[1244] | 1 | #LyX 1.6.7 created this file. For more info see http://www.lyx.org/ |
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[731] | 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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| 42 | \cite_engine basic |
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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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[807] | 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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[731] | 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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[1288] | 289 | Transformation to d-q coordinates |
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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 | For many applications, it is advantageous to consider altervative coordinate |
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| 294 | system denoted d-q as follows |
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| 295 | \begin_inset Formula \begin{eqnarray*} |
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| 296 | \left[\begin{array}{c} |
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| 297 | d\\ |
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| 298 | q\end{array}\right] & = & \left[\begin{array}{cc} |
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| 299 | \cos\vartheta & \sin\vartheta\\ |
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| 300 | -\sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} |
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| 301 | \alpha\\ |
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| 302 | \beta\end{array}\right]\\ |
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| 303 | \left[\begin{array}{c} |
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| 304 | \alpha\\ |
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| 305 | \beta\end{array}\right] & = & \left[\begin{array}{cc} |
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| 306 | \cos\vartheta & -\sin\vartheta\\ |
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| 307 | \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} |
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| 308 | d\\ |
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| 309 | q\end{array}\right]\end{eqnarray*} |
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| 310 | |
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| 311 | \end_inset |
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| 312 | |
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| 313 | Under this transformation, the whole model ( |
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| 314 | \begin_inset CommandInset ref |
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| 315 | LatexCommand ref |
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| 316 | reference "eq:model" |
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| 317 | |
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| 318 | \end_inset |
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| 319 | |
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| 320 | ) can be transformed into d-q coordinates. |
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| 321 | \end_layout |
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| 322 | |
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| 323 | \begin_layout Standard |
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| 324 | In this text, we will transform only one single quantity, |
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| 325 | \begin_inset Formula $L_{d}$ |
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| 326 | \end_inset |
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| 327 | |
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| 328 | and |
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| 329 | \begin_inset Formula $L_{q}$ |
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| 330 | \end_inset |
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| 331 | |
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| 332 | for which it holds |
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| 333 | \begin_inset Formula $L_{d}=kL_{q}$ |
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| 334 | \end_inset |
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| 335 | |
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| 336 | . |
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| 337 | Then, |
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| 338 | \begin_inset Formula \begin{eqnarray*} |
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| 339 | \left[\begin{array}{c} |
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| 340 | L_{\alpha}\\ |
