[731] | 1 | #LyX 1.6.4 created this file. For more info see http://www.lyx.org/ |
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| 2 | \lyxformat 345 |
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| 3 | \begin_document |
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| 4 | \begin_header |
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| 5 | \textclass scrartcl |
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| 6 | \begin_preamble |
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| 7 | \newcommand\blabl{} |
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| 8 | \end_preamble |
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| 9 | \use_default_options false |
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| 10 | \begin_modules |
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| 11 | theorems-ams |
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| 12 | theorems-ams-extended |
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| 13 | \end_modules |
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| 14 | \language english |
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| 15 | \inputencoding auto |
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| 16 | \font_roman lmodern |
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| 40 | \use_amsmath 1 |
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| 44 | \paperorientation portrait |
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| 45 | \secnumdepth 3 |
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| 46 | \tocdepth 3 |
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| 47 | \paragraph_separation indent |
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| 48 | \defskip medskip |
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| 49 | \quotes_language english |
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| 50 | \papercolumns 1 |
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| 51 | \papersides 1 |
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| 52 | \paperpagestyle default |
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| 53 | \tracking_changes false |
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| 54 | \output_changes false |
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| 55 | \author "" |
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| 56 | \author "" |
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| 57 | \end_header |
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| 58 | |
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| 59 | \begin_body |
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| 60 | |
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| 61 | \begin_layout Standard |
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| 62 | \begin_inset FormulaMacro |
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| 63 | \newcommand{\isa}[1]{i_{\alpha#1}} |
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| 64 | {i_{\alpha#1}} |
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| 65 | \end_inset |
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| 66 | |
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| 67 | |
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| 68 | \begin_inset FormulaMacro |
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| 69 | \newcommand{\isb}[1]{i_{\beta#1}} |
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| 70 | {i_{\beta#1}} |
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| 71 | \end_inset |
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| 72 | |
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| 73 | |
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| 74 | \begin_inset FormulaMacro |
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| 75 | \newcommand{\Dt}{\Delta t} |
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| 76 | {\Delta t} |
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| 77 | \end_inset |
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| 78 | |
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| 79 | |
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| 80 | \begin_inset FormulaMacro |
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| 81 | \newcommand{\om}{\omega} |
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| 82 | {\omega} |
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| 83 | \end_inset |
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| 84 | |
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| 85 | |
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| 86 | \begin_inset FormulaMacro |
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| 87 | \newcommand{\th}{\vartheta} |
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| 88 | {\vartheta} |
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| 89 | \end_inset |
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| 90 | |
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| 91 | |
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| 92 | \begin_inset FormulaMacro |
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| 93 | \newcommand{\usa}[1]{u_{\alpha#1}} |
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| 94 | {u_{\alpha#1}} |
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| 95 | \end_inset |
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| 96 | |
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| 97 | |
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| 98 | \begin_inset FormulaMacro |
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| 99 | \newcommand{\usb}[1]{u_{\beta#1}} |
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| 100 | {u_{\beta#1}} |
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| 101 | \end_inset |
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| 102 | |
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| 103 | |
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| 104 | \end_layout |
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| 105 | |
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| 106 | \begin_layout Title |
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| 107 | PMSM system description |
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| 108 | \end_layout |
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| 109 | |
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| 110 | \begin_layout Section |
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| 111 | Model of PMSM Drive |
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| 112 | \end_layout |
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| 113 | |
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| 114 | \begin_layout Standard |
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| 115 | Permanent magnet synchronous machine (PMSM) drive with surface magnets on |
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| 116 | the rotor is described by conventional equations of PMSM in the stationary |
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| 117 | reference frame: |
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| 118 | \begin_inset Formula \begin{align} |
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| 119 | \frac{d\isa{}}{dt} & =-\frac{R_{s}}{L_{s}}\isa{}+\frac{\Psi_{PM}}{L_{s}}\omega_{me}\sin\th+\frac{\usa{}}{L_{s}},\nonumber \\ |
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| 120 | \frac{d\isb{}}{dt} & =-\frac{R_{s}}{L_{s}}\isb{}-\frac{\Psi_{PM}}{L_{s}}\omega_{me}\cos\th+\frac{\usb{}}{L_{s}},\label{eq:simulator}\\ |
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| 121 | \frac{d\om}{dt} & =\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}\left(\isb{}\cos(\th)-\isa{}\sin(\th)\right)-\frac{B}{J}\om-\frac{p_{p}}{J}T_{L},\nonumber \\ |
