ae383lecturenotes_complexvariablesandlaplacetransform

8/19/2019 AE383LectureNotes_ComplexVariablesandLaplaceTransform

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Middle East Technical University Department of Aerospace Engineering

AEE 383 Lecture Notes Fall 2013 Dr. Ali Türker Kutay

1

Complex Variable

, √ 1, : :

||

∠ t a n− ̅ ̅

A complex number contains twice the information compared to a real number since it has two parts. The

two parts of a complex number can be expressed as the real and imaginary parts ( and ) if you use theCartesian coordinates as shown in the figure above. Or equivalently the same complex number can be

expressed in polar coordinates using magnitude and angle (

|| and

∠).

Complex Function is a function with a complex argument . In general the value (output) of a complex function will be a complexnumber as shown above. Here both and are real.Example:

1 1 ,

1 1

1 1

1 1

Let 1 , i.e., 1, 1: 1 1 1 1

1 11 1

1 1 1 1

10

110 310

is a single valued function since it maps any point on

plane to a single point on

plane. Reverseis not true! ∞ ⇒ 0 & 1.

Re

Im plane1

Re

Im plane

110 310

Re

Im

plane ||

∠


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Analytic Function

A complex function is said to be analytic if and all its derivatives exist.Example:

1 1 0 1

2 ℎ ℎ . Singular Points (Poles)

Points in the plane at which the function is not analytic are called singular points (poles). At poles orits derivatives approach infinity. A pole at is said to have multiplicity iflim→ ≠ 0≠ ∞ Example:

10 2 1 3 lim→− 3 l i m→− 10 2 1 3 ∞ lim→− 3 l i m→− 10 2 1 106

Hence 3 is a pole of order 2 (multiplicity 2) 0 is also a pole ( 1) 1 ( 1)

Zeros of a Function

If a function is analytic at andlim→ [ 1 ] ≠ 0≠ ∞

then is a zero of multiplicity .Example:

10 2 1 3 has a zero at 2 with 1. For large

≅ 10 Hence has four poles at 0,1,3,3 and one zero at 2 and three zeros at ∞.Laplace Transformation

Many common functions (step, sinusoidal functions, exponential functions, …) of a real valued

argument (e.g. time) can be transformed into algebraic functions of complex variable . The newfunction of will still be representing the original function of , but expressed in a different


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mathematical form. This is in a way similar to explaining the same thing in a different language.

Suppose that a certain subject is to be explained. This can be done in different languages and it may

be easier to do that in a particular language if for example that language has a richer vocabulary in

that subject. So if you use that language you can explain that topic using fewer words.

Differentiation and integration can be replaced by algebraic operations in the complex plane (linear

differential equations can be transformed into an algebraic equation in a complex variable . As youwill see later on this makes things (system analysis) a lot easier.Define

: a function of time such that 0 for < 0 : a complex variableℒ: an operational symbol for Laplace transformation

: Laplace transform of

Then the Laplace transformation of is given byℒ −

provided that there exists a real positive such that | −|


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ℒ −−

−+

[ 1 −+] 1

The result is a complex rational function with a simple pole at .Euler’s Theorem: c o s s i n Also

− c o s s i n Then

c o s 12 ( −)

s i n 12 ( −)

Example:

{0, < 0 sin , ≥ 0 where and are constants.

ℒ sin 2 ( −)−

2 (−− −+)

2 −− −+

2 [0 0 1 1 ] 2

The result is again a complex rational function with two poles at ± and no zeros.Similarly

ℒ cos Again there are two poles at ±, but this time there is also a zero at the origin.Note that in practice we almost never take the integral to find the Laplace transform of a function. Instead

we use Laplace transforms of common functions from Laplace transform tables and Laplace transform

properties. But still you need to know the definition, because I may ask you to find the transform of a function

directly using the definition in the exam.


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Inverse Laplace Transformation

ℒ− 12

+

−

where is a real constant greater than the real parts of all the singularities in .Important Properties of Laplace Transform

1. ℒ

2. ℒ ± ± 3.

ℒ lim→ 0 (differentiation)

ℒ

lim→ − − −

⋯ −

− −

−

− 0 − ′0 − ′′0 ⋯ −0 4. ℒ ∫ (integration)

… …

5.

