The Python Control Systems Toolbox contains a number of functions for plotting input/output responses in the time and frequency domain, root locus diagrams, and other standard charts used in control system analysis, for example:

bode_plot(sys)
nyquist_plot([sys1, sys2])
phase_plane_plot(sys, limits)
pole_zero_plot(sys)
root_locus_plot(sys)

While plotting functions can be called directly, the standard pattern used in the toolbox is to provide a function that performs the basic computation or analysis (e.g., computation of the time or frequency response) and returns and object representing the output data. A separate plotting function, typically ending in _plot is then used to plot the data, resulting in the following standard pattern:

response = ct.nyquist_response([sys1, sys2])
count = ct.response.count          # number of encirclements of -1
cplt = ct.nyquist_plot(response)  # Nyquist plot

Plotting commands return a :class:`~control.ControlPlot` object that provides access to the individual lines in the generated plot using cplt.lines, allowing various aspects of the plot to be modified to suit specific needs.

The plotting function is also available via the plot() method of the analysis object, allowing the following type of calls:

step_response(sys).plot()
frequency_response(sys).plot()
nyquist_response(sys).plot()
pp.streamlines(sys, limits).plot()
root_locus_map(sys).plot()

The remainder of this chapter provides additional documentation on how these response and plotting functions can be customized.

Input/output time responses are produced one of several python-control functions: :func:`~control.forced_response`, :func:`~control.impulse_response`, :func:`~control.initial_response`, :func:`~control.input_output_response`, :func:`~control.step_response`. Each of these return a :class:`~control.TimeResponseData` object, which contains the time, input, state, and output vectors associated with the simulation. Time response data can be plotted with the :func:`~control.time_response_plot` function, which is also available as the :func:`~control.TimeResponseData.plot` method. For example, the step response for a two-input, two-output can be plotted using the commands:

sys_mimo = ct.tf2ss(
    [[[1], [0.1]], [[0.2], [1]]],
    [[[1, 0.6, 1], [1, 1, 1]], [[1, 0.4, 1], [1, 2, 1]]], name="sys_mimo")
response = ct.step_response(sys)
response.plot()

which produces the following plot:

The :class:`~control.TimeResponseData` object can also be used to access the data from the simulation:

time, outputs, inputs = response.time, response.outputs, response.inputs
fig, axs = plt.subplots(2, 2)
for i in range(2):
    for j in range(2):
        axs[i, j].plot(time, outputs[i, j])

A number of options are available in the plot method to customize the appearance of input output data. For data produced by the :func:`~control.impulse_response` and :func:`~control.step_response` commands, the inputs are not shown. This behavior can be changed using the plot_inputs keyword. It is also possible to combine multiple lines onto a single graph, using either the overlay_signals keyword (which puts all outputs out a single graph and all inputs on a single graph) or the overlay_traces keyword, which puts different traces (e.g., corresponding to step inputs in different channels) on the same graph, with appropriate labeling via a legend on selected axes.

For example, using plot_input=True and overlay_signals=True yields the following plot:

ct.step_response(sys_mimo).plot(
  plot_inputs=True, overlay_signals=True,
  title="Step response for 2x2 MIMO system " +
  "[plot_inputs, overlay_signals]")

Input/output response plots created with either the :func:`~control.forced_response` or the :func:`~control.input_output_response` functions include the input signals by default. These can be plotted on separate axes, but also "overlaid" on the output axes (useful when the input and output signals are being compared to each other). The following plot shows the use of plot_inputs='overlay' as well as the ability to reposition the legends using the legend_map keyword:

timepts = np.linspace(0, 10, 100)
U = np.vstack([np.sin(timepts), np.cos(2*timepts)])
ct.input_output_response(sys_mimo, timepts, U).plot(
    plot_inputs='overlay',
    legend_map=np.array([['lower right'], ['lower right']]),
    title="I/O response for 2x2 MIMO system " +
    "[plot_inputs='overlay', legend_map]")

Another option that is available is to use the transpose keyword so that instead of plotting the outputs on the top and inputs on the bottom, the inputs are plotted on the left and outputs on the right, as shown in the following figure:

U1 = np.vstack([np.sin(timepts), np.cos(2*timepts)])
resp1 = ct.input_output_response(sys_mimo, timepts, U1)

U2 = np.vstack([np.cos(2*timepts), np.sin(timepts)])
resp2 = ct.input_output_response(sys_mimo, timepts, U2)

ct.combine_time_responses(
    [resp1, resp2], trace_labels=["Scenario #1", "Scenario #2"]).plot(
        transpose=True,
        title="I/O responses for 2x2 MIMO system, multiple traces "
        "[transpose]")

This figure also illustrates the ability to create "multi-trace" plots using the :func:`~control.combine_time_responses` function. The line properties that are used when combining signals and traces are set by the input_props, output_props and trace_props parameters for :func:`~control.time_response_plot`.

