Csl9 4 f15

Time Response of First Order System

April 14, 20231

Madhuri Bayya

April 14, 2023 2

What is the time response of a control system?

Time response:The time solution to differential equation.

April 14, 2023 3


Time response consists of:

1.Natural response2.Forced response

April 14, 2023 4


Time response PlotForced response orTransient response

Steady-state responseTime solution to

differential equation

Caused by the input

April 14, 2023 5

How to obtain time response?

Schematic

M)(tf

xM

Kx

)(tfKxxM Differential Equation

)(2 sFKXMXs Laplace Transform

Time Response Why Transfer Function?

Transfer Function

KMssF

sXsG

2

1

)(

)()(

April 14, 2023 6

How to obtain time response?

Why use Transfer Function?

1.Solution by inspection

2.Qualitative solution

April 14, 2023 7

How to obtain time response from transfer function?

Given:

How to obtain the time response?

First, we need to understand the concept of poles and zeros

April 14, 2023 8


First rule:The values of Laplace transform variable, s, that cause the transfer function to become infinite.

Second rule:Any roots of the denominator of the transfer function that are common to the roots of numerator

What are poles ?

April 14, 2023 9


Transfer Function

)3)(2)(5(

)3()(

sss

ssG

First rule:When s = -5, or s = -2 or s = -3, then, G(s)=infinity! Therefore, the poles of the transfer function G(s) are -5, -2 and -3.

)3)(2)(5(

)3()(

sss

ssG

Second rule:Although the term (s+3) can be cancelled out, the value -3 is still the poles of the transfer function G(s).

April 14, 2023 10


First rule:The values of Laplace transform variable, s, that cause the transfer function to become zero.

Second rule:Any roots of the numerator of the transfer function that are common to roots of the denominator

What are zeros ?

April 14, 2023 11


Transfer Function

)3)(2)(5(

)3()(

sss

ssG

First rule:When s = -3, then, G(s)= 0. Therefore, the zero of the transfer function G(s) is -3.

)3)(2)(5(

)3()(

sss

ssG

Second rule:Although the term (s+3) can be cancelled out, the value -3 is still the zero of the transfer function G(s).

April 14, 2023 12


Transfer Function

)3)(1)(1(

)7()(

sss

ssG

)9)(2)(11(

)3)(13()(

sss

sssG

Zero = -7Poles = -1, -1 and -3

Zero = -13 and -3Poles = -11, -2 and -9

April 14, 2023 13


Given:

How to obtain the time response?1. Find C(s)2. Expand the transfer function using partial

fraction expansion technique3. Perform inverse Laplace transform

April 14, 2023 14

How to obtain time response from transfer function? –Find C(s)

Given:

)(

)()(

sR

sC

Input

OutputsG

)()()( sRsGsC What is R(s)?

Test Input Signal

• Unlike electric networks, inputs to many practical systems are not exactly known ahead of time.

• Difficult to design a control system to satisfy all possible input signals

• For purpose of analysis and design some basic types of test inputs assumed, so that performance can be evaluated.

• Gives an idea of response to other complex inputs.• Response of complex signals can be determined by

superimposing those due to simple test signals

15Control Systems

General form of the standard test signals

r(t) = tn

R(s) = n!/sn+1

16Control Systems

Types of test signalsStep Input – Represents instantaneous change in reference input. Eg. Motor

n = 0

r(t) = A

R(s) = A/s

Ramp Input – Changes constantly with time

r(t) = At

n = 1

R(s) = A/s2

Types of test signalsParabolic Input – A signal that is one order faster than the ramp

n = 2r(t) = At2

R(s) = 2A/s3

Impulse Input – A signal which has zero value everywhere except at t=0, where its magnitude is infinite. Since perfect impulse cannot be achieved, it is usually approximated by a pulse of small width but unit area.

(t)=0; t 0 0

Mathematically, impulse function is derivative of step function

Test Signal Inputs

Test Signal r(t) R(s)

Stepposition

r(t) = A, t > 0 = 0, t < 0

R(s) = A/s

Rampvelocity

r(t) = At, t > 0 = 0, t < 0

R(s) = A/s2

Parabolicacceleration

r(t) = At2, t > 0 = 0, t < 0

R(s) = 2A/s3

19Control Systems

April 14, 2023 20

How to obtain time response from transfer function? –Find C(s)

Given:

R(s) unit step input implies A = 1

)5(

)2(1)(

)()()(

s

s

ssC

sRsGsC

April 14, 2023 21

How to obtain time response from transfer function? – Expand C(s)

Expand C(s) using Partial Fraction Expansion Technique

)5(

)2(1)(

s

s

ssC

)5(

)2(

5)(

ss

s

s

B

s

AsC

April 14, 2023 22

How to obtain time response from transfer function? – Expand C(s)

)5(

)2(

5)(

ss

s

s

B

s

AsC

5

2

)5(

)2(

0

ss

sA

5

3

)(

)2(

5

ss

sB

55

35

2)(

ss

sC

How to find A and B?

