Control System Quiz

Appendix

*

Control Theory Quiz

A set of questions with multiple answer choices is given here. These questions have been designed to capture important aspects of Control Theory covered in the book. For each of these questions, identify the best of the given answer choices. Compare your answers with the master key given at the end of the session. This quiz should serve the purpose of a self-appraisal test for the reader. 1. A linear time-invariant time-invariant system system initially at rest, when subjected subjected to a unit-step unit-step input, gives a response response y y((t ) = te  t t ; t > t > 0. The transfer function of the system is (A)

1

( s + 1)

(B)

2

1 s ( s

+ 1)

2

2. The syste system m shown shown in Fig. Q2 is of (A) zero order (B ) fi f irst order (C ) second order (D) third order 3. The impulse impulse respons responsee of an initially initially relaxed relaxed linear linear 2t t m (t ) where m (t ) is a unit step function. system is e 2 2t t To produce a response of te 2 ), the input must m (t ), be equal to (A) 2e 2 t m (t )

(B )

1

e 2 t m (t )

(C)

s

( s + 1)

(D)

2

1 s ( s

+ 1)

Displacement x (output) K M

Force F (input)

B

2 (c) e m (t ) (D ) e m (t ) Fig. Q2 4. The step step response response of a system with transfer transfer function function G( s s)) = 1/(t s + 1) attains more than 98% of its final value in time t t equal equal to (A) t (B) 2t (C) 3t (D) 4 t 5. The response response of the system of Fig. Fig. Q5a to to an input input r (t ) = 8m (t ) is shown in Fig. Q5b. The time-constant t is equal to (A) 0.535 msec (B) 0.32 msec (C) 0. 0 .09 msec (D) 11.2 5 msec 2 t

Appendix-B OLC.p65

 t  t

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Control Systems: Principles and Design y

r(t ) t s

3.6

y(t )

1 +1

0

0.32

t (msec)

(a)

(b) Fig. Q5

6. Closed-loop transfer function of a unity-feedback unity-feedback system is given by byY Y ( s)/ s)/ R( R( s s)) = 1/(t s + 1). Steady-state error to unit-ramp input is (A) • (B) t (C) 1 (D) 1 /t 7. The series series RLC circuit shown shown in Fig. Q7 is underdamped underdamped if 2

(A)

Ê R ˆ < ÁË 2 L˜ ¯

(B)

Ê R ˆ = ÁË 2 L˜ ¯

(C)

Ê R ˆ > ÁË 2 L˜ ¯

2

2

1

i(t )

R

L

LC 1

C

e0(t )

LC 1

Fig. Q7

LC

(D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 8. Closed-l Closed-loop oop transfer transfer function function of a unity-fee unity-feedback dback system is given by

Y (s ) R ( s )

=

w n2 s 2 + 2 z wn s

+

w n2

.

R1

R2

Steady-state error to unit-ramp input is e e1 e3 C 1 2 (A) • (B) 2z /w n (C) 1 (D ) 4/zw n 9. When two two networks networks shown in Fig. Fig. Q9 are E2(s) = H1(s) cascaded in tandem, the overall transfer function E 1( s) E 4( s s)/ )/ E E 1( s) s ) is Fig. Q9 (A) H 1( s s)) H 2( s) s ) (B ) H 1( s s)) + H + H 2( s s)) (C) H 1( s s)/ )/ H H 2( s s)) (D ) None of the answers in (A), (B), and (C) is correct  s t D 10. De Dead ad-t -tim imee model model e may be approximated by the transfer function 1-

(A) 1+

t D 2

t D 2

1+

s

(B) 1-

s

t D 2

t D 2

s

(C) s

1 - t D s

(D)

1 + t D s

C 2

e4

E 4( s) = H2(s) E 3( s)

1 + t D s 1 - t D s

11. If a system system has two real real and equal characteristic characteristic roots, roots, it is described as (A)) ha (A havi vin ng no da dam mpi ping ng (B)) bei (B ein ng un und der erda dam mpe ped d (C)) bei (C ein ng cr crit itiica callly da damp mped ed (D (D)) be beiing ov over erd dam ampe ped d 12. If the roots roots of a charact characteris eristic tic equatio equation n are given given by s by s1,2 = 3 ± j ± j2, 2, the values of damping z and damped natural frequency w d are

Appendix-B OLC.p65

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Appendix B: Control Theory Quiz

(A)

Ê ÁË

3 13

ˆ ¯

, 13 ˜

(B)

Ê ÁË

1 13

ˆ ¯

, 2 ˜

Ê ÁË

(C)

1 13

ˆ ¯

, 13 ˜

(D)

Ê ÁË

3 13

ˆ ¯

, 2 ˜

13. A liquid-level system system comprises comprises two noninteracting noninteracting tanks which which can be represented represented by two first-order first-order transfer functions in cascade having time constants t 1 = 100 sec and t 2 = 300 sec. The inflow q(t ) is the input and the level of the second tank is the output. The response of the output height to a step change in flow will be (A) having no damping

(B) underdamped

(C ) critically damped

14. An open-loop open-loop control control system system require requiress an operator operator to set a motorized valve setting setting c where 0 < c < 1, so that two fluids are mixed together. One stream has a constant flow of 10 m 3/hr and the outflowing stream should be 14 m 3/hr. The configuration is shown in Fig. Q14, where motorized valve dynamics are given. The value of valve setting to achieve the desired steady-state outflow is (A ) 0 . 2

(B ) 0.4

(C ) 0 .5

(D ) 0.7

(D ) overdamped Constant flow 10 m3/hr

1 0.5 0

c

20 0.4 0. 4 s + 1

Desired flow 14 m3/hr

Fig. Q14

15. A linear linear approximat approximation ion at the the operating operating point point ( x* x* = 0.5, u * = 1.5) to y to y(( x, x, u) = 2 x 2 x2 + xu + u2, is (A) d y = 3.5 d x + 3.5 d u (B) d y = 4 d x + 2 d u (C) d y = 2 d x + 3 d u (D) d y = 4.75 d x + 3.75 d u 16. In a level control system, system, the fluid flow flow is related to the inflow inflow rate by the following following transfer function: function: G( s) s) =

level H ( s) inf low Q ( s )

=

0 . 75 25 0 0 s + 1

Initially the tank is empty. The inflow control valve is suddenly opened to allow a flow rate of 0.5 m 3/sec into the tank. The level after 2500 sec is (A) 0. 0 .474 m (B ) 0.375 m (C) 0.75 m (D) 0.2 37 m 17. The syste system m of Fig. Fig. Q17 is is underdam underdamped ped of (A)

(B)

(C)

K M K M K

2

B > ÊÁ ˆ Ë 2M ˜ ¯

Displacement x (output) K

2

= ÊÁ ˆ Ë 2M ˜ ¯ B

2

B < ÊÁ ˆ Ë 2M ˜ ¯

B

M (D) None of the the answers answers in (A), (A), (B) and (C) (C) is correct correct 18. The dampin damping g ratio of of the system system of Fig. Fig. Q18 is is (A) (B) (C)

1

1

2

BM

Force F (input)

M

Fig. Q17

Displacement x (output)

1 M 2 B

1 B

Force F (input)

M

2 M (D) None of the answer answerss in (A), (B), (B), and(C) is correc correct. t.

B

Fig. Q18

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Control Systems: Principles and Design

19. For a secon second-or d-order der system system with with dampin damping g z > 1, the roots of the characterist characteristic ic equation are (A) real but not equal (B ) real and equal (C ) complex conjugate (D) imaginary 20. The plot in in Fig. Q20 Q20 shows shows the unit unit step step respons responsee of a first-order system. Transfer function of the system is (A) (C)

1

(B)

5 s + 1 2

(D)

2 15 s + 1 1

2 1.8 1.6 1.4 e d 1.2 u t i l 1 p m A 0.8

0.6

0.4 3 s + 1 15 s + 1 21. Thre Threee different different systems systems have have the following following poles: poles: p p1 0.2 = 0.2, p2 =  0.4, p3 = 0.6. What is the time0 constant of the fastest step response? 0 5 10 15 20 ( A) 0 . 2 (B) 0.6 Time (seconds) ( C) 1/ 1/ 0.2 (D) 1/0.6 Fig. Q20 22. Cons Consider ider the follo following wing stat statemen ements: ts: (i) The test signals signals step, ramp and parabola parabola are not useful useful for control-system control-system analysis analysis if actual inputs are are not step, ramp, and parabola, respectively. (ii) Transient and steady-state steady-state performance performance of control control systems is adequately given given by the response to to only one test signal, say step. (A) None of of th the ab above st statements is is tr true (B ) Statement (i (i) is is tr true bu but st statement (i (ii) is is fa false (C)) St (C Stat atem emen entt (i) (i) is fa fals lsee but but st stat atem emen entt (ii (ii)) is is tru truee (D)) Bo (D Both th th thee sta state tem men ents ts ar aree true true 23. Consider the op amp amp circuit shown in Fig. Q23. It can be used as (A) a lead compensator only ( B) a lag compensator only ( C) either a lead or a lag compensator (D) neither a lead nor a lag compensator

