Modelagem de Funções de Transferência de Plantas Industriais em Malha Aberta e Fechada utilizando Algoritmos Genéticos

❋❧♦r✐s✈❛❧❞♦ ❈❛r❞♦③♦ ❇♦♠✜♠ ❏✉♥✐♦r

Modelagem de Funções de Transferência de Plantas Industriais em Malha

Aberta e Fechada utilizando Algoritmos Genéticos

Dissertação apresentada ao Programa de Pós-Graduação

em Engenharia Elétrica da Universidade Federal de

Uberlândia, como requisito para o título de Mestre em

Ciências.

Área de concentração: Processamento da Informação

Linha de Pesquisa: Inteligência Artificial

Orientador: Dr. Keiji Yamanaka

Coorientador: Dr. Igor Santos Peretta

Uberlândia-MG

Dados Internacionais de Catalogação na Publicação (CIP) Sistema de Bibliotecas da UFU, MG, Brasil.

B695m

2017 Bomfim Junior, Florisvaldo Cardozo, 1980- Modelagem de funções de transferência de plantas industriais em malha aberta e fechada utilizando algoritmos genéticos / Florisvaldo Cardozo Bomfim Junior. - 2017.

51 f. : il.

Orientador: Keiji Yamanaka. Coorientador: Igor Santos Peretta.

Dissertação (mestrado) - Universidade Federal de Uberlândia, Programa de Pós-Graduação em Engenharia Elétrica.

Disponível em: http://dx.doi.org/10.14393/ufu.di.2018.25 Inclui bibliografia.

1. Engenharia elétrica - Teses. 2. Algoritmos genéticos - Teses. 3. Automação industrial - Teses. 4. Controle automático - Teses. I. Yamanaka, Keiji. II. Peretta, Igor Santos. III. Universidade Federal de Uberlândia. Programa de Pós-Graduação em Engenharia Elétrica. IV. Título.

❋❧♦r✐s✈❛❧❞♦ ❈❛r❞♦③♦ ❇♦♠✜♠ ❏✉♥✐♦r

Modelagem de Funções de Transferência de Plantas Industriais em

Malha Aberta e Fechada utilizando Algoritmos Genéticos

Dissertação apresentada ao Programa de Pós-Graduação

em Engenharia Elétrica da Universidade Federal de

Uberlândia, como requisito para o título de Mestre em

Ciências.

Área de concentração: Processamento da Informação

Linha de Pesquisa: Inteligência Artificial

Prof. Keiji Yamanaka, Dr.

FEELT/UFU (Orientador)

Prof. Igor Santos Peretta, Dr.

FEELT/UFU (Coorientador)

Prof. Edilberto Teixeira, Dr.

UNIUBE

Prof. Josué Silva de Morais, Dr.

FEELT/UFU

Uberlândia-MG

Agradecimentos

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶ ❋✉♥çã♦ tr❛♥s❧❛❞❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✷ ❋✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✸ ▼❛❧❤❛ ❛❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✹ ▼❛❧❤❛ ❢❡❝❤❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✺ ▼ét♦❞♦ ❞❡ ❩❡❛❣❧❡ ◆✐❝❤♦❧s✴❑✿●❛♥❤♦❀▲✿❆tr❛s♦ ✭θ✮❀❚✿❈♦♥st❛♥t❡ ❞❡ t❡♠♣♦ ✭τ✮❀ ✶✷

✻ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✕ Pr✐♠❡✐r❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✼ ▼ét♦❞♦ ❞❡ ❙✉♥❞❛r❡s❛♥ ❡ ❑r✐s❤♥❛s✇❛♠② ✲ Pr✐♠❡✐r❛ ❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✽ ▼ét♦❞♦ ❞❡ ◆✐s❤✐❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✾ ●rá✜❝♦ ❞❡ ❙♠✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✵ ❘❡s♣♦st❛ ❛♦ ❞❡❣r❛✉ ✉♥✐tár✐♦ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✶✶ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✶✷ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✶✸ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✹ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✶✺ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✻ ●rá✜❝♦ ❞❡ ❙♠✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✶✼ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✶✽ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✶✾ ●rá✜❝♦ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❛s ❝✉r✈❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✵ ❈✉r✈❛ ❞❡ r❡s♣♦st❛ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✶ ❘❡s♣♦st❛ r❡❛❧ ❛♦ ❞❡❣r❛✉ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✷ ❈✉r✈❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ✶a _{♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻}

✷✸ ❈✉r✈❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ✷a _{♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻}

✷✹ ❈✉r✈❛ ❞❡ ❛♣r♦①✐♠❛çã♦ ✸a _{♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼}

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

✶ ❙í♥t❡s❡ ❞✐r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷ ❚❛❜❡❧❛ ❞❡ ❝♦♠♣❛r❛çã♦ ❞❡ ♠ét♦❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✸ ❈♦♠♣❛r❛çã♦ ❞❡ r❡s✉❧t❛❞♦s ❞❡ ✷a _{♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶}

✹ ❘❡s✉❧t❛❞♦s ❞❡ ❛♣r♦①✐♠❛çã♦ ✸a _{♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶}

❙✉♠ár✐♦

1 Introdução 3

2 Revisão de Literatura 5

✷✳✻✳✹ ❆çã♦ ❉❡r✐✈❛t✐✈❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✼ ▼ét♦❞♦s ❞❡ s✐♥t♦♥✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✼✳✶ ❙í♥t❡s❡ ❉✐r❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✽ ❆❧❣♦r✐t♠♦ ●❡♥ét✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✽✳✶ ❋✉♥❝✐♦♥❛♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✽✳✷ ▼✉t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✽✳✸ ❙❡❧❡çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✽✳✹ ❘❡❝♦♠❜✐♥❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✽✳✺ ❆♣t✐❞ã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✾ ❙❇❳ ✭Simulated Binary Crossover✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

3 Trabalhos Relacionados 26

4 Resultados e Discussão 27

✹✳✶ ❈♦♥s✐❞❡r❛çõ❡s s♦❜r❡ ♦ ♠ét♦❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✷ ▲❡✈❛♥t❛♥❞♦ ❞❛❞♦s ♣❛r❛ ❛ s✐♠✉❧❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✸ ❆♣❧✐❝❛çã♦ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ Pr✐♠❡✐r❛

❖r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✸✳✶ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✹✳✸✳✷ ▼ét♦❞♦ ❞❡ ❙✉♥❞❛r❡s❛♥ ❡ ❑r✐s❤♥❛s✇❛♠② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✹✳✸✳✸ ▼ét♦❞♦ ❞❡ ◆✐s❤✐❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✹✳✹ ▼ét♦❞♦ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ❡♠ ♣r✐♠❡✐r❛

♦r❞❡♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✹✳✶ ❊str✉t✉r❛ ❞♦ ❝r♦♠♦ss♦♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✹✳✹✳✷ ❋✉♥çã♦ ❋✐t♥❡ss ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✹✳✸ ❆♣❧✐❝❛çã♦ ❞♦ ♠ét♦❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✹✳✹✳✹ ❈♦♠♣❛r❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✺ ❆♣❧✐❝❛çã♦ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ♣❛r❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❞❡ s❡❣✉♥❞❛

♦r❞❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✹✳✺✳✶ ▼ét♦❞♦ ❍❛rr✐♦t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✺✳✷ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✻ ❆♣r♦①✐♠❛çã♦ ❞❡ s✐st❡♠❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✉s❛♥❞♦ ♠ét♦❞♦ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ✸✾ ✹✳✻✳✶ ❈♦♠♣❛r❛çã♦ ❡♥tr❡ ♦s ♠ét♦❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✹✳✼ ❆♣r♦①✐♠❛çã♦ ❞❡ s✐st❡♠❛ ❞❡ ✸a _{♦r❞❡♠ ✉s❛♥❞♦ ♠ét♦❞♦ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ✳ ✳ ✹✶}

✹✳✽ ▼♦❞❡❧❛❣❡♠ ❞❡ ♣r♦❝❡ss♦s ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✽✳✶ ❉❡t❡r♠✐♥❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞♦ ❝r♦♠♦ss♦♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✹✳✾ ❆♣❧✐❝❛çã♦ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ r❡❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

Resumo

Em sistemas de controle, conhecer a função de transferência é de suma importância

para a calibração dos compensadores industriais (Controladores PID’s) e para determinar

o seu comportamento. Existem várias técnicas convencionais para a determinação dessas

funções de transferência, tais como Smith, Sundaresan e Krishnaswamy e Nishikawa para

sistemas de 1a_{ordem, e Harriot e Smith para sistemas de 2}a_ordem.

Este trabalho teve como objetivo determinar a função de transferência em sistemas de

malha aberta e malha fecha por meio do algoritmo genético e confrontar os modelos com os

obtidos nas formas convencionais. Para o desenvolvimento do algoritmo fez-se necessário

a aplicação de conhecimentos de automação e controle (transformada de Laplace,

Contro-ladores, diagrama de blocos e modelagem de sistemas lineares) usando como plataforma o

software matemático MatlabR. Os resultados foram confrontados e apresentaram melho-res melho-respostas quando comparado aos obtidos pelos métodos convencionais, comprovando

que o método aplicado atingiu as expectativas. Logo após, aplicou-se o sistema para a

de-terminação da função de transferência de uma planta real e os resultados comprovaram que

é uma ferramenta válida para aplicações reais.

Palavras-chave: Algoritmo Genético, Controle, Automação, Função de Transferência, Planta

Abstract

In control systems, knowing the transfer function is of paramount importance for the

calibration of industrial compensators (PID controllers) and to determine their behavior.

There are several conventional techniques for determining such transfer functions, such as

Smith, Sundaresan and Krishnaswamy and Nishikawa for first order systems, and Harriot

and Smith for second order systems.

This work aimed to determine the transfer function in open mesh and closed mesh

systems through the genetic algorithm and to compare the models with those obtained in

conventional forms. For the development of the algorithm it was necessary to apply

knowl-edge of automation and control (Laplace transform, Controllers, block diagram and linear

systems modeling) using as platform Matlab textregistered mathematical software. The

re-sults were compared and presented better responses, when compared to those obtained by

conventional methods, proving that the applied method reached expectations. Afterwards,

the system was applied to determine the transfer function of a real plant and the results

proved that it is a valid tool for real applications.