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| 122 | \frac{d\th}{dt} & =\omega_{me}.\nonumber \end{align} |
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| 123 | |
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| 124 | \end_inset |
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| 125 | |
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| 126 | Here, |
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| 127 | \begin_inset Formula $\isa{}$ |
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| 128 | \end_inset |
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| 129 | |
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| 130 | , |
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| 131 | \begin_inset Formula $\isb{}$ |
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| 132 | \end_inset |
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| 133 | |
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| 134 | , |
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| 135 | \begin_inset Formula $\usa{}$ |
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| 136 | \end_inset |
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| 137 | |
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| 138 | and |
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| 139 | \begin_inset Formula $\usb{}$ |
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| 140 | \end_inset |
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| 141 | |
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| 142 | represent stator current and voltage in the stationary reference frame, |
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| 143 | respectively; |
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| 144 | \begin_inset Formula $\om$ |
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| 145 | \end_inset |
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| 146 | |
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| 147 | is electrical rotor speed and |
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| 148 | \begin_inset Formula $\th$ |
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| 149 | \end_inset |
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| 150 | |
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| 151 | is electrical rotor position. |
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| 152 | |
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| 153 | \begin_inset Formula $R_{s}$ |
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| 154 | \end_inset |
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| 155 | |
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| 156 | and |
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| 157 | \begin_inset Formula $L_{s}$ |
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| 158 | \end_inset |
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| 159 | |
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| 160 | is stator resistance and inductance respectively, |
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| 161 | \begin_inset Formula $\Psi_{pm}$ |
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| 162 | \end_inset |
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| 163 | |
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| 164 | is the flux of permanent magnets on the rotor, |
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| 165 | \begin_inset Formula $B$ |
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| 166 | \end_inset |
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| 167 | |
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| 168 | is friction and |
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| 169 | \begin_inset Formula $T_{L}$ |
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| 170 | \end_inset |
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| 171 | |
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| 172 | is load torque, |
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| 173 | \begin_inset Formula $J$ |
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| 174 | \end_inset |
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| 175 | |
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| 176 | is moment of inertia, |
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| 177 | \begin_inset Formula $p_{p}$ |
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| 178 | \end_inset |
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| 179 | |
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| 180 | is the number of pole pairs, |
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| 181 | \begin_inset Formula $k_{p}$ |
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| 182 | \end_inset |
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| 183 | |
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| 184 | is the Park constant. |
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| 185 | \end_layout |
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| 186 | |
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| 187 | \begin_layout Standard |
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| 188 | The sensor-less control scenario arise when sensors of the speed and position |
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| 189 | ( |
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| 190 | \begin_inset Formula $\om$ |
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| 191 | \end_inset |
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| 192 | |
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| 193 | and |
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| 194 | \begin_inset Formula $\th$ |
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| 195 | \end_inset |
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| 196 | |
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| 197 | ) are missing (from various reasons). |
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| 198 | Then, the only observed variables are: |
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| 199 | \begin_inset Formula \begin{equation} |
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| 200 | y_{t}=\left[\begin{array}{c} |
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| 201 | \isa{}(t),\isb{}(t),\usa{}(t),\usb{}(t)\end{array}\right].\label{eq:obs}\end{equation} |
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| 202 | |
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| 203 | \end_inset |
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| 204 | |
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| 205 | Which are, however, observed only up to some precision. |
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| 206 | \end_layout |
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| 207 | |
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| 208 | \begin_layout Standard |
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| 209 | Discretization of the model ( |
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| 210 | \begin_inset CommandInset ref |
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| 211 | LatexCommand ref |
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| 212 | reference "eq:simulator" |
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| 213 | |
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| 214 | \end_inset |
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| 215 | |
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| 216 | ) was performed using Euler method with the following result: |
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| 217 | \begin_inset Formula \begin{align*} |
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| 218 | \isa{,t+1} & =(1-\frac{R_{s}}{L_{s}}\Dt)\isa{,t}+\frac{\Psi_{pm}}{L_{s}}\Dt\omega_{t}\sin\vartheta_{e,t}+\usa{,t}\frac{\Dt}{L_{s}},\\ |
