# freqplot.py - frequency domain plots for control systems # # Author: Richard M. Murray # Date: 24 May 09 # # This file contains some standard control system plots: Bode plots, # Nyquist plots and pole-zero diagrams. The code for Nichols charts # is in nichols.py. # # Copyright (c) 2010 by California Institute of Technology # All rights reserved. # # Redistribution and use in source and binary forms, with or without # modification, are permitted provided that the following conditions # are met: # # 1. Redistributions of source code must retain the above copyright # notice, this list of conditions and the following disclaimer. # # 2. Redistributions in binary form must reproduce the above copyright # notice, this list of conditions and the following disclaimer in the # documentation and/or other materials provided with the distribution. # # 3. Neither the name of the California Institute of Technology nor # the names of its contributors may be used to endorse or promote # products derived from this software without specific prior # written permission. # # THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS # "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT # LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS # FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL CALTECH # OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, # SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT # LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF # USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND # ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, # OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT # OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF # SUCH DAMAGE. # # $Id$ import math import matplotlib as mpl import matplotlib.pyplot as plt import numpy as np import warnings from math import nan from .ctrlutil import unwrap from .bdalg import feedback from .margins import stability_margins from .exception import ControlMIMONotImplemented from .statesp import StateSpace from .xferfcn import TransferFunction from . import config __all__ = ['bode_plot', 'nyquist_plot', 'gangof4_plot', 'singular_values_plot', 'bode', 'nyquist', 'gangof4'] # Default values for module parameter variables _freqplot_defaults = { 'freqplot.feature_periphery_decades': 1, 'freqplot.number_of_samples': 1000, 'freqplot.dB': False, # Plot gain in dB 'freqplot.deg': True, # Plot phase in degrees 'freqplot.Hz': False, # Plot frequency in Hertz 'freqplot.grid': True, # Turn on grid for gain and phase 'freqplot.wrap_phase': False, # Wrap the phase plot at a given value # deprecations 'deprecated.bode.dB': 'freqplot.dB', 'deprecated.bode.deg': 'freqplot.deg', 'deprecated.bode.Hz': 'freqplot.Hz', 'deprecated.bode.grid': 'freqplot.grid', 'deprecated.bode.wrap_phase': 'freqplot.wrap_phase', } # # Main plotting functions # # This section of the code contains the functions for generating # frequency domain plots # # # Bode plot # def bode_plot(syslist, omega=None, plot=True, omega_limits=None, omega_num=None, margins=None, method='best', *args, **kwargs): """Bode plot for a system Plots a Bode plot for the system over a (optional) frequency range. Parameters ---------- syslist : linsys List of linear input/output systems (single system is OK) omega : array_like List of frequencies in rad/sec to be used for frequency response dB : bool If True, plot result in dB. Default is false. Hz : bool If True, plot frequency in Hz (omega must be provided in rad/sec). Default value (False) set by config.defaults['freqplot.Hz'] deg : bool If True, plot phase in degrees (else radians). Default value (True) config.defaults['freqplot.deg'] plot : bool If True (default), plot magnitude and phase omega_limits : array_like of two values Limits of the to generate frequency vector. If Hz=True the limits are in Hz otherwise in rad/s. omega_num : int Number of samples to plot. Defaults to config.defaults['freqplot.number_of_samples']. margins : bool If True, plot gain and phase margin. method : method to use in computing margins (see :func:`stability_margins`) *args : :func:`matplotlib.pyplot.plot` positional properties, optional Additional arguments for `matplotlib` plots (color, linestyle, etc) **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- mag : ndarray (or list of ndarray if len(syslist) > 1)) magnitude phase : ndarray (or list of ndarray if len(syslist) > 1)) phase in radians omega : ndarray (or list of ndarray if len(syslist) > 1)) frequency in rad/sec Other Parameters ---------------- grid : bool If True, plot grid lines on gain and phase plots. Default is set by `config.defaults['freqplot.grid']`. initial_phase : float Set the reference phase to use for the lowest frequency. If set, the initial phase of the Bode plot will be set to the value closest to the value specified. Units are in either degrees or radians, depending on the `deg` parameter. Default is -180 if wrap_phase is False, 0 if wrap_phase is True. wrap_phase : bool or float If wrap_phase is `False` (default), then the phase will be unwrapped so that it is continuously increasing or decreasing. If wrap_phase is `True` the phase will be restricted to the range [-180, 180) (or [:math:`-\\pi`, :math:`\\pi`) radians). If `wrap_phase` is specified as a float, the phase will be offset by 360 degrees if it falls below the specified value. Default value is `False` and can be set using config.defaults['freqplot.wrap_phase']. The default values for Bode plot configuration parameters can be reset using the `config.defaults` dictionary, with module name 'bode'. Notes ----- 1. Alternatively, you may use the lower-level methods :meth:`LTI.frequency_response` or ``sys(s)`` or ``sys(z)`` or to generate the frequency response for a single system. 2. If a discrete time model is given, the frequency response is plotted along the upper branch of the unit circle, using the mapping ``z = exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt` is the discrete timebase. If timebase not specified (``dt=True``), `dt` is set to 1. Examples -------- >>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.") >>> mag, phase, omega = bode(sys) """ # Make a copy of the kwargs dictionary since we will modify it kwargs = dict(kwargs) # Check to see if legacy 'Plot' keyword was used if 'Plot' in kwargs: import warnings warnings.warn("'Plot' keyword is deprecated in bode_plot; use 'plot'", FutureWarning) # Map 'Plot' keyword to 'plot' keyword plot = kwargs.pop('Plot') # Get values for params (and pop from list to allow keyword use in plot) dB = config._get_param( 'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True) deg = config._get_param( 'freqplot', 'deg', kwargs, _freqplot_defaults, pop=True) Hz = config._get_param( 'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True) grid = config._get_param( 'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True) plot = config._get_param('freqplot', 'plot', plot, True) margins = config._get_param( 'freqplot', 'margins', margins, False) wrap_phase = config._get_param( 'freqplot', 'wrap_phase', kwargs, _freqplot_defaults, pop=True) initial_phase = config._get_param( 'freqplot', 'initial_phase', kwargs, None, pop=True) omega_num = config._get_param('freqplot', 'number_of_samples', omega_num) # If argument was a singleton, turn it into a tuple if not isinstance(syslist, (list, tuple)): syslist = (syslist,) omega, omega_range_given = _determine_omega_vector( syslist, omega, omega_limits, omega_num, Hz=Hz) if plot: # Set up the axes with labels so that multiple calls to # bode_plot will superimpose the data. This was implicit # before matplotlib 2.1, but changed after that (See # https://github.com/matplotlib/matplotlib/issues/9024). # The code below should work on all cases. # Get the current figure if 'sisotool' in kwargs: fig = kwargs.pop('fig') ax_mag = fig.axes[0] ax_phase = fig.axes[2] sisotool = kwargs.pop('sisotool') else: fig = plt.gcf() ax_mag = None ax_phase = None sisotool = False # Get the current axes if they already exist for ax in fig.axes: if ax.get_label() == 'control-bode-magnitude': ax_mag = ax elif ax.get_label() == 'control-bode-phase': ax_phase = ax # If no axes present, create them from scratch if ax_mag is None or ax_phase is None: plt.clf() ax_mag = plt.subplot(211, label='control-bode-magnitude') ax_phase = plt.subplot( 212, label='control-bode-phase', sharex=ax_mag) mags, phases, omegas, nyquistfrqs = [], [], [], [] for sys in syslist: if not sys.issiso(): # TODO: Add MIMO bode plots. raise ControlMIMONotImplemented( "Bode is currently only implemented for SISO systems.") else: omega_sys = np.asarray(omega) if sys.isdtime(strict=True): nyquistfrq = math.pi / sys.dt if not omega_range_given: # limit up to and including nyquist frequency omega_sys = np.hstack(( omega_sys[omega_sys < nyquistfrq], nyquistfrq)) else: nyquistfrq = None mag, phase, omega_sys = sys.frequency_response(omega_sys) mag = np.atleast_1d(mag) phase = np.atleast_1d(phase) # # Post-process the phase to handle initial value and wrapping # if initial_phase is None: # Start phase in the range 0 to -360 w/ initial phase = -180 # If wrap_phase is true, use 0 instead (phase \in (-pi, pi]) initial_phase = -math.pi if wrap_phase is not True else 0 elif isinstance(initial_phase, (int, float)): # Allow the user to override the default calculation if deg: initial_phase = initial_phase/180. * math.pi else: raise ValueError("initial_phase must be a number.") # Shift the phase if needed if abs(phase[0] - initial_phase) > math.pi: phase -= 2*math.pi * \ round((phase[0] - initial_phase) / (2*math.pi)) # Phase wrapping if wrap_phase is False: phase = unwrap(phase) # unwrap the phase elif wrap_phase is True: pass # default calculation OK elif isinstance(wrap_phase, (int, float)): phase = unwrap(phase) # unwrap the phase first if deg: wrap_phase *= math.pi/180. # Shift the phase if it is below the wrap_phase phase += 2*math.pi * np.maximum( 0, np.ceil((wrap_phase - phase)/(2*math.pi))) else: raise ValueError("wrap_phase must be bool or float.") mags.append(mag) phases.append(phase) omegas.append(omega_sys) nyquistfrqs.append(nyquistfrq) # Get the dimensions of the current axis, which we will divide up # TODO: Not current implemented; just use subplot for now if plot: nyquistfrq_plot = None if Hz: omega_plot = omega_sys / (2. * math.pi) if nyquistfrq: nyquistfrq_plot = nyquistfrq / (2. * math.pi) else: omega_plot = omega_sys if nyquistfrq: nyquistfrq_plot = nyquistfrq phase_plot = phase * 180. / math.pi if deg else phase mag_plot = mag if nyquistfrq_plot: # append data for vertical nyquist freq indicator line. # if this extra nyquist lime is is plotted in a single plot # command then line order is preserved when # creating a legend eg. legend(('sys1', 'sys2')) omega_nyq_line = np.array( (np.nan, nyquistfrq_plot, nyquistfrq_plot)) omega_plot = np.hstack((omega_plot, omega_nyq_line)) mag_nyq_line = np.array(( np.nan, 0.7*min(mag_plot), 1.3*max(mag_plot))) mag_plot = np.hstack((mag_plot, mag_nyq_line)) phase_range = max(phase_plot) - min(phase_plot) phase_nyq_line = np.array( (np.nan, min(phase_plot) - 0.2 * phase_range, max(phase_plot) + 0.2 * phase_range)) phase_plot = np.hstack((phase_plot, phase_nyq_line)) # # Magnitude plot # if dB: ax_mag.semilogx(omega_plot, 20 * np.log10(mag_plot), *args, **kwargs) else: ax_mag.loglog(omega_plot, mag_plot, *args, **kwargs) # Add a grid to the plot + labeling ax_mag.grid(grid and not margins, which='both') ax_mag.set_ylabel("Magnitude (dB)" if dB else "Magnitude") # # Phase plot # # Plot the data ax_phase.semilogx(omega_plot, phase_plot, *args, **kwargs) # Show the phase and gain margins in the plot if margins: # Compute stability margins for the system margin = stability_margins(sys, method=method) gm, pm, Wcg, Wcp = (margin[i] for i in (0, 1, 3, 4)) # Figure out sign of the phase at the first gain crossing # (needed if phase_wrap is True) phase_at_cp = phases[0][(np.abs(omegas[0] - Wcp)).argmin()] if phase_at_cp >= 0.: phase_limit = 180. else: phase_limit = -180. if Hz: Wcg, Wcp = Wcg/(2*math.pi), Wcp/(2*math.pi) # Draw lines at gain and phase limits ax_mag.axhline(y=0 if dB else 1, color='k', linestyle=':', zorder=-20) ax_phase.axhline(y=phase_limit if deg else math.radians(phase_limit), color='k', linestyle=':', zorder=-20) mag_ylim = ax_mag.get_ylim() phase_ylim = ax_phase.get_ylim() # Annotate the phase margin (if it exists) if pm != float('inf') and Wcp != float('nan'): if dB: ax_mag.semilogx( [Wcp, Wcp], [0., -1e5], color='k', linestyle=':', zorder=-20) else: ax_mag.loglog( [Wcp, Wcp], [1., 1e-8], color='k', linestyle=':', zorder=-20) if deg: ax_phase.semilogx( [Wcp, Wcp], [1e5, phase_limit + pm], color='k', linestyle=':', zorder=-20) ax_phase.semilogx( [Wcp, Wcp], [phase_limit + pm, phase_limit], color='k', zorder=-20) else: ax_phase.semilogx( [Wcp, Wcp], [1e5, math.radians(phase_limit) + math.radians(pm)], color='k', linestyle=':', zorder=-20) ax_phase.semilogx( [Wcp, Wcp], [math.radians(phase_limit) + math.radians(pm), math.radians(phase_limit)], color='k', zorder=-20) # Annotate the gain margin (if it exists) if gm != float('inf') and Wcg != float('nan'): if dB: ax_mag.semilogx( [Wcg, Wcg], [-20.*np.log10(gm), -1e5], color='k', linestyle=':', zorder=-20) ax_mag.semilogx( [Wcg, Wcg], [0, -20*np.log10(gm)], color='k', zorder=-20) else: ax_mag.loglog( [Wcg, Wcg], [1./gm, 1e-8], color='k', linestyle=':', zorder=-20) ax_mag.loglog( [Wcg, Wcg], [1., 1./gm], color='k', zorder=-20) if deg: ax_phase.semilogx( [Wcg, Wcg], [0, phase_limit], color='k', linestyle=':', zorder=-20) else: ax_phase.semilogx( [Wcg, Wcg], [0, math.radians(phase_limit)], color='k', linestyle=':', zorder=-20) ax_mag.set_ylim(mag_ylim) ax_phase.set_ylim(phase_ylim) if sisotool: ax_mag.text( 0.04, 0.06, 'G.M.: %.2f %s\nFreq: %.2f %s' % (20*np.log10(gm) if dB else gm, 'dB ' if dB else '', Wcg, 'Hz' if Hz else 'rad/s'), horizontalalignment='left', verticalalignment='bottom', transform=ax_mag.transAxes, fontsize=8 if int(mpl.