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.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
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.. "auto_examples/09_tds_control_manual/example_2_12.py"
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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_09_tds_control_manual_example_2_12.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_09_tds_control_manual_example_2_12.py:


Example 2.12 - Stability analysis of a Smith predictor with delay mismatch
==========================================================================

We will follow example 2.12 from :cite:`appeltans2023analysis` Section 2.6.2.
In there, for given plant and controller

.. math::

    H(s) = \frac{1}{s + 1}e^{-s\tau} , C(s) = \frac{s}{2} + 2,

the classical smith predictor is constructed. Assuming the delay mismatch
:math:`\delta` we arrive at the quasipolynomial

.. math::

    D_{cl}(s) = \frac{3}{2} s + 3 + (\frac{s}{2} + 2) e^{-s\tau} + 
        (\frac{-s}{2} - 2) e^{-s(\tau + \delta)},

which we will analyse for stability in :math:`(\tau, \delta)`-parameter
space.  

.. GENERATED FROM PYTHON SOURCE LINES 27-73




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

 .. code-block:: none

    /home/runner/work/tdcpy/tdcpy/src/tdcpy/common/quasipoly.py:136: SyntaxWarning: invalid escape sequence '\d'
      \dot{x}(t) = A_0*x(t - hA_0) + ... + A_{mA}*x(t - hA_{mA})






|

.. code-block:: Python


    import numpy as np
    import tdcpy
    import tdcpy.plot
    import matplotlib.pyplot as plt
    from tdcpy.common.quasipoly import qp_to_ndde
    from tdcpy.common.quasipoly import compress_qp, qp_to_ndde


    tau, delta = 1., 0.5

    coeffs = np.array([[3.,1.5],[2.0,0.5],[-2.0,-0.5]])
    delays = np.array([0.,tau,tau+delta])

    A, hA, H, hH = qp_to_ndde(coeffs,delays,ascending=True)

    ndde = tdcpy.NDDE(A=A,hA=hA,H=H,hH=hH)

    tau_grid = np.linspace(0, 8, 201)
    delta_grid = np.linspace(-8, 10, 451)
    Z = np.zeros((len(delta_grid), len(tau_grid)))

    # for i2 in range(0,len(tau_grid)-1):
    #     tau = tau_grid[i2]
    #     for i1 in range(0,len(delta_grid)):
    #         delta = delta_grid[i1]
    #         if np.abs(tau+delta)<1e-8:
    #             # case: tau+delta = 0
    #             if np.abs(tau)<1e-8:
    #                 tau = 1e-8
    #             hH[0],hH[1] = tau, 1e-8
    #             hA[1],hA[2] = tau, 1e-8
    #             ndde = tdcpy.NDDE(H=H,hH=hH,A=A,hA=hA)
    #             Z[i1,i2] = tdcpy.strong_spectral_abscissa(ndde)
    #         elif tau + delta < 0:
    #             # case: "real" delay cannot be negative
    #             Z[i1,i2] = -np.inf
    #         else:
    #             hH[0], hH[1] = tau, tau+delta
    #             hA[1], hA[2] = tau, tau+delta
    #             ndde = tdcpy.NDDE(H=H,hH=hH,A=A,hA=hA)
    #             Z[i1,i2] = tdcpy.strong_spectral_abscissa(ndde)

    # X, Y = np.meshgrid(tau_grid,delta_grid)
    # plt.contour(X,Y,Z,[0])
    # plt.plot(plt.xlim(),[0,0],'k-.')
    # plt.show()

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

   **Total running time of the script:** (0 minutes 0.003 seconds)


.. _sphx_glr_download_auto_examples_09_tds_control_manual_example_2_12.py:

.. only:: html

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

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

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

      :download:`Download Python source code: example_2_12.py <example_2_12.py>`

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

      :download:`Download zipped: example_2_12.zip <example_2_12.zip>`


.. only:: html

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

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