[1436] | 1 | #LyX 2.0 created this file. For more info see http://www.lyx.org/ |
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| 2 | \lyxformat 413 |
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| 3 | \begin_document |
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| 4 | \begin_header |
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| 5 | \textclass scrreprt |
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| 6 | \use_default_options true |
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| 32 | \use_amsmath 1 |
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| 36 | \cite_engine basic |
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| 39 | \paperorientation portrait |
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| 42 | \index Index |
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| 43 | \shortcut idx |
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| 44 | \color #008000 |
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| 45 | \end_index |
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| 46 | \secnumdepth 2 |
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| 47 | \tocdepth 2 |
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| 48 | \paragraph_separation indent |
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| 49 | \paragraph_indentation default |
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| 50 | \quotes_language german |
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| 51 | \papercolumns 1 |
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| 59 | \end_header |
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| 60 | |
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| 61 | \begin_body |
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| 62 | |
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| 63 | \begin_layout Title |
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| 64 | LQG s hyperstavem |
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| 65 | \end_layout |
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| 66 | |
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| 67 | \begin_layout Section |
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| 68 | Systém |
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| 69 | \end_layout |
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| 70 | |
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| 71 | \begin_layout Standard |
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| 72 | Jako systémem uvažujeme PMSM a předpokládáme jeho popis pomocí následujících |
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| 73 | diskrétních rovnic: |
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| 74 | \end_layout |
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| 75 | |
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| 76 | \begin_layout Standard |
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| 77 | \begin_inset Formula |
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| 78 | \begin{eqnarray} |
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| 79 | i_{\alpha,t+1} & = & ai_{\alpha,t}+b\omega_{t}\sin\vartheta_{t}+cu_{\alpha,t}\nonumber \\ |
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| 80 | i_{\beta,t+1} & = & ai_{\beta,t}-b\omega_{t}\cos\vartheta_{t}+cu_{\beta,t}\label{eq:system}\\ |
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| 81 | \omega_{t+1} & = & d\omega_{t}+e\left(i_{b,t}\cos\vartheta_{t}-i_{\alpha,t}\sin\vartheta_{t}\right)\nonumber \\ |
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| 82 | \vartheta_{t+1} & = & \vartheta_{t}+\Delta t\omega_{t}\nonumber |
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| 83 | \end{eqnarray} |
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| 84 | |
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| 85 | \end_inset |
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| 86 | |
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| 87 | kde |
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| 88 | \begin_inset Formula $i_{\alpha\beta}$ |
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| 89 | \end_inset |
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| 90 | |
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| 91 | představují proudy v osách |
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| 92 | \begin_inset Formula $\alpha$ |
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| 93 | \end_inset |
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| 94 | |
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| 95 | a |
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| 96 | \begin_inset Formula $\beta$ |
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| 97 | \end_inset |
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| 98 | |
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| 99 | , |
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| 100 | \begin_inset Formula $u_{\alpha\beta}$ |
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| 101 | \end_inset |
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| 102 | |
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| 103 | napětí v jednotlivých osách, |
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| 104 | \begin_inset Formula $\omega$ |
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| 105 | \end_inset |
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| 106 | |
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| 107 | je hodnota otáček a |
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| 108 | \begin_inset Formula $\vartheta$ |
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| 109 | \end_inset |
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| 110 | |
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| 111 | je poloha (úhel natočení). |
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| 112 | Konstantní na čase nezávislé parametry |
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| 113 | \begin_inset Formula $a,b,c,d,e$ |
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| 114 | \end_inset |
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| 115 | |
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| 116 | předpokládáme známé, |
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| 117 | \begin_inset Formula $\Delta t$ |
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| 118 | \end_inset |
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| 119 | |
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| 120 | je diskterizační časový krok a dolní indexy |
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| 121 | \begin_inset Formula $t$ |
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| 122 | \end_inset |
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| 123 | |
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| 124 | a |
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| 125 | \begin_inset Formula $t+1$ |
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| 126 | \end_inset |
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| 127 | |
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| 128 | představují diskrétní čas. |
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| 129 | \end_layout |
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| 130 | |
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| 131 | \begin_layout Standard |
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| 132 | Definujme stav systému v čase |
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| 133 | \begin_inset Formula $t$ |
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| 134 | \end_inset |
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| 135 | |
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| 136 | jako |
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| 137 | \begin_inset Formula |
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| 138 | \[ |
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| 139 | x_{t}=\left(i_{\alpha,t},i_{\beta,t},\omega_{t},\vartheta_{t}\right)^{T} |
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| 140 | \] |
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| 141 | |
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| 142 | \end_inset |
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| 143 | |
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| 144 | dále řízení v čase |
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| 145 | \begin_inset Formula $t$ |
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| 146 | \end_inset |
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| 147 | |
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| 148 | jako |
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| 149 | \begin_inset Formula |
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| 150 | \[ |
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| 151 | u_{t}=\left(u_{\alpha,t},u_{\beta,t}\right)^{T} |
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| 152 | \] |
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| 153 | |
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| 154 | \end_inset |
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| 155 | |
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| 156 | a výstup (měření) v čase |
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| 157 | \begin_inset Formula $t$ |
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| 158 | \end_inset |
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| 159 | |
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| 160 | jako |
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| 161 | \begin_inset Formula |
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| 162 | \[ |
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| 163 | y_{t}=\left(y_{\alpha,t},y_{\beta,t}\right)^{T} |
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| 164 | \] |
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| 165 | |
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| 166 | \end_inset |
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| 167 | |
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| 168 | příčemž význam použítých symbolů vychází z rovnic ( |
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| 169 | \begin_inset CommandInset ref |
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| 170 | LatexCommand ref |
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| 171 | reference "eq:system" |
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| 172 | |
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| 173 | \end_inset |
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| 174 | |
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| 175 | ). |
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| 176 | Když dále uvažujeme aditivní bílý gaussovský šum, získáme zápis systému |
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| 177 | ve tvaru |
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| 178 | \begin_inset Formula |
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| 179 | \begin{eqnarray} |
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| 180 | x_{t+1} & = & f(x_{t},u_{t})+v_{t}\nonumber \\ |
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| 181 | y_{t} & = & h(x_{t})+w_{t}\label{eq:systemrovnice} |
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| 182 | \end{eqnarray} |
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| 183 | |
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| 184 | \end_inset |
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| 185 | |
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| 186 | kde |
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| 187 | \begin_inset Formula $v_{t}$ |
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| 188 | \end_inset |
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| 189 | |
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| 190 | a |
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| 191 | \begin_inset Formula $w_{t}$ |
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| 192 | \end_inset |
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| 193 | |
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| 194 | představují náhodné veličiny s normálním rozdělením s nulovou střední hodnotou |
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| 195 | a kovariančními maticemi |
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| 196 | \begin_inset Formula $V$ |
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| 197 | \end_inset |
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| 198 | |
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| 199 | a |
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| 200 | \begin_inset Formula $W$ |
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| 201 | \end_inset |
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| 202 | |
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| 203 | v tomto pořadí a funkce |
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| 204 | \begin_inset Formula $f$ |
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| 205 | \end_inset |
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| 206 | |
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| 207 | a |
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| 208 | \begin_inset Formula $h$ |
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| 209 | \end_inset |
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| 210 | |
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| 211 | jsou definovány následovně: |
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| 212 | \end_layout |
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| 213 | |
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| 214 | \begin_layout Standard |
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| 215 | \begin_inset Formula |
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| 216 | \begin{eqnarray} |
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| 217 | f(x_{t},u_{t}) & = & \left(\begin{array}{c} |
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| 218 | ai_{\alpha,t}+b\omega_{t}\sin\vartheta_{t}+cu_{\alpha,t}\\ |
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| 219 | ai_{\beta,t}-b\omega_{t}\cos\vartheta_{t}+cu_{\beta,t}\\ |
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| 220 | d\omega_{t}+e\left(i_{\beta,t}\cos\vartheta_{t}-i_{\alpha,t}\sin\vartheta_{t}\right)\\ |
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| 221 | \vartheta_{t}+\Delta t\omega_{t} |
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| 222 | \end{array}\right)\nonumber \\ |
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| 223 | h(x_{t}) & = & \left(\begin{array}{c} |
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| 224 | i_{\alpha,t}\\ |
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| 225 | i_{\beta,t} |
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| 226 | \end{array}\right)\label{eq:systemsf} |
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| 227 | \end{eqnarray} |
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| 228 | |
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| 229 | \end_inset |
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| 230 | |
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| 231 | |
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| 232 | \end_layout |
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| 233 | |
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| 234 | \begin_layout Subsubsection |
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| 235 | Redukovaný model |
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| 236 | \end_layout |
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| 237 | |
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| 238 | \begin_layout Standard |
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| 239 | Z úsporných důvodů může být někdy výhodnější namísto popisu systému uvedeného |
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| 240 | výše (dále budeme označovat jako |
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| 241 | \emph on |
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| 242 | plný model |
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| 243 | \emph default |
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| 244 | ), který vychází z rovnic ( |
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| 245 | \begin_inset CommandInset ref |
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| 246 | LatexCommand ref |
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| 247 | reference "eq:systemrovnice" |
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| 248 | |
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| 249 | \end_inset |
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| 250 | |
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| 251 | ) a ( |
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| 252 | \begin_inset CommandInset ref |
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| 253 | LatexCommand ref |
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| 254 | reference "eq:systemsf" |
