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# phaseplot.py - generate 2D phase portraits
#
# Initial author: Richard M. Murray
# Creation date: 24 July 2011, converted from MATLAB version (2002);
# based on an original version by Kristi Morgansen
"""Generate 2D phase portraits.
This module contains functions for generating 2D phase plots. The base
function for creating phase plane portraits is `~control.phase_plane_plot`,
which generates a phase plane portrait for a 2 state I/O system (with no
inputs). Utility functions are available to customize the individual
elements of a phase plane portrait.
The docstring examples assume the following import commands::
>>> import numpy as np
>>> import control as ct
>>> import control.phaseplot as pp
"""
import math
import warnings
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint
from . import config
from .ctrlplot import ControlPlot, _add_arrows_to_line2D, _get_color, \
_process_ax_keyword, _update_plot_title
from .exception import ControlArgument
from .nlsys import NonlinearIOSystem, find_operating_point, \
input_output_response
__all__ = ['phase_plane_plot', 'phase_plot', 'box_grid']
# Default values for module parameter variables
_phaseplot_defaults = {
'phaseplot.arrows': 2, # number of arrows around curve
'phaseplot.arrow_size': 8, # pixel size for arrows
'phaseplot.arrow_style': None, # set arrow style
'phaseplot.separatrices_radius': 0.1 # initial radius for separatrices
}
def phase_plane_plot(
sys, pointdata=None, timedata=None, gridtype=None, gridspec=None,
plot_streamlines=True, plot_vectorfield=False, plot_equilpoints=True,
plot_separatrices=True, ax=None, suppress_warnings=False, title=None,
**kwargs
):
"""Plot phase plane diagram.
This function plots phase plane data, including vector fields, stream
lines, equilibrium points, and contour curves.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int or list of int
Time to simulate each streamline. If a list is given, a different
time can be used for each initial condition in `pointdata`.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot all elements in the given color (use ``plot_<element>`` =
{'color': c} to set the color in one element of the phase
plot (equilpoints, separatrices, streamlines, etc).
ax : matplotlib.axes.Axes, optional
The matplotlib axes to draw the figure on. If not specified and
the current figure has a single axes, that axes is used.
Otherwise, a new figure is created.
Returns
-------
cplt : `ControlPlot` object
Object containing the data that were plotted:
* cplt.lines: array of list of `matplotlib.artist.Artist`
objects:
- lines[0] = list of Line2D objects (streamlines, separatrices).
- lines[1] = Quiver object (vector field arrows).
- lines[2] = list of Line2D objects (equilibrium points).
* cplt.axes: 2D array of `matplotlib.axes.Axes` for the plot.
* cplt.figure: `matplotlib.figure.Figure` containing the plot.
See `ControlPlot` for more detailed information.
Other Parameters
----------------
arrows : int
Set the number of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrows']`.
arrow_size : float
Set the size of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_size']`.
arrow_style : matplotlib patch
Set the style of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_style']`.
dir : str, optional
Direction to draw streamlines: 'forward' to flow forward in time
from the reference points, 'reverse' to flow backward in time, or
'both' to flow both forward and backward. The amount of time to
simulate in each direction is given by the `timedata` argument.
plot_streamlines : bool or dict, optional
If True (default) then plot streamlines based on the pointdata
and gridtype. If set to a dict, pass on the key-value pairs in
the dict as keywords to `streamlines`.
plot_vectorfield : bool or dict, optional
If True (default) then plot the vector field based on the pointdata
and gridtype. If set to a dict, pass on the key-value pairs in
the dict as keywords to `phaseplot.vectorfield`.
plot_equilpoints : bool or dict, optional
If True (default) then plot equilibrium points based in the phase
plot boundary. If set to a dict, pass on the key-value pairs in the
dict as keywords to `phaseplot.equilpoints`.
plot_separatrices : bool or dict, optional
If True (default) then plot separatrices starting from each
equilibrium point. If set to a dict, pass on the key-value pairs
in the dict as keywords to `phaseplot.separatrices`.
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
title : str, optional
Set the title of the plot. Defaults to plot type and system name(s).