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| 219 | \isb{,t+1} & =(1-\frac{R_{s}}{L_{s}}\Dt)\isb{,t}-\frac{\Psi_{pm}}{L_{s}}\Dt\omega_{t}\cos\vartheta_{t}+\usb{,t}\frac{\Dt}{L_{s}},\\ |
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| 220 | \om_{t+1} & =(1--\frac{B}{J}\Dt)\om_{t}+\Dt\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right)-\frac{p_{p}}{J}T_{L}\Dt,\\ |
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| 221 | \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\end{align*} |
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| 222 | |
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| 223 | \end_inset |
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| 224 | |
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| 225 | In this work, we consider parameters of the model known, we can make the |
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| 226 | following substitutions to simplify notation, |
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| 227 | \begin_inset Formula $a=1-\frac{R_{s}}{L_{s}}\Dt$ |
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| 228 | \end_inset |
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| 229 | |
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| 230 | , |
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| 231 | \begin_inset Formula $b=\frac{\Psi_{pm}}{L_{s}}\Dt$ |
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| 232 | \end_inset |
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| 233 | |
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| 234 | , |
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| 235 | \begin_inset Formula $c=\frac{\Dt}{L_{s}}$ |
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| 236 | \end_inset |
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| 237 | |
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| 238 | , |
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| 239 | \begin_inset Formula $d=1-\frac{B}{J}\Dt$ |
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| 240 | \end_inset |
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| 241 | |
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| 242 | , |
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| 243 | \family roman |
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| 244 | \series medium |
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| 245 | \shape up |
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| 246 | \size normal |
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| 247 | \emph off |
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| 248 | \bar no |
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| 249 | \noun off |
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| 250 | \color none |
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| 251 | |
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| 252 | \begin_inset Formula $e=\Dt\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}$ |
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| 253 | \end_inset |
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| 254 | |
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| 255 | , which results in a simplified model: |
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| 256 | \family default |
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| 257 | \series default |
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| 258 | \shape default |
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| 259 | \size default |
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| 260 | \emph default |
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| 261 | \bar default |
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| 262 | \noun default |
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| 263 | \color inherit |
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| 264 | |
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| 265 | \begin_inset Formula \begin{align} |
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| 266 | \isa{,t+1} & =a\,\isa{,t}+b\omega_{t}\sin\vartheta_{t}+c\usa{,t},\nonumber \\ |
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| 267 | \isb{,t+1} & =a\,\isb{,t}-b\omega_{t}\cos\vartheta_{t}+c\usb{,t},\label{eq:model-simple}\\ |
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| 268 | \om_{t+1} & =d\om_{t}+e\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right),\nonumber \\ |
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| 269 | \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\nonumber \end{align} |
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| 270 | |
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| 271 | \end_inset |
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| 272 | |
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| 273 | |
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| 274 | \end_layout |
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| 275 | |
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| 276 | \begin_layout Standard |
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| 277 | The above equations can be aggregated into state |
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| 278 | \begin_inset Formula $x_{t}=[\isa{,t},\isb{,t},\om_{t},\th_{t}]$ |
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| 279 | \end_inset |
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| 280 | |
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| 281 | will be denoted as |
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| 282 | \begin_inset Formula $x_{t+1}=g(x_{t},u_{t})$ |
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| 283 | \end_inset |
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| 284 | |
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| 285 | . |
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| 286 | \end_layout |
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| 287 | |
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| 288 | \begin_layout Subsection |
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| 289 | Gaussian model of disturbances |
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| 290 | \end_layout |
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| 291 | |
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| 292 | \begin_layout Standard |
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| 293 | This model is motivated by the well known Kalman filter, which is optimal |
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| 294 | for linear system with Gaussian noise. |
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| 295 | Hence, we model all disturbances to have covariance matrices |
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| 296 | \begin_inset Formula $Q_{t}$ |
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| 297 | \end_inset |
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| 298 | |
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| 299 | and |
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| 300 | \begin_inset Formula $R_{t}$ |
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| 301 | \end_inset |
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| 302 | |
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| 303 | for the state |
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| 304 | \begin_inset Formula $x_{t}$ |
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| 305 | \end_inset |
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| 306 | |
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| 307 | and observations |
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| 308 | \begin_inset Formula $y_{t}$ |
