Urgent...Inverse Laplace of unit step response of ..........

[cmos babe]

Hi there, can anyone show me the steps of getting the inverse laplace transform of this equation which is the unit step reponse of prototype 2nd order system ?

 

[abionnnn]

Re: Urgent...Inverse Laplace of unit step response of ......

Your problem is simple you will need to use the method of partial fractions. The first step is to factorise (s²+2ζωs +ω²). You can get three different cases:

for ζ<1: (overdamped)

This is the simplest case, in which you have 2 real roots. You can then factorise the denomenator to be of the form: (s+x)(s+y). So you have:

ω²/(s(s+x)(s+y)) = A/s + B/(s+x) + C/(s+y)

where A,B,C are computable using the partial fractions method. (see **broken link removed** for example)

The inverse of the above system is the following:

u[t] (A + B e^(-xt) + C e^(-yt))

for ζ>1: (underdamped)

This will have imaginary roots, but the same method as before can be used. You just have to take care at the end since your resulting inverse is real even though it have imaginaries all over it.

for ζ=1: (critically damped)

This is a special case where you have x=y i.e.

ω²/(s(s+x)²) = A/s + B/(s+x) + C/(s+x)²

To solve for this, you will need to read up on partial fractions. But say you have A,B,C. The inverse (from memory) is then:

u[t] (A + B e^(-xt) + C t e^(-xt))

Hope this helped.

-abionnnn