.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/time_response/plot_01_hazen_servomechanism.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_time_response_plot_01_hazen_servomechanism.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_time_response_plot_01_hazen_servomechanism.py:


Hazen's theory of servomechanisms
=================================

Harold Hazen at MIT (1934) coined *servomechanism* and analysed
position-following systems as second-order dynamics,
:math:`\omega_n^2/(s^2 + 2\zeta\omega_n s + \omega_n^2)`. The damping ratio
:math:`\zeta` alone fixes the overshoot, and :math:`\zeta` with the natural
frequency :math:`\omega_n` sets peak and settling times
(:func:`~engineeringkit.control_systems.second_order_metrics`). The
simulated step and impulse responses
(:func:`~engineeringkit.control_systems.step_response`,
:func:`~engineeringkit.control_systems.impulse_response`) confirm the
formulas.

.. GENERATED FROM PYTHON SOURCE LINES 18-49



.. image-sg:: /api/gallery/control_systems/time_response/images/sphx_glr_plot_01_hazen_servomechanism_001.png
   :alt: plot 01 hazen servomechanism
   :srcset: /api/gallery/control_systems/time_response/images/sphx_glr_plot_01_hazen_servomechanism_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    zeta=0.2: overshoot 52.66% (formula 52.66%), settling 9.80 s (estimate 10.00 s)
    zeta=0.4: overshoot 25.38% (formula 25.38%), settling 4.20 s (estimate 5.00 s)
    zeta=0.6: overshoot  9.48% (formula  9.48%), settling 2.97 s (estimate 3.33 s)
    zeta=0.8: overshoot  1.52% (formula  1.52%), settling 1.88 s (estimate 2.50 s)






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import impulse_response, second_order_metrics, step_response, transfer_function

    wn = 2.0
    t = np.linspace(0, 10, 20001)
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
    for zeta in [0.2, 0.4, 0.6, 0.8]:
        servo = transfer_function([wn**2], [1, 2 * zeta * wn, wn**2])
        _, y = step_response(servo, t)
        _, h = impulse_response(servo, t)
        spec = second_order_metrics(zeta, wn)
        overshoot, peak_time = 100 * (y.max() - 1), t[np.argmax(y)]
        settled = t[np.flatnonzero(np.abs(y - 1) > 0.02)[-1]]
        print(
            f"zeta={zeta}: overshoot {overshoot:5.2f}% (formula {spec['percent_overshoot']:5.2f}%), "
            f"settling {settled:.2f} s (estimate {spec['settling_time_2pct']:.2f} s)"
        )
        assert abs(overshoot - spec["percent_overshoot"]) < 0.01 and abs(peak_time - spec["peak_time"]) < 1e-3
        # The impulse response is the derivative of the step response; it first crosses zero at the peak.
        assert abs(t[np.argmax(h < 0)] - spec["peak_time"]) < 1e-3
        axes[0].plot(t, y, label=rf"$\zeta$ = {zeta}")
        axes[1].plot(t, h, label=rf"$\zeta$ = {zeta}")
    axes[0].axhspan(0.98, 1.02, color="gray", alpha=0.2)
    axes[0].set(xlabel="time [s]", ylabel="step response")
    axes[1].set(xlabel="time [s]", ylabel="impulse response [1/s]")
    axes[0].legend()
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.664 seconds)


.. _sphx_glr_download_api_gallery_control_systems_time_response_plot_01_hazen_servomechanism.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/time_response/plot_01_hazen_servomechanism.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hazen_servomechanism.ipynb <plot_01_hazen_servomechanism.ipynb>`

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

      :download:`Download Python source code: plot_01_hazen_servomechanism.py <plot_01_hazen_servomechanism.py>`

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

      :download:`Download zipped: plot_01_hazen_servomechanism.zip <plot_01_hazen_servomechanism.zip>`


.. only:: html

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

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