# Figure 2.8: Step responses for a first-order, closed loop system with proportional and PI control

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Figure 2.8: Step responses for a first-order, closed loop system with proportional control (a) and PI control (b). The process transfer function is $P = 2/(s + 1)$. The controller gains for proportional control are $k_p = $$, 0.5, 1, and 2. The PI controller is designed using equation (2.28) with \zeta_c = 0.707 and \omega_c = 0.707$$, 1, and 2, which gives the controller parameters$k_p = 0$, 0.207, and 0.914 and$k_i = 0.25$, 0.50, and 2. # figure-2.8-PI_step_reesponses.py - step responses for P/PI controllers # RMM, 21 Jun 2021 # # Step responses for a first-order, closed loop system with proportional # control and PI control. The process transfer function is P = 2/(s + 1). # The controller gains for proportional control are k_p = 0, 0.5, 1, and # 2. The PI controller is designed using equation (2.28) with zeta_c = 0.707 # and omega_c = 0.707, 1, and 2, which gives the controller parameters k_p = # 0, 0.207, and 0.914 and k_i = 0.25, 0.50, and 2. # import numpy as np import matplotlib.pyplot as plt import control as ct # Process model b = 2; a = 1 P = ct.tf([b], [1, a]) # Set the simulation time vector time = np.linspace(0, 8, 100) # # Proportional control # # Choose gains to use kp_gains = [0, 0.5, 1, 2] for kp in kp_gains: Gyv = ct.tf([b], [1, a + b*kp]) Guv = ct.tf([-b*kp], [1, a + b*kp], dt=0) # force kp=0 to be cts time t, y = ct.step_response(Gyv, time) t, u = ct.step_response(Guv, time) if 'p_y_ax' not in locals(): p_y_ax = plt.subplot(3, 2, 1) plt.ylabel('Output$y$') plt.title('Proportional control') p_y_ax.plot(t, y) if 'p_u_ax' not in locals(): p_u_ax = plt.subplot(3, 2, 3) plt.ylabel('Input$u$') plt.xlabel('Normalized time$at$') p_u_ax.plot(t, u, label="kp = %0.3g" % kp) # Label proportional control curves p_u_ax.legend() # # PI control # # Figure out frequency of critical damping zeta = 0.707 wc = a / 2 / zeta # Plot results for different resonate frequencies wc_list = [wc, 1, 2] for wc in wc_list: kp = (2 * zeta * wc - a) / b ki = wc**2 / b Gyv = ct.tf([b, 0], [1, a + b*kp, b*ki]) Guv = -ct.tf([b*kp, b*ki], [1, a + b*kp, b*ki], dt=0) t, y = ct.step_response(Gyv, time) t, u = ct.step_response(Guv, time) if 'pi_y_ax' not in locals(): pi_y_ax = plt.subplot(3, 2, 2) plt.ylabel('Output$y$') plt.title('Proportional-integral control') pi_y_ax.plot(t, y) if 'pi_u_ax' not in locals(): pi_u_ax = plt.subplot(3, 2, 4) plt.ylabel('Input$u$') plt.xlabel('Normalized time$at\$')