Transformadas S e Z (21!11!2013)

Universidade Federal de Uberlândia Laboratório de Automação, Servomecanismos e Controle Prof. Aniel Silva de Morais e-mail: [email protected] Sala: 3N222/1E32

__________________________________________________________________________________________________________________________________

1 Controle Digital de Processos Prof. Aniel Silva de Morais Sala 3N222

Tabela de Transformadas S e Z

1 ( )f t ( )F s ( )F z

2 Impulso unitário ( )tδ 1 1

3 Impulso unitário atrasado de kT ( )( )1δ −k T −kTse −kz

4

Degrau unitário ( )1 t 1

s

1

z

z −

5

Degrau unitário atrasado de kT ( )( )1 1−k T

kTse

s

−

1

kzz

z−⋅

−

6

rampa unitária: t 2

1

s

( )21

Tz

z −

7

parábola unitária: 2t 3

1

s ( )

( )2

3

1

2 1

z zT

z

+⋅

−

8

ate− 1

s a+

aT

z

z e−−

9

1 ate−− ( )

a

s s a+ ( )

( )( )1

1

aT

aT

z e

z z e

−

−

−

− −

10

1 atat e−− + ( )

2

2

a

s s a+ ( ) ( )

( )( )

2

2

1 1

1

aT aT aT

aT

aT e z e aTe z

z e z

− − −

−

− + + − −

− −

11

atte− ( )2

1

s a+

( )2

aT

aT

Te z

z e

−

−−

12

at bte e− −− ( )( )

b a

s a s b

−+ +

( )( ) ( )

aT bT

aT bT

z e e

z e z e

− −

− −

−

− −

13

( )1 atat e−− ( )2

s

s a+ ( )

( )2

1aT

aT

z z e aT

z e

−

−

− +

−

14

( )sen tω 2 2s

ω+ ω

( )( )2

z sen T

z 2z cos T 1

⋅ ω− ⋅ ω +

15

( )cos tω 2 2

s

s + ω ( )( )

( )2

z z cos T

z 2z cos T 1

⋅ − ω− ⋅ ω +

16

( )ate sen t− ω ( )2 2s a

ω+ + ω

( )( )

aT

2 aT 2aT

z e sen T

z 2ze cos T e

−

− −

⋅ ω− ω +

17

( )cos− ωate t ( )2 2

s a

s a

++ + ω

( )( )( )

aT

2 aT 2aT

z z e cos T

z 2ze cos T e

−

− −

⋅ − ω

− ω +

18

ka ( )

1

lns a T+

z

z a−

19

kk a⋅ ( )( )2

1

lns a T+

( )2

a z

z a

⋅−

20 Para: ( ) ( ) ( )

kG s

s a s b=

+ + e ( )0

1 sTeB s

s

−−= ( ) ( )( )3

'0

a bT

aT bT

z eB G z k

z e z e

+ −

− −

+= ⋅− −

, onde ( )( )'

3

1 1

1

aT bT

a bT

e ekk

abe

− −

+ −

− −= ⋅

+

21 Para: ( ) ( )

kG s

s s c=

+ e ( )0

1 sTeB s

s

−−= ( ) ( )( )3

'0

1

cT

cT

z eB G z k

z z e

−

−

+= ⋅− −

, onde ( )'

3

1

1

cT

cT

ek Tk

ce

−

−

−⋅= ⋅+

22 Sinal amostrado: ( ) ( ) ( )*

0k

x t x kT t kT∞

== ⋅δ −∑ A transformada Z: ( ) ( )

0

k

k

X z x kT z∞

−

=

= ⋅∑

Universidade Federal de Uberlândia Laboratório de Automação, Servomecanismos e Controle Prof. Aniel Silva de Morais e-mail: [email protected] Sala: 3N222/1E32

__________________________________________________________________________________________________________________________________

2 Controle Digital de Processos Prof. Aniel Silva de Morais Sala 3N222

Lugar das Raízes 23

nσ = ζω 21d nω = ω ⋅ − ζ 1,2 ds j= −σ ± ⋅ω 21,2 n ns j 1= −ζω ± ⋅ω ⋅ − ζ

24

Mapeamento S-Z: ( )dj TsTz e e−σ± ω ⋅= =

( ) ( )Td dz e cos T j s i n T−σ⋅= ⋅ ω ± ⋅ ω

Tustin (Bilinear): 2 z 1

sT z 1

−= ⋅+

Euler: z 1

sT

−=

Backward: z 1

sz T

−=⋅

25 Sobre-passo: ( )2exp 1PS = − −πζ ζ cosζ φ= e tan dφ ω σ= 5%

33t = =τ σ 2%44t τ σ≅ =

Projeto de controladores por imposição de Pólos (Pole-Placement)

26 Controlador discreto ��

* ( )T z é a função de transferência de malha fechada ( ) ( )( )

( )0

1

1

T zC z

B G z T z= ⋅

−

27 Controlador Dead-Beat �� ( ) kT z Z −=

28

Controlador Dahlin �� ( ) 1 MF

MF

Tk

T

eT z z

z e

− ⋅σ−

− ⋅σ

−= ⋅ −

Processos Industriais

29

Atraso de transporte ou tempo morto

se− ⋅θ

Padé de 1ª Ordem ��

12

12

ss

es

− ⋅θ

θ− ⋅= θ+ ⋅

30

Teste em MA (malha aberta)�� ( )

1sk

G s es

− ⋅θ= ⋅τ +

y

ku

∆=∆

Controladores PID

31

Série�� ( ) 11 1

1d

Se pi f

sPID s k

s s

τ= ⋅ + ⋅ + τ τ +

32

Paralelo�� ( ) 1

1d

Pa pi f

sPID s k

s s

τ= + +τ τ +

33

Acadêmico�� ( ) 1

11

dAc p

i f

sPID s k

s s

τ= ⋅ + + τ τ +

34 Filtro da ação derivativa f dτ = α ⋅τ , onde,

1 1

3 20≤ α ≤

Fator de Incontrolabilidade �� I

θ=τ

Método Heurístico de Chien, Hrones e Reswick (CHR) para reposta mais rápida sem sobrepasso. 35

Pk iτ dτ

36 P ( )0,3 kτ θ⋅ ⋅ - -

37 PI ( )0,35 kτ θ⋅ ⋅ 1,16⋅τ -

38 PID ( )0,6 kτ θ⋅ ⋅ τ 0,5 θ⋅

Método Heurístico de Ziegler e Nichols (Z&N) 39

Pk iτ dτ

40 P ( )kτ θ⋅ - -

41 PI ( )0,9 kτ θ⋅ ⋅ 3,33 θ⋅ -

42 PID ( )1,2 kτ θ⋅ ⋅ 2 θ⋅ 0,5 θ⋅

Aproximação de sistema de ordem superior pela “regra da metade” de Skogestad

43 ( )

( )( )

0

00

1

1o

jj s

oi

i

T s

G s k es

θ

τ− ⋅

− += ⋅ ⋅

+

∏∏

Deseja-se eq. da Linha 30

Numeradores negativos Atrasos ordenados de acordo

com a magnitude. k=k0 � valor final

2010 2

ττ τ= +

200 0 0

32 i ji j

Tτθ θ τ

≥

= + + +∑ ∑

Transformadas S e Z (21!11!2013)

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