Additional customization is possible using the input_props, output_props, and trace_props keywords to set complementary line colors and styles for various signals and traces:

cplt = ct.step_response(sys_mimo).plot(
    plot_inputs='overlay', overlay_signals=True, overlay_traces=True,
    output_props=[{'color': c} for c in ['blue', 'orange']],
    input_props=[{'color': c} for c in ['red', 'green']],
    trace_props=[{'linestyle': s} for s in ['-', '--']])

Linear time invariant (LTI) systems can be analyzed in terms of their frequency response and python-control provides a variety of tools for carrying out frequency response analysis. The most basic of these is the :func:`~control.frequency_response` function, which will compute the frequency response for one or more linear systems:

sys1 = ct.tf([1], [1, 2, 1], name='sys1')
sys2 = ct.tf([1, 0.2], [1, 1, 3, 1, 1], name='sys2')
response = ct.frequency_response([sys1, sys2])

A Bode plot provide a graphical view of the response an LTI system and can be generated using the :func:`~control.bode_plot` function:

ct.bode_plot(response, initial_phase=0)

Computing the response for multiple systems at the same time yields a common frequency range that covers the features of all listed systems.

Bode plots can also be created directly using the :meth:`~control.FrequencyResponseData.plot` method:

sys_mimo = ct.tf(
    [[[1], [0.1]], [[0.2], [1]]],
    [[[1, 0.6, 1], [1, 1, 1]], [[1, 0.4, 1], [1, 2, 1]]], name="sys_mimo")
ct.frequency_response(sys_mimo).plot()

A variety of options are available for customizing Bode plots, for example allowing the display of the phase to be turned off or overlaying the inputs or outputs:

ct.frequency_response(sys_mimo).plot(
    plot_phase=False, overlay_inputs=True, overlay_outputs=True)

The :func:`~control.singular_values_response` function can be used to generate Bode plots that show the singular values of a transfer function:

ct.singular_values_response(sys_mimo).plot()

Different types of plots can also be specified for a given frequency response. For example, to plot the frequency response using a a Nichols plot, use plot_type='nichols':

response.plot(plot_type='nichols')

Another response function that can be used to generate Bode plots is the :func:`~control.gangof4` function, which computes the four primary sensitivity functions for a feedback control system in standard form:

proc = ct.tf([1], [1, 1, 1], name="process")
ctrl = ct.tf([100], [1, 5], name="control")
response = rect.gangof4_response(proc, ctrl)
ct.bode_plot(response)        # or response.plot()

Nyquist analysis can be done using the :func:`~control.nyquist_response` function, which evaluates an LTI system along the Nyquist contour, and the :func:`~control.nyquist_plot` function, which generates a Nyquist plot:

sys = ct.tf([1, 0.2], [1, 1, 3, 1, 1], name='sys')
nyquist_plot(sys)

The :func:`~control.nyquist_response` function can be used to compute the number of encirclements of the -1 point and can return the Nyquist contour that was used to generate the Nyquist curve.

By default, the Nyquist response will generate small semicircles around poles that are on the imaginary axis. In addition, portions of the Nyquist curve that are far from the origin are scaled to a maximum value, while the line style is changed to reflect the scaling, and it is possible to offset the scaled portions to separate out the portions of the Nyquist curve at \infty. A number of keyword parameters for both are available for :func:`~control.nyquist_response` and :func:`~control.nyquist_plot` to tune the computation of the Nyquist curve and the way the data are plotted:

sys = ct.tf([1, 0.2], [1, 0, 1]) * ct.tf([1], [1, 0])
nyqresp = ct.nyquist_response(sys)
nyqresp.plot(
    max_curve_magnitude=6, max_curve_offset=1,
    arrows=[0, 0.15, 0.3, 0.6, 0.7, 0.925], label='sys')
print("Encirclements =", nyqresp.count)

All frequency domain plotting functions will automatically compute the range of frequencies to plot based on the poles and zeros of the frequency response. Frequency points can be explicitly specified by including an array of frequencies as a second argument (after the list of systems):

sys1 = ct.tf([1], [1, 2, 1], name='sys1')
sys2 = ct.tf([1, 0.2], [1, 1, 3, 1, 1], name='sys2')
omega = np.logspace(-2, 2, 500)
ct.frequency_response([sys1, sys2], omega).plot(initial_phase=0)

Alternatively, frequency ranges can be specified by passing a list of the form [wmin, wmax], where wmin and wmax are the minimum and maximum frequencies in the (log-spaced) frequency range:

response = ct.frequency_response([sys1, sys2], [1e-2, 1e2])

The number of (log-spaced) points in the frequency can be specified using the omega_num keyword parameter.

