#! TODO: add module docstring
# modelsimp.py - tools for model simplification
#
# Author: Steve Brunton, Kevin Chen, Lauren Padilla
# Date: 30 Nov 2010
#
# This file contains routines for obtaining reduced order models
#
# Copyright (c) 2010 by California Institute of Technology
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#
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# $Id$

# Python 3 compatibility
from __future__ import print_function

# External packages and modules
import numpy as np
from .exception import ControlSlycot
from .lti import isdtime, isctime
from .statesp import StateSpace
from .statefbk import gram

__all__ = ['hsvd', 'balred', 'modred', 'era', 'markov', 'minreal']

# Hankel Singular Value Decomposition
#   The following returns the Hankel singular values, which are singular values
#of the matrix formed by multiplying the controllability and observability
#grammians
def hsvd(sys):
    """Calculate the Hankel singular values.

    Parameters
    ----------
    sys : StateSpace
        A state space system

    Returns
    -------
    H : array
        A list of Hankel singular values

    See Also
    --------
    gram

    Notes
    -----
    The Hankel singular values are the singular values of the Hankel operator.
    In practice, we compute the square root of the eigenvalues of the matrix
    formed by taking the product of the observability and controllability
    gramians.  There are other (more efficient) methods based on solving the
    Lyapunov equation in a particular way (more details soon).

    Examples
    --------
    >>> H = hsvd(sys)

    """
    # TODO: implement for discrete time systems
    if (isdtime(sys, strict=True)):
        raise NotImplementedError("Function not implemented in discrete time")

    Wc = gram(sys,'c')
    Wo = gram(sys,'o')
    WoWc = np.dot(Wo, Wc)
    w, v = np.linalg.eig(WoWc)

    hsv = np.sqrt(w)
    hsv = np.array(hsv)
    hsv = np.sort(hsv)
    # Return the Hankel singular values, high to low
    return hsv[::-1]

def modred(sys, ELIM, method='matchdc'):
    """
    Model reduction of `sys` by eliminating the states in `ELIM` using a given
    method.

    Parameters
    ----------
    sys: StateSpace
        Original system to reduce
    ELIM: array
        Vector of states to eliminate
    method: string
        Method of removing states in `ELIM`: either ``'truncate'`` or
        ``'matchdc'``.

    Returns
    -------
    rsys: StateSpace
        A reduced order model

    Raises
    ------
    ValueError
        Raised under the following conditions:

            * if `method` is not either ``'matchdc'`` or ``'truncate'``

            * if eigenvalues of `sys.A` are not all in left half plane
              (`sys` must be stable)

    Examples
    --------
    >>> rsys = modred(sys, ELIM, method='truncate')
    """

    #Check for ss system object, need a utility for this?

    #TODO: Check for continous or discrete, only continuous supported right now
        # if isCont():
        #    dico = 'C'
        # elif isDisc():
        #    dico = 'D'
        # else:
    if (isctime(sys)):
        dico = 'C'
    else:
        raise NotImplementedError("Function not implemented in discrete time")


    #Check system is stable
    if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
        raise ValueError("Oops, the system is unstable!")

    ELIM = np.sort(ELIM)
    # Create list of elements not to eliminate (NELIM)
    NELIM = [i for i in range(len(sys.A)) if i not in ELIM]
    # A1 is a matrix of all columns of sys.A not to eliminate
    A1 = sys.A[:, NELIM[0]].reshape(-1, 1)
    for i in NELIM[1:]:
        A1 = np.hstack((A1, sys.A[:,i].reshape(-1, 1)))
    A11 = A1[NELIM,:]
    A21 = A1[ELIM,:]
    # A2 is a matrix of all columns of sys.A to eliminate
    A2 = sys.A[:, ELIM[0]].reshape(-1, 1)
    for i in ELIM[1:]:
        A2 = np.hstack((A2, sys.A[:,i].reshape(-1, 1)))
    A12 = A2[NELIM,:]
    A22 = A2[ELIM,:]

    C1 = sys.C[:,NELIM]
    C2 = sys.C[:,ELIM]
    B1 = sys.B[NELIM,:]
    B2 = sys.B[ELIM,:]

    if method=='matchdc':
        # if matchdc, residualize

        # Check if the matrix A22 is invertible
        if np.linalg.matrix_rank(A22) != len(ELIM):
            raise ValueError("Matrix A22 is singular to working precision.")