"""
# Process arguments
params = kwargs.get('params', None)
sys = _create_system(sys, params)
pointdata = [-1, 1, -1, 1] if pointdata is None else pointdata
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Create axis if needed
user_ax = ax
fig, ax = _process_ax_keyword(user_ax, squeeze=True, rcParams=rcParams)
# Create copy of kwargs for later checking to find unused arguments
initial_kwargs = dict(kwargs)
# Utility function to create keyword arguments
def _create_kwargs(global_kwargs, local_kwargs, **other_kwargs):
new_kwargs = dict(global_kwargs)
new_kwargs.update(other_kwargs)
if isinstance(local_kwargs, dict):
new_kwargs.update(local_kwargs)
return new_kwargs
# Create list for storing outputs
out = np.array([[], None, None], dtype=object)
# Plot out the main elements
if plot_streamlines:
kwargs_local = _create_kwargs(
kwargs, plot_streamlines, gridspec=gridspec, gridtype=gridtype,
ax=ax)
out[0] += streamlines(
sys, pointdata, timedata, _check_kwargs=False,
suppress_warnings=suppress_warnings, **kwargs_local)
# Get rid of keyword arguments handled by streamlines
for kw in ['arrows', 'arrow_size', 'arrow_style', 'color',
'dir', 'params']:
initial_kwargs.pop(kw, None)
# Reset the gridspec for the remaining commands, if needed
if gridtype not in [None, 'boxgrid', 'meshgrid']:
gridspec = None
if plot_separatrices:
kwargs_local = _create_kwargs(
kwargs, plot_separatrices, gridspec=gridspec, ax=ax)
out[0] += separatrices(
sys, pointdata, _check_kwargs=False, **kwargs_local)
# Get rid of keyword arguments handled by separatrices
for kw in ['arrows', 'arrow_size', 'arrow_style', 'params']:
initial_kwargs.pop(kw, None)
if plot_vectorfield:
kwargs_local = _create_kwargs(
kwargs, plot_vectorfield, gridspec=gridspec, ax=ax)
out[1] = vectorfield(
sys, pointdata, _check_kwargs=False, **kwargs_local)
# Get rid of keyword arguments handled by vectorfield
for kw in ['color', 'params']:
initial_kwargs.pop(kw, None)
if plot_equilpoints:
kwargs_local = _create_kwargs(
kwargs, plot_equilpoints, gridspec=gridspec, ax=ax)
out[2] = equilpoints(
sys, pointdata, _check_kwargs=False, **kwargs_local)
# Get rid of keyword arguments handled by equilpoints
for kw in ['params']:
initial_kwargs.pop(kw, None)
# Make sure all keyword arguments were used
if initial_kwargs:
raise TypeError("unrecognized keywords: ", str(initial_kwargs))
if user_ax is None:
if title is None:
title = f"Phase portrait for {sys.name}"
_update_plot_title(title, use_existing=False, rcParams=rcParams)
ax.set_xlabel(sys.state_labels[0])
ax.set_ylabel(sys.state_labels[1])
plt.tight_layout()
return ControlPlot(out, ax, fig)
def vectorfield(
sys, pointdata, gridspec=None, ax=None, suppress_warnings=False,
_check_kwargs=True, **kwargs):
"""Plot a vector field in the phase plane.
This function plots a vector field for a two-dimensional state
space system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot the vector field in the given color.
ax : matplotlib.axes.Axes
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : Quiver
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Determine the points on which to generate the vector field
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the plotting limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use
color = _get_color(kwargs, ax=ax)
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Generate phase plane (quiver) data
vfdata = np.zeros((points.shape[0], 4))
sys._update_params(params)
for i, x in enumerate(points):
vfdata[i, :2] = x
vfdata[i, 2:] = sys._rhs(0, x, np.zeros(sys.ninputs))
with plt.rc_context(rcParams):
out = ax.quiver(
vfdata[:, 0], vfdata[:, 1], vfdata[:, 2], vfdata[:, 3],
angles='xy', color=color)
return out
def streamlines(
sys, pointdata, timedata=1, gridspec=None, gridtype=None, dir=None,
ax=None, _check_kwargs=True, suppress_warnings=False, **kwargs):
"""Plot stream lines in the phase plane.
