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## Copyright (C) 1996, 2000, 2002, 2004, 2005, 2006, 2007
## Auburn University. All rights reserved.
##
##
## This program 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 3 of the License, or (at
## your option) any later version.
##
## This program 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 this program; see the file COPYING. If not, see
## <http://www.gnu.org/licenses/>.
## -*- 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);
if (isa ([A, B; C, D], "single"))
meps = 2*eps("single")*norm ([A, B; C, D], "fro");
else
meps = 2*eps*norm ([A, B; C, D], "fro");
endif
## 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
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