Vibration & Control Unit 8: Controller Design Using Frequency Analysis

Frequency Response and Control Systems Overview

• Understanding frequency response is essential for controller design in control systems.

Bode Plot

• A graphical representation of a system's frequency response.
  • Frequency (log scale):
  • Magnitude in decibels is plotted as:
    $20 ext{log}A$
  • Phase angle in degrees.
  • Bode plots plot amplitude and phase against frequency on a logarithmic scale.

Polar Plot

• Another method to visualize system response.
  • Components:
  • Real axis (Re) and Imaginary axis (Im).
  • Magnitude (amplitude) and Phase angle (argument).
  • Negative phase angles indicate phase lag, plotted clockwise.

Response of Common Systems to a Sinusoidal Input

1. Obtaining G(jω):
   • Start with the system transfer function G(s) and replace s with $jω$.
   • Rearrange into the form:
     $a + jb$
   • Calculate magnitude and phase.
   • Can be plotted using Polar or Bode diagrams.

Pure Gain Response

• When G(s) = D:
  • G(jω) = D + 0j
  • Magnitude:
    • In dB:
      $20 ext{log}A = 20 ext{log}D$
  • Example: If D = 5,
    • Gain = 14 dB
    • Phase = 0°

Gain and Phase Calculation for Different Systems

Integrator Response

• Transfer Function:
  $G(s) = \frac{1}{s}$
• Responses:
  • For G(jω):
  • Follow similar steps to find magnitude and phase.
  • Magnitude and phase calculations vary with frequency:
    • Gain at different frequencies (0.1, 1, and 10 rad/s):
    • Gain (@ω=0.1) = 20 dB
    • Gain (@ω=1) = 0 dB
    • Gain (@ω=10) = -20 dB
    • Use formulas for calculating when frequency changes:
      $20 ext{log}|G(jω)|$

First Order System (Simple Lag)

• Transfer Function:
  $G(s) = \frac{1}{1 + \tau s}$
• G(jω) Representation:
  $G(jω) = \frac{1}{1 + \tau jω}$
• This system exhibits a lag response, impacting the overall phase and gain characteristics.

Summary of Key Points

• Bode Plots allow for easy interpretation of frequency response through logarithmic scaling.
• Polar Plots visually express the complex nature of gain and phase relationships.
• Integrator systems showcase a distinct variation in gain with frequency, demonstrating both theoretical and practical significance in control applications.