Class 2 - Mathematical Modeling Using Transfer Function Approach
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Transcript of 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
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 3
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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 4
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 5
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).
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 6
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 7
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
−−
−−
= =
+ + ⋅⋅⋅+ += = =
+ + ⋅⋅⋅+ +
=
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 8
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 9
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 10
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)
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 11
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.
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 12
Summary
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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
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 13
References
• http://en.wikipedia.org/wiki/Laplace_transform
amongst
others…
July – December 2008 prv/System Modeling Coursework/MIT-Manipal 14
And, before we break…
•
The impossible is the untried
Thanks for listening…