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| 341 | L_{\beta}\end{array}\right] & = & \left[\begin{array}{cc} |
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| 342 | \cos\vartheta & -\sin\vartheta\\ |
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| 343 | \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} |
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| 344 | L_{d}\\ |
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| 345 | L_{q}\end{array}\right].\\ |
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| 346 | & = & L_{d}\left[\begin{array}{cc} |
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| 347 | \cos\vartheta & -\sin\vartheta\\ |
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| 348 | \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} |
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| 349 | k\\ |
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| 350 | 1\end{array}\right]\\ |
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| 351 | & = & L\left[\begin{array}{cc} |
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| 352 | k\cos\vartheta & -\sin\vartheta\\ |
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| 353 | k\sin\vartheta & \cos\vartheta\end{array}\right]=L\left[\begin{array}{c} |
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| 354 | k_{c\vartheta}\\ |
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| 355 | k_{s\vartheta}\end{array}\right].\end{eqnarray*} |
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| 356 | |
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| 357 | \end_inset |
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| 358 | |
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| 359 | Then, model of the drive is changed to |
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| 360 | \begin_inset Formula \begin{align*} |
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| 361 | \isa{,t+1} & =(1-\frac{R_{s}}{L_{s}k_{c\vartheta}}\Dt)\isa{,t}+\frac{\Psi_{pm}}{L_{s}k_{c\vartheta}}\Dt\omega_{t}\sin\vartheta_{e,t}+\usa{,t}\frac{\Dt}{L_{s}k_{c\vartheta}},\\ |
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| 362 | \isb{,t+1} & =(1-\frac{R_{s}}{L_{s}k_{s\vartheta}}\Dt)\isb{,t}-\frac{\Psi_{pm}}{L_{s}k_{s\vartheta}}\Dt\omega_{t}\cos\vartheta_{t}+\usb{,t}\frac{\Dt}{L_{s}k_{s\vartheta}},\\ |
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| 363 | \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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| 364 | \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\end{align*} |
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| 365 | |
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| 366 | \end_inset |
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| 367 | |
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| 368 | Transformation to full d-q |
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| 369 | \begin_inset Formula \begin{eqnarray*} |
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| 370 | i_{d,t+1} & = & (1-\frac{R_{s}}{L_{d}}\Dt)i_{d,t}+\frac{L_{q}}{L_{d}}i_{q,t}\Dt\omega_{t}+u_{d,t}\frac{\Dt}{L_{d}},\\ |
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| 371 | i_{q,t+1} & = & -\frac{L_{d}}{L_{q}}\Dt\omega_{t}i_{d,t}+(1-\frac{R_{s}}{L_{q}}\Dt)i_{q,t}-\frac{\Psi_{pm}}{L_{q}}\Dt\omega_{t}+u_{q,t}\frac{\Dt}{L_{q}},\\ |
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| 372 | \omega_{t+1} & = & \underbrace{(1-\frac{B}{J}\Dt)}_{\approx1}\om_{t}+\Dt\frac{k_{p}p_{p}^{2}}{J}((L_{d}-L_{q})i_{d}+\Psi_{pm})i_{q}\\ |
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| 373 | \vartheta_{t+1} & = & \vartheta_{t}+\Delta t\omega_{t}\end{eqnarray*} |
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| 374 | |
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| 375 | \end_inset |
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| 376 | |
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| 377 | |
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| 378 | \end_layout |
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| 379 | |
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| 380 | \begin_layout Standard |
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| 381 | Observation: |
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| 382 | \begin_inset Formula \[ |
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| 383 | \left[\begin{array}{c} |
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| 384 | i_{\alpha}\\ |
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| 385 | i_{\beta}\end{array}\right]=\left[\begin{array}{cc} |
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| 386 | \cos\vartheta & -\sin\vartheta\\ |
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| 387 | \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} |
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| 388 | i_{d}\\ |
