.. 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_02_routh_stability_criterion.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_02_routh_stability_criterion.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_02_routh_stability_criterion.py:


Routh's stability criterion
===========================

Maxwell could not state a test for polynomials above fourth order. Edward
Routh's Adams Prize essay (1877) solved the problem with a table built from
the coefficients alone: the number of sign changes down its first column
(:func:`~engineeringkit.control_systems.routh_first_column`) equals the
number of roots in the right half-plane. For the closed loop of
:math:`K/(s(s+1)(s+2))`, :math:`s^3 + 3s^2 + 2s + K`, the table shows the
loop is stable only for :math:`0 < K < 6`.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import routh_first_column, routh_stable








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Sign changes count right-half-plane roots
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 23-36

.. code-block:: Python


    gains = np.linspace(0.05, 15.05, 151)  # avoids the marginal K = 6
    changes, unstable_roots = [], []
    for K in gains:
        column = routh_first_column([1, 3, 2, K])
        changes.append(int(np.sum(np.diff(np.sign(column)) != 0)))
        unstable_roots.append(int(np.sum(np.roots([1, 3, 2, K]).real > 0)))
    assert changes == unstable_roots
    assert all(routh_stable([1, 3, 2, K]) == (K < 6) for K in gains)

    for K in [2.0, 10.0]:
        print(f"K = {K:4.1f}: first column {np.round(routh_first_column([1, 3, 2, K]), 3)}")





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

 .. code-block:: none

    K =  2.0: first column [1.    3.    1.333 2.   ]
    K = 10.0: first column [ 1.     3.    -1.333 10.   ]




.. GENERATED FROM PYTHON SOURCE LINES 37-39

A fifth-order example
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-53

.. code-block:: Python


    p = np.poly([-1, -2, 0.5 + 1j, 0.5 - 1j, -3]).real
    column = routh_first_column(p)
    print("fifth order:", np.round(column, 3), "->", int(np.sum(np.diff(np.sign(column)) != 0)), "sign changes")
    assert int(np.sum(np.diff(np.sign(column)) != 0)) == 2

    fig, ax = plt.subplots()
    ax.step(gains, changes, where="mid", label="Routh sign changes")
    ax.plot(gains, unstable_roots, "o", ms=3, mfc="none", label="right-half-plane roots")
    ax.axvline(6, color="C3", ls="--", lw=0.8)
    ax.set(xlabel="loop gain $K$", ylabel="count", title="$s^3 + 3s^2 + 2s + K$")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/control_systems/stability/images/sphx_glr_plot_02_routh_stability_criterion_001.png
   :alt: $s^3 + 3s^2 + 2s + K$
   :srcset: /api/gallery/control_systems/stability/images/sphx_glr_plot_02_routh_stability_criterion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    fifth order: [ 1.     5.     5.75  -2.935 20.944  7.5  ] -> 2 sign changes





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

   **Total running time of the script:** (0 minutes 0.119 seconds)


.. _sphx_glr_download_api_gallery_control_systems_stability_plot_02_routh_stability_criterion.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_02_routh_stability_criterion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_routh_stability_criterion.ipynb <plot_02_routh_stability_criterion.ipynb>`

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

      :download:`Download Python source code: plot_02_routh_stability_criterion.py <plot_02_routh_stability_criterion.py>`

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

      :download:`Download zipped: plot_02_routh_stability_criterion.zip <plot_02_routh_stability_criterion.zip>`


.. only:: html

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

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