April 14, 2023 23

How to obtain time response from transfer function? – Inverse Laplace Transform

55

35

2)(

ss

sC

s5

2 Inverse Laplace Transform 5

2

55

3

s

Inverse Laplace Transform

te 5

5

3 tetc 5

5

3

5

2)(

Time response

April 14, 2023 24


Steady response

Transient response

April 14, 2023 25

What is time response of a control system?

Input pole generates force responseSystem pole generates natural responseSystem pole generates natural response in the form of e-αt .

Both zero and pole generates the amplitude of time response

April 14, 2023 26

What is time response of a control system? An Example

)5)(4)(2(

)3(

sss

s)(sC

ssR

1)(

By inspection:

)5()4()2()( 4321

s

K

s

K

s

K

s

KsC

What are poles and zeros?Poles = -2, -4 and -5Zero = -3

Force response Natural

response

Influence of poles on time response

April 14, 2023 27


)5)(4)(2(

)3(

sss

s)(sC

ssR

1)(

By inspection:)5()4()2(

)( 4321

s

K

s

K

s

K

s

KsC

Inverse Laplace Transform

ttt eKeKeKKtc 54

43

221)(

Time response – solution to differential equation

April 14, 2023 28


)5()4()2()( 4321

s

K

s

K

s

K

s

KsC

ttt eKeKeKKtc 54

43

221)(

How to evaluate K1, K2, K3 and K4?

)5)(4)(2(

)3()(

ssss

ssC

11

3

)5)(4)(2(

)3(

)5)(4)(2(

)3()(

001

sssss

s

ssss

ssKssC

April 14, 2023 29


40

3

)50)(40)(20(

30

)5)(4)(2(

)3(

)5)(4)(2(

)3()(

001

ss

sss

s

ssss

ssKssC

12

1

)52)(42(2

)32(

)5)(4(

)3(

)5)(4)(2(

)3)(2()()2(

222

sssss

s

ssss

ssKsCs

8

1

)54)(24)(4(

)34(

)5)(2(

)3(

)5)(4)(2(

)3)(4()()4(

441

ss

sss

s

ssss

ssKsCs

15

2

)45)(25(5

)35(

)4)(2(

)3(

)5)(4)(2(

)3)(5()()5(

551

ss

sss

s

ssss

ssKsCs

April 14, 2023 30


ttt eKeKeKKtc 54

43

221)(

ttt eeetc 54

3

2

15

2

8

1

12

1

40

3)(

April 14, 2023 31


April 14, 2023 32

What is the order of a control system?

The highest order of differential equations

The number of poles

April 14, 2023 33

What is the first order of a control system?

Solution Using Inspection

ateKKtc 21)(

)()(

ass

asC

)()( 21

as

K

s

KsC

1)()(

)(00

1

a

a

as

a

ass

sassCK

ss

1)(

)()()(2

a

a

s

a

ass

aassCasK

asas

atetc 1)(

One Pole Time Response

April 14, 2023 34

What is the first order of a control system?

atetc 1)(

Transfer Function

Pole Location

Time response

Time Response Plot

April 14, 2023 35

What is the first order of a control system? – Performance Parameters

Time constant

aTc

1

63% of

Final value

April 14, 2023 36


Rise Time

aT

aaT

TTT

r

r

r

2.2

11.031.2%10%90

Time to rise from 0.1 to 0.9 of final

value

April 14, 2023 37


Settling Time

aTs

4

Time to reach 2% of final value

April 14, 2023 38


Settling Time

aTs

4Time to reach

2% of final value

Rise Time

aTr

2.2

Time to rise from 0.1 to 0.9 of final value

Time constant

aTc

163% of

Final value

April 14, 2023 39

What is the first order of a control system? – An Example

Settling Time

sec50

44

aTs

Rise Time

sec50

2.22.2

aTr

Time constant

sec50

11

aTc50

50)(

s

sG

First Order because one pole: pole = a = -50

tetc 501)(

By inspection the solution of unit step input is:

ssR

1)(

April 14, 2023 40


50

200)(

s

sG

)1(4)( 50tetc

By inspection the solution of unit step input is:

ssR

1)(

But the numerator (200) is not equal to “a” (50).

50

504)(

s

sG

In general:

as

KsG

)( )1()( ate

a

Ktc

Notice that time constant, rise time and settling time are still the same. The performances only depend on the pole.

aTs

4

aTr

2.2

aTc

1

April 14, 2023 41


By experiment we get this curve!!

Find transfer function?

Step 1: Find the pole

Time constant is the time to reach 63% of final value

Final value = 0.7263% Final value = 0.63X0.72=0.45

Time constant = 0.13 sec

April 14, 2023 42


Step 1: Find the pole

Time constant = 0.13 sec

7.713.0

11

1

c

c

Ta

aT

)1()( atea

Ktc

General solution

54.5)7.7(72.072.072.0 aKa

K

7.7

54.5)(

sas

KsG

April 14, 2023 43


5

20)(

s

sG

What are the time constant, rise time, settling time and steady-state value?

April 14, 2023 44


April 14, 2023 45

Examples

• Discharge of electronic camera flash• Flow of fluid form a tank• Cooling of a cup of coffee

Csl9 4 f15

Engineering

Transcript of Csl9 4 f15