Third Order TF - Calculating A Specific Gain

Thread Starter

Alex Panebianco

Joined Mar 21, 2012
I'm currently stuck on a question in my Control Systems module, with relation to a third order transfer function.

For the problem, I'm given the Open Loop TF

\(G(s) = \frac{5K_a}{s(\frac{s}{2}+1)(\frac{s}{6}+1)}\)

and am working with a closed loop negative feedback system, giving the Closed Loop TF

\( G_{cl}=\frac{G(s)}{1+G(s)}\)

which I've simplified down to

\( G_{cl}=12\times\frac{5K_a}{s^3+8s^2+12s+60K_a} \)

I'm then asked to identify the gain required for this system to obtain a damping ratio of 0.6 from a Matlab generated root locus plot, however, my root locus plot appears to only allow a damping ratio of up to 0.577.

The Matlab commands I used were
>> num = 5;
>> den = [1/12 2/3 1 1];
>> Gc = tf(num,den);
>> Gcl = rltool(Gc)

Once the root locus interface had opened, I simply used the hold and drag functionality to try and find a point at which the damping was 0.6, which is where I encountered the problem of 0.577 apparently being the limit.

I'd like to ask if someone could please help me figure out where I've gone wrong along the way and give me a nudge in the right direction as to how to fix my problem.

Thank you in advance for any help offered.