## Copyright (C) 1997, 2000, 2002, 2003, 2004, 2005, 2007 ## Jose Daniel Munoz Frias ## ## ## 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} =} place (@var{sys}, @var{p}) ## @deftypefnx {Function File} {@var{K} =} place (@var{a}, @var{b}, @var{p}) ## Computes the matrix @var{K} such that if the state ## is feedback with gain @var{K}, then the eigenvalues of the closed loop ## system (i.e. @math{A-BK}) are those specified in the vector @var{p}. ## ## Version: Beta (May-1997): If you have any comments, please let me know. ## (see the file place.m for my address) ## @end deftypefn ## Author: Jose Daniel Munoz Frias ## Universidad Pontificia Comillas ## ICAIdea ## Alberto Aguilera, 23 ## 28015 Madrid, Spain ## ## E-Mail: daniel@dea.icai.upco.es ## ## Phone: 34-1-5422800 Fax: 34-1-5596569 ## ## Algorithm taken from "The Control Handbook", IEEE press pp. 209-212 ## ## code adaped by A.S.Hodel (a.s.hodel@eng.auburn.edu) for use in controls ## toolbox function K = place (argin1, argin2, argin3) if (nargin == 3) ## Ctmp is useful to use ss; it doesn't matter what the value of Ctmp is Ctmp = zeros (1, rows (argin1)); sys = ss (argin1, argin2, Ctmp); P = argin3; elseif (nargin == 2) sys = argin1; P = argin2; else print_usage (); endif ## check arguments if (! isstruct (sys)) error ("sys must be in system data structure format (see ss)"); endif sys = sysupdate (sys, "ss"); # make sure it has state space form up to date if (! is_controllable (sys)) error ("sys is not controllable"); elseif (min (size (P)) != 1) error ("P must be a vector") else P = P(:); # make P a column vector endif ## system must be purely continuous or discrete is_digital (sys); [n, nz, m, p] = sysdimensions (sys); nx = n+nz; # already checked that it's not a mixed system. if (m != 1) error ("sys has %d inputs; need only 1", m); endif ## takes the A and B matrix from the system representation [A, B] = sys2ss (sys); sp = length (P); if (nx == 0) error ("place: A matrix is empty (0x0)"); elseif (nx != length (P)) error ("A=(%dx%d), P has %d entries", nx, nx, length (P)) endif ## arguments appear to be compatible; let's give it a try! ## The second step is the calculation of the characteristic polynomial ofA PC = poly (A); ## Third step: Calculate the transformation matrix T that transforms the state ## equation in the controllable canonical form. ## first we must calculate the controllability matrix M: M = ctrb (A, B); ## second, construct the matrix W PCO = PC(nx:-1:1); PC1 = PCO; # Matrix to shift and create W row by row for n = 1:nx W(n,:) = PC1; PC1 = ; endfor T = M*W; ## finaly the matrix K is calculated PD = poly (P); # The desired characteristic polynomial PD = PD(nx+1:-1:2); PC = PC(nx+1:-1:2); K = (PD-PC)/T; ## Check if the eigenvalues of (A-BK) are the same specified in P Pcalc = eig (A-B*K); Pcalc = sortcom (Pcalc); P = sortcom (P); if (max ((abs(Pcalc)-abs(P))./abs(P) ) > 0.1) warning ("place: Pole placed at more than 10% relative error from specified"); endif endfunction %!shared A, B, C, P, Kexpected %! A = ; %! B = ; %! C = ; # C is useful to use ss; it doesn't matter what the value of C is %! P = [-1 -0.5]; %! Kexpected = [3.5 3.5]; %!assert (place (ss (A, B, C), P), Kexpected); %!assert (place (A, B, P), Kexpected);

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