__version__[0]) == 1 else 6) ax_phase.text( 0.04, 0.06, 'P.M.: %.2f %s\nFreq: %.2f %s' % (pm if deg else math.radians(pm), 'deg' if deg else 'rad', Wcp, 'Hz' if Hz else 'rad/s'), horizontalalignment='left', verticalalignment='bottom', transform=ax_phase.transAxes, fontsize=8 if int(mpl.__version__[0]) == 1 else 6) else: plt.suptitle( "Gm = %.2f %s(at %.2f %s), " "Pm = %.2f %s (at %.2f %s)" % (20*np.log10(gm) if dB else gm, 'dB ' if dB else '', Wcg, 'Hz' if Hz else 'rad/s', pm if deg else math.radians(pm), 'deg' if deg else 'rad', Wcp, 'Hz' if Hz else 'rad/s')) # Add a grid to the plot + labeling ax_phase.set_ylabel("Phase (deg)" if deg else "Phase (rad)") def gen_zero_centered_series(val_min, val_max, period): v1 = np.ceil(val_min / period - 0.2) v2 = np.floor(val_max / period + 0.2) return np.arange(v1, v2 + 1) * period if deg: ylim = ax_phase.get_ylim() ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], 45.)) ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], 15.), minor=True) else: ylim = ax_phase.get_ylim() ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], math.pi / 4.)) ax_phase.set_yticks(gen_zero_centered_series( ylim[0], ylim[1], math.pi / 12.), minor=True) ax_phase.grid(grid and not margins, which='both') # ax_mag.grid(which='minor', alpha=0.3) # ax_mag.grid(which='major', alpha=0.9) # ax_phase.grid(which='minor', alpha=0.3) # ax_phase.grid(which='major', alpha=0.9) # Label the frequency axis ax_phase.set_xlabel("Frequency (Hz)" if Hz else "Frequency (rad/sec)") if len(syslist) == 1: return mags[0], phases[0], omegas[0] else: return mags, phases, omegas # # Nyquist plot # # Default values for module parameter variables _nyquist_defaults = { 'nyquist.primary_style': ['-', '-.'], # style for primary curve 'nyquist.mirror_style': ['--', ':'], # style for mirror curve 'nyquist.arrows': 2, # number of arrors around curve 'nyquist.arrow_size': 8, # pixel size for arrows 'nyquist.encirclement_threshold': 0.05, # warning threshold 'nyquist.indent_radius': 1e-4, # indentation radius 'nyquist.indent_direction': 'right', # indentation direction 'nyquist.indent_points': 50, # number of points to insert 'nyquist.max_curve_magnitude': 20, # clip large values 'nyquist.max_curve_offset': 0.02, # offset of primary/mirror 'nyquist.start_marker': 'o', # marker at start of curve 'nyquist.start_marker_size': 4, # size of the maker } def nyquist_plot( syslist, omega=None, plot=True, omega_limits=None, omega_num=None, label_freq=0, color=None, return_contour=False, warn_encirclements=True, warn_nyquist=True, **kwargs): """Nyquist plot for a system Plots a Nyquist plot for the system over a (optional) frequency range. The curve is computed by evaluating the Nyqist segment along the positive imaginary axis, with a mirror image generated to reflect the negative imaginary axis. Poles on or near the imaginary axis are avoided using a small indentation. The portion of the Nyquist contour at infinity is not explicitly computed (since it maps to a constant value for any system with a proper transfer function). Parameters ---------- syslist : list of LTI List of linear input/output systems (single system is OK). Nyquist curves for each system are plotted on the same graph. plot : boolean If True, plot magnitude omega : array_like Set of frequencies to be evaluated, in rad/sec. omega_limits : array_like of two values Limits to the range of frequencies. Ignored if omega is provided, and auto-generated if omitted. omega_num : int Number of frequency samples to plot. Defaults to config.defaults['freqplot.number_of_samples']. color : string Used to specify the color of the line and arrowhead. return_contour : bool, optional If 'True', return the contour used to evaluate the Nyquist plot. **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- count : int (or list of int if len(syslist) > 1) Number of encirclements of the point -1 by the Nyquist curve. If multiple systems are given, an array of counts is returned. contour : ndarray (or list of ndarray if len(syslist) > 1)), optional The contour used to create the primary Nyquist curve segment, returned if `return_contour` is Tue. To obtain the Nyquist curve values, evaluate system(s) along contour. Additional Parameters --------------------- arrows : int or 1D/2D array of floats, optional Specify the number of arrows to plot on the Nyquist curve. If an integer is passed. that number of equally spaced arrows will be plotted on each of the primary segment and the mirror image. If a 1D array is passed, it should consist of a sorted list of floats between 0 and 1, indicating the location along the curve to plot an arrow. If a 2D array is passed, the first row will be used to specify arrow locations for the primary curve and the second row will be used for the mirror image. arrow_size : float, optional Arrowhead width and length (in display coordinates). Default value is 8 and can be set using config.defaults['nyquist.arrow_size']. arrow_style : matplotlib.patches.ArrowStyle, optional Define style used for Nyquist curve arrows (overrides `arrow_size`). encirclement_threshold : float, optional Define the threshold for generating a warning if the number of net encirclements is a non-integer value. Default value is 0.05 and can be set using config.defaults['nyquist.encirclement_threshold']. indent_direction : str, optional For poles on the imaginary axis, set the direction of indentation to be 'right' (default), 'left', or 'none'. indent_points : int, optional Number of points to insert in the Nyquist contour around poles that are at or near the imaginary axis. indent_radius : float, optional Amount to indent the Nyquist contour around poles on or near the imaginary axis. Portions of the Nyquist plot corresponding to indented portions of the contour are plotted using a different line style. label_freq : int, optiona Label every nth frequency on the plot. If not specified, no labels are generated. max_curve_magnitude : float, optional Restrict the maximum magnitude of the Nyquist plot to this value. Portions of the Nyquist plot whose magnitude is restricted are plotted using a different line style. max_curve_offset : float, optional When plotting scaled portion of the Nyquist plot, increase/decrease the magnitude by this fraction of the max_curve_magnitude to allow any overlaps between the primary and mirror curves to be avoided. mirror_style : [str, str] or False Linestyles for mirror image of the Nyquist curve. The first element is used for unscaled portions of the Nyquist curve, the second