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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{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon})
## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @dots{})
## @deftypefnx{Function File} {[@var{K}, @var{N}, @var{gamma}, @var{info}] =} hinfsyn (@var{P}, @var{nmeas}, @var{ncon}, @var{opt}, @dots{})
## H-infinity 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-infinity 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.
## @item @dots{}
## Optional pairs of keys and values. @code{'key1', value1, 'key2', value2}.
## @item opt
## Optional struct with keys as field names.
## Struct @var{opt} can be created directly or
## by function @command{options}. @code{opt.key1 = value1, opt.key2 = value2}.
## @end table
##
## @strong{Outputs}
## @table @var
## @item K
## State-space model of the H-infinity (sub-)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
## L-infinity 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{Option Keys and Values}
## @table @var
## @item 'method'
## String specifying the desired kind of controller:
## @table @var
## @item 'optimal', 'opt', 'o'
## Compute optimal controller using gamma iteration.
## Default selection for compatibility reasons.
## @item 'suboptimal', 'sub', 's'
## Compute (sub-)optimal controller. For stability reasons,
## suboptimal controllers are to be preferred over optimal ones.
## @end table
## @item 'gmax'
## The maximum value of the H-infinity norm of @var{N}.
## It is assumed that @var{gmax} is sufficiently large
## so that the controller is admissible. Default value is 1e15.
## @item 'gmin'
## Initial lower bound for gamma iteration. Default value is 0.
## @var{gmin} is only meaningful for optimal discrete-time controllers.
## @item 'tolgam'
## Tolerance used for controlling the accuracy of @var{gamma}
## and its distance to the estimated minimal possible
## value of @var{gamma}. Default value is 0.01.
## If @var{tolgam} = 0, then a default value equal to @code{sqrt(eps)}
## is used, where @var{eps} is the relative machine precision.
## For suboptimal controllers, @var{tolgam} is ignored.
## @item 'actol'
## Upper bound for the poles of the closed-loop system @var{N}
## used for determining if it is stable.
## @var{actol} >= 0 for stable systems.
## For suboptimal controllers, @var{actol} is ignored.
## @end table
##
## @strong{Block Diagram}
## @example
## @group
##
## gamma = min||N(K)|| N = lft (P, K)
## K inf
##
## +--------+
## w ----->| |-----> z
## | P(s) |
## u +---->| |-----+ v
## | +--------+ |
## | |
## | +--------+ |
## +-----| K(s) |<----+
## +--------+
##
## +--------+
## w ----->| N(s) |-----> z
## +--------+
## @end group
## @end example
##
## @strong{Algorithm}@*
## Uses SLICOT SB10FD, SB10DD and SB10AD by courtesy of
## @uref{http://www.slicot.org, NICONET e.V.}
##
## @seealso{augw, mixsyn}
## @end deftypefn
## Author: Lukas Reichlin <lukas.reichlin@gmail.com>
## Created: December 2009
## Version: 0.3
function [K, varargout] = hinfsyn (P, varargin)
## check input arguments
if (nargin == 0)
print_usage ();
endif
if (! isa (P, "lti"))
error ("hinfsyn: first argument must be an LTI system");
endif
if (nargin == 1 || (nargin > 1 && ! is_real_scalar (varargin{1}))) # hinfsyn (P, ...)
[nmeas, ncon] = __tito_dim__ (P, "hinfsyn");
elseif (nargin >= 3) # hinfsyn (P, nmeas, ncon, ...)
nmeas = varargin{1};
ncon = varargin{2};
varargin = varargin(3:end);
else
print_usage ();
endif
if (! is_real_scalar (nmeas))
error ("hinfsyn: second argument 'nmeas' invalid");
endif
if (! is_real_scalar (ncon))
error ("hinfsyn: third argument 'ncon' invalid");
endif
if (numel (varargin) > 0 && isstruct (varargin{1})) # hinfsyn (P, nmeas, ncon, opt, ...), hinfsyn (P, opt, ...)
