534 lines
16 KiB
Python
534 lines
16 KiB
Python
"""
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ltisys -- a collection of functions to convert linear time invariant systems
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from one representation to another.
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"""
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import numpy
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import numpy as np
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from numpy import (r_, eye, atleast_2d, poly, dot,
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asarray, prod, zeros, array, outer)
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from scipy import linalg
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from ._filter_design import tf2zpk, zpk2tf, normalize
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__all__ = ['tf2ss', 'abcd_normalize', 'ss2tf', 'zpk2ss', 'ss2zpk',
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'cont2discrete']
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def tf2ss(num, den):
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r"""Transfer function to state-space representation.
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Parameters
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----------
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num, den : array_like
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Sequences representing the coefficients of the numerator and
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denominator polynomials, in order of descending degree. The
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denominator needs to be at least as long as the numerator.
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Returns
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-------
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A, B, C, D : ndarray
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State space representation of the system, in controller canonical
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form.
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Examples
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--------
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Convert the transfer function:
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.. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
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>>> num = [1, 3, 3]
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>>> den = [1, 2, 1]
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to the state-space representation:
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.. math::
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\dot{\textbf{x}}(t) =
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\begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
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\begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
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\textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
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\begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
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>>> from scipy.signal import tf2ss
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>>> A, B, C, D = tf2ss(num, den)
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>>> A
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array([[-2., -1.],
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[ 1., 0.]])
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>>> B
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array([[ 1.],
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[ 0.]])
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>>> C
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array([[ 1., 2.]])
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>>> D
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array([[ 1.]])
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"""
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# Controller canonical state-space representation.
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# if M+1 = len(num) and K+1 = len(den) then we must have M <= K
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# states are found by asserting that X(s) = U(s) / D(s)
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# then Y(s) = N(s) * X(s)
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#
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# A, B, C, and D follow quite naturally.
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#
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num, den = normalize(num, den) # Strips zeros, checks arrays
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nn = len(num.shape)
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if nn == 1:
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num = asarray([num], num.dtype)
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M = num.shape[1]
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K = len(den)
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if M > K:
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msg = "Improper transfer function. `num` is longer than `den`."
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raise ValueError(msg)
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if M == 0 or K == 0: # Null system
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return (array([], float), array([], float), array([], float),
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array([], float))
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# pad numerator to have same number of columns has denominator
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num = np.hstack((np.zeros((num.shape[0], K - M), dtype=num.dtype), num))
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if num.shape[-1] > 0:
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D = atleast_2d(num[:, 0])
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else:
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# We don't assign it an empty array because this system
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# is not 'null'. It just doesn't have a non-zero D
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# matrix. Thus, it should have a non-zero shape so that
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# it can be operated on by functions like 'ss2tf'
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D = array([[0]], float)
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if K == 1:
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D = D.reshape(num.shape)
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return (zeros((1, 1)), zeros((1, D.shape[1])),
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zeros((D.shape[0], 1)), D)
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frow = -array([den[1:]])
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A = r_[frow, eye(K - 2, K - 1)]
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B = eye(K - 1, 1)
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C = num[:, 1:] - outer(num[:, 0], den[1:])
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D = D.reshape((C.shape[0], B.shape[1]))
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return A, B, C, D
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def _none_to_empty_2d(arg):
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if arg is None:
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return zeros((0, 0))
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else:
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return arg
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def _atleast_2d_or_none(arg):
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if arg is not None:
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return atleast_2d(arg)
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def _shape_or_none(M):
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if M is not None:
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return M.shape
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else:
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return (None,) * 2
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def _choice_not_none(*args):
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for arg in args:
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if arg is not None:
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return arg
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def _restore(M, shape):
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if M.shape == (0, 0):
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return zeros(shape)
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else:
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if M.shape != shape:
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raise ValueError("The input arrays have incompatible shapes.")
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return M
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def abcd_normalize(A=None, B=None, C=None, D=None):
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"""Check state-space matrices and ensure they are 2-D.