This function plots stream lines for a two-dimensional state space
system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int or list of int
Time to simulate each streamline. If a list is given, a different
time can be used for each initial condition in `pointdata`.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
dir : str, optional
Direction to draw streamlines: 'forward' to flow forward in time
from the reference points, 'reverse' to flow backward in time, or
'both' to flow both forward and backward. The amount of time to
simulate in each direction is given by the `timedata` argument.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : str
Plot the streamlines in the given color.
ax : matplotlib.axes.Axes
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
arrows : int
Set the number of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrows']`.
arrow_size : float
Set the size of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_size']`.
arrow_style : matplotlib patch
Set the style of arrows to plot along the streamlines. The default
value can be set in `config.defaults['phaseplot.arrow_style']`.
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Parse the arrows keyword
arrow_pos, arrow_style = _parse_arrow_keywords(kwargs)
# Determine the points on which to generate the streamlines
points, gridspec = _make_points(pointdata, gridspec, gridtype=gridtype)
if dir is None:
dir = 'both' if gridtype == 'meshgrid' else 'forward'
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use
color = _get_color(kwargs, ax=ax)
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Create reverse time system, if needed
if dir != 'forward':
revsys = NonlinearIOSystem(
lambda t, x, u, params: -np.asarray(sys.updfcn(t, x, u, params)),
sys.outfcn, states=sys.nstates, inputs=sys.ninputs,
outputs=sys.noutputs, params=sys.params)
else:
revsys = None
# Generate phase plane (streamline) data
out = []
for i, X0 in enumerate(points):
# Create the trajectory for this point
timepts = _make_timepts(timedata, i)
traj = _create_trajectory(
sys, revsys, timepts, X0, params, dir,
gridtype=gridtype, gridspec=gridspec, xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
# Plot the trajectory (if there is one)
if traj.shape[1] > 1:
with plt.rc_context(rcParams):
out += ax.plot(traj[0], traj[1], color=color)
# Add arrows to the lines at specified intervals
_add_arrows_to_line2D(
ax, out[-1], arrow_pos, arrowstyle=arrow_style, dir=1)
return out
def equilpoints(
sys, pointdata, gridspec=None, color='k', ax=None,
_check_kwargs=True, **kwargs):
"""Plot equilibrium points in the phase plane.
This function plots the equilibrium points for a planar dynamical system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : str
Plot the equilibrium points in the given color.
ax : matplotlib.axes.Axes
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Determine the points on which to generate the vector field
gridspec = [5, 5] if gridspec is None else gridspec
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Search for equilibrium points
equilpts = _find_equilpts(sys, points, params=params)
# Plot the equilibrium points
out = []
for xeq in equilpts:
with plt.rc_context(rcParams):
out += ax.plot(xeq[0], xeq[1], marker='o', color=color)
return out
def separatrices(
sys, pointdata, timedata=None, gridspec=None, ax=None,
_check_kwargs=True, suppress_warnings=False, **kwargs):
"""Plot separatrices in the phase plane.
This function plots separatrices for a two-dimensional state space
system.
Parameters
----------
sys : `NonlinearIOSystem` or callable(t, x, ...)
I/O system or function used to generate phase plane data. If a
function is given, the remaining arguments are drawn from the
`params` keyword.
pointdata : list or 2D array
List of the form [xmin, xmax, ymin, ymax] describing the
boundaries of the phase plot or an array of shape (N, 2)
giving points of at which to plot the vector field.
timedata : int or list of int
Time to simulate each streamline. If a list is given, a different
time can be used for each initial condition in `pointdata`.
gridtype : str, optional
The type of grid to use for generating initial conditions:
'meshgrid' (default) generates a mesh of initial conditions within
the specified boundaries, 'boxgrid' generates initial conditions
along the edges of the boundary, 'circlegrid' generates a circle of
initial conditions around each point in point data.
gridspec : list, optional
If the gridtype is 'meshgrid' and 'boxgrid', `gridspec` gives the
size of the grid in the x and y axes on which to generate points.
If gridtype is 'circlegrid', then `gridspec` is a 2-tuple
specifying the radius and number of points around each point in the
`pointdata` array.
params : dict or list, optional
Parameters to pass to system. For an I/O system, `params` should be
a dict of parameters and values. For a callable, `params` should be
dict with key 'args' and value given by a tuple (passed to callable).
color : matplotlib color spec, optional
Plot the separatrics in the given color. If a single color
specification is given, this is used for both stable and unstable
separatrices. If a tuple is given, the first element is used as
the color specification for stable separatrices and the second
elmeent for unstable separatrices.
ax : matplotlib.axes.Axes
Use the given axes for the plot, otherwise use the current axes.