varargin = horzcat (__opt2cell__ (varargin{1}), varargin(2:end));
endif
nkv = numel (varargin); # number of keys and values
if (rem (nkv, 2))
error ("hinfsyn: keys and values must come in pairs");
endif
## default arguments
gmax = 1e15;
gmin = 0;
tolgam = 0.01;
actol = eps; # tolerance for stability margin
method = "opt";
## handle keys and values
for k = 1 : 2 : nkv
key = lower (varargin{k});
val = varargin{k+1};
switch (key)
case "gmax"
if (! is_real_scalar (val) || val < 0)
error ("hinfsyn: 'gmax' must be a real-valued, non-negative scalar");
endif
gmax = val;
case "gmin"
if (! is_real_scalar (val) || val < 0)
error ("hinfsyn: 'gmin' must be a real-valued, non-negative scalar");
endif
gmin = val;
case "tolgam"
if (! is_real_scalar (val) || val < 0)
error ("hinfsyn: 'tolgam' must be a real-valued, non-negative scalar");
endif
tolgam = val;
case "actol"
if (! is_real_scalar (val) || val < 0)
error ("hinfsyn: 'actol' must be a real-valued, non-negative scalar");
endif
actol = val;
case "method"
## NOTE: I called this "method" because of the dark side,
## maybe something like "type" would make more sense ...
if (strncmpi (val, "s", 1))
method = "sub"; # sub-optimal
elseif (strncmpi (val, "o", 1) || strncmpi (val, "ric", 1))
method = "opt"; # optimal
else
error ("hinfsyn: invalid method '%s'", val);
endif
otherwise
warning ("hinfsyn: invalid property name '%s' ignored", key);
endswitch
endfor
[a, b, c, d, tsam] = ssdata (P);
## check assumption A1
m = columns (b);
p = rows (c);
m1 = m - ncon;
p1 = p - nmeas;
if (! isstabilizable (P(:, m1+1:m)))
error ("hinfsyn: (A, B2) must be stabilizable");
endif
if (! isdetectable (P(p1+1:p, :)))
error ("hinfsyn: (C2, A) must be detectable");
endif
## H-infinity synthesis
switch (method)
case "sub" # sub-optimal controller
if (isct (P)) # continuous-time plant
[ak, bk, ck, dk, rcond] = __sl_sb10fd__ (a, b, c, d, ncon, nmeas, gmax);
else # discrete-time plant
[ak, bk, ck, dk, rcond] = __sl_sb10dd__ (a, b, c, d, ncon, nmeas, gmax);
endif
case "opt" # optimal controller
if (isct (P)) # continuous-time plant
[ak, bk, ck, dk, ~, ~, ~, ~, ~, rcond] = __sl_sb10ad__ (a, b, c, d, ncon, nmeas, gmax, tolgam, -actol);
else # discrete-time plant
## NOTE: check whether it is an alternative to compute the bilinear transformation
## of P, use __sl_sb10ad__ for a continuous-time controller and then
## discretize the controller.
## estimate gamma
Pt = d2c (P, "tustin");
[at, bt, ct, dt] = ssdata (Pt);
[~, ~, ~, ~, ~, ~, ~, ~, gamma] = __sl_sb10ad__ (at, bt, ct, dt, ncon, nmeas, gmax, tolgam, -actol);
## gamma iteration - bisection method using __sl_sb10dd__
gmax = 1.2*gamma;
while (gmax > eps && (gmax - gmin)/gmax > tolgam)
gmid = (gmax + gmin)/2;
try
[ak, bk, ck, dk, rcond] = __sl_sb10dd__ (a, b, c, d, ncon, nmeas, gmid);
## check for stability
K = ss (ak, bk, ck, dk, tsam);
N = lft (P, K);
if (isstable (N, actol))
gmax = norm (N, inf);
else
gmin = gmid;
endif
catch # cannot find solution
gmin = gmid;
end_try_catch
endwhile
endif
otherwise
error ("hinfsyn: this should never happen");
endswitch
## controller
K = ss (ak, bk, ck, dk, tsam);
if (nargout > 1)
N = lft (P, K);
varargout{1} = N;
if (nargout > 2)
gamma = norm (N, inf);
varargout{2} = gamma;
if (nargout > 3)
varargout{3} = struct ("gamma", gamma, "rcond", rcond);
endif
endif
endif
endfunction
## sub-optimal controller, 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 = [ 1.0 -2.0 -3.0 0.0 0.0
%! 0.0 4.0 0.0 1.0 0.0
%! 5.0 -3.0 -4.0 0.0 1.0
%! 0.0 1.0 0.0 1.0 -3.0
%! 0.0 0.0 1.0 7.0 1.0];
%!
