Process Dynamics and Control - Formulario (Formulary)

PROCESS DYNAMICS

and CONTROL

LAPLACE TRANSFORM

[ () ] = [ () ] = ()

[ () + () ] = () + ()

−1

[ () + () ] = () + ()

2

(2) =

()

⎡ ⎤

= () − (0)

⎣ ⎦

2 2

⎡⎢ ⎤⎥

() (0)

= () − (0) −

2

⎣ ⎦

∞

⎡ ⎤ −

1 where

⎢ ⎥

∫ ( *) * = () () = ∫ ()

⎢ ⎥

⎣ ⎦

0 0

−θ

[ ]

→

() = ( − θ) () = ()

PARTIAL FRACTION EXPANSION

2

+5+1

() =

(

+3

)

(

+2

)

() = + +

+2 +3

2 (+2)(+3)+()(+2)+()(+3)

+5+1 =

(

+3

)

(

+2

)

(

+3

)

(

+2

)

2

+ 5 + 1 = ( + 2)( + 3) + ()( + 2) + ()( + 3) −θ

−

when there is a , omit it during the inverse transform and add it after by writing τ

θ

−

instead of .

τ

FIND TF

write material / energy balances → explicit disturbances and controlled variables → create a

differential equation like → coefficient of y must be 1 → laplace transform →

' + =

find TF in Laplace domain → use table to convert in time domain

THEOREMS (LIMITS)

lim () = lim ()

→ →

∞ 0

lim () = lim ()

→ →

0 ∞

The time needed to reach steady state: = 4 ÷ 5τ

LINEARIZATION ( ) ( )

' ≃ ' + '

1

2

nd

2 ORDER SYSTEM

() = 2 2

τ +2ζτ+1

at open loop:

( )

τ +τ /2

●​ 1 2

ζ = τ τ

1 2

●​ τ = τ τ

1 2

τ

●​ 0

τ =

1 2

ζ − ζ −1

0 0

τ

●​ 0

τ =

2 2

ζ + ζ