# Closed Loop system with a second order dominated response on the verge of instability

I have the forward path transfer function of C(s)G(s) = K/s(s+3)(s+7) and want to find K so that it is second order dominated on the verge of instability.

But i'm not sure how to find it when it is on the verge of instability, the only thing i could see when researching was that maybe its when C is very high?

I also found K when (both being second order dominated) when critically damped (I set the damping ratio to 1) and got K = 14.577 and when underdamped with a natural frequency of 2 rad/s and got K = 17.

Any help will be greatly appreciated and sorry for the bad formatting.

• Is that C(s)*G(s) in your words or do you mean something else? What amount of feedback is used (in order to calculate instability? May 12, 2020 at 11:28

A closed loop, third order transfer function: $$\small G(s)=\frac{K}{As^3+Bs^2+Cs+D}$$ will have a critically stable 2nd order denominator factor (i.e. steady-state sinusoidal response) if $$\\small BC=AD\$$.
$$\\small AD will be give a stable system, and $$\\small AD>BC\$$ will give an unstable system.
In your case, $$\small G(s)=\frac{K}{s^3+10s^2+21s+K}$$
Therefore, $$\\small K=210\$$ will give critical stability, with $$\small G(s)=\frac{210}{(s+10)(s^2+21)}$$ and the frequency of oscillation is, $$\\small\omega_n =\sqrt{21}=4.58 \:\mathrm{rad/sec}\$$