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If enough information on the system is provided, that is, enough
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properly-shaped arrays are passed to the function, the missing ones
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are built from this information, ensuring the correct number of
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rows and columns. Otherwise a ValueError is raised.
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Parameters
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----------
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A, B, C, D : array_like, optional
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State-space matrices. All of them are None (missing) by default.
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See `ss2tf` for format.
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Returns
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-------
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A, B, C, D : array
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Properly shaped state-space matrices.
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Raises
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------
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ValueError
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If not enough information on the system was provided.
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"""
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A, B, C, D = map(_atleast_2d_or_none, (A, B, C, D))
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MA, NA = _shape_or_none(A)
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MB, NB = _shape_or_none(B)
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MC, NC = _shape_or_none(C)
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MD, ND = _shape_or_none(D)
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p = _choice_not_none(MA, MB, NC)
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q = _choice_not_none(NB, ND)
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r = _choice_not_none(MC, MD)
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if p is None or q is None or r is None:
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raise ValueError("Not enough information on the system.")
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A, B, C, D = map(_none_to_empty_2d, (A, B, C, D))
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A = _restore(A, (p, p))
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B = _restore(B, (p, q))
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C = _restore(C, (r, p))
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D = _restore(D, (r, q))
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return A, B, C, D
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def ss2tf(A, B, C, D, input=0):
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r"""State-space to transfer function.
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A, B, C, D defines a linear state-space system with `p` inputs,
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`q` outputs, and `n` state variables.
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Parameters
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----------
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A : array_like
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State (or system) matrix of shape ``(n, n)``
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B : array_like
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Input matrix of shape ``(n, p)``
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C : array_like
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Output matrix of shape ``(q, n)``
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D : array_like
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Feedthrough (or feedforward) matrix of shape ``(q, p)``
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input : int, optional
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For multiple-input systems, the index of the input to use.
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Returns
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-------
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num : 2-D ndarray
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Numerator(s) of the resulting transfer function(s). `num` has one row
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for each of the system's outputs. Each row is a sequence representation
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of the numerator polynomial.
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den : 1-D ndarray
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Denominator of the resulting transfer function(s). `den` is a sequence
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representation of the denominator polynomial.
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Examples
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--------
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Convert the state-space representation:
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.. math::
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\dot{\textbf{x}}(t) =
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\begin{bmatrix} -2 & -1 \\ 1 & 0 \end{bmatrix} \textbf{x}(t) +
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\begin{bmatrix} 1 \\ 0 \end{bmatrix} \textbf{u}(t) \\
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\textbf{y}(t) = \begin{bmatrix} 1 & 2 \end{bmatrix} \textbf{x}(t) +
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\begin{bmatrix} 1 \end{bmatrix} \textbf{u}(t)
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>>> A = [[-2, -1], [1, 0]]
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>>> B = [[1], [0]] # 2-D column vector
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>>> C = [[1, 2]] # 2-D row vector
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>>> D = 1
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to the transfer function:
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.. math:: H(s) = \frac{s^2 + 3s + 3}{s^2 + 2s + 1}
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>>> from scipy.signal import ss2tf
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>>> ss2tf(A, B, C, D)
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(array([[1., 3., 3.]]), array([ 1., 2., 1.]))
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"""
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# transfer function is C (sI - A)**(-1) B + D
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# Check consistency and make them all rank-2 arrays
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A, B, C, D = abcd_normalize(A, B, C, D)
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nout, nin = D.shape
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if input >= nin:
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raise ValueError("System does not have the input specified.")
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# make SIMO from possibly MIMO system.
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B = B[:, input:input + 1]
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D = D[:, input:input + 1]
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try:
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den = poly(A)
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except ValueError:
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den = 1
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if (prod(B.shape, axis=0) == 0) and (prod(C.shape, axis=0) == 0):
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num = numpy.ravel(D)
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if (prod(D.shape, axis=0) == 0) and (prod(A.shape, axis=0) == 0):
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den = []
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return num, den
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num_states = A.shape[0]
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type_test = A[:, 0] + B[:, 0] + C[0, :] + D + 0.0
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num = numpy.empty((nout, num_states + 1), type_test.dtype)
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for k in range(nout):
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Ck = atleast_2d(C[k, :])
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num[k] = poly(A - dot(B, Ck)) + (D[k] - 1) * den
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return num, den
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def zpk2ss(z, p, k):
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"""Zero-pole-gain representation to state-space representation
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Parameters
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----------
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z, p : sequence
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Zeros and poles.
