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tzero.m
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1999-12-24
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## Copyright (C) 1996 Auburn University. All Rights Reserved
##
## This file is part of Octave.
##
## Octave is free software; you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by the
## Free Software Foundation; either version 2, or (at your option) any
## later version.
##
## Octave is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
## for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, write to the Free
## Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA.
## -*- 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_digit(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] = zgrownor(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