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| 255 | |
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| 256 | \end_inset |
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| 257 | |
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| 258 | ) použít jeho redukovanou verzi v následujícím tvaru: |
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| 259 | \end_layout |
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| 260 | |
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| 261 | \begin_layout Standard |
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| 262 | Vektor stavu |
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| 263 | \begin_inset Formula $x_{t}$ |
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| 264 | \end_inset |
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| 265 | |
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| 266 | bude mít jen dvě složky |
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| 267 | \begin_inset Formula |
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| 268 | \[ |
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| 269 | x_{t}=\left(\omega_{t},\vartheta_{t}\right)^{T} |
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| 270 | \] |
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| 271 | |
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| 272 | \end_inset |
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| 273 | |
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| 274 | a pro výstup (měření) |
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| 275 | \begin_inset Formula $y_{t}$ |
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| 276 | \end_inset |
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| 277 | |
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| 278 | využijeme toho, že proudy přímo měříme (i když ne zcela přesně) |
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| 279 | \begin_inset Formula |
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| 280 | \[ |
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| 281 | y_{t}=\left(i_{\alpha,t},i_{\beta,t}\right)^{T} |
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| 282 | \] |
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| 283 | |
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| 284 | \end_inset |
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| 285 | |
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| 286 | |
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| 287 | \end_layout |
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| 288 | |
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| 289 | \begin_layout Standard |
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| 290 | Rovnice systému ( |
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| 291 | \begin_inset CommandInset ref |
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| 292 | LatexCommand ref |
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| 293 | reference "eq:systemrovnice" |
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| 294 | |
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| 295 | \end_inset |
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| 296 | |
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| 297 | ) a ( |
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| 298 | \begin_inset CommandInset ref |
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| 299 | LatexCommand ref |
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| 300 | reference "eq:systemsf" |
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| 301 | |
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| 302 | \end_inset |
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| 303 | |
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| 304 | ) pak zapíšeme ve tvaru |
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| 305 | \begin_inset Formula |
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| 306 | \begin{eqnarray} |
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| 307 | x_{t+1} & = & f(x_{t},y_{t})+\overline{v}_{t}\nonumber \\ |
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| 308 | y_{t} & = & h(x_{t},y_{t},u_{t})+\overline{w}_{t}\label{eq:systemrovnice-reduk} |
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| 309 | \end{eqnarray} |
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| 310 | |
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| 311 | \end_inset |
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| 312 | |
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| 313 | |
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| 314 | \end_layout |
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| 315 | |
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| 316 | \begin_layout Standard |
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| 317 | \begin_inset Formula |
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| 318 | \begin{eqnarray} |
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| 319 | f(x_{t},y_{t}) & = & \left(\begin{array}{c} |
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| 320 | d\omega_{t}+e\left(i_{b,t}\cos\vartheta_{t}-i_{\alpha,t}\sin\vartheta_{t}\right)\\ |
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| 321 | \vartheta_{t}+\Delta t\omega_{t} |
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| 322 | \end{array}\right)\nonumber \\ |
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| 323 | h(x_{t},y_{t},u_{t}) & = & \left(\begin{array}{c} |
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| 324 | ai_{\alpha,t}+b\omega_{t}\sin\vartheta_{t}+cu_{\alpha,t}\\ |
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| 325 | ai_{\beta,t}-b\omega_{t}\cos\vartheta_{t}+cu_{\beta,t} |
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| 326 | \end{array}\right)\label{eq:systemsf-reduk} |
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| 327 | \end{eqnarray} |
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| 328 | |
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| 329 | \end_inset |
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| 330 | |
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| 331 | dále je pak třeba upravit kovarianční matice šumu |
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| 332 | \begin_inset Formula $V$ |
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| 333 | \end_inset |
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| 334 | |
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| 335 | a |
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| 336 | \begin_inset Formula $W$ |
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| 337 | \end_inset |
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| 338 | |
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| 339 | . |
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| 340 | Matici |
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| 341 | \begin_inset Formula $V$ |
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| 342 | \end_inset |
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| 343 | |
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| 344 | je nutno předpokládat v blokově diagonálním tvaru |
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| 345 | \begin_inset Formula |
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| 346 | \[ |
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| 347 | V=\left[\begin{array}{cc} |
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| 348 | V_{1} & 0\\ |
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| 349 | 0 & V_{2} |
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| 350 | \end{array}\right] |
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| 351 | \] |
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| 352 | |
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| 353 | \end_inset |
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| 354 | |
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| 355 | a jako nové kovarianční matice označíme |
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| 356 | \begin_inset Formula $\overline{V}=V_{2}$ |
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| 357 | \end_inset |
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| 358 | |
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| 359 | a |
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| 360 | \begin_inset Formula $\overline{W}=V_{1}+W$ |
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| 361 | \end_inset |
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| 362 | |
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| 363 | . |
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| 364 | \end_layout |
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| 365 | |
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| 366 | \begin_layout Section |
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| 367 | EKF |
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| 368 | \begin_inset CommandInset label |
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| 369 | LatexCommand label |
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| 370 | name "sub:EKF-popis" |
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| 371 | |
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| 372 | \end_inset |
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| 373 | |
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| 374 | |
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| 375 | \end_layout |
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| 376 | |
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| 377 | \begin_layout Standard |
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| 378 | Rozšířený Kalmanův filter (EKF) nahrazuje skutečný nelineární systém ( |
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| 379 | \begin_inset CommandInset ref |
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| 380 | LatexCommand ref |
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| 381 | reference "eq:systemrovnice" |
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| 382 | |
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| 383 | \end_inset |
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| 384 | |
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| 385 | a |
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| 386 | \begin_inset CommandInset ref |
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| 387 | LatexCommand ref |
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| 388 | reference "eq:systemsf" |
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| 389 | |
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| 390 | \end_inset |
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| 391 | |
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| 392 | případně |
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| 393 | \begin_inset CommandInset ref |
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| 394 | LatexCommand ref |
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| 395 | reference "eq:systemrovnice-reduk" |
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| 396 | |
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| 397 | \end_inset |
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| 398 | |
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| 399 | a |
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| 400 | \begin_inset CommandInset ref |
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| 401 | LatexCommand ref |
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| 402 | reference "eq:systemsf-reduk" |
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| 403 | |
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| 404 | \end_inset |
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| 405 | |
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| 406 | pro redukovaný model) lineárním |
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| 407 | \begin_inset Formula |
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| 408 | \begin{eqnarray} |
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| 409 | x_{t+1} & = & A_{t}x_{t}+B_{t}u_{t}+\tilde{v}_{t}\nonumber \\ |
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| 410 | y_{t} & = & C_{t}x_{t}+\tilde{w}_{t}\label{eq:linearizovany-system} |
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| 411 | \end{eqnarray} |
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| 412 | |
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| 413 | \end_inset |
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| 414 | |
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| 415 | kde do šumů |
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| 416 | \begin_inset Formula $\tilde{v}_{t}$ |
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| 417 | \end_inset |
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| 418 | |
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| 419 | a |
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| 420 | \begin_inset Formula $\tilde{w}_{t}$ |
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| 421 | \end_inset |
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| 422 | |
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| 423 | je možno zahrnout nepřesnosti linearizece tím, že se zvětší jejich kovariance |
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| 424 | oproti původním |
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| 425 | \begin_inset Formula $v_{t}$ |
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| 426 | \end_inset |
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| 427 | |
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| 428 | a |
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| 429 | \begin_inset Formula $w_{t}$ |
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| 430 | \end_inset |
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| 431 | |
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| 432 | respektive |
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| 433 | \begin_inset Formula $\overline{v}_{t}$ |
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| 434 | \end_inset |
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| 435 | |
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| 436 | a |
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| 437 | \begin_inset Formula $\overline{w}_{t}$ |
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| 438 | \end_inset |
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| 439 | |
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| 440 | v případě redukovaného modelu. |
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| 441 | Ostatní označení odpovídá nelineárním rovnicím PMSM ( |
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| 442 | \begin_inset CommandInset ref |
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| 443 | LatexCommand ref |
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| 444 | reference "eq:systemrovnice" |
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| 445 | |
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| 446 | \end_inset |
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| 447 | |
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| 448 | a |
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| 449 | \begin_inset CommandInset ref |
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| 450 | LatexCommand ref |
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| 451 | reference "eq:systemsf" |
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| 452 | |
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| 453 | \end_inset |
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| 454 | |
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| 455 | případně |
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| 456 | \begin_inset CommandInset ref |
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| 457 | LatexCommand ref |
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| 458 | reference "eq:systemrovnice-reduk" |
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| 459 | |
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| 460 | \end_inset |
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| 461 | |
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| 462 | a |
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| 463 | \begin_inset CommandInset ref |
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| 464 | LatexCommand ref |
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| 465 | reference "eq:systemsf-reduk" |
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| 466 | |
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| 467 | \end_inset |
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| 468 | |
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| 469 | ) s tím, že matice |
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| 470 | \begin_inset Formula $A$ |
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| 471 | \end_inset |
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| 472 | |
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| 473 | , |
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| 474 | \begin_inset Formula $B$ |