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| 309 | \end_inset |
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| 310 | |
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| 311 | respectively. |
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| 312 | \begin_inset Formula \begin{align*} |
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| 313 | x_{t+1} & \sim\mathcal{N}(g(x_{t}),Q_{t})\\ |
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| 314 | y_{t} & \sim\mathcal{N}([\isa{,t},\isb{,t}]',R_{t})\end{align*} |
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| 315 | |
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| 316 | \end_inset |
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| 317 | |
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| 318 | |
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| 319 | \end_layout |
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| 320 | |
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| 321 | \begin_layout Standard |
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| 322 | Under this assumptions, Bayesian estimation of the state, |
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| 323 | \begin_inset Formula $x_{t}$ |
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| 324 | \end_inset |
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| 325 | |
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| 326 | , can be approximated by so called Extended Kalman filter which approximates |
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| 327 | posterior density of the state by a Gaussian |
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| 328 | \begin_inset Formula \[ |
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| 329 | f(x_{t}|y_{1}\ldots y_{t})=\mathcal{N}(\hat{x}_{t},P_{t}).\] |
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| 330 | |
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| 331 | \end_inset |
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| 332 | |
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| 333 | Its sufficient statistics |
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| 334 | \begin_inset Formula $S_{t}=\left[\hat{x}_{t},P_{t}\right]$ |
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| 335 | \end_inset |
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| 336 | |
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| 337 | is evaluated recursively as follows: |
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| 338 | \begin_inset Formula \begin{eqnarray} |
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| 339 | \hat{x}_{t} & = & g(\hat{x}_{t-1})-K\left(y_{t}-h(\hat{x}_{t-1})\right).\label{eq:ekf_mean}\\ |
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[806] | 340 | R_{y} & = & CP_{t-1}C'+R_{t},\nonumber \\ |
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| 341 | K & = & P_{t-1}C'R_{y}^{-1},\nonumber \\ |
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[731] | 342 | P_{t} & = & A\left(P_{t-1}-P_{t-1}C'R_{y}^{-1}CP_{t-1}\right)A+Q_{t}.\label{eq:ekf_cov}\end{eqnarray} |
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| 343 | |
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| 344 | \end_inset |
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| 345 | |
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| 346 | where |
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| 347 | \begin_inset Formula $A=\frac{d}{dx_{t}}g(x_{t})$ |
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| 348 | \end_inset |
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| 349 | |
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| 350 | , |
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| 351 | \begin_inset Formula $C=\frac{d}{dx_{t}}h(x_{t})$ |
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| 352 | \end_inset |
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| 353 | |
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| 354 | , |
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| 355 | \begin_inset Formula $g(x_{t})$ |
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| 356 | \end_inset |
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| 357 | |
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| 358 | is model ( |
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| 359 | \begin_inset CommandInset ref |
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| 360 | LatexCommand ref |
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| 361 | reference "eq:model-simple" |
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| 362 | |
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| 363 | \end_inset |
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| 364 | |
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| 365 | ) and |
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| 366 | \begin_inset Formula $h(x_{t})$ |
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| 367 | \end_inset |
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| 368 | |
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| 369 | direct observation of |
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| 370 | \begin_inset Formula $y_{t}=[\isa{,t},\isb{,t}]$ |
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| 371 | \end_inset |
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| 372 | |
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| 373 | , i.e. |
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| 374 | \begin_inset Formula \[ |
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| 375 | A=\left[\begin{array}{cccc} |
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| 376 | a & 0 & b\sin\th & b\om\cos\th\\ |
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| 377 | 0 & a & -b\cos\th & b\om\sin\th\\ |
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| 378 | -e\sin\th & e\cos\th & d & -e(\isb{}\sin\th+\isa{}\cos\th)\\ |
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| 379 | 0 & 0 & \Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cccc} |
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| 380 | 1\\ |
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[806] | 381 | & 1\end{array}\right]\] |
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[731] | 382 | |
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| 383 | \end_inset |
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| 384 | |
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| 385 | |
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| 386 | \end_layout |
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| 387 | |
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| 388 | \begin_layout Standard |
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| 389 | Covariance matrices of the system |
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| 390 | \begin_inset Formula $Q$ |
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| 391 | \end_inset |
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| 392 | |
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| 393 | and |
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| 394 | \begin_inset Formula $R$ |
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| 395 | \end_inset |
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| 396 | |
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| 397 | are supposed to be known. |
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| 398 | \end_layout |
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| 399 | |
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[749] | 400 | \begin_layout Subsection |
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| 401 | Test system |
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| 402 | \end_layout |
