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1 ## Copyright (C) 1993, 1994, 1995 Auburn University. All rights reserved. |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by the |
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7 ## Free Software Foundation; either version 2, or (at your option) any |
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8 ## later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but WITHOUT |
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11 ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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12 ## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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13 ## for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, write to the Free |
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17 ## Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{retval}, @var{u}] =} is_stabilizable (@var{sys}, @var{tol}) |
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21 ## @deftypefnx {Function File} {[@var{retval}, @var{u}] =} is_stabilizable (@var{a}, @var{b}, @var{tol}) |
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22 ## Logical check for system stabilizability (i.e., all unstable modes are controllable). |
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23 ## |
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24 ## Test for stabilizability is performed via an ordered Schur decomposition |
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25 ## that reveals the unstable subspace of the system @var{a} matrix. |
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26 ## |
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27 ## Returns @code{retval} = 1 if the system, @var{a}, is stabilizable, |
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28 ## if the pair (@var{a}, @var{b}) is stabilizable, or 0 if not. |
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29 ## @var{u} = orthogonal basis of controllable subspace. |
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30 ## |
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31 ## Controllable subspace is determined by applying Arnoldi iteration with |
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32 ## complete re-orthogonalization to obtain an orthogonal basis of the |
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33 ## Krylov subspace. |
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34 ## @example |
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35 ## span ([b,a*b,...,a^ b]). |
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36 ## @end example |
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37 ## tol is a roundoff paramter, set to 200*eps if omitted. |
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38 ## @end deftypefn |
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39 |
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40 ## See also: size, rows, columns, length, ismatrix, isscalar, isvector |
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41 ## is_observable, is_stabilizable, is_detectable |
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42 |
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43 ## Author: A. S. Hodel <a.s.hodel@eng.auburn.edu> |
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44 ## Created: August 1993 |
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45 ## Updated by A. S. Hodel (scotte@eng.auburn.edu) Aubust, 1995 to use krylovb |
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46 ## Updated by John Ingram (ingraje@eng.auburn.edu) July, 1996 to accept systems |
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47 |
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48 function [retval, U] = is_stabilizable (a, b, tol) |
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49 |
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50 if(nargin < 1) usage("[retval,U] = is_stabilizable(a {, b ,tol})"); |
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51 elseif(isstruct(a)) |
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52 ## sustem passed. |
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53 if(nargin == 2) |
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54 tol = b; % get tolerance |
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55 elseif(nargin > 2) |
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56 usage("[retval,U] = is_stabilizable(sys{,tol})"); |
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57 endif |
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58 [a,b] = sys2ss(a); |
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59 else |
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60 ## a,b arguments sent directly. |
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61 if(nargin > 3) |
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62 usage("[retval,U] = is_stabilizable(a {, b ,tol})"); |
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63 endif |
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64 endif |
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65 |
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66 if(exist("tol")) |
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67 [retval,U] = is_controllable(a,b,tol); |
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68 else |
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69 [retval,U] = is_controllable(a,b); |
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70 tol = 1e2*rows(b)*eps; |
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71 endif |
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72 |
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73 if( !retval & columns(U) > 0) |
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74 ## now use an ordered Schur decomposition to get an orthogonal |
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75 ## basis of the unstable subspace... |
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76 n = rows(a); |
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77 [ua, s] = schur (-(a+eye(n)*tol), "A"); |
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78 k = sum( real(eig(a)) >= 0 ); # count unstable poles |
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79 |
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80 if( k > 0 ) |
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81 ua = ua(:,1:k); |
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82 ## now see if span(ua) is contained in span(U) |
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83 retval = (norm(ua - U*U'*ua) < tol); |
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84 else |
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85 retval = 1; # all poles stable |
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86 endif |
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87 endif |
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88 |
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89 endfunction |