C 2 R4

C 1

R2 R3

e1

R1 e2

Fig. Q23 24. Consider the the block diagram diagram shown in Fig. Q24. Q24. The transfer transfer function function between Y ( s) s) and W ( s) s ) is (A)

(C)

Appendix-B OLC.p65

D ( s) G ( s) N (s ) 1 + D ( s) G ( s) H ( s ) N (s ) 1 - D ( s) G ( s) H ( s )

18

(B)

N (s ) 1 + D ( s) G ( s) H ( s )

(D) None of the answers answers in (A), (B), and (C) is correct

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Appendix B: Control Theory Quiz W(s)

R(s)

N(s)

G(s)

D(s)

Y(s)

H(s)

Fig. Q24 25. The gear trai train n shown shown in Fig. Q25 (A) redu reduces ces the the speed speed and and the the torque torque Drive shaft (B) incre increases ases the speed speed and and the the torque torque (C) redu reduces ces the speed speed and and increases increases the the torque torque (D) incre increases ases the the speed and and reduces reduces the torque torque Load 26. Effect of back emf in an armature-control armature-controlled led dc servomoto servomotorr is shaft (A) to increase increase effective effective motor friction, thereby reducing reducing motor motor time-constant (B) to increase increase effective effective motor friction, thereby increasing increasing motor Fig. Q25 time-constant (C) to increase increase motor motor inertia, inertia, thereby thereby increasing increasing motor motor timeconstant (D) to increase increase motor inertia, thereby thereby reducing reducing motor motor time-constant time-constant 27. Electrical time-consta time-constant nt of an armature-controll armature-controlled ed dc servomotor is (A) equa equall to mechani mechanical cal time-c time-const onstant ant (B) smal smaller ler than than mechanical mechanical time-co time-consta nstant nt (C) large largerr than mechan mechanical ical time-c time-const onstant ant (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 28. Rat Ratio io of the the rotor rotor react reactanc ancee X X to to the rotor resistance resistance R R for a two-phase servomotor (A) is equal equal to that of of a normal normal induction induction motor motor (B) is less than than that that of a normal normal induction induction motor motor (C) is greater greater than than that of a normal normal induc induction tion motor motor (D) may be less less or greater greater than that that of a normal induction induction motor 29. A rotating load is connected to a field-controlled field-controlled dc motor motor with negligible negligible field inductance. inductance. A test results results in output steady-state steady-state speed (w ) of 2 rad/sec when an input voltage (e (e f ) of 100 V is applied to the motor terminals. Also the test shows that output speed reaches 0.632 ¥ 2 rad/sec within 0.75 sec. The transfer function w ( s)/ s)/ E E f ( s) s) of the motor is (A)

0.02

(B)

(0.375 s + 1)

0.02 0.75 s + 1

(C)

50

(D)

0.7 5 s + 1

30. The tran transfer sfer func function tion from from R R(( s) s ) to Y ( s) s ) of the system of Fig. Q30 is (A)

s (t s + 1) 2

t s (C)

KK2 ) s+ KK1

KK 1 2

t s

Appendix-B OLC.p65

+ (1 + + (1 +

KK2 ) s+ K K1

19

(B)

K1 s(t s + 1) 2

t s (D)

+ (1 +

KK2 ) s+ KK1

s(t s+ 1 + KK2 ) 2

t s

+ (1 +

KK2 ) s+ KK1

4/22/08, 3:45 PM

50

(0.375 s + 1)

20

Control Systems: Principles and Design W(s)

R(s)

E(s)

U(s) K 1

Y(s)

1 s

K t s + 1

K 2 s

Fig. Q30 31. The tran transfer sfer func function tion from W W (( s s)) to Y ( s) s) of the system of Fig. Q30 is (A ) 1

(C)

(B) KK 1

t s2

+ (1 +

KK2 ) s+ KK1

s (t s + 1) 2

t s

+ (1 +

KK2 ) s+ KK1

(D) None of the answers answers in (A), (B), (B), and (C) is correct correct

32. The tran transfer sfer func function tion betw between een R R(( s s)) and E and E ( s s)) of the system of Fig. Q30 is (A)

(C)

s(t s+ 1 + KK2 ) 2

t s

+ (1 +

KK2 ) s+ KK1

KK 1 2

t s

+ (1 +

KK2 ) s+ KK1

(B)

(D)

s (t s + 1) 2

t s

+ (1 +

KK2 ) s+ KK1

K1 s(t s + 1) 2

t s

+ (1 +

KK2 ) s+ KK1

33. The tran transfer sfer func function tion betw between een R R(( s s)) and U ( s s)) of the system of Fig. Q30 is (A)

(C)

s (t s + 1) 2

t s

+ (1 +

KK2 ) s+ KK1

KK 1 2

t s

+ (1 +

KK2 ) s+ KK1

(B)

(D)

s(t s+ 1 + KK2 ) 2

t s

+ (1 +

KK2 ) s+ KK1

K1 s(t s + 1) 2

t s

+ (1 +

KK2 ) s+ KK1

34. Consider Consider the the followi following ng statem statements ents:: (i) Sync Synchro hro is is a three three phase phase machi machine. ne. (ii) Rotor of synchro synchro transmiter is of of dumb bell type while that that of control transformer transformer is cylindrical. cylindrical. (A) None of of the above above statem statements ents is is true true (B) Statement (i) is true but statement (ii) is false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the stat statemen ements ts are are true true 35. Cons Consider ider the the followi following ng statem statements ents:: (i) AC devices (servomotor, (servomotor, synchro) synchro) are normally operated operated at higher frequencies frequencies (400 Hz, for example) example) and not at 50 Hz. (ii) The output of a synchro is demodulated before before feeding it to a dc motor for actuation of inertial inertial load. (A) None of of the above above statem statements ents is is true true (B) Statement (i) is true but statement (ii) is false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the stat statemen ements ts are are true true

Appendix-B OLC.p65

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36. A position control control system has a two-loop two-loop configuration. configuration. The minor loop loop is a velocity-feedback velocity-feedback loop realized through tachogenerator. Consider the following statements: (i) The tacho is is usually mounted on motor shaft shaft rather than than the load shaft. (ii) The minor minor loop is a positive positive-feed -feedback back loop. loop. (A) None of of the above above statem statements ents is is true true (B) Statement (i) is true but statement (ii) is false (C) Stat Statment ment (i) (i) is false false but statemen statementt (ii) is true true (D) Both the stat statemen ements ts are are true true 37. Feed Feedback back cont control rol syst systems ems are (A) insensitive to both forward- and feedback-path feedback-path parameter parameter changes changes (B) less sensitive sensitive to feedback-path feedback-path parameter parameter changes than than to forward-path forward-path parameter parameter charges (C) less sensitive sensitive to forward-path forward-path parameter parameter changes than to feedback-path feedback-path parameter parameter changes (D) equally sensitive sensitive to forward- and feedback-path feedback-path parameter parameter charges charges 38. In the syste system m of Fig. Fig. Q38, sensi sensitivi tivity ty of M M ( s) s) = Y ( s s)/ )/ R( R( s s)) with respect to parameter K K 1 is (A)

(B)

1 1 + K1 K 2

R(s)

Y(s) G(s)

K 1

1 1 + K1 G (s )

K 2

( C) 1 (D) None of of the answe answers rs in (A), (A), (B), (B), and (C) (C) is Fig. Q38 correct 39. In the syste system m of Fig. Fig. Q38, sensi sensitivit tivity y of M M ( s) s) = Y ( s s)/ )/ R( R( s) s) with respect to parameter K K 2 is (A)

(C)

1

1

(B)

1 + K 2 K 1

- K2 G ( s ) 1 + K2 G ( s )

1 + K2 G (s)

(D) None of the answers answers in (A), (B), (B), and (C) (C) is correct.

