# Copyright (C) 1996,1998 A. Scottedward Hodel 
#
# 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. 
# 
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# for more details.
# 
# You should have received a copy of the GNU General Public License 
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# Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA. 
 
function [g gmin gmax] = hinfnorm(sys,tol,gmin,gmax,ptol)
  # Usage: [g gmin gmax] = hinfnorm(sys[,tol,gmin,gmax,ptol])
  #
  # Computes the H infinity norm of a system data structure
  # sys = system data structure
  # tol = H infinity norm search tolerance (default: 0.001)
  # gmin = minimum value for norm search (default: 1e-9)
  # gmax = maximum value for norm search (default: 1e+9)
  # ptol: pole tolerance:
  #       if sys is continuous, poles with 
  #         |real(pole)| < ptol*||H|| (H is appropriate Hamiltonian)
  #         are considered to be on the imaginary axis.  
  #       if sys is discrete, poles with
  #         |abs(pole)-1| < ptol*||[s1,s2]|| (appropriate symplectic pencil)
  #         are considered to be on the unit circle
  #       Default: 1e-9
  #
  # References:
  # Doyle, Glover, Khargonekar, Francis, "State space solutions to standard
  #    H2 and Hinf control problems", IEEE TAC August 1989
  # Iglesias and Glover, "State-Space approach to discrete-time Hinf control,"
  #    Int. J. Control, vol 54, #5, 1991
  # Zhou, Doyle, Glover, "Robust and Optimal Control," Prentice-Hall, 1996

  if((nargin == 0) || (nargin > 4))
    usage("[g gmin gmax] = hinfnorm(sys[,tol,gmin,gmax,ptol])");
  elseif(!is_struct(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=",num2str(ptol), ...
      "), returning Inf"]);
    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||=",num2str(Dnrm)]);
  endif

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

  # check for scalar input arguments 2...5
  if( ! (is_scalar(tol) && is_scalar(gmin) 
	&& is_scalar(gmax) && is_scalar(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 = [In, -(B/Rg)*B' ; zeros(nz) , Ak' ];

      # 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 = [A + B*Rinv*D'*C,        B*Rinv*B'; ...
           -C'*(Ip + D*Rinv*D')*C, -(A + B*Rinv*D'*C)'];
      # 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