sistem kontrol i kuliah ii : transformasi laplace imron rosyadi, st email: pak.imron@gmail.com 1

Sistem Kontrol IKuliah II : Transformasi Laplace

Imron Rosyadi, STEmail: pak.imron@gmail.com

From Lecture #1

Control System Design Process

• Diagram on the next page gives a flowchart of the control system design process

4Original System

- Plant - Sensors

- Actuators

New System

Math Model of Controller

Key Activities of the“MAD” Control Engineer:

- Modeling - Analysis - Design

- ImplementationMath Model

of Plant

MeasurementModeling

Implementation - Physical controller - Coupling controller

with plant

Desired Performance

DevelopPerformance Specifications

Analysis

Design

Simulation

Control System Design Process

• Hidden in this chart are three important elements :

1. Modeling the system (using mathematics)2. Analysis techniques for describing and

understanding the system’s behavior3. Design techniques for developing control

algorithms to modify the system’s behavior

• Modeling, analysis, and design = the MAD control theorist

• A fourth key element is Implementation

Modeling is the key!

• The single most important element in a control system design and development process is the formulation of a model of the system.

• A framework for describing a system in a precise way makes it possible to develop rigorous techniques for analyzing the system and designing controllers for the system

Modeling

• Key Point: most systems of interesting in engineering can be described (approximately) by▫Linear▫Ordinary▫Constant-coefficient▫Differential equations

• Call these LODEs• Where we are going looks like this:

PhysicalReality

LODELaplace

Transform

Requirescalculus to solve

Requiresalgebra to solve

Review of Complex Number

Complex Numbers: Notation and Properties (1)• A complex number:

• The complex plane

Rectangular (Cartesian) coordinates

Polar coordinates

Due to Katie Johnson or Tyrone Vincent or someone

• Transformation between coordinates

Complex Numbers: Notation and Properties (2)

Complex Numbers: Notation and Properties (3)• Euler’s Formula:

• Note differentiation property

)sin()cos( xjxe jx

)sin()cos(

)cos()sin(

)sin()cos(

Exercise

• Show how Euler’s Formula is a parameterization of the unit circle

sincos je j

)tan(tantan

1sincos1

cossin1

Complex Numbers: Notation and Properties (4)

• Alternate notation for polar coordinates using Euler’s Formula

Compare to

Note: keep track of degrees and radians!

Complex Math – Review

• Complex multiplication and division: the hard way

Complex Math – Review• Complex multiplication and division: the easy

way Given:

Exercise

7.1099.0

0.767.331.4

6.30.761.4

7.336.3

Complex Math – Review• Complex conjugate:

• Some key results:

Given:

Define complex conjugate as

22 bass

))(())((

jbaxjbaxsxsx

Complex Math – Review• A function of a complex number is also a complex

number

• Example

Given: )}(Im{)}(Re{)( sGjsGsG

10)45(

2606.06442.0

1660.06224.0 j

Complex Math – Review• Derivatives of a function of complex numbers, G(s),

can be computed in the usual way

• Poles/Zeros

Complex Math – Review• Poles/Zeros at infinity

Laplace Transform

Laplace Transform Motivation

• Differential equations model dynamic systems

• Control system design requires simple methods for solving these equations!

• Laplace Transforms allow us to▫systematically solve linear time invariant (LTI)

differential equations for arbitrary inputs.▫easily combine coupled differential equations

into one equation.▫use with block diagrams to find

representations for systems that are made up of smaller subsystems.

The Laplace Transform Definition

• Laplace Transform exists if integral converges for any value of s▫Region of convergence is not as important

for inverting “one-sided” transforms

Laplace Transform Example (1)

• Example:

• Show that

Notation for “unit step”

Laplace Transform Example (2)

Laplace Transform of a Unit Step

• Find the Laplace Transform for the following function

otherwise0

dtesF stst

Exercise

• Find the Laplace Transform for the following function

otherwise0

The Laplace Transform Definition (Review)• Recall:

• The easiest way to use the Laplace Transform is by creating a table of Laplace Transform pairs. We can use several Laplace Transform properties to build the table.

The function with the simplest Laplace Transform (1)

• A special input (class) has a very simple Laplace Transform

• The impulse function:▫Has unit “energy”

▫Is zero except at t=0

Think of pulse in the limit

The function with the simplest Laplace Transform (2)

LT Properties: Scaling and Linearity

• Proof: Both properties inherited from linearity of integration and the Laplace Transform definition

sFsFtftf

saFtaf

2121 Due to Katie Johnson or Tyrone Vincent or someone

Example 1• Find the following Laplace Transforms

▫Hint: Use Euler’s Formula

Example 1 (2) tjtj eej

jsjsjtL

tjtj eet 2

jsjstL

LT Properties: Time and Frequency Shift

• Proof of frequency shift: Combine exponentials

sFetutf

Example 2

• Find the following Laplace Transforms

Example 2 (2)

LT Properties: Integration & Differentiation

• Proof of Differentiation Theorem: Integration by parts

vduuvudv

LT Properties: Integration & Differentiation (2)

fssFtfdt

Example 3

• Find Laplace Transform for

• What is the Laplace Transform of▫Derivative of a step?▫Derivative of sine?