element is used for portions that are scaled (using max_curve_magnitude). If `False` then omit completely. Default linestyle (['--', ':']) is determined by config.defaults['nyquist.mirror_style']. primary_style : [str, str], optional Linestyles for primary image of the Nyquist curve. The first element is used for unscaled portions of the Nyquist curve, the second element is used for portions that are scaled (using max_curve_magnitude). Default linestyle (['-', '-.']) is determined by config.defaults['nyquist.mirror_style']. start_marker : str, optional Matplotlib marker to use to mark the starting point of the Nyquist plot. Defaults value is 'o' and can be set using config.defaults['nyquist.start_marker']. start_marker_size : float, optional Start marker size (in display coordinates). Default value is 4 and can be set using config.defaults['nyquist.start_marker_size']. warn_nyquist : bool, optional If set to 'False', turn off warnings about frequencies above Nyquist. warn_encirclements : bool, optional If set to 'False', turn off warnings about number of encirclements not meeting the Nyquist criterion. Notes ----- 1. If a discrete time model is given, the frequency response is computed along the upper branch of the unit circle, using the mapping ``z = exp(1j * omega * dt)`` where `omega` ranges from 0 to `pi/dt` and `dt` is the discrete timebase. If timebase not specified (``dt=True``), `dt` is set to 1. 2. If a continuous-time system contains poles on or near the imaginary axis, a small indentation will be used to avoid the pole. The radius of the indentation is given by `indent_radius` and it is taken to the right of stable poles and the left of unstable poles. If a pole is exactly on the imaginary axis, the `indent_direction` parameter can be used to set the direction of indentation. Setting `indent_direction` to `none` will turn off indentation. If `return_contour` is True, the exact contour used for evaluation is returned. Examples -------- >>> sys = ss([[1, -2], [3, -4]], [[5], [7]], [[6, 8]], [[9]]) >>> count = nyquist_plot(sys) """ # Check to see if legacy 'Plot' keyword was used if 'Plot' in kwargs: warnings.warn("'Plot' keyword is deprecated in nyquist_plot; " "use 'plot'", FutureWarning) # Map 'Plot' keyword to 'plot' keyword plot = kwargs.pop('Plot') # Check to see if legacy 'labelFreq' keyword was used if 'labelFreq' in kwargs: warnings.warn("'labelFreq' keyword is deprecated in nyquist_plot; " "use 'label_freq'", FutureWarning) # Map 'labelFreq' keyword to 'label_freq' keyword label_freq = kwargs.pop('labelFreq') # Check to see if legacy 'arrow_width' or 'arrow_length' were used if 'arrow_width' in kwargs or 'arrow_length' in kwargs: warnings.warn( "'arrow_width' and 'arrow_length' keywords are deprecated in " "nyquist_plot; use `arrow_size` instead", FutureWarning) kwargs['arrow_size'] = \ (kwargs.get('arrow_width', 0) + kwargs.get('arrow_length', 0)) / 2 kwargs.pop('arrow_width', False) kwargs.pop('arrow_length', False) # Get values for params (and pop from list to allow keyword use in plot) omega_num_given = omega_num is not None omega_num = config._get_param('freqplot', 'number_of_samples', omega_num) arrows = config._get_param( 'nyquist', 'arrows', kwargs, _nyquist_defaults, pop=True) arrow_size = config._get_param( 'nyquist', 'arrow_size', kwargs, _nyquist_defaults, pop=True) arrow_style = config._get_param('nyquist', 'arrow_style', kwargs, None) indent_radius = config._get_param( 'nyquist', 'indent_radius', kwargs, _nyquist_defaults, pop=True) encirclement_threshold = config._get_param( 'nyquist', 'encirclement_threshold', kwargs, _nyquist_defaults, pop=True) indent_direction = config._get_param( 'nyquist', 'indent_direction', kwargs, _nyquist_defaults, pop=True) indent_points = config._get_param( 'nyquist', 'indent_points', kwargs, _nyquist_defaults, pop=True) max_curve_magnitude = config._get_param( 'nyquist', 'max_curve_magnitude', kwargs, _nyquist_defaults, pop=True) max_curve_offset = config._get_param( 'nyquist', 'max_curve_offset', kwargs, _nyquist_defaults, pop=True) start_marker = config._get_param( 'nyquist', 'start_marker', kwargs, _nyquist_defaults, pop=True) start_marker_size = config._get_param( 'nyquist', 'start_marker_size', kwargs, _nyquist_defaults, pop=True) # Set line styles for the curves def _parse_linestyle(style_name, allow_false=False): style = config._get_param( 'nyquist', style_name, kwargs, _nyquist_defaults, pop=True) if isinstance(style, str): # Only one style provided, use the default for the other style = [style, _nyquist_defaults['nyquist.' + style_name][1]] warnings.warn( "use of a single string for linestyle will be deprecated " " in a future release", PendingDeprecationWarning) if (allow_false and style is False) or \ (isinstance(style, list) and len(style) == 2): return style else: raise ValueError(f"invalid '{style_name}': {style}") primary_style = _parse_linestyle('primary_style') mirror_style = _parse_linestyle('mirror_style', allow_false=True) # If argument was a singleton, turn it into a tuple if not isinstance(syslist, (list, tuple)): syslist = (syslist,) # Determine the range of frequencies to use, based on args/features omega, omega_range_given = _determine_omega_vector( syslist, omega, omega_limits, omega_num, feature_periphery_decades=2) # If omega was not specified explicitly, start at omega = 0 if not omega_range_given: if omega_num_given: # Just reset the starting point omega[0] = 0.0 else: # Insert points between the origin and the first frequency point omega = np.concatenate(( np.linspace(0, omega[0], indent_points), omega[1:])) # Go through each system and keep track of the results counts, contours = [], [] for sys in syslist: if not sys.issiso(): # TODO: Add MIMO nyquist plots. raise ControlMIMONotImplemented( "Nyquist plot currently only supports SISO systems.") # Figure out the frequency range omega_sys = np.asarray(omega) # Determine the contour used to evaluate the Nyquist curve if sys.isdtime(strict=True): # Restrict frequencies for discrete-time systems nyquistfrq = math.pi / sys.dt if not omega_range_given: # limit up to and including nyquist frequency omega_sys = np.hstack(( omega_sys[omega_sys < nyquistfrq], nyquistfrq)) # Issue a warning if we are sampling above Nyquist if np.any(omega_sys * sys.dt > np.pi) and warn_nyquist: warnings.warn("evaluation above Nyquist frequency") # do indentations in s-plane where it is more convenient splane_contour = 1j * omega_sys # Bend the contour around any poles on/near the imaginary axis # TODO: smarter indent radius that depends on dcgain of system # and timebase of discrete system. if isinstance(sys, (StateSpace, TransferFunction)) \ and indent_direction != 'none': if sys.isctime(): splane_poles = sys.poles() splane_cl_poles = sys.feedback().poles() else: # map z-plane poles to s-plane, ignoring any at the