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| 389 | i_{q}\end{array}\right]+e_{t}\] |
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| 390 | |
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| 391 | \end_inset |
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| 392 | |
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| 393 | |
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| 394 | \end_layout |
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| 395 | |
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| 396 | \begin_layout Subsection |
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[731] | 397 | Gaussian model of disturbances |
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| 398 | \end_layout |
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| 399 | |
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| 400 | \begin_layout Standard |
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| 401 | This model is motivated by the well known Kalman filter, which is optimal |
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| 402 | for linear system with Gaussian noise. |
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| 403 | Hence, we model all disturbances to have covariance matrices |
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| 404 | \begin_inset Formula $Q_{t}$ |
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| 405 | \end_inset |
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| 406 | |
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| 407 | and |
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| 408 | \begin_inset Formula $R_{t}$ |
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| 409 | \end_inset |
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| 410 | |
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| 411 | for the state |
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| 412 | \begin_inset Formula $x_{t}$ |
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| 413 | \end_inset |
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| 414 | |
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| 415 | and observations |
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| 416 | \begin_inset Formula $y_{t}$ |
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| 417 | \end_inset |
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| 418 | |
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| 419 | respectively. |
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[879] | 420 | \begin_inset Formula \begin{align} |
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| 421 | x_{t+1} & \sim\mathcal{N}(g(x_{t}),Q_{t})\label{eq:model}\\ |
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| 422 | y_{t} & \sim\mathcal{N}([\isa{,t},\isb{,t}]',R_{t})\nonumber \end{align} |
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[731] | 423 | |
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| 424 | \end_inset |
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| 425 | |
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| 426 | |
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| 427 | \end_layout |
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| 428 | |
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| 429 | \begin_layout Standard |
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| 430 | Under this assumptions, Bayesian estimation of the state, |
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| 431 | \begin_inset Formula $x_{t}$ |
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| 432 | \end_inset |
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| 433 | |
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| 434 | , can be approximated by so called Extended Kalman filter which approximates |
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| 435 | posterior density of the state by a Gaussian |
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| 436 | \begin_inset Formula \[ |
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[848] | 437 | f(x_{t}|y_{1}\ldots y_{t})=\mathcal{N}(\hat{x}_{t},S_{t}).\] |
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[731] | 438 | |
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| 439 | \end_inset |
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| 440 | |
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| 441 | Its sufficient statistics |
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| 442 | \begin_inset Formula $S_{t}=\left[\hat{x}_{t},P_{t}\right]$ |
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| 443 | \end_inset |
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| 444 | |
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| 445 | is evaluated recursively as follows: |
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| 446 | \begin_inset Formula \begin{eqnarray} |
