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  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 ## Copyright (C) 2011 Lukas F. Reichlin ## ## This file is part of LTI Syncope. ## ## LTI Syncope 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. ## ## LTI Syncope 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 LTI Syncope. If not, see . ## -*- texinfo -*- ## @deftypefn{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{opt}, @dots{}) ## @deftypefnx{Function File} {[@var{Gr}, @var{info}] =} spamodred (@var{G}, @var{nr}, @var{opt}, @dots{}) ## ## Model order reduction by frequency weighted Singular Perturbation Approximation (SPA). ## The aim of model reduction is to find an @acronym{LTI} system @var{Gr} of order ## @var{nr} (nr < n) such that the input-output behaviour of @var{Gr} ## approximates the one from original system @var{G}. ## ## SPA is an absolute error method which tries to minimize ## @iftex ## @tex ## $$|| G - G_r ||_{\\infty} = min$$ ## $$|| V \\ (G - G_r) \\ W ||_{\\infty} = min$$ ## @end tex ## @end iftex ## @ifnottex ## @example ## ||G-Gr|| = min ## inf ## ## ||V (G-Gr) W|| = min ## inf ## @end example ## @end ifnottex ## where @var{V} and @var{W} denote output and input weightings. ## ## ## @strong{Inputs} ## @table @var ## @item G ## @acronym{LTI} model to be reduced. ## @item nr ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically according ## to the description of key @var{'order'}. ## @item @dots{} ## Optional pairs of keys and values. @code{"key1", value1, "key2", value2}. ## @item opt ## Optional struct with keys as field names. ## Struct @var{opt} can be created directly or ## by command @command{options}. @code{opt.key1 = value1, opt.key2 = value2}. ## @end table ## ## @strong{Outputs} ## @table @var ## @item Gr ## Reduced order state-space model. ## @item info ## Struct containing additional information. ## @table @var ## @item info.n ## The order of the original system @var{G}. ## @item info.ns ## The order of the @var{alpha}-stable subsystem of the original system @var{G}. ## @item info.hsv ## The Hankel singular values of the @var{alpha}-stable part of ## the original system @var{G}, ordered decreasingly. ## @item info.nu ## The order of the @var{alpha}-unstable subsystem of both the original ## system @var{G} and the reduced-order system @var{Gr}. ## @item info.nr ## The order of the obtained reduced order system @var{Gr}. ## @end table ## @end table ## ## ## @strong{Option Keys and Values} ## @table @var ## @item 'order', 'nr' ## The desired order of the resulting reduced order system @var{Gr}. ## If not specified, @var{nr} is chosen automatically such that states with ## Hankel singular values @var{info.hsv} > @var{tol1} are retained. ## ## @item 'left', 'output' ## @acronym{LTI} model of the left/output frequency weighting @var{V}. ## Default value is an identity matrix. ## ## @item 'right', 'input' ## @acronym{LTI} model of the right/input frequency weighting @var{W}. ## Default value is an identity matrix. ## ## @item 'method' ## Approximation method for the L-infinity norm to be used as follows: ## @table @var ## @item 'sr', 's' ## Use the square-root Singular Perturbation Approximation method. ## @item 'bfsr', 'p' ## Use the balancing-free square-root Singular Perturbation Approximation method. Default method. ## @end table ## ## @item 'alpha' ## Specifies the ALPHA-stability boundary for the eigenvalues ## of the state dynamics matrix @var{G.A}. For a continuous-time ## system, ALPHA <= 0 is the boundary value for ## the real parts of eigenvalues, while for a discrete-time ## system, 0 <= ALPHA <= 1 represents the ## boundary value for the moduli of eigenvalues. ## The ALPHA-stability domain does not include the boundary. ## Default value is 0 for continuous-time systems and ## 1 for discrete-time systems. ## ## @item 'tol1' ## If @var{'order'} is not specified, @var{tol1} contains the tolerance for ## determining the order of the reduced model. ## For model reduction, the recommended value of @var{tol1} is ## c*info.hsv(1), where c lies in the interval [0.00001, 0.001]. ## Default value is info.ns*eps*info.hsv(1). ## If @var{'order'} is specified, the value of @var{tol1} is ignored. ## ## @item 'tol2' ## The tolerance for determining the order of a minimal ## realization of the ALPHA-stable part of the given ## model. TOL2 <= TOL1. ## If not specified, ns*eps*info.hsv(1) is chosen. ## ## @item 'gram-ctrb' ## Specifies the choice of frequency-weighted controllability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'gram-obsv' ## Specifies the choice of frequency-weighted observability ## Grammian as follows: ## @table @var ## @item 'standard' ## Choice corresponding to a combination method [4] ## of the approaches of Enns [1] and Lin-Chiu [2,3]. Default method. ## @item 'enhanced' ## Choice corresponding to the stability enhanced ## modified combination method of [4]. ## @end table ## ## @item 'alpha-ctrb' ## Combination method parameter for defining the ## frequency-weighted controllability Grammian. ## abs(alphac) <= 1. ## If alphac = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphac = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'alpha-obsv' ## Combination method parameter for defining the ## frequency-weighted observability Grammian. ## abs(alphao) <= 1. ## If alphao = 0, the choice of ## Grammian corresponds to the method of Enns [1], while if ## alphao = 1, the choice of Grammian corresponds ## to the method of Lin and Chiu [2,3]. ## Default value is 0. ## ## @item 'equil', 'scale' ## Boolean indicating whether equilibration (scaling) should be ## performed on system @var{G} prior to order reduction. ## Default value is true if @code{G.scaled == false} and ## false if @code{G.scaled == true}. ## Note that for @acronym{MIMO} models, proper scaling of both inputs and outputs ## is of utmost importance. The input and output scaling can @strong{not} ## be done by the equilibration option or the @command{prescale} command ## because these functions perform state transformations only. ## Furthermore, signals should not be scaled simply to a certain range. ## For all inputs (or outputs), a certain change should be of the same ## importance for the model. ## @end table ## ## ## @strong{References}@* ## [1] Enns, D. ## @cite{Model reduction with balanced realizations: An error bound ## and a frequency weighted generalization}. ## Proc. 23-th CDC, Las Vegas, pp. 127-132, 1984. ## ## [2] Lin, C.-A. and Chiu, T.-Y. ## @cite{Model reduction via frequency-weighted balanced realization}. ## Control Theory and Advanced Technology, vol. 8, ## pp. 341-351, 1992. ## ## [3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. ## @cite{New results on frequency weighted balanced reduction ## technique}. ## Proc. ACC, Seattle, Washington, pp. 4004-4009, 1995. ## ## [4] Varga, A. and Anderson, B.D.O. ## @cite{Square-root balancing-free methods for the frequency-weighted ## balancing related model reduction}. ## (report in preparation) ## ## ## @strong{Algorithm}@* ## Uses SLICOT AB09ID by courtesy of ## @uref{http://www.slicot.org, NICONET e.V.} ## @end deftypefn ## Author: Lukas Reichlin ## Created: November 2011 ## Version: 0.1 function [Gr, info] = spamodred (varargin) [Gr, info] = __modred_ab09id__ ("spa", varargin{:}); endfunction ## TODO: add a test 

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