✶ ■♥tr♦❞✉çã♦

❊♠ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ✐♥❞✉str✐❛❧✱ ❝♦♥❤❡❝❡r ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ✉♠❛ ♣❧❛♥t❛✱ ♣♦❞❡ s❡r ❛ ♣❡❞r❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ s❡ ♦❜t❡r ✉♠ ❜♦♠ ❛❥✉st❡ ❞♦s ❝♦♠♣❡♥s❛❞♦r❡s ❞❛ ♠❛✲ ❧❤❛ ❞❡ ❝♦♥tr♦❧❡✳ ❊♠ ❞✐✈❡rs♦s t✐♣♦s ❞❡ ♠❛❧❤❛ ❞❡ ❝♦♥tr♦❧❡✱ t❛❧ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ♥ã♦ é ❝♦♥❤❡❝✐❞❛ ❞❡✈✐❞♦ ❛s ❞✐✜❝✉❧❞❛❞❡s ❞❡ ♠♦❞❡❧❛❣❡♠✳ ❉❡st❛ ❢♦r♠❛✱ ♦s ❝♦♠♣❡♥s❛❞♦r❡s sã♦ ❛❥✉st❛❞♦s ❞❡ ♠♦❞♦ ❡♠♣ír✐❝♦✱ t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ t❡♥t❛t✐✈❛ ❡ ❡rr♦✱ r❡s✉❧t❛♥❞♦ ❡♠ ♣❡r❞❛s ❞❡ ♠❛tér✐❛✲♣r✐♠❛✱ ❛❧é♠ ❞❡ ✈❛♣♦r✱ á❣✉❛ ❡ ❝♦♠❜✉stí✈❡✐s ✉s❛❞♦s ❝♦♠♦ ✈❛r✐á✈❡✐s ❞❡ ❝♦♥tr♦❧❡✳ ❆ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ é ❛ r❡❧❛çã♦ ❞❡ ♣❡❧♦ ♠❡♥♦s ❞✉❛s ✈❛r✐á✈❡✐s ❞❡ ♣r♦❝❡ss♦✱ s❡♥❞♦ ✉♠❛ ❞❡ ❡♥tr❛❞❛ ❡ ✉♠❛ ❞❡ s❛í❞❛✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ✉♠ ❛q✉❡❝❡❞♦r ✐♥❞✉str✐❛❧✱ ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❞❡ ❡♥tr❛❞❛✱ ♣♦❞❡✲s❡ ❛❞♦t❛r ❛ ✈❛③ã♦ ❞❡ ✈❛♣♦r ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❞❡ s❛í❞❛✱ ❛ t❡♠♣❡r❛t✉r❛ ❞❛ s✉❜stâ♥❝✐❛ q✉❡ ❞❡s❡❥❛♠♦s ❛q✉❡❝❡r✳ ❆t✉❛❧♠❡♥t❡✱ ❤á ✈ár✐❛s ❢♦r♠❛s ❞❡ ❡st✐♠❛r ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ✉♠ s✐st❡♠❛✱ s❡♥❞♦ q✉❡✱ ♥❛ ♠❛✐♦✲ r✐❛ ❞♦s ♠ét♦❞♦s ❡①✐st❡♥t❡s✱ ❛♣r♦①✐♠❛✲s❡ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ ❛tr❛s♦✳ ❉❡st❛ ❢♦r♠❛✱ ♠❡s♠♦ s❡ ♦ s✐st❡♠❛ ❢♦r ❞❡ s❡❣✉♥❞❛ ♦✉ ❞❡ t❡r❝❡✐r❛ ♦r❞❡♠✱ s❡rá r❡❞✉③✐❞♦ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ ❛tr❛s♦✳ ❊ss❛ ❞✐❢❡r❡♥ç❛ ♥❛ ❢✉♥çã♦ ♣♦❞❡ ❛❝❛rr❡t❛r ✉♠ ❝á❧❝✉❧♦ ❡q✉✐✈♦❝❛❞♦ ❞♦s ♣❛râ♠❡tr♦ ❞♦ ❝♦♠♣❡♥s❛❞♦r ✭ ❈♦♥tr♦❧❛❞♦r P■❉✶ _{✮✳ ❖✉tr❛ ❝❛r❛❝✲} t❡ríst✐❝❛ ❞♦s ♠ét♦❞♦s é q✉❡✱ ♣❛r❛ ❛ ♠❛✐♦r✐❛✱ t❡♠✲s❡ ❝♦♠♦ ❝❡♥ár✐♦ ✉♠ s✐st❡♠❛ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ✭❡str✉t✉r❛ ❞❡ ❝♦♥tr♦❧❡ ♦♥❞❡ ♥ã♦ ♦❝♦rr❡ ❛ r❡❛❧✐♠❡♥t❛çã♦ ✮ ❡ ♥ã♦ ❢❡❝❤❛❞❛ ✭❡str✉t✉r❛ ❞❡ ❝♦♥tr♦❧❡ ❡♠ q✉❡ ❤á ❛ ❝♦♠♣❛r❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❞❡s❡❥❛❞❛ ❝♦♠ ❛ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛ ✮✱ ♦ q✉❡ ❝❛✉s❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ♣❧❛♥t❛s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ✐♥t❡rr♦♠♣✐❞❛s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥tr♦❧❡ ❞❡ t❡♠♣❡r❛t✉r❛ ❞❡ ♣❡tró❧❡♦✳ ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ✉♠❛ ❢♦r♠❛ ♥ã♦ ❝♦♥✈❡♥❝✐♦♥❛❧ ❞❡ s❡ ❡st✐♠❛r ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ✉s❛♥❞♦ ❛ ❤❡✉ríst✐❝❛ ❞♦s ❛❧❣♦r✐t♠♦s ❣❡♥ét✐❝♦s ✭❆●✮✳ ❖ ❛❧❣♦r✐t♠♦ ♣r♦♣♦st♦ é ❝❛♣❛③ ❞❡ ❡st✐♠❛r ❢✉♥çõ❡s t❛♥t♦ ❡♠ s✐st❡♠❛s ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ❡ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛✳ ❖ ♠ét♦❞♦ ❛♣r❡s❡♥t❛❞♦ ❝♦♥s❡❣✉❡ ❡st✐♠❛r ✉♠❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ♠❛✐s ♣ró①✐♠❛ ❞❛ ❢✉♥çã♦ r❡❛❧✱ ♣♦ss✐❜✐❧✐t❛♥❞♦✱ ❛♦s ❡♥❣❡♥❤❡✐r♦s r❡s♣♦♥sá✈❡✐s✱ ✉♠ ♠❡❧❤♦r ❛❥✉st❡ ❞❡ s✉❛ ♠❛❧❤❛ ❞❡ ❝♦♥tr♦❧❡✳ ❉❡✈✐❞♦ à ❣r❛♥❞❡ ❝♦♥❝♦rrê♥❝✐❛ ❞❡ ♠❡r❝❛❞♦✱ ✉♠❛ ♣❧❛♥t❛ ❝♦♠ ♣❛râ♠❡tr♦s ♦t✐♠✐③❛❞♦s ♣♦❞❡ r❡♣r❡s❡♥t❛r ✉♠❛ ❡♥♦r♠❡ ✈❛♥✲ t❛❣❡♠ ♣❡r❛♥t❡ ❛♦s ❝♦♥❝♦rr❡♥t❡s✳

▼✉✐t❛s ♣❡sq✉✐s❛s ❛♥❛❧✐s❛r❛♠ ♦ ❡st❛❞♦ ❞❡ ❞❡s❡♠♣❡♥❤♦ ❞♦s ❧❛ç♦s ❞❡ ❝♦♥tr♦❧❡ ❡♠ ❞✐❢❡✲ r❡♥t❡s ✐♥❞ústr✐❛s ❞❡ ♣r♦❝❡ss♦✳ ❆ ♣r✐♥❝✐♣❛❧ ❝♦♥❝❧✉sã♦ ❢♦✐ q✉❡ ♠✉✐t❛s ✈❡③❡s ♦s ♣r✐♥❝í♣✐♦s

✶_{❊str✉t✉r❛ ❞❡ ❝♦♥tr♦❧❛❞♦r q✉❡ ♣♦ss✉✐ ❛çã♦ ✐♥t❡❣r❛❧✱ ❞❡r✐✈❛t✐✈❛ ❡ ♣r♦♣♦r❝✐♦♥❛❧✳}

❜ás✐❝♦s ❞❡ ❝♦♥tr♦❧❡ sã♦ ✐❣♥♦r❛❞♦s✱ ♦s ❛❧❣♦r✐t♠♦s ❞❡ ❝♦♥tr♦❧❡ sã♦ ✐♥❝♦rr❡t❛♠❡♥t❡ ❡s❝♦❧❤✐❞♦s ❡ s✐♥t♦♥✐③❛❞♦s✱ ❡♥q✉❛♥t♦ s❡♥s♦r❡s ❡ ❛t✉❛❞♦r❡s sã♦ ♠❛❧ s❡❧❡❝✐♦♥❛❞♦s ♦✉ ♠❛♥t✐❞♦s✳

✷ ❘❡✈✐sã♦ ❞❡ ▲✐t❡r❛t✉r❛

◆❡st❛ s❡çã♦ sã♦ ❞✐s❝✉t✐❞♦s ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛ ♣r♦♣♦st❛ ❛♣r❡s❡♥t❛❞❛ ♥❡st❡ tr❛❜❛❧❤♦✳

✷✳✶ ▼♦❞❡❧♦s ▼❛t❡♠át✐❝♦s

❖s s✐st❡♠❛s ❢ís✐❝♦s r❡❛✐s ❛♣r❡s❡♥t❛ ✉♠❛ ❛❧t❛ ❝♦♠♣❧❡①✐❞❛❞❡✱ ♦ q✉❡ ❞✐✜❝✉❧t❛ ♠✉✐t♦ ♦ s❡✉ ❡st✉❞♦✳ ❈♦♠ ✐ss♦ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❛ ♠♦❞❡❧❛❣❡♠ ❡stá ❞✐r❡t❛♠❡♥t❡ r❡❧❛❝✐♦♥❛❞❛ à ♥❛t✉r❡③❛ ❡ à ♣r❡❝✐sã♦ ❞♦ ❡st✉❞♦ q✉❡ s❡ ❞❡s❡❥❛ ❢❛③❡r r❡❢❡r❡♥t❡ ❛♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦ s✐st❡♠❛✳P❛r❛ t❛❧✱ ♣♦❞❡♠♦s ✉s❛r ♠ét♦❞♦s ♣❛r❛ ❛ s✉❛ ♦❜t❡♥çã♦ ❛tr❛✈és ❞❛ r❡s♣♦st❛ ❛♦ ❞❡❣r❛✉✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ❝✉r✈❛ ❞❡ r❡❛çã♦ ❛ ✉♠❛ ✈❛r✐❛çã♦ ♥❛ ❡♥tr❛❞❛ ❞♦ ♣r♦❝❡ss♦✳

✷✳✷ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡

❆ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ✭❊q✉❛çã♦ ✭✶✮✮ é ✉s❛❞❛ ❝♦♠♦ ❢❡rr❛♠❡♥t❛ ♣❛r❛ ❛ s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✷_{✱ ♣♦r r❡❛❧✐③❛r ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❢✉♥çõ❡s✱ ❝♦♠♦ s❡♥♦ ❡ ❝♦ss❡♥♦✱ ♣❛r❛} ✉♠ s✐st❡♠❛ ❛❧❣é❜r✐❝♦✳P♦r s❡r ✉♠ ♠❡✐♦ ❢❛❝✐❧✐t❛❞♦r ❞❡ r❡s♦❧✉çõ❡s rá♣✐❞❛s ✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮✱ ♣♦❞❡✲s❡ ❞❡✜♥✐✲❧❛ ❝♦♠♦✿

F(s) =

Z ∞

−∞

f(t)·e−s·t_{·dt ✭✶✮}

s❡♥❞♦✿

f(t) ✿ ✉♠❛ ❢✉♥çã♦ ♥♦ t❡♠♣♦ t❛❧ q✉❡f(t) = 0 ♣❛r❛ t <0❀

s ✿ ✉♠❛ ✈❛r✐á✈❡❧ ❝♦♠♣❧❡①❛✭s∈C✮❀ F(s) ✿ ❚r❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ f(t)❀

