# Calculate gain of a transfer function without root locus

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Hi,

Having a bit of trouble with this question: "Assuming a proportional controller is used, determine the gain to achieve a damping ratio of 0.5, for the following transfer function. Hence calculate the associated natural frequency and oscillation period. G(s) = -4(s+0.4) / s^2+1.6s+14."

I would normally try and solve this using root locus method, but the question explicitly says not to use root locus. The furthest I've been able to go is determining the zeros and poles of the transfer function as -0.4, and -1.6+/-3.66i respectively. Some additional data is given: "Data that may be required: (s-2.38)(s+4.14-2.60i)(s+4.14-2.60i)=s^3+5.9s^2+4.24s-56.82" I think I recognise the left hand side of that equation coming from the characteristic equation of a state space model but I'm really not sure how I could use this.

Can anyone lend a hand? Thanks.

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donpacino
Gold Member

Basically just plug a gain k in your system. Re-evaluate your closed loop transfer function. Then solve for the damping ratio you need.

Basically just plug a gain k in your system. Re-evaluate your closed loop transfer function. Then solve for the damping ratio you need.
It's not explicitly stated but I believe it's for the plant alone. I'm afraid I'm slightly confused by what you mean by reevaluating the transfer function, surely if I just add a proportional gain it's not going to change where the roots of the characteristic equation are?

donpacino
Gold Member
It's not explicitly stated but I believe it's for the plant alone. I'm afraid I'm slightly confused by what you mean by reevaluating the transfer function, surely if I just add a proportional gain it's not going to change where the roots of the characteristic equation are?
Root locus adds a gain in front of the transfer function, then takes negative feedback. Because of the neg feedback the gain effects the poles of the transfer function.

The same principal occurs here. You are doing VERY similar analysis to root locus. Simply apply negative feedback and see how the gain effects the poles.

LvW
I assume that G(s) is the plant transfer function - and the loop should be closed using a P-controller.
As a result, the closed-loop gain should have a dampiung ratio of 0.5.
This seems to be a clear description of the task
However, what is the information content of the "additional data"?
You are asking us - hence, you should know something about the meaning of these "data".