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| 447 | \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] | 448 | R_{y} & = & CP_{t-1}C'+R_{t},\nonumber \\ |
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| 449 | K & = & P_{t-1}C'R_{y}^{-1},\nonumber \\ |
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[848] | 450 | S_{t} & = & P_{t-1}-P_{t-1}C'R_{y}^{-1}CP_{t-1},\\ |
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| 451 | P_{t} & = & AS_{t}A'+Q_{t}.\label{eq:ekf_cov}\end{eqnarray} |
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[731] | 452 | |
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| 453 | \end_inset |
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| 454 | |
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| 455 | where |
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| 456 | \begin_inset Formula $A=\frac{d}{dx_{t}}g(x_{t})$ |
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| 457 | \end_inset |
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| 458 | |
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| 459 | , |
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| 460 | \begin_inset Formula $C=\frac{d}{dx_{t}}h(x_{t})$ |
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| 461 | \end_inset |
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| 462 | |
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| 463 | , |
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| 464 | \begin_inset Formula $g(x_{t})$ |
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| 465 | \end_inset |
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| 466 | |
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| 467 | is model ( |
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| 468 | \begin_inset CommandInset ref |
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| 469 | LatexCommand ref |
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| 470 | reference "eq:model-simple" |
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| 471 | |
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| 472 | \end_inset |
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| 473 | |
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| 474 | ) and |
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| 475 | \begin_inset Formula $h(x_{t})$ |
---|
| 476 | \end_inset |
---|
| 477 | |
---|
| 478 | direct observation of |
---|
| 479 | \begin_inset Formula $y_{t}=[\isa{,t},\isb{,t}]$ |
---|
| 480 | \end_inset |
---|
| 481 | |
---|
| 482 | , i.e. |
---|
| 483 | \begin_inset Formula \[ |
---|
| 484 | A=\left[\begin{array}{cccc} |
---|
| 485 | a & 0 & b\sin\th & b\om\cos\th\\ |
---|
| 486 | 0 & a & -b\cos\th & b\om\sin\th\\ |
---|
| 487 | -e\sin\th & e\cos\th & d & -e(\isb{}\sin\th+\isa{}\cos\th)\\ |
---|
| 488 | 0 & 0 & \Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cccc} |
---|
[848] | 489 | 1 & 0 & 0 & 0\\ |
---|
| 490 | 0 & 1 & 0 & 0\end{array}\right]\] |
---|
[731] | 491 | |
---|
| 492 | \end_inset |
---|
| 493 | |
---|
| 494 | |
---|
| 495 | \end_layout |
---|
| 496 | |
---|
| 497 | \begin_layout Standard |
---|
[879] | 498 | \begin_inset Formula \[ |
---|
| 499 | B=\left[\begin{array}{cc} |
---|
| 500 | c & 0\\ |
---|
| 501 | 0 & c\\ |
---|
| 502 | 0 & 0\\ |
---|
| 503 | 0 & 0\end{array}\right]\] |
---|
| 504 | |
---|
| 505 | \end_inset |
---|
| 506 | |
---|
| 507 | |
---|
| 508 | \end_layout |
---|
| 509 | |
---|
| 510 | \begin_layout Standard |
---|
[731] | 511 | Covariance matrices of the system |
---|
| 512 | \begin_inset Formula $Q$ |
---|
| 513 | \end_inset |
---|
| 514 | |
---|
| 515 | and |
---|
| 516 | \begin_inset Formula $R$ |
---|
| 517 | \end_inset |
---|
| 518 | |
---|
| 519 | are supposed to be known. |
---|
| 520 | \end_layout |
---|
| 521 | |
---|
[1309] | 522 | \begin_layout Subsubsection |
---|
| 523 | Reduced order version |
---|
| 524 | \end_layout |
---|
| 525 | |
---|
| 526 | \begin_layout Standard |
---|
| 527 | Equations ( |
---|
| 528 | \begin_inset CommandInset ref |
---|
| 529 | LatexCommand ref |
---|
| 530 | reference "eq:model" |
---|
| 531 | |
---|
| 532 | \end_inset |
---|
| 533 | |
---|
| 534 | ) ca be restructured by considering |
---|
| 535 | \begin_inset Formula $i_{s\alpha}$ |
---|
| 536 | \end_inset |
---|
| 537 | |
---|
| 538 | and |
---|
| 539 | \begin_inset Formula $i_{s\beta}$ |
---|
| 540 | \end_inset |
---|
| 541 | |
---|
| 542 | as external observations. |
---|
| 543 | Then the state variables are |
---|
| 544 | \begin_inset Formula $x_{t}=[\omega_{t},\vartheta_{t}]$ |
---|
| 545 | \end_inset |
---|
| 546 | |
---|
| 547 | and as follows: |
---|
| 548 | \begin_inset Formula \begin{align} |
---|
| 549 | \om_{t+1} & =d\om_{t}+e\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right),\label{eq:rord-state}\\ |
---|
| 550 | \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\nonumber \end{align} |
