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## Copyright (C) 2017-2018 Fabian Alexander Wilms <f.alexander.wilms@gmail.com>
##
## 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; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {@var{k} =} acker (@var{A}, @var{b}, @var{p})
## Calculates the state feedback matrix of a completely controllable SISO system
## using Ackermann's formula
##
## Given the state-space system
## @tex
## $$ \dot x = Ax + bu $$
## @end tex
## @ifnottex
## @example
## @group
## .
## x = @var{A}x + @var{b}u
## @end group
## @end example
## @end ifnottex
##
## and the desired eigenvalues of the closed loop in the vector @var{p},
## the state feedback vector k is calculated in the form
##
## @tex
## $$ k = (k_1 k_2 ... k_n) $$
## @end tex
## @ifnottex
## @example
## @group
## @var{k} = (k1 k2 ... kn)
## @end group
## @end example
## @end ifnottex
##
## such that the closed loop system matrix
##
## @tex
## $$ A - b\,k $$
## @end tex
## @ifnottex
## @example
## @group
## @var{A} - @var{b}@var{k}
## @end group
## @end example
## @end ifnottex
##
## has the eigenvalues given in @var{p}.
##
## @seealso{place}
## @end deftypefn
## This function uses equation (4) from the paper
## "Sliding mode control design based on Ackermann's formula",
## DOI: 10.1109/9.661072
## Author: Fabian Alexander Wilms <f.alexander.wilms@gmail.com>
## Created: May 2017
## Revised: March 2022, Torsten Lilge (compacter and faster code)
## Version: 0.2
function k = acker(A, b, p)
if (nargin != 3)
print_usage ();
endif
if (! is_real_square_matrix (A) || ! is_real_vector (b) || rows (A) != rows (b))
error ("acker: matrix A and vector b not conformal");
endif
if (! isnumeric (p) || ! isvector (p) || isempty (p) || (length (p) != size (A,1))) # p could be complex
error ("acker: p must be a vector of size of A, here %d", size (A,1));
endif
k = (inv (ctrb (A, b)))(end,:) * polyvalm (poly (p), A);
endfunction
%!test
%! # https://en.wikipedia.org/wiki/Ackermann's_formula#Example
%! A = [1 1; 1 2];
%! B = [1; 0];
%! P = roots ([1 11 30]);
%! K = acker(A,B,P);
%! assert (K, [14 57]);
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