# Figure 3.12: Frequency response computed by measuring the response of individual sinusoids

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Chapter | System Modeling |
---|---|

Figure number | 12 |

Figure title | A frequency response (gain only) computed by measuring # the response of individual sinusoids |

GitHub URL | https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.12-frequency response.py |

Requires | python-control |

**Figure 3.12**: A frequency response (gain only) computed by measuring the response of individual sinusoids. The figure on the left shows the response of the system as a function of time to a number of different unit magnitude inputs (at different frequencies). The figure on the right shows this same data in a different way, with the magnitude of the response plotted as a function of the input frequency. The filled circles correspond to the particular frequencies shown in the time responses.

# figure-3.12-frequency_response.py - frequency response computed by simulation # # Figure 3.12: A frequency response (gain only) computed by measuring # the response of individual sinusoids. The figure on the left shows # the response of the system as a function of time to a number of # different unit magnitude inputs (at different frequencies). The # figure on the right shows this same data in a different way, with # the magnitude of the response plotted as a function of the input # frequency. The filled circles correspond to the particular # frequencies shown in the time responses. # import numpy as np import matplotlib.pyplot as plt import control as ct ct.use_fbs_defaults() # System definition - third order, state space system A = [[-0.2, 2, 0], [-0.5, -0.2, 4], [0, 0, -10]] B = [0, 0, 1] C = [2.6, 0, 0] sys = ct.ss(A, B, C, 0) * 1.4 # state space object (with tweaked scale) # Set up the plotting grid to match the layout in the book fig = plt.figure(constrained_layout=True) gs = fig.add_gridspec(2, 2) # # (a) The response of the system as a function of time to a number of # different unit magnitude inputs (at different frequencies). # fig.add_subplot(gs[0, 0]) # first row, first column # List of frequencies for the time simulations (and frequency response points) omega_time = [0.1, 0.4, 1, 3] mag_time = [] # list to store magnitude of responses # Manual computation of the frequency response for omega in omega_time: # Compute out the time vector and inputs t = np.linspace(0, 50, 1000) u = np.sin(omega * t) # Simulate the system response = ct.forced_response(sys, t, u) # Plot the output plt.plot(response.time, response.outputs, 'b-') # Compute the magnitude of the response (avoiding initial transient) mag_time.append(max(response.outputs[500:])) # Add grid lines plt.xticks([0, 10, 20, 30, 40, 50]) plt.grid(which='major') # Label the plot plt.xlabel("Time [s]") plt.ylabel("Output, $y$") plt.title("Time domain simulations") # # (b) The same data in a different way, with the magnitude of the # response plotted as a function of the input frequency. The filled # circles correspond to the particular frequencies shown in the time # responses. # fig.add_subplot(gs[0, 1]) # first row, second column # List of frequencies to compute the frequency response omega_freq = [0.1, 0.2, 0.4, 1, 1.6, 3, 8, 10] mark_index = [1, 4, 6] # frequencies to mark on the plot mag_freq = [] # list to store magnitude of responses # Manual computation of the frequency response for omega in omega_freq: # Compute out the time vector and inputs t = np.linspace(0, 50, 1000) u = np.sin(omega * t) # Simulate the system response = ct.forced_response(sys, t, u) # Compute the magnitude of the response (avoiding initial transient) mag_freq.append(max(response.outputs[500:])) # Figure out which frequency points to mark omega_mark = np.array(omega_freq)[mark_index] mag_mark = np.array(mag_freq)[mark_index] # Plot the results plt.loglog(omega_freq, mag_freq, 'b-') plt.loglog(omega_mark, mag_mark, 'bo', markerfacecolor='none') plt.loglog(omega_time, mag_time, 'bo') plt.grid(axis='x', which='minor') plt.grid(axis='y', which='major') # Label the plot plt.xlabel("Frequency [rad/s]") plt.ylabel("Gain (log scale)") plt.title("Frequency response") plt.savefig("figure-3.12-frequency_response.png", bbox_inches='tight')