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1 # Copyright (C) 1996 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 function [zer, gain] = tzero(A,B,C,D) |
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20 # [zer{,gain}] = tzero(A,B,C,D) -or- |
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21 # [zer{,gain}] = tzero(Asys) |
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22 # Compute transmission zeros of a continuous |
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23 # . |
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24 # x = Ax + Bu |
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25 # y = Cx + Du |
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26 # |
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27 # or discrete |
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28 # x(k+1) = A x(k) + B u(k) |
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29 # y(k) = C x(k) + D u(k) |
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30 # |
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31 # system. |
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32 # |
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33 # outputs: |
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34 # zer: transmission zeros of the system |
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35 # gain: leading coefficient (pole-zero form) of SISO transfer function |
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36 # returns gain=0 if system is multivariable |
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37 # References: |
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38 # Hodel, "Computation of Zeros with Balancing," 1992 Lin. Alg. Appl. |
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39 |
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40 # R. Bruce Tenison July 4, 1994 |
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41 # A. S. Hodel Aug 1995: allow for MIMO and system data structures |
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245
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42 |
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43 # get A,B,C,D and Asys variables, regardless of initial form |
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44 if(nargin == 4) |
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45 Asys = ss2sys(A,B,C,D); |
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46 elseif( (nargin == 1) && (! is_struct(A))) |
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47 usage("[zer,gain] = tzero(A,B,C,D) or zer = tzero(Asys)"); |
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48 elseif(nargin != 1) |
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49 usage("[zer,gain] = tzero(A,B,C,D) or zer = tzero(Asys)"); |
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50 else |
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51 Asys = A; |
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52 [A,B,C,D] = sys2ss(Asys); |
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53 endif |
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54 |
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55 Ao = Asys; # save for leading coefficient |
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56 siso = is_siso(Asys); |
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57 digital = is_digital(Asys); # check if it's mixed or not |
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58 |
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59 # see if it's a gain block |
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60 if(isempty(A)) |
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61 zer = []; |
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62 gain = D; |
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63 return; |
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64 endif |
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65 |
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66 # First, balance the system via the zero computation generalized eigenvalue |
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67 # problem balancing method (Hodel and Tiller, Linear Alg. Appl., 1992) |
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68 |
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69 Asys = zgpbal(Asys); [A,B,C,D] = sys2ss(Asys); # balance coefficients |
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70 meps = 2*eps*norm([A, B; C, D],'fro'); |
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71 Asys = zgreduce(Asys,meps); [A, B, C, D] = sys2ss(Asys); # ENVD algorithm |
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72 if(!isempty(A)) |
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73 # repeat with dual system |
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74 Asys = ss2sys(A', C', B', D'); Asys = zgreduce(Asys,meps); |
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75 |
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76 # transform back |
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77 [A,B,C,D] = sys2ss(Asys); Asys = ss2sys(A', C', B', D'); |
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78 endif |
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79 |
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80 zer = []; # assume none |
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81 [A,B,C,D] = sys2ss(Asys); |
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82 if( !isempty(C) ) |
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83 [W,r,Pi] = qr([C, D]'); |
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84 [nonz,ztmp] = zgrownorm(r,meps); |
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85 if(nonz) |
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86 # We can now solve the generalized eigenvalue problem. |
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87 [pp,mm] = size(D); |
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88 nn = rows(A); |
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89 Afm = [A , B ; C, D] * W'; |
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90 Bfm = [eye(nn), zeros(nn,mm); zeros(pp,nn+mm)]*W'; |
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91 |
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92 jdx = (mm+1):(mm+nn); |
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93 Af = Afm(1:nn,jdx); |
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94 Bf = Bfm(1:nn,jdx); |
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95 zer = qz(Af,Bf); |
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96 endif |
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97 endif |
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98 |
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99 mz = length(zer); |
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100 [A,B,C,D] = sys2ss(Ao); # recover original system |
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101 #compute leading coefficient |
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102 if ( (nargout == 2) && siso) |
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103 n = rows(A); |
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104 if ( mz == n) |
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105 gain = D; |
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106 elseif ( mz < n ) |
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107 gain = C*(A^(n-1-mz))*B; |
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108 endif |
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109 else |
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110 gain = []; |
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111 endif |
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112 endfunction |
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113 |