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| 475 | \end_inset |
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| 476 | |
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| 477 | a |
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| 478 | \begin_inset Formula $C$ |
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| 479 | \end_inset |
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| 480 | |
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| 481 | vzniknou linearizací jako |
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| 482 | \begin_inset Formula |
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| 483 | \begin{eqnarray*} |
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| 484 | A_{t} & = & \frac{\partial f(x_{t},u_{t})}{\partial x_{t}}\\ |
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| 485 | B_{t} & = & \frac{\partial f(x_{t},u_{t})}{\partial u_{t}}\\ |
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| 486 | C_{t} & = & \frac{\partial h(x_{t})}{\partial x_{t}} |
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| 487 | \end{eqnarray*} |
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| 488 | |
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| 489 | \end_inset |
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| 490 | |
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| 491 | |
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| 492 | \end_layout |
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| 493 | |
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| 494 | \begin_layout Subsubsection* |
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| 495 | Matice derivací |
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| 496 | \end_layout |
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| 497 | |
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| 498 | \begin_layout Standard |
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| 499 | Konkrétně pro PMSM s funkcemi |
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| 500 | \begin_inset Formula $f$ |
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| 501 | \end_inset |
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| 502 | |
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| 503 | a |
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| 504 | \begin_inset Formula $h$ |
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| 505 | \end_inset |
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| 506 | |
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| 507 | danými vztahem ( |
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| 508 | \begin_inset CommandInset ref |
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| 509 | LatexCommand ref |
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| 510 | reference "eq:systemsf" |
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| 511 | |
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| 512 | \end_inset |
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| 513 | |
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| 514 | ) jsou matice |
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| 515 | \begin_inset Formula $A$ |
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| 516 | \end_inset |
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| 517 | |
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| 518 | , |
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| 519 | \begin_inset Formula $B$ |
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| 520 | \end_inset |
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| 521 | |
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| 522 | a |
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| 523 | \begin_inset Formula $C$ |
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| 524 | \end_inset |
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| 525 | |
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| 526 | následující: |
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| 527 | \begin_inset Formula |
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| 528 | \begin{eqnarray} |
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| 529 | A_{t} & = & \left[\begin{array}{cccc} |
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| 530 | a & 0 & b\sin\vartheta_{t} & b\omega_{t}\cos\vartheta_{t}\\ |
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| 531 | 0 & a & -b\cos\vartheta_{t} & b\omega_{t}\sin\vartheta_{t}\\ |
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| 532 | -e\sin\vartheta_{t} & e\cos\vartheta_{t} & d & -e\left(i_{\beta,t}\sin\vartheta_{t}+i_{\alpha,t}\cos\vartheta_{t}\right)\\ |
---|
| 533 | 0 & 0 & \Delta t & 1 |
---|
| 534 | \end{array}\right]\nonumber \\ |
---|
| 535 | B & = & \left[\begin{array}{cc} |
---|
| 536 | c & 0\\ |
---|
| 537 | 0 & c\\ |
---|
| 538 | 0 & 0\\ |
---|
| 539 | 0 & 0 |
---|
| 540 | \end{array}\right]\label{eq:matice-ekf-plny-stav}\\ |
---|
| 541 | C & = & \left[\begin{array}{cccc} |
---|
| 542 | 1 & 0 & 0 & 0\\ |
---|
| 543 | 0 & 1 & 0 & 0 |
---|
| 544 | \end{array}\right]\nonumber |
---|
| 545 | \end{eqnarray} |
---|
| 546 | |
---|
| 547 | \end_inset |
---|
| 548 | |
---|
| 549 | |
---|
| 550 | \end_layout |
---|
| 551 | |
---|
| 552 | \begin_layout Standard |
---|
| 553 | Pro redukovaný model jsou matice |
---|
| 554 | \begin_inset Formula $A$ |
---|
| 555 | \end_inset |
---|
| 556 | |
---|
| 557 | a |
---|
| 558 | \begin_inset Formula $C$ |
---|
| 559 | \end_inset |
---|
| 560 | |
---|
| 561 | ve tvaru |
---|
| 562 | \begin_inset Formula |
---|
| 563 | \begin{eqnarray} |
---|
| 564 | A_{t} & = & \left[\begin{array}{cc} |
---|
| 565 | d & -e\left(i_{\beta,t}\sin\vartheta_{t}+i_{\alpha,t}\cos\vartheta_{t}\right)\\ |
---|
| 566 | \Delta t & 1 |
---|
| 567 | \end{array}\right]\nonumber \\ |
---|
| 568 | C_{t} & = & \left[\begin{array}{cc} |
---|
| 569 | b\sin\vartheta_{t} & b\omega_{t}\cos\vartheta_{t}\\ |
---|
| 570 | -b\cos\vartheta_{t} & b\omega_{t}\sin\vartheta_{t} |
---|
| 571 | \end{array}\right]\label{eq:matice-ekf-red-stav} |
---|
| 572 | \end{eqnarray} |
---|
| 573 | |
---|
| 574 | \end_inset |
---|
| 575 | |
---|
| 576 | Matice |
---|
| 577 | \begin_inset Formula $B$ |
---|
| 578 | \end_inset |
---|
| 579 | |
---|
| 580 | pro redukovaný model uvedena není, protože pro samotný výpočet EKF není |
---|
| 581 | třeba a problematika lineárně kvadratického řízení pro redukovaný model |
---|
| 582 | bude rozebrána dále, viz část ( |
---|
| 583 | \begin_inset CommandInset ref |
---|
| 584 | LatexCommand ref |
---|
| 585 | reference "sub:LQ-řízení-pro-red-model" |
---|
| 586 | |
---|
| 587 | \end_inset |
---|
| 588 | |
---|
| 589 | ). |
---|
| 590 | \end_layout |
---|
| 591 | |
---|
| 592 | \begin_layout Subsubsection |
---|
| 593 | Rovnice EKF |
---|
| 594 | \end_layout |
---|
| 595 | |
---|
| 596 | \begin_layout Standard |
---|
| 597 | Následně lze užít algoritmu formálně shodného s klasickým Kalmanovým filtrem, |
---|
| 598 | kde místo lineárního systému je užit systém linearizovaný: |
---|
| 599 | \end_layout |
---|
| 600 | |
---|
| 601 | \begin_layout Standard |
---|
| 602 | |
---|
| 603 | \emph on |
---|
| 604 | predikce |
---|
| 605 | \emph default |
---|
| 606 | (time update) |
---|
| 607 | \begin_inset Formula |
---|
| 608 | \begin{eqnarray} |
---|
| 609 | \overline{\hat{x}}_{t} & = & f\left(\hat{x}_{t-1},u_{t-1}\right)\label{eq:rovnice-ekf-timeupd}\\ |
---|
| 610 | \overline{P}_{t} & = & A_{t-1}P_{t-1}A_{t-1}^{T}+V_{t-1}\nonumber |
---|
| 611 | \end{eqnarray} |
---|
| 612 | |
---|
| 613 | \end_inset |
---|
| 614 | |
---|
| 615 | |
---|
| 616 | \end_layout |
---|
| 617 | |
---|
| 618 | \begin_layout Standard |
---|
| 619 | |
---|
| 620 | \emph on |
---|
| 621 | korekce |
---|
| 622 | \emph default |
---|
| 623 | (data update) |
---|
| 624 | \begin_inset Formula |
---|
| 625 | \begin{eqnarray} |
---|
| 626 | S_{t} & = & C_{t}\overline{P}_{t}C_{t}^{T}+W_{t}\nonumber \\ |
---|
| 627 | K_{t} & = & \overline{P}_{t}C_{t}^{T}S_{t}^{-1}\nonumber \\ |
---|
| 628 | P_{t} & \text{=} & \left(I-K_{t}C_{t}\right)\overline{P}_{t}\label{eq:rovnice-ekf-dataupd}\\ |
---|
| 629 | \hat{y}_{t} & = & y_{t}-h(\overline{\hat{x}}_{t})\nonumber \\ |
---|
| 630 | \hat{x}_{t} & = & \overline{\hat{x}}_{t}+K_{t}\hat{y}_{t}\nonumber |
---|
| 631 | \end{eqnarray} |
---|
| 632 | |
---|
| 633 | \end_inset |
---|
| 634 | |
---|
| 635 | |
---|
| 636 | \end_layout |
---|
| 637 | |
---|
| 638 | \begin_layout Section |
---|
| 639 | Lineárně kvadratické řízení |
---|
| 640 | \end_layout |
---|
| 641 | |
---|
| 642 | \begin_layout Standard |
---|
| 643 | Tento algoritmus opět předpokládá lineární systém, kterým PMSM není. |
---|
| 644 | Chceme opět získat systém ve tvaru ( |
---|
| 645 | \begin_inset CommandInset ref |
---|
| 646 | LatexCommand ref |
---|
| 647 | reference "eq:linearizovany-system" |
---|
| 648 | |
---|
| 649 | \end_inset |
---|
| 650 | |
---|
| 651 | ) a je tedy nutné provést linearizaci. |
---|
| 652 | Nelze ale přímo použít matice získané v předchozí části ( |
---|
| 653 | \begin_inset CommandInset ref |
---|
| 654 | LatexCommand ref |
---|
| 655 | reference "sub:EKF-popis" |
---|
| 656 | |
---|
| 657 | \end_inset |
---|
| 658 | |
---|
| 659 | ) zabývající se EKF. |
---|
| 660 | Zde je nutné vycházet z Taylorova rozvoje a zohlednit i případné konstantní |
---|
| 661 | členy. |
---|
| 662 | Obecně pro funkci |
---|
| 663 | \begin_inset Formula $f(x)$ |
---|
| 664 | \end_inset |
---|
| 665 | |
---|
| 666 | má rozvoj do prvního řádu v nějakém bodě |
---|
| 667 | \begin_inset Formula $x_{0}$ |
---|
| 668 | \end_inset |
---|
| 669 | |
---|
| 670 | tvar |
---|
| 671 | \begin_inset Formula |
---|
| 672 | \[ |
---|
| 673 | f\left(x\right)\cong f\left(x_{0}\right)+\frac{\partial f}{\partial x}\left(x_{0}\right)\left(x-x_{0}\right) |
---|
| 674 | \] |
---|
| 675 | |
---|
| 676 | \end_inset |
---|
| 677 | |
---|
| 678 | kde parciální derivací |
---|
| 679 | \begin_inset Formula $f$ |
---|
| 680 | \end_inset |
---|
| 681 | |
---|
| 682 | dle |
---|
| 683 | \begin_inset Formula $x$ |
---|
| 684 | \end_inset |
---|
| 685 | |
---|
| 686 | je matice |
---|
| 687 | \begin_inset Formula $A$ |
---|
| 688 | \end_inset |
---|
| 689 | |
---|
| 690 | z předchozí části ( |
---|
| 691 | \begin_inset CommandInset ref |
---|
| 692 | LatexCommand ref |
---|
| 693 | reference "sub:EKF-popis" |
---|
| 694 | |
---|
| 695 | \end_inset |
---|
| 696 | |
---|
| 697 | ) o EKF vypočtená v bodě |
---|
| 698 | \begin_inset Formula $x_{0}$ |
---|
| 699 | \end_inset |
---|
| 700 | |
---|
| 701 | a tedy |
---|
| 702 | \begin_inset Formula |
---|
| 703 | \[ |
---|
| 704 | f\left(x\right)\cong Ax+\left(f\left(x_{0}\right)-Ax_{0}\right)=Ax+\gamma |
---|
| 705 | \] |
---|
| 706 | |
---|
| 707 | \end_inset |
---|
| 708 | |
---|
| 709 | kde vektor |
---|
| 710 | \begin_inset Formula $\gamma$ |
---|
| 711 | \end_inset |
---|
| 712 | |
---|
| 713 | představuje konstantní člen (nezávisí na |
---|
| 714 | \begin_inset Formula $x$ |
---|
| 715 | \end_inset |
---|
| 716 | |
---|
| 717 | ) a předchozí rovnice tedy není homogenní, jak bychom potřebovali jako výsledek |
---|
| 718 | linearizace ( |
---|
| 719 | \begin_inset CommandInset ref |
---|
| 720 | LatexCommand ref |
---|
| 721 | reference "eq:linearizovany-system" |
---|
| 722 | |
---|
| 723 | \end_inset |
---|
| 724 | |
---|
| 725 | ). |
---|
| 726 | Proto tedy zvětšíme matici |
---|
| 727 | \begin_inset Formula $A$ |
---|
| 728 | \end_inset |
---|
| 729 | |
---|
| 730 | o 1 (o jeden sloupec a řádek) a stejně tak zvětšíme i stav o 1 (přidáme |
---|
| 731 | konstantu) a předchozí rovnici získáme ve tvaru |
---|
| 732 | \begin_inset Formula |
---|
| 733 | \[ |
---|
| 734 | \left(\begin{array}{c} |
---|
| 735 | f\left(x\right)\\ |
---|
| 736 | 1 |
---|
| 737 | \end{array}\right)\cong\overline{A}\left(\begin{array}{c} |
---|
| 738 | x\\ |
---|
| 739 | 1 |
---|
| 740 | \end{array}\right) |
---|
| 741 | \] |
---|
| 742 | |
---|
| 743 | \end_inset |
---|
| 744 | |
---|
| 745 | kde |
---|
| 746 | \begin_inset Formula |
---|
| 747 | \[ |
---|
| 748 | \overline{A}=\left[\begin{array}{cc} |
---|
| 749 | A & \left(f\left(x_{0}\right)-Ax_{0}\right)\\ |
---|
| 750 | 0 & 1 |
---|
| 751 | \end{array}\right] |
---|
| 752 | \] |
---|
| 753 | |
---|
| 754 | \end_inset |
---|
| 755 | |
---|
| 756 | přičemž |
---|
| 757 | \begin_inset Formula $0$ |
---|
| 758 | \end_inset |
---|
| 759 | |
---|
| 760 | zde označuje nulový řádkový vektor vhodné velikosti. |
---|
| 761 | Tímto postupem lze již získat požadovaný lineární popis ( |
---|
| 762 | \begin_inset CommandInset ref |
---|
| 763 | LatexCommand ref |
---|
| 764 | reference "eq:linearizovany-system" |
---|
| 765 | |
---|
| 766 | \end_inset |
---|
| 767 | |
---|
| 768 | ), který současně zohledňuje i konstantní členy. |
---|
| 769 | \end_layout |
---|
| 770 | |
---|
| 771 | \begin_layout Subsubsection* |
---|
| 772 | Matice pro LQ |
---|
| 773 | \end_layout |
---|
| 774 | |
---|
| 775 | \begin_layout Standard |
---|
| 776 | Pro případ plného stavu je matice |
---|
| 777 | \begin_inset Formula $A_{t}$ |
---|
| 778 | \end_inset |
---|
| 779 | |
---|
| 780 | dána vztahem ( |
---|
| 781 | \begin_inset CommandInset ref |
---|
| 782 | LatexCommand ref |
---|
| 783 | reference "eq:matice-ekf-plny-stav" |
---|
| 784 | |
---|
| 785 | \end_inset |
---|
| 786 | |
---|
| 787 | ), kde jako hodnoty stavových veličin (složek vektoru |
---|
| 788 | \begin_inset Formula $x_{t}$ |
---|
| 789 | \end_inset |
---|
| 790 | |
---|
| 791 | ) použijeme hodnoty bodu |
---|
| 792 | \begin_inset Formula $x_{0}$ |
---|
| 793 | \end_inset |
---|
| 794 | |
---|
| 795 | , ve kterém linearizujeme. |
---|
| 796 | Konstantní člen |
---|
| 797 | \begin_inset Formula $\gamma=f\left(x_{0}\right)-A_{t}x_{0}$ |
---|
| 798 | \end_inset |
---|
| 799 | |
---|
| 800 | tedy vypočteme jako |
---|
| 801 | \begin_inset Formula |
---|
| 802 | \begin{eqnarray*} |
---|
| 803 | \gamma & = & \left(\begin{array}{c} |
---|
| 804 | -b\omega_{0}\vartheta_{0}\cos\vartheta_{0}\\ |
---|
| 805 | -b\omega_{0}\vartheta_{0}\sin\vartheta_{0}\\ |
---|
| 806 | e\vartheta_{0}\left(i_{\beta,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right)\\ |
---|
| 807 | 0 |
---|
| 808 | \end{array}\right) |
---|
| 809 | \end{eqnarray*} |
---|
| 810 | |
---|
| 811 | \end_inset |
---|
| 812 | |
---|
| 813 | kde dolní index |
---|
| 814 | \begin_inset Formula $0$ |
---|
| 815 | \end_inset |
---|
| 816 | |
---|
| 817 | neznačí nulový čas, ale bod linearizace |
---|
| 818 | \begin_inset Formula $x_{0}$ |
---|
| 819 | \end_inset |
---|
| 820 | |
---|
| 821 | . |
---|
| 822 | Matice |
---|
| 823 | \begin_inset Formula $\overline{A}_{t}$ |
---|
| 824 | \end_inset |
---|
| 825 | |
---|
| 826 | vypočtená v bodě |
---|
| 827 | \begin_inset Formula $x_{0}$ |
---|
| 828 | \end_inset |
---|
| 829 | |
---|
| 830 | (složky |
---|
| 831 | \begin_inset Formula $x_{0}$ |
---|
| 832 | \end_inset |
---|
| 833 | |
---|
| 834 | budou opět značeny dolním indexem |
---|
| 835 | \begin_inset Formula $0$ |
---|
| 836 | \end_inset |
---|
| 837 | |
---|
| 838 | ) pak je |
---|
| 839 | \begin_inset Formula |
---|
| 840 | \[ |
---|
| 841 | \left[\begin{array}{ccccc} |
---|
| 842 | a & 0 & b\sin\vartheta_{0} & b\omega_{0}\cos\vartheta_{0} & -b\omega_{0}\vartheta_{0}\cos\vartheta_{0}\\ |
---|
| 843 | 0 & a & -b\cos\vartheta_{0} & b\omega_{0}\sin\vartheta_{0} & -b\omega_{0}\vartheta_{0}\sin\vartheta_{0}\\ |
---|
| 844 | -e\sin\vartheta_{0} & e\cos\vartheta_{0} & d & -e\left(i_{\beta,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right) & e\vartheta_{0}\left(i_{b,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right)\\ |
---|
| 845 | 0 & 0 & \Delta t & 1 & 0\\ |
---|
| 846 | 0 & 0 & 0 & 0 & 1 |
---|
| 847 | \end{array}\right] |
---|
| 848 | \] |
---|
| 849 | |
---|
| 850 | \end_inset |
---|
| 851 | |
---|
| 852 | |
---|
| 853 | \end_layout |
---|
| 854 | |
---|
| 855 | \begin_layout Standard |
---|
| 856 | Matici |
---|
| 857 | \begin_inset Formula $B_{t}$ |
---|
| 858 | \end_inset |
---|
| 859 | |
---|
| 860 | derivací |
---|
| 861 | \begin_inset Formula $f(x_{t},u_{t})$ |
---|
| 862 | \end_inset |
---|
| 863 | |
---|
| 864 | dle vstupů |
---|
| 865 | \begin_inset Formula $u_{t}$ |
---|
| 866 | \end_inset |
---|
| 867 | |
---|
| 868 | lze volit konstantní a časově nezávislou ve tvaru ( |
---|
| 869 | \begin_inset CommandInset ref |
---|
| 870 | LatexCommand ref |
---|
| 871 | reference "eq:matice-ekf-plny-stav" |
---|
| 872 | |
---|
| 873 | \end_inset |
---|
| 874 | |
---|
| 875 | ), protože funkce |
---|
| 876 | \begin_inset Formula $f$ |
---|
| 877 | \end_inset |
---|
| 878 | |
---|
| 879 | je ve vstupech |
---|
| 880 | \begin_inset Formula $u$ |
---|
| 881 | \end_inset |
---|
| 882 | |
---|
| 883 | lineární. |
---|
| 884 | \end_layout |
---|
| 885 | |
---|
| 886 | \begin_layout Subsubsection* |
---|
| 887 | Ztrátová funkce |
---|
| 888 | \end_layout |
---|
| 889 | |
---|
| 890 | \begin_layout Standard |
---|
| 891 | Protože chceme využít lineárně kvadratického algoritmu, je třeba formulovat |
---|
| 892 | ztrátovou funkci jako aditivní a kvadratickou, obecně ve tvaru |
---|
| 893 | \begin_inset Formula |
---|
| 894 | \begin{equation} |
---|
| 895 | \mathrm{E}\left\{ x_{T}^{T}Q_{T}x_{T}+\sum_{t=0}^{T-1}\left(x_{t}^{T}Q_{t}x_{t}+u_{t}^{T}R_{t}u_{t}\right)\right\} \label{eq:lq-kvadraticka-ztrata} |