origin # because we don't need to indent for them zplane_poles = sys.poles() zplane_poles = zplane_poles[~np.isclose(abs(zplane_poles), 0.)] splane_poles = np.log(zplane_poles) / sys.dt zplane_cl_poles = sys.feedback().poles() zplane_cl_poles = zplane_cl_poles[ ~np.isclose(abs(zplane_poles), 0.)] splane_cl_poles = np.log(zplane_cl_poles) / sys.dt # # Check to make sure indent radius is small enough # # If there is a closed loop pole that is near the imaginary access # at a point that is near an open loop pole, it is possible that # indentation might skip or create an extraneous encirclement. # We check for that situation here and generate a warning if that # could happen. # for p_cl in splane_cl_poles: # See if any closed loop poles are near the imaginary axis if abs(p_cl.real) <= indent_radius: # See if any open loop poles are close to closed loop poles p_ol = splane_poles[ (np.abs(splane_poles - p_cl)).argmin()] if abs(p_ol - p_cl) <= indent_radius and \ warn_encirclements: warnings.warn( "indented contour may miss closed loop pole; " "consider reducing indent_radius to be less than " f"{abs(p_ol - p_cl):5.2g}", stacklevel=2) # # See if we should add some frequency points near imaginary poles # for p in splane_poles: # See if we need to process this pole (skip if on the negative # imaginary axis or not near imaginary axis + user override) if p.imag < 0 or abs(p.real) > indent_radius or \ omega_range_given: continue # Find the frequencies before the pole frequency below_points = np.argwhere( splane_contour.imag - abs(p.imag) < -indent_radius) if below_points.size > 0: first_point = below_points[-1].item() start_freq = p.imag - indent_radius else: # Add the points starting at the beginning of the contour assert splane_contour[0] == 0 first_point = 0 start_freq = 0 # Find the frequencies after the pole frequency above_points = np.argwhere( splane_contour.imag - abs(p.imag) > indent_radius) last_point = above_points[0].item() # Add points for half/quarter circle around pole frequency # (these will get indented left or right below) splane_contour = np.concatenate(( splane_contour[0:first_point+1], (1j * np.linspace( start_freq, p.imag + indent_radius, indent_points)), splane_contour[last_point:])) # Indent points that are too close to a pole for i, s in enumerate(splane_contour): # Find the nearest pole p = splane_poles[(np.abs(splane_poles - s)).argmin()] # See if we need to indent around it if abs(s - p) < indent_radius: # Figure out how much to offset (simple trigonometry) offset = np.sqrt(indent_radius ** 2 - (s - p).imag ** 2) \ - (s - p).real # Figure out which way to offset the contour point if p.real < 0 or (p.real == 0 and indent_direction == 'right'): # Indent to the right splane_contour[i] += offset elif p.real > 0 or (p.real == 0 and indent_direction == 'left'): # Indent to the left splane_contour[i] -= offset else: raise ValueError("unknown value for indent_direction") # change contour to z-plane if necessary if sys.isctime(): contour = splane_contour else: contour = np.exp(splane_contour * sys.dt) # Compute the primary curve resp = sys(contour) # Compute CW encirclements of -1 by integrating the (unwrapped) angle phase = -unwrap(np.angle(resp + 1)) encirclements = np.sum(np.diff(phase)) / np.pi count = int(np.round(encirclements, 0)) # Let the user know if the count might not make sense if abs(encirclements - count) > encirclement_threshold and \ warn_encirclements: warnings.warn( "number of encirclements was a non-integer value; this can" " happen is contour is not closed, possibly based on a" " frequency range that does not include zero.") # # Make sure that the enciriclements match the Nyquist criterion # # If the user specifies the frequency points to use, it is possible # to miss enciriclements, so we check here to make sure that the # Nyquist criterion is actually satisfied. # if isinstance(sys, (StateSpace, TransferFunction)): # Count the number of open/closed loop RHP poles if sys.isctime(): if indent_direction == 'right': P = (sys.poles().real > 0).sum() else: P = (sys.poles().real >= 0).sum() Z = (sys.feedback().poles().real >= 0).sum() else: if indent_direction == 'right': P = (np.abs(sys.poles()) > 1).sum() else: P = (np.abs(sys.poles()) >= 1).sum() Z = (np.abs(sys.feedback().poles()) >= 1).sum() # Check to make sure the results make sense; warn if not if Z != count + P and warn_encirclements: warnings.warn( "number of encirclements does not match Nyquist criterion;" " check frequency range and indent radius/direction", UserWarning, stacklevel=2) elif indent_direction == 'none' and any(sys.poles().real == 0) and \ warn_encirclements: warnings.warn( "system has pure imaginary poles but indentation is" " turned off; results may be meaningless", RuntimeWarning, stacklevel=2) counts.append(count) contours.append(contour) if plot: # 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=arrow_size, head_length=arrow_size) # Find the different portions of the curve (with scaled pts marked) reg_mask = np.logical_or( np.abs(resp) > max_curve_magnitude, splane_contour.real != 0) # reg_mask = np.logical_or( # np.abs(resp.real) > max_curve_magnitude, # np.abs(resp.imag) > max_curve_magnitude) scale_mask = ~reg_mask \ & np.concatenate((~reg_mask[1:], ~reg_mask[-1:])) \ & np.concatenate((~reg_mask[0:1], ~reg_mask[:-1])) # Rescale the points with large magnitude rescale = np.logical_and( reg_mask, abs(resp) > max_curve_magnitude) resp[rescale] *= max_curve_magnitude / abs(resp[rescale]) # Plot the regular portions of the curve (and grab the color) x_reg = np.ma.masked_where(reg_mask, resp.real) y_reg = np.ma.masked_where(reg_mask, resp.imag) p = plt.plot( x_reg, y_reg, primary_style[0], color=color, **kwargs) c = p[0].get_color() # Figure out how much to offset the curve: the offset goes from # zero at the start of the scaled section to max_curve_offset as # we move along the curve curve_offset = _compute_curve_offset( resp, scale_mask, max_curve_offset) # Plot the scaled sections of the curve (changing linestyle) x_scl = np.ma.masked_where(scale_mask, resp.real) y_scl = np.ma.masked_where(scale_mask, resp.imag) plt.plot( x_scl * (1 + curve_offset), y_scl * (1 + curve_offset), primary_style[1], color=c, **kwargs) # Plot the primary curve (invisible) for setting arrows x, y = resp.real.copy(), resp.imag.copy() x[reg_mask] *= (1 + curve_offset[reg_mask]) y[reg_mask] *= (1 + curve_offset[reg_mask]) p = plt.plot(x, y, linestyle='None', color=c, **kwargs) # Add arrows ax = plt.gca() _add_arrows_to_line2D( ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=1) # Plot the mirror image if mirror_style is not False: # Plot the regular and scaled segments plt.plot( x_reg, -y_reg, mirror_style[0], color=c, **kwargs) plt.plot( x_scl * (1 - curve_offset), -y_scl * (1 - curve_offset), mirror_style[1], color=c, **kwargs) # Add the arrows (on