aula de controle

FATEB – Engenharia Química com ênfase em processos de fabricação de celulose e papel

Controle e instrumentação de processos – Ivo Neitzel C EF P

M

Controle e Instrumentação de Processos

Engenharia Químicaênfase em processos de fabricação de celulose e papel

1a versão impressa não contém todo o material da versão FLASH

Ivo NeitzelFaculdade de Telêmaco Borba

CP09 M04 – P05 – v101 – build 11



M

Objetivos deste módulo:

Aplicações da Transformação de Laplace

Resposta de sistemas de controle em laço fechado & aberto


Resposta degrau – controle antecipativo (feedforward)

Resposta de freqüência



M




M

P2

perturbação

P2


M C EF P1

P2

variável controladavariável manipuladaação de controle

perturbação no equilíbrio = 0

desvio da perturbação

[file=TCP diagrama de blocos feedforward.vsd+wmf][ver=1.00][19.OUT.98]

-

P1



M

M C EF P1

P2

perturbação





-

( )2 11P M CFF V PY V G V G G G G= ⋅ + ⋅ ⋅ − ⋅ ⋅ ⋅

( )Y V G G G G G= ⋅ − ⋅ ⋅ ⋅


( )2 1P M CFF V PY V G G G G G= ⋅ − ⋅ ⋅ ⋅

2 1P M CFF V P

YG G G G G

V= − ⋅ ⋅ ⋅

EquivFF

YG

V=



M

M C EF P1

P2

perturbação





-

EquivFF

YG

V=

0Y para V= ∀

Comportamento idealEquivFFY G V= ⋅

0 G V= ⋅


2 10EquivFF P M CFF V PG G G G G G= − ⋅ ⋅ ⋅ =

V0Y para V= ∀0 EquivFFG V= ⋅

0EquivFFG =



M

M C EF P1

P2

perturbação





-

2 10P M CEquiv FF V PFF G GG G G G− ⋅ ⋅= ⋅ =

G G G G G⋅ ⋅ ⋅ =


1 2M CFF V P PG G G G G⋅ ⋅ ⋅ =

2

1

PCFF

M V P

GG

G G G=

⋅ ⋅

2

2

2

21

mt s

p

P

K eG

sτ

− ⋅⋅=

⋅ +1

1

1

11

mt s

p

P

K eG

sτ

− ⋅⋅=

⋅ +M mG K= V vG K=



M( )( )

2

1

2 1

1 2

1 e

1 e

m

m

t sp

CFF t s

p v m

K sG

K K K s

ττ

− ⋅

− ⋅

⋅ += × ×

⋅ ⋅ ⋅ +

( )2 12 1

1 2

1e

1

m mt t sp

CFF

p v m

K sG

K K K s

ττ

− − ⋅ ⋅ += × × ⋅ ⋅ ⋅ +


CPID ciG K

K

s= ⋅ 1DK s

+ ⋅ +

( )2 1

2

e1

1m m

C F

t t s

FD

cs

sKG

K

τ− − ⋅

= × × ⋅ +

⋅ +



M�( )2 1

" "

2

1e

1

m m

derivativoproporc

C

t t

ional

D s

FF cs

K sG K

τ− − ⋅

= × ×

++

��

( )2 12 11

e1

m mt t sp

CFF

K sG

K K K s

ττ

− − ⋅ += × × ⋅ ⋅ +


1 21p v mK K K sτ⋅ ⋅ +

2

1

p

cFF

p v m

KK

K K K⋅ ⋅ 1DFFK τ=

( )2 1

2

1e

1

m mDFF t t s

cFFCFFs

GK s

Kτ

− − ⋅ = × ×

⋅ +

⋅ +

Delay � tm2-tm1



M

O comportamento real de um sinal medido ...


Como descrever ?