        # Now precompute A22\A21 and A22\B2 (A22I = inv(A22))
        # We can solve two linear systems in one pass, since the
        # coefficients matrix A22 is the same. Thus, we perform the LU
        # decomposition (cubic runtime complexity) of A22 only once!
        # The remaining back substitutions are only quadratic in runtime.
        A22I_A21_B2 = np.linalg.solve(A22, np.concatenate((A21, B2), axis=1))
        A22I_A21 = A22I_A21_B2[:, :A21.shape[1]]
        A22I_B2 = A22I_A21_B2[:, A21.shape[1]:]

        Ar = A11 - np.dot(A12, A22I_A21)
        Br = B1 - np.dot(A12, A22I_B2)
        Cr = C1 - np.dot(C2, A22I_A21)
        Dr = sys.D - np.dot(C2, A22I_B2)
    elif method=='truncate':
        # if truncate, simply discard state x2
        Ar = A11
        Br = B1
        Cr = C1
        Dr = sys.D
    else:
        raise ValueError("Oops, method is not supported!")

    rsys = StateSpace(Ar,Br,Cr,Dr)
    return rsys

def balred(sys, orders, method='truncate', alpha=None):
    """
    Balanced reduced order model of sys of a given order.
    States are eliminated based on Hankel singular value.
    If sys has unstable modes, they are removed, the
    balanced realization is done on the stable part, then
    reinserted in accordance with the reference below.

    Reference: Hsu,C.S., and Hou,D., 1991,
    Reducing unstable linear control systems via real Schur transformation.
    Electronics Letters, 27, 984-986.

    Parameters
    ----------
    sys: StateSpace
        Original system to reduce
    orders: integer or array of integer
        Desired order of reduced order model (if a vector, returns a vector
        of systems)
    method: string
        Method of removing states, either ``'truncate'`` or ``'matchdc'``.
    alpha: float
        Redefines the stability boundary for eigenvalues of the system matrix A.
        By default for continuous-time systems, alpha <= 0 defines the stability
        boundary for the real part of A's eigenvalues and for discrete-time
        systems, 0 <= alpha <= 1 defines the stability boundary for the modulus
        of A's eigenvalues. See SLICOT routines AB09MD and AB09ND for more
        information.

    Returns
    -------
    rsys: StateSpace
        A reduced order model or a list of reduced order models if orders is a list

    Raises
    ------
    ValueError
        * if `method` is not ``'truncate'`` or ``'matchdc'``
    ImportError
        if slycot routine ab09ad, ab09md, or ab09nd is not found

    ValueError
        if there are more unstable modes than any value in orders

    Examples
    --------
    >>> rsys = balred(sys, orders, method='truncate')

    """
    if method!='truncate' and method!='matchdc':
        raise ValueError("supported methods are 'truncate' or 'matchdc'")
    elif method=='truncate':
        try:
            from slycot import ab09md, ab09ad
        except ImportError:
            raise ControlSlycot("can't find slycot subroutine ab09md or ab09ad")
    elif method=='matchdc':
        try:
            from slycot import ab09nd
        except ImportError:
            raise ControlSlycot("can't find slycot subroutine ab09nd")

    #Check for ss system object, need a utility for this?