40. A speed control control system is represented by the signal flow flow graph shown shown in Fig. Q40. Q40. The nominal nominal value of the parameter K is K is 10. Sensitivity of M M ( s s)) = w ( s)/ s )/w r ( s s)) to changes in K in K is is (A)

(C)

0.1 + 0.1 s + 11

(B)

0.1 - 0.1 s + 0.2

(D) None of the answers in (A), (A), (B), and (C) is correct.

s

w r ( s)

1

0.1 + 0.1 s + 0.2 s

K 10 s + 1

1

w ( s)

– 0.1 Fig. Q40 41. A speed control system system is represented represented by the bloc block k diagram of Fig. Q41. Q41. The system is is subjected to a step step OL disturbance W ( s). s ). w CL / (the steady-state speed under closed-loop operation/steady-state speed under w ss ss open-loop operation) is equal to

Appendix-B OLC.p65

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Control Systems: Principles and Design (s) W (s)

(A ) (B ) (C ) (D)

1 /2 1 2 None of the the answe answers rs in in (A), (A), (B), (B), and (C) w r ( s) = 0 is correct 42. The time time respons responsee of the syste system m of Fig. Fig. Q42a to an input r (t ) = 10 m (t ) is shown in Fig. Q42b. The gain K gain K is is equal to (A ) 1 0 (B ) 8 (C ) 4 (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct

10 10 s + 1

w

( s)

0.1

Fig. Q41

y

8 r

K t s

y

+1

t

O

(a)

(b)

Fig. Q42 43. An inertial inertial and a frictional load are driven by a dc motor motor with torque T M . The dynamic model of the system is T M (t ) = J

d w (t ) dt

+ Bw (t )

The steady-state speed of the motor for step input will be doubled when (A)) iner (A erti tiaa J J is is doubled (B)) fr (B fric icti tion on B B is doubled (C) bot both h the ine inerti rtiaa J J and and friction B friction B are doubled (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 44. Consider Consider the the followi following ng statem statements ents:: (i) If an open-loop system is unstable, unstable, applying feedback feedback will always improve its stability. stability. (ii) If an open-loop open-loop system is subject to parameter variations, variations, applying feedback will always improve robustness. Which of the following is the correct answer? (A) None of of the above above statem statements ents is is true true Y(s) R(s) 3 4 (B) Stat Statemen ementt (i) is true true but statemen statementt (ii) is false false s + 1 (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the statem statements ents are true true 45. The character characteristic istic equatio equation n of the the closed-loop closed-loop 10 system of Fig. Q45 is s + 10 (A) s2 + 11 s + 10 = 0 (B) s2 + 11 s + 130 = 0 Fig. Q45 (C) s2 + 10 s + 120 = 0 (D) s2 + 10 s + 12 = 0

Appendix-B OLC.p65

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Appendix B: Control Theory Quiz 46. Which of the following following describes describes the the step response of the closed-loop system of Fig. Q46? (A)) Un (A Unde derd rdam ampe ped d (B) Cri Critic ticall ally y dam damped ped (C) Ov Over erda damp mped ed (D) The output output does does not reach reach a steady-sta steady-state te value. value. 47. The system system of Fig. Fig. Q47 has has the steady-s steady-state tate error error (A) (C)

1

(B)

1.125 1.1

10 2 s + 2s + 1

Y(s)

Fig. Q46

0.1 1.125


1.125 W ( s) =

R ( s) =

R(s)

1 s

1 s

1.25

q r r

Y(s)

0.1

1

K

t ss + 1 t

2 s + 1

Fig. Q47

q

Fig. Q48

48. It is is desire desired d that that the the outpu outputt q changes by unit step with zero error at steady state (Fig. Q48). We can achieve it by injecting (A) q r = unit step (B) q r of step size (1+ K )/ )/ K K (C) q r of step size K size K /( /( K K +1) (D) None of the answers in (A), (B), and (C) is correct 49. Consider the the system of Fig. Q49. The steady-state error is zero for (A) K f = 0

(B) K f =

1

1.25 (D) None of the answers in (A), (B), and (C) is correct

(C) K f = 1.2 5

W ( s) =

1 s

K f

R(s) = 0

Y(s)

0.1

1.25

2 s + 1

Fig. Q49 50.. Sens 50 Sensit itiv ivit ity y S M s)) with respect to open-loop transfer function G ( s s)) G of the closed-loop transfer function M ( s (Fig. Q50) is s ( 2 s + 1) 1 (A) (B) 2 2 2 s + (3 K P + 1) s+ 3 KI 2 s + ( K P + 1) s + 3 KI (C)

Appendix-B OLC.p65

s ( 2 s + 1) 2

2s

+ ( K P + 1) s +

23

KI

(D)

s ( 2 s + 1) 2

2s

+ (3 K P + 1) s+ 3 KI

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24

Control Systems: Principles and Design W(s) R(s)

Y(s)

K ˆ 3 Ê G ( s) = Á K P + I ˜ s ¯ 2 s + 1 Ë

Fig. Q50 51. For For what what val value ue of of K is K is the time-constant of the system of Fig. Q51 equal to 0.2 seconds? (A) K K = =3 (B) K K = =5 (C) K K = =7 (D) K K = =9 52. Con Consid sider er the syst system em of Fig. Fig. Q52. Q52. The K The K P to move the time-constant to one sixth of its open-loop value is (A) (C)

R(s)

0.333

(B)

2 6

(D)

0.333

K

3 2 s + 1

Y(s)

0.333 6 2 0.333

R(s)

10

K P

Y(s)

s + 0.333

F ig . Q51 F ig . Q52 53. Consider the system of Fig. Q Q53. 53. The steady-stat steady-statee offset is zero when when K K P is ( A) z e r o (B) infinity (C) an any y va valu luee of K P (D) no val value ue of K K P W ( s) =

R(s) = 0

K P

0.3 2 s + 1

0.5 s Y(s)

Fig. Q53 54. A prop proporti ortional onal cont controll roller, er, K K P , is used with a first-order system G ( s) s) = K /( /(t s + 1) in unity-feedback structure. Increasing K Increasing K P will (A) increase the the time-constant time-constant and decrease decrease the steady-stat steady-statee error to to step inputs inputs (B) decrease the time-constant and decrease the steady-state steady-state error to step inputs (C) decrease the time-constant and increase increase the steady-state steady-state error error to step step inputs (D) increase the time-consta time-constant nt and increase increase the steady-state error to step inputs inputs 55. Cons Consider ider the syste system m of Fig. Q55. Q55. K K I that gives critical Y(s) R(s) K I 0.3 damping is s 2 s + 1 (A ) 0 . 4 1 7 (b) 0 .2 57 57 (C ) 1 (D) None of the answers answers in in (A), (B), (B), and (C) is Fig. Q55 correct

Appendix-B OLC.p65

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Appendix B: Control Theory Quiz 56. Consider the the system of Fig. Q56. The The steady-state steady-state offs offset et is zero zero for (A) K I = 0 (B) K I = • (C) an any y va valu luee of of K K I (D) no val value ue of K K I W (s ) = R(s) = 0

K I s

0.5 s

0.3 2 s + 1

Y(s)

Fig. Q56 57. An integ integral ral cont contro rolle ller r K K I / s s is used with a first-order system G( s) s ) = K /( /(t s + 1) in unity-feedback structure. Increasing K Increasing K I will (A) decrease damping damping with with no change change in steady-state steady-state offset offset to step commands. commands. (B) decrease damping damping with decrease in in steady-state steady-state offset offset to step step commands. commands. (C) increase damping damping with no change in steady-state steady-state offset offset to step step commands. commands. (D) increase damping damping with decrease in in steady-state steady-state offset offset to step step commands. commands. 58. The conditi condition on that all all the roots roots of the polyn polynomia omiall s ) = a0 s3 + a1 s2 + a2 s + a3; ai > 0 D( s) have negative real parts, is given by (A) a1 a3 > a0a2 (B) a1 a0 > a2a3 (C) a1a2 > a0a3 59. Cons Consider ider the the follow following ing statem statements ents:: (i) All the the roots roots of the polyno polynomial mial D1( s) s ) = 3 s4 + 10 s3 + 5 s2 + 2 have negative real parts. (ii) All the the roots roots of the polyno polynomial mial D2( s) s ) = s4 + 4 s 3 + 6 s 2 + 4 s  5 have negative real parts. Which of the following is the correct answer? (A) None of of the above above statem statements ents is is true true (B) Stat Statemen ementt (i) is true true but statemen statementt (ii) is false false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the statem statements ents are true true 60. The characteristic characteristic equation equation of a feedback control system system is given by

(D) a2 a0 > a1a3

2 s4 + s3 + 2 s2 + 5 s + 10 10 = 0 The number of roots in the right half of of s s-plane -plane are ( A) z e r o (B ) 1 (C) 2 61. Th Thee poly polyno nomi mial al D( s) s ) = s3 + 3 s 2 + 2 s + 6 has (A) tw two o roots roots in left left half half s-plane s-plane and one root in right half (B) two root rootss in righ rightt half half s-plane s-plane and one root in left half (C) tw two o ro root otss on on jw -axis -axis of s-plane s-plane and one root in right half (D) tw two o ro root otss on on jw -axis -axis of s-plane s-plane and one root in left half. 62. The characteristic characteristic equation of a feedback control control system system is is

(D) 3

s3 + a 1 s2 + a 2 s + K a a 2 = 0; a 1, a 2, > 0,

Appendix-B OLC.p65

25

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26


where K where K is is a variable positive scalar parameter. The system is stable for all values of K of K given given by (A) K K > > a 2 (B) K (B) K < < a 2 (C) K K > > a 1 (D) K K < < a 1 63. The open-loop open-loop transfer transfer function function of a unity feedback system is G ( s) s ) = K /[ /[ s s2( s s + 5)]; K K > >0 The system is unstable for (A) K K > >5 (B) K < K < 5 (C) K K > >0 (D) Al All the answers in (A), (B), and (C) are correct 64. A unity feedback feedback system system has open-loo open-loop p transfer transfer function function G ( s) s ) = K /[ /[ s( s( s + 1)( s s + 2)]; K K > >0 The value of K that K that results in oscillatory response to step input, is (A ) 2 (B ) 6 (C) 2 0 65. The first first two rows rows of Routh Routh tabulation of a third-order third-order system system are s 3