---|
| 896 | \end{equation} |
---|
| 897 | |
---|
| 898 | \end_inset |
---|
| 899 | |
---|
| 900 | kde |
---|
| 901 | \begin_inset Formula $\mathbf{E}$ |
---|
| 902 | \end_inset |
---|
| 903 | |
---|
| 904 | značí očekávanou hodnotu, |
---|
| 905 | \begin_inset Formula $Q_{t}$ |
---|
| 906 | \end_inset |
---|
| 907 | |
---|
| 908 | a |
---|
| 909 | \begin_inset Formula $R_{t}$ |
---|
| 910 | \end_inset |
---|
| 911 | |
---|
| 912 | jsou penalizační matice stavu systému (splnění požadavků řízení) a penalizace |
---|
| 913 | vstupů. |
---|
| 914 | \end_layout |
---|
| 915 | |
---|
| 916 | \begin_layout Standard |
---|
| 917 | Hlavním požadavkem na systém je dosažení požadované hodnoty otáček |
---|
| 918 | \begin_inset Formula $\overline{\omega}_{t}$ |
---|
| 919 | \end_inset |
---|
| 920 | |
---|
| 921 | v čase |
---|
| 922 | \begin_inset Formula $t$ |
---|
| 923 | \end_inset |
---|
| 924 | |
---|
| 925 | . |
---|
| 926 | Výše navržená ztráta ( |
---|
| 927 | \begin_inset CommandInset ref |
---|
| 928 | LatexCommand ref |
---|
| 929 | reference "eq:lq-kvadraticka-ztrata" |
---|
| 930 | |
---|
| 931 | \end_inset |
---|
| 932 | |
---|
| 933 | ) však vede na řízení pouze na nulovou hodnotu odpovídající |
---|
| 934 | \begin_inset Formula $\overline{\omega}\equiv0$ |
---|
| 935 | \end_inset |
---|
| 936 | |
---|
| 937 | , pro řízení na nenulové požadované otáčky je třeba modifikovat stav systému |
---|
| 938 | a zavést substituci |
---|
| 939 | \begin_inset Formula |
---|
| 940 | \[ |
---|
| 941 | \psi_{t}=\omega_{t}-\overline{\omega}_{t} |
---|
| 942 | \] |
---|
| 943 | |
---|
| 944 | \end_inset |
---|
| 945 | |
---|
| 946 | a veličinu |
---|
| 947 | \begin_inset Formula $\psi$ |
---|
| 948 | \end_inset |
---|
| 949 | |
---|
| 950 | pak již řídíme na nulovou hodnotu. |
---|
| 951 | Tuto substituci, která závisí na parametru |
---|
| 952 | \begin_inset Formula $\overline{\omega}$ |
---|
| 953 | \end_inset |
---|
| 954 | |
---|
| 955 | , je třeba zanést do všech rovnic. |
---|
| 956 | Ve stavu systému veličina |
---|
| 957 | \begin_inset Formula $\psi_{t}$ |
---|
| 958 | \end_inset |
---|
| 959 | |
---|
| 960 | nahradí veličinu |
---|
| 961 | \begin_inset Formula $\omega_{t}$ |
---|
| 962 | \end_inset |
---|
| 963 | |
---|
| 964 | . |
---|
| 965 | Dále je třeba zahrnout i všechny konstantní členy, které v důsledku substituce |
---|
| 966 | vzniknou. |
---|
| 967 | \end_layout |
---|
| 968 | |
---|
| 969 | \begin_layout Standard |
---|
| 970 | Penalizační matici stavu systému v ( |
---|
| 971 | \begin_inset CommandInset ref |
---|
| 972 | LatexCommand ref |
---|
| 973 | reference "eq:lq-kvadraticka-ztrata" |
---|
| 974 | |
---|
| 975 | \end_inset |
---|
| 976 | |
---|
| 977 | ) budeme uvažovat nezávislou na čase |
---|
| 978 | \begin_inset Formula $Q_{t}=Q$ |
---|
| 979 | \end_inset |
---|
| 980 | |
---|
| 981 | pro všechna |
---|
| 982 | \begin_inset Formula $t$ |
---|
| 983 | \end_inset |
---|
| 984 | |
---|
| 985 | a ve tvaru |
---|
| 986 | \begin_inset Formula |
---|
| 987 | \begin{equation} |
---|
| 988 | Q=\left[\begin{array}{ccccc} |
---|
| 989 | 0 & 0 & 0 & 0 & 0\\ |
---|
| 990 | 0 & 0 & 0 & 0 & 0\\ |
---|
| 991 | 0 & 0 & q & 0 & 0\\ |
---|
| 992 | 0 & 0 & 0 & 0 & 0\\ |
---|
| 993 | 0 & 0 & 0 & 0 & 0 |
---|
| 994 | \end{array}\right]\label{eq:matice-Q-lq} |
---|
| 995 | \end{equation} |
---|
| 996 | |
---|
| 997 | \end_inset |
---|
| 998 | |
---|
| 999 | kde |
---|
| 1000 | \begin_inset Formula $q$ |
---|
| 1001 | \end_inset |
---|
| 1002 | |
---|
| 1003 | je pevně zvolená konstanta a matice |
---|
| 1004 | \begin_inset Formula $Q$ |
---|
| 1005 | \end_inset |
---|
| 1006 | |
---|
| 1007 | má již rozměr 5x5, protože byl stav rozšířen o konstantní člen v důsledku |
---|
| 1008 | linearizace. |
---|
| 1009 | Koncovou matici |
---|
| 1010 | \begin_inset Formula $Q_{T}$ |
---|
| 1011 | \end_inset |
---|
| 1012 | |
---|
| 1013 | budeme uvažovat nulovou. |
---|
| 1014 | \end_layout |
---|
| 1015 | |
---|
| 1016 | \begin_layout Standard |
---|
| 1017 | Dalším požadavkem je omezení na napětí -- vstupy do systému, vyjádřené pomocí |
---|
| 1018 | maximálního napětí |
---|
| 1019 | \begin_inset Formula $U_{max}$ |
---|
| 1020 | \end_inset |
---|
| 1021 | |
---|
| 1022 | , které je schopen poskytnout napájecí zdroj. |
---|
| 1023 | Toto omezení můžeme zasat jako |
---|
| 1024 | \begin_inset Formula |
---|
| 1025 | \begin{equation} |
---|
| 1026 | \left\Vert u_{t}\right\Vert \leq U_{max}\label{eq:omezeni} |
---|
| 1027 | \end{equation} |
---|
| 1028 | |
---|
| 1029 | \end_inset |
---|
| 1030 | |
---|
| 1031 | případně jako omezení na každou složku vektoru |
---|
| 1032 | \begin_inset Formula $u_{t}$ |
---|
| 1033 | \end_inset |
---|
| 1034 | |
---|
| 1035 | zvlášť. |
---|
| 1036 | Tento požadavek nelze přímo zapsat jako kvadratickou funkci a proto je |
---|
| 1037 | třeba vhodně zvolit matici |
---|
| 1038 | \begin_inset Formula $R_{t}$ |
---|
| 1039 | \end_inset |
---|
| 1040 | |
---|
| 1041 | v ( |
---|
| 1042 | \begin_inset CommandInset ref |
---|
| 1043 | LatexCommand ref |
---|
| 1044 | reference "eq:lq-kvadraticka-ztrata" |
---|
| 1045 | |
---|
| 1046 | \end_inset |
---|
| 1047 | |
---|
| 1048 | ) aby dostatečně penalizovala příliš velké hodnoty řízení |
---|
| 1049 | \begin_inset Formula $u_{t}$ |
---|
| 1050 | \end_inset |
---|
| 1051 | |
---|
| 1052 | a dále počítat s tím, že při přesažení hodnoty |
---|
| 1053 | \begin_inset Formula $U_{max}$ |
---|
| 1054 | \end_inset |
---|
| 1055 | |
---|
| 1056 | dojde k ořezu. |
---|
| 1057 | Penalizační matici řízení opět volíme nezávislou na čase, tj. |
---|
| 1058 | |
---|
| 1059 | \begin_inset Formula $R_{t}=R$ |
---|
| 1060 | \end_inset |
---|
| 1061 | |
---|
| 1062 | pro všechna |
---|
| 1063 | \begin_inset Formula $t$ |
---|
| 1064 | \end_inset |
---|
| 1065 | |
---|
| 1066 | , a ve tvaru |
---|
| 1067 | \begin_inset Formula |
---|
| 1068 | \[ |
---|
| 1069 | R=\left[\begin{array}{cc} |
---|
| 1070 | r & 0\\ |
---|
| 1071 | 0 & r |
---|
| 1072 | \end{array}\right] |
---|
| 1073 | \] |
---|
| 1074 | |
---|
| 1075 | \end_inset |
---|
| 1076 | |
---|
| 1077 | kde |
---|
| 1078 | \begin_inset Formula $r$ |
---|
| 1079 | \end_inset |
---|
| 1080 | |
---|
| 1081 | je zvolená konstanta. |
---|
| 1082 | \end_layout |
---|
| 1083 | |
---|
| 1084 | \begin_layout Subsubsection* |
---|
| 1085 | Substituované rovnice |
---|
| 1086 | \end_layout |
---|
| 1087 | |
---|
| 1088 | \begin_layout Standard |
---|
| 1089 | V důsledku substituce |
---|
| 1090 | \begin_inset Formula $\psi_{t}=\omega_{t}-\overline{\omega}_{t}$ |
---|
| 1091 | \end_inset |
---|
| 1092 | |
---|
| 1093 | se rovnice ( |
---|
| 1094 | \begin_inset CommandInset ref |
---|
| 1095 | LatexCommand ref |
---|
| 1096 | reference "eq:system" |
---|
| 1097 | |
---|
| 1098 | \end_inset |
---|
| 1099 | |
---|
| 1100 | ) změní na |
---|
| 1101 | \begin_inset Formula |
---|
| 1102 | \begin{eqnarray} |
---|
| 1103 | i_{\alpha,t+1} & = & ai_{\alpha,t}+b\left(\psi_{t}+\overline{\omega}_{t}\right)\sin\vartheta_{t}+cu_{\alpha,t}\nonumber \\ |
---|
| 1104 | i_{\beta,t+1} & = & ai_{\beta,t}-b\left(\psi_{t}+\overline{\omega}_{t}\right)\cos\vartheta_{t}+cu_{\beta,t}\label{eq:system-s-psi}\\ |
---|
| 1105 | \psi_{t+1} & = & d\psi_{t}+e\left(i_{b,t}\cos\vartheta_{t}-i_{\alpha,t}\sin\vartheta_{t}\right)\nonumber \\ |
---|
| 1106 | \vartheta_{t+1} & = & \vartheta_{t}+\Delta t\left(\psi_{t}+\overline{\omega}_{t}\right)\nonumber |
---|
| 1107 | \end{eqnarray} |
---|
| 1108 | |
---|
| 1109 | \end_inset |
---|
| 1110 | |
---|
| 1111 | předpokládáme-li, že pro pro požadované otáčky |
---|
| 1112 | \begin_inset Formula $\overline{\omega}$ |
---|
| 1113 | \end_inset |
---|
| 1114 | |
---|
| 1115 | platí |
---|
| 1116 | \begin_inset Formula $\overline{\omega}_{t+1}=d\overline{\omega}_{t}$ |
---|
| 1117 | \end_inset |
---|
| 1118 | |
---|
| 1119 | . |
---|
| 1120 | |
---|
| 1121 | \end_layout |
---|
| 1122 | |
---|
| 1123 | \begin_layout Standard |
---|
| 1124 | Derivováním těchto rovnic dle nového stavu (substituovaného) |
---|
| 1125 | \begin_inset Formula $\left(i_{\alpha,t},i_{\beta,t},\psi_{t},\vartheta_{t}\right)^{T}$ |
---|
| 1126 | \end_inset |
---|
| 1127 | |
---|
| 1128 | získáme matici |
---|
| 1129 | \begin_inset Formula |
---|
| 1130 | \[ |
---|
| 1131 | A_{t}=\left[\begin{array}{cccc} |
---|
| 1132 | a & 0 & b\sin\vartheta_{t} & b\left(\psi_{t}+\overline{\omega}_{t}\right)\cos\vartheta_{t}\\ |
---|
| 1133 | 0 & a & -b\cos\vartheta_{t} & b\left(\psi_{t}+\overline{\omega}_{t}\right)\sin\vartheta_{t}\\ |
---|
| 1134 | -e\sin\vartheta_{t} & e\cos\vartheta_{t} & d & -e\left(i_{\beta,t}\sin\vartheta_{t}+i_{\alpha,t}\cos\vartheta_{t}\right)\\ |
---|
| 1135 | 0 & 0 & \Delta t & 1 |
---|
| 1136 | \end{array}\right] |
---|
| 1137 | \] |
---|
| 1138 | |
---|
| 1139 | \end_inset |
---|
| 1140 | |
---|
| 1141 | která je hodnotově stejná s maticí |
---|
| 1142 | \begin_inset Formula $A_{t}$ |
---|
| 1143 | \end_inset |
---|
| 1144 | |
---|
| 1145 | získanou na základě původního nesubstituovaného stavu (tj. |
---|
| 1146 | s |
---|
| 1147 | \begin_inset Formula $x^{(3)}=\omega$ |
---|
| 1148 | \end_inset |
---|
| 1149 | |
---|
| 1150 | ). |
---|
| 1151 | \end_layout |
---|
| 1152 | |
---|
| 1153 | \begin_layout Standard |
---|
| 1154 | Konstantní člen |
---|
| 1155 | \begin_inset Formula $\gamma=f\left(x_{0}\right)-A_{t}x_{0}$ |
---|
| 1156 | \end_inset |
---|
| 1157 | |
---|
| 1158 | je však již jiný a závisí na hodnotě |
---|
| 1159 | \begin_inset Formula $\overline{\omega}_{t}$ |
---|
| 1160 | \end_inset |
---|
| 1161 | |
---|
| 1162 | , která do něj vstupuje jako časově proměnný parametr. |
---|
| 1163 | \begin_inset Formula |
---|
| 1164 | \begin{eqnarray*} |
---|
| 1165 | \gamma_{\overline{\omega}_{t}} & = & \left(\begin{array}{c} |
---|
| 1166 | -b\omega_{0}\vartheta_{0}\cos\vartheta_{0}+b\overline{\omega}_{t}\sin\vartheta_{0}\\ |
---|
| 1167 | -b\omega_{0}\vartheta_{0}\sin\vartheta_{0}-b\overline{\omega}_{t}\cos\vartheta_{0}\\ |
---|
| 1168 | e\vartheta_{0}\left(i_{\beta,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right)\\ |
---|
| 1169 | \Delta t\overline{\omega}_{t} |
---|
| 1170 | \end{array}\right) |
---|
| 1171 | \end{eqnarray*} |
---|
| 1172 | |
---|
| 1173 | \end_inset |
---|
| 1174 | |
---|
| 1175 | Výsledná matice |
---|
| 1176 | \begin_inset Formula $\overline{A}_{t}$ |
---|
| 1177 | \end_inset |
---|
| 1178 | |
---|
| 1179 | je pak ve tvaru |
---|
| 1180 | \end_layout |
---|
| 1181 | |
---|
| 1182 | \begin_layout Standard |
---|
| 1183 | \begin_inset Formula |
---|
| 1184 | \[ |
---|
| 1185 | \left[\begin{array}{ccccc} |
---|
| 1186 | a & 0 & b\sin\vartheta_{0} & b\omega_{0}\cos\vartheta_{0} & -b\omega_{0}\vartheta_{0}\cos\vartheta_{0}+b\overline{\omega}_{t}\sin\vartheta_{0}\\ |
---|
| 1187 | 0 & a & -b\cos\vartheta_{0} & b\omega_{0}\sin\vartheta_{0} & -b\omega_{0}\vartheta_{0}\sin\vartheta_{0}-b\overline{\omega}_{t}\cos\vartheta_{0}\\ |
---|
| 1188 | -e\sin\vartheta_{0} & e\cos\vartheta_{0} & d & -e\left(i_{\beta,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right) & e\vartheta_{0}\left(i_{\beta,0}\sin\vartheta_{0}+i_{\alpha,0}\cos\vartheta_{0}\right)\\ |
---|
| 1189 | 0 & 0 & \Delta t & 1 & \Delta t\overline{\omega}_{t}\\ |
---|
| 1190 | 0 & 0 & 0 & 0 & 1 |
---|
| 1191 | \end{array}\right] |
---|
| 1192 | \] |
---|
| 1193 | |
---|
| 1194 | \end_inset |
---|
| 1195 | |
---|
| 1196 | |
---|
| 1197 | \end_layout |
---|
| 1198 | |
---|
| 1199 | \begin_layout Subsubsection* |
---|
| 1200 | Bellmanova funkce |
---|
| 1201 | \begin_inset CommandInset label |
---|
| 1202 | LatexCommand label |
---|
| 1203 | name "sub:BellmanDP" |
---|
| 1204 | |
---|
| 1205 | \end_inset |
---|
| 1206 | |
---|
| 1207 | |
---|
| 1208 | \end_layout |
---|
| 1209 | |
---|
| 1210 | \begin_layout Standard |
---|
| 1211 | Cílem úlohy je minimalizovat ztrátovou funkci ( |
---|
| 1212 | \begin_inset CommandInset ref |
---|
| 1213 | LatexCommand ref |
---|
| 1214 | reference "eq:lq-kvadraticka-ztrata" |
---|
| 1215 | |
---|
| 1216 | \end_inset |
---|
| 1217 | |
---|
| 1218 | ). |
---|
| 1219 | Klasickým postupem pro řešení této úlohy je užítí Bellmanovy funkce a algoritmu |
---|
| 1220 | dynamického programování: |
---|
| 1221 | \end_layout |
---|
| 1222 | |
---|
| 1223 | \begin_layout Standard |
---|
| 1224 | V koncovém čase |
---|
| 1225 | \begin_inset Formula $T$ |
---|
| 1226 | \end_inset |
---|
| 1227 | |
---|
| 1228 | položíme |
---|
| 1229 | \begin_inset Formula |
---|
| 1230 | \begin{equation} |
---|
| 1231 | V_{T}\left(x_{T}\right)=0\label{eq:bellVkonec} |
---|
| 1232 | \end{equation} |
---|
| 1233 | |
---|
| 1234 | \end_inset |
---|
| 1235 | |
---|
| 1236 | a dále počítáme zpět v čase |
---|
| 1237 | \begin_inset Formula |
---|
| 1238 | \begin{equation} |
---|
| 1239 | V_{t-1}\left(x_{t-1},u_{t-1}\right)=\min_{u_{t-1}}\mathrm{E}\left\{ x_{t}^{T}Q_{t}x_{t}+u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \label{eq:bellVrek} |
---|
| 1240 | \end{equation} |
---|
| 1241 | |
---|
| 1242 | \end_inset |
---|
| 1243 | |
---|
| 1244 | pro |
---|
| 1245 | \begin_inset Formula $t$ |
---|
| 1246 | \end_inset |
---|
| 1247 | |
---|
| 1248 | od |
---|
| 1249 | \begin_inset Formula $T-1$ |
---|
| 1250 | \end_inset |
---|
| 1251 | |
---|
| 1252 | do |
---|
| 1253 | \begin_inset Formula $1$ |
---|
| 1254 | \end_inset |
---|
| 1255 | |
---|
| 1256 | , kde střední hodnota je podmíněna |
---|
| 1257 | \begin_inset Formula $\mathcal{I}_{t}$ |
---|
| 1258 | \end_inset |
---|
| 1259 | |
---|
| 1260 | , které reprezentuje současně dostupnou informaci o systému zahrnující všechna |
---|
| 1261 | měření a řídící vstupy do času |
---|
| 1262 | \begin_inset Formula $t$ |
---|
| 1263 | \end_inset |
---|
| 1264 | |
---|
| 1265 | . |
---|
| 1266 | \end_layout |
---|
| 1267 | |
---|
| 1268 | \begin_layout Standard |
---|
| 1269 | Uvažovanou kvadratickou ztrátu za jeden časový krok |
---|
| 1270 | \begin_inset Formula |
---|
| 1271 | \[ |
---|
| 1272 | x_{t}^{T}Q_{t}x_{t}+u_{t}^{T}R_{t}u_{t} |
---|
| 1273 | \] |
---|
| 1274 | |
---|
| 1275 | \end_inset |
---|
| 1276 | |
---|
| 1277 | pří konkrétní volbě matice |
---|
| 1278 | \begin_inset Formula $Q$ |
---|
| 1279 | \end_inset |
---|
| 1280 | |
---|
| 1281 | ve tvaru ( |
---|
| 1282 | \begin_inset CommandInset ref |
---|
| 1283 | LatexCommand ref |
---|
| 1284 | reference "eq:matice-Q-lq" |
---|
| 1285 | |
---|
| 1286 | \end_inset |
---|
| 1287 | |
---|
| 1288 | ) přejde na |
---|
| 1289 | \begin_inset Formula |
---|
| 1290 | \[ |
---|
| 1291 | q\left(x_{t}^{(3)}-\overline{\omega}_{t}\right)+u_{t}^{T}R_{t}u_{t} |
---|
| 1292 | \] |
---|
| 1293 | |
---|
| 1294 | \end_inset |
---|
| 1295 | |
---|
| 1296 | kde horní index v závorce značí složku vektoru. |
---|
| 1297 | Pak je možno rovnici ( |
---|
| 1298 | \begin_inset CommandInset ref |
---|
| 1299 | LatexCommand ref |
---|
| 1300 | reference "eq:bellVrek" |
---|
| 1301 | |
---|
| 1302 | \end_inset |
---|
| 1303 | |
---|
| 1304 | ) dále zjednodušit |
---|
| 1305 | \end_layout |
---|
| 1306 | |
---|
| 1307 | \begin_layout Standard |
---|
| 1308 | \begin_inset Formula |
---|
| 1309 | \begin{eqnarray} |
---|
| 1310 | V_{t-1}\left(x_{t-1},u_{t-1}\right) & \text{=} & \min_{u_{t-1}}\mathrm{E}\left\{ x_{t}^{T}Q_{t}x_{t}+u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \nonumber \\ |
---|
| 1311 | & = & \min_{u_{t-1}}\left(\mathrm{E}\left\{ q\left(x_{t}^{(3)}-\overline{\omega}_{t}\right)\right\} +\mathrm{E}\left\{ u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \right)\nonumber \\ |
---|
| 1312 | & \text{=} & \min_{u_{t-1}}\left(q\mathrm{E}\left\{ \left(x_{t}^{(3)}\right)^{2}+\overline{\omega}_{t}^{2}+2x_{t}^{(3)}\overline{\omega}_{t}\right\} +\mathrm{E}\left\{ u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \right)\label{eq:eq:bellman-sPome}\\ |
---|
| 1313 | & = & \min_{u_{t-1}}\left(q\left(\mathrm{E}\left\{ \left(x_{t}^{(3)}\right)^{2}\right\} +\mathrm{E}\left\{ \overline{\omega}_{t}^{2}\right\} +\mathrm{E}\left\{ 2x_{t}^{(3)}\overline{\omega}_{t}\right\} \right)+\mathrm{E}\left\{ u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \right)\nonumber \\ |
---|
| 1314 | & = & \min_{u_{t-1}}\left(q\left(\left(\hat{x}_{t}^{(3)}\right)^{2}+\mathrm{Var}\left(x_{t}^{(3)}\right)+\overline{\omega}_{t}^{2}+2\hat{x}_{t}^{(3)}\overline{\omega}_{t}\right)+\mathrm{E}\left\{ u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \right)\nonumber \\ |
---|
| 1315 | & = & \min_{u_{t-1}}\left(q\left(\hat{x}_{t}^{(3)}-\overline{\omega}_{t}\right)+q\mathrm{Var}\left(x_{t}^{(3)}\right)+\mathrm{E}\left\{ u_{t}^{T}R_{t}u_{t}+V_{t}\left(x_{t},u_{t}\right)\mid\mathcal{I}_{t}\right\} \right)\nonumber |
---|
| 1316 | \end{eqnarray} |
---|
| 1317 | |
---|
| 1318 | \end_inset |
---|
| 1319 | |
---|
| 1320 | kde |
---|
| 1321 | \begin_inset Formula $\hat{x}$ |
---|
| 1322 | \end_inset |
---|
| 1323 | |
---|
| 1324 | označuje střední hodnotu |
---|
| 1325 | \begin_inset Formula $x$ |
---|
| 1326 | \end_inset |
---|
| 1327 | |
---|
| 1328 | a dále jsme využili toho, že |
---|
| 1329 | \begin_inset Formula $\overline{\omega}_{t}$ |
---|
| 1330 | \end_inset |
---|
| 1331 | |
---|
| 1332 | je daný parametr a tedy je pro výpočet střední hodnoty konstantou a vztahu |
---|
| 1333 | |
---|
| 1334 | \begin_inset Formula $\mathrm{Var}\left(x\right)=\mathrm{E}\left\{ x^{2}\right\} -\left(\mathrm{E}\left\{ x\right\} \right)^{2}$ |
---|
| 1335 | \end_inset |
---|
| 1336 | |
