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## -*- texinfo -*-
## @deftypefn {Function File} {[@var{zer}, @var{gain}] =} tzero (@var{a}, @var{b}, @var{c}, @var{d}, @var{opt})
## @deftypefnx {Function File} {[@var{zer}, @var{gain}] =} tzero (@var{sys}, @var{opt})
## Compute transmission zeros of a continuous system:
## @iftex
## @tex
## $$ \dot x = Ax + Bu $$
## $$ y = Cx + Du $$
## @end tex
## @end iftex
## @ifinfo
## @example
## .
## x = Ax + Bu
## y = Cx + Du
## @end example
## @end ifinfo
## or of a discrete one:
## @iftex
## @tex
## $$ x_{k+1} = Ax_k + Bu_k $$
## $$ y_k = Cx_k + Du_k $$
## @end tex
## @end iftex
## @ifinfo
## @example
## x(k+1) = A x(k) + B u(k)
## y(k)   = C x(k) + D u(k)
## @end example
## @end ifinfo
## 
## @strong{Outputs}
## @table @var
## @item zer
##  transmission zeros of the system
## @item gain
## leading coefficient (pole-zero form) of @acronym{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, @cite{Computation of Zeros with Balancing}, 1992 Lin. Alg. Appl.
## @end enumerate
## @end deftypefn

## Author: R. Bruce Tenison <btenison@eng.auburn.edu>
## Created: July 4, 1994
## A. S. Hodel Aug 1995: allow for MIMO and system data structures

function [zer, gain] = tzero (A, B, C, D)

  ## get A,B,C,D and Asys variables, regardless of initial form
  if (nargin == 4)
    Asys = ss (A, B, C, D);
  elseif (nargin == 1 && ! isstruct (A))
    error ("tzero: expecting argument to be system structure");
  elseif (nargin != 1)
    print_usage ();
  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)

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

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