M

Um sinal gerado para estudo

( ) ( )10

1

sink k

k

v t A tω=

= ⋅ ⋅∑

ω


Análise de Fourier

1ω

2ω

3ω 4

ω5ω

6ω

8ω

9ω

10ω

A4



M( ) ( )sink k

k

v t A tω= ⋅∑

( ) ( ) ( )sinv t A t dω ω ω= ⋅ ⋅ ⋅∫Sinal periódico

Sinal contínuo




M

Análise de freqüência


( ) ( )sinv t A tω= ⋅ ⋅ modelo ( )x t



M

( )( )

( )X s

G sV s

=

( )2 2

AV s

s

ωω⋅

=+

( ) ( )sinv t A tω= ⋅ ⋅ ( ) ( )2 2sinoidal

AX s G s

s

ωω⋅

= ⋅+

( )1

pKG s

sτ=

⋅ +

( )1

pKG s

sτ=

⋅ +


( ) ( )2 22 2

1 esin

11

t

senoidal px t A K tττ ω

ω φω τω τ

− ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ − + ++ ⋅

⋅

( ) ( )2 2

1sin

1senoidal px t A K tω φ

ω τ∞ = ⋅ ⋅ ⋅

+ ⋅⋅ −



M

( ) ( )sinv t A tω= ⋅ ⋅ P1 ( )x t



( ) ( )2 2

sin

1

pKx t A tω φ

ω τ

⋅

≡ ⋅ ⋅ ⋅ −

+



M



( ) ( )sinv t A tω= ⋅ ⋅ modelo ( )x t

( ) [ ] ( )sinx t A AR tω φ≡ ⋅ ⋅ ⋅ −



M

ℑ

β

z∈ℂ

z iα β= + ⋅z

i θ⋅= ⋅

2 2z α β= +

( )tgβ

θα

=


ℜ

α

β

θ

iz z e θ⋅= ⋅



M

1

1 1 1 1

iz i z e

θα β ⋅= + ⋅ = ⋅

2

2 2 2 2

iz i z e

θα β ⋅= + ⋅ = ⋅

( )1 2 1 2 1 2z z iα α β β+ = + + + ⋅


( )1 2

1 2 1 2

iz z z z e

θ θ⋅ +⋅ = ⋅ ⋅

1

1 1 1 1

iz i z e

θα β − ⋅= − ⋅ = ⋅

2

1 1 1z z z⋅ =



M

( ) [ ] ( )sinx t A AR tω φ≡ ⋅ ⋅ ⋅ −

( ) i tf t e dtω∞

− ⋅ ⋅

−∞

⋅ ⋅∫ ( )0

0

0i t i te dt f t e dtω ω

∞− ⋅ ⋅ − ⋅ ⋅

−∞

⋅ ⋅ + ⋅ ⋅∫ ∫T. Fourier


−∞ 0−∞

( )0

s tf t e dt

∞− ⋅⋅ ⋅∫T. Laplace

( ) ( ) ( ) iG s G s i i eG i θω α β ω ⋅→ = ⋅ ⇒ + ⋅ ⇒ ⋅⋅

( ) ( ) ( )sinGt A ix tω θω≡ ⋅ ⋅ ⋅⋅ +

G(s)



M

( ) ( )2 2sinoidal

AX s G s

s

ωω⋅

= ⋅+

( ) ( )2 2

1sin

1senoidal px t A K tω φ

ω τ∞ = ⋅ ⋅ ⋅

+ ⋅⋅ −

( )1

pKG s

sτ=

⋅ +

T. Laplace

Processo 1ª ordem


( ) ( )1 1

p ps i

i

K KG s G

sτ τω

ω== → =

⋅ + ⋅⋅ +⋅

( )11

1

1

p p iK KG i

i i iω

τ ω τ ωτ ωτ ω⋅

⋅ = = ×+ ⋅ ⋅

⋅+ −⋅ ⋅⋅ ⋅

−



M

( )( )

