## Finding transfer function gain without root locus

## Finding transfer function gain without root locus

(OP)

Hi,

Having a bit of trouble with this, assuming a proportional controller is used, I need to determine the gain to achieve a damping ratio of 0.5, for the following transfer function: G(s) = -4(s+0.4) / s^2+1.6s+14.

I need to solve this without using 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.

Having a bit of trouble with this, assuming a proportional controller is used, I need to determine the gain to achieve a damping ratio of 0.5, for the following transfer function: G(s) = -4(s+0.4) / s^2+1.6s+14.

I need to solve this without using 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.

## RE: Finding transfer function gain without root locus

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## RE: Finding transfer function gain without root locus

## RE: Finding transfer function gain without root locus

If the closed loop transfer function had a critically damped characteristic equation then it would be (s^2+2ζωs+(ω^2+k))=(s^2+2λs+λ^2)

but since in this example you want the desired characteristic equation to be

(s^2+2ζωs+(ω^2+k))=(s^2+λs+λ^2)

Now there are two equations and two unknowns, k an λ.

2ζω=λ=1.6

ω^2+k=λ^2

solve for k an λ

What bothers me is that I calculate a negative k.

Is this a realistic example?

If ω=sqrt(14) then

ζ=.214

Adding gain will only make the closed loop damping factor smaller.

Peter Nachtwey

Delta Computer Systems

http://www.deltamotion.com

## RE: Finding transfer function gain without root locus

Most control systems use more than the proportional gain.

The instructors always assume the open loop transfer function is known.

The results, the calculated gain doesn't rarely results in a usable response.

In 30+ years of controls I have never used the root locus.

The instructors are just wasting the students time and money.

The only place where I could see the root locus being used is for a mechanical governor or a hydraulic valve that uses a spring for control.

However, I doubt it because I have NEVER seen a mechanical system with a transfer function attached.

Peter Nachtwey

Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/

## RE: Finding transfer function gain without root locus

For example, distillation columns, stirred reactors, pH control with mixing/holding tanks, aircraft and ship board steering, satellite controls especially for multiple objects, paper machines... all are all high order processes. The various studies that requiring identification of the dominant interactions are done 5-10 years prior to the design phase.

Are root locus methods, Routh-Hurwitz commonly used in the design phase. The short answer is no. In part the control systems commonly dealt with are reduced to 2nd order systems intentionally. You don't need much analysis for that and the control schemes are easily developed.

It is a useful tool, but only when you need it. Were an engineer present a root locus plot to a project manager...it would be a good time to start looking for new employment...

## RE: Finding transfer function gain without root locus

But the tools exist now. However there are the practical aspects. Just because you can in a Mathematica work sheet doesn't mean you should or it is the best way.

This point is why the topic would make a good thread.

And the young engineer would be thinking why did they waste my time and money learning root-locus.

Peter Nachtwey

Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/

## RE: Finding transfer function gain without root locus

https://www.politesi.polimi.it/bitstream/10589/682...

## RE: Finding transfer function gain without root locus

At least he didn't waste time with root locus but anybody that mentions Z-N for motion control hasn't really tried it. It doesn't work.

2.1.5 about pole placement being inadequate is wrong.

What was good is his table of frequencies, amplitudes and speeds. So many of my customers screw that up because they haven't done the math ( physics ).

Peter Nachtwey

Delta Computer Systems

http://www.deltamotion.com

http://forum.deltamotion.com/