# 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