω


–40 db/dec20 db/dec

4

db

010

ω



91. Bode plots of an open-loop transfer function of a control system are shown in Figure P7.35.

The gain margin of the system is:

(a) K (b) �K(c) 1=K (d)�1=K

[IAS 1996]

92. For asymptotic Bode plot (dB versus log) of a transfer function with a factor of the form

1

s2þ2xonsþs2n

the corner frequency, slope in dB/dec and error at the corner frequency will be respectively

(a) xon, þ 40 and error is x dependent(b) on, �40 and error x is dependent(c) xon, þ20 and �3 dB(d) on, �20 and þdB

[IAS 1994]

0 K

Gain(db)

ω

Phase(degrees)

90°

180°

270°

ω



93. GA joð Þ,GB joð Þ and Gc joð Þ are three transfer functions whose phase variations are shown inFigure P7.36. The magnitude variation with respect to frequency is the same for both GA joð Þ andGc joð Þ; but the phase variation is as indicated by the curves labeled A and C in the figure. Thetransfer function GB joð Þ is such that xC joð Þ ¼ xA joð Þ þ wB joð Þ:

GA, GB and GC are respectively:

(a) Minimum-phase, all pass and non-minimum phase functions(b) Minimum-phase, non-minimum phase and all pass functions(c) All-pass, minimum phase and non-minimum phase functions(d) All-pass, non-minimum phase and minimum phase functions

[IAS 1994]

94. In the system shown in Figure P7.37, r tð Þ ¼ sino t:

The steady-state response, c(t), will exhibit a resonance peak at a frequency of:

(a) 4 rad=s (b) 2ffiffiffi2

prad=s

(c) 2 rad=s (d)ffiffiffi2

prad=s

[IAS 1993]


-

16s(s + 4)

c(t)+r(t)



95. A system has 12 poles and 2 zeros. Its high frequency asymptote in its magnitude plot has a slope of:

(a) �200 dB/decade (b) �240 dB/decade(c) �280 dB/decade (d) �320 dB/decade

½IES (EC) 1992�96. The root-locus of a unity-feedback system is shown in Figure P7.38. The open-loop transfer function

of the system is:

(a)K

s sþ 1ð Þ sþ 3ð Þ (b)K sþ 1ð Þs sþ 3ð Þ

(c)K sþ 3ð Þs sþ 1ð Þ (d)

Ks

sþ 1ð Þ sþ 3ð Þ

½IES (EE) 1999�97. The Nyquist plot of an open-loop system with the transfer function

GðsÞHðsÞ ¼ sþ 2

ðsþ 1Þðs� 1Þwill be of the form:

½IAS 1994�

−1−2−3

Im

s-plane


2

(a)

2

(b)

2

(c) (d)

2


98. The Nyquist plot of an open-loop stable system is shown in Figure P7.39. If (�1, 0) point is asindicated in the figure, then the closed loop system will be:

(a) Unstable with two RHP poles (b) Stable(c) Unstable with one pole in RHP (d) Unstable with three poles in RHP

½IAS 1995�99. Match List I with List II and select the correct answer using the codes given below the lists:

List I (Nyquist plot) List II (Transfer function)

A. 1.K

1þ sT1ð Þ 1þ sT2ð Þ

B. 2.K

1þ sT1ð Þ 1þ sT2ð Þ 1þ sT3ð Þ

C. 3.K

1þ sT1

D. 4.K

s 1þ sT1ð Þ

(−1, 0)

ω = ∞

ω = 0+

ω = 0−




(a) 4 3 1 5 (b) 3 1 2 5(c) 4 1 2 5 (d) 1 2 3 4

½IAS 1995�

100. Consider the followingNyquist plot (Figure P7.40). The feedback systemwill be stable, if and only ifthe critical point lies in the region:

(a) I (OP) (b) II (PQ)(c) III (QR) (d) IV (R to minus infinity)

½IES (EC) 1998�

101. The gain margin of a unity-negative feedback system having forward-transfer functionK

s sT þ 1ð Þis

(a) Infinity (b) KT(c) 1 (d) Zero

½IAS 2001�

102. TheNyquist plot of a systempasses through the (�1, j0) point in theGH plane. The phasemargin ofthe system is:

(a) >0 (b) Zero(c) <0 (d) Infinite

½IAS 2002�



103. The 3-dB cut-off frequency of a first-order low-pass filter isoc. If the input signal is 0.5 sinoct; theoutput will have a phase shift of:

(a) �45� (b) �90�(c) 45� (d) 90�

½IAS 2004�

104. Consider the following statements:

1. Nyquist criterion is in frequency domain.2. Bode plot is in frequency domain.3. Root-locus plot is in time domain.4. Routh Hermitz’s criterion is in time domain.

Which of the following is correct?

(a) 1, 2 and 3 are correct. (b) 2, 3 and 4 are correct.(c) 1 and 2 are correct. (d) All four are correct.