2

2

s 2

4

4

(D) 6 0

(A) The characteristic characteristic equation has one one root in right right half s-plane s-plane (B) The characte characterist ristic ic equation equation has two two roots on the the jw -axis -axis at s at s = ± j ± j (C) The characte characterist ristic ic equation equation has two two roots on the the jw -axis -axis at s at s = ± 2 j (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 66. Presence of transportion lag in the forward path path of a closed-loop control system system (A) decreases margin of stability (B) increases margin of stability (C) does not not affect affect its marg margin in of stabi stability lity 67. Cons Consider ider a unity-f unity-feedba eedback ck system system with plant plant G( s) s ) =

1 s

2

values of K K P for which the system is stable are (A) K P > 0 (B) un unstable for all K P > 0 68. The error error functi function on of a feedba feedback ck system system is E ( s) s) =

+ s +1

and cascade controller D controller D(( s s)) = K P +

(C) K P > 1

2 s

. The

(D) K P > 2

( s + 0.1)( s + 0.5) s( s+ 0.1)( s+ 0.5) + 0.5( s+ 1)

The steady-state value of e of e(t ) is (A ) 0 . 0 0 1 (B ) 0. 1 (C ) 0. 01 (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 69. Consider the system system of Fig. Fig. Q69. The steady-state steady-state error error is less than than 2% for step input for (A) K c > 49 (B) any value of K c (C) no val value ue of K (D) K c >50 K c

R(s)

K c

Y(s)

2 3

s

+

2

4s

+

5 s+ 2

Fig. Q69 70. A unity unity feedback feedback system system has has open-loop open-loop transfer function G( s) s ) = 9/[ s( s ( s s + 3)]. The system has (A)) da (A damp mpin ing g rati ratio o = 1/2, 1/2, and and nat natur ural al fre frequ quen ency cy = 9 (B) da damp mpin ing g rati ratio o = 1/6, 1/6, and and nat natur ural al fre frequ quen ency cy = 3 (C) da damp mpin ing g rati ratio o = 1/6, 1/6, and and nat natur ural al fre frequ quen ency cy = 9 (D)) da (D damp mpin ing g rati ratio o = 1/2, 1/2, and and natu natura rall freq freque uenc ncy y=3 71. A unity feedback system has open-loop open-loop transfer transfer function G( s s)) = 25/[ s( s ( s s + 6)]. The time t p at which the peak of the step-input response occurs, is (A) 11/7 sec (B ) 11/4 sec (C ) 11/14 sec (D) 11/2 8 sec

Appendix-B OLC.p65

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27


72. Peak overshoot overshoot of step-input step-input response of of an underdamped underdamped second-order second-order system is explicitly indicative indicative of (A) settling time (B ) ri r ise time (C) natural frequency (D) da d amping ratio 73. A unity feedback feedback system system has open-loo open-loop p transfer transfer function function G( s s)) = 25/[ s( s ( s s + 6)]. The peak overshoot in the stepinput response of the system is approximately equal to (A ) 5 % (B ) 10 % (C) 1 5 % (D) 2 0% 74. The step-input step-input response of the system of Fig. Q74a is shown in Fig. Q74b. Q74b. The value of of damping ratio ratio of the system is (A ) 0 . 3 9 (B) 0 .49 (C) 0. 59 (D) 0.69 y

0.003

K s ( s + p)

r

0.03

y

0 (a)

t

(b)

Fig. Q74 75. A unity unity feedback feedback system system with open-loop transfer function G( s s)) = 4/[ s( s( s + p p)] )] is critically damped. The value of the parameter p p is (A ) 4 (B ) 3 (C) 2 (D) 1 76. A unity feedback feedback system system has open-loop open-loop transfer transfer functio function n G ( s). s). The steady-state error is zero for (A) ste step p input input and typ type-1 e-1 G( s s)) (B) ramp input and type-1 G( s) s) (C) ste step p input input and typ type-0 e-0 G( s s)) (D) ramp input and type-0 G( s) s) 2 77. A unity feedback feedback system system with with forward forward path transfer transfer function function G ( s s)) = 1/[ s s ( s + 1)] is subjected to an input r (t ) = K = K 1 + K 2t t + +

1 2 t . 2

The steady-state error of the system is

(A) infinity ( B) 1 ( C ) z e ro ( D) None of the answers in (A), (B), and (C) is correct 78. Closed-loop transfer function of a unity feedback feedback system system is given by Y (s ) R ( s )

=

1

t s + 1

System K is System K (A) t (B) 1/t (C) 1 79. Closed-loop transfer function of of a unit unity y feedback system is given by v

Y (s ) R ( s )

=

(D)

•

w n2 s 2 + 2zwn s + w n2

System K System K is (A) w n /2 z (B) 1 (C) • (D) 2z /w n 80. Cons Consider ider the the position position control control system system of Fig. Q80. Q80. The value of of K K such that the steady-error is 10° for input 400t m (t ) rad/sec, is q r = 400t (A ) 1 0 4 . 5 q 20 q r K (B ) 114. 5 s (0. (0.1 1s + 1) (C ) 12 4. 5 (D) None of of the answer answerss in (A), (A), (B), and and (C) is is correct v

Fig. Q80

Appendix-B OLC.p65

27

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28


81. Consider the speed speed control system system shown in Fig. Q81. Parameter Parameter variations variations occurring during operating operating condi¢ = 0.9 K A. The value of K tions cause K cause K A to modify to K to K A K A that limits the change in steady-state motor speed due to parameter variations to 0.1%, is (A ) 2 5 (B ) 35 (C) 45 (D) 5 5 w r

= const

20 0.1 0. 1 s + 1

K A

w

0.1

Fig. Q81 82. Derivative error compensation compensation is employed in in feedback control systems systems to (A) incr increase ease the the effective effective damping damping in the system system (B) decre decrease ase the effect effective ive damping damping in the system system (C) impr improve ove the steady-s steady-state tate respons responsee of the system 83. Tachogenerato Tachogeneratorr feedback is sometimes sometimes used in position control systems systems to (A) incr increase ease the the effective effective damping damping in the system system (B) decre decrease ase the effect effective ive damping damping in the system system (C) impr improve ove the steady-s steady-state tate respons responsee of the system 84. Integral error error compensation compensation is employed in feedback control systems systems to to (A) imp improv rovee dam dampin ping g (B) impr improve ove spee speed d of of respo response nse (C) reduc reducee stead steady-st y-state ate erro error r 85. A temperature control control system is is found to have zero error error to a constant tracking tracking input, and an error error of 0.5°C to a tracking input that is linear in time, rising at the rate of 40°C/sec. What is the type number of the system? (A ) 0 (B) 1 . (C) 2 (D) Ty Type number cannot be determined from the given information 86. Poles of of second-order second-order models models with the same damping ratio ratio lie (A) on the real axis (B ) on a radial line from the origin (C) on the imaginary axis (D) None of the answers in (A), (B), and(C) is correct 87. The percenta percentage ge overshoot overshoot of a second-o second-order rder system system G ( s) =

K w n2

to a step input depends only on s 2 + 2zwn s + w n2 (A)) the val (A alu ue of the step inp nput ut (B)) th (B thee val alu ue of th thee da dam mpin ing g rat atio io (C) the val value ue of of the the gain gain K K (D) the param parameter eter w n 88. Consider the system system of Fig. Q88. Increasing proportional gain will (A) increase the the overshoot overshoot and decrease decrease the steady-state steady-state error to ramp inputs inputs (B) decrease the overshoot and increase increase the steady-state steady-state error error to ramp ramp inputs (C) increase the overshoot and and increase the steady-state steady-state error to ramp inputs inputs (D) decrease the overshoot and decrease decrease the steady-state steady-state error error to ramp ramp inputs 89. Consider the system of Fig. Q89. The steady-sta steady-state te offset is zero when (A) K P = •, K D = • Y(s) R(s) 1 (B) K P = 0, K 0, K D = any value K P s(s + 1) (C) K P = •, K D = any value (D) no va valu lues es of K K P and and K K D

Fig. Q88

Appendix-B OLC.p65

28

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29

Appendix B: Control Theory Quiz W (s ) = R(s) = 0

K P + KD s

s2

0.5 s Y(s)

