Phase plot examples¶
Code¶
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# RMM, 24 July 2011
#
# This file contains examples of phase portraits pulled from "Feedback
# Systems" by Astrom and Murray (Princeton University Press, 2008).
import os
import numpy as np
import matplotlib.pyplot as plt
from control.phaseplot import phase_plot
from numpy import pi
# Clear out any figures that are present
plt.close('all')
#
# Inverted pendulum
#
# Define the ODEs for a damped (inverted) pendulum
def invpend_ode(x, t, m=1., l=1., b=0.2, g=1):
return x[1], -b/m*x[1] + (g*l/m)*np.sin(x[0])
# Set up the figure the way we want it to look
plt.figure()
plt.clf()
plt.axis([-2*pi, 2*pi, -2.1, 2.1])
plt.title('Inverted pendulum')
# Outer trajectories
phase_plot(
invpend_ode,
X0=[[-2*pi, 1.6], [-2*pi, 0.5], [-1.8, 2.1],
[-1, 2.1], [4.2, 2.1], [5, 2.1],
[2*pi, -1.6], [2*pi, -0.5], [1.8, -2.1],
[1, -2.1], [-4.2, -2.1], [-5, -2.1]],
T=np.linspace(0, 40, 200),
logtime=(3, 0.7)
)
# Separatrices
phase_plot(invpend_ode, X0=[[-2.3056, 2.1], [2.3056, -2.1]], T=6, lingrid=0)
#
# Systems of ODEs: damped oscillator example (simulation + phase portrait)
#
def oscillator_ode(x, t, m=1., b=1, k=1):
return x[1], -k/m*x[0] - b/m*x[1]
# Generate a vector plot for the damped oscillator
plt.figure()
plt.clf()
phase_plot(oscillator_ode, [-1, 1, 10], [-1, 1, 10], 0.15)
#plt.plot([0], [0], '.')
# a=gca; set(a,'FontSize',20); set(a,'DataAspectRatio',[1,1,1])
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.title('Damped oscillator, vector field')
# Generate a phase plot for the damped oscillator
plt.figure()
plt.clf()
plt.axis([-1, 1, -1, 1]) # set(gca, 'DataAspectRatio', [1, 1, 1]);
phase_plot(
oscillator_ode,
X0=[
[-1, 1], [-0.3, 1], [0, 1], [0.25, 1], [0.5, 1], [0.75, 1], [1, 1],
[1, -1], [0.3, -1], [0, -1], [-0.25, -1], [-0.5, -1], [-0.75, -1], [-1, -1]
],
T=np.linspace(0, 8, 80),
timepts=[0.25, 0.8, 2, 3]
)
plt.plot([0], [0], 'k.') # 'MarkerSize', AM_data_markersize*3)
# set(gca, 'DataAspectRatio', [1,1,1])
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.title('Damped oscillator, vector field and stream lines')
#
# Stability definitions
#
# This set of plots illustrates the various types of equilibrium points.
#
def saddle_ode(x, t):
"""Saddle point vector field"""
return x[0] - 3*x[1], -3*x[0] + x[1]
# Asy stable
m = 1
b = 1
k = 1 # default values
plt.figure()
plt.clf()
plt.axis([-1, 1, -1, 1]) # set(gca, 'DataAspectRatio', [1 1 1]);
phase_plot(
oscillator_ode,
X0=[
[-1, 1], [-0.3, 1], [0, 1], [0.25, 1], [0.5, 1], [0.7, 1], [1, 1], [1.3, 1],
[1, -1], [0.3, -1], [0, -1], [-0.25, -1], [-0.5, -1], [-0.7, -1], [-1, -1],
[-1.3, -1]
],
T=np.linspace(0, 10, 100),
timepts=[0.3, 1, 2, 3],
parms=(m, b, k)
)
plt.plot([0], [0], 'k.') # 'MarkerSize', AM_data_markersize*3)
# plt.set(gca,'FontSize', 16)
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.title('Asymptotically stable point')
# Saddle
plt.figure()
plt.clf()
plt.axis([-1, 1, -1, 1]) # set(gca, 'DataAspectRatio', [1 1 1])
phase_plot(
saddle_ode,
scale=2,
timepts=[0.2, 0.5, 0.8],
X0=[
[-1, -1], [1, 1],
[-1, -0.95], [-1, -0.9], [-1, -0.8], [-1, -0.6], [-1, -0.4], [-1, -0.2],
[-0.95, -1], [-0.9, -1], [-0.8, -1], [-0.6, -1], [-0.4, -1], [-0.2, -1],
[1, 0.95], [1, 0.9], [1, 0.8], [1, 0.6], [1, 0.4], [1, 0.2],
[0.95, 1], [0.9, 1], [0.8, 1], [0.6, 1], [0.4, 1], [0.2, 1],
[-0.5, -0.45], [-0.45, -0.5], [0.5, 0.45], [0.45, 0.5],
[-0.04, 0.04], [0.04, -0.04]
],
T=np.linspace(0, 2, 20)
)
plt.plot([0], [0], 'k.') # 'MarkerSize', AM_data_markersize*3)
# set(gca,'FontSize', 16)
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.title('Saddle point')
# Stable isL
m = 1
b = 0
k = 1 # zero damping
plt.figure()
plt.clf()
plt.axis([-1, 1, -1, 1]) # set(gca, 'DataAspectRatio', [1 1 1]);
phase_plot(
oscillator_ode,
timepts=[pi/6, pi/3, pi/2, 2*pi/3, 5*pi/6, pi, 7*pi/6,
4*pi/3, 9*pi/6, 5*pi/3, 11*pi/6, 2*pi],
X0=[[0.2, 0], [0.4, 0], [0.6, 0], [0.8, 0], [1, 0], [1.2, 0], [1.4, 0]],
T=np.linspace(0, 20, 200),
parms=(m, b, k)
)
plt.plot([0], [0], 'k.') # 'MarkerSize', AM_data_markersize*3)
# plt.set(gca,'FontSize', 16)
plt.xlabel('$x_1$')
plt.ylabel('$x_2$')
plt.title('Undamped system\nLyapunov stable, not asympt. stable')
if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
plt.show()
|
Notes¶
1. The environment variable PYCONTROL_TEST_EXAMPLES is used for testing to turn off plotting of the outputs.