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Module « scipy.signal »

Fonction cont2discrete - module scipy.signal

Signature de la fonction cont2discrete

def cont2discrete(system, dt, method='zoh', alpha=None) 

Description

help(scipy.signal.cont2discrete)

Transform a continuous to a discrete state-space system.

Parameters
----------
system : a tuple describing the system or an instance of `lti`
    The following gives the number of elements in the tuple and
    the interpretation:

        * 1: (instance of `lti`)
        * 2: (num, den)
        * 3: (zeros, poles, gain)
        * 4: (A, B, C, D)

dt : float
    The discretization time step.
method : str, optional
    Which method to use:

        * gbt: generalized bilinear transformation
        * bilinear: Tustin's approximation ("gbt" with alpha=0.5)
        * euler: Euler (or forward differencing) method ("gbt" with alpha=0)
        * backward_diff: Backwards differencing ("gbt" with alpha=1.0)
        * zoh: zero-order hold (default)
        * foh: first-order hold (*versionadded: 1.3.0*)
        * impulse: equivalent impulse response (*versionadded: 1.3.0*)

alpha : float within [0, 1], optional
    The generalized bilinear transformation weighting parameter, which
    should only be specified with method="gbt", and is ignored otherwise

Returns
-------
sysd : tuple containing the discrete system
    Based on the input type, the output will be of the form

    * (num, den, dt)   for transfer function input
    * (zeros, poles, gain, dt)   for zeros-poles-gain input
    * (A, B, C, D, dt) for state-space system input

Notes
-----
By default, the routine uses a Zero-Order Hold (zoh) method to perform
the transformation. Alternatively, a generalized bilinear transformation
may be used, which includes the common Tustin's bilinear approximation,
an Euler's method technique, or a backwards differencing technique.

The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear
approximation is based on [2]_ and [3]_, the First-Order Hold (foh) method
is based on [4]_.

References
----------
.. [1] https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models

.. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf

.. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized
    bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754,
    2009.
    (https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf)

.. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control
    of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley,
    pp. 204-206, 1998.

Examples
--------
We can transform a continuous state-space system to a discrete one:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.signal import cont2discrete, lti, dlti, dstep

Define a continuous state-space system.

>>> A = np.array([[0, 1],[-10., -3]])
>>> B = np.array([[0],[10.]])
>>> C = np.array([[1., 0]])
>>> D = np.array([[0.]])
>>> l_system = lti(A, B, C, D)
>>> t, x = l_system.step(T=np.linspace(0, 5, 100))
>>> fig, ax = plt.subplots()
>>> ax.plot(t, x, label='Continuous', linewidth=3)

Transform it to a discrete state-space system using several methods.

>>> dt = 0.1
>>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']:
...    d_system = cont2discrete((A, B, C, D), dt, method=method)
...    s, x_d = dstep(d_system)
...    ax.step(s, np.squeeze(x_d), label=method, where='post')
>>> ax.axis([t[0], t[-1], x[0], 1.4])
>>> ax.legend(loc='best')
>>> fig.tight_layout()
>>> plt.show()

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