top of an invisible contour) x, y = resp.real.copy(), resp.imag.copy() x[reg_mask] *= (1 - curve_offset[reg_mask]) y[reg_mask] *= (1 - curve_offset[reg_mask]) p = plt.plot(x, -y, linestyle='None', color=c, **kwargs) _add_arrows_to_line2D( ax, p[0], arrow_pos, arrowstyle=arrow_style, dir=-1) # Mark the start of the curve if start_marker: plt.plot(resp[0].real, resp[0].imag, start_marker, color=c, markersize=start_marker_size) # Mark the -1 point plt.plot([-1], [0], 'r+') # Label the frequencies of the points if label_freq: ind = slice(None, None, label_freq) for xpt, ypt, omegapt in zip(x[ind], y[ind], omega_sys[ind]): # Convert to Hz f = omegapt / (2 * np.pi) # Factor out multiples of 1000 and limit the # result to the range [-8, 8]. pow1000 = max(min(get_pow1000(f), 8), -8) # Get the SI prefix. prefix = gen_prefix(pow1000) # Apply the text. (Use a space before the text to # prevent overlap with the data.) # # np.round() is used because 0.99... appears # instead of 1.0, and this would otherwise be # truncated to 0. plt.text(xpt, ypt, ' ' + str(int(np.round(f / 1000 ** pow1000, 0))) + ' ' + prefix + 'Hz') if plot: ax = plt.gca() ax.set_xlabel("Real axis") ax.set_ylabel("Imaginary axis") ax.grid(color="lightgray") # "Squeeze" the results if len(syslist) == 1: counts, contours = counts[0], contours[0] # Return counts and (optionally) the contour we used return (counts, contours) if return_contour else counts # Internal function to add arrows to a curve def _add_arrows_to_line2D( axes, line, arrow_locs=[0.2, 0.4, 0.6, 0.8], arrowstyle='-|>', arrowsize=1, dir=1, transform=None): """ Add arrows to a matplotlib.lines.Line2D at selected locations. Parameters: ----------- axes: Axes object as returned by axes command (or gca) line: Line2D object as returned by plot command arrow_locs: list of locations where to insert arrows, % of total length arrowstyle: style of the arrow arrowsize: size of the arrow transform: a matplotlib transform instance, default to data coordinates Returns: -------- arrows: list of arrows Based on https://stackoverflow.com/questions/26911898/ """ if not isinstance(line, mpl.lines.Line2D): raise ValueError("expected a matplotlib.lines.Line2D object") x, y = line.get_xdata(), line.get_ydata() arrow_kw = { "arrowstyle": arrowstyle, } color = line.get_color() use_multicolor_lines = isinstance(color, np.ndarray) if use_multicolor_lines: raise NotImplementedError("multicolor lines not supported") else: arrow_kw['color'] = color linewidth = line.get_linewidth() if isinstance(linewidth, np.ndarray): raise NotImplementedError("multiwidth lines not supported") else: arrow_kw['linewidth'] = linewidth if transform is None: transform = axes.transData # Compute the arc length along the curve s = np.cumsum(np.sqrt(np.diff(x) ** 2 + np.diff(y) ** 2)) arrows = [] for loc in arrow_locs: n = np.searchsorted(s, s[-1] * loc) # Figure out what direction to paint the arrow if dir == 1: arrow_tail = (x[n], y[n]) arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2])) elif dir == -1: # Orient the arrow in the other direction on the segment arrow_tail = (x[n + 1], y[n + 1]) arrow_head = (np.mean(x[n:n + 2]), np.mean(y[n:n + 2])) else: raise ValueError("unknown value for keyword 'dir'") p = mpl.patches.FancyArrowPatch( arrow_tail, arrow_head, transform=transform, lw=0, **arrow_kw) axes.add_patch(p) arrows.append(p) return arrows # # Function to compute Nyquist curve offsets # # This function computes a smoothly varying offset that starts and ends at # zero at the ends of a scaled segment. # def _compute_curve_offset(resp, mask, max_offset): # Compute the arc length along the curve s_curve = np.cumsum( np.sqrt(np.diff(resp.real) ** 2 + np.diff(resp.imag) ** 2)) # Initialize the offset offset = np.zeros(resp.size) arclen = np.zeros(resp.size) # Walk through the response and keep track of each continous component i, nsegs = 0, 0 while i < resp.size: # Skip the regular segment while i < resp.size and mask[i]: i += 1 # Increment the counter if i == resp.size: break # Keep track of the arclength arclen[i] = arclen[i-1] + np.abs(resp[i] - resp[i-1]) nsegs += 0.5 if i == resp.size: break # Save the starting offset of this segment seg_start = i # Walk through the scaled segment while i < resp.size and not mask[i]: i += 1 if i == resp.size: # See if we are done with this segment break # Keep track of the arclength arclen[i] = arclen[i-1] + np.abs(resp[i] - resp[i-1]) nsegs += 0.5 if i == resp.size: break # Save the ending offset of this segment seg_end = i # Now compute the scaling for this segment s_segment = arclen[seg_end-1] - arclen[seg_start] offset[seg_start:seg_end] = max_offset * s_segment/s_curve[-1] * \ np.sin(np.pi * (arclen[seg_start:seg_end] - arclen[seg_start])/s_segment) return offset # # Gang of Four plot # # TODO: think about how (and whether) to handle lists of systems def gangof4_plot(P, C, omega=None, **kwargs): """Plot the "Gang of 4" transfer functions for a system Generates a 2x2 plot showing the "Gang of 4" sensitivity functions [T, PS; CS, S] Parameters ---------- P, C : LTI Linear input/output systems (process and control) omega : array Range of frequencies (list or bounds) in rad/sec **kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional Additional keywords (passed to `matplotlib`) Returns ------- None """ if not P.issiso() or not C.issiso(): # TODO: Add MIMO go4 plots. raise ControlMIMONotImplemented( "Gang of four is currently only implemented for SISO systems.") # Get the default parameter values dB = config._get_param( 'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True) Hz = config._get_param( 'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True) grid = config._get_param( 'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True) # Compute the senstivity functions L = P * C S = feedback(1, L) T = L * S # Select a default range if none is provided # TODO: This needs to be made more intelligent if omega is None: omega = _default_frequency_range((P, C, S), Hz=Hz) # Set up the axes with labels so that multiple calls to # gangof4_plot will superimpose the data. See details in bode_plot. plot_axes = {'t': None, 's': None, 'ps': None, 'cs': None} for ax in plt.gcf().axes: label = ax.get_label() if label.startswith('control-gangof4-'): key = label[len('control-gangof4-'):] if key not in plot_axes: raise RuntimeError( "unknown gangof4 axis type '{}'".format(label)) plot_axes[key] = ax # if any of the axes are missing, start from scratch if any((ax is None for ax in plot_axes.values())): plt.clf() plot_axes = {'s': plt.subplot(221, label='control-gangof4-s'), 'ps': plt.subplot(222, label='control-gangof4-ps'), 'cs': plt.subplot(223, label='control-gangof4-cs'), 't': plt.subplot(224, label='control-gangof4-t')} # # Plot the four sensitivity functions # omega_plot = omega / (2. * math.pi) if Hz else omega # TODO: Need to add in the mag = 1 lines mag_tmp, phase_tmp, omega = S.frequency_response(omega) mag = np.squeeze(mag_tmp) if dB: plot_axes['s'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs) else: plot_axes['s'].loglog(omega_plot, mag, **kwargs) plot_axes['s'].set_ylabel("$|S|$" + " (dB)" if dB else "") plot_axes['s'].tick_params(labelbottom=False) plot_axes['s'].grid(grid, which='both') mag_tmp, phase_tmp, omega = (P * S).frequency_response(omega) mag = np.squeeze(mag_tmp) if dB: plot_axes['ps'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs) else: plot_axes['ps'].loglog(omega_plot, mag, **kwargs) plot_axes['ps'].tick_params(labelbottom=False) plot_axes['ps'].set_ylabel("$|PS|$" + " (dB)" if dB else "") plot_axes['ps'].grid(grid, which='both') mag_tmp, phase_tmp, omega = (C * S).frequency_response(omega) mag = np.squeeze(mag_tmp) if dB: plot_axes['cs'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs) else: plot_axes['cs'].loglog(omega_plot, mag, **kwargs) plot_axes['cs'].set_xlabel( "Frequency (Hz)" if Hz else "Frequency (rad/sec)") plot_axes['cs'].set_ylabel("$|CS|$" + " (dB)" if dB else "") plot_axes['cs'].grid(grid, which='both') mag_tmp, phase_tmp, omega = T.frequency_response(omega) mag = np.squeeze(mag_tmp) if dB: plot_axes['t'].semilogx(omega_plot, 20 * np.log10(mag), **kwargs) else: plot_axes['t'].loglog(omega_plot, mag, **kwargs) plot_axes['t'].set_xlabel( "Frequency (Hz)" if Hz else "Frequency (rad/sec)") plot_axes['t'].set_ylabel("$|T|$" + " (dB)" if dB else "") plot_axes['t'].grid(grid, which='both') plt.tight_layout() # # Singular values plot # def singular_values_plot(syslist, omega=None, plot=True, omega_limits=None, omega_num=None, *args, **kwargs): """Singular value plot for a system Plots a singular value plot for the system over a (optional) frequency range. Parameters ---------- syslist : linsys List of linear systems (single system is OK). omega : array_like List of frequencies in rad/sec to be used for frequency response. plot : bool If True (default), generate the singular values plot. omega_limits : array_like of two values Limits of the frequency vector to generate. If Hz=True the limits are in Hz otherwise in rad/s. omega_num : int Number of samples to plot. Default value (1000) set by config.defaults['freqplot.number_of_samples']. dB : bool If True, plot result in dB. Default value (False) set by config.defaults['freqplot.dB']. Hz : bool If True, plot frequency in Hz (omega must be provided in rad/sec). Default value (False) set by config.defaults['freqplot.Hz'] Returns ------- sigma : ndarray (or list of ndarray if len(syslist) > 1)) singular values omega : ndarray (or list of ndarray if len(syslist) > 1)) frequency in rad/sec Other Parameters ---------------- grid : bool If True, plot grid lines on gain and phase plots. Default is set by `config.defaults['freqplot.grid']`. Examples -------- >>> import numpy as np >>> den = [75, 1] >>> sys = TransferFunction( [[[87.8], [-86.4]], [[108.2], [-109.6]]], [[den, den], [den, den]]) >>> omega = np.logspace(-4, 1, 1000) >>> sigma, omega = singular_values_plot(sys, plot=True) >>> singular_values_plot(sys, 0.0, plot=False) (array([[197.20868123], [ 1.39141948]]), array([0.])) """ # Make a copy of the kwargs dictionary since we will modify it kwargs = dict(kwargs) # Get values for params (and pop from list to allow keyword use in plot) dB = config._get_param( 'freqplot', 'dB', kwargs, _freqplot_defaults, pop=True) Hz = config._get_param( 'freqplot', 'Hz', kwargs, _freqplot_defaults, pop=True) grid = config._get_param( 'freqplot', 'grid', kwargs, _freqplot_defaults, pop=True) plot = config._get_param( 'freqplot', 'plot', plot, True) omega_num = config._get_param('freqplot', 'number_of_samples', omega_num) # If argument was a singleton, turn it into a tuple if not isinstance(syslist, (list, tuple)): syslist = (syslist,) omega, omega_range_given = _determine_omega_vector( syslist, omega, omega_limits, omega_num, Hz=Hz) omega = np.atleast_1d(omega) if plot: fig = plt.gcf() ax_sigma = None # Get the current axes if they already exist for ax in fig.axes: if ax.get_label() == 'control-sigma': ax_sigma = ax # If no axes present, create them from scratch if ax_sigma is None: plt.clf() ax_sigma = plt.subplot(111, label='control-sigma') # color cycle handled manually as all singular values # of the same systems are expected to be of the same color color_cycle = plt.rcParams['axes.prop_cycle'].by_key()['color'] color_offset = 0 if len(ax_sigma.lines) > 0: last_color = ax_sigma.lines[-1].get_color() if last_color in color_cycle: color_offset = color_cycle.index(last_color) + 1 sigmas, omegas, nyquistfrqs = [], [], [] for idx_sys, sys in enumerate(syslist): omega_sys = np.asarray(omega) if sys.isdtime(strict=True): nyquistfrq = math.pi / sys.dt if not omega_range_given: # limit up to and including nyquist frequency omega_sys = np.hstack(( omega_sys[omega_sys < nyquistfrq], nyquistfrq)) omega_complex = np.exp(1j * omega_sys * sys.dt) else: nyquistfrq = None omega_complex = 1j*omega_sys fresp = sys(omega_complex, squeeze=False) fresp = fresp.transpose((2, 0, 1)) sigma = np.linalg.svd(fresp, compute_uv=False) sigmas.append(sigma.transpose()) # return shape is "channel first" omegas.append(omega_sys) nyquistfrqs.append(nyquistfrq) if plot: color = color_cycle[(idx_sys + color_offset) % len(color_cycle)] color = kwargs.pop('color', color) nyquistfrq_plot = None if Hz: omega_plot = omega_sys / (2. * math.pi) if nyquistfrq: nyquistfrq_plot = nyquistfrq / (2. * math.pi) else: omega_plot = omega_sys if nyquistfrq: nyquistfrq_plot = nyquistfrq sigma_plot = sigma if dB: ax_sigma.semilogx(omega_plot, 20 * np.log10(sigma_plot), color=color, *args, **kwargs) else: ax_sigma.loglog(omega_plot, sigma_plot, color=color, *args, **kwargs) if nyquistfrq_plot is not None: ax_sigma.axvline(x=nyquistfrq_plot, color=color) # Add a grid to the plot + labeling if plot: ax_sigma.grid(grid, which='both') ax_sigma.set_ylabel( "Singular Values (dB)" if dB else "Singular Values") ax_sigma.set_xlabel("Frequency (Hz)" if Hz else "Frequency (rad/sec)") if len(syslist) == 1: return sigmas[0], omegas[0] else: return sigmas, omegas # # Utility functions # # This section of the code contains some utility functions for # generating frequency domain plots # # Determine the frequency range to be used def _determine_omega_vector(syslist, omega_in, omega_limits, omega_num, Hz=None, feature_periphery_decades=None): """Determine the frequency range for a frequency-domain plot according to a standard logic. If omega_in and omega_limits are both None, then omega_out is computed on omega_num points according to a default logic defined by _default_frequency_range and tailored for the list of systems syslist, and omega_range_given is set to False. If omega_in is None but omega_limits is an array-like of 2 elements, then