.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/sp_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_sp_lqr.py>`
        to download the full example code.

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

.. _sphx_glr_examples_sp_lqr.py:


Discrete Time LQR
=================

.. GENERATED FROM PYTHON SOURCE LINES 5-26

.. code-block:: Python


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

    import condor as co


    class DblIntSampled(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]))
        dt = parameter()

        x = state(shape=A.shape[0])
        u = state(shape=B.shape[1])

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









.. GENERATED FROM PYTHON SOURCE LINES 27-30

To implement the periodic sampling, we use an event with
:attr:`~condor.contrib.Event.at_time` with a slice specification, starting at time
0 and occurring every ``dt`` time units without end.

.. GENERATED FROM PYTHON SOURCE LINES 30-38

.. code-block:: Python



    class SampleEvent(DblIntSampled.Event):
        at_time = slice(None, None, dt)

        update[u] = -K @ x









.. GENERATED FROM PYTHON SOURCE LINES 39-40

Now declare the trajectory analysis to simulate

.. GENERATED FROM PYTHON SOURCE LINES 40-65

.. code-block:: Python



    class DblIntSampledLQR(DblIntSampled.TrajectoryAnalysis):
        tf = 32.0

        initial[x] = [1.0, 0.1]
        # can initialize for aesthetics, but zero-crossing at t=0 updates it
        initial[u] = -K @ initial[x]

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

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

        class Options:
            adjoint_adaptive_max_step_size = False
            state_max_step_size = 0.5 / 8
            adjoint_max_step_size = 0.5 / 8


    dt = 0.5
    sim = DblIntSampledLQR(K=[0.5, 0.5], dt=dt)
    LTI_plot(sim)





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


    *

      .. image-sg:: /examples/images/sphx_glr_sp_lqr_001.png
         :alt: DblIntSampledLQR DblIntSampledLQRState.x
         :srcset: /examples/images/sphx_glr_sp_lqr_001.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/images/sphx_glr_sp_lqr_002.png
         :alt: DblIntSampledLQR DblIntSampledLQRState.u
         :srcset: /examples/images/sphx_glr_sp_lqr_002.png
         :class: sphx-glr-multi-img





.. GENERATED FROM PYTHON SOURCE LINES 66-68

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

.. GENERATED FROM PYTHON SOURCE LINES 68-84

.. code-block:: Python



    class SampledOptLQR(co.OptimizationProblem):
        K = variable(shape=DblIntSampledLQR.K.shape)
        params = parameter.create_from(DblIntSampled.parameter, K=K)
        sim = DblIntSampledLQR(**params)
        objective = sim.cost

        class Options:
            __implementation__ = co.implementations.ScipyCG


    lqr_sol = SampledOptLQR(dt=dt)

    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(
    [[0.66131423 1.32662666]]




.. GENERATED FROM PYTHON SOURCE LINES 85-86

Compare with the solution from the discrete algebraic Riccati equation:

.. GENERATED FROM PYTHON SOURCE LINES 86-97

.. code-block:: Python


    from scipy import linalg, signal

    Ad, Bd, _, _, _ = signal.cont2discrete(
        (DblIntSampledLQR.A, DblIntSampledLQR.B, None, None), dt
    )
    S = linalg.solve_discrete_are(Ad, Bd, DblIntSampledLQR.Q, DblIntSampledLQR.R)
    K = linalg.solve(Bd.T @ S @ Bd + DblIntSampledLQR.R, Bd.T @ S @ Ad)

    print(K)





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

 .. code-block:: none

    [[0.65140165 1.31420219]]




.. GENERATED FROM PYTHON SOURCE LINES 98-102

.. code-block:: Python


    sim_are = DblIntSampledLQR(K=K, dt=dt)
    LTI_plot(sim_are)




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


    *

      .. image-sg:: /examples/images/sphx_glr_sp_lqr_003.png
         :alt: DblIntSampledLQR DblIntSampledLQRState.x
         :srcset: /examples/images/sphx_glr_sp_lqr_003.png
         :class: sphx-glr-multi-img

    *

      .. image-sg:: /examples/images/sphx_glr_sp_lqr_004.png
         :alt: DblIntSampledLQR DblIntSampledLQRState.u
         :srcset: /examples/images/sphx_glr_sp_lqr_004.png
         :class: sphx-glr-multi-img





.. GENERATED FROM PYTHON SOURCE LINES 103-105

.. code-block:: Python


    plt.show()








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

   **Total running time of the script:** (0 minutes 23.155 seconds)


.. _sphx_glr_download_examples_sp_lqr.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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