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lqg.m

## Copyright (C) 1996, 1997, 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{k}, @var{q1}, @var{p1}, @var{ee}, @var{er}] =} lqg (@var{sys}, @var{sigw}, @var{sigv}, @var{q}, @var{r}, @var{in_idx})
## Design a linear-quadratic-gaussian optimal controller for the system
## @example
## dx/dt = A x + B u + G w       [w]=N(0,[Sigw 0    ])
##     y = C x + v               [v]  (    0   Sigv ])
## @end example
## or
## @example
## x(k+1) = A x(k) + B u(k) + G w(k)   [w]=N(0,[Sigw 0    ])
##   y(k) = C x(k) + v(k)              [v]  (    0   Sigv ])
## @end example
##
## @strong{Inputs}
## @table @var
## @item  sys
## system data structure
## @item  sigw
## @itemx  sigv
## intensities of independent Gaussian noise processes (as above)
## @item  q
## @itemx  r
## state, control weighting respectively.  Control @acronym{ARE} is
## @item  in_idx
## names or indices of controlled inputs (see @command{sysidx}, @command{cellidx})
##
## default: last dim(R) inputs are assumed to be controlled inputs, all
## others are assumed to be noise inputs.
## @end table
## @strong{Outputs}
## @table @var
## @item    k
## system data structure format @acronym{LQG} optimal controller (Obtain A, B, C
## matrices with @command{sys2ss}, @command{sys2tf}, or @command{sys2zp} as
## appropriate).
## @item    p1
## Solution of control (state feedback) algebraic Riccati equation.
## @item    q1
## Solution of estimation algebraic Riccati equation.
## @item    ee
## Estimator poles.
## @item    es
## Controller poles.
## @end table
## @seealso{h2syn, lqe, lqr}
## @end deftypefn

## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu>
## Created: August 1995
## revised for new system format August 1996

function [K, Q1, P1, Ee, Er] = lqg (sys, Sigw, Sigv, Q, R, input_list)

  if (nargin < 5 || nargin > 6)
    print_usage ();
  elseif (! isstruct (sys))
    error ("sys must be in system data structure");
  endif

  DIG = is_digital (sys);
  [A, B, C, D, tsam, n, nz, stname, inname, outname] = sys2ss (sys);
  [n, nz, nin, nout] = sysdimensions (sys);
  if (nargin == 5)
    ## construct default input_list
    input_list = (columns(Sigw)+1):nin;
  endif

  if (! (n+nz))
    error("lqg: 0 states in system");

  elseif (nin != columns (Sigw) + columns (R))
    error ("lqg: sys has %d inputs, dim(Sigw)=%d, dim(u)=%d",
         nin, columns (Sigw), columns (R));

  elseif (nout != columns (Sigv))
    error ("lqg: sys has %d outputs, dim(Sigv)=%d", nout, columns (Sigv));
  endif

  ## check for names of signals
  if (is_signal_list (input_list) || ischar (input_list))
    input_list = sysidx (sys, "in", input_list);
  endif

  if (length (input_list) != columns (R))
    error ("lqg: length(input_list)=%d, columns(R)=%d",
         length (input_list), columns (R));
  endif

  if (! issquare (Sigw))
    error ("lqg: Sigw is not square");
  endif

  if (! issquare (Sigv))
    error ("lqg: Sigv is not square");
  endif

  if (! issquare (Q))
    error ("lqg: Q is not square");
  endif

  if (! issquare (R))
    error ("lqg: Q is not square");
  endif

  ## permute (if need be)
  if (nargin == 6)
    all_inputs = sysreorder (nin, input_list);
    B = B(:,all_inputs);
    inname = inname (all_inputs);
  endif

  ## put parameters into correct variables
  m1 = columns (Sigw);
  m2 = m1+1;
  G = B(:,1:m1);
  B = B(:,m2:nin);

  ## now we can just do the design; call dlqr and dlqe, since all matrices
  ## are not given in Cholesky factor form (as in h2syn case)
  if (DIG)
    [Ks, P1, Er] = dlqr (A, B, Q, R);
    [Ke, Q1, jnk, Ee] = dlqe (A, G, C, Sigw, Sigv);
  else
    [Ks, P1, Er] = lqr (A, B, Q, R);
    [Ke, Q1, Ee] = lqe (A, G, C, Sigw, Sigv);
  endif
  Ac = A - Ke*C - B*Ks;
  Bc = Ke;
  Cc = -Ks;
  Dc = zeros (rows (Cc), columns (Bc));

  ## fix state names
  stname1 = strappend (stname, "_e");

  ## fix controller output names
  outname1 = strappend (inname(m2:nin), "_K");

  ## fix controller input names
  inname1 = strappend (outname, "_K");

  if (DIG)
    K = ss (Ac, Bc, Cc, Dc, tsam, n, nz, stname1, inname1, outname1,
          1:rows(Cc));
  else
    K = ss (Ac, Bc, Cc, Dc, tsam, n, nz, stname, inname1, outname1);
  endif

endfunction

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