Question

The differential equation for a control system is given as follows;

d3c/dt3 + 8d2c/dt2+ 31dc/dt + 48c = d2r/dt2 + 4dr/dt +25r

Determine the following, showing ALL relevant steps

a) The corresponding transfer function in its factorized form

b) Normalized transfer function

c) Magnitude slope plot

d) Phase slope plot

Answer 1

Transfer Function :

Apply Laplace Transform to the given differential equation

a)$L[\frac{d^3c}{dt^3}+\frac{8d^2c}{dt^2}+\frac{31dc}{dt}+48c] = L[\frac{d^2r}{dt^2}+\frac{4dr}{dt}+25r]$

b)C(s)[ s³+8s²+31s+48] = R(s)[s²+4s+25]

Transfer Function = Laplace Transform of Output/Laplace Transform of Input

c)=$\frac{C(s)}{R(s)}$

d)$\frac{C(s)}{R(s)}$ = $\frac{[s3+8s2+31s+48]}{[s2+4s+25]}$