( ) ( )1

1

1

pKG

ii

i i

τ ω

τω ωω

τ

⋅⋅ =

+ ⋅ ⋅ ⋅ −

⋅ ⋅

⋅ ⋅

−

2

1 1 1z z z⋅ =

( )11

1

1

p p iK KG i

i i iω

τ ω τ ωτ ωτ ω⋅

⋅ = = ×+ ⋅ ⋅

⋅+ −⋅ ⋅⋅ ⋅

−


( )( )( )22

1

1pKG

ii ω

τ

τ ω

ω

⋅ ⋅−⋅⋅ =

+ ⋅

2 2z α β= +

1 1 1z z z⋅ =



M( )( )( )21

1

pi

iKG

τω

ω

τ ω

− ⋅⋅=

+ ⋅

⋅⋅

( )( ) ( )22

11

p pKG i

Ki

τ ω

τ

τ ωω

ω− ⋅ ⋅

+ ⋅⋅+

+⋅ = ×

( )G i iα βω⋅ = + ⋅


( )G i iα βω⋅ = + ⋅

( ) 22G i α βω⋅ = +

( )( )( ) ( )( )

2 2

22

22

22

1 1

p pG

KKi

τ

τω

ω

ω τ ω+= +

+⋅

⋅ ⋅

⋅⋅



M

( )( ) ( )2

2 2

2

1

pG i Kτ ω

ω⋅

⋅ × +=

( )( )( ) ( )( )

2 2

22

22

22

1 1

p pG

KKi

τ

τω

ω

ω τ ω+= +

+⋅

⋅ ⋅

⋅⋅


( )( )( ) ( )( )2

22

2

1 1

pG i K

τ ω τω

ω+⋅ × +

⋅⋅=

+

( )( )( )

2 2

22

1

1

pG i K

τ ω

τω

ω⋅ = ×

+ ⋅

⋅+



M

( )( )( )

2 2

22

1

1

pG i K

τ ω

τω

ω⋅ = ×

+ ⋅

⋅+

( )( )21

pKG i ω

τ ω⋅ =

+ ⋅


( )21 τ ω+ ⋅

( ) ( )2 2

1

1

sinpsenoidalx A Kt tωω τ

φ∞ ⋅+

⋅⋅

= ⋅ ⋅ −



M

( )tgβ

θα

=

( )( ) ( )22

11

p pKG i

Ki

τ ω

τ

τ ωω

ω− ⋅ ⋅

+ ⋅⋅+

+⋅ = ×

( ) ( )21

p

p

tK

g

Kτ ω

τ ωθ

− ⋅ ⋅

+ ⋅=


( )21

p

τ ω+ ⋅( )tg θ τ ω= − ⋅



M

( )( )21

pKG i ω

τ ω⋅ =

+ ⋅


( )tg θ τ ω= − ⋅



M

restart;

G := 1/(tau*s+1);

Giw := subs(s = I*w, G);

MAPLE


modG := evalc(abs(Giw));

theta := evalc(argument(Giw));

with(plots);

loglogplot(subs(tau = 1, modG), w = 0.01 .. 100, view = 00.1 .. 1);

semilogplot((180/Pi)*subs(tau = 1, theta), w = 0.01 .. 100, view = -90 .. 0);



M

( )2 2 22 1

pKG s

s sτ τ ξ=

⋅ + ⋅ ⋅ ⋅ +

Processo 2ª ordem


2 1s sτ τ ξ⋅ + ⋅ ⋅ ⋅ +

( )24 4 2 2 2 2 2

1

2 1 4

G iw w w

ωτ τ ξ τ

⋅ =− + +

( ) 2tg wθ ξ τ= −



M

0.1ξ =

2ξ =


2ξ =

( ) ( )sinv t A tω= ⋅ ⋅ modelo( ) ( ) ( )sinx t A G i tω ω θ≡ ⋅ ⋅ ⋅ ⋅ +



M

0.1ξ =

2ξ =


2ξ =



M

( ) 1

2

1

1IPZ p

sG s K

s

ττ⋅ +

= ⋅⋅ +

( )21K τ ω⋅ + ⋅

Processo IPZ


( )( )

