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LCS L2

The document provides an overview of linear control systems and the Laplace transform. It discusses: 1) What the Laplace transform is and how it can be used to solve ordinary differential equations by converting them to algebraic equations. 2) Examples of taking the Laplace transform and inverse Laplace transform of simple functions. 3) How MATLAB can be used to calculate Laplace transforms. 4) Theorems of the Laplace transform and partial fraction expansion, which is used to simplify transformed functions. 5) How the Laplace transform can be applied to find solutions to differential equations.

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Mr. AK Raj
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150 views12 pages

LCS L2

The document provides an overview of linear control systems and the Laplace transform. It discusses: 1) What the Laplace transform is and how it can be used to solve ordinary differential equations by converting them to algebraic equations. 2) Examples of taking the Laplace transform and inverse Laplace transform of simple functions. 3) How MATLAB can be used to calculate Laplace transforms. 4) Theorems of the Laplace transform and partial fraction expansion, which is used to simplify transformed functions. 5) How the Laplace transform can be applied to find solutions to differential equations.

Uploaded by

Mr. AK Raj
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Linear Control System

Lecture 02
Outline

1 Laplace Transform
Laplace Transform
Laplace Transform using MATLAB
Theorems of the LaplaceTransform
Partial-Fraction Expansion
Differential Equation Solution using LaplaceTransform
Laplace Transform

A mathematical tool used to solve ordinary differential


equations.
Laplace transform
∫∞ − st dt
L [f(t)] = F(s) = 0− f(t)e
wher
e s = σ +jω
Inverse Laplace transform
∫ σ+j ∞
L −1 [F(s)]= 2πj 1 st
σ − j ∞ F (s)e ds
The Laplace transform has following two features:
1.Homogeneous and particular solution of ODE in one
operation.
2.Convert differential equation into an algebraic equation.
Laplace Transform
Example: Find the Laplace transform of
f (t)= Ae −a t u(t)
Example: Find the inverse Laplace transform of
1
F(s)= (s+3) 2
Outline

1 Laplace Transform
Laplace Transform
Laplace Transform using MATLAB
Theorems of the LaplaceTransform
Partial-Fraction Expansion
Differential Equation Solution using LaplaceTransform
Laplace Transform using MATLAB
Outline

1 Laplace Transform
Laplace Transform
Laplace Transform using MATLAB
Theorems of the LaplaceTransform
Partial-Fraction Expansion
Differential Equation Solution using LaplaceTransform
Theorems of the Laplace Transform
Outline

1 Laplace Transform
Laplace Transform
Laplace Transform using MATLAB
Theorems of the LaplaceTransform
Partial-Fraction Expansion
Differential Equation Solution using LaplaceTransform
Partial-Fraction Expansion

Convert the function to a sum of simpler terms.


F(s) = N(s)/D(s)

Case 1. Roots of the denominator of F (s) are real and distinct. Case 2.
Roots of the denominator of F (s) are real and repeated. Case 3. Roots
of the denominator of F (s) are complex or imaginary
Outline

1 Laplace Transform
Laplace Transform
Laplace Transform using MATLAB
Theorems of the LaplaceTransform
Partial-Fraction Expansion
Differential Equation Solution using LaplaceTransform
Differential Equation Solution using Laplace Transform
Example
d 2y
+ 12dy + 32y = 32u(t)
dt2 dt

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