½IES (EE) 1992�

105. The advantages of the Nyquist stability test are:

(a) It guides in stabilizing an unstable system.(b) It enables to predict closed-loop stability from open-loop results.(c) It is applicable to experimental results of frequency response to open-loop system.(d) All of the above

½IES (EE) 1992�

106. Which of the following is used for the Nyquist plot?

(a) Pole zero plot (b) Closed-loop function(c) Open-loop function (d) Characteristic equation

½IES (EE) 1992�

107. The constant M-circles corresponding to the magnitude (M) of the closed-loop transfer of a linearsystem for values of M greater than one lie in the G-plane and to the:

(a) Right of the M = 1 line (b) Left of the M = 1 line(c) Upper side of the M = +j1 line (d) Lowe side of the M = j1 line

½IES (EE) 1992�

108. Consider the Nyquist diagram (Figure P7.41) for given KG(s)H(s). The transfer function KG(s)H(s)has no poles and zeros in the right half of the plane. If (�1, j0) point is located first in region I and thenin region II, the change in stability of the system will be from:

(a) Unstable to stable (b) Stable to stable(c) Unstable to unstable (d) Stable to unstable


½IES (EE) 2002�

109. TheNyquist plot (Figure P7.42) of a control system is shown below. For this system,H(s) is equal to:

(a)K

s 1þ sT1ð Þ (b)K

s2 1þ sT1ð Þ(c)

K

s3 1þ sT1ð Þ 1þ sT2ð Þ (d)K

s2 1þ sT1ð Þ 1þ sT2ð Þ

½IES (EE) 2003�

110. The Nyquist plot for the closed-loop control system with the loop transfer function

G sð ÞH sð Þ ¼ 100

s sþ 10ð Þis plotted. Then the critical point (�1, j0) is:

(a) Never enclosed (b) Enclosed under certain conditions(c) Just touched (d) Enclosed

½IES (EE) 2004�

Real

Im




111. Consider the following Nyquist of a feedback system having open-transfer function

GH sð Þ ¼ sþ 1

s2 s� 2ð Þas shown in Figure P7.42.What is then number of closed-loop poles in the right half of the s-plane?

(a) 0 (b) 1(c) 2 (d) 3

½IES (EE) 2004�112. Consider the following statements for a counter-clockwise Nyquist path:

1. For a stable closed-loop system, theNyquist plot ofG(s)H(s) should encircle (�1, j0) point asmanytimes as there are poles ofG(s)H(s) in the right half of the s-plane, the encirclements, if there are any,must be made in the counter-clockwise direction.

2. If the loop-gain function G(s)H(s) is a stable function, for a stable closed-loop system is alwaysstable.

3. If the loop gain functionG(s)H(s) is a stable function. For a stable closed-loop system, the Nyquistplot of G(s)H(s) must not enclose the critical point (�1, j0).

Which of the statements given above is/are correct?

(a) Only 1 (b) 1 and 2(c) 1 and 3 (d) only 3

½IES (EE) 2004�113. Match List I (Nyquist plot of loop-transfer function of a control system) with List II (Gain margin in

dB), and select the correct answer using the code given below the lists.

List I List II

A. Does not intersect the axis between 0 and 1. >0(�1, j0)

B. Intersects the negative real axis 2. 1between 0 and (�1, j0)



C. Passes through (�1, j0) 3. < 0D. Encloses (�1, j0) 4. 0


(a) 2 4 1 3 (b) 3 1 4 2(c) 2 1 4 3 (d) 3 4 1 2

½IES (EE) 2005�

114. The open-loop transfer function of a unity-negative feedback system is

G sð Þ ¼ K sþ 10ð Þ sþ 20ð Þs3 sþ 100ð Þ sþ 200ð Þ

The polar plot of the system is:

½IES (EC) 1999�


115. The effect of adding poles and zeroes can be determined quickly by:

(a) Nicholas chart (b) Nyquist plot(c) Bode plot (d) Root locus

½IES (EC) 1992�116. The polar plot of a transfer function passes through the critical point (�1, 0). The gain margin is:

(a) Zero (b) �1 dB(c) 1 dB (d) Infinity

½IES (EE) 1999]

117. The polar plot of

G sð Þ ¼ 10

s sþ 1ð Þ2

intercepts the real axis at o ¼ o0. Then, the real parts and o0 are respectively given by:

(a) �5, 1 (b) �2.5, 1(c) �5, 0.5 (d) �5, 2

½IES (EC) 1992�118. For the transfer function


s sþ 1ð Þ 1þ 0:5sð Þthe phase cross-over frequency is:

(a) 0.5 rad/sec (b) 0.707 rad/sec(c) 1.732 rad/sec (d) 2 rad/sec

½IES (EC 1993�119. The gain-phase plot of an open-loop transfer function of four different systems labeled A, B, C and

D are shown in Figure P7.43. The correct sequence of the increasing order of stability of the foursystems will be:



(a) A, B, C, D (b) D, C, B, A(c) B, A, D, C (d) B, C, D, A

½IES (EC) 1994�

120. A portion of a polar plot of an open-loop transfer function is shown in Figure P7.44. The phasemargin and gain margin will be respectively:

(a) 30� and0:75 (b) 60� and 0.375(c) 60� and 0.75 (d) 60� and 1=0:75

½IES (EC) 1999�

121. A unity-feedback system has the following open-loop frequency response.

The gain margin and phase margin of the system are:

(a) 0 dB, �170� (b) 3.86 dB, �180�(c) 0 dB, �10� (d) 3.86 dB, �10�

½IES (EC) 1995�

120°

–0.375

–1

–0.75

Im G(s)H(s)

Re G(s)H(s)

Unit circle


o G joð Þj j ffG joð Þ2 7.5 �118�

3 4.8 �130�

4 3.15 �140�

5 2.25 �150�

6 1.70 �157�

8 1.00 �170�

10 0.64 �180�


122. The type and order of the system, whose Nyquist plot is shown in Figure P7.45, are respectively:

(a) 0, 1 (b) 1, 2(c) 0, 2 (d) 2, 1

½IES (EC) 1995�123. The radius and the center of M circles are given respectively by:

(a)M

M2 � 1;

�M2

M2 � 1; 0

� �(b)

M2

M2 � 1;

�M

M2 � 1; 0

� �

(c)M2

M� 1;

�M2

M� 1; 0

� �(d)

M2

M2 � 1;

�M2

M2 � 1; 0

� �½IES (EC) 1996�

124. Which one of the following is the polar plot of a typical type zero system with open-loop transferfunction

G joð Þ ¼ K

1þ joT1ð Þ 1þ joT2ð Þ

ω → ∞ ω = 040

Re

Im



½IES (EC) 1996�

125. The constant M circle for M = 1 is the:

(a) Straight line x ¼ �1=2 (b) Critical point (�j, j0)(c) Circle with r = 0.33 (d) Circle with r = 0.67

½IES (EC) 1999�

126. Consider the following statements: The gain margin and phase margin of an unstable system mayrespectively be:

1. Positive, positive 2. Positive, negative3. Negative, positive 4. Negative, negative


(a) 1 and 4 are correct. (b) 1 and 2 are correct.(c) 1, 2 and 3 are correct. (d) 2, 3 and 4 are correct.

½IES (EC) 1996�

127. The polar plot of an open-loop transfer function of a feedback-control system intersects the real axisat �2. The gain margin of the system is:

(a) �6 dB (b) 0 dB(c) 6 dB (d) 40 dB

½IES (EC) 1996�

128. Match List I (scientist) with List II (contribution in the area of), and select the correct answer usingthe codes given below the lists:List I List IIA. Bode 1. Asymptotic plotsB. Evans 2. Polar plots


C. Nyquist 3. Root locus techniques4. Constant M and N plots

Codes: A B C A B C

(a) 1 4 2 (b) 2 3 4(c) 3 1 4 (d) 1 3 2

½IES (EC) 2000�

129. The polar plot (for positive frequencies) for the open-loop transfer function of a unity-feedbackcontrol system is shown in Figure P7.46. The phase margin and the gain margin of the system arerespectively:

(a) 1508 and 4 (b) 1508 and 3/4(c) 308 and 4 (d) 30� and 3/4

½IES (EE) 2000�

130. Which of the following equations represents the constantmagnitude locus in theG-plane forM¼ 1;½x-axis in Re G(jo ) and y-axis in Im G(jo )�?(a) x ¼ �0.5 (b) x ¼ 0(c) x2 þ y2 ¼ 1 (d) ðxþ 1Þ2 þ y2 ¼ 1

½IES (EC) 2000�

131. Which of the following features is NOT associated with Nicholas chart?

(a) (0 dB,�1808) point on Nicholas chart represents the critical point (�1 þ j0).(b) It is symmetric about�1808.(c) The M loci are centered about (0 dB,�1808) point.(d) The frequency at the intersection of the G(jo ) locus andM¼þ3 dB locus give a bandwidth of

the closed-loop system.

½IES (EC) 2000�

–0.25

Im G(jω)

Re G(jω)

j1

–j1

1–1



132. The open-loop transfer function of a system is


1þ sð Þ 1þ 2sð Þ 1þ 3sð ÞThe phase crossover frequency oc is:

(a)ffiffiffi2

p(b) 1

(c) Zero (d)ffiffiffi3

p½IES (EC) 2001�

133. The open-loop transfer function of a unity-feedback control system is given as

GðsÞ ¼ 1

sð1þ sT1Þð1þ sT2ÞThe phase crossover frequency and the gain margin are respectively:

(a)1ffiffiffiffiffiffiffiffiffiffiT1T2

p andT1 þ T2

T1T2(b)

ffiffiffiffiffiffiffiffiffiffiT1T2

pand

T1 þ T2

T1T2

(c)1ffiffiffiffiffiffiffiffiffiffiT1T2

p andT1T2

T1 þ T2(d)

ffiffiffiffiffiffiffiffiffiffiT1T2

pand

T1T2

T1 þ T2 ½IES (EC) 2001�

134. A constant N-circle having center at �1=2 þ j0ð Þ in the G-plane, represents the phase angleequal to:(a) 180� (b) 90�(c) 45� (d) 0�

½IES (EC) 2001�

135. The constant M-circle represented by the equation x2 þ 2:25 xþ y2 ¼ �1:125; where x ¼Re G joð Þ½ � and y ¼ Im G joð Þ½ � has the value of M equal to:

(a) 1 (b) 2(c) 3 (d) 4

½IES (EC) 2001�

136. The Nyquist plot of


s2 1þ 0:5sð Þ 1þ sð ÞWill start (o ¼ 1) in the

(a) First quadrant and terminate (o ¼ 0) in the second quadrant(b) Fourth quadrant and terminate (o ¼ 0) in the second quadrant(c) Second quadrant and terminate (o ¼ 0) in the third quadrant(d) Third quadrant and terminate (o ¼ 1) in the first quadrant

½IES (EC) 2002�


137. Consider the following statements: Nichol?s chart gives information about:

1. Closed-loop frequency response2. The value of the peak magnitude of the closed-loop frequency response Mp3. The frequency at which Mp occurs

Which of the above statements are correct?

(a) 2 and 3 (b) 1 and 2(c) 1 and 3 (d) 1, 2 and 3

½IES (EC) 2002�138. Which of the following is the Nyquist diagram for the open-loop transfer function


s 1þ 0:1sð Þ 1þ 0:01sð Þ ?

½IES (EC) 2002�139. The constantN loci represented by the equation x2 þ xþ y2 ¼ 0, where and y ¼ Im G joð Þj j is for

the value of phase angle equal to:

(a) �45� (b) 0�(c) þ45� (d) þ90�

½IES (EC) 2002�

−1 Re

Im

(c)

−1 Re

Im

(d)

−1 Re

Im

(a)

−1 Re

Im

(b)


140. Which of the following techniques is utilized to determine the actual point at which the root locuscrosses the imaginary axis?

(a) Nyquist technique (b) Routh–Hurwitz criterion(c) Nichol?s criterion (d) Bode technique

½IES (EC) 2003�141. Consider the following techniques:

1. Bode plot 2. Nyquist plot3. Nichol?s plot 4. Routh–Hurwitz criterion

Which of these techniques are used to determine the relative stability of a closed-loop linear system?

(a) 1 and 2 (b) 1 and 4(c) 1, 2 and 3 (d) 2, 3 and 4

½IES (EC) 2003�142. The transfer function of a certain system is given by

G sð Þ ¼ s

1þ s

The Nyquist plot to the system is:

½IES (EE) 2001�

ω = 0 ω = 0

ω = ∞ω = ∞

Re G(s) Re G(s)

Im G(s) Im G(s)

(c) (d)

ω = 0

ω = 0

ω = ∞ ω = ∞

Re G(s) Re G(s)

Im G(s) Im G(s)

(a) (b)


143. The Nyquist plot, shown in Figure P7.47, matches with the transfer function:

(a)1

sþ 1ð Þ3 (b)1

sþ 1ð Þ2

(c)1

s2þ2sþ 2(d)

1

sþ 1

½IES (EC) 2003�144. The phase margin (PM) and the damping ratio xð Þ are related by:

(a) PM ¼90� � tan�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2x2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4x4

pq2

(b) PM ¼ tan�1 2xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2x2 þ


pq

(c) PM ¼90� þ tan�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2x2 þ


pq2

(d) PM ¼180� � tan�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2x2 þ


pq2

½IES (EC) 2003�145. The phase angle of the system

G sð Þ ¼ sþ 5

s2 þ 4sþ 9

varies between:

(a) 08 and 908 (b) 08 and�908(c) 08 and�1808 (d)�90 and�1808

½IES (EE) 2001�146. Which of the following statements is correct in respect of the theory of stability?

(a) Phase margin is the phase angle lagging, in short of 1808, at the frequency corresponding to again of 10.

(b) Gain margin is the value by which the gain falls short of unity, at a frequency corresponding to90� phase lag.

(c) Routh–Hurwitz criterion can determine the degree of stability.(d) Gain margin and phase margin are the measure of the degree of stability.

