findABCD.m

function [sys,K,Q,Ry,S,rcnd] = findABCD(s,n,l,R,meth,nsmpl,tol,printw)
%FINDABCD  Finds the system matrices and the Kalman gain of a discrete-time 
%          system, given the system order and the relevant part of the 
%          R factor of the concatenated block-Hankel matrices, using subspace 
%          identification techniques (MOESP and/or N4SID).
% 
%        [SYS,K] = FINDABCD(S,N,L,R,METH,NSMPL,TOL,PRINTW)  computes a state-
%        space realization SYS = (A,B,C,D) (an ss object), and the Kalman
%        predictor gain K (if NSMPL > 0). The model structure is:
%
%             x(k+1) = Ax(k) + Bu(k) + Ke(k),   k >= 1,
%             y(k)   = Cx(k) + Du(k) + e(k),
%
%        where x(k) and y(k) are vectors of length N and L, respectively.
%
%        [SYS,K,Q,Ry,S,RCND] = FINDABCD(S,N,L,R,METH,NSMPL,TOL,PRINTW)  also
%        returns the state, output, and state-output (cross-)covariance
%        matrices Q, Ry, and S (used for computing the Kalman gain), as well as
%        the vector RCND of length lr containing the reciprocal condition numbers
%        of the matrices involved in rank decisions, least squares or Riccati
%        equation solutions, where  
%           lr = 4,  if Kalman gain matrix K is not required, and
%           lr = 12, if Kalman gain matrix K is required.
%
%        S is the number of block rows in the block-Hankel matrices.
%
%        METH is an option for the method to use:
%        METH = 1 :  MOESP method with past inputs and outputs;
%             = 2 :  N4SID method;
%             = 3 :  combined method: A and C via MOESP, B and D via N4SID.
%        Default:    METH = 3.
%        Matrix R, computed by FINDR, should be determined with suitable arguments
%        METH and JOBD.  METH = 1 and JOBD = 1 must be used in findR, for METH = 1 
%        in FINDABCD;  METH = 1 must be used in FINDR, for METH = 3 in FINDABCD.
%
%        NSMPL is the total number of samples used for calculating the covariance
%        matrices and the Kalman predictor gain. This parameter is not needed if
%        the covariance matrices and/or the Kalman predictor gain matrix are not
%        desired. If NSMPL = 0, then K, Q, Ry, and S are not computed.
%        Default:    NSMPL = 0.
%
%        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.
%
%        The number of output arguments may vary, but should correspond to the 
%        input arguments, e.g.,
%        SYS = FINDABCD(S,N,L,R,METH)  or
%        [SYS,RCND] = FINDABCD(S,N,L,R,METH)  just return SYS, or SYS and RCND.
%
%        See also FINDAC, FINDBD, FINDBDK, FINDR, ORDER, SIDENT
%

%        RELEASE 2.0 of SLICOT System Identification Toolbox.
%        Based on SLICOT RELEASE 5.7, Copyright (c) 2002-2020 NICONET e.V.
%
%        V. Sima 18-01-2000.
%
%        Revisions:
%        V. Sima 02-03-2009.
%   

nin = nargin;  nout = nargout;
% 
if nin < 8;  printw = 0;  end;
if nin < 7 || isempty(tol);    tol   = 0;  end;  
if nin < 6 || isempty(nsmpl);  nsmpl = 0;  end;  
if nin < 5 || isempty(meth);   meth  = 3;  end;  
if nin < 4, 
   error('Wrong number of input arguments')
end   
%
% Compute all system matrices.
job = 1;
if nout == 1,
   [A,C,B,D] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
elseif nout >= 2,
   if nsmpl == 0,
      % Here K means rcnd.
      [A,C,B,D,K] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   elseif nout == 2,
      [A,C,B,D,K] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   elseif nout == 3,
      [A,C,B,D,K,Q] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   elseif nout == 4,
      [A,C,B,D,K,Q,Ry] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   elseif nout == 5,
      [A,C,B,D,K,Q,Ry,S] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   elseif nout == 6,
      [A,C,B,D,K,Q,Ry,S,rcnd] = sident(meth,job,s,n,l,R,tol,nsmpl,[],[],printw);
   else
      error('Wrong number of output arguments')
   end   
else
   error('Wrong number of output arguments')
end   
%
sys = ss(A,B,C,D,1);
%
% end findABCD