2 + 0. 0.4 4s + 1

Fig. Q89 90. Consider Consider the syste system m of Fig. Fig. Q90. Q90. R R(( s s)) and W ( s s)) are step step signals signals (A) Ste Steady ady-st -state ate erro errorr to R to R(( s s)) is zero and to W ( s) s ) is non-zero (B) Ste Steady ady-s -stat tatee error error to R to R(( s s)) is non-zero and to W ( s s)) is zero (C) Stead Steady-st y-state ate erro errorr to both both R R(( s) s) and W ( s) s ) is non-zero

W(s) R(s)

Y(s)

K s (t s + 1)

K P

K D s

Fig. Q90 91. Consider the system system of Fig. Fig. Q91. The steady-state steady-state error error is less than than 2% to step inputs inputs for (A)) an (A any y val value ue of K I (B) no values of K K I (C) K I > 50 (D) None of the answers is (A),(B), and (C) is correct

R(s)

3+

K I s

2 s3 + 4 s2 + 5 s+ 2

Y(s)

Fig. Q91 92. A unity feedback feedback system system has open-loo open-loop p transfer transfer function function G ( s) s) = 1/ s s 2. This uncompensated type-2 system possesses a satisfactory steady-state error for (A) step input signals ( B) ramp input signals ( C) both the step and ramp input signals (D) none of the step and ramp input signals 93. Cons Consider ider the the follow following ing statem statements ents:: (i) Many systems systems are are designed designed for peak peak overshoot overshoot in the range range 52 5%. (ii) Desired dominant dominant closed-loop closed-loop poles are usually a complex-conjugate complex-conjugate pair. pair. (A)) None of the (A the abo bove ve sta tate tem men entts is is tr true (B)) Sta (B tattem emeent (i) is tr true but sta tattem emen entt (ii (ii)) is is fa fals lsee (C) St State ateme ment nt (i) (i) is is false false but sta statem tement ent (ii (ii)) is tru truee (D) Bot Both h the the state statemen ments ts are are true true 94. The steady-state steady-state error due to step step commands commands can be eliminated from proportional proportional control systems systems with type-0 type-0 plants by

Appendix-B OLC.p65

29

4/22/08, 3:45 PM

30

95.

96. 97.

98.


(i) intention intentionally ally misadju misadjustin sting g reference reference input (ii) intr introduc oducing ing integral integral mode mode in the controller controller (A) No None ne of the ab abov ovee st stat ateeme ment ntss is is tr true ue (B)) St (B Stat atem emen entt (i) (i) is tr true but but sta tate tem men entt (i (ii) is fal falsse (C) Sta Statem tement ent (i) is fals falsee but but state statemen mentt (ii) (ii) is is true true (D) Bot Both h the the state statemen ments ts are are true true Consider Cons ider the the followi following ng statem statements ents:: (i) In a position position control system, system, the integrating integrating effect effect naturally appears appears in system system plant. (ii) We do not require integral integral control for position position control control systems to obtain obtain zero steady-state steady-state error to step disturbances in load. (A) No None ne of the ab abov ovee st stat ateeme ment ntss is is tr true ue (B)) St (B Stat atem emen entt (i) (i) is tr true but but sta tate tem men entt (i (ii) is fal falsse (C) Sta Statem tement ent (i) is fals falsee but but state statemen mentt (ii) (ii) is is true true (D) Bot Both h the the state statemen ments ts are are true true A type-1 plant plant is changed to type-2 feedback feedback system by the following following cascade control control action: ( A) P D (B) P I (C ) Either PD or PI (D) Neither PD nor PI A unity feedb feedback ack system system has open-lo open-loop op poles poles at s at s =  2 ± j ± j2, 2, s s =  1, and s and s = 0; and a zero at s at s =  3. The angles made by the root-locus asymptotes with the real axis, and the point of intersection of the asymptotes are, respectively, (A) (6 (60 0°,  60°, 180°) and  3/2 (B) (60°,  60°, 180°) and  2/3 (C ) (45°,  45°, 180°) and  2/3 (D) (45°,  45°, 180°) and  4/3 Which of the root locus locus plots shown in Fig. Fig. Q98 is the correct correct plot for a unity feedback feedback system with open-loop open-loop poles at s at s =  1 ± j ± j1, 1, and a zero at s at s =  2? (A) Fig. Q98a (B) Fig. Q98b (C) Fi Fig. Q98c (D) Fig. Q98d jw jw

jw jw

s

s

(a)

(b)

jw jw

jw

s

s

(c)

(d)

Fig. Q98 99. Which of the root locus locus plots shown in Fig. Fig. Q99 is the correct plot plot for a unity feedback system system with open-loop open-loop 2 transfer function G( s ) = = K / /[ [ ( + 5)]? s) K s s s (A) Fig. Q99a (B ) Fig. Q99b (C) Fi F ig. Q99c

Appendix-B OLC.p65

30

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31

Appendix B: Control Theory Quiz jw jw

jw jw

jw jw

s

s

(a)

s

(b)

(c)

Fig. Q99 100. A unity feedback system system has open-loop open-loop transfer transfer function G( s s)) = K = K ( s s + 1)( s s + 2) [ s s(( s s + 3) ( s + 4)]. For K = K = 10, the closed-loop poles are (A) all rea reall and and disti distinct nct (B) one real real and and two comp complex lex conju conjugate gate (C) all rea reall and and repea repeated ted (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 101. Root locus locus plot plot of a feedback feedback system system as as gain K gain K is is varied, is shown in Fig. Q101. The system response to step input is non-oscillatory non-oscillat ory for (A) 0 < K < 0.4 K < (B ) 0.4 < K K < <6 (C ) 6 < K K < <• (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 102. Cons Consider ider the the root locus locus plot shown shown in Fig. Fig. Q101. Q101.

jw

2

(i) Adding Adding a zero zero betw between een s s =  1 and s and s =  2 would move the root locus to the left. (ii)) Add (ii Adding ing a pole pole at at s s = 0 would move the root locus to the right. Which of the following is the correct answer? (A) None of of the above above statem statements ents is is true true (B) Statement (i) is true but statement (ii) is false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the stat statemen ements ts are are true true 103. Consider the the root locus plot plot of unity-feedback system with open-loop transfer function G( s) s) =

–

K = 0.4

K = 6 K •

Fig. Q101

K ( s + 5) s( s + 2)( s+ 4)( s2

+ 2 s + 2)

The meeting point of the asymptotes on the real axis occurs at ( A)  1 . 2 (B )  0.85 (C)  1.05 (D)  .75 104. Consider the root root locus plot plot of unity-feedback unity-feedback system with open-loop transfer fun function ction G ( s) s ) =

Appendix-B OLC.p65

31

K ( s + 5) s( s + 2)( s+ 4)( s2

s

1

–

+ 2 s + 2)

4/22/08, 3:45 PM

32


The break away point of the root loci on the real axis occurs at ( A)  3 (B )  4.5 (C )  5.5 (D) None of the answers is (A), (B), and (C) is correct 105. The transfe transferr function function of a lag compens compensator ator is D(( s D s)) =

1 + at s 1 + t s

;t > 0

Im

the value of a is given by (A) a = 1 (B ) a >1 (C) a < 1 (D) a is any constant 106. The root root locus plot plot of the character characteristi isticc equation equation

j1

45°

1+ KF 1+ KF ( s) s ) = 0

107.

108.

109.

110. 11 0.

Appendix-B OLC.p65

is given in Fig. Q106. The value of K at K at s s = ± j ± j1 1 is (A ) 4 (B ) 1 (C ) 10 (D) None of the answers answers in in (A), (B), and and (C) is correct Consider Consi der the the followin following g statement statements: s: (i) A lead compen compensator sator is nothing nothing but a PD controller with a filter. (ii) A lag compensa compensator tor has the the characteri characteristic sticss of a PI controller (A) None of of the above above statem statements ents is is true true (B) Stat Statemen ementt (i) is true true but statem statement ent (ii) (ii) is false (C) Stat Statemen ementt (i) is false false but statem statement ent (ii) (ii) is true (D) Both the stat statemen ements ts are are true true Figure Q108 shows shows the the Nyquist Nyquist plot of of a unity unity feedback system having open-loop transfer function G( s s)) with one pole in right half of s of s-plane. -plane. The feedback system is (A) st stab ablle (B)) un (B unst stab able le (C) mar margin ginall ally y stab stable le Figure Q109 shows shows the the Nyquist Nyquist plot of of a unity unity feedback system having open-loop transfer function of s-plane. -plane. The G( s s)) with one pole in right half of s feedback system is (A) stable (B ) unstable (C) mar margin ginall ally y stab stable le Pola Po larr plo plott of of G G( jw ) = 1/[ jw (1 + j + jw t )] )] (A) cros crosses ses the the negati negative ve real real axis axis (B) cros crosses ses the the negative negative imagin imaginary ary axis axis (C) cros crosses ses the the positive positive imagi imaginary nary axis axis (D) None of the answers answers in in (A), (B), and and (C) is correct