---|
| 551 | |
---|
| 552 | \end_inset |
---|
| 553 | |
---|
| 554 | and the onbservation equations are |
---|
| 555 | \begin_inset Formula \begin{align} |
---|
| 556 | \isa{,t+1} & =a\,\isa{,t}+b\omega_{t}\sin\vartheta_{t}+c\usa{,t},\nonumber \\ |
---|
| 557 | \isb{,t+1} & =a\,\isb{,t}-b\omega_{t}\cos\vartheta_{t}+c\usb{,t},\label{eq:rord-obs}\end{align} |
---|
| 558 | |
---|
| 559 | \end_inset |
---|
| 560 | |
---|
| 561 | These equations are used by the EKF to update estimates of mean values. |
---|
| 562 | The new matrices |
---|
| 563 | \begin_inset Formula $A$ |
---|
| 564 | \end_inset |
---|
| 565 | |
---|
| 566 | and |
---|
| 567 | \begin_inset Formula $C$ |
---|
| 568 | \end_inset |
---|
| 569 | |
---|
| 570 | are |
---|
| 571 | \begin_inset Formula \[ |
---|
| 572 | A=\left[\begin{array}{cc} |
---|
| 573 | d & -e(\isb{}\sin\th+\isa{}\cos\th)\\ |
---|
| 574 | \Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cc} |
---|
| 575 | b\sin\th & b\om\cos\th\\ |
---|
| 576 | -b\cos\th & b\om\sin\th\end{array}\right].\] |
---|
| 577 | |
---|
| 578 | \end_inset |
---|
| 579 | |
---|
| 580 | |
---|
| 581 | \end_layout |
---|
| 582 | |
---|
[749] | 583 | \begin_layout Subsection |
---|
| 584 | Test system |
---|
| 585 | \end_layout |
---|
| 586 | |
---|
| 587 | \begin_layout Standard |
---|
| 588 | A real PMSM system on which the algorithms will be tested has parameters: |
---|
| 589 | \end_layout |
---|
| 590 | |
---|
| 591 | \begin_layout Standard |
---|
| 592 | \begin_inset Formula \begin{eqnarray*} |
---|
| 593 | R_{s} & = & 0.28;\\ |
---|
| 594 | L_{s} & = & 0.003465;\\ |
---|
| 595 | \Psi_{pm} & = & 0.1989;\\ |
---|
| 596 | k_{p} & = & 1.5\\ |
---|
| 597 | p & = & 4.0;\\ |
---|
| 598 | J & = & 0.04;\\ |
---|
| 599 | \Delta t & = & 0.000125\end{eqnarray*} |
---|
| 600 | |
---|
| 601 | \end_inset |
---|
| 602 | |
---|
| 603 | which yields |
---|
| 604 | \begin_inset Formula \begin{eqnarray*} |
---|
| 605 | a & = & 0.9898\\ |
---|
| 606 | b & = & 0.0072\\ |
---|
| 607 | c & = & 0.0361\\ |
---|
| 608 | d & = & 1\\ |
---|
| 609 | e & = & 0.0149\end{eqnarray*} |
---|
| 610 | |
---|
| 611 | \end_inset |
---|
| 612 | |
---|
| 613 | The covaraince matrices |
---|
| 614 | \begin_inset Formula $Q$ |
---|
| 615 | \end_inset |
---|
| 616 | |
---|
| 617 | and |
---|
| 618 | \begin_inset Formula $R$ |
---|
| 619 | \end_inset |
---|
| 620 | |
---|
| 621 | are assumed to be known. |
---|
| 622 | For the initial tests, we can use the following values: |
---|
| 623 | \end_layout |
---|
| 624 | |
---|
| 625 | \begin_layout Standard |
---|
| 626 | \begin_inset Formula \begin{eqnarray*} |
---|
| 627 | Q & = & \mathrm{diag}(0.0013,0.0013,5e-6,1e-10),\\ |
---|
| 628 | R & = & \mathrm{diag}(0.0006,0.0006).\end{eqnarray*} |
---|
| 629 | |
---|
| 630 | \end_inset |
---|
| 631 | |
---|
| 632 | |
---|
| 633 | \end_layout |
---|
| 634 | |
---|
[806] | 635 | \begin_layout Standard |
---|
| 636 | Limits: |
---|
[807] | 637 | \begin_inset Formula \begin{align*} |
---|
| 638 | u_{\alpha,max} & =50V, & u_{\alpha,min} & =-50V,\\ |
---|
| 639 | u_{\beta,max} & =50V. & u_{\beta,min} & =-50V,\end{align*} |
---|
[806] | 640 | |
---|
| 641 | \end_inset |
---|
| 642 | |
---|
| 643 | |
---|
| 644 | \end_layout |
---|
| 645 | |
---|
[807] | 646 | \begin_layout Standard |
---|
| 647 | Perhaps better: |
---|
| 648 | \begin_inset Formula \[ |
---|
| 649 | u_{\alpha}^{2}+u_{\beta}^{2}<100^{2}.\] |
---|
| 650 | |
---|
| 651 | \end_inset |
---|
| 652 | |
---|
| 653 | |
---|
| 654 | \end_layout |
---|
| 655 | |
---|
[1288] | 656 | \begin_layout Standard |
---|
| 657 | |
---|
| 658 | \end_layout |
---|
| 659 | |
---|
[731] | 660 | \begin_layout Section |
---|
| 661 | Control |
---|
| 662 | \end_layout |
---|
| 663 | |
---|
| 664 | \begin_layout Standard |
---|
| 665 | The task is to reach predefined speed |
---|
| 666 | \begin_inset Formula $\overline{\omega}_{t}$ |
---|
| 667 | \end_inset |
---|
| 668 | |
---|
| 669 | . |
---|
| 670 | \end_layout |
---|
| 671 | |
---|
| 672 | \begin_layout Standard |
---|
| 673 | For simplicity, we will assume additive loss function: |
---|
[879] | 674 | \begin_inset Formula \begin{eqnarray*} |
---|
| 675 | l(x_{t},u_{t}) & = & (\omega_{t}-\overline{\omega}_{t})^{2}+q(\usa{,t}^{2}+\usb{,t}^{2}).\\ |
---|
| 676 | & = & (\omega_{t}-\overline{\omega}_{t})\Xi(\omega_{t}-\overline{\omega}_{t})+[\usa t,\usb t]\underbrace{\left[\begin{array}{cc} |
---|
| 677 | \upsilon & 0\\ |
---|
| 678 | 0 & \upsilon\end{array}\right]}_{\Upsilon}\left[\begin{array}{c} |
---|
| 679 | \usa t\\ |
---|
| 680 | \usb t\end{array}\right]\end{eqnarray*} |
---|
[731] | 681 | |
---|
| 682 | \end_inset |
---|
| 683 | |
---|
| 684 | Here, |
---|
[879] | 685 | \begin_inset Formula $\Upsilon$ |
---|
[731] | 686 | \end_inset |
---|
| 687 | |
---|
[879] | 688 | is the chosen penalization of the inputs, which remains to be tuned. |
---|
| 689 | \end_layout |
