1st Order System Response

This page summarizes step and frequency responses for first order system of the form:

• Response is within 2% of steady state value at 4τ.
• Bandwidth = 1/τ. High bandwidth means system responds fast.
• τ>0: stable.
• τ<0: unstable.

Step Response

Input: step A

Output:

Frequency Response

Input: Asin(ωt)

Output (steady state):

Output (steady state amplitude trend):

Change Response Speed

σ times faster:

• Gain (K): unchanged
• New time constant: τ/σ

Examples

• Example 1: Sine wave is input to a known system. Determine output.
• Example 2: Make a given system respond 10 times faster.
• Example 3: Make a system reach target value at specific time.
• Example 4: Estimate model parameters from output measurement.