---|
| 1337 | . |
---|
| 1338 | Tedy ve výpočtu Bellmanovy funkce |
---|
| 1339 | \begin_inset Formula $V$ |
---|
| 1340 | \end_inset |
---|
| 1341 | |
---|
| 1342 | v rovnici ( |
---|
| 1343 | \begin_inset CommandInset ref |
---|
| 1344 | LatexCommand ref |
---|
| 1345 | reference "eq:bellVrek" |
---|
| 1346 | |
---|
| 1347 | \end_inset |
---|
| 1348 | |
---|
| 1349 | ) můžeme náhodnou veličinu |
---|
| 1350 | \begin_inset Formula $x_{t}$ |
---|
| 1351 | \end_inset |
---|
| 1352 | |
---|
| 1353 | nahradit její střední hodnotou |
---|
| 1354 | \begin_inset Formula $\hat{x}_{t}$ |
---|
| 1355 | \end_inset |
---|
| 1356 | |
---|
| 1357 | , když navíc zahrneme do rovnice varianci třetí složky |
---|
| 1358 | \begin_inset Formula $x_{t}$ |
---|
| 1359 | \end_inset |
---|
| 1360 | |
---|
| 1361 | , tj. |
---|
| 1362 | varianci otáček stroje. |
---|
| 1363 | \end_layout |
---|
| 1364 | |
---|
| 1365 | \begin_layout Subsubsection* |
---|
| 1366 | Výpočet lineárně kvadratického řízení |
---|
| 1367 | \end_layout |
---|
| 1368 | |
---|
| 1369 | \begin_layout Standard |
---|
| 1370 | Pro samotný výpočet lineárně kvadratického řízení je užito následujících |
---|
| 1371 | rovnic |
---|
| 1372 | \begin_inset Formula |
---|
| 1373 | \begin{eqnarray*} |
---|
| 1374 | K_{T} & = & Q_{T}\\ |
---|
| 1375 | K_{t} & = & A_{t}^{T}\left(K_{t+1}-K_{t+1}B_{t}\left(B_{t}^{T}K_{t+1}B_{t}+R_{t}\right)^{-1}B_{t}^{T}K_{t+1}\right)A_{t}+Q_{t}\\ |
---|
| 1376 | L_{t} & = & -\left(B_{t}^{T}K_{t+1}B_{t}+R_{t}\right)^{-1}B_{t}^{T}K_{t+1}A_{t}\\ |
---|
| 1377 | u_{t} & = & L_{t}x_{t} |
---|
| 1378 | \end{eqnarray*} |
---|
| 1379 | |
---|
| 1380 | \end_inset |
---|
| 1381 | |
---|
| 1382 | Tyto rovnice by měly být napočítávány v čase zpět (od koncového času) až |
---|
| 1383 | do aktuálního času. |
---|
| 1384 | Protože ale systém vznikl linearizací v nějakém reprezentativním bodě, |
---|
| 1385 | který se s vývojem systému mění, je třeba celý výpočet znovu provést v |
---|
| 1386 | každém časovém kroku. |
---|
| 1387 | Proto je výhodnější si výpočet usnadnit například využitím ubíhajícího |
---|
| 1388 | horizontu. |
---|
| 1389 | |
---|
| 1390 | \end_layout |
---|
| 1391 | |
---|
| 1392 | \begin_layout Standard |
---|
| 1393 | Při výpočtu řízení |
---|
| 1394 | \begin_inset Formula $u_{t}$ |
---|
| 1395 | \end_inset |
---|
| 1396 | |
---|
| 1397 | pomocí matice |
---|
| 1398 | \begin_inset Formula $L_{t}$ |
---|
| 1399 | \end_inset |
---|
| 1400 | |
---|
| 1401 | je třeba dosadit za vektor |
---|
| 1402 | \begin_inset Formula $x_{t}$ |
---|
| 1403 | \end_inset |
---|
| 1404 | |
---|
| 1405 | správné hodnoty, konkrétně v důsledku nenulové požadované hodnoty |
---|
| 1406 | \begin_inset Formula $\overline{\omega}$ |
---|
| 1407 | \end_inset |
---|
| 1408 | |
---|
| 1409 | za třetí složku vektoru |
---|
| 1410 | \begin_inset Formula $x_{t}$ |
---|
| 1411 | \end_inset |
---|
| 1412 | |
---|
| 1413 | není dosazena hodnota |
---|
| 1414 | \begin_inset Formula $\omega_{t}$ |
---|
| 1415 | \end_inset |
---|
| 1416 | |
---|
| 1417 | , ale substituovaná |
---|
| 1418 | \begin_inset Formula $\psi_{t}=\omega_{t}-\overline{\omega}_{t}$ |
---|
| 1419 | \end_inset |
---|
| 1420 | |
---|
| 1421 | . |
---|
| 1422 | \end_layout |
---|
| 1423 | |
---|
| 1424 | \begin_layout Standard |
---|
| 1425 | Předchozí výpočet pomocí Riccatiho rovnice však není příliš vhodným z numerickýc |
---|
| 1426 | h důvodů ( |
---|
| 1427 | \series bold |
---|
| 1428 | nějaká reference |
---|
| 1429 | \series default |
---|
| 1430 | ). |
---|
| 1431 | Místo něj pro praktické výpočty použijeme algoritmus lineárně kvadratického |
---|
| 1432 | řízení založený na QR rozkladu ( |
---|
| 1433 | \series bold |
---|
| 1434 | reference |
---|
| 1435 | \series default |
---|
| 1436 | ). |
---|
| 1437 | Tento algoritmus má lepší numerické vlastnosti, umožňuje snadnější výpočet |
---|
| 1438 | maticové inverze (inverze pouze trojúhelníkové matice) a lze pomocí něj |
---|
| 1439 | implementovat i složitější kvadratickou ztrátovou funkci (nejen dva členy |
---|
| 1440 | pro penalizaci stavu a vstupů). |
---|
| 1441 | \end_layout |
---|
| 1442 | |
---|
| 1443 | \begin_layout Standard |
---|
| 1444 | Postup je založen na přepisu kvadratické ztráty do tvaru |
---|
| 1445 | \begin_inset Formula |
---|
| 1446 | \[ |
---|
| 1447 | x_{t+1}^{T}Q_{t}x_{t+1}+u_{t}^{T}R_{t}u_{t}=x_{t+1}^{T}\sqrt{Q_{t}}^{T}\sqrt{Q_{t}}x_{t+1}+u_{t}^{T}\sqrt{R_{t}}^{T}\sqrt{R_{t}}u_{t} |
---|
| 1448 | \] |
---|
| 1449 | |
---|
| 1450 | \end_inset |
---|
| 1451 | |
---|
| 1452 | kde |
---|
| 1453 | \begin_inset Formula $\sqrt{}$ |
---|
| 1454 | \end_inset |
---|
| 1455 | |
---|
| 1456 | je vhodná maticová odmocnina. |
---|
| 1457 | A tedy v každém časovém kroku |
---|
| 1458 | \begin_inset Formula $t$ |
---|
| 1459 | \end_inset |
---|
| 1460 | |
---|
| 1461 | minimalizujeme funkci |
---|
| 1462 | \begin_inset Formula |
---|
| 1463 | \[ |
---|
| 1464 | x_{t+1}^{T}\sqrt{Q_{t}}^{T}\sqrt{Q_{t}}x_{t+1}+u_{t}^{T}\sqrt{R_{t}}^{T}\sqrt{R_{t}}u_{t}+x_{t+1}^{T}\sqrt{S_{t}}^{T}\sqrt{S_{t}}x_{t+1} |
---|
| 1465 | \] |
---|
| 1466 | |
---|
| 1467 | \end_inset |
---|
| 1468 | |
---|
| 1469 | kde |
---|
| 1470 | \begin_inset Formula $S_{t}$ |
---|
| 1471 | \end_inset |
---|
| 1472 | |
---|
| 1473 | reprezentuje ztrátu v následujících časových krocích až do konce časového |
---|
| 1474 | horizontu. |
---|
| 1475 | Do tohoto kvadratického výrazu je možno dostadit model vývoje pro |
---|
| 1476 | \begin_inset Formula $x_{t+1}=A_{t}x_{t}+B_{t}u_{t}$ |
---|
| 1477 | \end_inset |
---|
| 1478 | |
---|
| 1479 | |
---|
| 1480 | \begin_inset Formula |
---|
| 1481 | \[ |
---|
| 1482 | \left(Ax_{t}+B_{t}u_{t}\right)^{T}\sqrt{Q_{t}}\sqrt{Q_{t}}\left(Ax_{t}+B_{t}u_{t}\right)+u_{t}^{T}\sqrt{R_{t}}^{T}\sqrt{R_{t}}u_{t}+\left(Ax_{t}+B_{t}u_{t}\right)^{T}\sqrt{S_{t}}^{T}\sqrt{S_{t}}\left(Ax_{t}+B_{t}u_{t}\right) |
---|
| 1483 | \] |
---|
| 1484 | |
---|
| 1485 | \end_inset |
---|
| 1486 | |
---|
| 1487 | a následně jej zapsat maticově ve tvaru |
---|
| 1488 | \begin_inset Formula |
---|
| 1489 | \[ |
---|
| 1490 | \left(\begin{array}{c} |
---|
| 1491 | u_{t}\\ |
---|
| 1492 | x_{t} |
---|
| 1493 | \end{array}\right)^{T}\left[\begin{array}{cc} |
---|
| 1494 | \sqrt{Q_{t}}B_{t} & \sqrt{Q_{t}}A_{t}\\ |
---|
| 1495 | \sqrt{R_{t}} & 0\\ |
---|
| 1496 | \sqrt{S_{t}}B_{t} & \sqrt{S_{t}}A_{t} |
---|
| 1497 | \end{array}\right]^{T}\underset{Z}{\underbrace{\left[\begin{array}{cc} |
---|
| 1498 | \sqrt{Q_{t}}B_{t} & \sqrt{Q_{t}}A_{t}\\ |
---|
| 1499 | \sqrt{R_{t}} & 0\\ |
---|
| 1500 | \sqrt{S_{t}}B_{t} & \sqrt{S_{t}}A_{t} |
---|
| 1501 | \end{array}\right]}}\left(\begin{array}{c} |
---|
| 1502 | u_{t}\\ |
---|
| 1503 | x_{t} |
---|
| 1504 | \end{array}\right) |
---|
| 1505 | \] |
---|
| 1506 | |
---|
| 1507 | \end_inset |
---|
| 1508 | |
---|
| 1509 | na matici |
---|
| 1510 | \begin_inset Formula $Z$ |
---|
| 1511 | \end_inset |
---|
| 1512 | |
---|
| 1513 | následně aplikujeme QR rozklad, to jest |
---|
| 1514 | \begin_inset Formula $Z=Q_{Z}R_{Z}$ |
---|
| 1515 | \end_inset |
---|
| 1516 | |
---|
| 1517 | a předchozí vztah upravíme na tvar |
---|
| 1518 | \begin_inset Formula |
---|
| 1519 | \[ |
---|
| 1520 | \left(\begin{array}{c} |
---|
| 1521 | u_{t}\\ |
---|
| 1522 | x_{t} |
---|
| 1523 | \end{array}\right)^{T}Z^{T}Z\left(\begin{array}{c} |
---|
| 1524 | u_{t}\\ |
---|
| 1525 | x_{t} |
---|
| 1526 | \end{array}\right)=\left(\begin{array}{c} |
---|
| 1527 | u_{t}\\ |
---|
| 1528 | x_{t} |
---|
| 1529 | \end{array}\right)^{T}R_{Z}^{T}Q_{Z}^{T}Q_{Z}R_{Z}\left(\begin{array}{c} |
---|
| 1530 | u_{t}\\ |
---|
| 1531 | x_{t} |
---|
| 1532 | \end{array}\right)=\left(\begin{array}{c} |
---|
| 1533 | u_{t}\\ |
---|
| 1534 | x_{t} |
---|
| 1535 | \end{array}\right)^{T}R_{Z}^{T}R_{Z}\left(\begin{array}{c} |
---|
| 1536 | u_{t}\\ |
---|
| 1537 | x_{t} |
---|
| 1538 | \end{array}\right) |
---|
| 1539 | \] |
---|
| 1540 | |
---|
| 1541 | \end_inset |
---|
| 1542 | |
---|
| 1543 | Matice |
---|
| 1544 | \begin_inset Formula $R_{Z}$ |
---|
| 1545 | \end_inset |
---|
| 1546 | |
---|
| 1547 | je v horním trojúhelníkovém tvaru, tedy blokově zapsáno |
---|
| 1548 | \begin_inset Formula |
---|
| 1549 | \[ |
---|
| 1550 | R_{Z}=\left[\begin{array}{cc} |
---|
| 1551 | R_{uu} & R_{ux}\\ |
---|
| 1552 | 0 & R_{xx} |
---|
| 1553 | \end{array}\right] |
---|
| 1554 | \] |
---|
| 1555 | |
---|
| 1556 | \end_inset |
---|
| 1557 | |
---|
| 1558 | Ztrátu nyní můžeme zapsat jako |
---|
| 1559 | \begin_inset Formula |
---|
| 1560 | \begin{eqnarray*} |
---|
| 1561 | \left(\begin{array}{c} |
---|
| 1562 | u_{t}\\ |
---|
| 1563 | x_{t} |
---|
| 1564 | \end{array}\right)^{T}R_{Z}^{T}R_{Z}\left(\begin{array}{c} |
---|
| 1565 | u_{t}\\ |
---|
| 1566 | x_{t} |
---|
| 1567 | \end{array}\right) & = & \left(\begin{array}{c} |
---|
| 1568 | R_{uu}u_{t}+R_{ux}x_{t}\\ |
---|
| 1569 | R_{xx}x_{t} |
---|
| 1570 | \end{array}\right)^{T}\left(\begin{array}{c} |
---|
| 1571 | R_{uu}u_{t}+R_{ux}x_{t}\\ |
---|
| 1572 | R_{xx}x_{t} |
---|
| 1573 | \end{array}\right)\\ |
---|
| 1574 | & = & \left(R_{uu}u_{t}+R_{ux}x_{t}\right)^{T}\left(R_{uu}u_{t}+R_{ux}x_{t}\right)+x_{t}^{T}R_{xx}^{T}R_{xx}x_{t} |
---|
| 1575 | \end{eqnarray*} |
---|
| 1576 | |
---|
| 1577 | \end_inset |
---|
| 1578 | |
---|
| 1579 | kterou, vzhledem k její kvadratičnosti a nezávislosti druhého členu na |
---|
| 1580 | \begin_inset Formula $u_{t}$ |
---|
| 1581 | \end_inset |
---|
| 1582 | |
---|
| 1583 | , zřejmě minimalizujeme volbou |
---|
| 1584 | \begin_inset Formula $u_{t}$ |
---|
| 1585 | \end_inset |
---|
| 1586 | |
---|
| 1587 | takovou, že |
---|
| 1588 | \begin_inset Formula $\left(R_{uu}u_{t}+R_{ux}x_{t}\right)=0$ |
---|
| 1589 | \end_inset |
---|
| 1590 | |
---|
| 1591 | a tedy volíme |
---|
| 1592 | \begin_inset Formula |
---|
| 1593 | \[ |
---|
| 1594 | u_{t}=-R_{uu}^{-1}R_{ux}x_{t} |
---|
| 1595 | \] |
---|
| 1596 | |
---|
| 1597 | \end_inset |
---|
| 1598 | |
---|
| 1599 | Matici |
---|
| 1600 | \begin_inset Formula $R_{xx}^{T}R_{xx}$ |
---|
| 1601 | \end_inset |
---|
| 1602 | |
---|
| 1603 | pak použijeme do předchozího časového kroku jako novou matici |
---|
| 1604 | \begin_inset Formula $S$ |
---|
| 1605 | \end_inset |
---|
| 1606 | |
---|
| 1607 | . |
---|
| 1608 | \end_layout |
---|
| 1609 | |
---|
| 1610 | \begin_layout Subsubsection |
---|
| 1611 | LQ řízení pro redukovaný model |
---|
| 1612 | \begin_inset CommandInset label |
---|
| 1613 | LatexCommand label |
---|
| 1614 | name "sub:LQ-řízení-pro-red-model" |
---|
| 1615 | |
---|
| 1616 | \end_inset |
---|
| 1617 | |
---|
| 1618 | |
---|
| 1619 | \end_layout |
---|
| 1620 | |
---|
| 1621 | \begin_layout Standard |
---|
| 1622 | Pro redukovaný systém samozřejmě platí vše uvedené v předchozím odstavci, |
---|
| 1623 | řízení je ale komplikovanější, protože ve funkci popisující vývoj systému |
---|
| 1624 | explicitně nevystupuje řízení |
---|
| 1625 | \begin_inset Formula $u_{t}$ |
---|
| 1626 | \end_inset |
---|
| 1627 | |
---|
| 1628 | . |
---|
| 1629 | Je tedy třeba vhodným způsobem tento problém vyřešit. |
---|
| 1630 | Jednou z možností je zřetězení dvou LQ regulátory. |
---|
| 1631 | V prvním kroku považovat za řízení proudy |
---|
| 1632 | \begin_inset Formula $i_{\alpha,\beta}$ |
---|
| 1633 | \end_inset |
---|
| 1634 | |
---|
| 1635 | a tedy tento první regulátor by na výstupu generoval požadované proudy |
---|
| 1636 | |
---|
| 1637 | \begin_inset Formula $\overline{i}_{\alpha,\beta}$ |
---|
| 1638 | \end_inset |
---|
| 1639 | |
---|
| 1640 | . |
---|
| 1641 | Druhý regulátor by pak na základě rovnic pro vývoj proudů a referenčních |
---|
| 1642 | hodnot proudů |
---|
| 1643 | \begin_inset Formula $\overline{i}_{\alpha,\beta}$ |
---|
| 1644 | \end_inset |
---|
| 1645 | |
---|
| 1646 | nalezl řízení |
---|
| 1647 | \begin_inset Formula $u_{\alpha,\beta}$ |
---|
| 1648 | \end_inset |
---|
| 1649 | |
---|
| 1650 | . |
---|
| 1651 | \end_layout |
---|
| 1652 | |
---|
| 1653 | \begin_layout Subsubsection* |
---|
| 1654 | Matice pro redukovaný model |
---|
| 1655 | \end_layout |
---|
| 1656 | |
---|
| 1657 | \begin_layout Standard |
---|
| 1658 | Protože ve funkci |
---|
| 1659 | \begin_inset Formula $f\left(x_{t},y_{t}\right)$ |
---|
| 1660 | \end_inset |
---|
| 1661 | |
---|
| 1662 | v rovnicích ( |
---|
| 1663 | \begin_inset CommandInset ref |
---|
| 1664 | LatexCommand ref |
---|
| 1665 | reference "eq:systemrovnice-reduk" |
---|
| 1666 | |
---|
| 1667 | \end_inset |
---|
| 1668 | |
---|
| 1669 | ) a ( |
---|
| 1670 | \begin_inset CommandInset ref |
---|
| 1671 | LatexCommand ref |
---|
| 1672 | reference "eq:systemsf-reduk" |
---|
| 1673 | |
---|
| 1674 | \end_inset |
---|
| 1675 | |
---|
| 1676 | ) explicitně nevystupuje řízení |
---|
| 1677 | \begin_inset Formula $u_{t}$ |
---|
| 1678 | \end_inset |
---|
| 1679 | |
---|
| 1680 | , je třeba zvolit trochu odlišný přístup, než pro plný model. |
---|
| 1681 | Řízení budeme navrhovat ve dvou krocích. |
---|
| 1682 | V prvním kroku budeme předpokládat, že vstupem jsou proudy |
---|
| 1683 | \begin_inset Formula $i_{\alpha\beta}$ |
---|
| 1684 | \end_inset |
---|
| 1685 | |
---|
| 1686 | a lineárně kvadratický algoritmus bude na svém výstupu produkovat požadované |
---|
| 1687 | hodnoty těchto proudů |
---|
| 1688 | \begin_inset Formula $\overline{i}_{\alpha\beta}$ |
---|
| 1689 | \end_inset |
---|
| 1690 | |
---|
| 1691 | . |
---|
| 1692 | V dalším kroku druhý lineárně kvadratický algoritmus na základě požadovaných |
---|
| 1693 | proudů |
---|
| 1694 | \begin_inset Formula $\overline{i}_{\alpha\beta}$ |
---|
| 1695 | \end_inset |
---|
| 1696 | |
---|
| 1697 | již navrhne hodnotu napětí |
---|
| 1698 | \begin_inset Formula $u_{\alpha\beta}$ |
---|
| 1699 | \end_inset |
---|
| 1700 | |
---|
| 1701 | . |
---|
| 1702 | \end_layout |
---|
| 1703 | |
---|
| 1704 | \begin_layout Standard |
---|
| 1705 | Dále provedeme ještě drobné zjednodušení a funkci |
---|
| 1706 | \begin_inset Formula $f\left(x_{t},y_{t}\right)$ |
---|
| 1707 | \end_inset |
---|
| 1708 | |
---|
| 1709 | rozdělíme na dvě části |
---|
| 1710 | \begin_inset Formula |
---|
| 1711 | \[ |
---|
| 1712 | f\left(x_{t},y_{t}\right)=\left(\begin{array}{c} |
---|
| 1713 | d\omega_{t}\\ |
---|
| 1714 | \vartheta_{t}+\Delta t\omega_{t} |
---|
| 1715 | \end{array}\right)+\left(\begin{array}{c} |
---|
| 1716 | e\left(i_{b,t}\cos\vartheta_{t}-i_{\alpha,t}\sin\vartheta_{t}\right)\\ |
---|
| 1717 | 0 |
---|
| 1718 | \end{array}\right) |
---|
| 1719 | \] |
---|
| 1720 | |
---|
| 1721 | \end_inset |
---|
| 1722 | |
---|
| 1723 | Matici |
---|
| 1724 | \begin_inset Formula $A_{t}$ |
---|
| 1725 | \end_inset |
---|
| 1726 | |
---|
| 1727 | pak položíme rovnou první maticí první, lineární, části systému |
---|
| 1728 | \begin_inset Formula |
---|
| 1729 | \[ |
---|
| 1730 | A=\left[\begin{array}{cc} |
---|
| 1731 | d & 0\\ |
---|
| 1732 | \Delta t & 1 |
---|
| 1733 | \end{array}\right] |
---|
| 1734 | \] |
---|
| 1735 | |
---|
| 1736 | \end_inset |
---|
| 1737 | |
---|
| 1738 | a matici |
---|
| 1739 | \begin_inset Formula $B_{t}$ |
---|
| 1740 | \end_inset |
---|
| 1741 | |
---|
| 1742 | pak získáme linearizací druhé části jako |
---|
| 1743 | \begin_inset Formula |
---|
| 1744 | \[ |
---|
| 1745 | B_{t}=\left[\begin{array}{cc} |
---|
| 1746 | -e\sin\vartheta_{t} & e\cos\vartheta_{t}\\ |
---|
| 1747 | 0 & 0 |
---|
| 1748 | \end{array}\right] |
---|
| 1749 | \] |
---|
| 1750 | |
---|
| 1751 | \end_inset |
---|
| 1752 | |
---|
| 1753 | Tento postup neodpovídá přesně postupu odvození derivací užitému pro plný |
---|
| 1754 | stav. |
---|
| 1755 | Jeho výhodou však je, že již není třeba přidávat konstantní členy jako |
---|
| 1756 | důsledek linearizace. |
---|
| 1757 | Snadněji se také zahrne požadavek na nenulovou referenční hodnotu |
---|
| 1758 | \begin_inset Formula $\overline{\omega}$ |
---|
| 1759 | \end_inset |
---|
| 1760 | |
---|
| 1761 | . |
---|
| 1762 | Následně je užito lineárně kvadratického algoritmu s výše popsanými maticemi. |
---|
| 1763 | \end_layout |
---|
| 1764 | |
---|
| 1765 | \begin_layout Standard |
---|
| 1766 | Ve druhém kroku pak na základě referenčních hodnot proudů |
---|
| 1767 | \begin_inset Formula $\overline{i}_{\alpha\beta}$ |
---|
| 1768 | \end_inset |
---|
| 1769 | |
---|
| 1770 | nalezneme požadované řízení |
---|
| 1771 | \begin_inset Formula $u_{\alpha\beta}$ |
---|
| 1772 | \end_inset |
---|
| 1773 | |
---|
| 1774 | . |
---|
| 1775 | Využijeme k tomu rovnic pro funkci |
---|
| 1776 | \begin_inset Formula $h(x_{t},y_{t},u_{t})$ |
---|
| 1777 | \end_inset |
---|
| 1778 | |
---|
| 1779 | viz ( |
---|
| 1780 | \begin_inset CommandInset ref |
---|
| 1781 | LatexCommand ref |
---|
| 1782 | reference "eq:systemsf-reduk" |
---|
| 1783 | |
---|
| 1784 | \end_inset |
---|
| 1785 | |
---|
| 1786 | ) |
---|
| 1787 | \begin_inset Formula |
---|
| 1788 | \[ |
---|
| 1789 | h(x_{t},y_{t},u_{t})=\left(\begin{array}{c} |
---|
| 1790 | ai_{\alpha,t}+b\omega_{t}\sin\vartheta_{t}+cu_{\alpha,t}\\ |
---|
| 1791 | ai_{\beta,t}-b\omega_{t}\cos\vartheta_{t}+cu_{\beta,t} |
---|
| 1792 | \end{array}\right) |
---|
| 1793 | \] |
---|
| 1794 | |
---|
| 1795 | \end_inset |
---|
| 1796 | |
---|
| 1797 | které jsou v proudech |
---|
| 1798 | \begin_inset Formula $i_{\alpha\beta}$ |
---|
| 1799 | \end_inset |
---|
| 1800 | |
---|
| 1801 | i napětích |
---|
| 1802 | \begin_inset Formula $u_{\alpha\beta}$ |
---|
| 1803 | \end_inset |
---|
| 1804 | |
---|
