is_dgkf.m

similer program for matlab

## Copyright (C) 1996, 1998, 2000, 2002, 2004, 2005, 2006, 2007
##               Auburn University.  All rights reserved.
##
## This file is part of Octave.
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## your option) any later version.
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## 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, see
## <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{retval}, @var{dgkf_struct} ] =} is_dgkf (@var{asys}, @var{nu}, @var{ny}, @var{tol} )
## Determine whether a continuous time state space system meets
## assumptions of @acronym{DGKF} algorithm.
## Partitions system into:
## @example
## [dx/dt]   [A  | Bw  Bu  ][w]
## [ z   ] = [Cz | Dzw Dzu ][u]
## [ y   ]   [Cy | Dyw Dyu ]
## @end example
## or similar discrete-time system.
## If necessary, orthogonal transformations @var{qw}, @var{qz} and nonsingular
## transformations @var{ru}, @var{ry} are applied to respective vectors
## @var{w}, @var{z}, @var{u}, @var{y} in order to satisfy @acronym{DGKF} assumptions.
## Loop shifting is used if @var{dyu} block is nonzero.
##
## @strong{Inputs}
## @table @var
## @item         asys
## system data structure
## @item           nu
## number of controlled inputs
## @item        ny
## number of measured outputs
## @item        tol
## threshold for 0; default: 200*@code{eps}.
## @end table
## @strong{Outputs}
## @table @var
## @item    retval
## true(1) if system passes check, false(0) otherwise
## @item    dgkf_struct
## data structure of @command{is_dgkf} results.  Entries:
## @table @var
## @item      nw
## @itemx     nz
## dimensions of @var{w}, @var{z}
## @item      a
## system @math{A} matrix
## @item      bw
## (@var{n} x @var{nw}) @var{qw}-transformed disturbance input matrix
## @item      bu
## (@var{n} x @var{nu}) @var{ru}-transformed controlled input matrix;
##
## @math{B = [Bw Bu]}
## @item      cz
## (@var{nz} x @var{n}) Qz-transformed error output matrix
## @item      cy
## (@var{ny} x @var{n}) @var{ry}-transformed measured output matrix
##
## @math{C = [Cz; Cy]}
## @item      dzu
## @item      dyw
## off-diagonal blocks of transformed system @math{D} matrix that enter
## @var{z}, @var{y} from @var{u}, @var{w} respectively
## @item      ru
## controlled input transformation matrix
## @item      ry
## observed output transformation matrix
## @item      dyu_nz
## nonzero if the @var{dyu} block is nonzero.
## @item      dyu
## untransformed @var{dyu} block
## @item      dflg
## nonzero if the system is discrete-time
## @end table
## @end table
## @code{is_dgkf} exits with an error if the system is mixed
## discrete/continuous.
##
## @strong{References}
## @table @strong
## @item [1]
## 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.
## @item [2]
## Maciejowksi, J.M., @cite{Multivariable Feedback Design}, Addison-Wesley, 1989.
## @end table
## @end deftypefn

## Author: A. S. Hodel <[e-mail]>
## Updated by John Ingram July 1996 to accept structured systems

## Revised by Kai P. Mueller April 1998 to solve the general H_infinity
## problem using unitary transformations Q (on w and z)
## and non-singular transformations R (on u and y) such
## that the Dzu and Dyw matrices of the transformed plant
##
##    ~
##    P  (the variable Asys here)
##
## become
##
##    ~            -1         T
##    D  = Q   D   R   = [ 0 I ]  or [ I ],
##     12   12  12  12
##
##    ~            T
##    D  = R   D   Q   = [ 0 I ] or [ I ].
##     21   21  21  21
##
## This transformation together with the algorithm in [1] solves
## the general problem (see [2] for example).

function [retval, dgkf_struct] = is_dgkf (Asys, nu, ny, tol)

  if (nargin < 3) || (nargin > 4)
    print_usage ();
  elseif (! isscalar (nu) || ! isscalar (ny))
    error ("is_dgkf: arguments 2 and 3 must be scalars")
  elseif (! isstruct (Asys))
    error ("Argument 1 must be a system data structure");
  endif
  if (nargin < 4)
    tol = 200*eps;
  elseif (! is_sample (tol))
    error ("is_dgkf: tol must be a positive scalar")
  endif

  retval = 1;           # assume passes test

  dflg = is_digital (Asys);
  [Anc, Anz, nin, nout ] = sysdimensions (Asys);

  if (Anz == 0 && Anc == 0)
    error ("is_dgkf: no system states");
  elseif (nu >= nin)
    error ("is_dgkf: insufficient number of disturbance inputs");
  elseif (ny >= nout)
    error ("is_dgkf: insufficient number of regulated outputs");
  endif

  nw = nin - nu;
  nz = nout - ny;

  nw1 = nw + 1;
  nz1 = nz + 1;

