How Do You Calculate K for a Closed Loop Damping Ratio in Higher Order Systems?

[vvl92]

From control systems:
I am asked to find the value of K that gives the closed loop damping ratio of 1/sqrt2.
The value for the complimentary sensitivity is
T(S)=(2KS +4K)/(s^3 +162S^2 +(320+2K)S +4K)
so how do I find the value for K?

I tried putting it in the general equation, but it won't fit since on the bottom, there is a s^3 term, and on the top, there is a s term.

If it is not allowed on physics forums to give a proper answer, then please can you just lead me in the right direction? My notes have nothing in and I can't find anything online!

Thanks!

 

[rude man]

I don't even know if a damping ratio is defined for a third-order system. I would ask what the definition of damping ratio is for such a system.

To quote Wikipedia: "The damping ratio is a parameter, usually denoted by ζ (zeta),[1] that characterizes the frequency response of a second order ordinary differential equation."

You could force the three poles to all be barely real but that would give you critical damping, not what is asked for. And factoring a 3rd order polynomial is a bear anyway ...

BTW I looked up "complementary sensitivity function" and it's just output/input of a noiseless system. So no help there.