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1 # Copyright (C) 1993, 1994, 1995 John W. Eaton |
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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, 675 Mass Ave, Cambridge, MA 02139, USA. |
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18 |
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19 function [retval,U] = is_stabilizable (a, b, tol) |
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20 |
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21 # Usage: [retval,U] = is_stabilizable (a {, b, tol}) |
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22 # |
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23 # Returns retval = 1 if the system, a, is stabilizable, if the pair (a, b) is |
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24 # stabilizable, or 0 if not. |
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25 # U = orthogonal basis of controllable subspace. |
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26 # |
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27 # Controllable subspace is determined by applying Arnoldi iteration with |
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28 # complete re-orthogonalization to obtain an orthogonal basis of the |
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29 # Krylov subspace. |
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30 # |
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31 # span ([b,a*b,...,a^ b]). |
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32 # |
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33 # tol is a roundoff paramter, set to 200*eps if omitted. |
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34 # |
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35 # See also: size, rows, columns, length, is_matrix, is_scalar, is_vector |
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36 # is_observable, is_stabilizable, is_detectable |
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37 |
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38 # Written by A. S. Hodel (scotte@eng.auburn.edu) August, 1993. |
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39 # Updated by A. S. Hodel (scotte@eng.auburn.edu) Aubust, 1995 to use krylovb |
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40 # Updated by John Ingram (ingraje@eng.auburn.edu) July, 1996 to accept systems |
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41 # $Revision: 2.0.0.0 $ |
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42 |
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43 if(nargin < 1) usage("[retval,U] = is_stabilizable(a {, b ,tol})"); |
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44 elseif(is_struct(a)) |
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45 # sustem passed. |
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46 if(nargin == 2) |
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47 tol = b; % get tolerance |
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48 elseif(nargin > 2) |
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49 usage("[retval,U] = is_stabilizable(sys{,tol})"); |
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50 endif |
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51 [a,b] = sys2ss(sys); |
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52 else |
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53 # a,b arguments sent directly. |
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54 if(nargin > 3) |
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55 usage("[retval,U] = is_stabilizable(a {, b ,tol})"); |
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56 endif |
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57 endif |
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58 |
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59 if(exist("tol")) |
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60 [retval,U] = is_controllable(a,b,tol); |
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61 else |
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62 [retval,U] = is_controllable(a,b); |
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63 tol = 1e2*rows(b)*eps; |
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64 endif |
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65 |
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66 if( !retval & columns(U) > 0) |
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67 # now use an ordered Schur decomposition to get an orthogonal |
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68 # basis of the unstable subspace... |
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69 n = rows(a); |
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70 [ua,s] = schur(-(a+eye(n)*tol),'A'); |
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71 k = sum( real(eig(a)) >= 0 ); # count unstable poles |
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72 |
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73 if( k > 0 ) |
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74 ua = ua(:,1:k); |
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75 # now see if span(ua) is contained in span(U) |
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76 retval = (norm(ua - U*U'*ua) < tol); |
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77 else |
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78 retval = 1; # all poles stable |
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79 endif |
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80 endif |
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81 |
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82 endfunction |
