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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.

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  • \$\begingroup\$ 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? \$\endgroup\$
    – Andy aka
    May 12, 2020 at 11:28

2 Answers 2

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I think you mainly need help understanding what it means for the system to be ‘second-order dominated on the verge of instability.’

It means that there is a pair of complex poles just barely to the left of the imaginary axis (or perhaps on the axis, which is marginally stable), with no other poles that are also close to the imaginary axis.

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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<BC\$ 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}\$

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