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control.nyquist_plot — Python Control Systems Library 0.10.2 documentation

control.nyquist_plot(data, omega=None, plot=None, label_freq=0, color=None, label=None, return_contour=None, title=None, ax=None, unit_circle=False, mt_circles=None, ms_circles=None, **kwargs)[source]

Nyquist plot for a system.

Generates a Nyquist plot for the system over a (optional) frequency range. The curve is computed by evaluating the Nyquist 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:

datalist of LTI or NyquistResponseData

List of linear input/output systems (single system is OK) or Nyquist responses (computed using nyquist_response). Nyquist curves for each system are plotted on the same graph.

omegaarray_like, optional

Set of frequencies to be evaluated, in rad/sec. Specifying omega as a list of two elements is equivalent to providing omega_limits.

unit_circlebool, optional

If True, display the unit circle, to read gain crossover frequency. The circle style is determined by config.defaults['nyquist.circle_style'].

mt_circlesarray_like, optional

Draw circles corresponding to the given magnitudes of sensitivity.

ms_circlesarray_like, optional

Draw circles corresponding to the given magnitudes of complementary sensitivity.

**kwargsmatplotlib.pyplot.plot keyword properties, optional

Additional keywords passed to matplotlib to specify line properties.

Returns:

cpltControlPlot object

Object containing the data that were plotted. See ControlPlot for more detailed information.

cplt.lines2D array of matplotlib.lines.Line2D

Array containing information on each line in the plot. The shape of the array is given by (nsys, 4) where nsys is the number of systems or Nyquist responses passed to the function. The second index specifies the segment type:

• lines[idx, 0]: unscaled portion of the primary curve

• lines[idx, 1]: scaled portion of the primary curve

• lines[idx, 2]: unscaled portion of the mirror curve

• lines[idx, 3]: scaled portion of the mirror curve

cplt.axes2D array of matplotlib.axes.Axes

Axes for each subplot.

cplt.figurematplotlib.figure.Figure

Figure containing the plot.

cplt.legend2D array of matplotlib.legend.Legend

Legend object(s) contained in the plot.

Other Parameters:

arrowsint 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. Default value is 2 and can be set using config.defaults['nyquist.arrows'].

arrow_sizefloat, optional

Arrowhead width and length (in display coordinates). Default value is 8 and can be set using config.defaults['nyquist.arrow_size'].

arrow_stylematplotlib.patches.ArrowStyle, optional

Define style used for Nyquist curve arrows (overrides arrow_size).

axmatplotlib.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.

blend_fractionfloat, optional

For portions of the Nyquist curve that are scaled to have a maximum magnitude of max_curve_magnitude, begin a smooth rescaling at this fraction of max_curve_magnitude. Default value is 0.15.

encirclement_thresholdfloat, 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_directionstr, optional

For poles on the imaginary axis, set the direction of indentation to be ‘right’ (default), ‘left’, or ‘none’.

indent_pointsint, optional

Number of points to insert in the Nyquist contour around poles that are at or near the imaginary axis.

indent_radiusfloat, 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.

labelstr or array_like of str, optional

If present, replace automatically generated label(s) with the given label(s). If data is a list, strings should be specified for each system.

label_freqint, optional

Label every nth frequency on the plot. If not specified, no labels are generated.

legend_locint or str, optional

Include a legend in the given location. Default is ‘upper right’, with no legend for a single response. Use False to suppress legend.

max_curve_magnitudefloat, 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. The default value is 20 and can be set using config.defaults['nyquist.max_curve_magnitude'].

max_curve_offsetfloat, 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. The default value is 0.02 and can be set using config.defaults['nyquist.max_curve_magnitude'].

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'].

omega_limitsarray_like of two values

Set limits for plotted frequency range. If Hz=True the limits are in Hz otherwise in rad/s. Specifying omega as a list of two elements is equivalent to providing omega_limits.

omega_numint, optional

Number of samples to use for the frequency range. Defaults to config.defaults['freqplot.number_of_samples']. Ignored if data is not a system or list of systems.

plotbool, optional

(legacy) If given, nyquist_plot returns the legacy return values of (counts, contours). If False, return the values with no plot.

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'].

rcParamsdict

Override the default parameters used for generating plots. Default is set by config.defaults['ctrlplot.rcParams'].

return_contourbool, optional

(legacy) If True, return the encirclement count and Nyquist contour used to generate the Nyquist plot.

show_legendbool, optional

Force legend to be shown if True or hidden if False. If None, then show legend when there is more than one line on the plot or legend_loc has been specified.

start_markerstr, 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_sizefloat, optional

Start marker size (in display coordinates). Default value is 4 and can be set using config.defaults['nyquist.start_marker_size'].

titlestr, optional

Set the title of the plot. Defaults to plot type and system name(s).

title_framestr, optional

Set the frame of reference used to center the plot title. If set to ‘axes’ (default), the horizontal position of the title will centered relative to the axes. If set to ‘figure’, it will be centered with respect to the figure (faster execution).

warn_nyquistbool, optional

If set to False, turn off warnings about frequencies above Nyquist.

warn_encirclementsbool, optional

If set to False, turn off warnings about number of encirclements not meeting the Nyquist criterion.

Notes

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.

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.

For those portions of the Nyquist plot in which the contour is indented to avoid poles, resulting in a scaling of the Nyquist plot, the line styles are according to the settings of the primary_style and mirror_style keywords. By default the scaled portions of the primary curve use a dashdot line style and the scaled portions of the mirror image use a dotted line style.

Examples

>>> G = ct.zpk([], [-1, -2, -3], gain=100)
>>> out = ct.nyquist_plot(G)

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