%! P = ss (A, B, C, D);
%! K = hinfsyn (P, 2, 2, "method", "sub", "gmax", 15);
%! M = [K.A, K.B; K.C, K.D];
%!
%! KA = [ -2.8043 14.7367 4.6658 8.1596 0.0848 2.5290
%! 4.6609 3.2756 -3.5754 -2.8941 0.2393 8.2920
%! -15.3127 23.5592 -7.1229 2.7599 5.9775 -2.0285
%! -22.0691 16.4758 12.5523 -16.3602 4.4300 -3.3168
%! 30.6789 -3.9026 -1.3868 26.2357 -8.8267 10.4860
%! -5.7429 0.0577 10.8216 -11.2275 1.5074 -10.7244];
%!
%! KB = [ -0.1581 -0.0793
%! -0.9237 -0.5718
%! 0.7984 0.6627
%! 0.1145 0.1496
%! -0.6743 -0.2376
%! 0.0196 -0.7598];
%!
%! KC = [ -0.2480 -0.1713 -0.0880 0.1534 0.5016 -0.0730
%! 2.8810 -0.3658 1.3007 0.3945 1.2244 2.5690];
%!
%! KD = [ 0.0554 0.1334
%! -0.3195 0.0333];
%!
%! M_exp = [KA, KB; KC, KD];
%!
%!assert (M, M_exp, 1e-4);
## sub-optimal controller, 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 = hinfsyn (P, 2, 2, "method", "sub", "gmax", 111.294);
%! M = [K.A, K.B; K.C, K.D];
%!
%! KA = [-18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857
%! 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644
%! -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910
%! -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351
%! -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169
%! -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590];
%!
%! KB = [ 16.9788 14.1648
%! -18.9215 -15.6726
%! 25.2046 21.2848
%! 20.1122 16.8322
%! 1.4104 1.2040
%! 5.3181 4.5149];
%!
%! KC = [ -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025
%! 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293];
%!
%! KD = [ 9.0317 7.5348
%! -3.4006 -2.8219];
%!
%! M_exp = [KA, KB; KC, KD];
%!
%!assert (M, M_exp, 1e-4);
## optimal controller, discrete-time case??? -- test for bisection method
%!shared M, M_exp, GAM_exp, GAM
%! 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);
%! [K, ~, GAM] = hinfsyn (P, 2, 2, "gmax", 1000, "tolgam", 1e-4);
%! M = [K.A, K.B; K.C, K.D];
%!
%! KA = [-18.0030 52.0376 26.0831 -0.4271 -40.9022 18.0857
%! 18.8203 -57.6244 -29.0938 0.5870 45.3309 -19.8644
%! -26.5994 77.9693 39.0368 -1.4020 -60.1129 26.6910
%! -21.4163 62.1719 30.7507 -0.9201 -48.6221 21.8351
%! -0.8911 4.2787 2.3286 -0.2424 -3.0376 1.2169
%! -5.3286 16.1955 8.4824 -0.2489 -12.2348 5.1590];
%!
%! KB = [ 16.9788 14.1648
%! -18.9215 -15.6726
%! 25.2046 21.2848
%! 20.1122 16.8322
%! 1.4104 1.2040
%! 5.3181 4.5149];
%!
%! KC = [ -9.1941 27.5165 13.7364 -0.3639 -21.5983 9.6025
%! 3.6490 -10.6194 -5.2772 0.2432 8.1108 -3.6293];
%!
%! KD = [ 9.0317 7.5348
%! -3.4006 -2.8219];
%!
%! M_exp = [KA, KB; KC, KD];
%! GAM_exp = 111.294;
%!
%!assert (M, M_exp, 1e-1);
%!assert (GAM, GAM_exp, 1e-3);