32

Re

j1

–

Fig. Q106 Im

w = 0

w = •

– 1

Re

-p lo lot G( jw ) -p

Fig. Q108 Im

w = • w = – 1

G( jw )- pl plo ot

w = 0 w = Re

Fig. Q109

4/22/08, 3:45 PM

33


111. Which of the polar plots plots shown in Fig. Q111 Q111 is the correct plot for for G G ( jw ) = 1/[( jw )2(1 + j + jwt )]? )]? (A) Fig. Q111a (B ) F Fiig. Q111b (C) Fi F ig. Q111c (D) Fig. Q111d Im

Im

w = 0

w =

w = •

•

Re

w 0

(a)

(b)

Im

Im 0

w

w

0

w = • Re

Re

w = •

(c)

(d)

Fig. Q111 112. Which of the Bode asy asymptotic mptotic plots plots shown in Fig. Q1 Q112 12 is the correct correct plot for for G G ( s s)) = K = K /[ /[ s s 2( s s + 5)]? (A) Fig. Q112 a (B ) Fi Fig. Q112 b (C) Fi F ig. Q112 c (D) Fig. Q112d dB

dB

– 6 – 6 dB/octave

– 40 – 40 dB/decade

– 12 12 dB/octave – 60 – 60 dB/decade

w = 5

log w

w = 5 (a)

dB

– 40 – 40 dB/decade

dB

– 6 – 6 dB/octave

– 12 12 dB/octave

– 60 60 dB/decade

w = 1 5

log w

(b)

log w (c)

w = 1 5 (d)

Fig. Q112

Appendix-B OLC.p65

33

4/22/08, 3:45 PM

log w

34


113. The Bode asymptotic asymptotic plot plot of a (A)) ar (A aree no no pol poles es at th thee ori origi gin n 114. The Bode asymptotic asymptotic plot of a (A) one pole and one zero

transfer tr ansfer function function is given given in Fig. Fig. (B)) is on (B onee pol polee at at the the or orig igin in transfer function transfer function is given in Fig. (B) two poles and one zero

+ 20 dB/deca dB/decade de

dB

– 12 12 dB/octave

dB

Q113. There There (C)) ar (C aree two two po pole less at at the the or orig igin in Q114. The transfer function function has (C ) one pole and two zeros

+ 40 dB/decade

log w

log w

– 6 dB/octave

F ig . Q113

F ig. Q114

115. The minimum-phase minimum-phase transfer function that corresponds corresponds to the Bode asymptotic plot shown in Fig. Q115, is (A) (C)

1 2 s

dB

(B) 2 s + 1

+1

+ 20 dB/decade

1

1

(D) s + 1 1 2 s + 1 2 116. A unity feedback feedback system system has open-loop open-loop transfer transfer function function G( s s). ). Polar plot of G of G( jw ) is shown in Fig. Q116. The gain margin (GM (GM ) and the phase margin ( F M ) of the feedback system are (A) GM GM = =  0.3; F M = M = 112.33° (B) GM GM = = 0.3; F M = M = 112.33° (C) GM GM = = 3.33; F M M = = 67.67° (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 117. A unity feedback system has open-loop open-loop transfer function G( s s)) = K /[ /[ s s(1 (1 + s t )]. The gain margin of the feedback system is (A) • (B ) 0 (C ) 1 (D) None of the the answers answers in (A), (A), (B), and and (C) is corre correct ct 118. A unity unity feedback feedback system system has open-loop transfer function G( s s). ). Bode plot of G( jw ) is shown in Fig. Q118. The feedback system has (A) posi positive tive phase phase margin margin and negative negative gain gain margin margin (B) posi positive tive phase phase margin margin and positi positive ve gain margin margin (C) negat negative ive phase phase margin margin and negativ negativee gain margin margin (D) negat negative ive phase phase margin and and positive positive gain margin margin

Appendix-B OLC.p65

34

w

=2

log w

Fig. Q115 Im 0.3 w

1

=• º 3 3 2. 1 1

Fig. Q116

4/22/08, 3:45 PM

w

=0 Re

Appendix B: Control Theory Quiz dB

119. The corner corner frequenci frequencies es in Bode plot plot of the transfer transfer function G( s s)) =

+ 10s + 100) s (50 s2 + 15 s+ 1)

6( s 2 2

are (A ) 1 , 5 0 (B ) 100, 1 (C) 10, 1 (D) None of the the answers answers in (A), (A), (B), (B), and (C) is correct 120. The low-fre low-frequenc quency y asymptote asymptote in in Bode plot plot of G( s s)) =

35

0

log w

f

180º – 180º

+ 10s + 100) 2 s (50 s2 + 15 s+ 1)

6( s 2

log w

has a slope of Fig. Q118 (A) 0 dB/decade (B)  2 0 dB/decade (C))  40 (C 40 dB/ dB/de deca cad de (D)) No (D None ne of th thee ans answ wer erss in in (A (A), (B (B)), an and (C) (C) is cor orre recct 121. The high-fr high-frequen equency cy asymptot asymptotee in Bode plot plot of G( s) s) =

(A ) (C) 122.

123.

124.

125.

126.. 126

127.

128.

Appendix-B OLC.p65

+ 10s + 100) s (50 s2 + 15 s+ 1)

6( s 2 2

has a slope of 0 dB /decade (B )  2 0 dB/decade  40 40 dB/decade (D) None of the answers in (A), (B), and(C) is correct If a syst systems ems gain is given given as |G |G( jw 0 )| = 0 dB, what will happen to an input signal at the frequency w 0? (A) It will be amplified (B ) It will be attenuated (C) Th Ther eree wil willl be be no no out outpu putt sig signa nall (D) No None ne of th thee ans answe wers rs in (A (A), ), (B (B), ), an and d (C) (C) is co corr rrec ectt An engine engineer er appli applies es the the input input r (t ) = 2 sint sint to to a chemical process and measures the output as y as y((t ) = 0.4 sin (t ( t   1.55). What are the gain and phase of the system? (A) ( 14dB,  89°) (B ) ( 8dB,  1.55°) (C ) (14 dB,  1.55°) (D) ( 8dB,  89°) A current current to pressure transducer has a gain of  10 dB dB and a phase of  60° at w = 10 rad/sec. An input signal of r of r (t ) = 5 cos(10t cos(10 t   p /2) /2) is injected. What is the output signal? (A)) 1. (A 1.58 58 co coss (2 (20 0 t t   5 p /6) (B (B)) 1. 1.5 58 co cos (1 (10 t t   p /6 /6)) (C) 1. 1.58 58 co coss (10 (10t t + + p /6) (D) Non Nonee of the the answ answers ers in (A), (A), (B) (B),, and and (C) (C) is cor correc rectt Bode magnitude magnitude plot of a system system has  20 dB gain at low frequencies. frequencies. The system system is (A) type-0 (B ) type-1 (C ) type-2 (D ) Nothing can be deduced about type number from the given information A PI contr control olle ler r K K P + K I / s s = 100 + 0.01/ s s has a high-frequency gain of 40dB. The values of K K P and and K K I that reduce high-frequency high-frequen cy gain to 20dB are (calculations based on asymptote plot in Bode coordinates) (i) (10,0.01) (ii) (1 ( 10,0.001) (A)) Bo (A Both th the sta tattem emen ents ts are tr tru ue (B)) Sta (B tattem emen entt (i (i) is is tru truee bu but (ii (ii)) is is fa fals lsee (C) Stat Statemen ementt (i) (i) is fals falsee but (ii) is true true (D) Both the state statement mentss are are false false t D s The frequ frequency ency respo response nse of the system system G ( s / s /(2t D) gives magnitude and phase of s)) = e  t s at frequency w = p /(2 (A) 2t D /p ,  90° (B) 2t D /p , 180° (C ) 2t D /p ,0° (D) None of the answers in (A), (B) and (C) is correct A PD controller controller has has transfer transfer function function (0.1 (0.1 + 0.01 s s). ). The frequency at which the magnitude is 20dB is (calculations based on asymptotic plot is Bode coordinates) (A ) 1 0 (B ) 1000 (C ) 1 ( D) 100

35

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36


129. Frequency response of of a system system is given given in the the following table. Frequen cy (rad/sec)

Gain

0 .1 1 10 100

P ha s e ( d e g)

5 1.5 0.2 5 0. 1

0  90  180  235

Gain margin is (A ) 4 (B) 0 .2 5 (C) 5 130. Frequency response o off a sys system tem is given given in the the following table. Frequency (rad /s ec)

(D) 1. 5

Gain

1 10 100 1 00 0

P ha s e ( d e g)

