Step Response of an LTI system

[qwertydump]

Hi lads, need help with this question from a signals and systems course.





LTI system is defined as follows



$$H(s) = \frac{1}{ s^2 - 2rs cos(theta) + r^2}$$



where r = 20pi, theta = 1.47.



Find an expression for the step response of the system





I think what I should do is multiply H(s) by 1/s cos it's a step response

and then mix and match it with the laplace tables to get the inverse transform.



So my attempt was:



$$\frac{1}{s^{2}-2rs cos(th) + r^{2}}$$



= $$\frac{1}{s^{2}-2rscos(th)+r^{2}cos^{2}(th) + r^{2}- r^{2}cos^{2}(th)}$$



= $$\frac{1}{(s-rcos(th))^{2}+r^{2}-r^{2}cos^{2}(th)}$$



= $$\frac{1}{(s-rcos(th))^{2}+r^{2}(1-cos^{2}(th))}$$



= $$\frac{1}{(s-rcos(th))^{2}+r^{2}sin^{2}(th) }$$



= $$\frac{1}{(s-rcos(th))^{2}+(rsin(th))^{2}}$$



This was great cos then I could match it to a transform on my laplace transform table



this one:



$$\frac{A(s+a) + Bw}{(s+a)^{2}+w^{2}}$$



= $$e^{-at}[Acos(wt)+Bsin(wt)]$$



but the problem was that I realized that A & B would be zero and i'd get zero as a result and also I hadn't multiplied my equation by 1/s and that screws up everything. I haven't a clue what else to do then.



so ... am I approching this thing all wrong?



any guidance or help very much appreciated

please bear in mind that I don't really know if I should be trying to do the laplace transform or what.



Berty

 

[Valkarie]

Yes, you are approaching the problem correctly. You need to multiply your equation with 1/s in order to get the step response. The expression for the step response is then: e^{-at}[Acos(wt)+Bsin(wt)] where A and B are constants that depend on r and theta, and a and w are parameters related to r and theta as follows: a = r cos(theta) w = r sin(theta) You can then plug in the values of r and theta to solve for A and B, and then you will have the expression for the step response.