MIE404: Control Systems

Lecture 3Laplace Transforms & Block Diagrams

9/16/2015 MIE404 – Fall 2015 1

Lecture Overview

• Last Time• Modelling Mechanical and Electrical Systems

• This Lecture• Review of Laplace Transforms• Transfer Functions• Block Diagrams

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Laplace Transforms - Motivation

• Take and ODE (of any order)

• To an algebraic equation

• Benefits

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ܯ ሷݔ ݐ + ሶݔ ݐ + ݔ ݐ = (ݐ)

ଶݏܯ ݏ + ݏ ݏ + = (ݏ)ܨ

Laplace Transform - Definition

Given a function f(t) in the time domain, we define its Laplace transform, F(s), as

F(s) is the frequency domain representation of f(t).

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Finding the Laplace Transform• Example 1• Heaviside Step Function

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ݑ ݐ = ቊ0, ݐ < 01, ݐ 0

Finding the Laplace Transform

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• Example 1• Decaying Exponential

ݐ = ቊ 0, ݐ < 0௧ , ݐ 0

Common Laplace Transforms

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Common Laplace Transforms

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t0

į(t)

Properties of Laplace Transforms• Laplace transforms have many properties we will

exploit in the course• Linearity

• Differentiation

• Integration

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Converting Models to Laplace Domain• Consider our Mass-Spring Damper System

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f(t)

x(t)

M

b

k

(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ

Converting Models to Laplace Domain

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Transfer Functions for Coupled Systems

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x1(t)

M1

b

k1

f(t)M2k2

x2(t)

Transfer Functions for Coupled Systems

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Block Diagrams

• Block Diagrams

• Why are they useful?

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Block Diagrams• Examples

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f(t)

x(t)

M

b

k

(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ

1ଶݏܯ + ݏ +

(ݏ)ܨ (ݏ)

Block Diagrams• Examples

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f(t)

x(t)

M

b

k

(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ

1ݏ

ሷݔ ሶݔ +ݔ

ܯ

ܯ

1ݏ

1ܯ

Building Up A Block Diagram• Example

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(ݐ) = ܯ ሷ(ݐ)ݔ + ሶ(ݐ)ݔ + (ݐ)ݔ

Building Up Block Diagram• Example – Coupled System

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x1(t)

M1

b

k1

f(t)M2k2

x2(t)

Building Up Block Diagram• Example Coupled System

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Summary• This Time

• Review of Laplace Transforms

• Transfer Functions

• Block Diagrams

• Next Lecture• Modelling - Electro-Mechanical Systems• More on Laplace Transforms – Using to solve for system

dynamic response.

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