( )

2

1

2

2

1

1IPZ

pKG s

τ ω

τ ω

⋅ + ⋅

+ ⋅=

( ) ( )1 2

2

1 21

tg wτ τ

θτ τ ω−

= ×+ ⋅ ⋅



M


( )( )

( )

2

1

2

2

1

1IPZ

pKG s

τ ω

τ ω

⋅ + ⋅

+ ⋅=



M

( ) ( )1 2

2

1 21

tg wτ τ

θτ τ ω−

= ×+ ⋅ ⋅




M

Processo: tempo morto

( ) t

tm

m seG s − ⋅=

( ) 1G s =


( ) 1tmG s =

( ) mtg tθ ω− ⋅=



M

Analisando o comportamento do nível de um tanque

Resposta de freqüência

( )entra saiF Fdy

dt A

−=


dt A=

( )0

1 1G s

A s= ×

( ) 2sin 2 sin

10entraF t t

ππ

⋅ = ⋅ ⋅ + ⋅ 0saiF =

1A =



M

( )0

1G s

s=

( )00

1 i iG i

i i i

iω

ω ω ω ω−

⋅ = = = = +⋅ ⋅

−⋅2

1 1i−


( )0

1G i ω

ω⋅ =

( )2

0 2

1 10

iG i ω

ω ω ω− ⋅ = + = =

90θ = − °



M

( ) ( )sinv t A tω= ⋅ ⋅modelo

( ) ( ) ( )sinx t A G i tω ω θ≡ ⋅ ⋅ ⋅ ⋅ +


( )0

1G i ω

ω⋅ =



MMETHOD RK4

STARTTIME = 0

STOPTIME=100

DT = 0.002

d/dt(y)=(Fentra-Fsai)/Area

init(y) = 0


init(y) = 0

Area = 1

Fsai = 0

freq = 1

freqw = 2*PI*freq

Fentra = sin(freqw*time)+sin(freqw*time/10)



M

Fourier




M

Fentra

Y

Amplitude de Fentra

Amplitude de Y para f = 0.1 Hz

( )0

11.5915

2 10G i ω

π⋅ = =

⋅


Amplitude de Y para f = 1 Hz

( )0

10.1591

2 1G i ω

π⋅ = =

⋅ ⋅



M


Laço fechado - feedback



M

( ) ( )sinv t A tω= ⋅ ⋅

( ) ( ) ( )ω ω φ= ⋅ ⋅ ⋅ ⋅ +


( )( ) 1

p

p v c m

GX s

V s G G G G=+ ⋅ ⋅ ⋅

( ) ( ) ( )sinEQVx t A G i tω ω φ= ⋅ ⋅ ⋅ ⋅ +



M

( )( ) 1

p

p v c m

GX s

V s G G G G=+ ⋅ ⋅ ⋅ c cG K=p

p

KG

s=

Kω ⋅


( )2 2 2 2 2

p

p v c m

EQV

K

K Ki

KG

Kωω

ω⋅ =

⋅ ⋅ ⋅

⋅

+



M




MSem controle


Com controle P



MMETHOD EULER

STARTTIME = 0

STOPTIME=1000

DT = 0.001

d/dt(y)=(Fentra-Fsai)/Area

init(y) = 0

Area = 1

ym = Km*y

erro = SP - ym

acao = Kc*erro


acao = Kc*erro

vazao = Kv*acao

Km = 1

Kv = 1

SP = 0

Kc = 2

Fss = 0

Fentra = Fss + vazao

Fsai = Fss + sin(0.1*time)+sin(10*time)



M




M

servo


Baixas freqüências “atravessam” o sistema ...



M

Exemplo

Laço fechado - feedback




M


Madonna “anterior”

Fsai = Fss + #Fdesvio(time)



M




M

+




M


aula de controle

Documents