Returns
-------
out : list of Line2D objects
Other Parameters
----------------
rcParams : dict
Override the default parameters used for generating plots.
Default is set by `config.defaults['ctrlplot.rcParams']`.
suppress_warnings : bool, optional
If set to True, suppress warning messages in generating trajectories.
Notes
-----
The value of `config.defaults['separatrices_radius']` is used to set the
offset from the equlibrium point to the starting point of the separatix
traces, in the direction of the eigenvectors evaluated at that
equilibrium point.
"""
# Process keywords
rcParams = config._get_param('ctrlplot', 'rcParams', kwargs, pop=True)
# Get system parameters
params = kwargs.pop('params', None)
# Create system from callable, if needed
sys = _create_system(sys, params)
# Parse the arrows keyword
arrow_pos, arrow_style = _parse_arrow_keywords(kwargs)
# Determine the initial states to use in searching for equilibrium points
gridspec = [5, 5] if gridspec is None else gridspec
points, _ = _make_points(pointdata, gridspec, 'meshgrid')
# Find the equilibrium points
equilpts = _find_equilpts(sys, points, params=params)
radius = config._get_param('phaseplot', 'separatrices_radius')
# Create axis if needed
if ax is None:
ax = plt.gca()
# Set the axis limits
xlim, ylim, maxlim = _set_axis_limits(ax, pointdata)
# Figure out the color to use for stable, unstable subspaces
color = _get_color(kwargs)
match color:
case None:
stable_color = 'r'
unstable_color = 'b'
case (stable_color, unstable_color) | [stable_color, unstable_color]:
pass
case single_color:
stable_color = unstable_color = single_color
# Make sure all keyword arguments were processed
if _check_kwargs and kwargs:
raise TypeError("unrecognized keywords: ", str(kwargs))
# Create a "reverse time" system to use for simulation
revsys = NonlinearIOSystem(
lambda t, x, u, params: -np.array(sys.updfcn(t, x, u, params)),
sys.outfcn, states=sys.nstates, inputs=sys.ninputs,
outputs=sys.noutputs, params=sys.params)
# Plot separatrices by flowing backwards in time along eigenspaces
out = []
for i, xeq in enumerate(equilpts):
# Plot the equilibrium points
with plt.rc_context(rcParams):
out += ax.plot(xeq[0], xeq[1], marker='o', color='k')
# Figure out the linearization and eigenvectors
evals, evecs = np.linalg.eig(sys.linearize(xeq, 0, params=params).A)
# See if we have real eigenvalues (=> evecs are meaningful)
if evals[0].imag > 0:
continue
# Create default list of time points
if timedata is not None:
timepts = _make_timepts(timedata, i)
# Generate the traces
for j, dir in enumerate(evecs.T):
# Figure out time vector if not yet computed
if timedata is None:
timescale = math.log(maxlim / radius) / abs(evals[j].real)
timepts = np.linspace(0, timescale)
# Run the trajectory starting in eigenvector directions
for eps in [-radius, radius]:
x0 = xeq + dir * eps
if evals[j].real < 0:
traj = _create_trajectory(
sys, revsys, timepts, x0, params, 'reverse',
gridtype='boxgrid', xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
color = stable_color
linestyle = '--'
elif evals[j].real > 0:
traj = _create_trajectory(
sys, revsys, timepts, x0, params, 'forward',
gridtype='boxgrid', xlim=xlim, ylim=ylim,
suppress_warnings=suppress_warnings)
color = unstable_color
linestyle = '-'
# Plot the trajectory (if there is one)
if traj.shape[1] > 1:
with plt.rc_context(rcParams):
out += ax.plot(
traj[0], traj[1], color=color, linestyle=linestyle)
# Add arrows to the lines at specified intervals
with plt.rc_context(rcParams):
_add_arrows_to_line2D(
ax, out[-1], arrow_pos, arrowstyle=arrow_style,
dir=1)
return out
#
# User accessible utility functions
#
# Utility function to generate boxgrid (in the form needed here)
def boxgrid(xvals, yvals):
"""Generate list of points along the edge of box.
points = boxgrid(xvals, yvals) generates a list of points that
corresponds to a grid given by the cross product of the x and y values.