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| 403 | |
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| 404 | \begin_layout Standard |
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| 405 | A real PMSM system on which the algorithms will be tested has parameters: |
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| 406 | \end_layout |
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| 407 | |
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| 408 | \begin_layout Standard |
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| 409 | \begin_inset Formula \begin{eqnarray*} |
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| 410 | R_{s} & = & 0.28;\\ |
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| 411 | L_{s} & = & 0.003465;\\ |
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| 412 | \Psi_{pm} & = & 0.1989;\\ |
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| 413 | k_{p} & = & 1.5\\ |
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| 414 | p & = & 4.0;\\ |
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| 415 | J & = & 0.04;\\ |
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| 416 | \Delta t & = & 0.000125\end{eqnarray*} |
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| 417 | |
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| 418 | \end_inset |
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| 419 | |
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| 420 | which yields |
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| 421 | \begin_inset Formula \begin{eqnarray*} |
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| 422 | a & = & 0.9898\\ |
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| 423 | b & = & 0.0072\\ |
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| 424 | c & = & 0.0361\\ |
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| 425 | d & = & 1\\ |
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| 426 | e & = & 0.0149\end{eqnarray*} |
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| 427 | |
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| 428 | \end_inset |
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| 429 | |
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| 430 | The covaraince matrices |
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| 431 | \begin_inset Formula $Q$ |
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| 432 | \end_inset |
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| 433 | |
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| 434 | and |
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| 435 | \begin_inset Formula $R$ |
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| 436 | \end_inset |
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| 437 | |
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| 438 | are assumed to be known. |
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| 439 | For the initial tests, we can use the following values: |
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| 440 | \end_layout |
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| 441 | |
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| 442 | \begin_layout Standard |
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| 443 | \begin_inset Formula \begin{eqnarray*} |
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| 444 | Q & = & \mathrm{diag}(0.0013,0.0013,5e-6,1e-10),\\ |
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| 445 | R & = & \mathrm{diag}(0.0006,0.0006).\end{eqnarray*} |
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| 446 | |
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| 447 | \end_inset |
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| 448 | |
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| 449 | |
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| 450 | \end_layout |
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| 451 | |
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[806] | 452 | \begin_layout Standard |
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| 453 | Limits: |
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| 454 | \begin_inset Formula \begin{eqnarray*} |
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| 455 | u_{\alpha,max} & = & 50V,\\ |
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| 456 | u_{\beta,max} & = & 50V.\end{eqnarray*} |
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| 457 | |
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| 458 | \end_inset |
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| 459 | |
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| 460 | |
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| 461 | \end_layout |
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| 462 | |
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[731] | 463 | \begin_layout Section |
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| 464 | Control |
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| 465 | \end_layout |
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| 466 | |
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| 467 | \begin_layout Standard |
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| 468 | The task is to reach predefined speed |
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| 469 | \begin_inset Formula $\overline{\omega}_{t}$ |
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| 470 | \end_inset |
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| 471 | |
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| 472 | . |
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| 473 | \end_layout |
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| 474 | |
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| 475 | \begin_layout Standard |
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| 476 | For simplicity, we will assume additive loss function: |
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| 477 | \begin_inset Formula \[ |
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| 478 | l(x_{t},u_{t})=(\omega_{t}-\overline{\omega}_{t})^{2}+\psi(\usa{,t}^{2}+\usb{,t}^{2}).\] |
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| 479 | |
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| 480 | \end_inset |
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| 481 | |
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| 482 | Here, |
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| 483 | \begin_inset Formula $\psi$ |
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| 484 | \end_inset |
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| 485 | |
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[806] | 486 | is the chosen penalization of the inputs, in this case |
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| 487 | \begin_inset Formula $\psi=0$ |
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| 488 | \end_inset |
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| 489 | |
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| 490 | . |
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[731] | 491 | |
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| 492 | \end_layout |
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| 493 | |
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| 494 | \begin_layout Standard |
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| 495 | Following the standard dynamic programming approach, optimization of the |
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| 496 | loss function can be done recursively, as follows: |
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| 497 | \begin_inset Formula \[ |
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| 498 | V(x_{t-1},u_{t-1})=\arg\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(x_{t},u_{t})\right\} ,\] |