❆ tr❛♥s❢♦r♠❛❞❛ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ f(t) ❡①✐st❡ s❡ f(t) é s❡❝❝✐♦♥❛❧♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠

t♦❞♦ ✐♥t❡r✈❛❧♦ ✜♥✐t♦ ♥❛ r❡❣✐ã♦ t > 0 ❡ s❡ ❛ ❢✉♥çã♦ ❢♦r ❞❛ ♦r❞❡♠ ❡①♣♦♥❡♥❝✐❛❧ q✉❛♥❞♦ t

t❡♥❞❡ ❛ ✐♥✜♥✐t♦✳✭▼❆❨❆✱ P✳❆✳ ❡t ❛❧✱ ✷✵✶✶✮

✷_{❙✐st❡♠❛s ✐♥✈❛r✐❛♥t❡s ♥♦ t❡♠♣♦s✳}

✷✳✷✳✶ ❋✉♥çã♦ ❞❡❣r❛✉

❊ss❡ t✐♣♦ ❞❡ ❢✉♥çã♦ é ♠✉✐t♦ ✉s❛❞♦ ❡♠ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✱ ❝♦❧♦❝❛♥❞♦ ❛ ♣❧❛♥t❛ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ❡ ❛♣❧✐❝❛♥❞♦ ✉♠ ❞❡❣r❛✉ ✉♥✐tár✐♦ ❞❡ ❛♠♣❧✐t✉❞❡ ✶ ❡♠ s✉❛ ❡♥tr❛❞❛✳ ❈♦♠♦ q✉❛❧q✉❡r ❢✉♥çã♦ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✶ é ✐❣✉❛❧ ❛ ❡❧❛ ♠❡s♠❛✱ t❡♠♦s ❡♥tã♦ q✉❡ ❛ ❝✉r✈❛ ❞❡ r❡♣♦st❛ ❛♦ ❞❡❣r❛✉ ♥❛ s❛í❞❛ ❝♦rr❡s♣♦♥❞❡ à ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ♣r♦❝✉r❛❞❛✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮

❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ ❞❡❣r❛✉✿

u(t) =

 



✶✱ t❃❂✵

✵✱ t❁✵ ✭✷✮

❆♦ ❢❛③❡r ❛ ✐♥t❡❣r❛çã♦ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ é ✈á❧✐❞❛ ❡♠ t♦❞♦ ♣❧❛♥♦ ✬s✬ ❡①❝❡t♦ ♥♦ ♣ó❧♦ s = 0✳ ❋✐s✐❝❛♠❡♥t❡✱ ✉♠❛ ❢✉♥çã♦ ❞❡❣r❛✉ ♦❝♦rr❡♥❞♦ ❡♠ t = 0

❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ s✐♥❛❧ ❝♦♥st❛♥t❡ ✐♥s❡r✐❞♦ s✉❜✐t❛♠❡♥t❡ ♥❛ ❡♥tr❛❞❛ ❞♦ s✐st❡♠❛✱ ♥♦ ✐♥st❛♥t❡

t ✐❣✉❛❧ ❛ ③❡r♦✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮

✷✳✷✳✷ ❋✉♥çã♦ ❚r❛♥s❧❛❞❛❞❛

❙✉♣♦♥❞♦ q✉❡ f(t) é ③❡r♦ ♣❛r❛ t < 0 ♦✉ f(t−α) = 0 ♣❛r❛ t < a✳ ❆s ❢✉♥çõ❡s f(t) ❡

f(t−α) sã♦ ♠♦str❛❞❛s ❣r❛✜❝❛♠❡♥t❡ ♥❛ ✜❣✉r❛ ✶✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮

❋✐❣✉r❛ ✶✿ ❋✉♥çã♦ tr❛♥s❧❛❞❛❞❛

❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ✉♠❛ ❢✉♥çã♦ tr❛♥s❧❛❞❛❞❛ é ❞❛❞❛ ♣♦r✿

L_{{f(t−α)}=e}−α·s

·F(s) ✭✸✮

✷✳✷✳✸ ❚❡♦r❡♠❛ ❞❛ ❞✐❢❡r❡♥❝✐❛çã♦

◆❛ ❛♣❧✐❝❛çã♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡♠ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ✶a _{♦r❞❡♠ t❡♠♦s}

❛ s❡❣✉✐♥t❡ ❢♦r♠✉❧❛çã♦✿

L

δf(t)

δt

=F(s)·s−F(0) ✭✹✮

❯♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❡stá ♥❛ s♦❧✉çã♦ ❛♥❛❧ít✐❝❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❝♦♠✉♥s✱ ♦ q✉❡ ♠♦str❛ q✉❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ✉♠ ❞❡r✐✈❛❞♦ é ✉♠ t❡r♠♦ ❛❧❣é❜r✐❝♦✳ ❆ss✐♠✱ ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ é tr❛♥s❢♦r♠❛❞❛ ❡♠ ✉♠❛ ❡q✉❛çã♦ ❛❧❣é❜r✐❝❛✱ q✉❡ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ r❡s♦❧✈✐❞❛ ✉s❛♥❞♦ r❡❣r❛s ❞❡ á❧❣❡❜r❛✳

✷✳✷✳✹ ❚❡♦r❡♠❛ ❞❛ ■♥t❡❣r❛çã♦

◆❛ ❛♣❧✐❝❛çã♦ ❞❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❛♣❧❛❝❡ ❞❡ ✉♠❛ ✐♥t❡❣r❛❧✱ ❡♥❝♦♥tr❛♠♦s ❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ tr❛♥s❢♦r♠❛❞❛ F(s)✳

L

Z ∞

0

f(t)·dt

= F(s)

s −

Z

F(0) ✭✺✮

✷✳✷✳✺ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✜♥❛❧

❖ ✈❛❧♦r ❞❡ ❛❝♦♠♦❞❛çã♦ ❞♦ tr❛♥s✐❡♥t❡ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❛♣❧✐❝❛♥❞♦ ❛ ❡①♣r❡ssã♦ ♦ ❧✐♠✐t❡ ❞♦ t❡r♠♦ ❵s✬ t❡♥❞❡♥❞♦ ❛ ③❡r♦✸_{✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮}

❱f inal =Lims→₀

Z ∞

0

δf(t)

δt ·e

−s·t_·dt

=Lims→₀{F(s)·s−F(0)}

❱f inal =Lims→0F(0)·s

✭✻✮

✷✳✷✳✻ ❚❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✐♥✐❝✐❛❧

❖ ✈❛❧♦r ✐♥✐❝✐❛❧ ❞❡ ✉♠ s✐st❡♠❛ ♣♦❞❡ s❡r ❞❡t❡r♠✐♥❛❞♦ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♦ ✈❛❧♦r ✜♥❛❧ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❧✐♠✐t❡ t❡♥❞❡♥❞♦ ✬s✬ ❛ ✐♥✜♥✐t♦✳

✸_{❈♦♠♦ é ✉♠❛ ❢✉♥çã♦ ♥❛ ❢r❡q✉ê♥❝✐❛✱ s❡✉ ✈❛❧♦r ✜♥❛❧ é ❡♥❝♦♥tr❛❞♦ q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ s t❡♥❞❡ ❛ ③❡r♦✱}

r❡♣r❡s❡♥t❛♥❞♦ q✉❡ ❛t✐♥❣✐✉ ♦ ✈❛❧♦r ✜♥❛❧ ❛♣ós ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠❛ ❡♥tr❛❞❛nq✉❛❧q✉❡r✳

❱inicial =Lims→∞

Z ∞

0

δf(t)

δt ·e

−s·t_·dt

=Lims→∞{F(s)·s−F(0)}

❱inicial =Lims→∞F(∞)·s

✭✼✮

✷✳✷✳✼ ▼♦❞❡❧♦s ❞❡ ❡♥tr❛❞❛ ❡ s❛í❞❛ ❡ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛

❆ ❞❡t❡r♠✐♥❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ♣♦❞❡ s❡r ❛❧❝❛♥ç❛❞♦ ♣❡❧♦ ❛❣r✉♣❛♠❡♥t♦ ❞❡ ✸ ✈❛r✐á✈❡✐s✿ ✶✳ ❊♥tr❛❞❛ ✭ ❝❛✉s❛ ✮

✷✳ ❙❛í❞❛ ✭ ❡❢❡✐t♦ ✮

✸✳ ■♥t❡r♠❡❞✐ár✐❛ ✭❢✉♥çã♦ ❞❡s❡❥❛❞❛✮

P❛r❛ ♠♦❞❡❧♦s ❞✐♥â♠✐❝♦s ❧✐♥❡❛r❡s ✉t✐❧✐③❛❞♦s ♥♦ ❝♦♥tr♦❧❡ ❞❡ ♣r♦❝❡ss♦✱ é ♣♦ssí✈❡❧ ❡❧✐♠✐♥❛r ✈❛r✐á✈❡✐s ✐♥t❡r♠❡❞✐ár✐❛s ❛♥❛❧ít✐❝❛s ♣❛r❛ ♣r♦❞✉③✐r ✉♠ ♠♦❞❡❧♦input-output✱ ❞❡ ♠♦❞♦ q✉❡ ❛s

✈❛r✐á✈❡✐s ✐♥t❡r♠❡❞✐ár✐❛s sã♦ ❝♦♥s✐❞❡r❛❞♦s ♥♦ ♠♦❞❡❧♦✱ ♠❡s♠♦ q✉❡ ♥ã♦ s❡❥❛♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❝❛❧❝✉❧❛❞❛s✳ P♦rt❛♥t♦✱ ♥ã♦ ❤á ♠❛✐s s✉♣♦s✐çõ❡s ❡ s✐♠♣❧✐✜❝❛çõ❡s ❡♥✈♦❧✈✐❞❛s ♥❛ ♠♦❞❡❧❛❣❡♠ ❞❡ input-output ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s✳ ❯♠❛ ♠❛♥❡✐r❛ ♠✉✐t♦ ❝♦♠✉♠ ❞❡ ❛♣r❡s❡♥t❛r ♠♦❞❡❧♦s input-output✱ q✉❡ ❛♣r❡s❡♥t❛♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❝♦♥s✐❞❡rá✈❡❧ ♥♦ ❝♦♥tr♦❧❡ ❞❡ ♣r♦❝❡ss♦✱ é ❛

❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✳✭▼❆❨❆✱ P✳❆✳ ❡t ❛❧✱ ✷✵✶✶✮

✷✳✸ ❋✉♥çã♦ ❞❡ ❚r❛♥s❢❡rê♥❝✐❛

P❛r❛ ❢❛❧❛r♠♦s ❞❡ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ t❡♠♦s q✉❡ ❝♦♥s✐❞❡r❛r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ♣❛râ♠❡tr♦s ❝♦♥❝❡♥tr❛❞♦s ♣❡rt❡♥❝❡♥t❡s à ❝❧❛ss❡ ❞❡ ❛♦ s✐st❡♠❛s ▲■❚✹_{✱ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s} ✉♠❛ ❡♥tr❛❞❛ ❡ ✉♠❛ s❛í❞❛✳ ❙✉♣♦♥❞♦ q✉❡ ♦ s✐st❡♠❛ ❛ s❡❣✉✐r ❝✉♠♣r❛ ❡st❛s ❝♦♥❞✐çõ❡s✱ ♦ ♠♦❞❡❧♦ ♠❛t❡♠át✐❝♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ✜❣✉r❛ ✷✳✭▼❆❨❆✱ P✳❆✳ ❡t ❛❧✱ ✷✵✶✶✮