Pole/zero maps and root locus diagrams provide insights into system response based on the locations of system poles and zeros in the complex plane. The :func:`~control.pole_zero_map` function returns the poles and zeros and can be used to generate a pole/zero plot:

sys = ct.tf([1, 2], [1, 2, 3], name='SISO transfer function')
response = ct.pole_zero_map(sys)
ct.pole_zero_plot(response)

A root locus plot shows the location of the closed loop poles of a system as a function of the loop gain:

ct.root_locus_map(sys).plot()

The grid in the left hand plane shows lines of constant damping ratio as well as arcs corresponding to the frequency of the complex pole. The grid can be turned off using the grid keyword. Setting grid to False will turn off the grid but show the real and imaginary axis. To completely remove all lines except the root loci, use grid='empty'.

On systems that support interactive plots, clicking on a location on the root locus diagram will mark the pole locations on all branches of the diagram and display the gain and damping ratio for the clicked point below the plot title:

Root locus diagrams are also supported for discrete time systems, in which case the grid is show inside the unit circle:

sysd = sys.sample(0.1)
ct.root_locus_plot(sysd)

Lists of systems can also be given, in which case the root locus diagram for each system is plotted in different colors:

sys1 = ct.tf([1], [1, 2, 1], name='sys1')
sys2 = ct.tf([1, 0.2], [1, 1, 3, 1, 1], name='sys2')
ct.root_locus_plot([sys1, sys2], grid=False)

Insight into nonlinear systems can often be obtained by looking at phase plane diagrams. The :func:`~control.phase_plane_plot` function allows the creation of a 2-dimensional phase plane diagram for a system. This functionality is supported by a set of mapping functions that are part of the phaseplot module.

The default method for generating a phase plane plot is to provide a 2D dynamical system along with a range of coordinates and time limit:

sys = ct.nlsys(
    lambda t, x, u, params: np.array([[0, 1], [-1, -1]]) @ x,
    states=['position', 'velocity'], inputs=0, name='damped oscillator')
axis_limits = [-1, 1, -1, 1]
T = 8
ct.phase_plane_plot(sys, axis_limits, T)

By default, the plot includes streamlines generated from starting points on limits of the plot, with arrows showing the flow of the system, as well as any equilibrium points for the system. A variety of options are available to modify the information that is plotted, including plotting a grid of vectors instead of streamlines and turning on and off various features of the plot.

To illustrate some of these possibilities, consider a phase plane plot for an inverted pendulum system, which is created using a mesh grid:

def invpend_update(t, x, u, params):
    m, l, b, g = params['m'], params['l'], params['b'], params['g']
    return [x[1], -b/m * x[1] + (g * l / m) * np.sin(x[0]) + u[0]/m]
invpend = ct.nlsys(invpend_update, states=2, inputs=1, name='invpend')

ct.phase_plane_plot(
    invpend, [-2*pi, 2*pi, -2, 2], 5,
    gridtype='meshgrid', gridspec=[5, 8], arrows=3,
    plot_equilpoints={'gridspec': [12, 9]},
    params={'m': 1, 'l': 1, 'b': 0.2, 'g': 1})
plt.xlabel(r"$\theta$ [rad]")
plt.ylabel(r"$\dot\theta$ [rad/sec]")

This figure shows several features of more complex phase plane plots: multiple equilibrium points are shown, with saddle points showing separatrices, and streamlines generated along a 5x8 mesh of initial conditions. At each mesh point, a streamline is created that goes 5 time units forward and backward in time. A separate grid specification is used to find equilibrium points and separatrices (since the course grid spacing of 5x8 does not find all possible equilibrium points). Together, the multiple features in the phase plane plot give a good global picture of the topological structure of solutions of the dynamical system.

