I have a homework question to generate the Laplace transfer function from unit step response data. The tf should be 10.108/(s^2+2.128s+14.44). I have measured parameters from a plot of data giving: overshoot(os)% = 28.2%, settling time = 6.5s and peak time = 0.9s. Steady state value = 0.7. From this I have calculated damping factor using z = -ln(0.282)/(sqrt(pi^2+ln^2(0.282)) = 0.374. I then use this to calculate natural frequency wn = 4/(z x 6.5) = 1.645. But these numbers are not the correct values which should be z = 0.28 and wn = 3.8.

The lecturer has checked my calculations and also my plot measurements and all are correct. So, I have no idea what I am doing wrong. Is it because the system has gain less than 1? Do these equation not apply to a gain <1 system? Or is it something to do with the system being open loop?

 

The problem with using settling time is that it isn't particularly well defined.

My approach would be to use the first two peaks in the transient response. One needs to know (i.e. measure) both the peak values and the time of their occurrence.

From these four values [and having particular regard to the steady state final value] you can readily deduce both the damped frequency and the damping factor.

See the attachment for details of what I mean.

 

I have a homework question to generate the Laplace transfer function from unit step response data. The tf should be 10.108/(s^2+2.128s+14.44). I have measured parameters from a plot of data giving: overshoot(os)% = 28.2%, settling time = 6.5s and peak time = 0.9s. Steady state value = 0.7. From this I have calculated damping factor using z = -ln(0.282)/(sqrt(pi^2+ln^2(0.282)) = 0.374. I then use this to calculate natural frequency wn = 4/(z x 6.5) = 1.645. But these numbers are not the correct values which should be z = 0.28 and wn = 3.8.

The lecturer has checked my calculations and also my plot measurements and all are correct. So, I have no idea what I am doing wrong. Is it because the system has gain less than 1? Do these equation not apply to a gain <1 system? Or is it something to do with the system being open loop?

I got the same values that you did, but that's based on the measurements that you gave me. I would suggest you check your overshoot calculations. For the "correct" values, the overshoot would have to be 40%. Are you sure your measured peak wasn't 0.98?