❋✐❣✉r❛ ✷✿ ❋✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛

qny(t)(n)+qn−₁y(t)(

n−₁₎

+...+q2y(t)

′′

+q1y(t)

′

+q0y(t) =

pmu(t)(m)+pm−₁u(t)(

m−₁₎

+...+p1u(t)

′

+p0u(t)

✭✽✮

✹_{▲✐♥❡❛r ■♥✈❛r✐❛♥t❡ ♥♦ ❚❡♠♣♦✳}

❡♠ q✉❡✿

y(n)(t) = δ

n_y(t)

δtn y(t)

(n−1)

= δy(t)

n−₁ δtn−₁ ...y(t)

′

= δy(t)

δt

u(m)(t) = δu(t)

m

δtm +u(t) m−₁

= δu(t)

m−₁ δtm−1 ...u(t)

′

= δu(t)

δt u(t)

✭✾✮

❡ ♦s ❝♦❡✜❝✐❡♥t❡s qi ❝♦♠ i = 0,1,2,3...n ❡ pj ❝♦♠ j = 0,1,2,3...m sã♦ ❝♦♥st❛♥t❡s ❡ ❞❡♣❡♥❞❡♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ❞♦s ♣❛râ♠❡tr♦s ❞♦ s✐st❡♠❛ ❝♦♥s✐❞❡r❛❞♦✳ ●❡r❛❧♠❡♥t❡ n >= m✱

s❡♥❞♦ n ❛ ♦r❞❡♠ ❞❛ ❡q✉❛çã♦ q✉❡ t❛♠❜é♠ r❡♣r❡s❡♥t❛ ♦ s❡✉ s✐st❡♠❛✳ ❈♦♥s✐❞❡r❛♥❞♦ ♦s

✈❛❧♦r❡s ✐♥✐❝✐❛✐s ♥✉❧♦s✱ t❡♠♦s✱ ❝♦♠♦ r❡s✉❧t❛❞♦✿

(qnsn+qn−1s

n−₁

+...+q2s2+q1s+q0)Y(s) =

(pmsm+pm−1s

m−₁

+...+p2s2+p1s+p0)U(s)

✭✶✵✮

❉❡✜♥✐♥❞♦

Q(s) = qnsn+qn−₁s

n−₁

+...+q2s2+q1s+q0 ✭✶✶✮

❡

P(s) =pmsm+pm−1s

m−1

+...+p2s2+p1s+p0 ✭✶✷✮

♣♦❞❡♠♦s ❡s❝r❡✈❡r✿

Q(s)Y(s) = P(s)U(s) ✭✶✸✮

✏❉❡♥♦♠✐♥❛✲s❡ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ G(s) ❞❡ ✉♠ s✐st❡♠❛ ❧✐♥❡❛r✱ ❞❡ ♣❛✲

râ♠❡tr♦s ❝♦♥❝❡♥tr❛❞♦s✱ ✐♥✈❛r✐❛♥t❡s ♥♦ t❡♠♣♦ ❡ ❞❡ ❡♥tr❛❞❛ ❡ s❛í❞❛ ú♥✐❝❛s✱ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛s tr❛♥s❢♦r♠❛❞❛s ❞❡ ▲❛♣❧❛❝❡ ❞❛ ✈❛r✐á✈❡❧ ❞❡ s❛í❞❛ ❡ ❞❛ ✈❛r✐á✈❡❧ ❞❡ ❡♥tr❛❞❛✱ s✉♣♦♥❞♦ ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ♥✉❧❛s✳✑ ✭▼❆❨❆✱ P✳❆✳ ❡t ❛❧✱ ✷✵✶✶✮

G(s) = P(s)

Q(s) ✭✶✹✮

✷✳✸✳✶ P♦❧♦

❯♠ ♣♦❧♦ é ❞❡✜♥✐❞♦ ❝♦♠♦ ✉♠❛ r❛✐③ ❞♦ ❞❡♥♦♠✐♥❛❞♦r ❞❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✱ ♦✉ s❡❥❛✱ é ✉♠❛ r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ ❝❛r❛❝t❡ríst✐❝♦✳ ■♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐♥â♠✐❝♦ ❞♦ s✐st❡♠❛ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦s ♣♦❧♦s✱ t❛✐s ❝♦♠♦✿

✶✳ ❆ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛❀

✷✳ ❖ ♣♦t❡♥❝✐❛❧ ♣❛r❛ tr❛♥s✐❡♥t❡s ♣❡r✐ó❞✐❝♦s❀

❉❡st❛ ❢♦r♠❛✱ ♥ã♦ ♣♦❞❡♠♦s ❞❡✐①❛r ❞❡ ♣❡r❝❡❜❡r q✉❡ ♦ ❝♦♥tr♦❧❡ ❞❡ r❡❛❧✐♠❡♥t❛çã♦ ❛❢❡t❛ ♦s ♣♦❧♦s✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ❝♦♥tr♦❧❡ ♣r♦♣♦r❝✐♦♥❛❧ ♠❛✐s ✐♥t❡❣r❛❧✳

✷✳✸✳✷ ❩❡r♦s

❯♠ ③❡r♦ é ✉♠❛ r❛✐③ ❞♦ ♥✉♠❡r❛❞♦r ❞❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✳ ❩❡r♦s ♥ã♦ ✐♥✢✉❡♥❝✐❛♠ ♦s ❡①♣♦❡♥t❡s✳

✷✳✸✳✸ ❖r❞❡♠ ❞❡ ◆✉♠❡r❛❞♦r ❡ ❉❡♥♦♠✐♥❛❞♦r

❖s s✐st❡♠❛s ❢ís✐❝♦s ❡stã♦ ❡♠ ❝♦♥❢♦r♠✐❞❛❞❡ ❝♦♠ ✉♠❛ ❧✐♠✐t❛çã♦ ❡s♣❡❝í✜❝❛ ❡♥tr❡ ❛s ♦r❞❡♥s ❞♦ ♥✉♠❡r❛❞♦r ❡ ❞♦ ❞❡♥♦♠✐♥❛❞♦r✱ ♦✉ s❡❥❛✱ ❛ ♦r❞❡♠ ❞♦ ❞❡♥♦♠✐♥❛❞♦r ❞❡✈❡ s❡r ♠❛✐♦r q✉❡ ❛ ♦r❞❡♠ ❞♦ ♥✉♠❡r❛❞♦r✱ ♦✉ ♠❡❧❤♦r ❞✐③❡♥❞♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣♦❧♦s s❡♠♣r❡ ❞❡✈❡ s❡r ♠❛✐♦r ❞♦ q✉❡ ♦ ❞❡ ③❡r♦s✱ ♣♦✐s ❛ ❢✉♥çã♦ F(s) t♦r♥❛✲s❡ ✐❧✐♠✐t❛❞❛ q✉❛♥❞♦ ♦ ⑤s⑤ t❡♥❞❡ ♣❛r❛ ✐♥✜♥✐t♦✳

✷✳✸✳✹ ●❛♥❤♦ ♥♦ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦

❖ ❣❛♥❤♦ ❞❡ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦ é ♦ ✈❛❧♦r ❞❛ ❞✐✈✐sã♦ ❞❛ ✈❛r✐❛çã♦ ❞❛ s❛í❞❛ ❝♦♠ ❛ ✈❛r✐✲ ❛çã♦ ❞❛ ❡♥tr❛❞❛ ♣❛r❛ t♦❞♦s ♦s s✐st❡♠❛s✳ ❖ ❣❛♥❤♦ ♥♦ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦ é ♥♦r♠❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛❞♦ ♣♦rK ❡ ♣♦❞❡♠♦s r❡♣r❡s❡♥tá✲❧♦ ♣♦r ♠❡✐♦ ❞❛ ❊q✉❛çã♦ ✶✺✳

K = δ❙❛í❞❛

δEntrada ✭✶✺✮

✷✳✹ ❉✐❛❣r❛♠❛ ❞❡ ❇❧♦❝♦s

➱ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❢♦r♠❛ ❞❡ ❜❧♦❝♦s✺_{❡♥tr❡ ❛s ✈❛r✐á✈❡✐s ❞♦ s✐st❡♠❛✳ P♦❞❡♠ s❡r ✉s❛✲} ❞♦s ♣❛r❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ s✐st❡♠❛s✱ s❡♥❞♦ ✉♠❛ ❢♦r♠❛ ❝♦♥✈❡♥✐❡♥t❡ ❞❡ r❡♣r❡s❡♥t❛r ❛s ❡q✉❛✲ çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞✐♥â♠✐❝♦ ❞❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❡ ♥ã♦ ❧✐♥❡❛r❡s✳ ◆♦s

✺_{❋♦r♠❛ ❞❡ ❛♣r❡s❡♥t❛çã♦ ✜❣✉r❛t✐✈❛✳}

❝❛s♦s ❡♠ q✉❡ ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ é ✐❣✉❛❧ ❛ ✶✱ ♦ ❜❧♦❝♦ ♣♦❞❡ s❡r ❞✐s♣❡♥s❛❞♦✳✭▼❆❨❆✱ P✳❆✳ ❡t ❛❧✱ ✷✵✶✶✮

✷✳✺ ▼ét♦❞♦s ●rá✜❝♦s ❞❡ ■❞❡♥t✐✜❝❛çã♦ ❞♦ Pr♦❝❡ss♦ ✭❈✉r✈❛ ❞❡ r❡✲

❛çã♦✮

❉❡ ❛❝♦r❞♦ ❝♦♠ ❑❛ts✉❤✐r♦✱ ✷✵✶✺✱ ♦ ♠ét♦❞♦ ❞❛ ❝✉r✈❛ ❞❡ r❡❛çã♦ é ✉♠ ❞♦s ♣r♦❝❡ss♦s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♠♦❞❡❧♦s ❞✐♥â♠✐❝♦s✱ ❢♦r♥❡❝❡♥❞♦ ❢✉♥çõ❡s ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❛❞❡q✉❛❞❛s ♣❛r❛ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s✳ P❛r❛ ❛ ❛♣❧✐❝❛çã♦ ❞♦s ♠ét♦❞♦s✱ ❞❡✈❡♠♦s ❡①❡❝✉t❛r ♦s s❡❣✉✐♥t❡s ♣r♦❝❡❞✐♠❡♥t♦s✿

✶✳ P❡r♠✐t❛ q✉❡ ♦ ♣r♦❝❡ss♦ ❛t✐♥❥❛ ♦ ❡st❛❞♦ ❡st❛❝✐♦♥ár✐♦✳

✷✳ ■♥tr♦❞✉③❛ ✉♠❛ ú♥✐❝❛ ♠✉❞❛♥ç❛ ❞❡ ❡t❛♣❛ ♥❛ ✈❛r✐á✈❡❧ ❞❡ ❡♥tr❛❞❛✳

✸✳ ❘❡❝♦❧❤❛ ❞❛❞♦s ❞❡ r❡s♣♦st❛ ❞❡ ❡♥tr❛❞❛ ❡ s❛í❞❛ ❛té ♦ ♣r♦❝❡ss♦ ✈♦❧t❛r ❛♦ ❡st❛❞♦ ❡st❛✲ ❝✐♦♥ár✐♦✳