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| 499 | |
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| 500 | \end_inset |
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| 501 | |
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| 502 | where |
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| 503 | \begin_inset Formula $V(x_{t},u_{t})$ |
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| 504 | \end_inset |
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| 505 | |
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| 506 | is the Bellman function. |
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| 507 | Since the model evolution is stochastic, we can reformulate it in terms |
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| 508 | of sufficient statistics, |
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| 509 | \begin_inset Formula $S$ |
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| 510 | \end_inset |
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| 511 | |
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| 512 | as follows: |
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| 513 | \begin_inset Formula \[ |
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| 514 | V(S_{t-1})=\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(S_{t})\right\} .\] |
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| 515 | |
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| 516 | \end_inset |
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| 517 | |
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| 518 | |
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| 519 | \end_layout |
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| 520 | |
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| 521 | \begin_layout Standard |
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| 522 | Representation of the Bellman function depends on chosen approximation. |
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| 523 | \end_layout |
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| 524 | |
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[751] | 525 | \begin_layout Subsection |
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| 526 | Test Scenarios |
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| 527 | \end_layout |
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| 528 | |
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[731] | 529 | \begin_layout Standard |
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[751] | 530 | With almost full information, design of the control strategy should be almost |
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| 531 | trivial: |
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| 532 | \begin_inset Formula \begin{eqnarray*} |
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| 533 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
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| 534 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,0.01]).\end{eqnarray*} |
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| 535 | |
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| 536 | \end_inset |
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| 537 | |
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| 538 | |
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| 539 | \end_layout |
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| 540 | |
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| 541 | \begin_layout Standard |
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[753] | 542 | The difficulty arise with growing initial covariance matrix: |
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[751] | 543 | \begin_inset Formula \begin{eqnarray*} |
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| 544 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
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[806] | 545 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,1]).\end{eqnarray*} |
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[751] | 546 | |
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| 547 | \end_inset |
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| 548 | |
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| 549 | |
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| 550 | \end_layout |
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| 551 | |
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| 552 | \begin_layout Standard |
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| 553 | Or even worse: |
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| 554 | \begin_inset Formula \begin{eqnarray*} |
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| 555 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
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[806] | 556 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,10]).\end{eqnarray*} |
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[751] | 557 | |
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| 558 | \end_inset |
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| 559 | |
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| 560 | |
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[753] | 561 | \end_layout |
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| 562 | |
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| 563 | \begin_layout Standard |
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| 564 | The requested value |
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[806] | 565 | \begin_inset Formula $\overline{\omega}_{t}=1.0015.$ |
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[753] | 566 | \end_inset |
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| 567 | |
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| 568 | |
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[806] | 569 | \end_layout |
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| 570 | |
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| 571 | \begin_layout Conjecture |
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| 572 | It is sufficient to consider hyper-state |
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| 573 | \begin_inset Formula $H=[\hat{i}_{\alpha},\hat{i}_{\beta},\hat{\omega},\hat{\vartheta},P(3,3),P(4,4)]$ |
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| 574 | \end_inset |
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| 575 | |
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| 576 | . |
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| 577 | \end_layout |
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| 578 | |
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| 579 | \begin_layout Conjecture |
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[731] | 580 | \begin_inset CommandInset bibtex |
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| 581 | LatexCommand bibtex |
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| 582 | bibfiles "bibtex/vs,bibtex/vs-world,bibtex/world_classics,bibtex/world,new_bib_PS" |
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| 583 | options "plain" |
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| 584 | |
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| 585 | \end_inset |
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| 586 | |
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| 587 | |
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| 588 | \end_layout |
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| 589 | |
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| 590 | \end_body |
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| 591 | \end_document |
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