Phase plots can be built up by hand using a variety of helper functions that are part of the :mod:`~control.phaseplot` (pp) module:

import control.phaseplot as pp

def oscillator_update(t, x, u, params):
    return [x[1] + x[0] * (1 - x[0]**2 - x[1]**2),
            -x[0] + x[1] * (1 - x[0]**2 - x[1]**2)]
oscillator = ct.nlsys(
    oscillator_update, states=2, inputs=0, name='nonlinear oscillator')

ct.phase_plane_plot(oscillator, [-1.5, 1.5, -1.5, 1.5], 0.9)
pp.streamlines(
    oscillator, np.array([[0, 0]]), 1.5,
    gridtype='circlegrid', gridspec=[0.5, 6], dir='both')
pp.streamlines(
    oscillator, np.array([[1, 0]]), 2*pi, arrows=6, color='b')
plt.gca().set_aspect('equal')

The following helper functions are available:

.. autosummary::
   ~control.phaseplot.equilpoints
   ~control.phaseplot.separatrices
   ~control.phaseplot.streamlines
   ~control.phaseplot.vectorfield

The :func:`~control.phase_plane_plot` function calls these helper functions based on the options it is passed.

Note that unlike other plotting functions, phase plane plots do not involve computing a response and then plotting the result via a plot() method. Instead, the plot is generated directly be a call to the :func:`~control.phase_plane_plot` function (or one of the :mod:`~control.phaseplot` helper functions.

A set of common options are available to customize control plots in various ways. The following general rules apply:

• If a plotting function is called multiple times with data that generate control plots with the same shape for the array of subplots, the new data will be overlaid with the old data, with a change in color(s) for the new data (chosen from the standard matplotlib color cycle). If not overridden, the plot title and legends will be updated to reflect all data shown on the plot.

• If a plotting function is called and the shape for the array of subplots does not match the currently displayed plot, a new figure is created. Note that only the shape is checked, so if two different types of plotting commands that generate the same shape of subplots are called sequentially, the :func:`matplotlib.pyplot.figure` command should be used to explicitly create a new figure.

• The ax keyword argument can be used to direct the plotting function to use a specific axes or array of axes. The value of the ax keyword must have the proper number of axes for the plot (so a plot generating a 2x2 array of subplots should be given a 2x2 array of axes for the ax keyword).

• The color, linestyle, linewidth, and other matplotlib line property arguments can be used to override the default line properties. If these arguments are absent, the default matplotlib line properties are used and the color cycles through the default matplotlib color cycle.

  The :func:`~control.bode_plot`, :func:`~control.time_response_plot`, and selected other commands can also accept a matplotlib format string (e.g., 'r--'). The format string must appear as a positional argument right after the required data argument.

  Note that line property arguments are the same for all lines generated as part of a single plotting command call, including when multiple responses are passed as a list to the plotting command. For this reason it is often easiest to call multiple plot commands in sequence, with each command setting the line properties for that system/trace.

• The label keyword argument can be used to override the line labels that are used in generating the title and legend. If more than one line is being plotted in a given call to a plot command, the label argument value should be a list of labels, one for each line, in the order they will appear in the legend.

  For input/output plots (frequency and time responses), the labels that appear in the legend are of the form "<output name>, <input name>, <trace name>, <system name>". The trace name is used only for multi-trace time plots (for example, step responses for MIMO systems). Common information present in all traces is removed, so that the labels appearing in the legend represent the unique characteristics of each line.

  For non-input/output plots (e.g., Nyquist plots, pole/zero plots, root locus plots), the default labels are the system name.

  If label is set to False, individual lines are still given labels, but no legend is generated in the plot. (This can also be accomplished by setting legend_map to False).

  Note: the label keyword argument is not implemented for describing function plots or phase plane plots, since these plots are primarily intended to be for a single system. Standard matplotlib commands can be used to customize these plots for displaying information for multiple systems.

• The legend_loc, legend_map and show_legend keyword arguments can be used to customize the locations for legends. By default, a minimal number of legends are used such that lines can be uniquely identified and no legend is generated if there is only one line in the plot. Setting show_legend to False will suppress the legend and setting it to True will force the legend to be displayed even if there is only a single line in each axes. In addition, if the value of the legend_loc keyword argument is set to a string or integer, it will set the position of the legend as described in the :func:`matplotlib.legend` documentation. Finally, legend_map can be set to an array that matches the shape of the subplots, with each item being a string indicating the location of the legend for that axes (or None for no legend).

• The rcParams keyword argument can be used to override the default matplotlib style parameters used when creating a plot. The default parameters for all control plots are given by the ct.rcParams dictionary and have the following values:

  Key	Value
  'axes.labelsize'	'small'
  'axes.titlesize'	'small'
  'figure.titlesize'	'medium'
  'legend.fontsize'	'x-small'
  'xtick.labelsize'	'small'
  'ytick.labelsize'	'small'

  Only those values that should be changed from the default need to be specified in the rcParams keyword argument. To override the defaults for all control plots, update the ct.rcParams dictionary entries.