✹✳ ❊①❡❝✉t❡ ♦s ❝á❧❝✉❧♦s ❞❛ ❝✉r✈❛ ❞❡ r❡❛çã♦ ❞♦ ♣r♦❝❡ss♦ ❣rá✜❝♦✳

❊ss❡ ♣r♦❝❡ss♦ ♥♦s ♣❡r♠✐t❡ ❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❝♦♠ ❛tr❛s♦✱ ♦✉ s❡❣✉♥❞❛ ♦r❞❡♠ s❡♠ ❛tr❛s♦ ❞♦ t✐♣♦✿

F(s) = K·e

−θ·s

τ·s+ 1 ✭✶✻✮

✷✳✺✳✶ ❚✐♣♦s ❞❡ ▼❛❧❤❛s

❊♠ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡ ♣♦❞❡✲s❡ ❝♦❧♦❝❛r ✉♠❛ ♠❛❧❤❛ ❞❡ ❝♦♥tr♦❧❡ ❡♠ ❞✉❛s ❝♦♥❞✐çõ❡s✿ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ ✭✜❣✉r❛ ✸✮ ❡ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ✭✜❣✉r❛ ✹✮✳ ❊♠ s✐st❡♠❛s ❞❡ ♠❛❧❤❛ ❛❜❡rt❛ ♥ã♦ ❤á ❛t✉❛çã♦ ❞♦ ❝♦♠♣❡♥s❛❞♦r✻_{✱ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ ❞❡ s❛í❞❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s} ❞✐③❡r q✉❡ ♥ã♦ ❡①✐st❡ ✉♠❛ r❡tr♦❛❧✐♠❡♥t❛çã♦✳ ❏á ♦ s✐st❡♠❛ ❞❡ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ❝♦♠♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛ ✭P❱✼_{✮ ❝♦♠ ❛ ✈❛r✐á✈❡❧ ❞❡ r❡❢❡rê♥❝✐❛ ✭❙P}✽_{✮ ❡ ❣❡r❛ ✉♠ ✈❛❧♦r ❞❡ ❡rr♦} ✭❊q✉❛çã♦ ✭✶✼✮✮✱ q✉❡ s❡rá ✉s❛❞♦ ♣❛r❛ ❝♦♠♣❡♥s❛r ❛ s❛í❞❛✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮

✻_{❈♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥tr♦❧❛❞♦r❡s P■❉✬s✱ ❋✉③③② ❞❡♥tr❡ ♦✉tr♦s✳}

✼_{❱❛r✐á✈❡❧ q✉❡ ❞❡s❡❥❛♠♦s ❝♦♥tr♦❧❛r✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✱ t❡♠♣❡r❛t✉r❛✱ ♥í✈❡❧✱ ♣r❡ssã♦✱ ✈❛③ã♦ ❡ ♦✉tr❛s✳} ✽_{❘❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r q✉❡ ❞❡s❡❥❛♠♦s ♣❛r❛ ❛ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛ P❱✳}

e(t) =sp−pv ✭✶✼✮

❋✐❣✉r❛ ✸✿ ▼❛❧❤❛ ❛❜❡rt❛

❋✐❣✉r❛ ✹✿ ▼❛❧❤❛ ❢❡❝❤❛❞❛

✷✳✺✳✷ ❩❡❛❣❧❡ ◆✐❝❤♦❧s ✕ Pr✐♠❡✐r❛ ❖r❞❡♠

❆ ♣r✐♠❡✐r❛ té❝♥✐❝❛ ❛❞❛♣t❛t✐✈❛ é ♦ ♠ét♦❞♦ ❞❡ ❩❡❣❧❡ ◆✐❝❤♦❧s✭✶✾✹✷✮✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❛♣❧✐❝❛r ✉♠❛ r❡t❛ t❛♥❣❡♥t❡ à ❝✉r✈❛ ❡♠ ✉♠ ♣♦♥t♦ ❞❡ ✐♥✢❡①ã♦ ✭✜❣✉r❛ ✺✮✱ ♣❛r❛ s❡ ❞❡t❡r♠✐♥❛r ❛s ❝♦♥st❛♥t❡s ❞❡ ❣❛♥❤♦✱ t❡♠♣♦ ❡ ❛tr❛s♦✳✭▼❛r❧✐♥✱ ❚✳ ❊✳✱✷✵✶✹✮

❋✐❣✉r❛ ✺✿ ▼ét♦❞♦ ❞❡ ❩❡❛❣❧❡ ◆✐❝❤♦❧s✴❑✿●❛♥❤♦❀▲✿❆tr❛s♦ ✭θ✮❀❚✿❈♦♥st❛♥t❡ ❞❡ t❡♠♣♦ ✭τ✮❀

❆tr❛✈és ❞❛ ❞✐✈✐sã♦ ❞❛ ✈❛r✐❛çã♦ ❞❡ s❛í❞❛ ✭P❱✮ ❡ ❡♥tr❛❞❛ ✭▼❱✮ ❝♦♥s❡❣✉✐♠♦s ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞♦ ❣❛♥❤♦ ❞♦ s✐st❡♠❛ ❛ s❡r ❛♥❛❧✐s❛❞♦✱ q✉❡ ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛ ❊q✉❛çã♦ ✭✶✽✮✳

Y(t) = K·u(t)·[1−e−(1−θ)·t_]

Y(t)′

=K·u(t)·e

−(1−θ)·t

τ

K = δY(t)

δu(t)

✭✶✽✮

✷✳✺✳✸ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✕ Pr✐♠❡✐r❛ ❖r❞❡♠

❖ ❉r✳ ❈❡❝✐❧ ❙♠✐t❤ ✭✶✾✼✷✮ ♣r♦♣ôs q✉❡ ♦s ✈❛❧♦r❡s ❞❡ θ ❡ ❞❡ τ s❡❥❛♠ s❡❧❡❝✐♦♥❛❞♦s ❞❡

t❛❧ ♠♦❞♦ q✉❡ ♦ ♠♦❞❡❧♦ ❡ ❛s r❡s♣♦st❛s r❡❛✐s ❝♦✐♥❝✐❞❛♠ ❡♠ ❞♦✐s ♣♦♥t♦s q✉❡ ❛♣r❡s❡♥t❛♠ ❡❧❡✈❛❞❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦✳ ❖s ✈❛❧♦r❡s ✐♥t❡r♠❡❞✐ár✐♦s ❞❡t❡r♠✐♥❛❞♦s ❛ ♣❛rt✐r ❞♦ ❣rá✜❝♦ sã♦ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ✈❛❧♦r ❛♣❧✐❝❛❞♦ ♥❛ ❡♥tr❛❞❛ ❡ ❛ ♠❛❣♥✐t✉❞❡ ❞♦ ❡st❛❞♦ ❞❡ ♠✉❞❛♥ç❛ ♥❛ s❛í❞❛✱ s❡♥❞♦ ❛❞♦t❛❞♦s ❞♦✐s t❡♠♣♦s✱ ❡♠ q✉❡ ♦ ♣r✐♠❡✐r♦ é q✉❛♥❞♦ ❛ s❛í❞❛ ❛t✐♥❣❡ ✷✽✪✱ ❡ ♦ s❡❣✉♥❞♦✱ ✻✸✪ ❞♦ ✈❛❧♦r ✜♥❛❧ ❞❛ s❛í❞❛ ✭❋✐❣✉r❛ ✻✮✳✭▼❛r❧✐♥✱ ❚✳ ❊✳✱✷✵✶✹✮

❋✐❣✉r❛ ✻✿ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✕ Pr✐♠❡✐r❛ ❖r❞❡♠

K = δY(t)

δu(t)

Y(θ+τ) = u(t)·(1−e−₁

) = 0.623·u(t)

Y ·(θ+ τ

3) = u(t)·(1−e

−1

3) = 0.283·u(t)

t2 =θ+τ

t1 =θ+ τ

3

τ = 3

2 ·(t2−t1)

θ =t2−τ

✭✶✾✮

✷✳✺✳✹ ▼ét♦❞♦ ❞❡ ❙✉♥❞❛r❡s❛♥ ❡ ❑r✐s❤♥❛s✇❛♠② ✲ Pr✐♠❡✐r❛ ❖r❞❡♠

❊st❡ ♠ét♦❞♦ t❛♠❜é♠ ❡✈✐t❛ ❛ ✉t✐❧✐③❛çã♦ ❞♦ ♣♦♥t♦ ❞❡ ✐♥✢❡①ã♦ ♣❛r❛ ❡st✐♠❛r ❛ ❝♦♥st❛♥t❡ ❞❡ t❡♠♣♦ τ ❡ ❞❡ ❛tr❛s♦ ❞❡ tr❛♥s♣♦rt❡θ✳ ❊❧❡s ♣r♦♣✉s❡r❛♠ q✉❡ ❞♦✐s t❡♠♣♦s✱ t1 ❡t2 ✭✜❣✉r❛

✼✮✱ s❡❥❛♠ ❡st✐♠❛❞♦s ❛ ♣❛rt✐r ❞❛ ❝✉r✈❛ ❞❡ r❡s♣♦st❛ ❛ ✉♠ ❞❡❣r❛✉✱ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦s ✸✺✳✸✪ ❡ ✽✺✳✸✪ ❞❛ r❡s♣♦st❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳✭▼❛r❧✐♥✱ ❚✳ ❊✳✱✷✵✶✹✮

❋✐❣✉r❛ ✼✿ ▼ét♦❞♦ ❞❡ ❙✉♥❞❛r❡s❛♥ ❡ ❑r✐s❤♥❛s✇❛♠② ✲ Pr✐♠❡✐r❛ ❖r❞❡♠

K = δY(t)

δu(t)

t2−θ= 1.91723·τ

t1−θ= 0.43541·τ

τ = 0.67233.(t2−t1)

θ = 1.29273·t1−0.292737·t2

✭✷✵✮

✷✳✺✳✺ ▼ét♦❞♦ ❞❡ ◆✐s❤✐❦❛✇❛ ✲ Pr✐♠❡✐r❛ ❖r❞❡♠

❈♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❛s ❝♦♥st❛♥t❡s ✉s❛♥❞♦ ♦ ❝á❧❝✉❧♦ ❞❛s ár❡❛s ✭✜❣✉r❛ ✽✮✳✭◆■❙❍■❑❆❲❆✱ ❍✳✱✷✵✵✼✮✳

❋✐❣✉r❛ ✽✿ ▼ét♦❞♦ ❞❡ ◆✐s❤✐❦❛✇❛

Kp =

δy(t)

δu(t)′

A0 =

Z ∞

0 e−1

τ·t·dt

t0 = A0 δy(t)

A1 =

Z t0 0

(1−e−1

τ·t)·dt

τ = A1

0.367·δy(t)