---|
| 690 | |
---|
| 691 | \begin_layout Standard |
---|
| 692 | |
---|
| 693 | \series bold |
---|
| 694 | Note |
---|
| 695 | \series default |
---|
| 696 | : classical notation of penalization matrices is |
---|
| 697 | \begin_inset Formula $Q$ |
---|
[806] | 698 | \end_inset |
---|
| 699 | |
---|
[879] | 700 | and |
---|
| 701 | \begin_inset Formula $R$ |
---|
| 702 | \end_inset |
---|
| 703 | |
---|
| 704 | , but it conflicts wit |
---|
| 705 | \begin_inset Formula $Q$ |
---|
| 706 | \end_inset |
---|
| 707 | |
---|
| 708 | and |
---|
| 709 | \begin_inset Formula $R$ |
---|
| 710 | \end_inset |
---|
| 711 | |
---|
| 712 | in ( |
---|
| 713 | \begin_inset CommandInset ref |
---|
| 714 | LatexCommand ref |
---|
| 715 | reference "eq:model" |
---|
| 716 | |
---|
| 717 | \end_inset |
---|
| 718 | |
---|
| 719 | ). |
---|
[731] | 720 | \end_layout |
---|
| 721 | |
---|
| 722 | \begin_layout Standard |
---|
| 723 | Following the standard dynamic programming approach, optimization of the |
---|
| 724 | loss function can be done recursively, as follows: |
---|
| 725 | \begin_inset Formula \[ |
---|
| 726 | V(x_{t-1},u_{t-1})=\arg\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(x_{t},u_{t})\right\} ,\] |
---|
| 727 | |
---|
| 728 | \end_inset |
---|
| 729 | |
---|
| 730 | where |
---|
| 731 | \begin_inset Formula $V(x_{t},u_{t})$ |
---|
| 732 | \end_inset |
---|
| 733 | |
---|
| 734 | is the Bellman function. |
---|
| 735 | Since the model evolution is stochastic, we can reformulate it in terms |
---|
| 736 | of sufficient statistics, |
---|
| 737 | \begin_inset Formula $S$ |
---|
| 738 | \end_inset |
---|
| 739 | |
---|
| 740 | as follows: |
---|
| 741 | \begin_inset Formula \[ |
---|
| 742 | V(S_{t-1})=\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(S_{t})\right\} .\] |
---|
| 743 | |
---|
| 744 | \end_inset |
---|
| 745 | |
---|
| 746 | |
---|
| 747 | \end_layout |
---|
| 748 | |
---|
| 749 | \begin_layout Standard |
---|
| 750 | Representation of the Bellman function depends on chosen approximation. |
---|
| 751 | \end_layout |
---|
| 752 | |
---|
[751] | 753 | \begin_layout Subsection |
---|
[879] | 754 | LQG control |
---|
| 755 | \end_layout |
---|
| 756 | |
---|
| 757 | \begin_layout Standard |
---|
| 758 | Control of linear state-space model with Gaussian noise |
---|
| 759 | \begin_inset Formula \begin{eqnarray*} |
---|
| 760 | x_{t} & = & Ax_{t-1}+Bu_{t}+Q^{\frac{1}{2}}v_{t},\\ |
---|
| 761 | y_{t} & = & Cx_{t}+Du_{t}+R^{\frac{1}{2}}w_{t}.\end{eqnarray*} |
---|
| 762 | |
---|
| 763 | \end_inset |
---|
| 764 | |
---|
| 765 | to minimize loss function |
---|
| 766 | \begin_inset Formula \[ |
---|
[880] | 767 | L_{t}=(x_{t}-\overline{x}_{t})'\Xi(x_{t}-\overline{x}_{t})+u_{t}'\Upsilon u_{t}.\] |
---|
[879] | 768 | |
---|
| 769 | \end_inset |
---|
| 770 | |
---|
| 771 | |
---|
| 772 | \end_layout |
---|
| 773 | |
---|
| 774 | \begin_layout Standard |
---|
| 775 | Optimal solution in the sense of dynamic programming on horizon |
---|
| 776 | \begin_inset Formula $t+h$ |
---|
| 777 | \end_inset |
---|
| 778 | |
---|
| 779 | is: |
---|
| 780 | \begin_inset Newline newline |
---|
| 781 | \end_inset |
---|
| 782 | |
---|
| 783 | |
---|
| 784 | \begin_inset Formula \begin{eqnarray*} |
---|
[880] | 785 | u_{t} & = & L_{t}\left(\hat{x}_{t}-\overline{x}_{t}\right),\\ |
---|
[879] | 786 | L_{t} & = & -(B'S_{t+1}B+\Upsilon)^{-1}B'S_{t+1}A,\\ |
---|
| 787 | S_{t} & = & A'(S_{t+1}-S_{t+1}B(B'S_{t+1}B+\Upsilon)^{-1}B'S_{t+1})A+\Xi,\end{eqnarray*} |
---|
| 788 | |
---|
| 789 | \end_inset |
---|
| 790 | |
---|
| 791 | This solution is certainty equivalent, i.e. |
---|
| 792 | only the first moment, |
---|
| 793 | \begin_inset Formula $\hat{x}$ |
---|
| 794 | \end_inset |
---|
| 795 | |
---|
| 796 | , of the Kalman filter is used. |
---|
| 797 | \end_layout |
---|
| 798 | |
---|
| 799 | \begin_layout Subsection |
---|
[1244] | 800 | PI control |
---|
| 801 | \end_layout |
---|
| 802 | |
---|
| 803 | \begin_layout Standard |
---|
| 804 | The classical control is based on transformation to |
---|
| 805 | \begin_inset Formula $dq$ |
---|
| 806 | \end_inset |
---|
| 807 | |
---|
| 808 | reference frame: |
---|
| 809 | \begin_inset Formula \begin{eqnarray*} |
---|
| 810 | i_{d} & = & i_{\alpha}\cos(\vartheta)+i_{\beta}\sin(\vartheta),\\ |
---|
[1248] | 811 | i_{q} & = & i_{\beta}\cos(\vartheta)-i_{\alpha}\sin(\vartheta).\end{eqnarray*} |
---|
[1244] | 812 | |
---|
| 813 | \end_inset |
---|
| 814 | |
---|
| 815 | Desired |
---|
| 816 | \begin_inset Formula $i_{q}$ |
---|
| 817 | \end_inset |
---|
| 818 | |
---|
| 819 | current, |
---|
| 820 | \begin_inset Formula $\overline{i}_{q}$ |
---|
| 821 | \end_inset |
---|
| 822 | |
---|
| 823 | , is derived using PI controller |
---|
| 824 | \begin_inset Formula \[ |
---|
| 825 | \overline{i}_{q}=PI(\overline{\omega}-\omega,P_{i},I_{i}).\] |
---|
| 826 | |
---|
