_lti_conversion.py

"""
ltisys -- a collection of functions to convert linear time invariant systems
from one representation to another.
"""
import numpy
import numpy as np
from numpy import (r_, eye, atleast_2d, poly, dot,
                   asarray, prod, zeros, array, outer)
from scipy import linalg

from ._filter_design import tf2zpk, zpk2tf, normalize


__all__ = ['tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk',
           'cont2discrete']


def tf2ss(num, den):
    r"""Transfer function to state-space representation.

    Parameters
    ----------
    num, den : array_like
        Sequences representing the coefficients of the numerator and
        denominator polynomials, in order of descending degree. The
        denominator needs to be at least as long as the numerator.

    Returns
    -------
    A, B, C, D : ndarray
        State space representation of the system, in controller canonical
        form.

    Examples
    --------
    Convert the transfer function:

    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}

    >>> num = [1, 3, 3]
    >>> den = [1, 2, 1]

    to the state-space representation:

    .. math::

        \dot{\textbf{x}}(t) =
        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\

        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)

    >>> from scipy.signal import tf2ss
    >>> A, B, C, D = tf2ss(num, den)
    >>> A
    array([[-2., -1.],
           [ 1.,  0.]])
    >>> B
    array([[ 1.],
           [ 0.]])
    >>> C
    array([[ 1.,  2.]])
    >>> D
    array([[ 1.]])
    """
    # Controller canonical state-space representation.
    #  if M+1 = len(num) and K+1 = len(den) then we must have M <= K
    #  states are found by asserting that X(s) = U(s) / D(s)
    #  then Y(s) = N(s) * X(s)
    #
    #   A, B, C, and D follow quite naturally.
    #
    num, den = normalize(num, den)   # Strips zeros, checks arrays
    nn = len(num.shape)
    if nn == 1:
        num = asarray([num], num.dtype)
    M = num.shape[1]
    K = len(den)
    if M > K:
        msg = "Improper transfer function. `num` is longer than `den`."
        raise ValueError(msg)
    if M == 0 or K == 0:  # Null system
        return (array([], float), array([], float), array([], float),
                array([], float))

    # pad numerator to have same number of columns has denominator
    num = r_['-1', zeros((num.shape[0], K - M), num.dtype), num]

    if num.shape[-1] > 0:
        D = atleast_2d(num[:, 0])

    else:
        # We don't assign it an empty array because this system
        # is not 'null'. It just doesn't have a non-zero D
        # matrix. Thus, it should have a non-zero shape so that
        # it can be operated on by functions like 'ss2tf'
        D = array([[0]], float)

    if K == 1:
        D = D.reshape(num.shape)

        return (zeros((1, 1)), zeros((1, D.shape[1])),
                zeros((D.shape[0], 1)), D)

    frow = -array([den[1:]])
    A = r_[frow, eye(K - 2, K - 1)]
    B = eye(K - 1, 1)
    C = num[:, 1:] - outer(num[:, 0], den[1:])
    D = D.reshape((C.shape[0], B.shape[1]))

    return A, B, C, D


def _none_to_empty_2d(arg):
    if arg is None:
        return zeros((0, 0))
    else:
        return arg


def _atleast_2d_or_none(arg):
    if arg is not None:
        return atleast_2d(arg)


def _shape_or_none(M):
    if M is not None:
        return M.shape
    else:
        return (None,) * 2


def _choice_not_none(*args):
    for arg in args:
        if arg is not None:
            return arg


def _restore(M, shape):
    if M.shape == (0, 0):
        return zeros(shape)
    else:
        if M.shape != shape:
            raise ValueError("The input arrays have incompatible shapes.")
        return M


def abcd_normalize(A=None, B=None, C=None, D=None):
    """Check state-space matrices and ensure they are 2-D.

    If enough information on the system is provided, that is, enough
    properly-shaped arrays are passed to the function, the missing ones
    are built from this information, ensuring the correct number of
    rows and columns. Otherwise a ValueError is raised.

