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.. _sphx_glr_auto_examples_09_tds_control_manual_example_2_01.py:


Example 2.1 - Stability analysis of retarded DDE
================================================

We consider the following retarded delay differential equation (RDDE) from
:cite:`verheyden2008efficient` Section 6.1:

.. math::

    \dot{x}(t) =
        \begin{bmatrix}
            -1 & 0 & 0 & 0 \\
            0 & 1 & 0 & 0 \\
            0 & 0 & -10 & -4 \\
            0 & 0 & 4 & -10
        \end{bmatrix} x(t)
        + 
        \begin{bmatrix}
            3 & 3 & 3 & 3\\
            0 & -1.5 & 0 & 0 \\
            0 & 0 & 3 & -5 \\
            0 & 5 & 5 & 5
        \end{bmatrix} x(t-1).

We will follow the steps from :cite:`appeltans2023analysis` Section 2.2 to
achive the same results as presented there, i.e. we will

1. create the `RDDE` matrix representation
2. compute characteristic roots via the `tdcpy.roots` function
3. plot the computed characteristic roots using `tdcpy.plot.eigen_plot` function

.. bibliography::
    :filter: {"auto_examples/tds_control_manual/example_2_1"} & docnames

.. GENERATED FROM PYTHON SOURCE LINES 40-62



.. image-sg:: /auto_examples/09_tds_control_manual/images/sphx_glr_example_2_01_001.png
   :alt: example 2 01
   :srcset: /auto_examples/09_tds_control_manual/images/sphx_glr_example_2_01_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    import tdcpy
    import tdcpy.plot

    # Create RDDE matrix representation
    A0 = np.array([[-1, 0, 0, 0],
                   [0, 1, 0, 0],
                   [0, 0, -10, -4],
                   [0, 0, 4, -10]])
    A1 = np.array([[3, 3, 3, 3],
                   [0, -1.5, 0, 0],
                   [0, 0, 3, -5],
                   [0, 5, 5, 5]])

    rdde = tdcpy.RDDE(A=[A0, A1], hA=[0, 1])
    cr, info = tdcpy.roots(rdde, r=-2.5)

    tdcpy.plot.eigen_plot(cr)
    plt.show()



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