| 827 | \end_inset |
---|
| 828 | |
---|
| 829 | This current needs to be achieved through voltages |
---|
| 830 | \begin_inset Formula $u_{d},u_{q}$ |
---|
| 831 | \end_inset |
---|
| 832 | |
---|
| 833 | which are again obtained from a PI controller |
---|
| 834 | \begin_inset Formula \begin{eqnarray*} |
---|
| 835 | u_{d} & = & PI(-i_{d},P_{u},I_{u}),\\ |
---|
| 836 | u_{q} & = & PI(\overline{i}_{q}-i_{q},P_{u},I_{u}).\end{eqnarray*} |
---|
| 837 | |
---|
| 838 | \end_inset |
---|
| 839 | |
---|
| 840 | These are compensated (for some reason) as follows: |
---|
| 841 | \begin_inset Formula \begin{eqnarray*} |
---|
| 842 | u_{d} & = & u_{d}-L_{S}\omega\overline{i}_{q},\\ |
---|
| 843 | u_{q} & = & u_{q}+\Psi_{pm}\omega.\end{eqnarray*} |
---|
| 844 | |
---|
| 845 | \end_inset |
---|
| 846 | |
---|
| 847 | Conversion to |
---|
| 848 | \begin_inset Formula $u_{\alpha},u_{\beta}$ |
---|
| 849 | \end_inset |
---|
| 850 | |
---|
| 851 | is |
---|
| 852 | \begin_inset Formula \begin{align*} |
---|
| 853 | u_{\alpha} & =|U|\cos(\phi), & u_{\beta} & =|U|\sin(\phi)\\ |
---|
| 854 | |U| & =\sqrt{u_{d}^{2}+u_{q}^{2}}, & \phi & =\begin{cases} |
---|
| 855 | \arctan(\frac{u_{q}}{u_{d}})+\vartheta & u_{d}\ge0\\ |
---|
| 856 | \arctan(\frac{u_{q}}{u_{d}})+\pi+\vartheta & u_{d}<0\end{cases}\end{align*} |
---|
| 857 | |
---|
| 858 | \end_inset |
---|
| 859 | |
---|
| 860 | |
---|
| 861 | \end_layout |
---|
| 862 | |
---|
| 863 | \begin_layout Standard |
---|
| 864 | PI controller is defined as follows: |
---|
| 865 | \begin_inset Formula \begin{eqnarray*} |
---|
| 866 | x & = & PI(\epsilon,P,I)\\ |
---|
| 867 | & = & P\epsilon+I(S_{t-1}+\epsilon)\\ |
---|
| 868 | S_{t} & = & S_{t-1}+\epsilon\end{eqnarray*} |
---|
| 869 | |
---|
| 870 | \end_inset |
---|
| 871 | |
---|
| 872 | Constants for the system: |
---|
| 873 | \begin_inset Formula \[ |
---|
| 874 | P_{i}=3,\,\, I_{i}=0.00375,\,\, P_{u}=20,\,\, I_{u}=0.5.\] |
---|
| 875 | |
---|
| 876 | \end_inset |
---|
| 877 | |
---|
| 878 | The requested values for |
---|
| 879 | \begin_inset Formula $\omega$ |
---|
| 880 | \end_inset |
---|
| 881 | |
---|
| 882 | should be kept in interval |
---|
| 883 | \begin_inset Formula $<-30,30>$ |
---|
| 884 | \end_inset |
---|
| 885 | |
---|
| 886 | . |
---|
| 887 | \end_layout |
---|
| 888 | |
---|
| 889 | \begin_layout Subsection |
---|
[1261] | 890 | Cautious LQG control |
---|
| 891 | \end_layout |
---|
| 892 | |
---|
| 893 | \begin_layout Standard |
---|
| 894 | Uncertainty in |
---|
| 895 | \begin_inset Formula $A$ |
---|
| 896 | \end_inset |
---|
| 897 | |
---|
| 898 | . |
---|
| 899 | \end_layout |
---|
| 900 | |
---|
| 901 | \begin_layout Standard |
---|
| 902 | Sigma points: |
---|
| 903 | \begin_inset Formula $x^{(i)}=\hat{x}+hv_{i}$ |
---|
| 904 | \end_inset |
---|
| 905 | |
---|
| 906 | , |
---|
| 907 | \begin_inset Formula $v_{i}$ |
---|
| 908 | \end_inset |
---|
| 909 | |
---|
| 910 | are eigenvectors of |
---|
| 911 | \begin_inset Formula $P$ |
---|
| 912 | \end_inset |
---|
| 913 | |
---|
| 914 | . |
---|
| 915 | \end_layout |
---|
| 916 | |
---|
| 917 | \begin_layout Standard |
---|
| 918 | \begin_inset Formula \begin{eqnarray*} |
---|
| 919 | E\{x'A(x)QA(x)x\} & = & \frac{1}{n}\sum x'A(x^{(i)})QA^{(i)}(x)x\\ |
---|
| 920 | & = & x'Zx'\end{eqnarray*} |
---|
| 921 | |
---|
| 922 | \end_inset |
---|
| 923 | |
---|
| 924 | |
---|
| 925 | \end_layout |
---|
| 926 | |
---|
| 927 | \begin_layout Standard |
---|
| 928 | Uncented transform... |
---|
| 929 | \end_layout |
---|
| 930 | |
---|
| 931 | \begin_layout Subsection |
---|
[879] | 932 | Poor-man's dual LQG control |
---|
| 933 | \end_layout |
---|
| 934 | |
---|
| 935 | \begin_layout Standard |
---|
| 936 | Various heuristic solutions to dual extension of LQG has been proposed. |
---|
| 937 | Most of them is based on approximation of the loss function |
---|
| 938 | \begin_inset Formula \[ |
---|
| 939 | L_{t}=(x_{t}-\overline{x}_{t})'\Xi(x_{t}-\overline{x}_{t})+(u_{t}-\overline{u}_{t})'\Upsilon(u_{t}-\overline{u}_{t})+DUAL\_TERM.\] |
---|
| 940 | |
---|
| 941 | \end_inset |
---|
| 942 | |
---|
| 943 | where DUAL_TERM is typically a function of |
---|
| 944 | \begin_inset Formula $P_{t+2}$ |
---|
| 945 | \end_inset |
---|
| 946 | |
---|
| 947 | . |
---|
| 948 | |
---|
| 949 | \end_layout |
---|
| 950 | |
---|
| 951 | \begin_layout Standard |
---|
| 952 | To be continued... |
---|
| 953 | \end_layout |
---|
| 954 | |
---|
| 955 | \begin_layout Subsection |
---|
[751] | 956 | Test Scenarios |
---|
| 957 | \end_layout |
---|
| 958 | |
---|
[731] | 959 | \begin_layout Standard |
---|
[751] | 960 | With almost full information, design of the control strategy should be almost |
---|
| 961 | trivial: |
---|
| 962 | \begin_inset Formula \begin{eqnarray*} |
---|
| 963 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
---|
| 964 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,0.01]).\end{eqnarray*} |
