.. 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_01_watt_centrifugal_governor.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_01_watt_centrifugal_governor.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_01_watt_centrifugal_governor.py:


Watt's centrifugal governor and proportional droop
==================================================

James Watt's flyball governor (1788) closed the steam valve as the engine
sped up: the throttle moved *in proportion* to the speed error. Model the
engine as :math:`J\dot\omega = \tau - b\omega - \tau_{load}` and the governor
as :math:`\tau = K_p(\omega_{ref} - \omega)`, a proportional
:class:`~engineeringkit.control_systems.PID`. A proportional governor holds
speed far better than an open throttle, but it always leaves a steady
*droop*, :math:`\Delta\omega = \tau_{load}/(b + K_p)`, which shrinks with
gain but never vanishes.

.. GENERATED FROM PYTHON SOURCE LINES 16-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import PID, dc_gain, feedback, step_response, transfer_function

    J, b = 2.0, 0.5  # inertia [kg m^2], friction [N m s]
    engine = transfer_function([1], [J, b])  # speed per unit torque








.. GENERATED FROM PYTHON SOURCE LINES 25-30

Response to a sudden load
-------------------------

The load torque enters at the plant input, so the speed disturbance is
:math:`G/(1 + GC)`.

.. GENERATED FROM PYTHON SOURCE LINES 30-46

.. code-block:: Python


    t = np.linspace(0, 40, 1601)
    load = 1.0
    fig, ax = plt.subplots()
    for kp in [0.0, 2.0, 8.0, 32.0]:
        to_speed = feedback(engine, PID(kp).transfer_function()) if kp else engine
        _, dw = step_response(to_speed, t)
        droop = -load * dc_gain(to_speed)
        assert np.isclose(droop, -load / (b + kp))
        assert np.isclose(-load * dw[-1], droop, rtol=1e-3)
        ax.plot(t, -load * dw, label=f"$K_p$ = {kp:g}: droop {droop:.3f} rad/s")
        print(f"K_p = {kp:5.1f}: steady speed change {droop:.4f} rad/s")
    ax.set(xlabel="time [s]", ylabel="speed change [rad/s]", title="Engine speed after a 1 N m load step")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/control_systems/feedback/images/sphx_glr_plot_01_watt_centrifugal_governor_001.png
   :alt: Engine speed after a 1 N m load step
   :srcset: /api/gallery/control_systems/feedback/images/sphx_glr_plot_01_watt_centrifugal_governor_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    K_p =   0.0: steady speed change -2.0000 rad/s
    K_p =   2.0: steady speed change -0.4000 rad/s
    K_p =   8.0: steady speed change -0.1176 rad/s
    K_p =  32.0: steady speed change -0.0308 rad/s





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

   **Total running time of the script:** (0 minutes 0.145 seconds)


.. _sphx_glr_download_api_gallery_control_systems_feedback_plot_01_watt_centrifugal_governor.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_01_watt_centrifugal_governor.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_watt_centrifugal_governor.ipynb <plot_01_watt_centrifugal_governor.ipynb>`

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

      :download:`Download Python source code: plot_01_watt_centrifugal_governor.py <plot_01_watt_centrifugal_governor.py>`

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

      :download:`Download zipped: plot_01_watt_centrifugal_governor.zip <plot_01_watt_centrifugal_governor.zip>`


.. only:: html

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

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