| 1805 | lineární a lze opět použít lineárně kvadratický algoritmus. |
---|
| 1806 | Členy |
---|
| 1807 | \begin_inset Formula $\pm b\omega_{t}\begin{array}{c} |
---|
| 1808 | \sin\\ |
---|
| 1809 | \cos |
---|
| 1810 | \end{array}\vartheta_{t}$ |
---|
| 1811 | \end_inset |
---|
| 1812 | |
---|
| 1813 | zde pak vystupují jako konstanty a projeví se jako korekce vynásobená konstanto |
---|
| 1814 | u |
---|
| 1815 | \begin_inset Formula $\frac{1}{c}$ |
---|
| 1816 | \end_inset |
---|
| 1817 | |
---|
| 1818 | odečtená od výsledku. |
---|
| 1819 | \end_layout |
---|
| 1820 | |
---|
| 1821 | \begin_layout Section |
---|
| 1822 | LQG s hyperstavem |
---|
| 1823 | \end_layout |
---|
| 1824 | |
---|
| 1825 | \begin_layout Standard |
---|
| 1826 | Následující postup s hyperstavem vychází s článku (Kim2006) [Stochastic |
---|
| 1827 | Feedback Controller Design Considering the Dual Effect, Kim J., Rock S. |
---|
| 1828 | M., 2006]. |
---|
| 1829 | V tomto článku je však narozdíl od následujícího postupu používán spojitý |
---|
| 1830 | čas. |
---|
| 1831 | \end_layout |
---|
| 1832 | |
---|
| 1833 | \begin_layout Standard |
---|
| 1834 | Jedná se o analogii s LQG v předchozí části, s tím rozdílem, že použijem |
---|
| 1835 | EKF algoritmus v jistém smyslu jakoby dvakrát. |
---|
| 1836 | Protože tímto přístupem již značně narůstá dimenzionalita úlohy je z výpočetníc |
---|
| 1837 | h důvodů výhodnější užití redukovaného modelu, i přes komplikace, které |
---|
| 1838 | způsobuje při řízení. |
---|
| 1839 | \end_layout |
---|
| 1840 | |
---|
| 1841 | \begin_layout Subsubsection* |
---|
| 1842 | Hyperstav |
---|
| 1843 | \end_layout |
---|
| 1844 | |
---|
| 1845 | \begin_layout Standard |
---|
| 1846 | Vyjdeme z redukovaného stavu |
---|
| 1847 | \begin_inset Formula |
---|
| 1848 | \[ |
---|
| 1849 | x_{t}=\left(\omega_{t},\vartheta_{t}\right)^{T} |
---|
| 1850 | \] |
---|
| 1851 | |
---|
| 1852 | \end_inset |
---|
| 1853 | |
---|
| 1854 | a na něj formálně aplikujeme EKF. |
---|
| 1855 | Tím získáme, kromě odhadu stavu |
---|
| 1856 | \begin_inset Formula $x_{t}$ |
---|
| 1857 | \end_inset |
---|
| 1858 | |
---|
| 1859 | i odhad jeho variance v podobě matice |
---|
| 1860 | \begin_inset Formula |
---|
| 1861 | \[ |
---|
| 1862 | P=\left[\begin{array}{cc} |
---|
| 1863 | P_{\omega} & P_{\omega\vartheta}\\ |
---|
| 1864 | P_{\omega\vartheta} & P_{\vartheta} |
---|
| 1865 | \end{array}\right] |
---|
| 1866 | \] |
---|
| 1867 | |
---|
| 1868 | \end_inset |
---|
| 1869 | |
---|
| 1870 | a současně rovnice EKF ( |
---|
| 1871 | \begin_inset CommandInset ref |
---|
| 1872 | LatexCommand ref |
---|
| 1873 | reference "eq:rovnice-ekf-timeupd" |
---|
| 1874 | |
---|
| 1875 | \end_inset |
---|
| 1876 | |
---|
| 1877 | ) a ( |
---|
| 1878 | \begin_inset CommandInset ref |
---|
| 1879 | LatexCommand ref |
---|
| 1880 | reference "eq:rovnice-ekf-dataupd" |
---|
| 1881 | |
---|
| 1882 | \end_inset |
---|
| 1883 | |
---|
| 1884 | ) představují předpis pro výpočet |
---|
| 1885 | \begin_inset Formula $P$ |
---|
| 1886 | \end_inset |
---|
| 1887 | |
---|
| 1888 | : |
---|
| 1889 | \end_layout |
---|
| 1890 | |
---|
| 1891 | \begin_layout Standard |
---|
| 1892 | \begin_inset Formula |
---|
| 1893 | \begin{eqnarray} |
---|
| 1894 | \overline{P} & = & APA^{T}+V\nonumber \\ |
---|
| 1895 | S & = & C\overline{P}C^{T}+W\nonumber \\ |
---|
| 1896 | K & = & \overline{P}C^{T}S^{-1}\label{eq:ekf-stav}\\ |
---|
| 1897 | P^{+} & \text{=} & \left(I-KC\right)\overline{P}\nonumber |
---|
| 1898 | \end{eqnarray} |
---|
| 1899 | |
---|
| 1900 | \end_inset |
---|
| 1901 | |
---|
| 1902 | kde jsou z důvodu jednoduššího zápisu vynechány časové indexy |
---|
| 1903 | \begin_inset Formula $t$ |
---|
| 1904 | \end_inset |
---|
| 1905 | |
---|
| 1906 | a místo nic je užit horní index |
---|
| 1907 | \begin_inset Formula $+$ |
---|
| 1908 | \end_inset |
---|
| 1909 | |
---|
| 1910 | pro hodnotu v následujícím čase |
---|
| 1911 | \begin_inset Formula $t+1$ |
---|
| 1912 | \end_inset |
---|
| 1913 | |
---|
| 1914 | . |
---|
| 1915 | \end_layout |
---|
| 1916 | |
---|
| 1917 | \begin_layout Standard |
---|
| 1918 | Nyní definujeme |
---|
| 1919 | \emph on |
---|
| 1920 | hyperstav |
---|
| 1921 | \emph default |
---|
| 1922 | |
---|
| 1923 | \begin_inset Formula $\xi_{t}$ |
---|
| 1924 | \end_inset |
---|
| 1925 | |
---|
| 1926 | v čase |
---|
| 1927 | \begin_inset Formula $t$ |
---|
| 1928 | \end_inset |
---|
| 1929 | |
---|
| 1930 | jako |
---|
| 1931 | \begin_inset Formula |
---|
| 1932 | \[ |
---|
| 1933 | \xi_{t}=\left(\omega_{t},\vartheta_{t},P_{\omega},P_{\omega\vartheta},P_{\vartheta}\right)^{T} |
---|
| 1934 | \] |
---|
| 1935 | |
---|
| 1936 | \end_inset |
---|
| 1937 | |
---|
| 1938 | Na hyperstav již aplikujeme algoritmus pro LQG, jak byl popsán v předchozí |
---|
| 1939 | části. |
---|
| 1940 | Problém však představuje nalezení matice derivací |
---|
| 1941 | \begin_inset Formula $A$ |
---|
| 1942 | \end_inset |
---|
| 1943 | |
---|
| 1944 | , protože je třeba derivovat maticové rovnice pro výpočet EKF ( |
---|
| 1945 | \begin_inset CommandInset ref |
---|
| 1946 | LatexCommand ref |
---|
| 1947 | reference "eq:ekf-stav" |
---|
| 1948 | |
---|
| 1949 | \end_inset |
---|
| 1950 | |
---|
| 1951 | ) pro stavu |
---|
| 1952 | \begin_inset Formula $x$ |
---|
| 1953 | \end_inset |
---|
| 1954 | |
---|
| 1955 | . |
---|
| 1956 | Jedním ze způsobů jak je to možné provést je derivovat každou z rovnic |
---|
| 1957 | ( |
---|
| 1958 | \begin_inset CommandInset ref |
---|
| 1959 | LatexCommand ref |
---|
| 1960 | reference "eq:ekf-stav" |
---|
| 1961 | |
---|
| 1962 | \end_inset |
---|
| 1963 | |
---|
| 1964 | ) dle jednotlivých složek vektoru |
---|
| 1965 | \begin_inset Formula $\xi$ |
---|
| 1966 | \end_inset |
---|
| 1967 | |
---|
| 1968 | : |
---|
| 1969 | \begin_inset Formula |
---|
| 1970 | \begin{eqnarray} |
---|
| 1971 | \frac{\partial\overline{P}}{\partial\xi_{i}} & = & \frac{\partial A}{\partial\xi_{i}}PA^{T}+A\frac{\partial P}{\partial\xi_{i}}A^{T}+AP\frac{\partial A^{T}}{\partial\xi_{i}}\nonumber \\ |
---|
| 1972 | \frac{\partial S}{\partial\xi_{i}} & = & \frac{\partial C}{\partial\xi_{i}}\overline{P}C^{T}+C\frac{\partial\overline{P}}{\partial\xi_{i}}C^{T}+C\overline{P}\frac{\partial C^{T}}{\partial\xi_{i}}\nonumber \\ |
---|
| 1973 | \frac{\partial K}{\partial\xi_{i}} & = & \frac{\partial\overline{P}}{\partial\xi_{i}}C^{T}S^{-1}+\overline{P}\frac{\partial C^{T}}{\partial\xi_{i}}S^{-1}-\overline{P}C^{T}S^{-1}\frac{\partial S}{\partial\xi_{i}}S^{-1}\label{eq:ekf-stav-derivace}\\ |
---|
| 1974 | \frac{\partial P^{+}}{\partial\xi_{i}} & \text{=} & \frac{\partial\overline{P}}{\partial\xi_{i}}-\frac{\partial K}{\partial\xi_{i}}C\overline{P}-K\frac{\partial C}{\partial\xi_{i}}\overline{P}-KC\frac{\partial\overline{P}}{\partial\xi_{i}}\nonumber |
---|
| 1975 | \end{eqnarray} |
---|
| 1976 | |
---|
| 1977 | \end_inset |
---|
| 1978 | |
---|
| 1979 | kde |
---|
| 1980 | \begin_inset Formula $\frac{\partial}{\partial\xi_{i}}$ |
---|
| 1981 | \end_inset |
---|
| 1982 | |
---|
| 1983 | představuje zápis derivace dle |
---|
| 1984 | \begin_inset Formula $i$ |
---|
| 1985 | \end_inset |
---|
| 1986 | |
---|
| 1987 | -té složky vektoru |
---|
| 1988 | \begin_inset Formula $\xi$ |
---|
| 1989 | \end_inset |
---|
| 1990 | |
---|
| 1991 | a matice |
---|
| 1992 | \begin_inset Formula $V$ |
---|
| 1993 | \end_inset |
---|
| 1994 | |
---|
| 1995 | a |
---|
| 1996 | \begin_inset Formula $W$ |
---|
| 1997 | \end_inset |
---|
| 1998 | |
---|
| 1999 | uvažujeme jako konstanty v |
---|
| 2000 | \begin_inset Formula $\xi$ |
---|
| 2001 | \end_inset |
---|
| 2002 | |
---|
| 2003 | . |
---|
| 2004 | Matice linearizovaného vývoje hyperstavu |
---|
| 2005 | \begin_inset Formula $A_{hyp}$ |
---|
| 2006 | \end_inset |
---|
| 2007 | |
---|
| 2008 | bude mít nyní blokový tvar |
---|
| 2009 | \begin_inset Formula |
---|
| 2010 | \[ |
---|
| 2011 | A_{hyp}=\left[\begin{array}{ccccc} |
---|
| 2012 | A_{1} & A_{2} & 0 & 0 & 0\\ |
---|
| 2013 | \left(\frac{\partial P^{+}}{\partial\omega}\right)^{sl} & \left(\frac{\partial P^{+}}{\partial\vartheta}\right)^{sl} & \left(\frac{\partial P^{+}}{\partial P_{\omega}}\right)^{sl} & \left(\frac{\partial P^{+}}{\partial P_{\omega\vartheta}}\right)^{sl} & \left(\frac{\partial P^{+}}{\partial P_{\vartheta}}\right)^{sl} |
---|
| 2014 | \end{array}\right] |
---|
| 2015 | \] |
---|
| 2016 | |
---|
| 2017 | \end_inset |
---|
| 2018 | |
---|
| 2019 | kde |
---|
| 2020 | \begin_inset Formula $A_{i}$ |
---|
| 2021 | \end_inset |
---|
| 2022 | |
---|
| 2023 | představuje |
---|
| 2024 | \begin_inset Formula $i$ |
---|
| 2025 | \end_inset |
---|
| 2026 | |
---|
| 2027 | -tý sloupec matice |
---|
| 2028 | \begin_inset Formula $A$ |
---|
| 2029 | \end_inset |
---|
| 2030 | |
---|
| 2031 | , zápis |
---|
| 2032 | \begin_inset Formula $0$ |
---|
| 2033 | \end_inset |
---|
| 2034 | |
---|
| 2035 | je sloupec nul vhodné délky a parciální derivace |
---|
| 2036 | \begin_inset Formula $P^{+}$ |
---|
| 2037 | \end_inset |
---|
| 2038 | |
---|
| 2039 | dle složky |
---|
| 2040 | \begin_inset Formula $\xi_{i}$ |
---|
| 2041 | \end_inset |
---|
| 2042 | |
---|
| 2043 | v závorce s horním indexem |
---|
| 2044 | \emph on |
---|
| 2045 | sl |
---|
| 2046 | \emph default |
---|
| 2047 | |
---|
| 2048 | \begin_inset Formula $\left(\frac{\partial P^{+}}{\partial\xi_{i}}\right)^{sl}$ |
---|
| 2049 | \end_inset |
---|
| 2050 | |
---|
| 2051 | je myšlena v tom smyslu, že po vypočtení příslušné derivace |
---|
| 2052 | \begin_inset Formula $\frac{\partial P^{+}}{\partial\xi_{i}}$ |
---|
| 2053 | \end_inset |
---|
| 2054 | |
---|
| 2055 | z rovnice ( |
---|
| 2056 | \begin_inset CommandInset ref |
---|
| 2057 | LatexCommand ref |
---|
| 2058 | reference "eq:ekf-stav-derivace" |
---|
| 2059 | |
---|
| 2060 | \end_inset |
---|
| 2061 | |
---|
| 2062 | ) jsou z této matice vybrány 3 z jejích 4 prvků tvořící horní nebo dolní |
---|
| 2063 | trojúhelník a zapísány je ve smyslu tvorby vektoru hyperstavu do sloupce: |
---|
| 2064 | \begin_inset Formula |
---|
| 2065 | \[ |
---|
| 2066 | \frac{\partial P^{+}}{\partial\xi_{i}}=\left[\begin{array}{cc} |
---|
| 2067 | \frac{\partial P_{\omega}^{+}}{\partial\xi_{i}} & \frac{\partial P_{\omega\vartheta}^{+}}{\partial\xi_{i}}\\ |
---|
| 2068 | \frac{\partial P_{\omega\vartheta}^{+}}{\partial\xi_{i}} & \frac{\partial P_{\vartheta}^{+}}{\partial\xi_{i}} |
---|
| 2069 | \end{array}\right] |
---|
| 2070 | \] |
---|
| 2071 | |
---|
| 2072 | \end_inset |
---|
| 2073 | |
---|
| 2074 | |
---|
| 2075 | \begin_inset Formula |
---|
| 2076 | \[ |
---|
| 2077 | \left(\frac{\partial P^{+}}{\partial\xi_{i}}\right)^{sl}=\left(\begin{array}{ccc} |
---|
| 2078 | \frac{\partial P_{\omega}^{+}}{\partial\xi_{i}} & \frac{\partial P_{\omega\vartheta}^{+}}{\partial\xi_{i}} & \frac{\partial P_{\vartheta}^{+}}{\partial\xi_{i}}\end{array}\right)^{T} |
---|
| 2079 | \] |
---|
| 2080 | |
---|
| 2081 | \end_inset |
---|
| 2082 | |
---|
| 2083 | |
---|
| 2084 | \end_layout |
---|
| 2085 | |
---|
| 2086 | \begin_layout Standard |
---|
| 2087 | Matici |
---|
| 2088 | \begin_inset Formula $A_{hyp}$ |
---|
| 2089 | \end_inset |
---|
| 2090 | |
---|
| 2091 | vzniklou předchozím postupem již můžeme použít v algoritmu EKF pro hyperstav. |
---|
| 2092 | Jako matici pozorování |
---|
| 2093 | \begin_inset Formula $C_{hyp}$ |
---|
| 2094 | \end_inset |
---|
| 2095 | |
---|
| 2096 | použijeme původní matici |
---|
| 2097 | \begin_inset Formula $C$ |
---|
| 2098 | \end_inset |
---|
| 2099 | |
---|
| 2100 | pouze doplněnou nulami na vhodný rozměr. |
---|
| 2101 | Pro lineárně kvadratické řízení platí opět totéž, co pro jednoduché (tj. |
---|
| 2102 | bez hyperstavu) a matici |
---|
| 2103 | \begin_inset Formula $A_{hyp}$ |
---|
| 2104 | \end_inset |
---|
| 2105 | |
---|
| 2106 | je tedy třeba rozšířit zahrnutím konstantních členů, dále je třeba ošetřit |
---|
| 2107 | substitucí řízení na nenulové požadované otáčky |
---|
| 2108 | \begin_inset Formula $\overline{\omega}$ |
---|
| 2109 | \end_inset |
---|
| 2110 | |
---|
| 2111 | . |
---|
| 2112 | |
---|
| 2113 | \end_layout |
---|
| 2114 | |
---|
| 2115 | \begin_layout Standard |
---|
| 2116 | Protože uvažujeme redukovaný model je třeba užít zřetězení dvou LQ regulátorů. |
---|
| 2117 | Výhodou využití hyperstavu ale je, že máme k dispozici i odhady variancí |
---|
| 2118 | |
---|
| 2119 | \begin_inset Formula $P$ |
---|
| 2120 | \end_inset |
---|
| 2121 | |
---|
| 2122 | původního stavu a tedy je možno zahrnout do kritéria například penalizaci |
---|
| 2123 | |
---|
| 2124 | \begin_inset Formula $P_{\omega}$ |
---|
| 2125 | \end_inset |
---|
| 2126 | |
---|
| 2127 | , která vystupuje v Bellmanově funkci viz vzorec ( |
---|
| 2128 | \begin_inset CommandInset ref |
---|
| 2129 | LatexCommand ref |
---|
| 2130 | reference "eq:eq:bellman-sPome" |
---|
| 2131 | |
---|
| 2132 | \end_inset |
---|
| 2133 | |
---|
| 2134 | ). |
---|
| 2135 | \end_layout |
---|
| 2136 | |
---|
| 2137 | \begin_layout Subsubsection* |
---|
| 2138 | Plný model |
---|
| 2139 | \end_layout |
---|
| 2140 | |
---|
| 2141 | \begin_layout Standard |
---|
| 2142 | Analogicky lze postupovat i pro plný model, všechny odpovídající matice |
---|
| 2143 | však budou podstatně větší, protože velikost hyperstavu narůstá řádově |
---|
| 2144 | kvadraticky. |
---|
| 2145 | |
---|
| 2146 | \end_layout |
---|
| 2147 | |
---|
| 2148 | \begin_layout Standard |
---|
| 2149 | Tedy pro stav |
---|
| 2150 | \begin_inset Formula |
---|
| 2151 | \[ |
---|
| 2152 | x_{t}=\left(i_{\alpha,t},i_{\beta,t},\omega_{t},\vartheta_{t}\right)^{T} |
---|
| 2153 | \] |
---|
| 2154 | |
---|
| 2155 | \end_inset |
---|
| 2156 | |
---|
| 2157 | vypočteme z EKF kovarianční matici |
---|
| 2158 | \begin_inset Formula |
---|
| 2159 | \[ |
---|
| 2160 | P=\left[\begin{array}{cccc} |
---|
| 2161 | P_{5} & P_{6} & P_{8} & P_{11}\\ |
---|
| 2162 | P_{6} & P_{7} & P_{9} & P_{12}\\ |
---|
| 2163 | P_{8} & P_{9} & P_{10} & P_{13}\\ |
---|
| 2164 | P_{11} & P_{12} & P_{13} & P_{14} |
---|
| 2165 | \end{array}\right] |
---|
| 2166 | \] |
---|
| 2167 | |
---|
| 2168 | \end_inset |
---|
| 2169 | |
---|
| 2170 | a definujeme |
---|
| 2171 | \emph on |
---|
| 2172 | hyperstav |
---|
| 2173 | \emph default |
---|
| 2174 | |
---|
| 2175 | \begin_inset Formula $\xi_{t}$ |
---|
| 2176 | \end_inset |
---|
| 2177 | |
---|
| 2178 | v čase |
---|
| 2179 | \begin_inset Formula $t$ |
---|
| 2180 | \end_inset |
---|
| 2181 | |
---|
| 2182 | jako |
---|
| 2183 | \begin_inset Formula |
---|
| 2184 | \[ |
---|
| 2185 | \xi_{t}=\left(i_{\alpha,t},i_{\beta,t},\omega_{t},\vartheta_{t},P_{5},P_{6},P_{7},P_{8},P_{9},P_{10},P_{11},P_{12},P_{13},P_{14}\right)^{T} |
---|
| 2186 | \] |
---|
| 2187 | |
---|
| 2188 | \end_inset |
---|
| 2189 | |
---|
| 2190 | Rovnice pro výpočet matice |
---|
| 2191 | \begin_inset Formula $P$ |
---|
| 2192 | \end_inset |
---|
| 2193 | |
---|
| 2194 | , a tedy i jejích prvků |
---|
| 2195 | \begin_inset Formula $P_{i}$ |
---|
| 2196 | \end_inset |
---|
| 2197 | |
---|
| 2198 | , jsou formálně shodné s rovnicemi pro redukovaný model, pouze rozměry vystupují |
---|
| 2199 | cích matic jsou větší. |
---|
| 2200 | A matice |
---|
| 2201 | \begin_inset Formula $A_{hyp}$ |
---|
| 2202 | \end_inset |
---|
| 2203 | |
---|
| 2204 | je ve tvaru |
---|
| 2205 | \begin_inset Formula |
---|
| 2206 | \[ |
---|
| 2207 | A_{hyp}=\left[\begin{array}{c} |
---|
| 2208 | \begin{split}A\quad & \quad0\end{split} |
---|
| 2209 | \\ |
---|
| 2210 | \left(\frac{\partial P^{+}}{\partial P_{i}}\right)_{i\in\left\{ 1\ldots14\right\} }^{sl} |
---|
| 2211 | \end{array}\right] |
---|
| 2212 | \] |
---|
| 2213 | |
---|
| 2214 | \end_inset |
---|
| 2215 | |
---|
| 2216 | |
---|
| 2217 | \end_layout |
---|
| 2218 | |
---|
| 2219 | \begin_layout Section |
---|
| 2220 | Experimenty |
---|
| 2221 | \end_layout |
---|
| 2222 | |
---|
| 2223 | \begin_layout Subsection* |
---|
| 2224 | Použité nastavení experimentů |
---|
| 2225 | \end_layout |
---|
| 2226 | |
---|
| 2227 | \begin_layout Standard |
---|
| 2228 | Pro simulování chování PMSM byly použity dva typy simulátorů. |
---|
| 2229 | Prvním byla pouze jednoduchá implementace rovnic popisujících PMSM. |
---|
| 2230 | Druhou testovanou možností bylo využití simulátoru PMSM ( |
---|
| 2231 | \series bold |
---|
| 2232 | reference |
---|
| 2233 | \series default |
---|
| 2234 | ). |
---|
| 2235 | \end_layout |
---|
| 2236 | |
---|
| 2237 | \begin_layout Standard |
---|
| 2238 | Testování probíhalo na horizontu 120000 časových vzorků, což odpovídá 15 |
---|
| 2239 | \emph on |
---|
| 2240 | s |
---|
| 2241 | \emph default |
---|
| 2242 | . |
---|
| 2243 | Ve všech případech byly užity odhadovací a řídící algoritmy předpokládající |
---|
| 2244 | stejnou indukčnost v osách |
---|
| 2245 | \emph on |
---|
| 2246 | d |
---|
| 2247 | \emph default |
---|
| 2248 | a |
---|
| 2249 | \emph on |
---|
| 2250 | q |
---|
| 2251 | \emph default |
---|
| 2252 | . |
---|
| 2253 | ( |
---|
| 2254 | \emph on |
---|
| 2255 | pro různé indukčnosti jsou složitější rovnice, tedy mnohem složitější matice |
---|
| 2256 | derivací a velmi těžko se napočítávají kompenzace v důsledku konstantních |
---|
| 2257 | členů |
---|
| 2258 | \emph default |
---|
| 2259 | ). |
---|
| 2260 | Testování probíhalo na různých profilech požadovaných (referenčních) otáček: |
---|
| 2261 | \end_layout |
---|
| 2262 | |
---|
| 2263 | \begin_layout Description |
---|
