CHEE319 Notes 2011 Lecture3

8/19/2019 CHEE319 Notes 2011 Lecture3

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Solution ofSolution of ODEsODEs usingusing LaplaceLaplace TransformsTransforms

Process Dynamics and ControlProcess Dynamics and Control


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Common TransformsCommon Transforms

UsefulUseful Laplace Laplace TransformsTransforms

1. Exponential1. Exponential

2. Cosine2. Cosine


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UsefulUseful Laplace Laplace TransformsTransforms

3. Sine3. Sine


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Operators

1. Derivative of a function, ,

2. Integral of a function


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OperatorsOperators3. Delayed function


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Input Signals1. Constant1. Constant

2. Step2. Step

3. Ramp function3. Ramp function


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Input Signals Input Signals

4. Rectangular Pulse4. Rectangular Pulse

5.5. Unit impulse


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LaplaceLaplace TransformsTransforms

Final Value Theorem

Limitations:Limitations:

Initial Value Theorem


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Solution ofSolution of ODEsODEs

Cruise Control Example

"" Taking theTaking the Laplace Laplace transform of the ODEtransform of the ODE yields (recalling theyields (recalling the LaplaceLaplace

transform is a linear operator)transform is a linear operator)

Force of Engine (u)

Friction

Speed (v)


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Solution ofSolution of ODEsODEs

!! Solve by inverseSolve by inverse LaplaceLaplace transform:transform: (tables)(tables)

"" Solution is obtained by aSolution is obtained by a getting the inversegetting the inverse LaplaceLaplace transformtransform

from a tablefrom a table

"" AlternativelyAlternatively we can use partial fraction expansion to computewe can use partial fraction expansion to compute

the solution usingthe solution using simple inverse transformssimple inverse transforms


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Solution of LinearSolution of Linear ODEsODEs

!! DC Motor DC Motor

!! System dynamics describes (negligible inductance)System dynamics describes (negligible inductance)


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LaplaceLaplace TransformTransform

!! Expressing in terms of angular velocityExpressing in terms of angular velocity

"" TakingTaking LaplaceLaplace TransformsTransforms

"" SolvingSolving

"" Note that this function can be written as Note that this function can be written as


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LinearLinear ODEsODEs

We deal with rational functions of the form where degree of > degree of

is called the characteristic polynomial of the function

The roots of are the poles of the function

Theorem:

Every polynomial with real coefficients can be factored into the product of only two types of factors

" powers of linear terms and/or

" powers of irreducible quadratic terms,


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Partial fraction ExpansionsPartial fraction Expansions

1. has real and distinct factors

expand as

2. has real but repeated factor

expanded


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Partial Fraction ExpansionPartial Fraction Expansion

Heaviside expansion

For a rational function of the form

Constants are given by


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ExampleExample

The polynomialThe polynomial

has rootshas roots

It can be factored asIt can be factored as

By partial fraction expansionBy partial fraction expansion


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ByBy HeavisideHeaviside

"" becomes becomes

""By inverseBy inverse laplacelaplace


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Heaviside expansion




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ExampleExample


has rootshas roots




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ByBy HeavisideHeaviside

"" becomes becomes

""By inverseBy inverse laplacelaplace


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Heaviside expansion




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PartialPartial Fraction ExpansionFraction Expansion

ExampleExample


has rootshas roots




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ByBy HeavisideHeaviside,,

which yieldswhich yields

Taking the inverseTaking the inverse laplacelaplace


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PartialPartial Fraction ExpansionFraction Expansion

ExampleExample


has rootshas roots

It can be factored as (It can be factored as ( ))

Solving for A and B,Solving for A and B,


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Equating similar powers of s in,Equating similar powers of s in,

yieldsyields

hencehence

GivingGiving

Taking the inverseTaking the inverse laplacelaplace


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Partial Fraction ExpansionsPartial Fraction Expansions

Algorithm for Solution of ODEs

" Take Laplace Transform of both sides of ODE

" Solve for

"Factor the characteristic polynomial# Find the roots (roots or poles function in Matlab)

# Identify factors and multiplicities

" Perform partial fraction expansion

" Inverse Laplace using Tables of Laplace Transforms


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!! For a givenFor a given functionfunction

!! The polynomialThe polynomial has three distinct types of has three distinct types of rootsroots

"" Real rootsReal roots

## yields exponential termsyields exponential terms

## yieldsyields constant termsconstant terms

"" Complex rootsComplex roots

## yields exponentially weighted sinusoidalyields exponentially weighted sinusoidal

signalssignals

## yieldsyields pure sinusoidal pure sinusoidal signalsignal

!! A lot of information is obtained from the roots of A lot of information is obtained from the roots of

CHEE319 Notes 2011 Lecture3

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