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This is the system with its step response below.

enter image description hereenter image description here

I'm asked to find the transfer function. Note that this is a question from an exam sheet and a computer can't be used.

I can say that it has to do with an underdamped system but will my transfer function have a zero as well? More specifically, will b1 in the following be zero? $$T(s)=\frac{b_1s+b_0}{s^2+2ζω_ns+ω_n^2}=\frac{b_1s+b_0}{(s+a)^2+ω^2}$$

I can also tell that b0 is equal to ωn^2 since the steady state error is zero and that ω=2.

To sum up, I'm having trouble with b1 and a if my speculations are correct.

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  • \$\begingroup\$ What's the input step size? \$\endgroup\$ – Chu Jul 23 '17 at 22:36
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    \$\begingroup\$ 1 but I guess it's multiplied by that 2 in the diagram. \$\endgroup\$ – John Katsantas Jul 23 '17 at 22:37
  • \$\begingroup\$ Ok, so r(t)= unit step. The initial slope of the response is not zero (at least, as far as I can see), so there may well be a zero. But I would try to fit a standard (no zeros) 2nd order using the log_dec method first and check the goodness of fit. \$\endgroup\$ – Chu Jul 23 '17 at 22:54
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Well, complementing on the comment by Chu, I would:

  1. Assume that the system is a second order one.
  2. Suggest that, from the plot, actually the initial slope is zero (it seems to smoothly ramp up in the beginning), so there is a two degree difference between numerator and denominator. That means no zero.
  3. Then use standard overshoot and settling time equations for second order responses to derive the damping ratio and natural frequency.
  4. Last, the steady-state gain is unity.
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