Figure 3.2: Illustration of a state model

Chapter	System Modeling
Figure number	3.2
Figure title	Illustration of a state model
GitHub URL	https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.2-state model.py
Requires	python-control

Figure 3.2: Illustration of a state model. A state model gives the rate of change of the state as a function of the state. The plot on the left shows the evolution of the state as a function of time. The plot on the right, called a phase portrait, shows the evolution of the states relative to each other, with the velocity of the state denoted by arrows.

# figure-3.2-state_mode.py - illustration of a state model
# RMM, 2 Jul 2021
#
# Figure 3.2: Illustration of a state model. A state model gives the rate of
# change of the state as a function of the state. The plot on the left shows
# the evolution of the state as a function of time. The plot on the right,
# called a phase portrait, shows the evolution of the states relative to
# each other, with the velocity of the state denoted by arrows.
#

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

#
# Spring mass system with nonlinear dampling
#
# This function gives the dynamics for a dampled oscillator with nonlinear
# damping.  The states of the system are
#
#   x[0]        position
#   x[1]        velocity
#
# The nonlinear damping is implemented as a change in the linear damping
# coefficient at a small velocity.  This is intended to roughly correspond
# to some sort of stiction (and give an interesting phase portrait).  The
# default parameters for the system are given by
#
#   m = 1       mass, kg
#   k = 1       spring constant, N/m
#   b1 = 1      damping constant near origin, N-sec/m
#   b2 = 0.01   damping constant away from origin, N-sec/m
#   dth = 0.5   threshold for switching between damping
#
# This corresponds to a fairly lightly damped oscillator away from the origin.

def _nlspringmass(t, x, u, params):
    m = params.get('m', 1)
    k = params.get('k', 1)
    b1 = params.get('b1', 2)
    b2 = params.get('b2', 0.01)
    dth = params.get('dth', 0.2)

    # Compute the friction force
    if abs(x[1]) < dth:
        Fb = b1 * x[1];
    elif x[1] < 0:
        Fb = -b1 * dth \
            + b2 * (x[1] + dth);
    else:
        Fb = b1 * dth \
            + b2 * (x[1] - dth);

    # Return the time derivative of the state
    return np.array([x[1], -k/m * x[0] - Fb/m])
nlspringmass = ct.nlsys(_nlspringmass, None, states=2, inputs=0, outputs=2)

#
# (a) Simulation of the nonlinear spring mass system
#
plt.subplot(2, 2, 1)

t = np.linspace(0, 16, 100)
resp = ct.input_output_response(nlspringmass, t, 0, [2, 0])
y = resp.outputs

plt.plot(t, y[0], '-', t, y[1], '--')
plt.xlabel('Time $t$ [s]')
plt.ylabel(r'Position $q$ [m], velocity $\dot q$̇ [m/s]')
plt.title('Time plot')
plt.legend(['Position $q$', 'Velocity $v$'])

#
# (b) Generate a phase plot for the damped oscillator
#
ax = plt.subplot(2, 2, 2)

cplt = ct.phase_plane_plot(
    nlspringmass,               # dynamics
    [-1, 1, -1, 1],             # bounds of the plot
    gridspec=[8, 8],            # number of points for vectorfield
    plot_vectorfield=True,      # plot vectorfield
    plot_streamlines=False,     # plot streamlines separately
    plot_separatrices=False,    # leave off separatrices
    ax=ax
)
ct.phaseplot.streamlines(       # Plot streamlines from selected points
    nlspringmass,
    np.array([[-1, 0.4], [0.1, 1], [1, -0.4], [-0.1, -1]]),
    10, ax=ax
)

plt.xlabel('Position $q$ [m]')
plt.ylabel(r'Velocity $\dot q$ [m/s]')
plt.title('Phase portrait')
plt.axis([-1, 1, -1, 1])

plt.tight_layout()