| 2264 | ( |
---|
| 2265 | \begin_inset Formula $0$ |
---|
| 2266 | \end_inset |
---|
| 2267 | |
---|
| 2268 | ) nulové požadované otáčky pro všechna |
---|
| 2269 | \begin_inset Formula $t$ |
---|
| 2270 | \end_inset |
---|
| 2271 | |
---|
| 2272 | |
---|
| 2273 | \end_layout |
---|
| 2274 | |
---|
| 2275 | \begin_layout Description |
---|
| 2276 | ( |
---|
| 2277 | \begin_inset Formula $\pm1$ |
---|
| 2278 | \end_inset |
---|
| 2279 | |
---|
| 2280 | ) trojúhelníkové pulzy v rozmezí |
---|
| 2281 | \begin_inset Formula $\pm1$ |
---|
| 2282 | \end_inset |
---|
| 2283 | |
---|
| 2284 | |
---|
| 2285 | \end_layout |
---|
| 2286 | |
---|
| 2287 | \begin_layout Description |
---|
| 2288 | ( |
---|
| 2289 | \begin_inset Formula $\pm10$ |
---|
| 2290 | \end_inset |
---|
| 2291 | |
---|
| 2292 | ) trojúhelníkové pulzy v rozmezí |
---|
| 2293 | \begin_inset Formula $\pm10$ |
---|
| 2294 | \end_inset |
---|
| 2295 | |
---|
| 2296 | |
---|
| 2297 | \end_layout |
---|
| 2298 | |
---|
| 2299 | \begin_layout Description |
---|
| 2300 | ( |
---|
| 2301 | \begin_inset Formula $\pm200$ |
---|
| 2302 | \end_inset |
---|
| 2303 | |
---|
| 2304 | ) trojúhelníkové pulzy v rozmezí |
---|
| 2305 | \begin_inset Formula $\pm200$ |
---|
| 2306 | \end_inset |
---|
| 2307 | |
---|
| 2308 | |
---|
| 2309 | \end_layout |
---|
| 2310 | |
---|
| 2311 | \begin_layout Standard |
---|
| 2312 | Dále pak byl testován i vliv špatného počátečního odhadu polohy (úhlu natočení) |
---|
| 2313 | |
---|
| 2314 | \begin_inset Formula $\vartheta_{0}$ |
---|
| 2315 | \end_inset |
---|
| 2316 | |
---|
| 2317 | . |
---|
| 2318 | |
---|
| 2319 | \end_layout |
---|
| 2320 | |
---|
| 2321 | \begin_layout Standard |
---|
| 2322 | Při použití simulátoru PMSM se velmi často vyskytovaly nedostatky způsobené |
---|
| 2323 | úbytky napětí, proto byla testována i verze upraveného simulátoru, která |
---|
| 2324 | se snažila úbytky kompenzovat. |
---|
| 2325 | |
---|
| 2326 | \end_layout |
---|
| 2327 | |
---|
| 2328 | \begin_layout Standard |
---|
| 2329 | Testovány byly celkem čtyři různé modely: redukovaný model s hyperstavem |
---|
| 2330 | i bez něj a plný model s hyperstavem a bez něj. |
---|
| 2331 | \end_layout |
---|
| 2332 | |
---|
| 2333 | \begin_layout Subsection* |
---|
| 2334 | Jednoduchý simulátor PMSM na základě rovnic |
---|
| 2335 | \end_layout |
---|
| 2336 | |
---|
| 2337 | \begin_layout Standard |
---|
| 2338 | Průběhy otáček |
---|
| 2339 | \begin_inset Formula $\omega$ |
---|
| 2340 | \end_inset |
---|
| 2341 | |
---|
| 2342 | a polohy |
---|
| 2343 | \begin_inset Formula $\vartheta$ |
---|
| 2344 | \end_inset |
---|
| 2345 | |
---|
| 2346 | pro simulace na jednoduchém simulátoru zachycují grafy na obrázcích ( |
---|
| 2347 | \begin_inset CommandInset ref |
---|
| 2348 | LatexCommand ref |
---|
| 2349 | reference "fig_grafym_ref0" |
---|
| 2350 | |
---|
| 2351 | \end_inset |
---|
| 2352 | |
---|
| 2353 | ), ( |
---|
| 2354 | \begin_inset CommandInset ref |
---|
| 2355 | LatexCommand ref |
---|
| 2356 | reference "fig_grafym_ref1" |
---|
| 2357 | |
---|
| 2358 | \end_inset |
---|
| 2359 | |
---|
| 2360 | ), ( |
---|
| 2361 | \begin_inset CommandInset ref |
---|
| 2362 | LatexCommand ref |
---|
| 2363 | reference "fig_grafym_ref10" |
---|
| 2364 | |
---|
| 2365 | \end_inset |
---|
| 2366 | |
---|
| 2367 | ) a ( |
---|
| 2368 | \begin_inset CommandInset ref |
---|
| 2369 | LatexCommand ref |
---|
| 2370 | reference "fig_grafym_ref200" |
---|
| 2371 | |
---|
| 2372 | \end_inset |
---|
| 2373 | |
---|
| 2374 | ). |
---|
| 2375 | \end_layout |
---|
| 2376 | |
---|
| 2377 | \begin_layout Standard |
---|
| 2378 | \begin_inset Float figure |
---|
| 2379 | wide false |
---|
| 2380 | sideways false |
---|
| 2381 | status collapsed |
---|
| 2382 | |
---|
| 2383 | \begin_layout Plain Layout |
---|
| 2384 | \align center |
---|
| 2385 | \begin_inset Tabular |
---|
| 2386 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 2387 | <features tabularvalignment="middle"> |
---|
| 2388 | <column alignment="center" valignment="top" width="0"> |
---|
| 2389 | <column alignment="center" valignment="top" width="0"> |
---|
| 2390 | <row> |
---|
| 2391 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2392 | \begin_inset Text |
---|
| 2393 | |
---|
| 2394 | \begin_layout Plain Layout |
---|
| 2395 | \begin_inset Graphics |
---|
| 2396 | filename grafy/lqplots/pm2r0s1.eps |
---|
| 2397 | scale 40 |
---|
| 2398 | |
---|
| 2399 | \end_inset |
---|
| 2400 | |
---|
| 2401 | |
---|
| 2402 | \end_layout |
---|
| 2403 | |
---|
| 2404 | \end_inset |
---|
| 2405 | </cell> |
---|
| 2406 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2407 | \begin_inset Text |
---|
| 2408 | |
---|
| 2409 | \begin_layout Plain Layout |
---|
| 2410 | \begin_inset Graphics |
---|
| 2411 | filename grafy/lqplots/pm2r0s2.eps |
---|
| 2412 | scale 40 |
---|
| 2413 | |
---|
| 2414 | \end_inset |
---|
| 2415 | |
---|
| 2416 | |
---|
| 2417 | \end_layout |
---|
| 2418 | |
---|
| 2419 | \end_inset |
---|
| 2420 | </cell> |
---|
| 2421 | </row> |
---|
| 2422 | <row> |
---|
| 2423 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2424 | \begin_inset Text |
---|
| 2425 | |
---|
| 2426 | \begin_layout Plain Layout |
---|
| 2427 | 1 -- redukovaný model |
---|
| 2428 | \end_layout |
---|
| 2429 | |
---|
| 2430 | \end_inset |
---|
| 2431 | </cell> |
---|
| 2432 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2433 | \begin_inset Text |
---|
| 2434 | |
---|
| 2435 | \begin_layout Plain Layout |
---|
| 2436 | 2 -- redukovaný model s hyperstavem |
---|
| 2437 | \end_layout |
---|
| 2438 | |
---|
| 2439 | \end_inset |
---|
| 2440 | </cell> |
---|
| 2441 | </row> |
---|
| 2442 | <row> |
---|
| 2443 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2444 | \begin_inset Text |
---|
| 2445 | |
---|
| 2446 | \begin_layout Plain Layout |
---|
| 2447 | \begin_inset Graphics |
---|
| 2448 | filename grafy/lqplots/pm2r0s3.eps |
---|
| 2449 | scale 40 |
---|
| 2450 | |
---|
| 2451 | \end_inset |
---|
| 2452 | |
---|
| 2453 | |
---|
| 2454 | \end_layout |
---|
| 2455 | |
---|
| 2456 | \end_inset |
---|
| 2457 | </cell> |
---|
| 2458 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2459 | \begin_inset Text |
---|
| 2460 | |
---|
| 2461 | \begin_layout Plain Layout |
---|
| 2462 | \begin_inset Graphics |
---|
| 2463 | filename grafy/lqplots/pm2r0s4.eps |
---|
| 2464 | scale 40 |
---|
| 2465 | |
---|
| 2466 | \end_inset |
---|
| 2467 | |
---|
| 2468 | |
---|
| 2469 | \end_layout |
---|
| 2470 | |
---|
| 2471 | \end_inset |
---|
| 2472 | </cell> |
---|
| 2473 | </row> |
---|
| 2474 | <row> |
---|
| 2475 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 2476 | \begin_inset Text |
---|
| 2477 | |
---|
| 2478 | \begin_layout Plain Layout |
---|
| 2479 | 3 -- plný model |
---|
| 2480 | \end_layout |
---|
| 2481 | |
---|
| 2482 | \end_inset |
---|
| 2483 | </cell> |
---|
| 2484 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2485 | \begin_inset Text |
---|
| 2486 | |
---|
| 2487 | \begin_layout Plain Layout |
---|
| 2488 | 4 -- plný model s hyperstavem |
---|
| 2489 | \end_layout |
---|
| 2490 | |
---|
| 2491 | \end_inset |
---|
| 2492 | </cell> |
---|
| 2493 | </row> |
---|
| 2494 | </lyxtabular> |
---|
| 2495 | |
---|
| 2496 | \end_inset |
---|
| 2497 | |
---|
| 2498 | |
---|
| 2499 | \begin_inset Caption |
---|
| 2500 | |
---|
| 2501 | \begin_layout Plain Layout |
---|
| 2502 | Průběhy otáček |
---|
| 2503 | \begin_inset Formula $\omega$ |
---|
| 2504 | \end_inset |
---|
| 2505 | |
---|
| 2506 | a polohy |
---|
| 2507 | \begin_inset Formula $\vartheta$ |
---|
| 2508 | \end_inset |
---|
| 2509 | |
---|
| 2510 | pro simulace na jednoduchém simulátoru s nulovými požadovanými otáčkami |
---|
| 2511 | \end_layout |
---|
| 2512 | |
---|
| 2513 | \end_inset |
---|
| 2514 | |
---|
| 2515 | |
---|
| 2516 | \begin_inset CommandInset label |
---|
| 2517 | LatexCommand label |
---|
| 2518 | name "fig_grafym_ref0" |
---|
| 2519 | |
---|
| 2520 | \end_inset |
---|
| 2521 | |
---|
| 2522 | |
---|
| 2523 | \end_layout |
---|
| 2524 | |
---|
| 2525 | \begin_layout Plain Layout |
---|
| 2526 | |
---|
| 2527 | \end_layout |
---|
| 2528 | |
---|
| 2529 | \end_inset |
---|
| 2530 | |
---|
| 2531 | |
---|
| 2532 | \end_layout |
---|
| 2533 | |
---|
| 2534 | \begin_layout Standard |
---|
| 2535 | \begin_inset Float figure |
---|
| 2536 | wide false |
---|
| 2537 | sideways false |
---|
| 2538 | status collapsed |
---|
| 2539 | |
---|
| 2540 | \begin_layout Plain Layout |
---|
| 2541 | \align center |
---|
| 2542 | \begin_inset Tabular |
---|
| 2543 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 2544 | <features tabularvalignment="middle"> |
---|
| 2545 | <column alignment="center" valignment="top" width="0"> |
---|
| 2546 | <column alignment="center" valignment="top" width="0"> |
---|
| 2547 | <row> |
---|
| 2548 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2549 | \begin_inset Text |
---|
| 2550 | |
---|
| 2551 | \begin_layout Plain Layout |
---|
| 2552 | \begin_inset Graphics |
---|
| 2553 | filename grafy/lqplots/pm2r1s1.eps |
---|
| 2554 | scale 40 |
---|
| 2555 | |
---|
| 2556 | \end_inset |
---|
| 2557 | |
---|
| 2558 | |
---|
| 2559 | \end_layout |
---|
| 2560 | |
---|
| 2561 | \end_inset |
---|
| 2562 | </cell> |
---|
| 2563 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2564 | \begin_inset Text |
---|
| 2565 | |
---|
| 2566 | \begin_layout Plain Layout |
---|
| 2567 | \begin_inset Graphics |
---|
| 2568 | filename grafy/lqplots/pm2r1s2.eps |
---|
| 2569 | scale 40 |
---|
| 2570 | |
---|
| 2571 | \end_inset |
---|
| 2572 | |
---|
| 2573 | |
---|
| 2574 | \end_layout |
---|
| 2575 | |
---|
| 2576 | \end_inset |
---|
| 2577 | </cell> |
---|
| 2578 | </row> |
---|
| 2579 | <row> |
---|
| 2580 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2581 | \begin_inset Text |
---|
| 2582 | |
---|
| 2583 | \begin_layout Plain Layout |
---|
| 2584 | 1 -- redukovaný model |
---|
| 2585 | \end_layout |
---|
| 2586 | |
---|
| 2587 | \end_inset |
---|
| 2588 | </cell> |
---|
| 2589 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2590 | \begin_inset Text |
---|
| 2591 | |
---|
| 2592 | \begin_layout Plain Layout |
---|
| 2593 | 2 -- redukovaný model s hyperstavem |
---|
| 2594 | \end_layout |
---|
| 2595 | |
---|
| 2596 | \end_inset |
---|
| 2597 | </cell> |
---|
| 2598 | </row> |
---|
| 2599 | <row> |
---|
| 2600 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2601 | \begin_inset Text |
---|
| 2602 | |
---|
| 2603 | \begin_layout Plain Layout |
---|
| 2604 | \begin_inset Graphics |
---|
| 2605 | filename grafy/lqplots/pm2r1s3.eps |
---|
| 2606 | scale 40 |
---|
| 2607 | |
---|
| 2608 | \end_inset |
---|
| 2609 | |
---|
| 2610 | |
---|
| 2611 | \end_layout |
---|
| 2612 | |
---|
| 2613 | \end_inset |
---|
| 2614 | </cell> |
---|
| 2615 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2616 | \begin_inset Text |
---|
| 2617 | |
---|
| 2618 | \begin_layout Plain Layout |
---|
| 2619 | \begin_inset Graphics |
---|
| 2620 | filename grafy/lqplots/pm2r1s4.eps |
---|
| 2621 | scale 40 |
---|
| 2622 | |
---|
| 2623 | \end_inset |
---|
| 2624 | |
---|
| 2625 | |
---|
| 2626 | \end_layout |
---|
| 2627 | |
---|
| 2628 | \end_inset |
---|
| 2629 | </cell> |
---|
| 2630 | </row> |
---|
| 2631 | <row> |
---|
| 2632 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 2633 | \begin_inset Text |
---|
| 2634 | |
---|
| 2635 | \begin_layout Plain Layout |
---|
| 2636 | 3 -- plný model |
---|
| 2637 | \end_layout |
---|
| 2638 | |
---|
| 2639 | \end_inset |
---|
| 2640 | </cell> |
---|
| 2641 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2642 | \begin_inset Text |
---|
| 2643 | |
---|
| 2644 | \begin_layout Plain Layout |
---|
| 2645 | 4 -- plný model s hyperstavem |
---|
| 2646 | \end_layout |
---|
| 2647 | |
---|
| 2648 | \end_inset |
---|
| 2649 | </cell> |
---|
| 2650 | </row> |
---|
| 2651 | </lyxtabular> |
---|
| 2652 | |
---|
| 2653 | \end_inset |
---|
| 2654 | |
---|
| 2655 | |
---|
| 2656 | \begin_inset Caption |
---|
| 2657 | |
---|
| 2658 | \begin_layout Plain Layout |
---|
| 2659 | Průběhy otáček |
---|
| 2660 | \begin_inset Formula $\omega$ |
---|
| 2661 | \end_inset |
---|
| 2662 | |
---|
| 2663 | a polohy |
---|
| 2664 | \begin_inset Formula $\vartheta$ |
---|
| 2665 | \end_inset |
---|
| 2666 | |
---|
| 2667 | pro simulace na jednoduchém simulátoru s profilem požadovaných otáček |
---|
| 2668 | \begin_inset Formula $\pm1$ |
---|
| 2669 | \end_inset |
---|
| 2670 | |
---|
| 2671 | |
---|
| 2672 | \end_layout |
---|
| 2673 | |
---|
| 2674 | \end_inset |
---|
| 2675 | |
---|
| 2676 | |
---|
| 2677 | \begin_inset CommandInset label |
---|
| 2678 | LatexCommand label |
---|
| 2679 | name "fig_grafym_ref1" |
---|
| 2680 | |
---|
| 2681 | \end_inset |
---|
| 2682 | |
---|
| 2683 | |
---|
| 2684 | \end_layout |
---|
| 2685 | |
---|
| 2686 | \end_inset |
---|
| 2687 | |
---|
| 2688 | |
---|
| 2689 | \end_layout |
---|
| 2690 | |
---|
| 2691 | \begin_layout Standard |
---|
| 2692 | \begin_inset Float figure |
---|
| 2693 | wide false |
---|
| 2694 | sideways false |
---|
| 2695 | status collapsed |
---|
| 2696 | |
---|
| 2697 | \begin_layout Plain Layout |
---|
| 2698 | \align center |
---|
| 2699 | \begin_inset Tabular |
---|
| 2700 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 2701 | <features tabularvalignment="middle"> |
---|
| 2702 | <column alignment="center" valignment="top" width="0"> |
---|
| 2703 | <column alignment="center" valignment="top" width="0"> |
---|
| 2704 | <row> |
---|
| 2705 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2706 | \begin_inset Text |
---|
| 2707 | |
---|
| 2708 | \begin_layout Plain Layout |
---|
| 2709 | \begin_inset Graphics |
---|
| 2710 | filename grafy/lqplots/pm2r10s1.eps |
---|
| 2711 | scale 40 |
---|
| 2712 | |
---|
| 2713 | \end_inset |
---|
| 2714 | |
---|
| 2715 | |
---|
| 2716 | \end_layout |
---|
| 2717 | |
---|
| 2718 | \end_inset |
---|
| 2719 | </cell> |
---|
| 2720 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2721 | \begin_inset Text |
---|
| 2722 | |
---|
| 2723 | \begin_layout Plain Layout |
---|
| 2724 | \begin_inset Graphics |
---|
| 2725 | filename grafy/lqplots/pm2r10s2.eps |
---|
| 2726 | scale 40 |
---|
| 2727 | |
---|
| 2728 | \end_inset |
---|
| 2729 | |
---|
| 2730 | |
---|
| 2731 | \end_layout |
---|
| 2732 | |
---|
| 2733 | \end_inset |
---|
| 2734 | </cell> |
---|
| 2735 | </row> |
---|
| 2736 | <row> |
---|
| 2737 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2738 | \begin_inset Text |
---|
| 2739 | |
---|
| 2740 | \begin_layout Plain Layout |
---|
| 2741 | 1 -- redukovaný model |
---|
| 2742 | \end_layout |
---|
| 2743 | |
---|
| 2744 | \end_inset |
---|
| 2745 | </cell> |
---|
| 2746 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2747 | \begin_inset Text |
---|
| 2748 | |
---|
| 2749 | \begin_layout Plain Layout |
---|
| 2750 | 2 -- redukovaný model s hyperstavem |
---|
| 2751 | \end_layout |
---|
| 2752 | |
---|
| 2753 | \end_inset |
---|
| 2754 | </cell> |
---|
| 2755 | </row> |
---|
| 2756 | <row> |
---|
| 2757 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2758 | \begin_inset Text |
---|
| 2759 | |
---|
| 2760 | \begin_layout Plain Layout |
---|
| 2761 | \begin_inset Graphics |
---|
| 2762 | filename grafy/lqplots/pm2r10s3.eps |
---|
| 2763 | scale 40 |
---|
| 2764 | |
---|
| 2765 | \end_inset |
---|
| 2766 | |
---|
| 2767 | |
---|
| 2768 | \end_layout |
---|
| 2769 | |
---|
| 2770 | \end_inset |
---|
| 2771 | </cell> |
---|
| 2772 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2773 | \begin_inset Text |
---|
| 2774 | |
---|
| 2775 | \begin_layout Plain Layout |
---|
| 2776 | \begin_inset Graphics |
---|
| 2777 | filename grafy/lqplots/pm2r10s4.eps |
---|
| 2778 | scale 40 |
---|
| 2779 | |
---|
| 2780 | \end_inset |
---|
| 2781 | |
---|
| 2782 | |
---|
| 2783 | \end_layout |
---|
| 2784 | |
---|
| 2785 | \end_inset |
---|
| 2786 | </cell> |
---|
| 2787 | </row> |
---|
| 2788 | <row> |
---|
| 2789 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 2790 | \begin_inset Text |
---|
| 2791 | |
---|
| 2792 | \begin_layout Plain Layout |
---|
| 2793 | 3 -- plný model |
---|
| 2794 | \end_layout |
---|
| 2795 | |
---|
| 2796 | \end_inset |
---|
| 2797 | </cell> |
---|
| 2798 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2799 | \begin_inset Text |
---|
| 2800 | |
---|
| 2801 | \begin_layout Plain Layout |
---|
| 2802 | 4 -- plný model s hyperstavem |
---|
| 2803 | \end_layout |
---|
| 2804 | |
---|
| 2805 | \end_inset |
---|
| 2806 | </cell> |
---|
| 2807 | </row> |
---|
| 2808 | </lyxtabular> |
---|
| 2809 | |
---|
| 2810 | \end_inset |
---|
| 2811 | |
---|
| 2812 | |
---|
| 2813 | \begin_inset Caption |
---|
| 2814 | |
---|
| 2815 | \begin_layout Plain Layout |
---|
| 2816 | Průběhy otáček |
---|
| 2817 | \begin_inset Formula $\omega$ |
---|
| 2818 | \end_inset |
---|
| 2819 | |
---|
| 2820 | a polohy |
---|
| 2821 | \begin_inset Formula $\vartheta$ |
---|
| 2822 | \end_inset |
---|
| 2823 | |
---|
| 2824 | pro simulace na jednoduchém simulátoru s profilem požadovaných otáček |
---|
| 2825 | \begin_inset Formula $\pm10$ |
---|
| 2826 | \end_inset |
---|
| 2827 | |
---|
| 2828 | |
---|
| 2829 | \end_layout |
---|
| 2830 | |
---|
| 2831 | \end_inset |
---|
| 2832 | |
---|
| 2833 | |
---|
| 2834 | \begin_inset CommandInset label |
---|
| 2835 | LatexCommand label |
---|
| 2836 | name "fig_grafym_ref10" |
