.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/frequency/plot_04_nichols_chart.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_control_systems_frequency_plot_04_nichols_chart.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_control_systems_frequency_plot_04_nichols_chart.py:


The Nichols chart (1947)
========================

Nathaniel Nichols, at the MIT Radiation Laboratory during the war, plotted
the open-loop gain in decibels against its phase. On those axes the
closed-loop gain of a unity-feedback loop, :math:`|L/(1+L)|`, is a fixed
family of contours (:func:`~engineeringkit.control_systems.nichols_closed_loop`),
so one curve shows both. The highest contour the locus touches is the
closed-loop resonant peak :math:`M_r`
(:func:`~engineeringkit.control_systems.nichols`), and reshaping the locus
away from the critical point :math:`(-180^\circ, 0\text{ dB})` lowers it.
The chart was published in James, Nichols and Phillips's *Theory of
Servomechanisms* (1947). Here a lead compensator
(:func:`~engineeringkit.control_systems.design_lead`) lifts the phase of
:math:`L = 4/(s(s+1)(s+2))` near crossover and pulls the resonant peak
down.

.. GENERATED FROM PYTHON SOURCE LINES 21-47



.. image-sg:: /api/gallery/control_systems/frequency/images/sphx_glr_plot_04_nichols_chart_001.png
   :alt: Nichols chart with closed-loop M-contours
   :srcset: /api/gallery/control_systems/frequency/images/sphx_glr_plot_04_nichols_chart_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    uncompensated: Mr = 14.69 dB at 1.187 rad/s
    lead compensated: Mr = 5.54 dB at 1.930 rad/s






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import bode, complementary_sensitivity, design_lead, nichols, transfer_function
    from engineeringkit.control_systems.visualizers import plot_nichols

    loop = transfer_function([4], [1, 3, 2, 0])
    design = design_lead(loop, 50.0)
    w = np.logspace(-2, 2, 4000)

    ax = plot_nichols(loop, w, label="uncompensated")
    plot_nichols(design.compensated_loop, w, ax=ax, label="lead compensated", contours=None)
    ax.set(xlim=(-270, -60), ylim=(-30, 30), title="Nichols chart with closed-loop M-contours")

    for name, L in (("uncompensated", loop), ("lead compensated", design.compensated_loop)):
        n = nichols(L, w)
        # The resonant peak read from the chart is the peak of the closed-loop Bode magnitude.
        _, mag, _ = bode(complementary_sensitivity(L), w)
        print(f"{name}: Mr = {n['resonant_peak_db']:.2f} dB at {n['resonant_frequency']:.3f} rad/s")
        assert np.isclose(n["resonant_peak_db"], mag.max(), atol=1e-6)
        i = np.argmax(n["closed_gain_db"])
        ax.plot(n["phase_deg"][i], n["gain_db"][i], "ks", mfc="none")

    before, after = nichols(loop, w)["resonant_peak_db"], nichols(design.compensated_loop, w)["resonant_peak_db"]
    assert after < before - 3
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.190 seconds)


.. _sphx_glr_download_api_gallery_control_systems_frequency_plot_04_nichols_chart.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/control_systems/frequency/plot_04_nichols_chart.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_04_nichols_chart.ipynb <plot_04_nichols_chart.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_04_nichols_chart.py <plot_04_nichols_chart.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_nichols_chart.zip <plot_04_nichols_chart.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_