Control System 2011

8/3/2019 Control System 2011

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Cont ro l Syst emContents

Chapter Topic PageNo

Chapter-1 IntroductionTheory at a glance

Previous Years GATE Questions

Previous Years IES Questions

Previous Years GATE Answer

Previous Years IES Answer

22

8

8

10

10

Chapter-2 Transfer Function, Block Diagramsand Signal Flow GraphsTheory at a glance





1212

19

21

24

25

Chapter-3 Mathematical Modeling ControlSystemTheory at a glance


35Previous Years IES Answer

28

28

33

35

Chapter-4 Time Response Analysis ofControl SystemTheory at a glance





37

37

49

52

59

62


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Indias No 1 Control System

IES Academy Contents

www.iesacademy.com Email: [email protected] Page-1________________________________________________________________________25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290

Chapter-5 Frequency AnalysisTheory at a glance



6969

74

76Chapter-6 Stability Analysis of Control

SystemTheory at a glance





77

77

92

98

110

118

Chapter-7 Root Locus TechniqueTheory at a glance


Previous Years IES QuestionsPrevious Years GATE Answer


132132

136

138143

146

Chapter-8 CompensatorsTheory at a glance





150150

155

156

160

161

Chapter-9 Industrial ControllersTheory at a glance

Previous Years GATE QuestionsPrevious Years IES Questions



165165

171172

176

177

Chapter-10 Introduction to State SpaceVariableTheory at a glance





180

180

188

190

192

195


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IES Academy Chapter 1


I n t roduc t i onContents for this chapter

1. Introduction2. OpenLoop System

3. Mathematical Model for Open-loop Control System

4. Closed-Loop Control System

5. Mathematical Model for Closed-Loop System

6. Comparison of Open Loop and Closed Loop

7. Laplace Transform

8. Basic Laplace Transform Theorem

9. Summary

Theory a t a Glanc e(For IES, GATE, PSU & JTO)

1. IntroductionControl system is a combination of elements arranged in a planned manner. Where each

element causes an effect to produce a desire output.

Example of control systems

1. System for the control of position.

2. System for the control of velocity.

2. OpenLoop System1. No feedback in open loop system is used.

2. Control system (open-loop) depends only on the accuracy of input calibration.

Example of open-loop control system

1. Traffic signal light

2. Electric lift

3. Automatic washing machine

3. Mathematical Model for Open-loop Control System

C= G

R


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Where, G = gain of system

C = o/p of system

R = input

Points:

1. Feedback system is not used for improving stability.2. An open-loop system may become unstable when we used negative feedback.

4. Closed-Loop Control SystemIn a closed loop control system the output has an effect on control action through a feedback.

Example of closed-loop system:1. D.C. Motor speed control

2. Radar tracking system

3. Auto pilot system

5. Mathematical Model for Closed-Loop System

C G=

R 1+GH

1* Here feedback is negative

2. This form is also called control canonical form

From figure

C(S)= G(S) =

E(S)As a Forward path transfer function

B(S)=H(S)=

C(S)As a feedback transfer function

The o/p of summing point

[ ]E(S) R(S) B(S)= ;

C(S)= R(S) B(S)G(S) ;

C(S)= R(S) C(S) H(S)

G(S);

C(S) = R(S) G(S) G(S) C(S) H(S) ;

[ ]C(S) 1+G(S) H(S) = R(S) G(S)


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C(S) G(S)=

R(S) 1+G(S) H(S)

6. Comparison of Open Loop and Closed LoopOpen Loop System Closed Loop System

1. The accuracy of an open loop system

depends on the calibration of the i/p.

1. As the error between the reference input

and the output is continuously measured-

through feedback.

2. The open loop system is more stable. 2. The closed loop system is less stable.

3. It is less accurate. 3. It is more accurate.

4. It is cheap and less complex. 4. It is expensive and more complex circuit.

5. Effect of Noise and disturbance is morein open loop control system. 5. Effect of Noise and disturbance is less inclosed loop control system.

7. Laplace TransformLaplace transformation is very great tool in control system.

The mathematical expression for laplace transforms

st

0

F(t) F(S)

F(S) F(t) e dt

=

=

L

The term laplace transform of F(t) is used for the letter LF(t).

8. Basic Laplace Transform TheoremBasic theorems of laplace transform are given below

Theorem 1: Multiplication by a constantLet k be a constant and F(S) be the laplace transform of F(t), then

[ ]Kf(t) KF(S)= Theorem 2: Sum and difference

Let F1(S) and F2(S) be the laplace transform of ( ) ( )1 2f t f t , then

( ) ( )1 2 1 2f t f t = F (S) F (S)


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Theorem 3: Differentiation

[ ]

22 1

2

dF(t)i. = SF(S) -F(0)

dt

d F(t)ii. = S F(S) - F(0) -f (0)

dt

L

L

1 dF(0)where, F (0) =dt

In general, for higher order derivatives or F(t)

1 2 (1) ( 1)( ) ( ) ( ) (0) (0)

=

nn n n n

n

d F ts F S s f O s f f

dtL

Where, F1(0) denotes the ith order derivative of f(t) with respect to t1,

Theorem 4: Integration

1

1 2

2 2

F(S) F (0)i. F(t) = +

S S

F(S) F (0) F (0)ii. F(t) = + +

S S S

L

L

Theorem 5: Shift in time

The laplace transform of F(t) delayed by time T is equal to the laplace transform F(t)multiplied by eST that is

[ ] -STsF(t T) u (t T) = e F(S)L Where US(tT) denotes the unit step function that is shifted in time to the right by T.

