Figure 3.2: Illustration of a state model - Revision history

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Murray: Created page with "{{Figure |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-stat..."

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Created page with "{{Figure |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-stat..."

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{{Figure
|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
}}
[[Image:figure-3.2-state_model.png]]

'''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.

<nowiki>
# 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(x, t, u=0, m=1, k=1, b1=2, b2=0.01, 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])

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

t = np.linspace(0, 16, 100)
y = sp.integrate.odeint(nlspringmass, [2, 0], t)

plt.plot(t, y[:, 0], '-', t, y[:, 1], '--')
plt.xlabel('Time $t$ [s]')
plt.ylabel('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
#
plt.subplot(2, 2, 2)

ct.phase_plot(
nlspringmass, # dynamics
(-1, 1, 8), (-1, 1, 8), # X, Y range, npoints
scale=0.2,
X0=[[-1, 0.4], [0.1, 1], [1, -0.4], [-0.1, -1]])

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

plt.tight_layout()
</nowiki>

@@ Line 5: / Line 5: @@
 |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.2-state_model.py
 }}
-[[Image:figure-3.2-state_model.png|480px]]
+[[Image:figure-3.2-state_model.png|640px]]
 '''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.

@@ Line 5: / Line 5: @@
 |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.2-state_model.py
 }}
-[[Image:figure-3.2-state_model.png|width=480px]]
+[[Image:figure-3.2-state_model.png|480px]]
 '''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.

@@ Line 36: / Line 36: @@
+#
 # 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
+#
@@ Line 48: / Line 48: @@
 # This corresponds to a fairly lightly damped oscillator away from the origin.
-def nlspringmass(x, t, u=0, m=1, k=1, b1=2, b2=0.01, dth=0.2):
+def _nlspringmass(t, x, u, params):
      # Compute the friction force
      if abs(x[1]) < dth:
@@ Line 61: / Line 67: @@
      # Return the time derivative of the state
      return np.array([x[1], -k/m * x[0] - Fb/m])
+#
@@ Line 68: / Line 75: @@
 t = np.linspace(0, 16, 100)
-y = sp.integrate.odeint(nlspringmass, [2, 0], t)
+resp = ct.input_output_response(nlspringmass, t, 0, [2, 0])
-plt.plot(t, y[:, 0], '-', t, y[:, 1], '--')
+plt.plot(t, y[0], '-', t, y[1], '--')
 plt.xlabel('Time $t$ [s]')
-plt.ylabel('Position $q$ [m], velocity $\dot q$̇ [m/s]')
+plt.ylabel(r'Position $q$ [m], velocity $\dot q$̇ [m/s]')
 plt.title('Time plot')
 plt.legend(['Position $q$', 'Velocity $v$'])
@@ Line 79: / Line 87: @@
 # (b) Generate a phase plot for the damped oscillator
+#
-plt.subplot(2, 2, 2)
+ax = plt.subplot(2, 2, 2)
-ct.phase_plot(
+cplt = ct.phase_plane_plot(
      nlspringmass,               # dynamics
-     (-1, 1, 8), (-1, 1, 8),     # X, Y range, npoints
+     [-1, 1, -1, 1],             # bounds of the plot
-     scale=0.2,
+    gridspec=[8, 8],            # number of points for vectorfield
-     X0=[[-1, 0.4], [0.1, 1], [1, -0.4], [-0.1, -1]])
+     plot_vectorfield=True,      # plot vectorfield
 plt.xlabel('Position $q$ [m]')
-plt.ylabel('Velocity $\dot q$ [m/s]')
+plt.ylabel(r'Velocity $\dot q$ [m/s]')
 plt.title('Phase portrait')
 plt.axis([-1, 1, -1, 1])

@@ Line 2: / Line 2: @@
 |Chapter=System Modeling
 |Figure number=3.2
-|Sort key=312
+|Sort key=302
 |Figure title=Illustration of a state model
 |GitHub URL=https://github.com/murrayrm/fbs2e-python/blob/main/figure-3.2-state_model.py