½IES (EC) 2004�

ω = 0

ω = ∞ Re

Im



147. The radius of constant-N circle for N ¼ 1 is:

(a) 2 (b)ffiffiffi2

p(c) 1 (d) 1ffiffi

2p

½IES (EC) 1999�

148. The forward-path transfer function of a unity feedback system is given by

G sð Þ ¼ 1

1þ sð Þ2

What is the phase margin for this system?(a) �p rad (b) 0 rad(c) p=2 rad (d) p rad

½IES (EC) 2004�

149. All the constant-N circles in the G-plane cross the real axis at the fixed points. Which are thesepoints?

(a) �1 and origin (b) Origin and þ1(c) �0.5 and þ0.5 (d) �1 and þ1

½IES (EC) 2004�

150. What is the value of M for the constant-M circle represented by the equation 8x2 þ 18xþ 8y2 þ9 ¼ 0; where x ¼ Re GðjoÞj j and y ¼ Iml GðjoÞj j?(a) 0.5 (b) 2(c) 3 (d) 8

½IES (EC) 2004�

151. If the gain of the loop system is doubled, the gain margin of the system is:

(a) Not affected (b) Doubled(c) Halved (d) One fourth of original value

½IES (EC) 2005�

152. Which of the following methods can determine the closed-loop system resonance frequency ofoperation?

(a) Root locus method (b) Nyquist method(c) Bode plot (d) M and N circle method

½IES (EC) 2005�

153. For a stable closed-loop system, the gain at the phase cross-over frequency should always be:

(a) <20 dB (b) >6 dB(c) <6 dB (d) <0 dB

½IES (EC) 2005�


154. Match List I (frequency response) with List II (time response), and select the correct answer using thecode given below the lists:

List I List IIA. Bandwidth 1. OvershootB. Phase margin 2. StabilityC. Response peak 3. Speed of time responseD. Gain margin 4. Damping ratio


(a) 3 2 1 4 (b) 1 4 3 2(c) 3 4 1 2 (d) 1 2 3 4

½IES (EC) 2005�155. The open-loop transfer function of a feedback control system is


sþ 1ð Þ3


(a) 2 (b) 4(c) 8 (d) 16

½GATE (EC) 1991�156. The Nyquist plot of a phase transfer function G joð ÞH joð Þ of a system encloses the (�1, 0) point.


(a) Less than zero (b) Zero(c) Greater than zero (d) Infinity

½GATE (EC) 1998�157. The gain margin (in dB) of a system having the loop transfer function

G sð ÞH sð Þ ¼ffiffiffi2

p

s sþ 1ð Þis:

(a) 0 (b) 3(c) 6 (d) 1

½GATE (EC) 1999�158. The phase margin (in degrees) of a system having the loop transfer function

G sð ÞH sð Þ ¼ 2ffiffiffi3

ps sþ 1ð Þ

is:

(a) 458 (b) �308(c) 608 (d) 308

½GATE (EC) 1999�


159. The constant-M loci is symmetrical with respect to:

(a) Real axis and imaginary axis (b) M ¼ 1 straight line and the real axis(c) M ¼ 1 straight line and the imaginary axis (d) M ¼ 1 straight line

½IES (EC) 1998�160. A system with the transfer function

G sð Þ ¼ s

1þ s

is subjected to a sinusoidal input r tð Þ ¼ sinot in steady-state, the phase angle of the output relativeto the input at o ¼ 0 and o ¼ 1 will be respectively:

(a) 0� and�90� (b) 0� and 0�

(c) 90� and 0� (d) 90� and �90�

½IES (EE) 2000�161. Consider the following Nyquist plots of different control systems:

–1 –1O O

ω

ω

Im Im

Re Re

(a) (b)

–1 –1O O

ω

ω

ImIm

Re Re

(c) (d)


Which of these plots represents a stable system?

(a) 1 alone (b) 2, 3 and 4(c) 1, 3 and 4 (d) 1, 2 and 4

½IES (EE) 2000�

162. The system with the open-loop transfer function


s s2 þ sþ 1ð Þhas a gain margin of:

(a) �6 dB (b) 0 dB(c) 3.5 dB (d) 6 dB

½GATE (EC) 2002�

163. The gain margin and the phase margin of a feedback system with

G sð ÞH sð Þ ¼ s

sþ 100ð Þ3

are:

(a) 0 dB, 08 (b) 1;1(c) 1; 0� (d) 88:5dB; 1

½GATE (EC) 2003�

164. If the Nyquist plot of an open-loop transfer function of a feedback control system is shown in theleft half of the s-plane (Figure P7.48), then the number of closed-loop poles in the right half of thes-plane will be:

(a) Zero (b) 1(c) 2 (d) 3

½IES (EE) 1999�

ω→0′

ω = −∞

−1

Re

GH plane

Im



165. Which of the following polar diagrams corresponds to a lag network?

½GATE (EC) 2005�

166. The gain and phase crossover frequencies in rad/sec are respectively:

(a) 0.632 and 1.26 (b) 0.632 and 0.485(c) 0.485 and 0.632 (d) 1.26 mA and 0.632 V

½GATE (EC) 2005�

167. The polar plot of

G sð Þ ¼ 1þ s

1þ 4s

ω = ∞ω = 0

Re

Im

(a)

Im

ω = ∞ ω = 0

Re

(d)

Im

ω = ∞ ω = 0

Re

(b)

Im

ω = ∞ω = 0

Re

(c)


for 0 � o � 1 in the G-plane is :

½IES (EE) 1999�

168. TheNyquist plot ofG joð ÞH joð Þ for a closed control system passes through the (�1, j0) point in theGH-plane. The gain margin of the system in dB is equal to:

(a) Infinite (b) Greater than zero(c) Less than zero (d) Zero

½GATE (EC) 2006�

169. A unity-feedback system has the open-loop transfer function

G sð Þ ¼ 1

sþ 2ð Þ s� 1ð Þ sþ 3ð ÞThe Nyquist plot of G encircles the origin:

(a) Never (b) Once(c) Twice (d) Thrice

½GATE (EE) 1992�


170. The polar plot of a type �1, 3 pole, open-loop system is shown in Figure P7.49. The closed-loopsystem is:

(a) Always stable(b) Marginally stable(c) Unstable with one pole on the right half s-plane(d) Unstable with two poles on the right half s-plane

½GATE (EE) 2001�

171. The Nyquist plot of a loop transfer function G(s)H(s) of a closed-loop control system passes throughthe point (�1, j0) in the G(s)H(s) plane. The phase margin of the system is:

(a) 08 (b) 458(c) 908 (d) 1808

½GATE (EE) 2002�172. The open-loop transfer function of a unity feedback control system is given as

G sð Þ ¼ asþ 1

s2

The value of a to give a phase margin of 45� is equal to:

(a) 0.141 (b) 0.441(c) 0.841 (d) 1.141

½GATE (EE) 2004�

173. The gain margin of a unity-feedback control system with the open-loop transfer function

G sð Þ ¼ sþ 1

s2

is:

(a) 0 (b) 1=ffiffiffi2

p(c)

ffiffiffi2

p(d) 1

½GATE (EE) 2005�



174. 207. In the GH (s) plane, the Nyquist plot of the loop-transfer function

G sð ÞH sð Þ ¼ pe�0:25

s

passes through the negative real axis at the point:

(a) (�0.25, j0) (b) (�0.5, j0)(c) (�1, j0) (d) (�2, j0)

½GATE (EE) 2005�

175. If the compensated system shown in Figure P7.50 has a phase margin of 60� at the crossoverfrequency of 1 rad/sec, then the value of the gain K is:

(a) 0.366 (b) 0.732(c) 1.366 (d) 2.738

½GATE (EE) 2005�

176. The Nyquist plot for a control system is shown in Figure P7.51

the bode plot for the same system will be


Figure P7.51 Nyquist plot for Objective Question 176.


177. The Nyquist plot of the open-loop transfer function G(s)H(s) is shown in Figure P7.52. It indicatesthat:

(a) The open-loop system is unstable, but the closed-loop system is stable.(b) Both open-loop and closed-loop systems are unstable.(c) Open-loop system is stable, but closed-loop system is unstable.(d) Both open-loop and closed-loop systems are stable.

½IES (EE) 1997�

ω1ω2

ω2

ω1

ω1 ω1

|G|

|G|

|G||G|

−40 dB/dec−40 dB/dec

−40 dB/dec

−40 dB/dec

−20 dB/dec

−20 dB/dec

−20 dB/dec

−60 dB/dec −60 dB/dec

(c) (d)

(b)

0

(a)

J Im

GH plane

(−1, j0)ω = +∞

ω = 0+

ω = −∞Re



178. Which of the following statements regarding the stability of a feedback control system is correct?

(a) Gain margin (GM) gives the complete information about the relative stability of the gain system.(b) Phase margin (PM) gives the complete information about the stability of the system.(c) GM and PM together give information about the relative stability of the system.(d) Gain cross-over and phase cross-over frequencies give the required information about the relative

stability of the system.½IES (EC) 1998�

179. Match List I (Plot/diagram/chart) with List II (Characteristic), and select the correct answer usingthe codes given below the lists:

List I List IIA. Constant-M loci 1. Constant gain and phase shift loci of the closed-loop systemB. Constant-N loci 2. Plot of loop gain with variation of oC. Nichol?s chart 3. Circles of constant gain for closed-loop transfer functionD. Nyquist plot 4. Circles of constant phase shift of closed-loop transfer function


(a) 3 4 2 1 (b) 3 4 1 2(c) 4 3 2 1 (d) 4 3 1 2

½IES (EE) 1998�


Chapter 8

1. A system is described by

X ¼ 0 1�2 �3

� �Xþ 0

i

� �u

The system is(a) Controllable and observable (b) Uncontrollable and observable(c) Controllable and unobservable (d) Uncontrollable and unobservable

½IAS 1995�2. Which one of the following methods is NOT used for the analysis of nonlinear control systems?

(a) Phase plane method (b) Decreasing function method(c) Liapunov’s method (d) Piecewise linear

½IES 2003�3. The state equations of a system are given by _x ¼ Xþ u; y ¼ X. The system is

(a) Controllable and observable(b) Controllable but not completely observable(c) Neither controllable nor completely observable(d) Not completely controllable but observable

½IES 2005�4. The state representation of a second-order system is _x1 ¼ �x1 þ u; _x2 ¼ x1 � 2x2 þ u. Consider the

following statements regarding the above system:

1. The system is completely state controllable.2. If x1 is the output, then the system is completely output controllable.3. If x2 is the output, then the system is completely output controllable.