θ =t0−τ

✭✷✶✮

✷✳✺✳✻ ■♥t❡❣r❛çã♦ ♣♦r ➪r❡❛ ✲ ▼ét♦❞♦ ❚r❛♣❡③♦✐❞❛❧

➱ ✉♠ ♣r♦❝❡ss♦ ♥✉♠ér✐❝♦ ❞❡ ✐♥t❡❣r❛çã♦ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦✱ ❞❡✜♥✐r ♦ ✈❛❧♦r ❞❛ ✐♥t❡❣r❛❧ ❞❡♥tr♦ ❞❡ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡t❡r♠✐♥❛❞♦✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ár❡❛ s♦❜ ❛ ❝✉r✈❛ q✉❡ ❞❡✜♥❡ ❛ ❢✉♥çã♦ ❢✭t✮ ♥♦ ✐♥t❡r✈❛❧♦tn ❛ t(n+1)✳✭❆❘❊◆❆▲❊❙✱ ❙✳ ❡t ❛❧✱ ✷✵✵✽✮

❖ ♠ét♦❞♦ tr❛♣❡③♦✐❞❛❧ é ❝♦♥s✐❞❡r❛❞♦ ✉♠❛ ✐♥t❡r♣♦❧❛çã♦ ❧✐♥❡❛r✱ ♥❛ q✉❛❧ ❛ ❢✉♥çã♦ f(t)

é r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ p(t) ✭❡q✉❛çã♦ ✷✷✮✳ ❉❡st❛ ❢♦r♠❛✱ ❛ ár❡❛ s♦❜ ❛ r❡t❛

r❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❞❛ ✐♥t❡❣r❛❧✳

A=

Z t(n+1)

tn

f(t)·dt = ∆t

2 ·(f(t(n+1)) +f(t)) ✭✷✷✮

❡♠ q✉❡ ∆t ✿ ■♥t❡r✈❛❧♦ ❡♥tr❡ ♦s ❞♦✐s t❡♠♣♦s ❛♠♦str❛❞♦s✳

✷✳✺✳✼ ▼ét♦❞♦ ❍❛rr✐♦t ✲ ❙❡❣✉♥❞❛ ❖r❞❡♠ ❙♦❜r❡ ❆♠♦rt❡❝✐❞♦

❍❛rr✐♦t ♣❧♦t♦✉ ❛ r❡s♣♦st❛ ♥❛ ❢♦r♠❛ ❢r❛❝✐♦♥ár✐❛ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✭s❡♠ t❡♠♣♦ ♠♦rt♦✮✳ ❊♥❝♦♥tr♦✉ q✉❡ t♦❞❛s ❛s ❝✉r✈❛s s❡ ✐♥t❡rs❡❝t❛♠ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ ✼✸✪ ❞♦ ✈❛❧♦r ✜♥❛❧ ❞♦ ❡st❛❞♦ ❡stá✈❡❧✱ ❡♠ q✉❡ t

τ1+τ2 é ✐❣✉❛❧ ❛ ✶✳✸✳✭▼❛r❧✐♥✱ ❚✳ ❊✳✱✷✵✶✹✮

Kp =

Y(t)′ u(t)′

τtot =

T23%

1.3

τ1 =τrat·τtot

τ2 =τtot−τ1

✭✷✸✮

G(s) = Kp

(τ1·s+ 1)(τ2·s+ 1) ✭✷✹✮

✷✳✺✳✽ ▼ét♦❞♦ ❞❡ ❙♠✐t❤ ✕ ❙❡❣✉♥❞❛ ❖r❞❡♠

❖ ♠ét♦❞♦ ❞❡ ❙♠✐t❤ ❝♦♥s✐st❡ ♥❛ ♦❜t❡♥çã♦ ❞❡ ❞♦✐s ♦r✐✉♥❞♦s ❞❛ r❡s♣♦st❛ ❛♦ ❞❡❣r❛✉ ✉♥✐✲ tár✐♦ ✭▼❛r❧✐♥✱ ❚✳ ❊✳✱✷✵✶✹✮✳ ❆ss✉♠❡ q✉❡ ♦ ♠♦❞❡❧♦ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ♣♦❞❡ s❡r ❞❛❞♦ ♣♦r✿

G(s) = K·e

−θ·s

τ2_·s2_{+ 2·ζ·τ.s+ 1} ✭✷✺✮

❚♦♠❛♠♦s ♦s s❡❣✉✐♥t❡s ♣r♦❝❡❞✐♠❡♥t♦s ♣❛r❛ ♦ ❝á❧❝✉❧♦✿

✶✳ ❉❡t❡r♠✐♥❛rt20✭✷✵✪ ❞♦ ✈❛❧♦r ✜♥❛❧✮ ❡t60✭✻✵✪ ❞♦ ✈❛❧♦r ✜♥❛❧✮✱ ❞❛ r❡s♣♦st❛ ❛♦ ❞❡❣r❛✉✳

✷✳ ❉❡t❡r♠✐♥❛r ζ ❡ t60

τ ✱ ✉s❛♥❞♦ ❛ ✜❣✉r❛ ✾✳

✸✳ ❉❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ τ✱ ❞❛❞♦ q✉❡t60 é ❝♦♥❤❡❝✐❞♦✳

✹✳ ❈❛❧❝✉❧❛r ♦ ✈❛❧♦r ❞♦ ❣❛♥❤♦ ❞❛ ♣❧❛♥t❛ ✉s❛♥❞♦ ❛ ❊q✉❛çã♦ ✭✷✻✮✳

Kp =

Y(t)′

u(t)′ ✭✷✻✮

❋✐❣✉r❛ ✾✿ ●rá✜❝♦ ❞❡ ❙♠✐t❤

✷✳✻ ❙✐st❡♠❛ ❞❡ ❈♦♥tr♦❧❡

P❛r❛ ❞✐s❝♦rr❡r s♦❜r❡ ♦ q✉❡ é ✉♠ ❝♦♥tr♦❧❡ ❞❡ ♣r♦❝❡ss♦✱ ✐♥✐❝✐❛❧♠❡♥t❡ ✈❛♠♦s ❛♣r❡s❡♥t❛r ✉♠ ❡①❡♠♣❧♦✱ q✉❡ é ♦ ❝♦♥tr♦❧❡ ❞❡ t❡♠♣❡r❛t✉r❛✱ ♦♥❞❡ t❡♠♦s ❝♦♠♦ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛ ✭P❱✮ ❛ t❡♠♣❡r❛t✉r❛ ❛❢❡r✐❞❛ ♣♦r ✉♠ P❚✲✶✵✵✾ _{❛ ♥♦ss❛ ✈❛r✐á✈❡❧ ♠❛♥✐♣✉❧❛❞❛ ✭▼❱✮ é ❛} ❝♦rr❡♥t❡ ❡❧étr✐❝❛ q✉❡ r❡❛❧✐③❛ ♦ ❛q✉❡❝✐♠❡♥t♦ ❞❛ á❣✉❛ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ r❡s✐stê♥❝✐❛✱ ❝❛s♦ ❛ ♠❡s♠❛ ❡st❡❥❛ ❞✐❢❡r❡♥t❡ ❞❛ ✈❛r✐á✈❡❧ ❞❡s❡❥❛❞❛ ✭❙P✮✳ P♦rt❛♥t♦✱ ❞❛❞❛s ❛s ✈❡③❡s q✉❡ ❛ P❱ ✜❝❛r ❞✐✈❡r❣❡♥t❡ ❞♦ ✈❛❧♦r ❞❡s❡❥❛❞♦✱ ♦ ❝♦♥tr♦❧❛❞♦r ✐rá ❛t✉❛r ♥❛ ✈❛r✐á✈❡❧ ♠❛♥✐♣✉❧❛❞❛✳ ❉❡st❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ é ✉♠❛ ✐♥t❡r❝♦♥❡①ã♦ ❞❡ ❝♦♠♣♦♥❡♥t❡s ❢♦r♠❛♥❞♦ ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ❞❡ s✐st❡♠❛ q✉❡ ♣r♦❞✉③✐rá ✉♠❛ r❡s♣♦st❛ ❞❡s❡❥❛❞❛✳ ❖ ♣r♦❝❡ss♦ ♣♦❞❡ ❡st❛r ❡♠ ❞♦✐s t✐♣♦s ❞❡ ♠❛❧❤❛s ❞❡ ❝♦♥tr♦❧❡✿ ❛❜❡rt❛ ❡ ❢❡❝❤❛❞❛✳ ❯♠ s✐st❡♠❛ ❡♠ ♠❛❧❤❛ ❛❜❡rt❛ é ✉♠❛ ❝♦♥✜❣✉r❛çã♦ ♦♥❞❡ ♥ã♦ ❤á ❝♦♠♣❛r❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛ ❝♦♠ ❛ ❞❡s❡❥❛❞❛✳ ❙❡♥❞♦ ❛ss✐♠ ❡ss❡ t✐♣♦ ❞❡ ♣❧❛♥t❛ ♥ã♦ ❡stá ♣r♦t❡❣✐❞❛ ❝♦♥tr❛ ♣❡rt✉r❜❛çõ❡s ❡①t❡r♥❛s✱ t❛✐s ❝♦♠♦ t❡♠♣❡r❛t✉r❛ ❡ ❛❧t❡r❛çõ❡s ♥❛ ✈❛r✐á✈❡❧ ♠❛♥✐♣✉❧❛❞❛✳ ❆♦ ❝♦♥trár✐♦ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ♠❛❧❤❛ ❛❜❡rt❛✱ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♥tr♦❧❡ ❞❡ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ✉t✐❧✐③❛ ✉♠❛ ♠❡❞✐❞❛ ❛❞✐❝✐♦♥❛❧ ♣❛r❛ ❝♦♠♣❛r❛r ❛ s❛í❞❛ r❡❛❧ ❝♦♠ ❛ r❡s♣♦st❛ ❞❡s❡❥❛❞❛✳ ❊ss❛ ♠❡❞✐❞❛ é ❝❤❛♠❛❞❛ ❞❡ s✐♥❛❧ ❞❡ r❡tr♦ ❛❧✐♠❡♥t❛çã♦✱ s❡♥❞♦ q✉❡ ❡ss❡ t✐♣♦ ❞❡ ❝♦♥❝❡✐t♦ t❡♠ s✐❞♦ ✉s❛❞♦ ❝♦♠♦ ❛❧✐❝❡r❝❡ ♣❛r❛ ❛♥á❧✐s❡ ❡ ♣r♦❥❡t♦ ❞❡ s✐st❡♠❛s ❞❡ ❝♦♥tr♦❧❡✳✭❑❆❚❙❯❍■❘❖✱ ❖✳ ❡t ❛❧✱ ✷✵✶✺✮

✾_{➱ ✉♠ t✐♣♦ ❞❡ t❡r♠♦rr❡s✐stê♥❝✐❛ q✉❡ ♠❡❞❡ ❛ t❡♠♣❡r❛t✉r❛ ♣❡❧❛ ❝♦rr❡❧❛çã♦ ❞❛ s✉❛ r❡s✐stê♥❝✐❛ ❡❧étr✐❝❛}

