.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/feedback/plot_04_ziegler_nichols_tuning.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_feedback_plot_04_ziegler_nichols_tuning.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_feedback_plot_04_ziegler_nichols_tuning.py:


Ziegler and Nichols tune a PID from the edge of instability
===========================================================

John Ziegler and Nathaniel Nichols of Taylor Instruments (1942) gave plant
engineers a recipe: turn up a proportional gain until the loop oscillates
steadily, note that *ultimate gain* :math:`K_u` and period :math:`T_u`, and
read P, PI or PID settings from a table
(:func:`~engineeringkit.control_systems.ziegler_nichols`). The ultimate
point is exactly the gain margin and phase-crossover frequency of the loop
(:func:`~engineeringkit.control_systems.stability_margins`). For
:math:`1/(s+1)^3`, :math:`K_u = 8` and :math:`\omega_u = \sqrt3` rad/s.

.. GENERATED FROM PYTHON SOURCE LINES 16-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import poles, stability_margins, transfer_function, ziegler_nichols

    plant = transfer_function([1], [1, 3, 3, 1])
    margins = stability_margins(plant)
    ku, tu = margins["gain_margin"], 2 * np.pi / margins["phase_cross_frequency"]
    print(f"ultimate gain {ku:.4f}, ultimate period {tu:.4f} s")
    assert np.isclose(ku, 8.0, rtol=1e-4) and np.isclose(tu, 2 * np.pi / np.sqrt(3), rtol=1e-4)





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

 .. code-block:: none

    ultimate gain 8.0000, ultimate period 3.6276 s




.. GENERATED FROM PYTHON SOURCE LINES 28-30

Tuned step responses
--------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-46

.. code-block:: Python


    t = np.linspace(0, 60, 3001)
    fig, ax = plt.subplots()
    for kind in ["P", "PI", "PID"]:
        pid = ziegler_nichols(ku, tu, kind)
        assert np.all(poles(pid.closed_loop(plant)).real < 0)
        r = pid.simulate(plant, t)
        ax.plot(r.t, r.y, label=f"{kind}: kp={pid.kp:.2f}, ki={pid.ki:.2f}, kd={pid.kd:.2f}")
        print(kind, {k: round(v, 3) for k, v in r.summary().items()})
        if kind != "P":
            assert abs(r.y[-1] - 1) < 0.01
    ax.axhline(1, color="k", ls="--", lw=0.8)
    ax.set(xlabel="time [s]", ylabel="output", title=r"Ziegler-Nichols settings for $1/(s+1)^3$")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/control_systems/feedback/images/sphx_glr_plot_04_ziegler_nichols_tuning_001.png
   :alt: Ziegler-Nichols settings for $1/(s+1)^3$
   :srcset: /api/gallery/control_systems/feedback/images/sphx_glr_plot_04_ziegler_nichols_tuning_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    P {'final': 0.8, 'peak': 1.234}
    PI {'final': 0.999, 'peak': 1.561}
    PID {'final': 1.0, 'peak': 1.406}





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

   **Total running time of the script:** (0 minutes 0.189 seconds)


.. _sphx_glr_download_api_gallery_control_systems_feedback_plot_04_ziegler_nichols_tuning.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/feedback/plot_04_ziegler_nichols_tuning.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_ziegler_nichols_tuning.ipynb <plot_04_ziegler_nichols_tuning.ipynb>`

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

      :download:`Download Python source code: plot_04_ziegler_nichols_tuning.py <plot_04_ziegler_nichols_tuning.py>`

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

      :download:`Download zipped: plot_04_ziegler_nichols_tuning.zip <plot_04_ziegler_nichols_tuning.zip>`


.. only:: html

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

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