½IES 1993�5. The system matrix of a continuous-time system, described in the state variable form, is

A ¼x 0 00 y �10 1 �2

24

35

The system is stable for all values of x and y satisfying

(a) x < 1=2; y < 1=2 (b) x > 1=2; y > 0(c) x < 0; y < 2 (d) x < 0; y < 1=2

½IES 1993�6. The state equation of a system is

X ¼ 0 1�20 9

� �Xþ 0

1

� �


The poles of this system are located at

(a)�1;�9 (b) � 1;�20(c) � 4;�5 (d) � 9;�20

½IES 1995�7. The z-transform of a signal is given by

CðzÞ ¼ 1z�1ð1� z4Þ4ð1� z�1Þ2

Its final value is

(a) 1=4 (b) Zero(c) 1:0 (d) Infinity

½GATE 1999�

8. The transfer function, YðsÞ=UðsÞ, of a system described by the state equations xðtÞ ¼ –2xðtÞ þ 2uðtÞand yðtÞ ¼ 0:5xðtÞ is(a) 0:5=ðs� 2Þ (b) 1=ðs� 2Þ(c) 0:5=ðsþ 2Þ (d) 1=ðsþ 2Þ

½GATE 2002�

9. The state variable equations of a system are:

1. x1 ¼ �3x1�x2 þ 22. x2 ¼ 2x1

The system is

(a) Controllable but not observable (b) Observable but not controllable(c) Neither controllable nor observable (d) Controllable and observable

½GATE 2004�

10. Given

A ¼ 1 00 1

� �the state transition matrix is given by

(a)0 e�t

e�t 0

� �(b)

et 00 et

� �

(c)e�t 00 e�t

� �(d)

0 et

et 0

� �½GATE 2004�

11. If

5 0 20 3 02 0 1

24

35


The inverse of A is

(a)1 0 �20 1=3 0�2 0 5

24

35 (b)

5 0 20 �1=3 02 0 1

24

35

(c)1=5 0 1=20 1=3 01=2 0 1

24

35 (d)

1=5 0 1=20 1=3 0

�1=2 0 1

24

35

½GATE 1998�12. Given the homogenous state space equation

X ¼ �3 10 �2

� �the steady state value of Xss ¼ lim

t!1xðtÞ, given the initial value of Xð0Þ ¼ ½10;�10�T, is

(a) Xss ¼ 00

� �(b) Xss ¼ �3

�2

� �

(c) Xss ¼ �1010

� �(d) Xss ¼ 1

1� �

½GATE 2002�13. The matrix of any state-space equation for the transfer function CðsÞ=RðsÞ of the system, shown in

Figure P8.22, is

(a)�1 00 �1

� �(b)

0 10 �1

� �(c) ½�1� (d) ½3�

½GATE 1994�14. A system is described by the state equation XX ¼ AXþ BU. The output is given by Y ¼ CX,

where

A ¼ �4 �13 �1

� �; B ¼ 1

1

� �; C ¼ ½1; 0�

The transfer function GðsÞ of the system is

(a)s

s2 þ 5sþ 7(b)

1

s2 þ 5sþ 7

(c)s

s2 þ 3sþ 2(d)

1

s2 þ 3sþ 2 ½GATE 1995�



15. A state variable system

XðtÞ ¼ 0 10 �3

� �XðtÞ þ 1

0

� �uðtÞ

with the initial condition X(0)¼ ½�1 3�T and the unit step input u(t) has the state transition matrix

(a)1

1

3ð1� e�3tÞ

0 e�3t

24

35 (b)

11

3ðe�t � e�3tÞ

0 e�t

24

35

(c)1

1

3ðe�t � e�3tÞ

0 e�3t

24

35 (d)

1 ð1� e�tÞ0 e�t

" #½GATE 2005�

16. A state variable system

XðtÞ ¼ 0 10 �3

� �XðtÞ þ 1

0

� �uðtÞ

with the initial condition X(0) ¼ ½�1 3�T and the unit step input u(t) has the state transitionequation

(a) XðtÞ ¼ t � e�t

e�t

� �(b) XðtÞ ¼ t � e�t

3e�3t

� �

(c) XðtÞ ¼ t � e�3t

3e�3t

� �(d) XðtÞ ¼ t � e�3t

e�t

� �½GATE 2005�

17. The value of matrix A in X ¼ AX for the system described by the differential equation yþ2yþ 3y ¼ 0 is

(a)1 0�2 �1

� �(b)

1 0�1 �2

� �

(c)0 1�2 �1

� �(d)

0 1�3 �2

� �½IES 1998�

18. The minimum number of states necessary to describe the network shown in Figure P8.23 in a statevariable form is



(a) 2 (b) 3(c) 4 (d) 6

½IES 1998�19. A system is represented by yþ 2yþ 5yþ 6y ¼ 5x. Its state variables are x1 ¼ y; x2 ¼ y and

x3 ¼ y. Then the coefficient matrix A will be

(a)0 1 00 0 1�6 �5 �2

24

35 (b)

