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

From FBSwiki
Jump to navigation Jump to search
Chapter Feedback Principles
Figure number 8
Figure title Step responses for a first-order, closed loop system with proportional # control and PI control.
GitHub URL https://github.com/murrayrm/fbs2e-python/blob/main/figure-2.8-PI step responses.py
Requires python-control

Figure-2.8-PI step response.png

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 . The controller gains for proportional control are 0, 0.5, 1, and 2. The PI controller is designed using equation (2.28) with 0.707 and 0.707, 1, and 2, which gives the controller parameters 0, 0.207, and 0.914 and 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 $at$')
    pi_u_ax.plot(t, u, label="wc = %0.3g" % wc)

# Label PI curves
pi_u_ax.legend()

# Overalll figure labeling
plt.tight_layout()