.. 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_04_hurwitz_determinants.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_04_hurwitz_determinants.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_04_hurwitz_determinants.py:


Hurwitz's determinant test
==========================

Aurel Stodola, designing turbine regulators in Zürich, asked Adolf Hurwitz
for a practical stability condition. Hurwitz (1895), unaware of Routh's
work, gave it as determinants: build the matrix
:math:`H_{ij} = a_{2j-i}` from the coefficients, and every root lies in the
left half-plane exactly when all leading minors are positive
(:func:`~engineeringkit.control_systems.hurwitz_determinants`). At the
boundary :math:`\Delta_{n-1}` passes through zero as a pair of roots
crosses the imaginary axis.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import hurwitz_determinants








.. GENERATED FROM PYTHON SOURCE LINES 22-27

A quartic family
----------------

The roots of :math:`(s+1)^4 + K` are :math:`-1 + K^{1/4}e^{i(\pi/4 + k\pi/2)}`,
so the rightmost pair reaches the imaginary axis at :math:`K = 4`.

.. GENERATED FROM PYTHON SOURCE LINES 27-50

.. code-block:: Python


    gains = np.linspace(0.1, 8, 400)
    deltas = np.array([hurwitz_determinants(np.poly([-1, -1, -1, -1]) + [0, 0, 0, 0, K]) for K in gains])
    rightmost = np.array([np.roots(np.poly([-1, -1, -1, -1]) + [0, 0, 0, 0, K]).real.max() for K in gains])
    stable_by_hurwitz = np.all(deltas > 0, axis=1)
    assert np.array_equal(stable_by_hurwitz, rightmost < 0)
    boundary = gains[np.argmin(np.abs(deltas[:, 2]))]
    print(f"Delta_3 vanishes near K = {boundary:.3f} (exact 4)")
    assert abs(boundary - 4) < 0.02

    fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
    for k in range(4):
        axes[0].plot(gains, deltas[:, k], label=rf"$\Delta_{k + 1}$")
    axes[0].axhline(0, color="k", lw=0.8)
    axes[0].axvline(4, color="C3", ls="--", lw=0.8)
    axes[0].set(xlabel="$K$", ylabel="Hurwitz minor", ylim=(-60, 120))
    axes[0].legend()
    axes[1].plot(gains, rightmost)
    axes[1].axhline(0, color="k", lw=0.8)
    axes[1].set(xlabel="$K$", ylabel="largest real part of a root")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/control_systems/stability/images/sphx_glr_plot_04_hurwitz_determinants_001.png
   :alt: plot 04 hurwitz determinants
   :srcset: /api/gallery/control_systems/stability/images/sphx_glr_plot_04_hurwitz_determinants_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Delta_3 vanishes near K = 4.001 (exact 4)





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

   **Total running time of the script:** (0 minutes 0.263 seconds)


.. _sphx_glr_download_api_gallery_control_systems_stability_plot_04_hurwitz_determinants.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_04_hurwitz_determinants.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_hurwitz_determinants.ipynb <plot_04_hurwitz_determinants.ipynb>`

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

      :download:`Download Python source code: plot_04_hurwitz_determinants.py <plot_04_hurwitz_determinants.py>`

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

      :download:`Download zipped: plot_04_hurwitz_determinants.zip <plot_04_hurwitz_determinants.zip>`


.. only:: html

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

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