0 1 00 0 1�2 �5 �6

24

35

(c)0 0 10 1 0�6 �5 �2

24

35 (d)

0 0 10 1 0�2 �5 �6

24

35

½IES 1999�

20. The state equation of a linear system is given by X ¼ AXþ BU, where

A ¼ 0 2�2 0

� �and B ¼ 0

1

� �

The state transition matrix of the system is

(a)e2t 00 e2t

� �(b)

e�2t 00 e2t

� �

(c)sin 2t cos 2t� cos 2t sin 2t

� �(d)

cos 2t sin 2t� sin 2t cos 2t

� �½IES 1999�

21. Consider the single-input, single-output system with its state variable representation:

X ¼�1 0 00 �2 00 0 �3

24

35Xþ

110

24

35U; Y ¼ ½1 0 2�X

The system is

(a) Neither controllable nor observable (b) Controllable but not observative(c) Uncontrollable but observable (d) Both controllable and observable

½IES 2001�

22. A particular control system is described by the following state equations:

X ¼ 0 1�2 �3

� �Xþ 0

1

� �U and Y ¼ ½2 0�X


The transfer function of this system is

(a)YðsÞUðsÞ ¼

1

2s2 þ 3sþ 1(b)

YðsÞUðsÞ ¼

2

2s2 þ 3sþ 1

(c)YðsÞUðsÞ ¼

1

s2 þ 3sþ 2(d)

YðsÞUðsÞ ¼

2

s2 þ 3sþ 2½IES 2001�

23. A transfer function of a control system does not have pole-zero cancellation. Which one of thefollowing statements is true?

(a) System is neither controllable nor observable(b) System is completely controllable and observable(c) System is observable but uncontrollable(d) System is controllable but unobservable

½IES 2002�

24. Consider the following statements:

1. A discrete-time system is said to be stable, if and only if its response for until impulses �ðtÞ decayswith K

2.Routh–Herwitz testingmay be applied to determine the stability of the discrete-data systemusingbilinear transformation

Z ¼ 1þ v

1� v

Adiscrete data system is unstable if any of the roots of the characteristics equation lies within the unitcircle on the complex plane.

Which of these statements is/are correct?

(a) 1 and 2 (b) 1 and 3(c) 3 only (d) 2 and 3

½IES 2003�25. For a unity-feedback system, the origin of the s-plane is mapped in the z-plane by the transformation

z ¼ eST to which one of the following?

(a) Origin (b) 1þ j0(c) �1þ j0 (d) 0þ j1

½IES 2004�26. A linear time-invariant discrete-time system is described by the vector-matrix difference equation

xðkþ 1Þ ¼ FxðkÞ þ GuðkÞwhere xðkÞ is the state vector, F is an ðn� nÞ constant matrix, G is an ðn� rÞ constant matrix andu(k) is the control vector. The state transition matrix of the system is given by the Z-transform of

(a) ZI� F (b) ðZI� FÞZ(c) ðZI� FÞ�1G (d) ðZI� FÞ–1Z

½GATE 1991�


27. The Z-transform of the time functionX1k¼0

�ðn� kÞ

is

(a)Z� 1

Z(b)

Z

Z� 1

(c)Z

ðZ� 1Þ2 (d)ðZ� 1Þ2

Z½GATE 1998�

28. The block diagram of a sampled-data system is shown in Figure P8.24

(a)GRðzÞ

1þ GHðzÞ (b)GðzÞRðzÞ1þ GHðzÞ

(c)GRðzÞ

1þ GðzÞHðzÞ (d)GðzÞRðzÞ

1þ GðzÞHðzÞ½IES 1998�

29. Consider the following statements regarding hold circuits for the reconstruction of sampled signals:

1. Hold circuits are essentially low-pass filters.2. A first-order hold circuit introduces less phase-lag in comparison to zero hold circuit3. A zero-order hold has a flat gain-frequency response over the frequency range 0 � v � 2�=T,

where T is the sampling period.

Which of the following is correct?

(a) 3 alone (b) 1 and 2(c) 2 and 3 (d) 1 alone

½IES 1999�

30. The overall pulse transfer function of the system shown in Figure P8.25 is




(a)1� expð�1ÞZ� expð�1Þ (b)

Z½1þ expð�1Þ�ðZ� 1Þ½Zþ expð�1Þ�

(c)1þ expð�1ÞZþ expð�1Þ (d)

Z½1� expð�1Þ�ðZ� 1Þ½Z� expð�1Þ�

½IES 1999�

31. The system matrix of a discrete system is given by

A ¼ 0 1�3 �5

� �The characteristic equation is given by

(a) z2 þ 5zþ 3 ¼ 0 (b) z2 � 3z� 5 ¼ 0

(c) z2 þ 3zþ 5 ¼ 0 (d) z2 þ zþ 2 ¼ 0½IES 2001�


control system 400 objective questions from gate & ies

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