ECE317 : Feedback and Control

Lecture:ODE solution via Laplace transform

Dr. Richard Tymerski

Dept. of Electrical and Computer Engineering

Portland State University

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Course roadmap

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Laplace transform

Transfer function

Block Diagram

Linearization

Models for systems

• electrical

• mechanical

• example system

Modeling Analysis Design

Stability

• Pole locations

• Routh-Hurwitz

Time response

• Transient

• Steady state (error)

Frequency response

• Bode plot

Design specs

Frequency domain

Bode plot

Compensation

Design examples

Matlab & PECS simulations & laboratories

Laplace transform (review)

• One of most important math tools in the course!

• Definition: For a function f(t) (f(t)=0 for t<0),

• We denote Laplace transform of f(t) by F(s).

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f(t)

t0

F(s)

(s: complex variable)A:=B (A is defined by B.)

Laplace transform table

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Inverse Laplace

Transform

(u(t) is often omitted.)

Advantages of s-domain (review)

• We can transform an ordinary differential equation into an algebraic equation which is easy to solve.

(This lecture)

• It is easy to analyze and design interconnected (series, feedback etc.) systems.

• Frequency domain information of signals can be easily dealt with.

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An advantage of Laplace transform

• We can transform an ordinary differential equation (ODE) into an algebraic equation (AE).

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ODE AE

Partial fraction

expansionSolution to ODE

t-domain s-domain

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Example 1 (distinct roots)

ODE with initial conditions (ICs)

1. Laplace transform

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distinct roots

Properties of Laplace transformDifferentiation (review)

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t-domain

s-domain

Example 1 (cont’d)

2. Partial fraction expansion

Multiply both sides by s(s+1)(s+2) :

Compare coefficients:

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unknowns

Example 1 (cont’d)

3. Inverse Laplace transform

If we are interested in only the final value of y(t), apply the Final Value Theorem, without explicitly computing y(t):

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(You may omit u(t).)

Example 2 (repeated roots)

ODE with zero initial conditions (ICs)

1. Laplace transform

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Repeated roots

Example 2 (cont’d)

2. Partial fraction expansion

Multiply both sides by

Compare coefficients:

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unknowns

Example 2 (cont’d)

3. Inverse Laplace transform

If we are interested in only the final value of y(t), apply the Final Value Theorem, without explicitly computing y(t):

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(u(t) omitted.)

Properties of Laplace transformFrequency shift theorem (review)

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Proof.t-domain

s-domain

Ex.

Example 3 (complex roots)

ODE with zero initial conditions (ICs)

1. Laplace transform

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Complex roots

Example 3 (cont’d)

2. Partial fraction expansion

Multiply both sides by

Compare coefficients:

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unknowns

Example 3 (cont’d)

3. Inverse Laplace transform

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Laplace transform table

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Frequency shift theorem

Summary

• Solution to ODE via Laplace transform1. Laplace transform

2. Partial fraction expansion

3. Inverse Laplace transform

• Next, modeling of physical systems in s-domain

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