.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/stability/plot_01_maxwell_on_governors.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_stability_plot_01_maxwell_on_governors.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_stability_plot_01_maxwell_on_governors.py:


Maxwell's "On Governors": stability from characteristic roots
=============================================================

Governors fitted with integral action to remove droop sometimes "hunted",
oscillating with growing amplitude. James Clerk Maxwell (1868) linearised
the engine-governor equations and showed that motion is stable exactly when
every root of the characteristic polynomial has a negative real part
(:func:`~engineeringkit.control_systems.poles`). For the cubic
:math:`s^3 + a_1 s^2 + a_2 s + a_3` he found the condition
:math:`a_1 a_2 > a_3`: too much integral gain destabilises the governor.

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import feedback, poles, step_response, transfer_function

    a1, a2 = 2.0, 3.0  # engine damping and governor stiffness








.. GENERATED FROM PYTHON SOURCE LINES 23-29

Sweep the integral gain
-----------------------

Unity feedback around :math:`k/(s(s^2 + a_1 s + a_2))` gives the cubic
above with :math:`a_3 = k`; Maxwell predicts instability for
:math:`k > a_1 a_2 = 6`.

.. GENERATED FROM PYTHON SOURCE LINES 29-43

.. code-block:: Python


    gains = np.linspace(0.5, 12, 231)
    growth = np.array([poles(feedback(transfer_function([k], [1, a1, a2, 0]))).real.max() for k in gains])
    critical = gains[np.argmax(growth > 0)]
    print(f"first unstable gain on the grid: {critical:.2f} (Maxwell: {a1 * a2:.2f})")
    assert np.all(growth[gains < a1 * a2 - 0.05] < 0) and np.all(growth[gains > a1 * a2 + 0.05] > 0)

    fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
    axes[0].plot(gains, growth)
    axes[0].axhline(0, color="k", lw=0.8)
    axes[0].axvline(a1 * a2, color="C3", ls="--", label="$a_1 a_2$")
    axes[0].set(xlabel="integral gain $k$", ylabel="largest real part of a root [1/s]")
    axes[0].legend()




.. image-sg:: /api/gallery/control_systems/stability/images/sphx_glr_plot_01_maxwell_on_governors_001.png
   :alt: plot 01 maxwell on governors
   :srcset: /api/gallery/control_systems/stability/images/sphx_glr_plot_01_maxwell_on_governors_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    first unstable gain on the grid: 6.00 (Maxwell: 6.00)

    <matplotlib.legend.Legend object at 0x7f0aee8fe840>



.. GENERATED FROM PYTHON SOURCE LINES 44-46

Hunting appears past the limit
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 46-56

.. code-block:: Python


    t = np.linspace(0, 30, 1501)
    for k in [3.0, 6.0, 7.0]:
        _, y = step_response(feedback(transfer_function([k], [1, a1, a2, 0])), t)
        axes[1].plot(t, y, label=f"$k$ = {k:g}")
    axes[1].set(xlabel="time [s]", ylabel="speed response", ylim=(-1, 3))
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.290 seconds)


.. _sphx_glr_download_api_gallery_control_systems_stability_plot_01_maxwell_on_governors.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/stability/plot_01_maxwell_on_governors.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_maxwell_on_governors.ipynb <plot_01_maxwell_on_governors.ipynb>`

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

      :download:`Download Python source code: plot_01_maxwell_on_governors.py <plot_01_maxwell_on_governors.py>`

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

      :download:`Download zipped: plot_01_maxwell_on_governors.zip <plot_01_maxwell_on_governors.zip>`


.. only:: html

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

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