omega_out is computed with the function np.logspace on omega_num points within the interval [min, max] = [omega_limits[0], omega_limits[1]], and omega_range_given is set to True. If omega_in is not None, then omega_out is set to omega_in, and omega_range_given is set to True Parameters ---------- syslist : list of LTI List of linear input/output systems (single system is OK) omega_in : 1D array_like or None Frequency range specified by the user omega_limits : 1D array_like or None Frequency limits specified by the user omega_num : int Number of points to be used for the frequency range (if the frequency range is not user-specified) Hz : bool, optional If True, the limits (first and last value) of the frequencies are set to full decades in Hz so it fits plotting with logarithmic scale in Hz otherwise in rad/s. Omega is always returned in rad/sec. Returns ------- omega_out : 1D array Frequency range to be used omega_range_given : bool True if the frequency range was specified by the user, either through omega_in or through omega_limits. False if both omega_in and omega_limits are None. """ omega_range_given = True if omega_in is None: if omega_limits is None: omega_range_given = False # Select a default range if none is provided omega_out = _default_frequency_range( syslist, number_of_samples=omega_num, Hz=Hz, feature_periphery_decades=feature_periphery_decades) else: omega_limits = np.asarray(omega_limits) if len(omega_limits) != 2: raise ValueError("len(omega_limits) must be 2") omega_out = np.logspace(np.log10(omega_limits[0]), np.log10(omega_limits[1]), num=omega_num, endpoint=True) else: omega_out = np.copy(omega_in) return omega_out, omega_range_given # Compute reasonable defaults for axes def _default_frequency_range(syslist, Hz=None, number_of_samples=None, feature_periphery_decades=None): """Compute a default frequency range for frequency domain plots. This code looks at the poles and zeros of all of the systems that we are plotting and sets the frequency range to be one decade above and below the min and max feature frequencies, rounded to the nearest integer. If no features are found, it returns logspace(-1, 1) Parameters ---------- syslist : list of LTI List of linear input/output systems (single system is OK) Hz : bool, optional If True, the limits (first and last value) of the frequencies are set to full decades in Hz so it fits plotting with logarithmic scale in Hz otherwise in rad/s. Omega is always returned in rad/sec. number_of_samples : int, optional Number of samples to generate. The default value is read from ``config.defaults['freqplot.number_of_samples']. If None, then the default from `numpy.logspace` is used. feature_periphery_decades : float, optional Defines how many decades shall be included in the frequency range on both sides of features (poles, zeros). The default value is read from ``config.defaults['freqplot.feature_periphery_decades']``. Returns ------- omega : array Range of frequencies in rad/sec Examples -------- >>> from matlab import ss >>> sys = ss("1. -2; 3. -4", "5.; 7", "6. 8", "9.") >>> omega = _default_frequency_range(sys) """ # Set default values for options number_of_samples = config._get_param( 'freqplot', 'number_of_samples', number_of_samples) feature_periphery_decades = config._get_param( 'freqplot', 'feature_periphery_decades', feature_periphery_decades, 1) # Find the list of all poles and zeros in the systems features = np.array(()) freq_interesting = [] # detect if single sys passed by checking if it is sequence-like if not hasattr(syslist, '__iter__'): syslist = (syslist,) for sys in syslist: try: # Add new features to the list if sys.isctime(): features_ = np.concatenate( (np.abs(sys.poles()), np.abs(sys.zeros()))) # Get rid of poles and zeros at the origin toreplace = np.isclose(features_, 0.0) if np.any(toreplace): features_ = features_[~toreplace] elif sys.isdtime(strict=True): fn = math.pi * 1. / sys.dt # TODO: What distance to the Nyquist frequency is appropriate? freq_interesting.append(fn * 0.9) features_ = np.concatenate((sys.poles(), sys.zeros())) # Get rid of poles and zeros on the real axis (imag==0) # * origin and real < 0 # * at 1.: would result in omega=0. (logaritmic plot!) toreplace = np.isclose(features_.imag, 0.0) & ( (features_.real <= 0.) | (np.abs(features_.real - 1.0) < 1.e-10)) if np.any(toreplace): features_ = features_[~toreplace] # TODO: improve features_ = np.abs(np.log(features_) / (1.j * sys.dt)) else: # TODO raise NotImplementedError( "type of system in not implemented now") features = np.concatenate((features, features_)) except NotImplementedError: pass # Make sure there is at least one point in the range if features.shape[0] == 0: features = np.array([1.]) if Hz: features /= 2. * math.pi features = np.log10(features) lsp_min = np.rint(np.min(features) - feature_periphery_decades) lsp_max = np.rint(np.max(features) + feature_periphery_decades) if Hz: lsp_min += np.log10(2. * math.pi) lsp_max += np.log10(2. * math.pi) if freq_interesting: lsp_min = min(lsp_min, np.log10(min(freq_interesting))) lsp_max = max(lsp_max, np.log10(max(freq_interesting))) # TODO: Add a check in discrete case to make sure we don't get aliasing # (Attention: there is a list of system but only one omega vector) # Set the range to be an order of magnitude beyond any features if number_of_samples: omega = np.logspace( lsp_min, lsp_max, num=number_of_samples, endpoint=True) else: omega = np.logspace(lsp_min, lsp_max, endpoint=True) return omega # # Utility functions to create nice looking labels (KLD 5/23/11) # def get_pow1000(num): """Determine exponent for which significand of a number is within the range [1, 1000). """ # Based on algorithm from http://www.mail-archive.com/ # matplotlib-users@lists.sourceforge.net/msg14433.html, accessed 2010/11/7 # by Jason Heeris 2009/11/18 from decimal import Decimal from math import floor dnum = Decimal(str(num)) if dnum == 0: return 0 elif dnum < 0: dnum = -dnum return int(floor(dnum.log10() / 3)) def gen_prefix(pow1000): """Return the SI prefix for a power of 1000. """ # Prefixes according to Table 5 of [BIPM 2006] (excluding hecto, # deca, deci, and centi). if pow1000 < -8 or pow1000 > 8: raise ValueError( "Value is out of the range covered by the SI prefixes.") return ['Y', # yotta (10^24) 'Z', # zetta (10^21) 'E', # exa (10^18) 'P', # peta (10^15) 'T', # tera (10^12) 'G', # giga (10^9) 'M', # mega (10^6) 'k', # kilo (10^3) '', # (10^0) 'm', # milli (10^-3) r'$\mu$', # micro (10^-6) 'n', # nano (10^-9) 'p', # pico (10^-12) 'f', # femto (10^-15) 'a', # atto (10^-18) 'z', # zepto (10^-21) 'y'][8 - pow1000] # yocto (10^-24) def find_nearest_omega(omega_list, omega): omega_list = np.asarray(omega_list) return omega_list[(np.abs(omega_list - omega)).argmin()] # Function aliases bode = bode_plot nyquist = nyquist_plot gangof4 = gangof4_plot