.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/nonlinear/plot_01_andronov_phase_plane.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_nonlinear_plot_01_andronov_phase_plane.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_nonlinear_plot_01_andronov_phase_plane.py:


Andronov, Vitt and Khaikin's phase plane (1937)
===============================================

Aleksandr Andronov's school in Gorky studied self-sustained oscillations by
eliminating time and drawing the motion of :math:`\ddot x = f(x, \dot x)` as
a curve in the plane :math:`(x, \dot x)`
(:func:`~engineeringkit.control_systems.phase_portrait`). A relay servo,
:math:`\ddot x = -M\,\mathrm{sign}(x)`, is not linear anywhere, yet its
phase plane is read at a glance. Each half plane holds parabolas
:math:`\dot x^2/2 + M|x| = E`, which join into closed orbits: the servo
oscillates for ever at the amplitude it started with. Feeding back the
rate, :math:`\ddot x = -M\,\mathrm{sign}(x + \tau\dot x)`, tilts the
switching line. Every orbit then loses energy at each switch, reaches the
line and *slides* along it, :math:`\dot x = -x/\tau`, into the origin.

.. GENERATED FROM PYTHON SOURCE LINES 19-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.control_systems import phase_portrait

    M = 1.0
    t = np.linspace(0, 12, 12001)
    fig, axs = plt.subplots(1, 2, figsize=(10, 4.2), sharey=True)




.. image-sg:: /api/gallery/control_systems/nonlinear/images/sphx_glr_plot_01_andronov_phase_plane_001.png
   :alt: plot 01 andronov phase plane
   :srcset: /api/gallery/control_systems/nonlinear/images/sphx_glr_plot_01_andronov_phase_plane_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 29-31

Pure relay: a family of closed orbits
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-40

.. code-block:: Python


    for x0 in (0.4, 0.7, 1.0):
        orbit = phase_portrait(lambda x, v: -M * np.sign(x), x0, 0.0, t)
        energy = orbit.v**2 / 2 + M * np.abs(orbit.x)
        print(f"x0 = {x0}: energy varies by {np.ptp(energy):.1e} around {M * x0}")
        assert np.allclose(energy, M * x0, rtol=2e-3)
        orbit.plot(ax=axs[0], field=x0 == 1.0)
    axs[0].set(title=r"$\ddot x = -M\,\mathrm{sign}(x)$: closed orbits")





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

 .. code-block:: none

    x0 = 0.4: energy varies by 6.0e-04 around 0.4
    x0 = 0.7: energy varies by 7.9e-04 around 0.7
    x0 = 1.0: energy varies by 9.4e-04 around 1.0

    [Text(0.5, 1.0, '$\\ddot x = -M\\,\\mathrm{sign}(x)$: closed orbits')]



.. GENERATED FROM PYTHON SOURCE LINES 41-43

Rate feedback: switching, then sliding
--------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-64

.. code-block:: Python


    tau = 0.4
    slide = phase_portrait(lambda x, v: -M * np.sign(x + tau * v), 1.0, 0.0, t)
    energy = slide.v**2 / 2 + M * np.abs(slide.x)
    assert np.all(np.diff(energy) <= 1e-6)  # loses energy at each switch, gains none (to the RK4 step error)

    # Once on the switching line the motion obeys x + tau v = 0, so x decays as exp(-t / tau).
    on_line = np.abs(slide.x + tau * slide.v) < 2e-3
    start = np.flatnonzero(~on_line)[-1] + 1
    later = (t > t[start]) & (t < t[start] + 2)
    print(f"sliding from t = {t[start]:.2f} s at x = {slide.x[start]:.4f}")
    assert np.allclose(slide.x[later], slide.x[start] * np.exp(-(t[later] - t[start]) / tau), atol=1e-4)
    slide.plot(ax=axs[1])
    line = np.linspace(-1.2, 1.2, 2)
    axs[1].plot(line, -line / tau, "r--", lw=1, label=r"switching line $x + \tau\dot x = 0$")
    axs[1].legend(loc="lower left", fontsize=8)
    axs[1].set(title=rf"$\ddot x = -M\,\mathrm{{sign}}(x + {tau}\,\dot x)$: sliding to rest")
    print("final state with rate feedback:", slide.summary())
    assert abs(slide.x[-1]) < 1e-6 and abs(slide.v[-1]) < 1e-6
    fig.tight_layout()
    plt.show()




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

 .. code-block:: none

    sliding from t = 2.41 s at x = -0.1088
    final state with rate feedback: {'final_x': 1.430924759760228e-15, 'final_v': 3.0964814046186007e-16, 'max_abs_x': 1.0}





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

   **Total running time of the script:** (0 minutes 0.999 seconds)


.. _sphx_glr_download_api_gallery_control_systems_nonlinear_plot_01_andronov_phase_plane.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/nonlinear/plot_01_andronov_phase_plane.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_andronov_phase_plane.ipynb <plot_01_andronov_phase_plane.ipynb>`

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

      :download:`Download Python source code: plot_01_andronov_phase_plane.py <plot_01_andronov_phase_plane.py>`

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

      :download:`Download zipped: plot_01_andronov_phase_plane.zip <plot_01_andronov_phase_plane.zip>`


.. only:: html

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

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