  [A, B, C, D] = sys2ss (Asys);
  ## scale input/output for numerical reasons
  if (norm (C, "fro") * norm (B, "fro") == 0)
    error ("||C||*||B|| = 0; no dynamic connnection from inputs to outputs");
  endif
  xx = sqrt (norm (B, Inf) / norm (C, Inf));
  B = B / xx;  C = C * xx;

  ## partition matrices
  Bw = B(:,1:nw);
  Bu = B(:,nw1:nin);

  Cz = C(1:nz,:);
  Cy = C(nz1:nout,:);

  Dzw = D(1:nz,1:nw);
  Dzu = D(1:nz,nw1:nin);

  Dyw = D(nz1:nout,1:nw);
  Dyu = D(nz1:nout,nw1:nin);

  ## Check for loopo shifting
  Dyu_nz = (norm (Dyu, Inf) != 0);
  if (Dyu_nz)
    warning ("is_dgkf: D22 nonzero; performing loop shifting");
  endif

  ## 12 - rank condition at w = 0
  xx =[A, Bu; Cz, Dzu];
  [nr, nc] = size (xx);
  irank = rank (xx);
  if (irank != nc)
    retval = 0;
    warning ("rank([A Bu; Cz Dzu]) = %d, need %d; n=%d, nz=%d, nu=%d",
             irank, nc, Anc+Anz, nz, nu);
    warning (" *** 12-rank condition violated at w = 0");
  endif

  ## 21 - rank condition at w = 0
  xx =[A, Bw; Cy, Dyw];
  [nr, nc] = size (xx);
  irank = rank (xx);
  if (irank != nr)
    retval = 0;
    warning ("rank([A Bw; Cy Dyw]) = %d, need %d; n=%d, ny=%d, nw=%d",
             irank, nr, Anc+Anz, ny, nw);
    warning (" *** 21-rank condition violated at w = 0");
  endif

  ## can Dzu be transformed to become [0 I]' or [I]?
  ## This ensures a normalized weight
  [Qz, Ru] = qr (Dzu);
  irank = rank (Ru);
  if (irank != nu)
    retval = 0;
    warning ("*** rank(Dzu(%d x %d) = %d", nz, nu, irank);
    warning ("*** Dzu does not have full column rank");
  endif
  if (nu >= nz)
    Qz = Qz(:,1:nu)';
  else
    Qz = [Qz(:,(nu+1):nz), Qz(:,1:nu)]';
  endif
  Ru = Ru(1:nu,:);

  ## can Dyw be transformed to become [0 I] or [I]?
  ## This ensures a normalized weight
  [Qw, Ry] = qr (Dyw');
  irank = rank (Ry);
  if (irank != ny)
    retval = 0;
    warning ("*** rank(Dyw(%d x %d) = %d", ny, nw, irank);
    warning (" *** Dyw does not have full row rank");
  endif

  if (ny >= nw)
    Qw = Qw(:,1:ny);
  else
    Qw = [Qw(:,(ny+1):nw), Qw(:,1:ny)];
  endif
  Ry = Ry(1:ny,:)';

  ## transform P by Qz/Ru and Qw/Ry
  Bw  = Bw*Qw;
  Bu  = Bu/Ru;
  B   = [Bw, Bu];
  Cz  = Qz*Cz;
  Cy  = Ry\Cy;
  C   = [Cz; Cy];
  Dzw = Qz*Dzw*Qw;
  Dzu = Qz*Dzu/Ru;
  Dyw = Ry\Dyw*Qw;

  ## pack the return structure
  dgkf_struct.nw = nw;
  dgkf_struct.nu = nu;
  dgkf_struct.nz = nz;
  dgkf_struct.ny = ny;
  dgkf_struct.A  = A;
  dgkf_struct.Bw = Bw;
  dgkf_struct.Bu = Bu;
  dgkf_struct.Cz = Cz;
  dgkf_struct.Cy = Cy;
  dgkf_struct.Dzw = Dzw;
  dgkf_struct.Dzu = Dzu;
  dgkf_struct.Dyw = Dyw;
  dgkf_struct.Dyu = Dyu;
  dgkf_struct.Ru = Ru;
  dgkf_struct.Ry = Ry;
  dgkf_struct.Dyu_nz = Dyu_nz;
  dgkf_struct.dflg = dflg;

endfunction