System Identification of 2nd Order System

This page describes a method to identify system parameters for second order systems. Step response is used to identify the parameters. The system is assumed to be underdamped.

Overview

The general form of the transfer function of a second order system is

where

• K is the gain
• τ is the natrual period of oscillation
• ζ is the damping ratio
• ωn is the natural frequency (rad/sec)

Note that

The problem is to find these parameters given an input signal and its output signal.

Step Response

Input: apply a step with magnitude A

Output:

Measure the output's steady state value, y_ss. Then the gain is given by,

Measure the output's maximum value, y_max. Then the maximum percent overshoot is given by,

Then the damping ratio is given by,

Measure the peak-to-peak duration, Δt. Then the damped natural frequency is given by

Then the the natural frequency is given by