Parameters
----------
xvals, yvals : 1D array_like
Array of points defining the points on the lower and left edges of
the box.
Returns
-------
grid : 2D array
Array with shape (p, 2) defining the points along the edges of the
box, where p is the number of points around the edge.
"""
return np.array(
[(x, yvals[0]) for x in xvals[:-1]] + # lower edge
[(xvals[-1], y) for y in yvals[:-1]] + # right edge
[(x, yvals[-1]) for x in xvals[:0:-1]] + # upper edge
[(xvals[0], y) for y in yvals[:0:-1]] # left edge
)
# Utility function to generate meshgrid (in the form needed here)
# TODO: add examples of using grid functions directly
def meshgrid(xvals, yvals):
"""Generate list of points forming a mesh.
points = meshgrid(xvals, yvals) generates a list of points that
corresponds to a grid given by the cross product of the x and y values.
Parameters
----------
xvals, yvals : 1D array_like
Array of points defining the points on the lower and left edges of
the box.
Returns
-------
grid : 2D array
Array of points with shape (n * m, 2) defining the mesh.
"""
xvals, yvals = np.meshgrid(xvals, yvals)
grid = np.zeros((xvals.shape[0] * xvals.shape[1], 2))
grid[:, 0] = xvals.reshape(-1)
grid[:, 1] = yvals.reshape(-1)
return grid
# Utility function to generate circular grid
def circlegrid(centers, radius, num):
"""Generate list of points around a circle.
points = circlegrid(centers, radius, num) generates a list of points
that form a circle around a list of centers.
Parameters
----------
centers : 2D array_like
Array of points with shape (p, 2) defining centers of the circles.
radius : float
Radius of the points to be generated around each center.
num : int
Number of points to generate around the circle.
Returns
-------
grid : 2D array
Array of points with shape (p * num, 2) defining the circles.
"""
centers = np.atleast_2d(np.array(centers))
grid = np.zeros((centers.shape[0] * num, 2))
for i, center in enumerate(centers):
grid[i * num: (i + 1) * num, :] = center + np.array([
[radius * math.cos(theta), radius * math.sin(theta)] for
theta in np.linspace(0, 2 * math.pi, num, endpoint=False)])
return grid
#
# Internal utility functions
#
# Create a system from a callable
def _create_system(sys, params):
if isinstance(sys, NonlinearIOSystem):
if sys.nstates != 2:
raise ValueError("system must be planar")
return sys
# Make sure that if params is present, it has 'args' key
if params and not params.get('args', None):
raise ValueError("params must be dict with key 'args'")
_update = lambda t, x, u, params: sys(t, x, *params.get('args', ()))
_output = lambda t, x, u, params: np.array([])
return NonlinearIOSystem(
_update, _output, states=2, inputs=0, outputs=0, name="_callable")
# Set axis limits for the plot
def _set_axis_limits(ax, pointdata):
# Get the current axis limits
if ax.lines:
xlim, ylim = ax.get_xlim(), ax.get_ylim()
else:
# Nothing on the plot => always use new limits
xlim, ylim = [np.inf, -np.inf], [np.inf, -np.inf]
# Short utility function for updating axis limits
def _update_limits(cur, new):
return [min(cur[0], np.min(new)), max(cur[1], np.max(new))]
# If we were passed a box, use that to update the limits
if isinstance(pointdata, list) and len(pointdata) == 4:
xlim = _update_limits(xlim, [pointdata[0], pointdata[1]])
ylim = _update_limits(ylim, [pointdata[2], pointdata[3]])
elif isinstance(pointdata, np.ndarray):
pointdata = np.atleast_2d(pointdata)
xlim = _update_limits(
xlim, [np.min(pointdata[:, 0]), np.max(pointdata[:, 0])])
ylim = _update_limits(
ylim, [np.min(pointdata[:, 1]), np.max(pointdata[:, 1])])
# Keep track of the largest dimension on the plot
maxlim = max(xlim[1] - xlim[0], ylim[1] - ylim[0])
# Set the new limits
ax.autoscale(enable=True, axis='x', tight=True)
ax.autoscale(enable=True, axis='y', tight=True)
