Identification of a damped PT2 system

Introduction

The identification of the system parameters from a real dynamical system is often required for model control or simulations. In many cases the dynamical system is a PT2 system with the transfer function H(s).

In case of D<1 the system is underdamped and there exists a simple way to extract the parameters k, D and wn from the step-response of the system (see PT2-System Wikipedia (german)).

In case of D>1 up to now i do not have found any simple solution to do that job and this article describes a simple method to determine the two time constans T1 and T2 of the transfer function

Method to get T1 and T2 with the measured step-response

1. Capture the step-response y(t) and normalize it

Capture the step-response y(t) and normalize so that you get the step-response with an unit-step input signal (step from 0 to 1).

2. Determine steady-state gain k

You can measure the gain k in steady-state (in the example above k=2).

3. Measure t25 and t75

Measure the timestamps where the step-response y(t) reaches 25% and 75% of the steady-state value.

4. Calculate the time-constants T1 and T2

Now calculate the time-constants of the transfer function H(s) with the following method

Example with ltSpice

Following example demonstrates the functionality of this approach. The step-response of a passive RC filter of 2nd order was simulated, k, t25 and t75 measured with the cursors, then T1 und T2 were calculated using ltSpice parameters and then the step-response of the analog circuit was compared with the identified transfer function H(s).

Conclusion

Many times i identified T1 and T2 of overdamped PT2 systems by fast try and error or some numerical methods. But the results were bad or it was a huge effort to get good results.

The method presented in this article allows now to determine transfer function with two simple measurements and returns the both time-constants with a high accuracy.

Feel free to test this method and give me some feedback of your experience.

I used the computer algebra system maxima to do the derivation of this equations with numerical methods (not analytical).