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hinfnorm.m

## Copyright (C) 1996, 1998, 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{g}, @var{gmin}, @var{gmax}] =} hinfnorm (@var{sys}, @var{tol}, @var{gmin}, @var{gmax}, @var{ptol})
## Computes the 
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## H-infinity
## @end ifinfo
## norm of a system data structure.
##
## @strong{Inputs}
## @table @var
## @item sys
## system data structure
## @item tol
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## H-infinity
## @end ifinfo
## norm search tolerance (default: 0.001)
## @item gmin
## minimum value for norm search (default: 1e-9)
## @item gmax
## maximum value for norm search (default: 1e+9)
## @item ptol
## pole tolerance:
## @itemize @bullet
## @item if sys is continuous, poles with
## @iftex
## @tex
## $ \vert {\rm real}(pole) \vert < ptol \Vert H \Vert $
## @end tex
## @end iftex
## @ifinfo
## @math{ |real(pole))| < ptol*||H|| }
## @end ifinfo
## (@var{H} is appropriate Hamiltonian)
## are considered to be on the imaginary axis.
##
## @item if sys is discrete, poles with
## @iftex
## @tex
## $ \vert { \rm pole } - 1 \vert < ptol \Vert [ s_1 s_2 ] \Vert $
## @end tex
## @end iftex
## @ifinfo
## @math{|abs(pole)-1| < ptol*||[s1,s2]||}
## @end ifinfo
## (appropriate symplectic pencil)
## are considered to be on the unit circle.
##
## @item Default value: 1e-9
## @end itemize
## @end table
##
## @strong{Outputs}
## @table @var
## @item g
## Computed gain, within @var{tol} of actual gain.  @var{g} is returned as Inf
## if the system is unstable.
## @item gmin
## @itemx gmax
## Actual system gain lies in the interval [@var{gmin}, @var{gmax}].
## @end table
##
## References:
## Doyle, Glover, Khargonekar, Francis, @cite{State-space solutions to standard}
## @iftex
## @tex
## $ { \cal H }_2 $ @cite{and} $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## @cite{H-2 and H-infinity}
## @end ifinfo
## @cite{control problems}, @acronym{IEEE} @acronym{TAC} August 1989;
## Iglesias and Glover, @cite{State-Space approach to discrete-time}
## @iftex
## @tex
## $ { \cal H }_\infty $
## @end tex
## @end iftex
## @ifinfo
## @cite{H-infinity}
## @end ifinfo
## @cite{control}, Int. J. Control, vol 54, no. 5, 1991;
## Zhou, Doyle, Glover, @cite{Robust and Optimal Control}, Prentice-Hall, 1996.
## @end deftypefn

function [g, gmin, gmax] = hinfnorm (sys, tol, gmin, gmax, ptol)

  if (nargin == 0 || nargin > 4)
    print_usage ();
  elseif (! isstruct (sys))
    error ("Sys must be a system data structure");
  endif

  ## set defaults where applicable
  if (nargin < 5)
    ptol = 1e-9;        # pole tolerance
  endif
  if (nargin < 4)
    gmax = 1e9;         # max gain value
  endif

  dflg = is_digital (sys);
  sys = sysupdate (sys, "ss");
  [A, B, C, D] = sys2ss (sys);
  [n, nz, m, p] = sysdimensions (sys);

  ## eigenvalues of A must all be stable
  if (! is_stable (sys))
    warning ("hinfnorm: unstable system (is_stable, ptol=%g), returning Inf",
           ptol);
    g = Inf;
  endif

  Dnrm = norm (D);
  if (nargin < 3)
    gmin = max (1e-9, Dnrm);      # min gain value
  elseif (gmin < Dnrm)
    warning ("hinfnorm: setting Gmin=||D||=%g", Dnrm);
  endif

  if (nargin < 2)
    tol = 0.001;        # convergence measure for gmin, gmax
  endif

  ## check for scalar input arguments 2...5
  if (! isscalar (tol) && isscalar (gmin)
      && isscalar (gmax) && isscalar (ptol))
    error ("hinfnorm: tol, gmin, gmax, ptol must be scalars");
  endif

  In = eye (n+nz);
  Im = eye (m);
  Ip = eye (p);
  ## find the Hinf norm via binary search
  while (gmax/gmin - 1 > tol)
    g = (gmax+gmin)/2;

    if (dflg)
      ## multiply g's through in formulas to avoid extreme magnitudes...
      Rg = g^2*Im - D'*D;
      Ak = A + (B/Rg)*D'*C;
      Ck = g^2*C'*((g^2*Ip-D*D')\C);

      ## set up symplectic generalized eigenvalue problem per Iglesias & Glover
      s1 = [Ak , zeros(nz); -Ck, In];
      s2 = ;

      ## guard against roundoff again: zero out extremely small values
      ## prior to balancing
      s1 = s1 .* (abs(s1) > ptol*norm(s1,"inf"));
      s2 = s2 .* (abs(s2) > ptol*norm(s2,"inf"));
      [cc, dd, s1, s2] = balance (s1, s2);
      [qza, qzb, zz, pls] = qz (s1, s2, "S"); # ordered qz decomposition
      eigerr = abs (abs(pls)-1);
      normH = norm ([s1, s2]);
      Hb = [s1, s2];

      ## check R - B' X B condition (Iglesias and Glover's paper)
      X = zz((nz+1):(2*nz),1:nz)/zz(1:nz,1:nz);
      dcondfailed = min (real (eig (Rg - B'*X*B)) < ptol);
    else
      Rinv = inv(g*g*Im - (D' * D));
      H = ;
      ## guard against roundoff: zero out extremely small values prior
      ## to balancing
      H = H .* (abs (H) > ptol * norm (H, "inf"));
      [DD, Hb] = balance (H);
      pls = eig (Hb);
      eigerr = abs (real (pls));
      normH = norm (H);
      dcondfailed = 0;          # digital condition; doesn't apply here
    endif
    if ((min (eigerr) <= ptol * normH) | dcondfailed)
      gmin = g;
    else
      gmax = g;
    endif
  endwhile
endfunction

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