Example 3 (2)

ssstdtut LL

1)0(1)(

tduL Impulse!

1)sin(22

tdL Cosine!

Exercise• What is the Laplace Transform of

)cos(22

tdL -Sine!

Initial Value Theorem

Final Value Theorem

Inverse Laplace and LODE solution

Inverse Laplace Transform

Partial Fraction Idea -1

Recall: Laplace differentiation theorem (1)

• The differentiation theorem

• Higher order derivatives

Differentiation Theorem (revisited)

• Differentiation Theorem when initial conditions are zero

dsfssFstf

Solving differential equations: a simple example (1)• Consider

0,1 tdt

Solving differential equations: a simple example (2)

• Solution Summary▫Use differentiation theorem to take Laplace

Transform of differential equation▫Solve for the unknown Laplace Transform

Function▫Find the inverse Laplace Transform

0,0 txttx

Example 1• Find the Laplace Transform for the solution to

120300

00...0

sXxssXxsxsXs

fsffssFstfdt

dL nnnn

Notation:

30,100,123 xxtxxx

- Partial Fraction Expansions• In general, LODEs can be transformed into a function

that is expressed as a ratio of polynomials• In a partial fraction expansion we try to break it into its

parts, so we can use a table to go back to the time domain:

• Three ways of finding coefficients▫ Put partial fraction expansion over common

denominator and equate coefficients of s (Example 1)▫ Residue formula ▫ Equate both sides for several values of s (not

covered)

- Partial Fraction Expansions• Have to consider that in general we can

encounter:

▫Real, distinct roots▫Real repeated roots▫Complex conjugate pair roots (2nd order

terms)▫Repeated complex conjugate roots

22211 ))(()()()(

Example 1, Part 2• Given X(s), find x(t).

• This Laplace Transform function is not immediately familiar, but it is made up of parts that are.

• Factor denominator, then use partial fraction expansion:

Finding A, B, and C

• To solve, re-combine RHS and equate numerator coefficients (“Equate coefficients” method)

Final Step• Example 1 completed:

• Since

• By inspection,

KtKe t 0,

1 2 teetx tt

Residue Formula (1)• The residue formula allows us to find one

coefficient at a time by multiplying both sides of the equation by the appropriate factor.

• Returning to Example 1:

Residue Formula (2)• For Laplace Transform with non-repeating roots,

• The general residue formula is:

Example 2• Find the solution to the following differential

equation:

Example 2 (2)

0,2)( 2 teetx tt

0)(21)(3)(

0)(20)(300)(2

sXssXssXs

sXxssXxsxsXs

3)()23( 2 ssXss

)2()1()2)(1(

sXsA 112

Inverse Laplace Transform with Repeated Roots• We have discussed taking the inverse Laplace

transform of functions with non-repeated, real roots using partial fraction expansion.

• Now we will consider partial fraction expansion rules for functions with repeated (real) roots:▫# of constants = order of repeated roots

• Example:

)3()3(3)3(

Repeated real roots in Laplace transform table• The easiest way to take an inverse Laplace

transform is to use a table of Laplace transform pairs.

)(2)sin(

2)sin(

Repeated Real Roots

Repeated Imaginary Roots(also use cosine term)

Repeated Complex Roots(also use cosine term)

Example with repeated roots• Example: find x(t)

• Take Laplace Transform of both sides:

exxx t

Example with repeated roots (2)

•Terms with repeated roots:

Example with repeated roots (3) C = 1B = 2

Exercise 1• Find the solution to the following differential

equation 0)0(1)0(044 xxxxx

sXssXssXs

sXxssXxsxsXs

040400

Above …

Inverse Laplace and LODE solutions

- Partial fraction expansions - LODE solution examples

* Real roots* Real, repeated roots

Next:* Complex roots

A complex conjugate pair is actually two distinct, simple first order poles, so can find residues and combine in the usual way:

Inverse Laplace Transform with Complex Roots

• To simplify your algebra, don’t use first-order denominators such as

• Instead, rename variables

• So that

21 KKB 21 11 KjKjC

Laplace Transform Pairs for Complex Roots

• More Laplace transform pairs (complex roots):

• Also, see the table in your textbook and most other control systems textbooks.

)()sin(

)()cos(

Return to example from above:74

Example with complex roots• Example: find x(t)

• Laplace Transform

Example with complex roots (2)

Exercise 2• Find solution to the following differential

equation

0)0(1)0(084 xxxxx

tetetx

sXssXssXs

sXxssXxsxsXs

tt 2sin2cos

080400

Resume Lecture #2

•Review of Complex Number•Laplace Transform•Inverse Laplace Transform•Solving LODE

sistem kontrol i kuliah ii : transformasi laplace imron rosyadi, st email: pak.imron@gmail.com 1

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