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| 965 | |
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| 966 | \end_inset |
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| 967 | |
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| 968 | |
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| 969 | \end_layout |
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| 970 | |
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| 971 | \begin_layout Standard |
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[753] | 972 | The difficulty arise with growing initial covariance matrix: |
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[751] | 973 | \begin_inset Formula \begin{eqnarray*} |
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| 974 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
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[806] | 975 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,1]).\end{eqnarray*} |
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[751] | 976 | |
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| 977 | \end_inset |
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| 978 | |
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| 979 | |
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| 980 | \end_layout |
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| 981 | |
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| 982 | \begin_layout Standard |
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| 983 | Or even worse: |
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| 984 | \begin_inset Formula \begin{eqnarray*} |
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| 985 | \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ |
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[806] | 986 | P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,10]).\end{eqnarray*} |
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[751] | 987 | |
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| 988 | \end_inset |
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| 989 | |
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[879] | 990 | ==\SpecialChar \- |
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| 991 | = |
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[753] | 992 | \end_layout |
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| 993 | |
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| 994 | \begin_layout Standard |
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| 995 | The requested value |
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[806] | 996 | \begin_inset Formula $\overline{\omega}_{t}=1.0015.$ |
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[753] | 997 | \end_inset |
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| 998 | |
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| 999 | |
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[806] | 1000 | \end_layout |
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| 1001 | |
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| 1002 | \begin_layout Conjecture |
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| 1003 | It is sufficient to consider hyper-state |
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| 1004 | \begin_inset Formula $H=[\hat{i}_{\alpha},\hat{i}_{\beta},\hat{\omega},\hat{\vartheta},P(3,3),P(4,4)]$ |
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| 1005 | \end_inset |
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| 1006 | |
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| 1007 | . |
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| 1008 | \end_layout |
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| 1009 | |
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| 1010 | \begin_layout Conjecture |
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[731] | 1011 | \begin_inset CommandInset bibtex |
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| 1012 | LatexCommand bibtex |
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| 1013 | bibfiles "bibtex/vs,bibtex/vs-world,bibtex/world_classics,bibtex/world,new_bib_PS" |
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| 1014 | options "plain" |
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| 1015 | |
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| 1016 | \end_inset |
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| 1017 | |
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| 1018 | |
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| 1019 | \end_layout |
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| 1020 | |
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| 1021 | \end_body |
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| 1022 | \end_document |
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