• Then the transfer function G(S) of the plant is given as

Automation and control Engineering (EEA 3010)

Unit 1, topic -2

Transfer Function and stability of LTI systems

Transfer Function

• Transfer Function is defined as the ratio of Laplace transform of the output to the Laplace transform of the input. Considering all initial conditions to zero.

• Where is the Laplace operator.

Plant or Process u(t) y(t)

) (

) (

)

( )

(

S Y

t y

and S

U t

u If











2

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Transfer Function

• Then the transfer function G(S) of the plant is given as

G(S) Y(S)

U(S)

) (

) ) (

( U S

S S Y

G 

Why Laplace Transform?

• By use of Laplace transform we can convert many common functions into algebraic function of complex variable s.

• For example

Or

• Where s is a complex variable (complex frequency) and is given as

2

sin



 

  t s



a e

s

 



1





 j s  

4

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Laplace Transform of Derivatives

• Not only common function can be converted into simple algebraic expressions but calculus operations can also be converted into algebraic expressions.

• For example

) 0 ( )

) (

( sX S x

dt t

dx  



x dx S

X t s

x

d ( )

) ( )

) (

( 0

2

0

2

2

  



Laplace Transform of Derivatives

• In general

• Where is the initial condition of the system.

) ( )

( )

) (

( 



0  

0

n n

x x

s S

X s

dt t x

d 



) x(0

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Laplace Transform of Integrals

) ( )

( X S

dt s t

x  1





• The time domain integral becomes division by

s in frequency domain.

Calculation of the Transfer Function

dt t B dx

dt t C dy

dt t x

A d ( ) ( ) ( )



2 

2

• Consider the following equation where y(t) is input of the system and x(t) is the output.

• or

• 𝐴 𝑥(𝑡) = 𝐶𝑦(𝑡) − 𝐵𝑥(𝑡)

• Taking the Laplace transform on either sides

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Calculation of the Transfer Function , contd..

• Considering Initial conditions to zero in order to find the transfer function of the system

• Rearranging the above equation

) ( )

( )

(s CsY s BsX s

X

As²  

) ( ]

)[

(

) ( )

( )

(

s CsY Bs

As s

X

s CsY s

BsX s

X As









2 2

B As

C Bs

As Cs s

Y s X

 

 ₂  )

( ) (

Transfer Function

• In general

• Where x is the input of the system and y is the output of the system.

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Transfer Function, contd..

• When order of the denominator polynomial is greater than the numerator polynomial the transfer function is said to be ‘proper’.

• Otherwise ‘improper’

Transfer Function, contd…

• Transfer function helps us to check

– The stability of the system

– Time domain and frequency domain characteristics of the system

– Response of the system for any given input

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Stability of Control System

• There are several meanings of stability, in general there are two kinds of stability definitions in control system study.

– Absolute Stability – Relative Stability

Stability of Control System, contd

• Roots of denominator polynomial of a transfer function are called ‘poles’.

• And the roots of numerator polynomials of a transfer function are called ‘zeros’.

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Stability of Control System, contd..

• Poles of the system are represented by ‘x ’ and zeros of the system are represented by

‘o’.

• System order is always equal to number of poles of the transfer function.

• Following transfer function represents n

order plant .

Stability of Control System

• Pole is also defined as “the frequency at which system becomes infinite”. Hence the name pole where field is infinite.

• And zero is the frequency at which system becomes 0.

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Stability of Control Systems

• The poles and zeros of the system are plotted in s-plane to check the stability of the system.

s-plane

LHP RHP



 j



 j

s  

Recall

Stability of Control Systems

• If all the poles of the system lie in left half plane the system is said to be Stable.

• If any of the poles lie in right half plane the system is said to be unstable.

• If pole(s) lie on imaginary axis the system is said to be marginally stable.

18

s-plane

LHP RHP



 j

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Stability of Control Systems

• For example

• Then the only pole of the system lie at

10 3

1  

 

 A B and C

B As

s C

G( ) , if ,

3 pole

LHP RHP





j

-3X

Stability

• The system is said to be stable if for any bounded input the output of the system is also bounded (BIBO).

• Thus the for any bounded input the output either remain constant or decrease with time.

20

u(t)

t 1

Unit Step Input

Plant

y(t)

t Output

1

overshoot

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• If for any bounded input the output is not bounded the system is said to be unstable.

u(t)

t 1

Unit Step Input

Plant

y(t)

t

eat

• For example

3 1 )

( ) ) (

1(   

s s

U s s Y

G 3

1 )

( ) ) (

2(   

s s

U s s Y

G

-4 -2 0 2 4

-4 -3 -2 -1 0 1 2 3 4

Pole-Zero Map

Real Axis

Imaginary Axis

-4 -2 0 2 4

-4 -3 -2 -1 0 1 2 3 4

Pole-Zero Map

Real Axis

Imaginary Axis

stable unstable

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3 1 )

( ) ) (

1(   

s s

U s s Y

G

3 1 )

( ) ) (

2(   

s s

U s s Y

G

) ( )

(

3 1 )

( ) ) (

(

3

1 1

1 1

t u e

t y

s s

U s s Y

G

 t











 

  



3 1 )

( ) ) (

(

3

1 1

2 1

s s

U s s Y

G

 t



 

 ^{ }

  



stable

unstable

0 1 2 3 4

0 0.2 0.4 0.6 0.8 1

exp(-3t)*u(t)

2 4 6 8 10

12x 10¹² exp(3t)*u(t)

• Whenever one or more than one poles are in RHP the solution of dynamic equations contains increasing exponential terms.

• Such as .

• That makes the response of the system unbounded and hence the overall response of the system is unstable.

e³t

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