# Copyright (C) 1996 Auburn University.  All Rights Reserved
#
# This file is part of Octave. 
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## -*- texinfo -*-
##@deftypefn {Function File} {} tzero (@var{a}, @var{b}, @var{c}, @var{d}@{, @var{opt}@})
##@deftypefnx {Function File} {} tzero (@var{sys}@{,@var{opt}@})
## Compute transmission zeros of a continuous
##@example
##.
##x = Ax + Bu
##y = Cx + Du
##@end example
##or discrete
##@example
##x(k+1) = A x(k) + B u(k)
##y(k)   = C x(k) + D u(k)
##@end example
##system.
##@strong{Outputs}
##@table @var
##@item zer
## transmission zeros of the system
##@item gain
##leading coefficient (pole-zero form) of SISO transfer function
##returns gain=0 if system is multivariable
##@end table
##@strong{References}
##@enumerate
##@item Emami-Naeini and Van Dooren, Automatica, 1982.
##@item Hodel, "Computation of Zeros with Balancing," 1992 Lin. Alg. Appl.
##@end enumerate
##@end deftypefn


function [zer, gain] = tzero(A,B,C,D)
  # R. Bruce Tenison July 4, 1994
  # A. S. Hodel Aug 1995: allow for MIMO and system data structures

  # get A,B,C,D and Asys variables, regardless of initial form
  if(nargin == 4)
    Asys = ss2sys(A,B,C,D);
  elseif( (nargin == 1) && (! is_struct(A)))
    usage("[zer,gain] = tzero(A,B,C,D) or zer = tzero(Asys)");
  elseif(nargin != 1)
    usage("[zer,gain] = tzero(A,B,C,D) or zer = tzero(Asys)");
  else
    Asys = A;
    [A,B,C,D] = sys2ss(Asys);
  endif

  Ao = Asys;			# save for leading coefficient
  siso = is_siso(Asys);
  digital = is_digital(Asys);	# check if it's mixed or not

  # see if it's a gain block
  if(isempty(A))
    zer = [];
    gain = D;
    return;
  endif

  # First, balance the system via the zero computation generalized eigenvalue
  # problem balancing method (Hodel and Tiller, Linear Alg. Appl., 1992)

  Asys = zgpbal(Asys); [A,B,C,D] = sys2ss(Asys);   # balance coefficients
  meps = 2*eps*norm([A, B; C, D],'fro');
  Asys = zgreduce(Asys,meps);  [A, B, C, D] = sys2ss(Asys); # ENVD algorithm
  if(!isempty(A))
    # repeat with dual system
    Asys = ss2sys(A', C', B', D');   Asys = zgreduce(Asys,meps);

    # transform back
    [A,B,C,D] = sys2ss(Asys);    Asys = ss2sys(A', C', B', D');
  endif

  zer = [];			# assume none
  [A,B,C,D] = sys2ss(Asys);
  if( !isempty(C) )
    [W,r,Pi] = qr([C, D]');
    [nonz,ztmp] = zgrownorm(r,meps);
    if(nonz)
      # We can now solve the generalized eigenvalue problem.
      [pp,mm] = size(D);
      nn = rows(A);
      Afm = [A , B ; C, D] * W';
      Bfm = [eye(nn), zeros(nn,mm); zeros(pp,nn+mm)]*W';

      jdx = (mm+1):(mm+nn);
      Af = Afm(1:nn,jdx);
      Bf = Bfm(1:nn,jdx);
      zer = qz(Af,Bf);
    endif
  endif
  
  mz = length(zer);
  [A,B,C,D] = sys2ss(Ao);		# recover original system
  #compute leading coefficient
  if ( (nargout == 2) && siso)
    n = rows(A);
    if ( mz == n)
      gain = D;
    elseif ( mz < n )
      gain = C*(A^(n-1-mz))*B;
    endif
  else
    gain = [];
  endif
endfunction