  The default values for style parameters for control plots can be restored using :func:`~control.reset_rcParams`.

• The title keyword can be used to override the automatic creation of the plot title. The default title is a string of the form "<Type> plot for <syslist>" where <syslist> is a list of the sys names contained in the plot (which can be updated if the plotting is called multiple times). Use title=False to suppress the title completely. The title can also be updated using the :func:`~control.ControlPlot.set_plot_title` method for the returned control plot object.

  The plot title is only generated if ax is None.

The following code illustrates the use of some of these customization features:

P = ct.tf([0.02], [1, 0.1, 0.01])   # servomechanism
C1 = ct.tf([1, 1], [1, 0])          # unstable
L1 = P * C1
C2 = ct.tf([1, 0.05], [1, 0])       # stable
L2 = P * C2

plt.rcParams.update(ct.rcParams)
fig = plt.figure(figsize=[7, 4])
ax_mag = fig.add_subplot(2, 2, 1)
ax_phase = fig.add_subplot(2, 2, 3)
ax_nyquist = fig.add_subplot(1, 2, 2)

ct.bode_plot(
    [L1, L2], ax=[ax_mag, ax_phase],
    label=["$L_1$ (unstable)", "$L_2$ (unstable)"],
    show_legend=False)
ax_mag.set_title("Bode plot for $L_1$, $L_2$")
ax_mag.tick_params(labelbottom=False)
fig.align_labels()

ct.nyquist_plot(L1, ax=ax_nyquist, label="$L_1$ (unstable)")
ct.nyquist_plot(
    L2, ax=ax_nyquist, label="$L_2$ (stable)",
    max_curve_magnitude=22, legend_loc='upper right')
ax_nyquist.set_title("Nyquist plot for $L_1$, $L_2$")

fig.suptitle("Loop analysis for servomechanism control design")
plt.tight_layout()

As this example illustrates, python-control plotting functions and Matplotlib plotting functions can generally be intermixed. One type of plot for which this does not currently work is pole/zero plots with a continuous time omega-damping grid (including root locus diagrams), due to the way that axes grids are implemented. As a workaround, the :func:`~control.pole_zero_subplots` command can be used to create an array of subplots with different grid types, as illustrated in the following example:

ax_array = ct.pole_zero_subplots(2, 1, grid=[True, False])
sys1 = ct.tf([1, 2], [1, 2, 3], name='sys1')
sys2 = ct.tf([1, 0.2], [1, 1, 3, 1, 1], name='sys2')
ct.root_locus_plot([sys1, sys2], ax=ax_array[0, 0])
cplt = ct.root_locus_plot([sys1, sys2], ax=ax_array[1, 0])
cplt.set_plot_title("Root locus plots (w/ specified axes)")

Alternatively, turning off the omega-damping grid (using grid=False or grid='empty') allows use of Matplotlib layout commands.

Response functions take a system or list of systems and return a response object that can be used to retrieve information about the system (e.g., the number of encirclements for a Nyquist plot) as well as plotting (via the plot method).

.. autosummary::
   :toctree: generated/

   ~control.describing_function_response
   ~control.frequency_response
   ~control.forced_response
   ~control.gangof4_response
   ~control.impulse_response
   ~control.initial_response
   ~control.input_output_response
   ~control.nyquist_response
   ~control.pole_zero_map
   ~control.root_locus_map
   ~control.singular_values_response
   ~control.step_response

.. autosummary::
   :toctree: generated/

   ~control.bode_plot
   ~control.describing_function_plot
   ~control.nichols_plot
   ~control.nyquist_plot
   ~control.phase_plane_plot
   ~control.phaseplot.equilpoints
   ~control.phaseplot.separatrices
   ~control.phaseplot.streamlines
   ~control.phaseplot.vectorfield
   ~control.pole_zero_plot
   ~control.root_locus_plot
   ~control.singular_values_plot
   ~control.time_response_plot

These additional functions can be used to manipulate response data or returned values from plotting routines.

.. autosummary::
   :toctree: generated/

   ~control.combine_time_responses
   ~control.pole_zero_subplots
   ~control.reset_rcParams

The following classes are used in generating response data.

.. autosummary::
   :toctree: generated/

   ~control.ControlPlot
   ~control.DescribingFunctionResponse
   ~control.FrequencyResponseData
   ~control.FrequencyResponseList
   ~control.NyquistResponseData
   ~control.PoleZeroData
   ~control.TimeResponseData
   ~control.TimeResponseList