Question

Consider a process with transfer function 3e-2s (0.5s - 1) Gp($) = (10s + 1)(5s +1)(2S + 1) Find an approximate model of first order transfer function plus dead time (FOPDT) using: i. ii. Taylor's expansion Skogestad's Half Rule method.

Answer 1

1) by Taylor series expansion

We know that from Taylor series expansion of dead Time

e^{​​​​​​-td*s} = 1- td*s where td is dead time

And e^-td*s = (1/e^{​​​​​​td*s}) = 1/(1+td*s)

So the given transfer function is approximate as first order with dead time as = K*e^{​​​​​​-td*s} /(1+ts)

t is the time constant

So we keep the largest time constant as the first order time constant t= 10

K = -3

So to calculate td

-0.5s+1 = e^{​​​​​​-0.5s}

1/(5s+1) ~ e^{​​​​​​-5s}

1/(2s+1) ~ e^{​​​​​-2s}

So the given transfer function look like

Gps = -3*e^{​​​​​-2s}​​​*e^{​​​​​​-0.5s}*e^​​​-5s *e ^-2s /(10s+1)

Gps = -3e^{​​​​-9.5s} /(10s+1)

B)

By skogested approximation

K = same as in the given transfer function = -3

t = time constant = largest time constant + (second largest time constant/2) =10+(5/2) = 12.5

Dead time = (second largest time constant /2) + dead time in given transfer function+ all left time constant in given transfer function + 0.5

= (5/2)+ 2 + 2+0.5 = 7

Therefore Gps = -3e^{​​​​-7s} /(12.5s+1)