csl10 9 f15 extra

ELECTRICAL ELECTRONICS COMMUNICATION INSTRUMENTATION

CONTROL SYSTEMS

B.Madhuri

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Time response of a second order control system?

Second Order SystemSystem with two poles

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Second order system

• Two independent energy storage elements– Exchange of energy through mechanical mass and

fluid capacitance (tank) through piston– Exchange between electrical inductance and

mechanical inertia in electric motor

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Time response of a second order control system

Over-damped Time Response

a

acbb

2

42

2,1

146.1

854.7

2

459

)1(2

)9)(1(499

2

1

2

2,1

tt eKeKKtc 21321)(

General solution

2

3

1

21

21 ))((

9)(

s

K

s

K

s

K

ssssC

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Time response of a second order control system?

Under-damped Time Response

a

acbb

2

42

2,1

81

81

2

322

)1(2

)9)(1(422

2

1

2

2,1

j

j

j

)cos()( d

tdAetc

General solution

nd j

j

2,1

2,1 81

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Time response of a second order control system

Under-damped Time Response

)cos()( d

tdAetc

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Time response of a second order control system

Un-damped Time Response

a

acbb

2

42

2,1

3

3

2

36

)1(2

)9)(1(400

2

1

2

2,1

j

j

j

)cos()( Atc

General solution

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Time response of a second order control system

Critically-damped Time Response

a

acbb

2

42

2,1

3

2

6

)1(2

)9)(1(466

2,1

2

2,1

tt teKeKtc 11

21)(

General solution

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Time response of second order control system - Summary

Over-damped

Under-damped

Un-damped

Critically-damped

Poles are real and distinct

Poles are real and imaginary

Poles are imaginary

Poles are real and repeated

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Time response of second order control system - Summary

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Time response of second order control system - Example

a

acbb

2

42

2,1

23.1352,2

)175(410

)1(2

)200)(1(41010

1

2,1

2

2,1

j

j

The poles are real and imaginary.Therefore UNDER-DAMPED.

)cos()( d

tdAetcGeneral solution of under-damped

)23.13cos()( 5 tAetc

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Time response of second order control system - Example

2

2)(

s

sG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response

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Time response of second order control system - Example

)6)(3(

5)(

sssG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response

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Time response of second order control system? - Example

)20)(10(

)7(10)(

ss

ssG

What are poles and zeroes?Plot on s-planeWrite the general expression of the step response?State the nature of response.

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Time response of second order control system

So far, we based on the poles of the second order system

a

acbb

2

42

2,1

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Time response of second order control system

We are going to learn two quantities that will help us

analyze second order system

Natural Frequency

nDamping

Ratio

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Time response of second order control system- Natural Frequency

Natural frequency of a second order system is the frequency of oscillation of the system without

damping

Natural Frequency n

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Time response of second order control system- Natural frequency

bass

bsG

2)( Natural frequency of a

second order system is the frequency of

oscillation of the system without damping

bass

bsG

2)(

bs

bsG

2)( Poles:

njs

bjs

bs

bs

02Therefore,

2n

n

b

b

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Time response of second order control system- Damping Ratio

Damping Ratio

(rad/sec)Frequency Natural

FrequencyDecay lExponentia

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Time response of second order control system- Damping Ratio

Damping Ratio

n

a

2

bass

bsG

2)(

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Time response of second order control system

bass

bsG

2)(

22

2

2)(

nn

n

sssG

General Form

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Time response of second order control system

362.4

36)(

2

sssG Find damping ratio and

the natural frequency?

6

362

n

n

Natural Frequency

35.0)6(2

2.4

2.42

n

Damping Ratio

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Time response of second order control system

22

2

2)(

nn

n

sssG

a

acbbs

2

42

2,1

Find poles in general form.

1

2

422

22,1

22

2,1

nn

nnn

s

s

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Time response of second order control system.

122,1 nns

Case 1: 0 njs 2,1

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Time response of second order control system

122,1 nns

Case 2: 10

22,1 1 nn js

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Time response of second order control system

122,1 nns

Case 3: 1 ns 2,1

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Time response of second order control system

122,1 nns

Case 4: 112

2,1 nns

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Time response of second order control system

1

1

10

0

OVERDAMPED

CRITICALLY DAMPED

UNDER DAMPED

UNDAMPED

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Time response of second order control system- Example

40012

400)(

2

sssG

90090

900)(

2

sssG

22530

225)(

2

sssG

625

625)(

2 s

sG

3.0)20(2

12

20400

nUNDERDAMPED

5.1)30(2

90

30900

nOVER-DAMPED

1)15(2

30

15225

nCRITICALLY-DAMPED

0)25(2

0

25625

nUN-DAMPED

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Time response of second order control system. Special Case: Performance of under-damped System

Under-damped:

10 Four performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time

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Time response of second order control system. Special Case: Performance of under-damped System

Rise TimeGeneral

definition: The time required for the waveform to go from 0.1 of the final value to 0.9 of the final value

rnT

Underdamped case: The time taken to reach the final value the first time.

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Time response of second order control system. Special Case: Performance of under-damped System

Peak Time

The time required to reach the first maximum peak

21

n

pT

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Time response of second order control system. Special Case: Performance of under-damped System

Percent Overshoot

The amount that the waveform overshoots the steady-state

21

100%

eOS

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Time response of second order control system. Special Case: Performance of under-damped System

Settling Time

The time required for the transient damped oscillations to reach and stay within 2% of the steady-state value

nsT

4

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Time response of second order control system. Special Case: Performance of under-damped System – Example

36116

361)(

2

sssG Find: OSTTT rpsn ,%,,,,

19361 n

421.0)19(2

16

5.0)19(421.0

44

nsT

182.0421.01191 22

n

pT

%3.23100100%22

421.01

421.0

1

eeOS

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Time response of second order control system. Special Case: Performance of under-damped System – Example

36116

361)(

2

sssG

Find:rT

19n

421.0

079.019

501.1

501.1

r

nr

T

T

Or interpolate

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Time response of second order control system? –Lessons

April 14, 2023 37

11

10

0

OVERDAMPED

CRITICALLY DAMPED

UNDER DAMPED

UNDAMPED

WHERE DO WE GET

22

2

2)(

nn

n

sssG

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Time response of second order control system? -

April 14, 2023 38

What Interpretations?

Performances of second order under-damped control system are determined by two quantities called natural frequency and damping ratioFour performance parameters:1.Rise time2.Peak time3.Percent overshoot4.Settling time

21

n

pT

nsT

421100%

eOS

nr

fT

)(

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Time response of second order control system? -

April 14, 2023 39

a

acbb

2

42

2,1

22

2

2)(

nn

n

sssG

22,1 1 nn js

dd js 2,1

For under-damped system

)cos(

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Time response of second order control system? -

April 14, 2023 40

nsT

4

Line of constant settling time

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Time response of second order control system?

April 14, 2023 41

21

n

pT

Line of constant peak time

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Time response of second order control system?

April 14, 2023 42

21100%

eOS)cos(

Line of constant %OS

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Time response of second order control system?

April 14, 2023 43

Constant Settling Time

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Time response of second order control system?

April 14, 2023 44

Constant Peak Time

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Time response of second order control system?

April 14, 2023 45

Constant Percent Overshoot