Formulas of Rise time, settling time, and other step-response characteristics for arbitrary-order transfer function

How the parameters of transients are estimated (as in the picture) from an arbitrary linear transfer function (formula is given) ?

https://i.stack.imgur.com/nVv1P.png

https://i.stack.imgur.com/Aoecz.gif

Answer 1

[Star Strider]

If you are estimating the transfer function from data, use the System Identification Toolbox functions, specifically iddata and ssest, since it is more robust than tfest, although tfest might be preferable in this instance. Then (if necessary) use the tf function to turn the state space realisation into a transfer function, and then use sys.Numerator and sys.Denominator (for example) to get the coefficients.

[Andrew Sol]

The question is not how to get the transfer function, but how to relate the parameters of the transfer function of an arbitrary order (provided that it is stable) with the characteristics of the transient process (overshoot, control time, etc.). Please read the question carefully.

[Star Strider]

I read it carefully. It initially appears to be a system identification and parameter estimation problem:

‘How the parameters of transients are estimated (as in the picture) from an arbitrary linear transfer function (formula is given) ?’

Once the parameters of the transfer function itself are known, it can be modeled as a simplified differential equation (initially in Laplace space, then inverted), and the parameters estimated from it as described in the Wikipedia article on Step response. I am in no way claiming that for an arbitrary stable transfer function it would be a trivial computation, however it would (theoretically, since I have never actually attempted this) be possible.

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Answer 2

[Sulaymon Eshkabilov]

Use step(), e.g.:

 SYS = tf(1, [1 1 2]);
step(SYS)
% Move the cursor over the plot and use the right mouse button options,
% -> characteristics -> Peak time, Settling time, etc.

[Andrew Sol]

This is a numerical calculation. I am wondering how to relate the parameters of the transfer function to the parameters of the transient.