Frequency Response Analysis of Linear Systems

Introduction to Frequency Response

Frequency response analysis examines how a system responds to sinusoidal inputs of varying frequencies. This analysis provides insights into the system’s dynamic characteristics, such as bandwidth, resonance frequencies, and stability margins. It is a powerful tool for designing and analyzing controllers.

Obtaining the Frequency Response

To obtain the frequency response of a system in Python, we can use the freqresp function from the control package. This function calculates the magnitude and phase response of the system over a range of frequencies.

Example: Frequency response of a first-order system

import control as co
import matplotlib.pyplot as plt
import numpy as np

# Create a first-order transfer function model
sys = co.tf(1,[1,10])

# Define the frequency range
w = np.logspace(-2, 2, 1000)

# Calculate the frequency response
mag, phase, omega = co.freqresp(sys, w)

# Plot the Bode plot
plt.figure()
plt.subplot(211)
plt.semilogx(omega, 20*np.log10(mag))
plt.title('Bode Plot')
plt.ylabel('Magnitude (dB)')
plt.grid(True)

plt.subplot(212)
plt.semilogx(omega, np.degrees(phase))
plt.xlabel('Frequency (rad/s)')
plt.ylabel('Phase (degrees)')
plt.grid(True)

plt.show()

Interpreting the Bode Plot

The Bode plot is a graphical representation of the frequency response, showing the magnitude and phase as a function of frequency. It is particularly useful for analyzing systems with multiple poles and zeros.

• Magnitude Plot: The magnitude plot displays the gain of the system at different frequencies. It shows how the amplitude of the output changes with respect to the input amplitude.
• Phase Plot: The phase plot displays the phase shift introduced by the system at different frequencies. It shows how the timing of the output is affected by the input.

Importance in Control Design

Frequency response analysis is crucial in control design because it helps us understand how a system will behave under different operating conditions.

• Stability: By examining the phase margin and gain margin from the Bode plot, we can determine the stability of a closed-loop system.
• Performance: We can assess the system’s bandwidth, rise time, and settling time by analyzing the frequency response characteristics.
• Robustness: Frequency response techniques can be used to design controllers that are robust to model uncertainties and disturbances.

Python Tools

The control package in Python provides a comprehensive set of tools for frequency response analysis, including functions for plotting Bode plots, Nyquist plots, and Nichols charts.

Advanced Frequency Domain Analysis

TBA