    #TODO: Check for continous or discrete, only continuous supported right now
        # if isCont():
        #    dico = 'C'
        # elif isDisc():
        #    dico = 'D'
        # else:
    dico = 'C'

    job = 'B' # balanced (B) or not (N)
    equil = 'N'  # scale (S) or not (N)
    if alpha is None:
        if dico == 'C':
            alpha = 0.
        elif dico == 'D':
            alpha = 1.

    rsys = [] #empty list for reduced systems

    #check if orders is a list or a scalar
    try:
        order = iter(orders)
    except TypeError: #if orders is a scalar
        orders = [orders]

    for i in orders:
        n = np.size(sys.A,0)
        m = np.size(sys.B,1)
        p = np.size(sys.C,0)
        if method == 'truncate':
            #check system stability
            if np.any(np.linalg.eigvals(sys.A).real >= 0.0):
                #unstable branch
                Nr, Ar, Br, Cr, Ns, hsv = ab09md(dico,job,equil,n,m,p,sys.A,sys.B,sys.C,alpha=alpha,nr=i,tol=0.0)
            else:
                #stable branch
                Nr, Ar, Br, Cr, hsv = ab09ad(dico,job,equil,n,m,p,sys.A,sys.B,sys.C,nr=i,tol=0.0)
            rsys.append(StateSpace(Ar, Br, Cr, sys.D))

        elif method == 'matchdc':
            Nr, Ar, Br, Cr, Dr, Ns, hsv = ab09nd(dico,job,equil,n,m,p,sys.A,sys.B,sys.C,sys.D,alpha=alpha,nr=i,tol1=0.0,tol2=0.0)
            rsys.append(StateSpace(Ar, Br, Cr, Dr))

    #if orders was a scalar, just return the single reduced model, not a list
    if len(orders) == 1:
        return rsys[0]
    #if orders was a list/vector, return a list/vector of systems
    else:
        return rsys

def minreal(sys, tol=None, verbose=True):
    '''
    Eliminates uncontrollable or unobservable states in state-space
    models or cancelling pole-zero pairs in transfer functions. The
    output sysr has minimal order and the same response
    characteristics as the original model sys.

    Parameters
    ----------
    sys: StateSpace or TransferFunction
        Original system
    tol: real
        Tolerance
    verbose: bool
        Print results if True

    Returns
    -------
    rsys: StateSpace or TransferFunction
        Cleaned model
    '''
    sysr = sys.minreal(tol)
    if verbose:
        print("{nstates} states have been removed from the model".format(
                nstates=len(sys.pole()) - len(sysr.pole())))
    return sysr

def era(YY, m, n, nin, nout, r):
    """
    Calculate an ERA model of order `r` based on the impulse-response data `YY`.

    .. note:: This function is not implemented yet.

    Parameters
    ----------
    YY: array
        `nout` x `nin` dimensional impulse-response data
    m: integer
        Number of rows in Hankel matrix
    n: integer
        Number of columns in Hankel matrix
    nin: integer
        Number of input variables
    nout: integer
        Number of output variables
    r: integer
        Order of model

    Returns
    -------
    sys: StateSpace
        A reduced order model sys=ss(Ar,Br,Cr,Dr)

    Examples
    --------
    >>> rsys = era(YY, m, n, nin, nout, r)
    """
    raise NotImplementedError('This function is not implemented yet.')

def markov(Y, U, m):
    """
    Calculate the first `M` Markov parameters [D CB CAB ...]
    from input `U`, output `Y`.

    Parameters
    ----------
    Y: array_like
        Output data
    U: array_like
        Input data
    m: int
        Number of Markov parameters to output

    Returns
    -------
    H: ndarray
        First m Markov parameters

    Notes
    -----
    Currently only works for SISO

    Examples
    --------
    >>> H = markov(Y, U, m)
    """

    # Convert input parameters to matrices (if they aren't already)
    Ymat = np.array(Y)
    Umat = np.array(U)
    n = np.size(U)

    # Construct a matrix of control inputs to invert
    UU = Umat
    for i in range(1, m-1):
        # TODO: second index on UU doesn't seem right; could be neg or pos??
        newCol = np.vstack((0, np.reshape(UU[0:n-1, i-2], (-1, 1))))
        UU = np.hstack((UU, newCol))
    Ulast = np.vstack((0, np.reshape(UU[0:n-1, m-2], (-1, 1))))
    for i in range(n-1, 0, -1):
        Ulast[i] = np.sum(Ulast[0:i-1])
    UU = np.hstack((UU, Ulast))

    # Invert and solve for Markov parameters
    H = np.linalg.lstsq(UU, Y)[0]

    return H