figure-2.12,14-static_nlsys.py

# figure-2.12,14-static_nlsys.py - static nonlinear feedback system
# RMM, 21 Jun 2021
#
# Figure 2.12: Responses of a static nonlinear system. The left figure shows
# the in- put/output relations of the open loop systems and the right figure
# shows responses to the input signal (2.38). The ideal response is shown
# with solid bold lines. The nominal response of the nonlinear system is
# shown using dashed bold lines and the responses for different parameter
# values are shown using thin lines. Notice the large variability in the
# responses.
#
# Figure 2.14: Responses of the systems with integral feedback (ki =
# 1000). The left figure shows the input/output relationships for the closed
# loop systems, and the center figure shows responses to the input signal
# (2.38) (compare to the corresponding responses in Figure 2.12. The right
# figure shows the individual errors (solid lines) and the approximate error
# given by equation (2.42) (dashed line).
#
# Intial code contributed by Adam Matic, 26 May 2021.
#

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

# Static nonlinearity
def F(u, alpha, beta):
    return alpha * (u + beta * (u ** 3))

# Reference signal
t = np.linspace(0, 6, 300)
r = np.sin(t) + np.sin(np.pi * t) + np.sin((np.pi**2) * t)

#
# Open loop response
#

# Set up the plots for Figure 2.12
fig, [ax1, ax2] = plt.subplots(1, 2, figsize=(6, 3))

# Generate the input/output curves and system responses
for a in [0.1, 0.2, 0.5]:
    for b in [0, 0.5, 1, 2]:
        y = F(r, a, b)
        ax1.plot(r, y, 'r', linewidth=0.5);
        ax2.plot(t, y, 'r', linewidth=0.5)

# Generate the nominal response
y = F(r, 0.2, 1)
ax1.plot(r, y, 'b--', linewidth=1.5);
ax2.plot(t, y, 'b--', linewidth=1.5)

# Left plot labels
ax1.set_title("I/O relationships")
ax1.set_xlabel("Input $u$")
ax1.set_ylabel("Output $y$")

# Draw reference line, set axis limits
ax1.plot(y, y, 'k-', linewidth=1.5)
ax1.set_ylim(-3, 3)
ax1.set_xlim(-2.5, 2.5)

# Right plot labels
ax2.set_title("Output signals")
ax2.set_xlabel("Time $t$")
ax2.set_ylabel("Output $y$")

# Draw reference line, set axis limits
ax2.plot(t, r, 'k-', linewidth=1.5)
ax2.set_ylim(-1, 5)
ax2.set_xlim(0, 2)

plt.tight_layout()

#
# Closed loop response
#

# Create an I/O system representing the static nonlinearity
P = ct.nlsys(
    updfcn=None,
    outfcn=lambda t, x, u, params: F(u, params['a'], params['b']),
    inputs=['u'], outputs=['y'], name='P')

# Integral controller
ki = 1000
C = ct.tf([ki], [1, 0])

# Closed loop system
sys = ct.feedback(P * C, 1)

# Set up the plots for Figure 2.12
fig, [ax1, ax2, ax3] = plt.subplots(1, 3, figsize=(6, 3))

# Generate the input/output curves and system responses
for a in [0.1, 0.2, 0.5]:
    for b in [0, 0.5, 1, 2]:
        # Simulate the system dynamics
        t, y = ct.input_output_response(sys, t, r, params={'a':a, 'b':b})
        
        ax1.plot(r, y, 'r', linewidth=0.5);
        ax2.plot(t, y, 'r', linewidth=0.5)
        ax3.plot(t, r-y, 'r', linewidth=0.5)

# Left plot labels
ax1.set_title("I/O relationships")
ax1.set_xlabel("Input $u$")
ax1.set_ylabel("Output $y$")

# Draw reference line, set axis limits
ax1.plot(y, y, 'k-', linewidth=1.5)
ax1.set_ylim(-3, 3)
ax1.set_xlim(-2.5, 2.5)

# Center plot labels
ax2.set_title("Output signals")
ax2.set_xlabel("Time $t$")
ax2.set_ylabel("Output $y$")

# Draw reference line, set axis limits
ax2.plot(t, r, 'k-', linewidth=1)
ax2.set_ylim(-1, 5)
ax2.set_xlim(0, 2)

# Right plot labels
ax3.set_title("Error")
ax3.set_xlabel("Time $t$")
ax3.set_ylabel("Error $e$")

# Draw bounding line, set axis limits
rdot = np.diff(r)/(t[1] - t[0])     # Approximation of derivative
bmin = 0.1                          # See FBS2e, below equation (2.40)
ax3.plot(t[:-1], rdot/(bmin * ki), 'b--', linewidth=1.5)
ax3.set_xlim(0, 2)

plt.tight_layout()