findx0BD.m

function [x0,B,D,V,rcnd] = findx0BD(A,C,y,u,withx0,withd,tol,printw)
%FINDX0BD  Estimates the initial state and/or the matrices B and D of a
%          discrete-time linear system, given the (estimated) system
%          matrices A, C, and a set of input/output data.
%
%        [X0,B,D] = FINDX0BD(A,C,Y,U,WITHX0,WITHD,TOL,PRINTW)  estimates the
%        initial state X0 and the matrices B and D of a discrete-time 
%        system using the system matrices A, C, output data Y and the input 
%        data U. The model structure is :
%
%             x(k+1) = Ax(k) + Bu(k),   k >= 1,
%             y(k)   = Cx(k) + Du(k),
%
%        The vectors y(k) and u(k) are transposes of the k-th rows of Y and U,
%        respectively.
%
%        WITHX0 is a switch for estimating the initial state x0.
%        WITHX0 = 1: estimate x0;
%               = 0: do not estimate x0.
%        Default:    WITHX0 = 1.
%
%        WITHD  is a switch for estimating the matrix D.
%        WITHD  = 1: estimate the matrix D;
%               = 0: do not estimate the matrix D.
%        Default:    WITHD = 1.
%
%        TOL is the tolerance used for estimating the rank of matrices. 
%        If  TOL > 0,  then the given value of  TOL  is used as a lower bound
%        for the reciprocal condition number.
%        Default:    prod(size(matrix))*epsilon_machine where epsilon_machine
%                    is the relative machine precision.
%
%        PRINTW is a switch for printing the warning messages.
%        PRINTW = 1: print warning messages;
%               = 0: do not print warning messages.
%        Default:    PRINTW = 0.
%
%        [x0,B,D,V,rcnd] = FINDX0BD(A,C,Y,U)  also returns the orthogonal
%        matrix V which reduces the system state matrix A to a real Schur
%        form, as well as some estimates of the reciprocal condition numbers
%        of the matrices involved in rank decisions.
%
%            B = FINDX0BD(A,C,Y,U,0,0)  returns B only, and
%        [B,D] = FINDX0BD(A,C,Y,U,0)    returns B and D only. 
%
%        See also FINDBD, INISTATE
%

%        RELEASE 2.0 of SLICOT System Identification Toolbox.
%        Based on SLICOT RELEASE 5.7, Copyright (c) 2002-2020 NICONET e.V.
%
%        V. Sima 13-05-2000.
%
%        For efficiency, most errors are checked in the mexfile findBD. 
%
%        Revisions:
%        V. Sima, July 2000, Mar. 2009.
%
ni = nargin;  nout = nargout;
if ni < 4,
   error('FINDX0BD needs at least 4 input parameters')
end
if nout < 1,
   error('FINDX0BD needs at least 1 output parameters')
end
%
if ni < 8, printw = 0; end
if ni < 7 || isempty(tol);    tol = 0; end
if ni < 6 || isempty(withd);  withd = 1;
elseif withd ~= 0 && withd ~= 1,  error('withd could only be 0 or 1'); 
end
if ni < 5 || isempty(withx0); withx0 = 1; end
job = withd + 1;
%
if withx0 == 1,
   if withd == 1,
      [x0,B,D,Vl,rcndl] = findBD(withx0,1,job,A,C,y,u,tol,printw);
      if nout > 3,  V = Vl;  end
      if nout > 4,  rcnd = rcndl;  end
   else
      [x0,B,Vl,rcndl] = findBD(withx0,1,job,A,C,y,u,tol,printw);
      if nout > 2,  D = Vl;  end
      if nout > 3,  V = rcndl;  end
   end
else
   % Below, x0 means B, and B means D or V !
   if withd == 1,
      [x0,B,Vl,rcndl] = findBD(withx0,1,job,A,C,y,u,tol,printw);
      if nout > 2,  D = Vl;  end
      if nout > 3,  V = rcndl;  end
   else
      [x0,B,Vl] = findBD(withx0,1,job,A,C,y,u,tol,printw);
      if nout > 2,  D = Vl;  end
   end
end
%
% end findx0BD