.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/control_systems/state_space/plot_02_luenberger_observer.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_state_space_plot_02_luenberger_observer.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_state_space_plot_02_luenberger_observer.py:


Luenberger's observer (1964)
============================

State feedback needs every state, but sensors measure only some of them.
David Luenberger (1964) built the missing states from a copy of the model
driven by the same input and corrected by the output error,
:math:`\dot{\hat x} = A\hat x + Bu + L(y - C\hat x)`. The estimation error
:math:`e = x - \hat x` then obeys :math:`\dot e = (A - LC)e` whatever the
input, and if :math:`(A, C)` is observable the gain :math:`L` places those
error poles anywhere (:func:`~engineeringkit.control_systems.observer_gain`).

A lightly damped mass on a spring is measured by its position only; the
observer recovers the velocity.

.. GENERATED FROM PYTHON SOURCE LINES 18-50

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import linalg
    from scipy.integrate import solve_ivp

    from engineeringkit.control_systems import is_observable, observer_gain

    A = np.array([[0.0, 1.0], [-4.0, -0.2]])  # natural frequency 2 rad/s, damping ratio 0.05
    B = np.array([[0.0], [1.0]])
    C = np.array([[1.0, 0.0]])
    assert is_observable(A, C)
    error_poles = [-3.0, -4.0]
    L = observer_gain(A, C, error_poles)
    print("observer gain L =", L.ravel().round(4).tolist())
    assert np.allclose(np.sort(np.linalg.eigvals(A - L @ C).real), [-4, -3])


    def u(t):
        return np.sin(1.3 * t)


    def rhs(t, z):
        x, xhat = z[:2], z[2:]
        y = C @ x
        return np.concatenate((A @ x + B[:, 0] * u(t), A @ xhat + B[:, 0] * u(t) + L @ (y - C @ xhat)))


    x0, xhat0 = np.array([1.0, 0.5]), np.zeros(2)
    t = np.linspace(0, 6, 601)
    sol = solve_ivp(rhs, (0, 6), np.concatenate((x0, xhat0)), t_eval=t, rtol=1e-10, atol=1e-12)
    x, xhat = sol.y[:2], sol.y[2:]





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

 .. code-block:: none

    observer gain L = [6.8, 6.64]




.. GENERATED FROM PYTHON SOURCE LINES 51-53

The error follows the placed poles exactly
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 53-71

.. code-block:: Python


    predicted = np.array([linalg.expm((A - L @ C) * tk) @ (x0 - xhat0) for tk in t]).T
    assert np.allclose(x - xhat, predicted, atol=1e-7)
    # Velocity error falls by e^-3 per second or faster once the transient settles.
    print(f"velocity error at t = 0, 2, 4 s: {np.abs(x[1] - xhat[1])[[0, 200, 400]].round(5).tolist()}")
    assert np.abs(x[1] - xhat[1])[400] < 1e-4

    fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True, figsize=(7, 6))
    ax1.plot(t, x[0], "k-", label="position (measured)")
    ax1.plot(t, xhat[0], "C0--", label="estimate")
    ax1.legend()
    ax2.plot(t, x[1], "k-", label="velocity (not measured)")
    ax2.plot(t, xhat[1], "C0--", label="estimate")
    ax2.set(xlabel="time [s]")
    ax2.legend()
    ax1.set(title=f"Observer error poles {error_poles}")

    plt.show()



.. image-sg:: /api/gallery/control_systems/state_space/images/sphx_glr_plot_02_luenberger_observer_001.png
   :alt: Observer error poles [-3.0, -4.0]
   :srcset: /api/gallery/control_systems/state_space/images/sphx_glr_plot_02_luenberger_observer_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    velocity error at t = 0, 2, 4 s: [0.5, 0.01856, 5e-05]





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

   **Total running time of the script:** (0 minutes 0.053 seconds)


.. _sphx_glr_download_api_gallery_control_systems_state_space_plot_02_luenberger_observer.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/state_space/plot_02_luenberger_observer.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_luenberger_observer.ipynb <plot_02_luenberger_observer.ipynb>`

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

      :download:`Download Python source code: plot_02_luenberger_observer.py <plot_02_luenberger_observer.py>`

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

      :download:`Download zipped: plot_02_luenberger_observer.zip <plot_02_luenberger_observer.zip>`


.. only:: html

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

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