.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/ct_lqr.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_ct_lqr.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_ct_lqr.py:


Continuous Time LQR
===================

.. GENERATED FROM PYTHON SOURCE LINES 5-44

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from _sgm_test_util import LTI_plot

    import condor as co


    class DblInt(co.ODESystem):
        A = np.array([[0.0, 1.0], [0.0, 0.0]])
        B = np.array([[0.0], [1.0]])

        K = parameter(shape=(1, B.shape[0]))

        x = state(shape=A.shape[0])
        dynamic_output.u = -K @ x

        dot[x] = A @ x + B @ u


    class DblIntLQR(DblInt.TrajectoryAnalysis):
        tf = 32.0

        initial[x] = [1.0, 0.1]

        Q = np.eye(2)
        R = np.eye(1)

        cost = trajectory_output(integrand=(x.T @ Q @ x + u.T @ R @ u) / 2)

        class Options:
            state_rtol = 1e-8
            adjoint_rtol = 1e-8


    ct_sim = DblIntLQR(K=[1.0, 0.1])
    LTI_plot(ct_sim)





.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/images/sphx_glr_ct_lqr_001.png
         :alt: DblIntLQR DblIntLQRState.x
         :srcset: /examples/images/sphx_glr_ct_lqr_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/images/sphx_glr_ct_lqr_002.png
         :alt: DblIntLQR DblIntLQRDynamicOutput.u
         :srcset: /examples/images/sphx_glr_ct_lqr_002.png
         :class: sphx-glr-multi-img





.. GENERATED FROM PYTHON SOURCE LINES 45-47

Determine the optimal gain by embedding the trajectory analysis in an optimization
problem:

.. GENERATED FROM PYTHON SOURCE LINES 47-63

.. code-block:: Python



    class CtOptLQR(co.OptimizationProblem):
        K = variable(shape=DblIntLQR.K.shape)

        sim = DblIntLQR(K=K)
        objective = sim.cost

        class Options:
            __implementation__ = co.implementations.ScipyCG


    lqr_sol = CtOptLQR()

    print(lqr_sol.K)





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

 .. code-block:: none

    /opt/hostedtoolcache/Python/3.12.11/x64/lib/python3.12/site-packages/condor/implementations/iterative.py:481: RuntimeWarning: Method CG cannot handle bounds.
      min_out = minimize(
    [[0.99997598 1.73199404]]




.. GENERATED FROM PYTHON SOURCE LINES 64-65

Compare with the solution from the continuous algebraic Riccati equation:

.. GENERATED FROM PYTHON SOURCE LINES 65-73

.. code-block:: Python


    from scipy import linalg

    S = linalg.solve_continuous_are(DblIntLQR.A, DblIntLQR.B, DblIntLQR.Q, DblIntLQR.R)
    K = linalg.solve(DblIntLQR.R, DblIntLQR.B.T @ S)

    print(K)





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

 .. code-block:: none

    [[1.         1.73205081]]




.. GENERATED FROM PYTHON SOURCE LINES 74-78

.. code-block:: Python


    sim_are = DblIntLQR(K=K)
    LTI_plot(sim_are)




.. rst-class:: sphx-glr-horizontal


    *

      .. image-sg:: /examples/images/sphx_glr_ct_lqr_003.png
         :alt: DblIntLQR DblIntLQRState.x
         :srcset: /examples/images/sphx_glr_ct_lqr_003.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/images/sphx_glr_ct_lqr_004.png
         :alt: DblIntLQR DblIntLQRDynamicOutput.u
         :srcset: /examples/images/sphx_glr_ct_lqr_004.png
         :class: sphx-glr-multi-img





.. GENERATED FROM PYTHON SOURCE LINES 79-81

.. code-block:: Python


    plt.show()








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

   **Total running time of the script:** (0 minutes 2.405 seconds)


.. _sphx_glr_download_examples_ct_lqr.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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

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

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

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

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


.. only:: html

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

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