10 2 1 0. 1

 50  100  155  2 35

Phase margin is ( A)  2 5 ° (B ) +25° (C)  55° 131. Nyquist plot plot and Bode magnitude plot plot of two systems aare re given in Fig. Q131. (A) Both the syst systems ems are typetype-0 0 (B) Both the syst systems ems are typetype-1 1 (C) Syst System em I is type-0 type-0 and syste system m II is type-1 type-1 (D) Syst System em II is type-0 type-0 and and system system I is type-1 type-1 Im (G)

dB

w = w =•

(D) +55°

20dB/decade – 20dB/decade

w = w =• log w

Re (G) 20dB/decade – 20dB/decade w System I

System II

Fig. Q131 132. Consider the the following following statements for an underdamped underdamped second-order second-order system: (i) Peak overshoot overshoot in step-input step-input response response reduces as damping is increased from 0.2 to 0.6. (ii) Resonance peak in frequency response response reduces as damping is increased increased from 0.2 0.2 to 0.6. Which of the following is the correct answer? (A) None of of the ab above statements is true (B ) Statement (i) is true but st statement (ii) is false (C)) St (C Stat atem emen entt (i) (i) is fa fals lsee but but st stat atem emen entt (ii (ii)) is is tru truee (D)) Bo (D Both th th thee sta state teme ment ntss are are tr true ue 133. Unda Undamped mped natur natural al freq frequency uency w n and resonance frequency w r of a unity feedback system with open-loop transfer function

Appendix-B OLC.p65

36

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37


w n2 G( s) s) = ; z < s ( s + 2z w n ) are related as (A) w n = w r (C) w n < w r

1 2

,

(B) w n > w r (D) None of the answers answers in (A), (B), (B), and (C) is correct correct

w n2 134. For a unity feedback feedback system system with open-loop transfer function G ( s) s) = , z < 0.7; s ( s + 2 z w n ) (i) phase margin margin is explicitly indicative of damping damping ratio; ratio; (ii) resonance peak is explicitly indicative of damping damping ratio. ratio. Which of the following answers is correct? (A) None of the above st statements is true (B ) Statement (i) is true but statement (i (ii) is false (C)) St (C Stat atem emen entt (i) (i) is fa fals lsee but but st stat atem emen entt (ii (ii)) is is tru truee (D)) Bo (D Both th th thee sta state teme ment ntss are are tr true ue 135. Open-loop transfer function of a unity feedback feedback system system is G( s) s ) = K = K /[ /[ s( s ( s s + 5)]. The gain K gain K that that results in a phase margin of 50° is approximately ( A) 1 5 (B ) 2 0 (C) 2 5 (D) 30 136. Consider a unity-feedback unity-feedback system system with open-loop transfer function G( s s)) =

K s ( s

+ 1)

cause: (A) the gain crossover frequency to reduce (C ) the system to respond more slowly 137.

138. 139. 140.

141.

142. 14 2.

Appendix-B OLC.p65

. Increase in gain gain K K will

(B ) the gain crossover frequency to increase (D) None of the statement in (A), (B), and (C) is correct The closed-loop closed-loop dynamics of a system is of second-order. second-order. To improve improve the damping, damping, we should should (A) decrease the phase margin (B ) increase the phase margin (C ) decrease the gain margin (D) increase the gain margin The vertica verticall axis of the Nichols Nichols chart chart repres represents ents (A) open-loop ga gain (B) open-loop ph phase (C ) closed-loop ga gain (D) closed-loop ph phase M -Contours -Contours on the Nichols chart represent (A) open-loop ga gain (B) open-loop ph phase (C ) closed-loop ga gain (D) closed-loop ph phase Consider Cons ider the the followi following ng stateme statements nts (i) The resonan resonance ce peak of the freque frequency ncy respons responsee of G of G/(1+ G) is given by the M the M -contour -contour on the Nichols chart which is tangent to the Nichols plot of G of G . (ii) The resonanc resonancee peak of the frequenc frequency y response response of G of G/(1+ G) is the peak value of the frequency response of G/(1+ G) on Bode magnitude plot. (A) Both the statements are true (B) Statement (i) is true but (ii) is false (C) Statement (i) is false but (i (ii) is true (D) Both the st statements are false The closed-loop closed-loop dynamics of a system system is of second-order. second-order. The The resonance peak of the the closed-loop frequency frequency response is related to (i) overs overshoot hoot of time time resp response onse (ii) (i i) da damp mpin ing g ratio ratio (A) Both the statements are true (B) Statement (i) is true but (ii) is false (C) Statement (i) is false but (i (ii) is true (D) Both the st statements are false Thee bandw Th bandwid idth th of G of G/(1+ /(1+G G) (i) is the the frequ frequency ency at which which the  3 dB dB M M -contour -contour on Nichols chart intersects the Nichols plot of G. (ii) is the frequenc frequency y at which the Bode Bode magnitude magnitude plot plot of G of G/(1+ /(1+G G) has a gain gain of  3 dB. (A) Both th the st statements are tr true (B) Statement (i (i) is is tr true bu but (i (ii) is is false (C) Sta Statem tement ent (i) is fal false se but (i (ii) i) is tru truee (D) Bot Both h the the st state atemen ments ts are fal false se

37

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38


143. A unity-feedback unity-feedback system has open-loop transfer function G( s) s) =

w n2 s (s + 2z w n )

(i) The peak overs overshoot hoot of step-i step-input nput respons responsee of G of G/(1+ /(1+G G) is indicative of damping (ii) The res resonan onance ce peak of frequenc frequency y response response of G of G /(1+ /(1+G G) is indicative of damping. (A) Both th the st statements ar are tr true (B) Statement (i (i) is is tr true bu but (i (ii) is is fa false (C) Sta Statem tement ent (i) is fal false se but (ii (ii)) is is true true (D) Bot Both h the the sta statem tement entss are are fal false se 144.. For the sys 144 system tem 1/( s s2 + 0.1 s + 1), the resonance frequency w r and natural frequency w n are related as (A) w r = w n (B) w r > w n (C) w r < w n (D) None of the the answers answers in (A), (A), (B), (B), and (C) is is correct correct 145. Maxim Maximum um phase-l phase-lead ead of the the compensa compensator tor D( D( s) s ) = (0.5 s + 1) 1) (0 (0.0 .05 5 s + 1), is

146.. 146

147.

148.

149.

(A) 52 52° at 4 rad/sec (B) 52° at 10 rad/sec (C ) 55° at 12 rad/sec (D ) None of the answers in (A), (B), and (C) is correct In cont control rol sy syste stems ms (i) reduction in bandwidth bandwidth results in sluggish sluggish response; (ii) reduction in bandwidth bandwidth results results in better signal/noise signal/noise ratio. Which of the following is the correct answer? (A) None of of the above above statem statements ents is is true true (B) Statement (i) is true but statement (ii) is false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the stat statemen ements ts are are true true Consider Cons ider the the followi following ng stateme statements: nts: (i) Lead compensation compensation is suitable for for systems having having unsatisfactory unsatisfactory transient transient response. It also also provides a limlimited improvement in steady-state response. (ii) Lag compensation compensation is suitable for systems systems with satisfactory satisfactory transient transient response but but unsatisfactory unsatisfactory steadystate response. Which of the following is the correct answer? (A) None of the the stateme statements nts is is true true (B) Statement (i) is true but statement (ii) is false (C) Statement (i) is false but but statement statement (ii) (ii) is true (D) Both the stat statemen ements ts are are true true Consider Cons ider the the followi following ng stateme statements: nts: (i) The natu natural ral freq frequen uency cy w n must be large for good performance; however, bandwidth considerations impose a limit on increasing w n. (ii) When design design consid considerati erations ons impose impose a limit limit on w b, a PD compensator with a filter is suitable. (A) No None ne of the ab abov ovee stat atm ment ntss is tr true ue (B)) St (B Stat atem emen entt (i) is tru ruee but stat ateeme ment nt (ii ii)) is fal alsse (C) Sta Statem tement ent (i) is fals falsee but but state statemen mentt (ii) (ii) is is true true (D) Bot Both h the the state statemen ments ts are are true true Consider the sampled-data sampled-data system system shown in Fig. Fig. Q149. Steady-state Steady-state error for for unit-ramp input is (A) (C)

1

(B)

K T 1

T K

(D) None of the answers answers in (A), (B), and (C) is correct

K r T

Zeroorder hold

K s(t s + 1)

y

Fig. Q149

Appendix-B OLC.p65

38

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39


150. In a digit digital al control system, selection selection of a large ssampling ampling period period (A) increases the stability margin (B) decreases the stability margin (C ) has no effect on stability (D) has an effect on stability that depends on plant parameters 151. In a digital digital control scheme, selection selection of a large sampling sampling interval interval (A) improves th the st steady-state pe performance (B) deteriorates th the st steady-state pe performance (C)) ha (C hass no ef effe fect ct on st stea eady dy-s -sta tate te pe perf rfor orma manc ncee (D)) ha (D hass an ef effe fect ct on st stea eady dy-s -sta tate te pe perf rfor orma manc ncee th that at depends on plant parameters 152. In a sampled-data sampled-data control system, system, delay introduced by sampling sampling and reconstruction reconstruction process is approximately equal to (A) sampling interval (B) tw twice the sampling interval (C ) half the sampling interval (D) None of the answers in (A), (B), and (C) is correct 153. Given a state state variab variable le model model

 N Under the transformation

N

=

= )N + >u

y =

?N

+ du

2N

; 2 a constant nonsingular matrix, the model becomes

 N

=

y =

>

u

+ du >

= 2 ` >;

(C) ) = 2 `)2; > = 2> ; ? = ?2 (D) ) = 2 ` )2; 154. A state variable formulation formulation of a system system is given by the equations equations

>

=

)

>

=

`

? N

+

= 2 ` )2;

(A)

= 2)2 `;

) N

2

>; ?