    Parameters
    ----------
    A, B, C, D : array_like, optional
        State-space matrices. All of them are None (missing) by default.
        See `ss2tf` for format.

    Returns
    -------
    A, B, C, D : array
        Properly shaped state-space matrices.

    Raises
    ------
    ValueError
        If not enough information on the system was provided.

    """
    A, B, C, D = map(_atleast_2d_or_none, (A, B, C, D))

    MA, NA = _shape_or_none(A)
    MB, NB = _shape_or_none(B)
    MC, NC = _shape_or_none(C)
    MD, ND = _shape_or_none(D)

    p = _choice_not_none(MA, MB, NC)
    q = _choice_not_none(NB, ND)
    r = _choice_not_none(MC, MD)
    if p is None or q is None or r is None:
        raise ValueError("Not enough information on the system.")

    A, B, C, D = map(_none_to_empty_2d, (A, B, C, D))
    A = _restore(A, (p, p))
    B = _restore(B, (p, q))
    C = _restore(C, (r, p))
    D = _restore(D, (r, q))

    return A, B, C, D


def ss2tf(A, B, C, D, input=0):
    r"""State-space to transfer function.

    A, B, C, D defines a linear state-space system with `p` inputs,
    `q` outputs, and `n` state variables.

    Parameters
    ----------
    A : array_like
        State (or system) matrix of shape ``(n, n)``
    B : array_like
        Input matrix of shape ``(n, p)``
    C : array_like
        Output matrix of shape ``(q, n)``
    D : array_like
        Feedthrough (or feedforward) matrix of shape ``(q, p)``
    input : int, optional
        For multiple-input systems, the index of the input to use.

    Returns
    -------
    num : 2-D ndarray
        Numerator(s) of the resulting transfer function(s). `num` has one row
        for each of the system's outputs. Each row is a sequence representation
        of the numerator polynomial.
    den : 1-D ndarray
        Denominator of the resulting transfer function(s). `den` is a sequence
        representation of the denominator polynomial.

    Examples
    --------
    Convert the state-space representation:

    .. math::

        \dot{\textbf{x}}(t) =
        \begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
        \begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\

        \textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
        \begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)

    >>> A = [[-2, -1], [1, 0]]
    >>> B = [[1], [0]]  # 2-D column vector
    >>> C = [[1, 2]]    # 2-D row vector
    >>> D = 1

    to the transfer function:

    .. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}

    >>> from scipy.signal import ss2tf
    >>> ss2tf(A, B, C, D)
    (array([[1., 3., 3.]]), array([ 1.,  2.,  1.]))
    """
    # transfer function is C (sI - A)**(-1) B + D

    # Check consistency and make them all rank-2 arrays
    A, B, C, D = abcd_normalize(A, B, C, D)

    nout, nin = D.shape
    if input >= nin:
        raise ValueError("System does not have the input specified.")

    # make SIMO from possibly MIMO system.
    B = B[:, input:input + 1]
    D = D[:, input:input + 1]

    try:
        den = poly(A)
    except ValueError:
        den = 1

    if (prod(B.shape, axis=0) == 0) and (prod(C.shape, axis=0) == 0):
        num = numpy.ravel(D)
        if (prod(D.shape, axis=0) == 0) and (prod(A.shape, axis=0) == 0):
            den = []
        return num, den

    num_states = A.shape[0]
    type_test = A[:, 0] + B[:, 0] + C[0, :] + D + 0.0
    num = numpy.empty((nout, num_states + 1), type_test.dtype)
    for k in range(nout):
        Ck = atleast_2d(C[k, :])
        num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den

    return num, den


def zpk2ss(z, p, k):
    """Zero-pole-gain representation to state-space representation

    Parameters
    ----------
    z, p : sequence
        Zeros and poles.
    k : float
        System gain.

    Returns
    -------
    A, B, C, D : ndarray
        State space representation of the system, in controller canonical
        form.

    """
    return tf2ss(*zpk2tf(z, p, k))


def ss2zpk(A, B, C, D, input=0):
    """State-space representation to zero-pole-gain representation.

