Class 2 - Mathematical Modeling Using Transfer Function Approach

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Blog: www.godsfavouritechild.wordpress.comWeb address: http://www.esnips.com/web/SystemModelingClassNotes

Class 2: Mathematical Modeling of systems using transfer function approach

mailto:[email protected]

mailto:[email protected]

http://www.godsfavouritechild.wordpress.com/

http://www.esnips.com/web/SystemModelingClassNotes

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

•

I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

•

For best results, it is always suggested you read the source material.


Why mathematical model is required?

•

It is required to understand, analyse

and control the system

•

Fundamental physical laws of science and engineering are used for modeling.–

Electrical Systems: Ohms, Kirchoffs

and Lenz law

–

Mechanical Systems: Thermodynamic and Newton’s law


Introduction to mathematical modeling

•

The mathematical model is usually in the form of differential equations. A differential equation can describe relationship between input and output

•

For a linear system, Laplace transform can be used to find the solutions of the differential equations

•

Using Laplace Transforms, we can represent the real system using transfer functions.


Introduction

•

Deriving reasonable mathematical models

is the most important part of the entire analysis.

•

The principle of causality

is assumed throughout this course.•

A trade-off exists between simplicity

and accuracy.

•

Linear Systems.–

Superposition

applies.

•

Linear Time-Invariant

(vs. Time-Varying) Systems.•

Differential equations with constant coefficients

(vs. coefficients of

functions of time).


Transfer function approach

•

The physical system, the input and the output signals can be separated and easily visualized.

•

Transfer function in the Laplace domain is that relation which algebraically relates the input and output of a control system, for the case when the initial conditions are zero.


Transfer function approach

( ) ( 1) ( ) ( 1). .

0 1 1 0 1 1

Consider the linear time-invariant system defined by the differential equation:

( ) where input and output

Transfer

n n n n

n n n na y a y a y a y b x b x b x b x n mx y

− −

− −+ + ⋅⋅⋅ + + = + + ⋅⋅⋅+ + ≥= =

zero initial conditions1

0 1 11

0 1 1

[ ]function ( )[ ]

( ) ( ) ( ) ( )

( ( )) number of order of the system.

m mm m

n nn n

L yG sL x

b s b s b s bY s N sX s a s a s a s a D s

Order D s

−−

−−

= =

+ + ⋅⋅⋅+ += = =

+ + ⋅⋅⋅+ +

=


Characteristics of transfer function

•

It is mathematical model to express the relationship between the output and the input variable.

•

It is independent of the magnitude and nature of the input or driving function.

•

It includes the units. However it does not provide any information concerning the physical structure.

•

If it is known, the output or response can be studied for various forms of inputs.

•

If it is unknown, it may be established experimentally by introducing known inputs and studying the output. Once established, it gives

a full

description of the dynamic characteristics.


Example: Impulse response function

0 0

1

Convolution Integral. ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

where ( ) ( ) 0 for 0.

Let ( ) ( ).( ) 1

( ) ( )

[ ( )] ( ): Impulse-Response Function.

The transfer func

t t

Y s G s X s

y t x g t dt g x t dt

g t x t t

x t tX sY s G s

L G s g t

τ τ τ τ

δ

−

=

⇒ = − = −

= = <

=⇒ =⇒ =

=

∫ ∫

tion and the impulse-response function contain the same complete information about the system dynamics.


Laplace transform

•

Laplace transforms provide a method for representing and analyzing linear systems using algebraic methods.

•

In systems that begin undeflected

and at rest the Laplace can directly replace the d/dt

operator in differential equations. It is a superset of

the phasor

representation in that it has both a complex part, for the steady state response, but also a real part, representing the transient part.

•

As with the other representations the Laplace s is related to the rate of change in the system.

D=ss = σ

+ jω

(if the initial conditions/derivatives are all zero at t=0s)


Procedure to apply Laplace transform

•

Convert the system transfer function, or differential equation, to the s-domain by replacing ’D’

with ’s’. (Note: If any of the initial

conditions are non-zero these must be also be added.)•

Convert the input function(s) to the s-domain using the transform tables.

•

Algebraically combine the input and transfer function to find an output function.

•

Use partial fractions to reduce the output function to simpler components.

•

Convert the output equation back to the time-domain using the tables.


Summary

•

The understanding of the system is best obtained in linear systems is from deriving the transfer function of the system.

•

In order to make the analytical complexity easy to handle the transformation of time domain to a different domain is necessary. Laplace transformation helps in such transformation. T

•

he transfer function is obtained by identifying the input and output variables from the system and deducing using Laplace transforms


References

• http://en.wikipedia.org/wiki/Laplace_transform

amongst

others…


And, before we break…

•

The impossible is the untried

Thanks for listening…

Class 2 - Mathematical Modeling Using Transfer Function Approach

Documents

Transcript of Class 2 - Mathematical Modeling Using Transfer Function Approach