How to Solve for s1 | Experts' Guide

[shayaan_musta]

Hello experts,

I have attached two images below. Kindly tell me how to find s1 (the equation in red box)?

Dominant closed loops poles:
Those poles which lie near to the origin.

The standard form of second order system:
s²+2ξω_ns+ω_n²

 

[berkeman]

Hello experts,

I have attached two images below. Kindly tell me how to find s1 (the equation in red box)?

Dominant closed loops poles:
Those poles which lie near to the origin.

The standard form of second order system:
s²+2ξω_ns+ω_n²

Can you say more about what part of the textbook derivation you are not following? Which step does not make sense? That would help us to guide you better.

 

[shayaan_musta]

It is from the book "Modern Control Engineering 5th Edition by Katsuhiko Ogata". Page no. 380 and problem no. A-6-17.

It is available in PDF.

 

[berkeman]

It is from the book "Modern Control Engineering 5th Edition by Katsuhiko Ogata". Page no. 380 and problem no. A-6-17.

It is available in PDF.

That's not what I was asking. I was asking you to go through the derivation for us step by step, up until the part that you are not understanding...

 

[shayaan_musta]

Ohhh sorry. My bad.

Actually, I am not getting the starting point. Because I don't know where does β go and from where does x come in equation (red box) because 6-32 and 6-33 are the sources of the equation (red box). Since, 6-32 and 6-33 has beta but the equation (red box) does not have beta. So I am not sure how to start derivation.