❝♦♠ ❛ t❡♠♣❡r❛t✉r❛✳ ❆ ♠❛✐♦r✐❛ ❞❡st❡s s❡♥s♦r❡s é ❢❡✐t❛ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❡s♣✐r❛❧ ❞❡ ✜♦ ✜♥♦ ♠♦♥t❛❞❛ ♥✉♠ s✉♣♦rt❡ ❝❡râ♠✐❝♦ ♦✉ ❞❡ ✈✐❞r♦✳

✷✳✻✳✶ ❈♦♥tr♦❧❛❞♦r❡s P■❉

❆ ❝❧❛ss✐✜❝❛çã♦ ❞♦s ❝♦♥tr♦❧❛❞♦r❡s P■❉ é r❡❛❧✐③❛❞❛ ❛ ♣❛rt✐r ❞❛ ❛çã♦ ❞❡ ❝♦♥tr♦❧❡ q✉❡ ♦ ♠❡s♠♦ ❡①❡r❝❡✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ é ♦ t✐♣♦ ❞❡ ♦♣❡r❛çã♦ ♠❛t❡♠át✐❝❛ q✉❡ é r❡❛❧✐③❛❞❛ s♦❜r❡ ♦ ❡rr♦ ♣❛r❛ s❡ ❞❡t❡r♠✐♥❛r ♦ s✐♥❛❧ ❞❡ s❛í❞❛✳ P♦❞❡♠♦s ❝❧❛ss✐✜❝á✲❧♦s ❡♠ três ❛çõ❡s q✉❡ sã♦✿ Pr♦♣♦r❝✐♦♥❛❧✭Kp✮✱ ■♥t❡❣r❛❧✭Ki

s ✮ ❡ ❉❡r✐✈❛t✐✈❛✭Kd·s✮ ✭❊q✉❛çã♦ ✭✷✼✮✮✳ ▼✉✐t♦s ♣r♦❝❡ss♦s ✐♥❞✉str✐❛✐s sã♦ ❝♦♥tr♦❧❛❞♦s ❛ ♣❛rt✐r ❞❡ ❝♦♥tr♦❧❛❞♦r❡s P■❉✱ ❙✉❛ ♣♦♣✉❧❛r✐❞❛❞❡ é ❛tr✐❜✉í❞❛ ♣❛r❝✐❛❧♠❡♥t❡ ❛♦ s❡✉ ❜♦♠ ❞❡s❡♠♣❡♥❤♦ ❡♠ ✉♠❛ ❛♠♣❧❛ ❢❛✐①❛ ❞❡ ❝♦♥❞✐çõ❡s ❞❡ ♦♣❡r❛çã♦ ❡ à s✉❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦✳

G(s) =Kp+

Ki

s +Kd·s ✭✷✼✮

✷✳✻✳✷ ❆çã♦ Pr♦♣♦r❝✐♦♥❛❧

❖ ❝♦♥tr♦❧❛❞♦r ♣r♦♣♦r❝✐♦♥❛❧ ✭P✮ é ♦ ♠❛✐s s✐♠♣❧❡s ❞♦s ❝♦♥tr♦❧❛❞♦r❡s✱ ♣♦✐s s✉❛ ❛çã♦ é ❛ ❞❡ ♠✉❧t✐♣❧✐❝❛r ♦ ✈❛❧♦r ❞♦ ❡rr♦ ♣♦r ✉♠ ❣❛♥❤♦ ♣r♦♣♦r❝✐♦♥❛❧✭❊q✉❛çã♦ ✭✷✽✮✮✳ ❙✉❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ é ❞❛❞❛ ♣♦r ✿

G(s)p =Kp·e(t) ✭✷✽✮

❡♠ q✉❡✿

Kp✿●❛♥❤♦ Pr♦♣♦r❝✐♦♥❛❧ e(t)✿❊rr♦

❖ ❝♦♥tr♦❧❛❞♦r ♣r♦♣♦r❝✐♦♥❛❧ ❡♠ ♠❛❧❤❛ ❢❡❝❤❛❞❛ ❛♣r❡s❡♥t❛ ♦ ❡rr♦ ❞❡ r❡❣✐♠❡ ♣❡r♠❛♥❡♥t❡✱ ♣❛r❛ ✉♠ s✐st❡♠❛ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ❛♣r❡s❡♥t❛❞♦ ❛ s❡❣✉✐r✿

Y(s)

R(s) =

Kp·K·e−θ·s

τ·s+ 1 ✭✷✾✮

❋❡❝❤❛♥❞♦ ❛ ♠❛❧❤❛✱ t❡♠♦s✿

Y(s)

R(s) =

Kp·K·e−θ·s

τ·s+ 1 +Kp ·K ·e−θ·s

✭✸✵✮ ❆♣❧✐❝❛♥❞♦ ♦ t❡♦r❡♠❛ ❞♦ ✈❛❧♦r ✜♥❛❧ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❞❡ ❡st❛❜✐❧✐③❛çã♦ ❞❛ ✈❛r✐á✈❡❧ ❝♦♥tr♦❧❛❞❛✱ t❡♠♦s✿

Lims→₀s·

1

s ·

Kp·K·e−θ·s

τ ·s+ 1 +Kp·K·e−θ·s

= Kp·K

1 +Kp·K ✭✸✶✮ ❈♦♥❝❧✉í♠♦s q✉❡✿

✶✳ ❆ ♦r❞❡♠ ❞♦ s✐st❡♠❛ ♥ã♦ ♠✉❞❛❀

✷✳ ❙❡ ♦ ❣❛♥❤♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛✉♠❡♥t❛✱ ❛ ❢r❡q✉ê♥❝✐❛ ❞❡ ♦s❝✐❧❛çã♦ ❛✉♠❡♥t❛ ❡ ♦ ❢❛t♦r ❞❡ ❛♠♦rt❡❝✐♠❡♥t♦ ❞✐♠✐♥✉✐✳

✷✳✻✳✸ ❆çã♦ ■♥t❡❣r❛❧

❆ ❛çã♦ ✐♥t❡❣r❛❧ ♣r♦❞✉③ ✉♠ s✐♥❛❧ ❞❡ s❛í❞❛ q✉❡ é ♣r♦♣♦r❝✐♦♥❛❧ à ♠❛❣♥✐t✉❞❡ ❡ à ❞✉r❛çã♦ ❞♦ ❡rr♦✱ ♦✉ s❡❥❛✱ ❛♦ ❡rr♦ ❛❝✉♠✉❧❛❞♦✳ ■ss♦ ❢♦r♥❡❝❡ ✉♠❛ ❛❧t❡r♥❛t✐✈❛ ♣❛r❛ ❝♦rr✐❣✐r ♦ ❡rr♦ ❞❡

off-set ❣❡r❛❞♦ ♣❡❧❛ ❛çã♦ ✐♥t❡❣r❛❧ ❡ ❛❝❡❧❡r❛ ❛ r❡s♣♦st❛ ❞♦ s✐st❡♠❛✱ ♣❡r♠✐t✐♥❞♦ ❝❤❡❣❛r ❛♦

✈❛❧♦r ❞❡ r❡❢❡rê♥❝✐❛✳

✷✳✻✳✹ ❆çã♦ ❉❡r✐✈❛t✐✈❛

❆ ❛çã♦ ❞❡r✐✈❛t✐✈❛ ❢♦r♥❡❝❡ ✉♠❛ ❝♦rr❡çã♦ ❛♥t❡❝✐♣❛❞❛ ❞♦ ❡rr♦✱ ❞✐♠✐♥✉✐♥❞♦ ♦ t❡♠♣♦ ❞❡ r❡s♣♦st❛ ❡ ♠❡❧❤♦r❛♥❞♦ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ s✐st❡♠❛✳ ❈♦♥t✉❞♦ ♥ã♦ é ❛❝♦♥s❡❧❤á✈❡❧ ✉sá✲❧❛ ❡♠ s✐st❡♠❛s ❝♦♠ ♠✉✐t❛ ✈✐❜r❛çã♦✳

✷✳✼ ▼ét♦❞♦s ❞❡ s✐♥t♦♥✐❛

❙ã♦ ♠ét♦❞♦s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦s ♣❛râ♠❡tr♦s ❞❡ ❣❛♥❤♦✿❞❡ ✐♥t❡❣r❛çã♦✱ ❞❡r✐✈❛çã♦ ❡ ♣r♦♣♦rçã♦✳

✷✳✼✳✶ ❙í♥t❡s❡ ❉✐r❡t❛

❊st❡ ♠ét♦❞♦ ❜✉s❝❛ ❞❡✜♥✐r ❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛ ❞❡ ♠❛❧❤❛ ❛❜❡rt❛ ♣❛r❛ ✐♠♣♦r ❛ r❡s♣♦st❛ ❞❡ ♠❛❧❤❛ ❢❡❝❤❛❞❛✳ ❙❡♥❞♦ ❛ss✐♠✱ é ♥❡❝❡ssár✐♦ ✈❡r✐✜❝❛r s❡ ♦ ❝♦♥tr♦❧❛❞♦r r❡s✉❧t❛♥t❡ é r❡❛❧✐③á✈❡❧✱ ♦✉ s❡❥❛✱ s❡ ♥ã♦ ♣♦ss✉✐ ✉♠ t❡♠♣♦ ♠♦rt♦ ♣♦s✐t✐✈♦ ♦✉ t❡♠♦s ❞❡ ❞✐❢❡r❡♥❝✐❛çã♦ ♣✉r❛ ❝♦♠ ♠❛✐s ③❡r♦s q✉❡ ♣♦❧♦s ♥❛ ❢✉♥çã♦ ❞❡ tr❛♥s❢❡rê♥❝✐❛✳

❉❡♠♦♥str❛çã♦✿

❉❛❞❛ ❛ ❢✉♥çã♦ ❡♠ ❢r❡q✉ê♥❝✐❛✿

F(s) = Kp·e

−θ·s

τ·s+ 1 ✭✸✷✮

❆❞✐❝✐♦♥❛♥❞♦ ✉♠ ❝♦♥tr♦❧❛❞♦r ●✭s✮ ❡♠ sér✐❛ ❝♦♠ ❛ ❢✉♥çã♦ ❡ ❢❡❝❤❛♥❞♦ ✉♠❛ r❡❛❧✐♠❡♥t❛çã♦ ♥❡❣❛t✐✈❛✱ t❡♠♦s✿

G(s) = τ

(τd+θ)·Kp

+ 1

(τd+θ)·Kp.s ✭✸✸✮ ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❝♦♥str✉✐r ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿

❚❛❜❡❧❛ ✶✿ ❙í♥t❡s❡ ❞✐r❡t❛

▼♦❞✉❧❛r P❛râ♠❡tr♦s ❞♦ ❈♦♥tr♦❧❛❞♦r P■❉_K

p Ti Td

G(s) = K τ·s+1

τ

K·τc τ ✵

G(s) = K·e−θ·s

τ·_s+1

τ

K·_(τc+θ) τ ✵

✷✳✽ ❆❧❣♦r✐t♠♦ ●❡♥ét✐❝♦

❙ã♦ ❛❧❣♦r✐t♠♦s ❞❡ ♦t✐♠✐③❛çã♦ ♥✉♠ér✐❝❛✱ ✐♥s♣✐r❛❞♦s t❛♥t♦ ♥❛ s❡❧❡çã♦ ♥❛t✉r❛❧ q✉❛♥t♦ ♥❛ ❣❡♥ét✐❝❛ ♥❛t✉r❛❧✱ ♣♦❞❡♥❞♦ s❡r ❛♣❧✐❝❛❞♦ ❛ ✉♠❛ ❣❛♠❛ ❞❡ ♣r♦❜❧❡♠❛s✱ s❡♥❞♦ ✉s❛❞♦s ♣❛r❛ ❛❥✉❞❛r ❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ❞♦ ❞✐❛ ❛ ❞✐❛✳

