System Identification of 1st Order System

This page describes two methods to identify system parameters for first order systems. The first method uses a step response and the second method uses frequency response.

Overview

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

where K is the DC gain and τ is the time constant. The problem is to find these two 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 DC gain is given by,

Measure the time when the output is 63.2% of steady state value. That time is the time constant τ.

Time constants:

• 1τ: 1-e^-1 = 0.63212
• 2τ: 1-e^-2 = 0.86467
• 3τ: 1-e^-3 = 0.95021

Frequency Response

Input: apply a sinusoid Asin(ωt)

Output (steady state):

Measure the horizontal time shift, Δt. Then the time constant is given by,

Measure the output's amplitude, A'. Then the DC gain is given by,