= ?2

(B)

)

2>; ?

?

=

=

?2 1

?2

È-1 0˘ È x1 ˘ È1˘ Í 0 - 3˙ Í x ˙ + Í1˙ u Î ˚ Î 2˚ Î ˚ È x1 ˘ y = [1 [ 1 0] Í ˙ Î x2 ˚

È x1 ˘ Í x ˙ Î 2˚

=

The transfer function of the system is (A) (C)

1 ( s + 1) ( s

(B)

+ 3)

1 s

1 s

+1


+3

155. A state variable variable formulation formulation of a system is given by the equations equations

È-1 0˘ È x1 ˘ È1˘ = x2(0) = 0 Í 0 - 3˙ Í x ˙ + Í1˙ u ; x1(0) = x Î ˚ Î 2˚ Î ˚ È x1 ˘ y = [1 [ 1 0] Í ˙ Î x2 ˚

È x1 ˘ Í x ˙ Î 2˚

=

The response y response y((t ) to unit-step input is (A) 1 + e t

(B)

3t t (1  e 3 )

3 (D) None of the answers answers in (A), (B), (B), and (C) is correct correct

(C ) 1 e  t t

Appendix-B OLC.p65

1

39

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40


156. The eigen eigenvalue valuess of the matri matrix x =

)

È 0 1 0˘ Í0 0 1˙ Í ˙ ÍÎ0 - 3 - 4˙˚

are (A) 0,  1,  3 (C ) 0, 0,  4 157.. Giv 157 Given en the sys system tem

(B) 0,  3,  4 (D) No None of the answers in (A), (B), and (C) is correct

=

 N

È0 Í1 Í ÍÎ0

y = [ 0

1

- 2 0˘ - 24˙˙ - 9˙˚

0

1 ]N

0 0

N

+

È3˘ Í1˙ u Í ˙ ÍÎ0 ˙˚

The characteristic equation of the system is (A) s3 + 20 s2 + 24 s + 9 = 0 (B) s3 + 9 s 2 + 24 s + 20 = 0 (C) s3 + 24 s2 + 9 s + 20 = 0 (D) None of of the answer answerss in (A), (A), (B), (B), and (C) is corr correct ect 158. A state variabl variablee model of a syste system m is given given by

È x1 ˘ Í x ˙ Î 2˚

=

È 1 1˘ È x1 ˘ È0 ˘ Í- 2 -1˙ Í x ˙ + Í1 ˙ u Î ˚Î 2˚ Î ˚

y = [1 [1

0]

È x1 ˘ Í x ˙ Î 2˚

The system is (A) controllable and observable (C ) observable but uncontrollable 159.. Th 159 Thee transfe transferr functi function on

(B ) controllable but unobservable (D ) uncontrollable and unobservable

G ( s) s) = ?( s1  )) 1> of the system =

 N

y =

)N+ > u ?N +

du

has pole-zero cancellation. The system (A) is uncontrollable and unobservable (C ) is controllable but unobservable 160.. Th 160 Thee transfe transferr functi function on

(B) is observable but uncontrollable (D ) may be any one of (A), (B), and (C)

G ( s) s ) = ?( s1  )) 1> of the system

 N

= )N + >u

y = ?N + du has no pole-zero cancellation. The system (A) is controllable and observable (C ) is controllable but unobservable

Appendix-B OLC.p65

40

(B) is observable but uncontrollable (D ) may be any one of (A), (B), and (C)

4/22/08, 3:45 PM

41

Appendix B: Control Theory Quiz 161.. Con 161 Consid sider er the the syst system em )

=

1 0˘ È 0 Í 0 0 1˙ ; > = Í ˙ ÍÎ - 6 -11 - 6˙˚

È0 ˘ Í0 ˙ ; ? = [4 Í ˙ ÍÎ1 ˙˚

5

1]

The transfer function of the system has pole-zero cancellation. The system is (A) controllable and observable (B) uncontrollable and unobservable (C ) controllable but unobservable (D) observable but uncontrollable 162. Con Consid sider er the the syst system em )

=

È0 - 2 ˘ È1˘ Í1 - 3˙ ; > = Í1˙ ; ? = [0 Î ˚ Î˚

1]

The transfer function of the system has pole-zero cancellation. The system is (A) controllable and observable (B) uncontrollable and unobservable (C ) controllable but unobservable (D) observable but uncontrollable 163. In a control system system design exercise exercise for a marine marine engine, an engineer engineer is using the state-space state-space framework. framework. The system model is given by

 N

=

È13 / 6 4 / 3 ˘ Í-4 / 3 -7 / 6 ˙ N + >u; > Î ˚

=

È0 ˘ Í0.5˙ Î ˚

The engineer decides to experiment with a different method of actuation for the system and this leads to the different input matrix >1

=

È-1/ 6˘ Í 1/ 3 ˙ Î ˚

(A) Both the syste systems ms are are controll controllable able (B) Syst System em with inpu inputt matrix matrix > is controllable, and it becomes uncontrollable when > is changed to >1 (C) Syst System em with inpu inputt matrix matrix > is uncontrollable and it becomes controllable when > is changed to >1 (D) Both the the systems systems are uncont uncontrolla rollable ble 164. Give Given n the stat statee variable variable model model {),>,?} of a single-input, single-output system. The asymptotic stability is determined from (A) ),> and ? matrices (B) ) and > matrices (C) ) and ? matrices (D) ) matrix

Master Key 1. 5. 9. 1 3. 1 7. 2 1. 2 5. 2 9. 3 3. 3 7. 4 1. 4 5. 4 9.

Appendix-B OLC.p65

(C ) (A ) (D) (D) ( A) (D) (C ) (B ) (D) (C ) ( A) (B ) (B )

2. 6. 10. 14. 1 8. 2 2. 2 6. 30. 34. 38. 42. 46. 50.

41

(C ) (B ) ( A) ( A) (D ) (A ) ( A) (C ) ( C) (C ) (C ) (A ) (D )

3. 7. 11 . 15 . 19 . 23. 27. 31 . 35 . 39 . 43 . 47 . 51 .

(C ) (A ) (C ) (A ) (A ) (C ) (B ) (B ) (D ) (C ) (D) (C ) (A )

4. 8. 12 . 16. 2 0. 2 4. 2 8. 32 . 36. 40. 44. 48. 52 .

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(D) (B) (D) (D) (C) (B) (B) (A) (B ) (B) (C) (B) ( A)

42 5 3. 5 7. 6 1. 6 5. 6 9. 7 3. 7 7. 8 1. 8 5. 8 9. 9 3. 9 7. 101. 105. 109. 113. 117. 12 1. 12 5. 12 9. 133. 137. 141. 145. 149. 153. 157. 161.

Appendix-B OLC.p65

Control Systems: Principles and Design (B ) (A ) (D) (B ) (C ) (B ) (B ) (D) (B ) (C ) (D) (B ) ( A) (C ) (B ) ( A) ( A) (C ) (A ) (A ) (B ) (B ) (A ) (D) (C ) (B ) (B ) (C )

54. 5 8. 62 . 66. 70. 74. 78. 82 . 86. 90. 94. 9 8. 102 . 106. 110. 114. 118. 12 2 . 12 6. 130. 134. 138. 142 . 146. 150. 154. 158. 162 .

42

( B) (C ) (D ) (A ) (D ) (C ) (B ) ( A) (B ) (B ) ( D) (B ) (D) ( A) (D) (C) (B) (D) ( A) (B) (D ) ( A) ( A) (D) (B) (B ) ( A) (D )

55 . 59 . 63 . 67 . 71 . 75 . 79 . 83 . 87 . 91 . 9 5. 99. 103. 107. 111. 115. 119. 12 3. 12 7. 131. 135. 139. 143. 147. 151. 155. 159. 163.

(A ) (A ) (D ) (C ) (C ) (A ) (A ) (A ) (B ) (D ) ( B) (C) (D ) (D) (C) (D) (D) (A) (B ) (C) (C ) (C ) ( A) (D ) (C) (C ) (D) (B )

56. 60. 64. 68. 72 . 76. 80. 84. 88. 92 . 96. 100. 104. 108. 11 2 . 116 . 12 0. 12 4. 12 8. 132 . 136. 140. 14 4. 148 . 15 2 . 156. 160. 164.

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(C) (C) (B) (D) (D) ( A) (B) (C) ( A) (D) (B ) ( A) (D) ( A) (B) (C) (C) (D) (B) (D) (B) ( A) (C) (D) (C) ( A) ( A) (D)

Control System Quiz

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