# scipy.signal.cont2discrete

scipy.signal.cont2discrete(system, dt, method='zoh', alpha=None)

  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()