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## Copyright (C) 2009-2014 Lukas F. Reichlin
##
## This file is part of LTI Syncope.
##
## LTI Syncope 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.
##
## LTI Syncope 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 LTI Syncope. If not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn{Function File} {[@var{K}, @var{N}, @var{info}] =} h2syn (@var{P}, @var{nmeas}, @var{ncon})
## H-2 control synthesis for @acronym{LTI} plant.
##
## @strong{Inputs}
## @table @var
## @item P
## Generalized plant. Must be a proper/realizable @acronym{LTI} model.
## @item nmeas
## Number of measured outputs v. The last @var{nmeas} outputs of @var{P} are connected to the
## inputs of controller @var{K}. The remaining outputs z (indices 1 to p-nmeas) are used
## to calculate the H-2 norm.
## @item ncon
## Number of controlled inputs u. The last @var{ncon} inputs of @var{P} are connected to the
## outputs of controller @var{K}. The remaining inputs w (indices 1 to m-ncon) are excited
## by a harmonic test signal.
## @end table
##
## @strong{Outputs}
## @table @var
## @item K
## State-space model of the H-2 optimal controller.
## @item N
## State-space model of the lower LFT of @var{P} and @var{K}.
## @item info
## Structure containing additional information.
## @item info.gamma
## H-2 norm of @var{N}.
## @item info.rcond
## Vector @var{rcond} contains estimates of the reciprocal condition
## numbers of the matrices which are to be inverted and
## estimates of the reciprocal condition numbers of the
## Riccati equations which have to be solved during the
## computation of the controller @var{K}. For details,
## see the description of the corresponding SLICOT routine.
## @end table
##
## @strong{Block Diagram}
## @example
## @group
##
## gamma = min||N(K)|| N = lft (P, K)
## K 2
##
## +--------+
## w ----->| |-----> z
## | P(s) |
## u +---->| |-----+ v
## | +--------+ |
## | |
## | +--------+ |
## +-----| K(s) |<----+
## +--------+
##
## +--------+
## w ----->| N(s) |-----> z
## +--------+
## @end group
## @end example
##
## @strong{Algorithm}@*
## Uses SLICOT SB10HD and SB10ED by courtesy of
## @uref{http://www.slicot.org, NICONET e.V.}
##
## @seealso{augw, lqr, dlqr, kalman}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: December 2009
## Version: 0.3
function [K, varargout] = h2syn (P, nmeas, ncon)
## check input arguments
if (nargin != 1 && nargin != 3)
print_usage ();
endif
if (! isa (P, "lti"))
error ("h2syn: first argument must be an LTI system");
endif
if (nargin == 1)
[nmeas, ncon] = __tito_dim__ (P, "h2syn");
endif
if (! is_real_scalar (nmeas))
error ("h2syn: second argument 'nmeas' invalid");
endif
if (! is_real_scalar (ncon))
error ("h2syn: third argument 'ncon' invalid");
endif
[a, b, c, d, tsam] = ssdata (P);
## check assumptions A1 - A3
m = columns (b);
p = rows (c);
m1 = m - ncon;
p1 = p - nmeas;
d11 = d(1:p1, 1:m1);
if (isct (P) && any (d11(:)))
warning ("h2syn: setting matrice D11 to zero");
d(1:p1, 1:m1) = 0;
endif
if (! isstabilizable (P(:, m1+1:m)))
error ("h2syn: (A, B2) must be stabilizable");
endif
if (! isdetectable (P(p1+1:p, :)))
error ("h2syn: (C2, A) must be detectable");
endif
## H-2 synthesis
if (isct (P)) # continuous plant
[ak, bk, ck, dk, rcond] = __sl_sb10hd__ (a, b, c, d, ncon, nmeas);
else # discrete plant
[ak, bk, ck, dk, rcond] = __sl_sb10ed__ (a, b, c, d, ncon, nmeas);
endif
## controller
K = ss (ak, bk, ck, dk, tsam);
if (nargout > 1)
N = lft (P, K);
varargout{1} = N;
if (nargout > 2)
varargout{2} = struct ("gamma", norm (N, 2), "rcond", rcond);
endif
endif
endfunction
## continuous-time case
%!shared M, M_exp
%! A = [-1.0 0.0 4.0 5.0 -3.0 -2.0
%! -2.0 4.0 -7.0 -2.0 0.0 3.0
%! -6.0 9.0 -5.0 0.0 2.0 -1.0
%! -8.0 4.0 7.0 -1.0 -3.0 0.0
%! 2.0 5.0 8.0 -9.0 1.0 -4.0
%! 3.0 -5.0 8.0 0.0 2.0 -6.0];
%!
%! B = [-3.0 -4.0 -2.0 1.0 0.0
%! 2.0 0.0 1.0 -5.0 2.0
%! -5.0 -7.0 0.0 7.0 -2.0
%! 4.0 -6.0 1.0 1.0 -2.0
%! -3.0 9.0 -8.0 0.0 5.0
%! 1.0 -2.0 3.0 -6.0 -2.0];
%!