ax.set_xlim(xlim)
ax.set_ylim(ylim)
return xlim, ylim, maxlim
# Find equilibrium points
def _find_equilpts(sys, points, params=None):
equilpts = []
for i, x0 in enumerate(points):
# Look for an equilibrium point near this point
xeq, ueq = find_operating_point(sys, x0, 0, params=params)
if xeq is None:
continue # didn't find anything
# See if we have already found this point
seen = False
for x in equilpts:
if np.allclose(np.array(x), xeq):
seen = True
if seen:
continue
# Save a new point
equilpts += [xeq.tolist()]
return equilpts
def _make_points(pointdata, gridspec, gridtype):
# Check to see what type of data we got
if isinstance(pointdata, np.ndarray) and gridtype is None:
pointdata = np.atleast_2d(pointdata)
if pointdata.shape[1] == 2:
# Given a list of points => no action required
return pointdata, None
# Utility function to parse (and check) input arguments
def _parse_args(defsize):
if gridspec is None:
return defsize
elif not isinstance(gridspec, (list, tuple)) or \
len(gridspec) != len(defsize):
raise ValueError("invalid grid specification")
return gridspec
# Generate points based on grid type
match gridtype:
case 'boxgrid' | None:
gridspec = _parse_args([6, 4])
points = boxgrid(
np.linspace(pointdata[0], pointdata[1], gridspec[0]),
np.linspace(pointdata[2], pointdata[3], gridspec[1]))
case 'meshgrid':
gridspec = _parse_args([9, 6])
points = meshgrid(
np.linspace(pointdata[0], pointdata[1], gridspec[0]),
np.linspace(pointdata[2], pointdata[3], gridspec[1]))
case 'circlegrid':
gridspec = _parse_args((0.5, 10))
if isinstance(pointdata, np.ndarray):
# Create circles around each point
points = circlegrid(pointdata, gridspec[0], gridspec[1])
else:
# Create circle around center of the plot
points = circlegrid(
np.array(
[(pointdata[0] + pointdata[1]) / 2,
(pointdata[0] + pointdata[1]) / 2]),
gridspec[0], gridspec[1])
case _:
raise ValueError(f"unknown grid type '{gridtype}'")
return points, gridspec
def _parse_arrow_keywords(kwargs):
# Get values for params (and pop from list to allow keyword use in plot)
# TODO: turn this into a utility function (shared with nyquist_plot?)
arrows = config._get_param(
'phaseplot', 'arrows', kwargs, None, pop=True)
arrow_size = config._get_param(
'phaseplot', 'arrow_size', kwargs, None, pop=True)
arrow_style = config._get_param('phaseplot', 'arrow_style', kwargs, None)
# Parse the arrows keyword
if not arrows:
arrow_pos = []
elif isinstance(arrows, int):
N = arrows
# Space arrows out, starting midway along each "region"
arrow_pos = np.linspace(0.5/N, 1 + 0.5/N, N, endpoint=False)
elif isinstance(arrows, (list, np.ndarray)):
arrow_pos = np.sort(np.atleast_1d(arrows))
else:
raise ValueError("unknown or unsupported arrow location")
# Set the arrow style
if arrow_style is None:
arrow_style = mpl.patches.ArrowStyle(
'simple', head_width=int(2 * arrow_size / 3),
head_length=arrow_size)
return arrow_pos, arrow_style
# TODO: move to ctrlplot?
def _create_trajectory(
sys, revsys, timepts, X0, params, dir, suppress_warnings=False,
gridtype=None, gridspec=None, xlim=None, ylim=None):
# Comput ethe forward trajectory
if dir == 'forward' or dir == 'both':
fwdresp = input_output_response(
sys, timepts, X0=X0, params=params, ignore_errors=True)
if not fwdresp.success and not suppress_warnings:
warnings.warn(f"{X0=}, {fwdresp.message}")
# Compute the reverse trajectory
if dir == 'reverse' or dir == 'both':
revresp = input_output_response(
revsys, timepts, X0=X0, params=params, ignore_errors=True)
if not revresp.success and not suppress_warnings:
warnings.warn(f"{X0=}, {revresp.message}")
# Create the trace to plot
if dir == 'forward':
traj = fwdresp.states
elif dir == 'reverse':
traj = revresp.states[:, ::-1]
elif dir == 'both':
traj = np.hstack([revresp.states[:, :1:-1], fwdresp.states])