Module « scipy.linalg »

Fonction solve_continuous_are - module scipy.linalg

Signature de la fonction solve_continuous_are

def solve_continuous_are(a, b, q, r, e=None, s=None, balanced=True) 

Description

solve_continuous_are.__doc__

    Solves the continuous-time algebraic Riccati equation (CARE).

    The CARE is defined as

    .. math::

          X A + A^H X - X B R^{-1} B^H X + Q = 0

    The limitations for a solution to exist are :

        * All eigenvalues of :math:`A` on the right half plane, should be
          controllable.

        * The associated hamiltonian pencil (See Notes), should have
          eigenvalues sufficiently away from the imaginary axis.

    Moreover, if ``e`` or ``s`` is not precisely ``None``, then the
    generalized version of CARE

    .. math::

          E^HXA + A^HXE - (E^HXB + S) R^{-1} (B^HXE + S^H) + Q = 0

    is solved. When omitted, ``e`` is assumed to be the identity and ``s``
    is assumed to be the zero matrix with sizes compatible with ``a`` and
    ``b``, respectively.

    Parameters
    ----------
    a : (M, M) array_like
        Square matrix
    b : (M, N) array_like
        Input
    q : (M, M) array_like
        Input
    r : (N, N) array_like
        Nonsingular square matrix
    e : (M, M) array_like, optional
        Nonsingular square matrix
    s : (M, N) array_like, optional
        Input
    balanced : bool, optional
        The boolean that indicates whether a balancing step is performed
        on the data. The default is set to True.

    Returns
    -------
    x : (M, M) ndarray
        Solution to the continuous-time algebraic Riccati equation.

    Raises
    ------
    LinAlgError
        For cases where the stable subspace of the pencil could not be
        isolated. See Notes section and the references for details.

    See Also
    --------
    solve_discrete_are : Solves the discrete-time algebraic Riccati equation

    Notes
    -----
    The equation is solved by forming the extended hamiltonian matrix pencil,
    as described in [1]_, :math:`H - \lambda J` given by the block matrices ::

        [ A    0    B ]             [ E   0    0 ]
        [-Q  -A^H  -S ] - \lambda * [ 0  E^H   0 ]
        [ S^H B^H   R ]             [ 0   0    0 ]

    and using a QZ decomposition method.

    In this algorithm, the fail conditions are linked to the symmetry
    of the product :math:`U_2 U_1^{-1}` and condition number of
    :math:`U_1`. Here, :math:`U` is the 2m-by-m matrix that holds the
    eigenvectors spanning the stable subspace with 2-m rows and partitioned
    into two m-row matrices. See [1]_ and [2]_ for more details.

    In order to improve the QZ decomposition accuracy, the pencil goes
    through a balancing step where the sum of absolute values of
    :math:`H` and :math:`J` entries (after removing the diagonal entries of
    the sum) is balanced following the recipe given in [3]_.

    .. versionadded:: 0.11.0

    References
    ----------
    .. [1]  P. van Dooren , "A Generalized Eigenvalue Approach For Solving
       Riccati Equations.", SIAM Journal on Scientific and Statistical
       Computing, Vol.2(2), :doi:`10.1137/0902010`

    .. [2] A.J. Laub, "A Schur Method for Solving Algebraic Riccati
       Equations.", Massachusetts Institute of Technology. Laboratory for
       Information and Decision Systems. LIDS-R ; 859. Available online :
       http://hdl.handle.net/1721.1/1301

    .. [3] P. Benner, "Symplectic Balancing of Hamiltonian Matrices", 2001,
       SIAM J. Sci. Comput., 2001, Vol.22(5), :doi:`10.1137/S1064827500367993`

    Examples
    --------
    Given `a`, `b`, `q`, and `r` solve for `x`:

    >>> from scipy import linalg
    >>> a = np.array([[4, 3], [-4.5, -3.5]])
    >>> b = np.array([[1], [-1]])
    >>> q = np.array([[9, 6], [6, 4.]])
    >>> r = 1
    >>> x = linalg.solve_continuous_are(a, b, q, r)
    >>> x
    array([[ 21.72792206,  14.48528137],
           [ 14.48528137,   9.65685425]])
    >>> np.allclose(a.T.dot(x) + x.dot(a)-x.dot(b).dot(b.T).dot(x), -q)
    True