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k : float
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System gain.
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Returns
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-------
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A, B, C, D : ndarray
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State space representation of the system, in controller canonical
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form.
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"""
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return tf2ss(*zpk2tf(z, p, k))
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def ss2zpk(A, B, C, D, input=0):
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"""State-space representation to zero-pole-gain representation.
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A, B, C, D defines a linear state-space system with `p` inputs,
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`q` outputs, and `n` state variables.
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Parameters
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----------
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A : array_like
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State (or system) matrix of shape ``(n, n)``
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B : array_like
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Input matrix of shape ``(n, p)``
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C : array_like
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Output matrix of shape ``(q, n)``
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D : array_like
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Feedthrough (or feedforward) matrix of shape ``(q, p)``
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input : int, optional
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For multiple-input systems, the index of the input to use.
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Returns
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-------
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z, p : sequence
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Zeros and poles.
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k : float
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System gain.
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"""
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return tf2zpk(*ss2tf(A, B, C, D, input=input))
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def cont2discrete(system, dt, method="zoh", alpha=None):
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"""
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Transform a continuous to a discrete state-space system.
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Parameters
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----------
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system : a tuple describing the system or an instance of `lti`
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The following gives the number of elements in the tuple and
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the interpretation:
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* 1: (instance of `lti`)
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* 2: (num, den)
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* 3: (zeros, poles, gain)
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* 4: (A, B, C, D)
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dt : float
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The discretization time step.
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method : str, optional
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Which method to use:
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* gbt: generalized bilinear transformation
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* bilinear: Tustin's approximation ("gbt" with alpha=0.5)
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* euler: Euler (or forward differencing) method ("gbt" with alpha=0)
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* backward_diff: Backwards differencing ("gbt" with alpha=1.0)
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* zoh: zero-order hold (default)
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* foh: first-order hold (*versionadded: 1.3.0*)
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* impulse: equivalent impulse response (*versionadded: 1.3.0*)
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alpha : float within [0, 1], optional
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The generalized bilinear transformation weighting parameter, which
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should only be specified with method="gbt", and is ignored otherwise
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Returns
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-------
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sysd : tuple containing the discrete system
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Based on the input type, the output will be of the form
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* (num, den, dt) for transfer function input
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* (zeros, poles, gain, dt) for zeros-poles-gain input
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* (A, B, C, D, dt) for state-space system input
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Notes
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-----
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By default, the routine uses a Zero-Order Hold (zoh) method to perform
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the transformation. Alternatively, a generalized bilinear transformation
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may be used, which includes the common Tustin's bilinear approximation,
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an Euler's method technique, or a backwards differencing technique.
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The Zero-Order Hold (zoh) method is based on [1]_, the generalized bilinear
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approximation is based on [2]_ and [3]_, the First-Order Hold (foh) method
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is based on [4]_.
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Discretization#Discretization_of_linear_state_space_models
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.. [2] http://techteach.no/publications/discretetime_signals_systems/discrete.pdf
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.. [3] G. Zhang, X. Chen, and T. Chen, Digital redesign via the generalized
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bilinear transformation, Int. J. Control, vol. 82, no. 4, pp. 741-754,
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2009.
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(https://www.mypolyuweb.hk/~magzhang/Research/ZCC09_IJC.pdf)
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.. [4] G. F. Franklin, J. D. Powell, and M. L. Workman, Digital control
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of dynamic systems, 3rd ed. Menlo Park, Calif: Addison-Wesley,
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pp. 204-206, 1998.