    A, B, C, D defines a linear state-space system with `p` inputs,
    `q` outputs, and `n` state variables.

    Parameters
    ----------
    A : array_like
        State (or system) matrix of shape ``(n, n)``
    B : array_like
        Input matrix of shape ``(n, p)``
    C : array_like
        Output matrix of shape ``(q, n)``
    D : array_like
        Feedthrough (or feedforward) matrix of shape ``(q, p)``
    input : int, optional
        For multiple-input systems, the index of the input to use.

    Returns
    -------
    z, p : sequence
        Zeros and poles.
    k : float
        System gain.

    """
    return tf2zpk(*ss2tf(A, B, C, D, input=input))


def 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()

    """
    if len(system) == 1:
        return system.to_discrete()
    if len(system) == 2:
        sysd = cont2discrete(tf2ss(system[0], system[1]), dt, method=method,
                             alpha=alpha)
        return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
    elif len(system) == 3:
        sysd = cont2discrete(zpk2ss(system[0], system[1], system[2]), dt,
                             method=method, alpha=alpha)
        return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
    elif len(system) == 4:
        a, b, c, d = system
    else:
        raise ValueError("First argument must either be a tuple of 2 (tf), "
                         "3 (zpk), or 4 (ss) arrays.")

    if method == 'gbt':
        if alpha is None:
            raise ValueError("Alpha parameter must be specified for the "
                             "generalized bilinear transform (gbt) method")
        elif alpha < 0 or alpha > 1:
            raise ValueError("Alpha parameter must be within the interval "
                             "[0,1] for the gbt method")

    if method == 'gbt':
        # This parameter is used repeatedly - compute once here
        ima = np.eye(a.shape[0]) - alpha*dt*a
        ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a)
        bd = linalg.solve(ima, dt*b)

        # Similarly solve for the output equation matrices
        cd = linalg.solve(ima.transpose(), c.transpose())
        cd = cd.transpose()
        dd = d + alpha*np.dot(c, bd)

    elif method == 'bilinear' or method == 'tustin':
        return cont2discrete(system, dt, method="gbt", alpha=0.5)

    elif method == 'euler' or method == 'forward_diff':
        return cont2discrete(system, dt, method="gbt", alpha=0.0)

    elif method == 'backward_diff':
        return cont2discrete(system, dt, method="gbt", alpha=1.0)

    elif method == 'zoh':
        # Build an exponential matrix
        em_upper = np.hstack((a, b))

        # Need to stack zeros under the a and b matrices
        em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])),
                              np.zeros((b.shape[1], b.shape[1]))))

        em = np.vstack((em_upper, em_lower))
        ms = linalg.expm(dt * em)

        # Dispose of the lower rows
        ms = ms[:a.shape[0], :]

        ad = ms[:, 0:a.shape[1]]
        bd = ms[:, a.shape[1]:]

        cd = c
        dd = d

    elif method == 'foh':
        # Size parameters for convenience
        n = a.shape[0]
        m = b.shape[1]

        # Build an exponential matrix similar to 'zoh' method
        em_upper = linalg.block_diag(np.block([a, b]) * dt, np.eye(m))
        em_lower = zeros((m, n + 2 * m))
        em = np.block([[em_upper], [em_lower]])

        ms = linalg.expm(em)

        # Get the three blocks from upper rows
        ms11 = ms[:n, 0:n]
        ms12 = ms[:n, n:n + m]
        ms13 = ms[:n, n + m:]

        ad = ms11
        bd = ms12 - ms13 + ms11 @ ms13
        cd = c
        dd = d + c @ ms13

    elif method == 'impulse':
        if not np.allclose(d, 0):
            raise ValueError("Impulse method is only applicable"
                             "to strictly proper systems")

        ad = linalg.expm(a * dt)
        bd = ad @ b * dt
        cd = c
        dd = c @ b * dt

    else:
        raise ValueError("Unknown transformation method '%s'" % method)

    return ad, bd, cd, dd, dt