Edit Figure: Figure 2.9: Responses to a unit step change in the reference signal for different values of the design parameters

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[[Image:figure-2.9-secord_stepresp.png]] '''Figure 2.9:''' Responses to a unit step change in the reference signal for different values of the design parameters <math>\omega_\text{c}</math> and <math>\zeta_\text{c}</math>. The left figure shows responses for fixed <math>\zeta_\text{c} = </math>0.707 and <math>\omega_\text{c} =</math> 1, 2, and 5. The right figure shows responses for <math>\omega_\text{c} = </math> 2 and <math>\zeta_\text{c = }</math> 0.5, 0.707, and 1. The process parameters are <math>a = b = 1</math>. The initial value of the control signal is <math>k_\text{p}</math>. <nowiki> # figure-2.9-secord_stepresp.py - step responses for second order systems # RMM, 21 Jun 2021 # # Responses to a unit step change in the reference signal for different # values of the design parameters \omega_c and \zeta_c. The left column # shows responses for fixed \zeta_c = 0.707 and \omega_c = 1, 2, and 5. The # right figure column responses for \omega_c = 2 and \zeta_c = 0.5, 0.707, # and 1. The process parameters are a = b = 1. The initial value of the # control signal is kp. # import numpy as np import matplotlib.pyplot as plt import control as ct # Process model b = 1; a = 1 P = ct.tf([b], [1, a]) # Set the simulation time vector time = np.linspace(0, 6, 100) # # Omega sweep # # Choose gains to use wc_list = [1, 2, 5] zc = 0.707 for wc in wc_list: kp = (2 * zc * wc - a) / b ki = wc**2 C = ct.tf([kp, ki], [1, 0]) Gyr = P*C / (1 + P*C) Gur = C / (1 + P*C) t, y = ct.step_response(Gyr, time) t, u = ct.step_response(Gur, time) if 'w_y_ax' not in locals(): w_y_ax = plt.subplot(3, 2, 1) plt.ylabel('Output $y$') plt.title("Sweep $\omega_c$, $\zeta_c = %g$" % zc) w_y_ax.plot(t, y) if 'w_u_ax' not in locals(): w_u_ax = plt.subplot(3, 2, 3) plt.ylabel('Input $u$') plt.xlabel('Normalized time $\omega_c t$') w_u_ax.plot(t, u, label="$\omega_c = %g$" % wc) # Label the omega sweep curves w_u_ax.legend(loc="upper right") # # Zeta sweep # # Figure out frequency of critical damping wc = 2 zc_list = [0.5, 0.707, 1] # Plot results for different resonate frequencies for zc in zc_list: kp = (2 * zc * wc - a) / b ki = wc**2 C = ct.tf([kp, ki], [1, 0]) Gyr = P*C / (1 + P*C) Gur = C / (1 + P*C) t, y = ct.step_response(Gyr, time) t, u = ct.step_response(Gur, time) if 'z_y_ax' not in locals(): z_y_ax = plt.subplot(3, 2, 2) plt.ylabel('Output $y$') plt.title("Sweep $\zeta_c$, $\omega_c = %g$" % wc) z_y_ax.plot(t, y) if 'z_u_ax' not in locals(): z_u_ax = plt.subplot(3, 2, 4) plt.ylabel('Input $u$') plt.xlabel('Normalized time $\omega_c t$') z_u_ax.plot(t, u, label="$\zeta_c = %g$" % zc) # Label the zeta sweep curves z_u_ax.legend(loc="upper right") # Overalll figure labeling plt.tight_layout() </nowiki>

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