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Examples
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--------
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We can transform a continuous state-space system to a discrete one:
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>>> import numpy as np
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>>> import matplotlib.pyplot as plt
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>>> from scipy.signal import cont2discrete, lti, dlti, dstep
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Define a continuous state-space system.
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>>> A = np.array([[0, 1],[-10., -3]])
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>>> B = np.array([[0],[10.]])
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>>> C = np.array([[1., 0]])
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>>> D = np.array([[0.]])
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>>> l_system = lti(A, B, C, D)
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>>> t, x = l_system.step(T=np.linspace(0, 5, 100))
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>>> fig, ax = plt.subplots()
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>>> ax.plot(t, x, label='Continuous', linewidth=3)
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Transform it to a discrete state-space system using several methods.
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>>> dt = 0.1
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>>> for method in ['zoh', 'bilinear', 'euler', 'backward_diff', 'foh', 'impulse']:
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... d_system = cont2discrete((A, B, C, D), dt, method=method)
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... s, x_d = dstep(d_system)
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... ax.step(s, np.squeeze(x_d), label=method, where='post')
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>>> ax.axis([t[0], t[-1], x[0], 1.4])
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>>> ax.legend(loc='best')
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>>> fig.tight_layout()
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>>> plt.show()
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"""
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if len(system) == 1:
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return system.to_discrete()
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if len(system) == 2:
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sysd = cont2discrete(tf2ss(system[0], system[1]), dt, method=method,
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alpha=alpha)
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return ss2tf(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
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elif len(system) == 3:
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sysd = cont2discrete(zpk2ss(system[0], system[1], system[2]), dt,
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method=method, alpha=alpha)
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return ss2zpk(sysd[0], sysd[1], sysd[2], sysd[3]) + (dt,)
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elif len(system) == 4:
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a, b, c, d = system
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else:
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raise ValueError("First argument must either be a tuple of 2 (tf), "
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"3 (zpk), or 4 (ss) arrays.")
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if method == 'gbt':
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if alpha is None:
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raise ValueError("Alpha parameter must be specified for the "
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"generalized bilinear transform (gbt) method")
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elif alpha < 0 or alpha > 1:
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raise ValueError("Alpha parameter must be within the interval "
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"[0,1] for the gbt method")
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if method == 'gbt':
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# This parameter is used repeatedly - compute once here
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ima = np.eye(a.shape[0]) - alpha*dt*a
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ad = linalg.solve(ima, np.eye(a.shape[0]) + (1.0-alpha)*dt*a)
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bd = linalg.solve(ima, dt*b)
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# Similarly solve for the output equation matrices
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cd = linalg.solve(ima.transpose(), c.transpose())
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cd = cd.transpose()
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dd = d + alpha*np.dot(c, bd)
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elif method == 'bilinear' or method == 'tustin':
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return cont2discrete(system, dt, method="gbt", alpha=0.5)
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elif method == 'euler' or method == 'forward_diff':
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return cont2discrete(system, dt, method="gbt", alpha=0.0)
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elif method == 'backward_diff':
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return cont2discrete(system, dt, method="gbt", alpha=1.0)
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elif method == 'zoh':
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# Build an exponential matrix
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em_upper = np.hstack((a, b))
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# Need to stack zeros under the a and b matrices
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em_lower = np.hstack((np.zeros((b.shape[1], a.shape[0])),
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np.zeros((b.shape[1], b.shape[1]))))
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em = np.vstack((em_upper, em_lower))
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ms = linalg.expm(dt * em)
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# Dispose of the lower rows
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ms = ms[:a.shape[0], :]
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ad = ms[:, 0:a.shape[1]]
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bd = ms[:, a.shape[1]:]
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cd = c
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dd = d
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elif method == 'foh':
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# Size parameters for convenience
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n = a.shape[0]
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m = b.shape[1]
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# Build an exponential matrix similar to 'zoh' method
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em_upper = linalg.block_diag(np.block([a, b]) * dt, np.eye(m))
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em_lower = zeros((m, n + 2 * m))
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em = np.block([[em_upper], [em_lower]])
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ms = linalg.expm(em)
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# 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
|