%! C = [ 1.0 -1.0 2.0 -4.0 0.0 -3.0
%! -3.0 0.0 5.0 -1.0 1.0 1.0
%! -7.0 5.0 0.0 -8.0 2.0 -2.0
%! 9.0 -3.0 4.0 0.0 3.0 7.0
%! 0.0 1.0 -2.0 1.0 -6.0 -2.0];
%!
%! D = [ 0.0 0.0 0.0 -4.0 -1.0
%! 0.0 0.0 0.0 1.0 0.0
%! 0.0 0.0 0.0 0.0 1.0
%! 3.0 1.0 0.0 1.0 -3.0
%! -2.0 0.0 1.0 7.0 1.0];
%!
%! P = ss (A, B, C, D);
%! K = h2syn (P, 2, 2);
%! M = [K.A, K.B; K.C, K.D];
%!
%! KA = [ 88.0015 -145.7298 -46.2424 82.2168 -45.2996 -31.1407
%! 25.7489 -31.4642 -12.4198 9.4625 -3.5182 2.7056
%! 54.3008 -102.4013 -41.4968 50.8412 -20.1286 -26.7191
%! 108.1006 -198.0785 -45.4333 70.3962 -25.8591 -37.2741
%! -115.8900 226.1843 47.2549 -47.8435 -12.5004 34.7474
%! 59.0362 -101.8471 -20.1052 36.7834 -16.1063 -26.4309];
%!
%! KB = [ 3.7345 3.4758
%! -0.3020 0.6530
%! 3.4735 4.0499
%! 4.3198 7.2755
%! -3.9424 -10.5942
%! 2.1784 2.5048];
%!
%! KC = [ -2.3346 3.2556 0.7150 -0.9724 0.6962 0.4074
%! 7.6899 -8.4558 -2.9642 7.0365 -4.2844 0.1390];
%!
%! KD = [ 0.0000 0.0000
%! 0.0000 0.0000];
%!
%! M_exp = [KA, KB; KC, KD];
%!
%!assert (M, M_exp, 1e-4);
## discrete-time case
%!shared M, M_exp
%! A = [-0.7 0.0 0.3 0.0 -0.5 -0.1
%! -0.6 0.2 -0.4 -0.3 0.0 0.0
%! -0.5 0.7 -0.1 0.0 0.0 -0.8
%! -0.7 0.0 0.0 -0.5 -1.0 0.0
%! 0.0 0.3 0.6 -0.9 0.1 -0.4
%! 0.5 -0.8 0.0 0.0 0.2 -0.9];
%!
%! B = [-1.0 -2.0 -2.0 1.0 0.0
%! 1.0 0.0 1.0 -2.0 1.0
%! -3.0 -4.0 0.0 2.0 -2.0
%! 1.0 -2.0 1.0 0.0 -1.0
%! 0.0 1.0 -2.0 0.0 3.0
%! 1.0 0.0 3.0 -1.0 -2.0];
%!
%! C = [ 1.0 -1.0 2.0 -2.0 0.0 -3.0
%! -3.0 0.0 1.0 -1.0 1.0 0.0
%! 0.0 2.0 0.0 -4.0 0.0 -2.0
%! 1.0 -3.0 0.0 0.0 3.0 1.0
%! 0.0 1.0 -2.0 1.0 0.0 -2.0];
%!
%! D = [ 1.0 -1.0 -2.0 0.0 0.0
%! 0.0 1.0 0.0 1.0 0.0
%! 2.0 -1.0 -3.0 0.0 1.0
%! 0.0 1.0 0.0 1.0 -1.0
%! 0.0 0.0 1.0 2.0 1.0];
%!
%! P = ss (A, B, C, D, 1); # value of sampling time doesn't matter
%! K = h2syn (P, 2, 2);
%! M = [K.A, K.B; K.C, K.D];
%!
%! KA = [-0.0551 -2.1891 -0.6607 -0.2532 0.6674 -1.0044
%! -1.0379 2.3804 0.5031 0.3960 -0.6605 1.2673
%! -0.0876 -2.1320 -0.4701 -1.1461 1.2927 -1.5116
%! -0.1358 -2.1237 -0.9560 -0.7144 0.6673 -0.7957
%! 0.4900 0.0895 0.2634 -0.2354 0.1623 -0.2663
%! 0.1672 -0.4163 0.2871 -0.1983 0.4944 -0.6967];
%!
%! KB = [-0.5985 -0.5464
%! 0.5285 0.6087
%! -0.7600 -0.4472
%! -0.7288 -0.6090
%! 0.0532 0.0658
%! -0.0663 0.0059];
%!
%! KC = [ 0.2500 -1.0200 -0.3371 -0.2733 0.2747 -0.4444
%! 0.0654 0.2095 0.0632 0.2089 -0.1895 0.1834];
%!
%! KD = [-0.2181 -0.2070
%! 0.1094 0.1159];
%!
%! M_exp = [KA, KB; KC, KD];
%!
%!assert (M, M_exp, 1e-4);