figure-6.1-superposition.py

# superposition.py - superposition of homogeneous and particular solutions
# RMM, 19 Apr 2024

import matplotlib.pyplot as plt
import numpy as np
import control as ct
import fbs                      # FBS plotting customizations

# Spring mass system
from springmass import springmass       # use spring mass dynamics
sys = springmass / springmass(0).real   # normalize the response to 1
X0 = [2, -1]                            # initial condition

# Create input vectors
tvec = np.linspace(0, 60, 100)
u1 = 0 * tvec
u2 = np.hstack([tvec[0:50]/tvec[50], 1 - tvec[0:50]/tvec[50]])

# Run simulations for the different cases
homogeneous = ct.forced_response(sys, tvec, u1, X0=X0)
particular = ct.forced_response(sys, tvec, u2)
complete = ct.forced_response(sys, tvec, u1 + u2, X0=X0)

# Plot results
fig, axs = plt.subplots(3, 3, figsize=[8, 4], layout='tight')
for i, resp in enumerate([homogeneous, particular, complete]):
    axs[i, 0].plot(resp.time, resp.inputs)
    axs[0, 0].set_title("Input $u$")
    axs[i, 0].set_ylim(-2, 2)
    
    axs[i, 1].plot(
        resp.time, resp.states[0], 'b',
        resp.time, resp.states[1], 'r--')
    axs[0, 1].set_title("States $x_1$, $x_2$")
    axs[i, 1].set_ylim(-2, 2)

    axs[i, 2].plot(resp.time, resp.outputs)
    axs[0, 2].set_title("Output $y$")
    axs[i, 2].set_ylim(-2, 2)

# Label the plots
axs[0, 0].set_ylabel("Homogeneous")
axs[1, 0].set_ylabel("Particular")
axs[2, 0].set_ylabel("Complete")
for i in range(3):
    axs[2, i].set_xlabel("Time $t$ [s]")

# Save the figure
fbs.savefig('figure-6.1-superposition.png')