.. 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_01_nyquist_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_frequency_plot_01_nyquist_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_frequency_plot_01_nyquist_stability_criterion.py:


Nyquist's stability criterion
=============================

Long-distance telephone repeaters with feedback sometimes "sang". Harry
Nyquist (1932) showed how to predict this from the measured open-loop
frequency response alone. Trace :math:`L(j\omega)` for all frequencies
(:func:`~engineeringkit.control_systems.nyquist`): the closed loop has
:math:`Z = N + P` unstable poles, where :math:`N` counts clockwise
encirclements of :math:`-1`
(:func:`~engineeringkit.control_systems.nyquist_encirclements`) and
:math:`P` the unstable open-loop poles.

.. GENERATED FROM PYTHON SOURCE LINES 16-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import feedback, nyquist_encirclements, poles, transfer_function
    from engineeringkit.control_systems.visualizers import plot_nyquist








.. GENERATED FROM PYTHON SOURCE LINES 23-27

A stable plant at two gains
---------------------------

:math:`K/(s+1)^3` crosses the negative real axis at :math:`-K/8`.

.. GENERATED FROM PYTHON SOURCE LINES 27-38

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    for K in [4.0, 12.0]:
        loop = transfer_function([K], [1, 3, 3, 1])
        N = nyquist_encirclements(loop)
        Z = int(np.sum(poles(feedback(loop)).real > 0))
        print(f"K = {K:4.1f}: N = {N}, P = 0, closed-loop unstable poles Z = {Z}")
        assert Z == N + 0
        plot_nyquist(loop, ax=axes[0], label=f"K = {K:g}, N = {N}")
    axes[0].set(xlim=(-3, 13), ylim=(-8, 8), title=r"$K/(s+1)^3$")




.. image-sg:: /api/gallery/control_systems/frequency/images/sphx_glr_plot_01_nyquist_stability_criterion_001.png
   :alt: $K/(s+1)^3$
   :srcset: /api/gallery/control_systems/frequency/images/sphx_glr_plot_01_nyquist_stability_criterion_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    K =  4.0: N = 0, P = 0, closed-loop unstable poles Z = 0
    K = 12.0: N = 2, P = 0, closed-loop unstable poles Z = 2

    [(-3.0, 13.0), (-8.0, 8.0), Text(0.5, 1.0, '$K/(s+1)^3$')]



.. GENERATED FROM PYTHON SOURCE LINES 39-44

An unstable plant stabilised by feedback
----------------------------------------

:math:`2/(s-1)` has one unstable pole (:math:`P = 1`); one
counter-clockwise encirclement (:math:`N = -1`) gives :math:`Z = 0`.

.. GENERATED FROM PYTHON SOURCE LINES 44-56

.. code-block:: Python


    loop = transfer_function([2], [1, -1])
    N = nyquist_encirclements(loop)
    Z = int(np.sum(poles(feedback(loop)).real > 0))
    print(f"2/(s-1): N = {N}, P = 1, Z = {Z}")
    assert N == -1 and Z == N + 1
    plot_nyquist(loop, ax=axes[1], label="$2/(s-1)$")
    axes[1].set(title="Unstable open loop, stable closed loop")
    axes[1].set_aspect("equal")
    fig.tight_layout()

    plt.show()




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

 .. code-block:: none

    2/(s-1): N = -1, P = 1, Z = 0





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

   **Total running time of the script:** (0 minutes 0.233 seconds)


.. _sphx_glr_download_api_gallery_control_systems_frequency_plot_01_nyquist_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/frequency/plot_01_nyquist_stability_criterion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_nyquist_stability_criterion.ipynb <plot_01_nyquist_stability_criterion.ipynb>`

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

      :download:`Download Python source code: plot_01_nyquist_stability_criterion.py <plot_01_nyquist_stability_criterion.py>`

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

      :download:`Download zipped: plot_01_nyquist_stability_criterion.zip <plot_01_nyquist_stability_criterion.zip>`


.. only:: html

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

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