Time shift

ℒ − where

ℒ Note that in general we omit

with the assumption that

0 for

< 0.

In this course we use onesided Laplace transform,

where we take the integral

in the definition from 0 to∞. So we assume thatthe function to be

transformed is defined in

that range. To make sure

that this is the case we

multiply the function with

unit step function as

shown here.

} linearity


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−

−

−

−

−+

⇒ 0 ∞ ⇒ ∞

− −

−

6.

Initial value theorem: lim→ l i m→ 7.

Final value theorem:

lim→ lim→ Provided that is analytic on the imaginary axis.Example:

10

3 1 7

lim→ lim→ 107 : ℎ Example:

, ℒ− sin lim→ l i m→ sin → ± 1, ℎ has two poles on the imaginary axis, , , thus FVT is not applicable. If we apply FVT:lim→ 0

which is WRONG!


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8.

Complex shifting:

ℒ− − −

−+ , ̅ ̅ ℒ 9. Real convolution:

ℒ , ℒ Convolution integral:

∗ ∗ ℒ ∗

Very useful in finding the inverse Laplace transform!

(This is an example that many complex operations in time domain correspond to simpler operations

in Laplace domain. We have two functions, and . In time domain we take the convolutionintegral of these two functions. This operation is equal to simply multiplying the Laplace transforms

of these two functions.)

Example:

1⏟

, − ℒ− ∗ −−

−

−

1 −

00

1

f (t )

e-at

e-at

f (t )

Proof of commutative property:

̅ ⇒ ̅

0 ⇒ ̅ ⇒ ̅ 0 ̅ ̅ ̅


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How Laplace Transform Makes System Analysis Easier

Let’s remember how we do system analysis on the following flow chart:

In the class on the mass spring damper system we found the response to step input force. Remember that

the solution was pretty difficult to get (I just showed the end result in class) even for this simple system with

the simple input function. This was an example of going straight down from the left column following the

red arrows. By using Laplace transform we can get to the same result more easily (follow the green path).

Example: Let’s go back to the previous example. Find the response of the spring mass damper system to

initial conditions:

̈ ̇ 0, 0 , ̇ Take the Laplace transform of the above equation:ℒ ̈ ̇ ℒ̈ ℒ̇ ℒ 0

Then using the differentiation property we get

ℒ ̈ ̇ ̇ 0 ̇ 0 ̇ ̇

External inputs

Use laws of physics

Linear

D namic

Linear ODE’s

Classical

techni ues

Response in

time

Algebraic

equations

Algebraic

techni ues

Response in

domain

ℒ Take Laplace transform

ℒ

ℒ− Take inverse Laplace transform


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Note that the above function is a complex rational function with two poles at the roots of the denominator

polynomial:

, ± √ 4

2

and a zero at

̇ Response to initial conditions can be found by taking the inverse Laplace transform of :

.. ℒ− We will do the Laplace inversion later. To find the response to external forcing we again take the Laplace

transform of the original ODE, this time with on the right hand side, but with zero I.C.’s:

ℒ ̈ ̇ ℒ

Using the linearity and differentiation properties we get

1

Obviously the result depends on the input force applied. We need to take the Laplace transform of the input

applied and insert it into the above expression and then take the inverse Laplace transform as before.If is the unit step function, we already know from a previous example that

ℒ 1 Therefore.. 1 1

and

.. ℒ− { 1 } The good thing here is that if we want to compute the response of the system to a different input function,

all we need to do is to just take the Laplace transform of that function and obtain by simply inserting into the above function. For example let sin Then from a Laplace transform table we get

.. then becomes

..


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We can then invert the above function. This is much easier than solving the system differential equation in

time domain with the sinusoidal input.

Transfer Function

For a general input function the response in

domain can be found as

1 = Note that the function depends on the physical properties of the system (values of , , and ) anddoes NOT depend on input applied or the initial conditions of the system. This function is called the transfer

function of the system. In general transfer functions always depend on only the physical properties of the

system. For linear time invariant (LTI) dynamic systems transfer functions are always complex rational

functions. In the most general case a transfer function will have the following form:

− − ⋯ − − ⋯

− − ⋯ − − ⋯ − − ⋯ − − ⋯

where

,

, and

. For real physical systems

is always

≤ .

ae383lecturenotes_complexvariablesandlaplacetransform

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