Theorem 6: Complex shifting

The laplace transform of F(t) multiplied byte , where is a constant is equal to the laplace

transform F(S), with S replaced by ( )S that is

t

e F(t) = F(S

)

L

Theorem 7: Initial-value theoremIf the laplace transform of F(t) is F(S), then

0lim ( ) lim ( )

=

t SF t SF S


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Theorem 8: Final value theorem

0

0

lim ( ) lim ( )

lim ( ) lim ( )

t S

t S

F t S F t

F t SF S

=

=

L

Point to be RememberIf the denominator of SF(S) has any root having real part as zero or positive, then final value

theorem is not valid. [GATE 2007]

USEFUL TRANSFORM (LAPLACE) PAIR

1 F(t) F(S) = LF(t)2 (t) unit impulse 1

3 U(t)1

S

4 U(tT)1 st

eS

5 t2

1

S

6

2

2

t 3

1

S

7n

t 1nn

s +

8 ate

1

s a+

9 ate 1

s a

10 atte ( )

2

1

s a+

11 atte ( )2

1

s a

12

13 h tt e ( )1

/+

+n

n s a


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14 sin t2 2

+s

15 cos t

e t

( )2 2

+

+ +

s

s

16 sin t

e t( )

2 2

+ +s

17 sin h t2 2

s

18 cos h t2 2

s

s

9. Summary1. Open loop control system no feedback used.

2. In closed loop control system we used feedback.

3. Open loop system is more stable.

4. Closed loop system is more accurate.

5. Final value theorem can not used if denominators of SF(S) have real part as a zero or

positive.


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ASKED OBJECTIVE QUESTIONS (GATE, IES)

Previous Years GATE Quest ions

Basic Laplace Transform Theorem

GATE-1. If the Laplace Transform of a signal y(t) is1

( ) ,( 1)

=

Y ss s

then its final

value is: [GATE-2007]

(a) -1 (b) 0 (c) 0 (d) Unbounded

GATE-2. The unit impulse response of a system is ( ) , 0= t f t e t [GATE-2006]

For this system, the steady-state value of the output for unit step inputis equal to

(a) -1 (b) 0 (c) 1 (d)

Previous Years IES Quest ions

Closed-Loop Control SystemIES-1. When a human being tries to approach an object, his brain acts as

(a) An error measuring device (b) A controller [IES-1999]

(c) An actuator (d) An amplifier

IES-2. Assertion (A): Feedback control systems offer more accurate control

over open-loop systems. [IES-2000]

Reason (R): The feedback path establishes a link for input and output

comparison and subsequent error correction.

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true but R is NOT the correct explanation of A

(c) A is true but R is false

(d) A is false but R is true

IES-3. Consider the following statements: [IES-2000]

1. The effect of feedback is to reduce the system error

2. Feedback increases the gain of the system in one frequency rangebut decreases in another

3. Feedback can cause a system that is originally stable to become

unstable

Which of these statements are correct?

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

IES-4. Consider the following statements which respect to feedback control

systems: [IES-2006]


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1. Accuracy cannot be obtained by adjusting loop gain.

2. Feedback decreases overall gain.

3. Introduction of noise due to sensor reduces overall accuracy.

4. Introduction of feedback may lead to the possibility of instability of

closed loop system.

Which of the statements given above are correct?(a) 1, 2, 3 and 4 (b) Only 1, 2 and 4

(c) Only 1 and 3 (d) Only 2, 3 and 4

IES-5. A negative-feedback closed-loop system is supplied to an input of 5V.

The system has a forward gain of 1 and a feedback gain of a 1. What is

the output voltage? [IES-2009]

(a) 1.0 V (b) 1.5 V (c) 2.0 V (d) 2.5 V

Basic Laplace Transform Theorem

IES-6. Consider the function F(s) =2 2s

+where F(s) is the Laplace transform

of f(t). What is the steady-state value of f(t) ? [IES-2009](a) Zero (b) One (c) Two (d) A value between -1 and +1

IES-7. The transfer function of a linear-time-invariant system is given as

( )1

s 1+. What is the steady-state value of the unit-impulse response?

[IES-2009]

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


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Answ ers w i th Ex p lanat ion

(Object ive)

Previous Years GATE Answ er

GATE-1. Ans. (d)1

( )( 1)

Final value of Y(s)

1 1 1-1 -1 -1LT ( ( )) LT = LT( 1) 1

( ) ( )

value =

t

=+

= + +

=

Y sS S

Y sS S S S

tY t e u t

final

GATE-2. Ans. (c) Unit impulse response of a system is

( ) t 0= t f t e

1( )

1=

+f s

S

O/P for unit step I/P1 1

1=

+S S

1

( 1)=

+S S

1( ( )) lim = 1

( 1)0t

C s SS Ss

= =

+

Previous Years IES Answ er

IES-1.Ans. (b)

IES-2. Ans. (a)

IES-3. Ans. (d) Feedback is applied to reduce

the system error. Consider the

example.

( )( )

( )( ) ( )

C s G s

R s 1 G s H s

1 1s 11 s 1

1s 1

=

+= =

+

Thus, we see that the closed loop system is unstable while the open loop system is

stable.

IES-4. Ans. (d)


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IES-5. Ans. (d) Output voltage = inA

V1 AB+

( )1

5x 2.5V1 1x1x

= =+

IES-6. Ans. (d) This is the Laplace transform of sin t.

So, f(t) = sin t

Steady-state value of f(t) is undetermined because poles of F(s) are not in LHS ofs-plane. Therefore, steady-state value will vary between - 1 and + 1.

IES-7. Ans. (a) Steady state value =( )s 0

1lims 1

s 1 += 0

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Control System 2011