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Reference Books1. Modern Control Engineering,

Katsuhiko OgataPrentice Hall, 1992.

2. Modern Control Systems,Dorf R.C.,Seventh Edition, Addison-Wesley, 1995.

3. Automatic Control Systems,Benjamin C. Kuo

Sixth Edition, Prentice Hall India, 1996.

4. Feedback and Control Systems,Distefano III, Stubberud, Williams

Schaum Publishing Co., 1967.

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Laplace Transform in

Control EngineeringDr.P.V.Manivannan

Mechanical Engineering Department

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Definition of Laplace Transform

Complex variable:

- Real part

- Imaginary part

s j

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Graphical representation of s

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Laplace transforms of StandardFunctions (Input signals)

1. Unit step function: f(t) = u(t), 0 for t < 0, 1 for t 0

At t = , e-st approaches zero (0),

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Laplace transforms of StandardFunctions (Input signals)

Ramp function r(t) = t, for t 0;Unit

Integrating by parts and evaluating gives

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Laplace transforms of StandardFunctions (Input signals)

2. Exponential function: f(t) = eat, t 0

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Laplace transforms of Standard Functions

Find the Laplace transform of: sintandcost

cos sin

cos sin

j t

j t

e t j t

e t j t

1cos ( )

21

sin ( )

2

j j

j j

e e

e e

j

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0

2 2

2 2

( )

1 1 1 1

2 ( ) 2 ( )

( )

, :

cos

( )

j t j t ste e e dt

j s j j s j

s

Similarly we can write

st

s

1L(si n t ) =

2j

L

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Laplace transforms of StandardFunctions (Input signals)

Shifted unit step functionf(t) = u(t 3); 0 for t< 3 & 1 for t> 3

-as

-3s

L{f (t ) . u (t -a )} = e L {f (t )}

e

= s

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Laplace transform of Pulse function

-as

( ) ( ) ( )

e

A s

sin propert

bs

f t A u t a u t b

e

s

u g

-as

L f ( t )=

eL{u (t -a )} =

s

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Laplace Transform Theorms

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Laplace Transform theorms

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Laplace Transform theorms

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Inverse Laplace Transform

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Inverse Laplace Transform

Inverse Laplace of a function (inLaplace domain) can be found

easily by Partial Fraction method& Inverse Laplace Table

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Example Problem:1

( )u t a

-as-1 e

L s

3 42( ) ( )s sG s e es

We know that

3 4

2

s se e

s s

-1 -1

L L

2 ( 3) ( 4)u t u t

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Example Problem:2

Given Function:2

1( )( 5)

sG s es

2

1We also know that,

(s+a)

at

te

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Example Problem:2

Here,2

1( ) [ ( )]( 5)

g t G ss

-1L

and , as 'a'= 1as se e

5( 1)( 1) ( 1) tg t t e

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Example Problem:2

5( 1)

2

5( 1)

1( 1) . u(t-1)

( 5)

( ) ( 1) . u(t-1)

s t

t

e t es

g t t e

-1L

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Example Problem:3

Example problem:4

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Example problem:4(Simple Roots)

Example problem:4

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Example problem:4(Simple Roots)

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Example Problem:5

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Example Problem:5

E l P bl 6

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Example Problem:6

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S l i Diff ti l E ti

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Solving Differential Equation

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2 2

3 A Bs+C= +

s+1( 1)( 9) s 9

Y

s s

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Finding Solution for Electrical

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Finding Solution for ElectricalSystem

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Current waveform

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Example Problem:7

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Time Domain Response ofdifferent systems

First Order system time

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First Order system timeresponse for step input

First Order system time

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First Order system timeresponse

First Order system time

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First Order system timeresponse

First Order system time

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First Order system timeresponse

Second Order system time response

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Second Order system time response

Second Order system time

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Second Order system timeresponse

Second Order system time

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Second Order system timeresponse

Second Order system time

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Second Order system timeresponse

Example Problem:1

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Example Problem:1

E l P bl 2

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Example Problem:2

Find K and afor the closed-loop system below suchthat the transient response to a step input satisfies:%OS 5% & ts 4 seconds

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Laplace Transform of Periodic Function

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Laplace Transform of Periodic Function

Laplace Transform of Periodic Function

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Laplace Transform of Periodic Function

Laplace Transform of Periodic Function

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Laplace Transform of Periodic Function

( )=t.[u(t) - u(t-1)] , has period of T = 2f t

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Laplace Transform of Periodic

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p

Function

( ) sin . [ ( ) u(t- )], period (p) =f t t u t

Laplace Transform of Periodic

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p

Function

Inverse Laplace transform of

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pPeriodic function

(1 )1( ) (where T is a constant)( 1)(1 )

s T

sT

eGiven that G s s e

As we know that Laplace transform of Periodic wave will

have term:1/ (1-e

-sT

),we will consider remaining part alone

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Thank you