❆ ✐❞❡✐❛ ❞❡ ✉s❛r ✉♠❛ ♣♦♣✉❧❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦ ❞❡ ❡♥❣❡♥❤❛r✐❛ ♣rát✐❝❛ ❢♦✐ ❝♦♥s✐❞❡r❛❞❛✱ ✈ár✐❛s ✈❡③❡s✱ ❞✉r❛♥t❡ ❛s ❞é❝❛❞❛s ❞❡ ✶✾✺✵ ❡ ✶✾✻✵✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ❆● ❢♦✐ ✐♥✈❡♥t❛❞♦ ♣♦r ❏♦❤♥ ❍♦❧❧❛♥❞✱ ❡♠ ✶✾✻✵✳ ❖s s❡✉s ♠♦t✐✈♦s ♣❛r❛ ❞❡s❡♥✲ ✈♦❧✈❡r t❛✐s ♦s ❛❧❣♦r✐t♠♦s ❢♦r❛♠ ♠✉✐t♦ ❛❧é♠ ❞♦ t✐♣♦ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♥✈❡♥❝✐♦✲ ♥❛✐s✳✭❈❖▲❊❨✱ ❉❛✈✐❞ ❆✳✱✶✾✾✾✮

❈♦♥st✐t✉✐çã♦ ❞♦s ❛❧❣♦r✐t♠♦s ❣❡♥ét✐❝♦s✿

✶✳ ❯♠ ♥ú♠❡r♦ ♦✉ ♣♦♣✉❧❛çã♦ ❞❡ s✉♣♦s✐çõ❡s ❞❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛❀

✷✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❝❛❧❝✉❧❛r ♦ q✉ã♦ sã♦ ❜♦♥s ♦✉ r✉✐♥s ❛s s♦❧✉çõ❡s ✐♥❞✐✈✐❞✉❛✐s ❞❡♥tr♦ ❞❛ ♣♦♣✉❧❛çã♦❀

✸✳ ❯♠ ♠ét♦❞♦ ♣❛r❛ ♠✐st✉r❛r ❢r❛❣♠❡♥t♦s ♣❛r❛ ❢♦r♠❛r ♠❡❧❤♦r❡s s♦❧✉çõ❡s❀

✹✳ ❯♠ ♦♣❡r❛❞♦r ❞❡ ♠✉t❛çã♦ ♣❛r❛ ❡✈✐t❛r ❛ ♣❡r❞❛ ♣❡r♠❛♥❡♥t❡ ❞❡ ❞✐✈❡rs✐❞❛❞❡ ❞❡♥tr♦

❞❛s s♦❧✉çõ❡s✳

✷✳✽✳✶ ❋✉♥❝✐♦♥❛♠❡♥t♦

❊♠ ✈❡③ ❞❡ ♣❛rt✐r ❞❡ ✉♠ ú♥✐❝♦ ♣♦♥t♦ ✭♦✉ ❛❞✐✈✐♥❤❛r✮ ❞❡♥tr♦ ❞♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛✱ ❆●✬s sã♦ ✐♥✐❝✐❛❧✐③❛❞♦s ❝♦♠ ♣♦♣✉❧❛çã♦ ❞❡ s✉♣♦s✐çõ❡s✳ ❊st❡s sã♦ ❣❡r❛❧♠❡♥t❡ ❛❧❡❛tór✐♦s ❡ ❡s♣❛✲ ❧❤❛❞♦s ♣♦r t♦❞♦ ♦ ❡s♣❛ç♦ ❞❡ ❜✉s❝❛✳ ❯♠ ❛❧❣♦r✐t♠♦ tí♣✐❝♦ ✉s❛ três ♦♣❡r❛❞♦r❡s ✲ s❡❧❡çã♦✱ r❡❝♦♠❜✐♥❛çã♦ ❡ ♠✉t❛çã♦ ✭❡s❝♦❧❤✐❞♦s ❡♠ ♣❛rt❡ ♣♦r ❛♥❛❧♦❣✐❛ ❝♦♠ ♦ ♠✉♥❞♦ ♥❛t✉r❛❧✮ ✲ ❛ ✜♠ ❞❡ ❞✐r❡❝✐♦♥❛r ❛ ♣♦♣✉❧❛çã♦✱ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ sér✐❡ ❞❡ ❡t❛♣❛s ❞❡ t❡♠♣♦ ♦✉ ❣❡r❛çõ❡s✱ ♣❛r❛ ❛ ❝♦♥✈❡r❣ê♥❝✐❛ ♥♦ ót✐♠♦ ❣❧♦❜❛❧✳ ◆♦r♠❛❧♠❡♥t❡✱ ❡ss❛s s✉♣♦s✐çõ❡s ✐♥✐❝✐❛✐s sã♦ ♠❛♥t✐❞❛s ❝♦♠♦ ❝♦❞✐✜❝❛çõ❡s ❜✐♥ár✐❛s ✭♦✉ str✐♥❣s✮ ❞❛s ✈❛r✐á✈❡✐s ✈❡r❞❛❞❡✐r❛s✱ ❡♠❜♦r❛ ✉♠ ♥ú♠❡r♦ ❝r❡s❝❡♥t❡ ❞❡ ❆●✬s ✉s❡ ✧✈❛❧♦r❡s r❡❛✐s✧✭♦✉ s❡❥❛✱ ❜❛s❡ ❞❡❝✐♠❛❧✮✱ ♦✉ ❝♦❞✐✜❝❛çõ❡s q✉❡ ❢♦r❛♠ ❡s❝♦❧❤✐❞❛s ♣❛r❛ ✐♠✐t❛r✱ ❞❡ ❛❧❣✉♠❛ ♠❛♥❡✐r❛✱ ❛ ❡str✉t✉r❛ ❞❡ ❞❛❞♦s ♥❛t✉r❛✐s ❞♦ ♣r♦❜❧❡♠❛✳✭❈❖▲❊❨✱ ❉❛✈✐❞ ❆✳✱✶✾✾✾✮

◗✉❛♥t♦ ❛♦s três ♣r✐♥❝✐♣❛✐s ♦♣❡r❛❞♦r❡s ❞❛ ♣♦♣✉❧❛çã♦✿

✶✳ ❆ s❡❧❡çã♦ t❡♥t❛ ♣r❡ss✐♦♥❛r ❛ ♣♦♣✉❧❛çã♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ s❡♠❡❧❤❛♥t❡ ❛♦ ❞❛ s❡❧❡çã♦ ♥❛t✉r❛❧ ❡♥❝♦♥tr❛❞❛ ♥♦s s✐st❡♠❛s ❜✐♦❧ó❣✐❝♦s✳

✷✳ ❘❡❝♦♠❜✐♥❛çã♦ ♣❡r♠✐t❡ q✉❡ ❛s s♦❧✉çõ❡s tr♦q✉❡♠ ✐♥❢♦r♠❛çõ❡s ❞❡ ❢♦r♠❛ s❡♠❡❧❤❛♥t❡ à q✉❡ é ✉s❛❞❛ ♣♦r ✉♠ ♦r❣❛♥✐s♠♦ ♥❛t✉r❛❧ s✉❜♠❡t✐❞♦ à r❡♣r♦❞✉çã♦ s❡①✉❛❧✳

✸✳ ▼✉t❛çã♦ é ✉s❛❞❛ ♣❛r❛ ♠✉❞❛r ❛❧❡❛t♦r✐❛♠❡♥t❡ ✭flip✮ ♦ ✈❛❧♦r ❞❡ ❜✐ts ✐♥❞✐✈✐❞✉❛✐s ❞❡♥tr♦

❞❡ str✐♥❣s ✐♥❞✐✈✐❞✉❛✐s✳ ❖ ✉s♦ tí♣✐❝♦ é ✉s❛❞♦ ❝♦♠ ♠✉✐t❛ ♠♦❞❡r❛çã♦✳

❆♣ós✱ ♦s ♦♣❡r❛❞♦r❡s ❞❡ s❡❧❡çã♦✱ r❡❝♦♠❜✐♥❛çã♦ ❡ ♠✉t❛çã♦ ❢♦r❛♠ ❛♣❧✐❝❛❞❛s à ♣♦♣✉❧❛çã♦ ✐♥✐❝✐❛❧✳ ❯♠❛ ♥♦✈❛ ♣♦♣✉❧❛çã♦ s❡rá ❢♦r♠❛❞❛ ❡ ❛❞✐❝✐♦♥❛❞♦ ♠❛✐s ✶ ♥♦ ❝♦♥t❛❞♦r ❞❛ ❣❡r❛çã♦✳ ❊st❡ ♣r♦❝❡ss♦ ❞❡ s❡❧❡çã♦✱crossover ❡ ♠✉t❛çã♦ é ❝♦♥t✐♥✉❛❞♦ ❛té ❝❡rt♦ ♥ú♠❡r♦ ❞❡ ❣❡r❛çõ❡s

t❡r❡♠ ❞❡❝♦rr✐❞♦ ♦✉ ❛❧❣✉♠❛ ❢♦r♠❛ ❞❡ ♦ ❝r✐tér✐♦ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ s❡r ❝✉♠♣r✐❞♦✳

✷✳✽✳✷ ▼✉t❛çã♦

◆♦ ♠✉♥❞♦ ♥❛t✉r❛❧✱ ✈ár✐♦s ♣r♦❝❡ss♦s ♣♦❞❡♠ ❝❛✉s❛r ♠✉t❛çã♦✳ ❆s t❛①❛s ❞❡ ❜❛❝tér✐❛s sã♦ ❛♣r♦①✐♠❛❞❛♠❡♥t❡2.10−₃ ♣♦r ❣❡♥♦♠❛ ♣♦r ❣❡r❛çã♦ ✭❋❯✾✵✱ ❇❆✾✻✱ ♣✶✾✳✮✳ P♦r ♠❡✐♦ ❞❡ ✉♠❛

r❡♣r❡s❡♥t❛çã♦ ❜✐♥ár✐❛✱ ❛ ♠✉t❛çã♦ é ♣❛rt✐❝✉❧❛r♠❡♥t❡ ❢á❝✐❧ ❞❡ ✐♠♣❧❡♠❡♥t❛r✳ ❆ ❝❛❞❛ ♥♦✈❛ ❣❡r❛çã♦ t♦❞❛ ❛ ♣♦♣✉❧❛çã♦ é ✈❛rr✐❞❛✱ ❝♦♠ ❝❛❞❛ ♣♦s✐çã♦ ❞❡ ❜✐t ❡♠ ❝❛❞❛ ❝♦r❞❛ ✈✐s✐t❛❞❛✱ ❡