---|
| 2837 | |
---|
| 2838 | \end_inset |
---|
| 2839 | |
---|
| 2840 | |
---|
| 2841 | \end_layout |
---|
| 2842 | |
---|
| 2843 | \end_inset |
---|
| 2844 | |
---|
| 2845 | |
---|
| 2846 | \end_layout |
---|
| 2847 | |
---|
| 2848 | \begin_layout Standard |
---|
| 2849 | \begin_inset Float figure |
---|
| 2850 | wide false |
---|
| 2851 | sideways false |
---|
| 2852 | status collapsed |
---|
| 2853 | |
---|
| 2854 | \begin_layout Plain Layout |
---|
| 2855 | \align center |
---|
| 2856 | \begin_inset Tabular |
---|
| 2857 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 2858 | <features tabularvalignment="middle"> |
---|
| 2859 | <column alignment="center" valignment="top" width="0"> |
---|
| 2860 | <column alignment="center" valignment="top" width="0"> |
---|
| 2861 | <row> |
---|
| 2862 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2863 | \begin_inset Text |
---|
| 2864 | |
---|
| 2865 | \begin_layout Plain Layout |
---|
| 2866 | \begin_inset Graphics |
---|
| 2867 | filename grafy/lqplots/pm2r200s1.eps |
---|
| 2868 | scale 40 |
---|
| 2869 | |
---|
| 2870 | \end_inset |
---|
| 2871 | |
---|
| 2872 | |
---|
| 2873 | \end_layout |
---|
| 2874 | |
---|
| 2875 | \end_inset |
---|
| 2876 | </cell> |
---|
| 2877 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2878 | \begin_inset Text |
---|
| 2879 | |
---|
| 2880 | \begin_layout Plain Layout |
---|
| 2881 | \begin_inset Graphics |
---|
| 2882 | filename grafy/lqplots/pm2r200s2.eps |
---|
| 2883 | scale 40 |
---|
| 2884 | |
---|
| 2885 | \end_inset |
---|
| 2886 | |
---|
| 2887 | |
---|
| 2888 | \end_layout |
---|
| 2889 | |
---|
| 2890 | \end_inset |
---|
| 2891 | </cell> |
---|
| 2892 | </row> |
---|
| 2893 | <row> |
---|
| 2894 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2895 | \begin_inset Text |
---|
| 2896 | |
---|
| 2897 | \begin_layout Plain Layout |
---|
| 2898 | 1 -- redukovaný model |
---|
| 2899 | \end_layout |
---|
| 2900 | |
---|
| 2901 | \end_inset |
---|
| 2902 | </cell> |
---|
| 2903 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2904 | \begin_inset Text |
---|
| 2905 | |
---|
| 2906 | \begin_layout Plain Layout |
---|
| 2907 | 2 -- redukovaný model s hyperstavem |
---|
| 2908 | \end_layout |
---|
| 2909 | |
---|
| 2910 | \end_inset |
---|
| 2911 | </cell> |
---|
| 2912 | </row> |
---|
| 2913 | <row> |
---|
| 2914 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 2915 | \begin_inset Text |
---|
| 2916 | |
---|
| 2917 | \begin_layout Plain Layout |
---|
| 2918 | \begin_inset Graphics |
---|
| 2919 | filename grafy/lqplots/pm2r200s3.eps |
---|
| 2920 | scale 40 |
---|
| 2921 | |
---|
| 2922 | \end_inset |
---|
| 2923 | |
---|
| 2924 | |
---|
| 2925 | \end_layout |
---|
| 2926 | |
---|
| 2927 | \end_inset |
---|
| 2928 | </cell> |
---|
| 2929 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2930 | \begin_inset Text |
---|
| 2931 | |
---|
| 2932 | \begin_layout Plain Layout |
---|
| 2933 | \begin_inset Graphics |
---|
| 2934 | filename grafy/lqplots/pm2r200s4.eps |
---|
| 2935 | scale 40 |
---|
| 2936 | |
---|
| 2937 | \end_inset |
---|
| 2938 | |
---|
| 2939 | |
---|
| 2940 | \end_layout |
---|
| 2941 | |
---|
| 2942 | \end_inset |
---|
| 2943 | </cell> |
---|
| 2944 | </row> |
---|
| 2945 | <row> |
---|
| 2946 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 2947 | \begin_inset Text |
---|
| 2948 | |
---|
| 2949 | \begin_layout Plain Layout |
---|
| 2950 | 3 -- plný model |
---|
| 2951 | \end_layout |
---|
| 2952 | |
---|
| 2953 | \end_inset |
---|
| 2954 | </cell> |
---|
| 2955 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 2956 | \begin_inset Text |
---|
| 2957 | |
---|
| 2958 | \begin_layout Plain Layout |
---|
| 2959 | 4 -- plný model s hyperstavem |
---|
| 2960 | \end_layout |
---|
| 2961 | |
---|
| 2962 | \end_inset |
---|
| 2963 | </cell> |
---|
| 2964 | </row> |
---|
| 2965 | </lyxtabular> |
---|
| 2966 | |
---|
| 2967 | \end_inset |
---|
| 2968 | |
---|
| 2969 | |
---|
| 2970 | \begin_inset Caption |
---|
| 2971 | |
---|
| 2972 | \begin_layout Plain Layout |
---|
| 2973 | Průběhy otáček |
---|
| 2974 | \begin_inset Formula $\omega$ |
---|
| 2975 | \end_inset |
---|
| 2976 | |
---|
| 2977 | a polohy |
---|
| 2978 | \begin_inset Formula $\vartheta$ |
---|
| 2979 | \end_inset |
---|
| 2980 | |
---|
| 2981 | pro simulace na jednoduchém simulátoru s profilem požadovaných otáček |
---|
| 2982 | \begin_inset Formula $\pm200$ |
---|
| 2983 | \end_inset |
---|
| 2984 | |
---|
| 2985 | |
---|
| 2986 | \end_layout |
---|
| 2987 | |
---|
| 2988 | \end_inset |
---|
| 2989 | |
---|
| 2990 | |
---|
| 2991 | \begin_inset CommandInset label |
---|
| 2992 | LatexCommand label |
---|
| 2993 | name "fig_grafym_ref200" |
---|
| 2994 | |
---|
| 2995 | \end_inset |
---|
| 2996 | |
---|
| 2997 | |
---|
| 2998 | \end_layout |
---|
| 2999 | |
---|
| 3000 | \end_inset |
---|
| 3001 | |
---|
| 3002 | |
---|
| 3003 | \end_layout |
---|
| 3004 | |
---|
| 3005 | \begin_layout Standard |
---|
| 3006 | Následující tabulka shrnuje dosažené průměrné ztráty (z 10 běhů) pro jednotlivé |
---|
| 3007 | modely a profily referenčních otáček na jednoduchém simulátoru. |
---|
| 3008 | Jako ztráty jsou zde uvažovány pouze součty kvadrátů odchylek požadovaných |
---|
| 3009 | a skutečných otáček. |
---|
| 3010 | \end_layout |
---|
| 3011 | |
---|
| 3012 | \begin_layout Standard |
---|
| 3013 | \align center |
---|
| 3014 | \begin_inset Tabular |
---|
| 3015 | <lyxtabular version="3" rows="5" columns="5"> |
---|
| 3016 | <features tabularvalignment="middle"> |
---|
| 3017 | <column alignment="center" valignment="top" width="0"> |
---|
| 3018 | <column alignment="center" valignment="top" width="0"> |
---|
| 3019 | <column alignment="center" valignment="top" width="0"> |
---|
| 3020 | <column alignment="center" valignment="top" width="0"> |
---|
| 3021 | <column alignment="center" valignment="top" width="0"> |
---|
| 3022 | <row> |
---|
| 3023 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3024 | \begin_inset Text |
---|
| 3025 | |
---|
| 3026 | \begin_layout Plain Layout |
---|
| 3027 | |
---|
| 3028 | \series bold |
---|
| 3029 | použitý model |
---|
| 3030 | \backslash |
---|
| 3031 | požadované otáčky |
---|
| 3032 | \end_layout |
---|
| 3033 | |
---|
| 3034 | \end_inset |
---|
| 3035 | </cell> |
---|
| 3036 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3037 | \begin_inset Text |
---|
| 3038 | |
---|
| 3039 | \begin_layout Plain Layout |
---|
| 3040 | \begin_inset Formula $0$ |
---|
| 3041 | \end_inset |
---|
| 3042 | |
---|
| 3043 | |
---|
| 3044 | \end_layout |
---|
| 3045 | |
---|
| 3046 | \end_inset |
---|
| 3047 | </cell> |
---|
| 3048 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3049 | \begin_inset Text |
---|
| 3050 | |
---|
| 3051 | \begin_layout Plain Layout |
---|
| 3052 | \begin_inset Formula $\pm1$ |
---|
| 3053 | \end_inset |
---|
| 3054 | |
---|
| 3055 | |
---|
| 3056 | \end_layout |
---|
| 3057 | |
---|
| 3058 | \end_inset |
---|
| 3059 | </cell> |
---|
| 3060 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3061 | \begin_inset Text |
---|
| 3062 | |
---|
| 3063 | \begin_layout Plain Layout |
---|
| 3064 | \begin_inset Formula $\pm10$ |
---|
| 3065 | \end_inset |
---|
| 3066 | |
---|
| 3067 | |
---|
| 3068 | \end_layout |
---|
| 3069 | |
---|
| 3070 | \end_inset |
---|
| 3071 | </cell> |
---|
| 3072 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3073 | \begin_inset Text |
---|
| 3074 | |
---|
| 3075 | \begin_layout Plain Layout |
---|
| 3076 | \begin_inset Formula $\pm200$ |
---|
| 3077 | \end_inset |
---|
| 3078 | |
---|
| 3079 | |
---|
| 3080 | \end_layout |
---|
| 3081 | |
---|
| 3082 | \end_inset |
---|
| 3083 | </cell> |
---|
| 3084 | </row> |
---|
| 3085 | <row> |
---|
| 3086 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3087 | \begin_inset Text |
---|
| 3088 | |
---|
| 3089 | \begin_layout Plain Layout |
---|
| 3090 | 1 -- redukovaný model |
---|
| 3091 | \end_layout |
---|
| 3092 | |
---|
| 3093 | \end_inset |
---|
| 3094 | </cell> |
---|
| 3095 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3096 | \begin_inset Text |
---|
| 3097 | |
---|
| 3098 | \begin_layout Plain Layout |
---|
| 3099 | 2611 |
---|
| 3100 | \end_layout |
---|
| 3101 | |
---|
| 3102 | \end_inset |
---|
| 3103 | </cell> |
---|
| 3104 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3105 | \begin_inset Text |
---|
| 3106 | |
---|
| 3107 | \begin_layout Plain Layout |
---|
| 3108 | 2731 |
---|
| 3109 | \end_layout |
---|
| 3110 | |
---|
| 3111 | \end_inset |
---|
| 3112 | </cell> |
---|
| 3113 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3114 | \begin_inset Text |
---|
| 3115 | |
---|
| 3116 | \begin_layout Plain Layout |
---|
| 3117 | 28640 |
---|
| 3118 | \end_layout |
---|
| 3119 | |
---|
| 3120 | \end_inset |
---|
| 3121 | </cell> |
---|
| 3122 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3123 | \begin_inset Text |
---|
| 3124 | |
---|
| 3125 | \begin_layout Plain Layout |
---|
| 3126 | 10815000 |
---|
| 3127 | \end_layout |
---|
| 3128 | |
---|
| 3129 | \end_inset |
---|
| 3130 | </cell> |
---|
| 3131 | </row> |
---|
| 3132 | <row> |
---|
| 3133 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3134 | \begin_inset Text |
---|
| 3135 | |
---|
| 3136 | \begin_layout Plain Layout |
---|
| 3137 | 2 -- redukovaný model s hyperstavem |
---|
| 3138 | \end_layout |
---|
| 3139 | |
---|
| 3140 | \end_inset |
---|
| 3141 | </cell> |
---|
| 3142 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3143 | \begin_inset Text |
---|
| 3144 | |
---|
| 3145 | \begin_layout Plain Layout |
---|
| 3146 | 2377 |
---|
| 3147 | \end_layout |
---|
| 3148 | |
---|
| 3149 | \end_inset |
---|
| 3150 | </cell> |
---|
| 3151 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3152 | \begin_inset Text |
---|
| 3153 | |
---|
| 3154 | \begin_layout Plain Layout |
---|
| 3155 | 2480 |
---|
| 3156 | \end_layout |
---|
| 3157 | |
---|
| 3158 | \end_inset |
---|
| 3159 | </cell> |
---|
| 3160 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3161 | \begin_inset Text |
---|
| 3162 | |
---|
| 3163 | \begin_layout Plain Layout |
---|
| 3164 | 3070 |
---|
| 3165 | \end_layout |
---|
| 3166 | |
---|
| 3167 | \end_inset |
---|
| 3168 | </cell> |
---|
| 3169 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3170 | \begin_inset Text |
---|
| 3171 | |
---|
| 3172 | \begin_layout Plain Layout |
---|
| 3173 | 439740 |
---|
| 3174 | \end_layout |
---|
| 3175 | |
---|
| 3176 | \end_inset |
---|
| 3177 | </cell> |
---|
| 3178 | </row> |
---|
| 3179 | <row> |
---|
| 3180 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3181 | \begin_inset Text |
---|
| 3182 | |
---|
| 3183 | \begin_layout Plain Layout |
---|
| 3184 | 3 -- plný model |
---|
| 3185 | \end_layout |
---|
| 3186 | |
---|
| 3187 | \end_inset |
---|
| 3188 | </cell> |
---|
| 3189 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3190 | \begin_inset Text |
---|
| 3191 | |
---|
| 3192 | \begin_layout Plain Layout |
---|
| 3193 | 3579 |
---|
| 3194 | \end_layout |
---|
| 3195 | |
---|
| 3196 | \end_inset |
---|
| 3197 | </cell> |
---|
| 3198 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3199 | \begin_inset Text |
---|
| 3200 | |
---|
| 3201 | \begin_layout Plain Layout |
---|
| 3202 | 3163 |
---|
| 3203 | \end_layout |
---|
| 3204 | |
---|
| 3205 | \end_inset |
---|
| 3206 | </cell> |
---|
| 3207 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3208 | \begin_inset Text |
---|
| 3209 | |
---|
| 3210 | \begin_layout Plain Layout |
---|
| 3211 | 4268 |
---|
| 3212 | \end_layout |
---|
| 3213 | |
---|
| 3214 | \end_inset |
---|
| 3215 | </cell> |
---|
| 3216 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3217 | \begin_inset Text |
---|
| 3218 | |
---|
| 3219 | \begin_layout Plain Layout |
---|
| 3220 | 11168 |
---|
| 3221 | \end_layout |
---|
| 3222 | |
---|
| 3223 | \end_inset |
---|
| 3224 | </cell> |
---|
| 3225 | </row> |
---|
| 3226 | <row> |
---|
| 3227 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3228 | \begin_inset Text |
---|
| 3229 | |
---|
| 3230 | \begin_layout Plain Layout |
---|
| 3231 | 4 -- plný model s hyperstavem |
---|
| 3232 | \end_layout |
---|
| 3233 | |
---|
| 3234 | \end_inset |
---|
| 3235 | </cell> |
---|
| 3236 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3237 | \begin_inset Text |
---|
| 3238 | |
---|
| 3239 | \begin_layout Plain Layout |
---|
| 3240 | 3240 |
---|
| 3241 | \end_layout |
---|
| 3242 | |
---|
| 3243 | \end_inset |
---|
| 3244 | </cell> |
---|
| 3245 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3246 | \begin_inset Text |
---|
| 3247 | |
---|
| 3248 | \begin_layout Plain Layout |
---|
| 3249 | 3797 |
---|
| 3250 | \end_layout |
---|
| 3251 | |
---|
| 3252 | \end_inset |
---|
| 3253 | </cell> |
---|
| 3254 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3255 | \begin_inset Text |
---|
| 3256 | |
---|
| 3257 | \begin_layout Plain Layout |
---|
| 3258 | 3389 |
---|
| 3259 | \end_layout |
---|
| 3260 | |
---|
| 3261 | \end_inset |
---|
| 3262 | </cell> |
---|
| 3263 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3264 | \begin_inset Text |
---|
| 3265 | |
---|
| 3266 | \begin_layout Plain Layout |
---|
| 3267 | 61902 |
---|
| 3268 | \end_layout |
---|
| 3269 | |
---|
| 3270 | \end_inset |
---|
| 3271 | </cell> |
---|
| 3272 | </row> |
---|
| 3273 | </lyxtabular> |
---|
| 3274 | |
---|
| 3275 | \end_inset |
---|
| 3276 | |
---|
| 3277 | |
---|
| 3278 | \end_layout |
---|
| 3279 | |
---|
| 3280 | \begin_layout Standard |
---|
| 3281 | Z této tabulky je patrné, že pro redukovaný model dosahuje verze s hyperstavem |
---|
| 3282 | nižší ztráty, což je více patrné zvláště při vyšších otáčkách. |
---|
| 3283 | Naproti tomu nelze říci, že by verze s hyperstavem byla lepší pro plný |
---|
| 3284 | model. |
---|
| 3285 | Výhoda užití hyperstavu se však projeví, když máme špatný počáteční odhad |
---|
| 3286 | polohy |
---|
| 3287 | \begin_inset Formula $\vartheta_{0}$ |
---|
| 3288 | \end_inset |
---|
| 3289 | |
---|
| 3290 | . |
---|
| 3291 | Uvažujme tedy počáteční odhad |
---|
| 3292 | \begin_inset Formula $\vartheta_{0}=1,5$ |
---|
| 3293 | \end_inset |
---|
| 3294 | |
---|
| 3295 | , zatímco skutečná poloha je |
---|
| 3296 | \begin_inset Formula $0$ |
---|
| 3297 | \end_inset |
---|
| 3298 | |
---|
| 3299 | (hodnota |
---|
| 3300 | \begin_inset Formula $1,5$ |
---|
| 3301 | \end_inset |
---|
| 3302 | |
---|
| 3303 | je volena, protože je dostatečně daleko od |
---|
| 3304 | \begin_inset Formula $0$ |
---|
| 3305 | \end_inset |
---|
| 3306 | |
---|
| 3307 | , ale ještě nedosahuje |
---|
| 3308 | \begin_inset Formula $\frac{\pi}{2}$ |
---|
| 3309 | \end_inset |
---|
| 3310 | |
---|
| 3311 | , kdy hrozí nebezpečí otáčení na opačnou stranu). |
---|
| 3312 | Na grafech obrázek ( |
---|
| 3313 | \begin_inset CommandInset ref |
---|
| 3314 | LatexCommand ref |
---|
| 3315 | reference "fig_grafym_pocth34" |
---|
| 3316 | |
---|
| 3317 | \end_inset |
---|
| 3318 | |
---|
| 3319 | ) je možné pozorovat počátek běhu, při profilu požadovaných otáček |
---|
| 3320 | \begin_inset Formula $\pm200$ |
---|
| 3321 | \end_inset |
---|
| 3322 | |
---|
| 3323 | , ze kterého je zřejmé lepší zvládnutí špatného počátečního odhadu polohy |
---|
| 3324 | při užití hyperstavu. |
---|
| 3325 | Průměrné ztráty pak jsou: |
---|
| 3326 | \begin_inset Formula $1,1035\cdot10^{7}$ |
---|
| 3327 | \end_inset |
---|
| 3328 | |
---|
| 3329 | pro plný model bez hyperstavu a |
---|
| 3330 | \family roman |
---|
| 3331 | \series medium |
---|
| 3332 | \shape up |
---|
| 3333 | \size normal |
---|
| 3334 | \emph off |
---|
| 3335 | \bar no |
---|
| 3336 | \strikeout off |
---|
| 3337 | \uuline off |
---|
| 3338 | \uwave off |
---|
| 3339 | \noun off |
---|
| 3340 | \color none |
---|
| 3341 | \lang english |
---|
| 3342 | |
---|
| 3343 | \begin_inset Formula $4,4955\cdot10^{4}$ |
---|
| 3344 | \end_inset |
---|
| 3345 | |
---|
| 3346 | |
---|
| 3347 | \family default |
---|
| 3348 | \series default |
---|
| 3349 | \shape default |
---|
| 3350 | \size default |
---|
| 3351 | \emph default |
---|
| 3352 | \bar default |
---|
| 3353 | \strikeout default |
---|
| 3354 | \uuline default |
---|
| 3355 | \uwave default |
---|
| 3356 | \noun default |
---|
| 3357 | \color inherit |
---|
| 3358 | \lang czech |
---|
| 3359 | pro plný model s hyperstavem. |
---|
| 3360 | \end_layout |
---|
| 3361 | |
---|
| 3362 | \begin_layout Standard |
---|
| 3363 | \begin_inset Float figure |
---|
| 3364 | wide false |
---|
| 3365 | sideways false |
---|
| 3366 | status collapsed |
---|
| 3367 | |
---|
| 3368 | \begin_layout Plain Layout |
---|
| 3369 | \align center |
---|
| 3370 | \begin_inset Tabular |
---|
| 3371 | <lyxtabular version="3" rows="2" columns="2"> |
---|
| 3372 | <features tabularvalignment="middle"> |
---|
| 3373 | <column alignment="center" valignment="top" width="0"> |
---|
| 3374 | <column alignment="center" valignment="top" width="0"> |
---|
| 3375 | <row> |
---|
| 3376 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3377 | \begin_inset Text |
---|
| 3378 | |
---|
| 3379 | \begin_layout Plain Layout |
---|
| 3380 | \begin_inset Graphics |
---|
| 3381 | filename grafy/lqplots/jinath0r200m3.eps |
---|
| 3382 | scale 40 |
---|
| 3383 | |
---|
| 3384 | \end_inset |
---|
| 3385 | |
---|
| 3386 | |
---|
| 3387 | \end_layout |
---|
| 3388 | |
---|
| 3389 | \end_inset |
---|
| 3390 | </cell> |
---|
| 3391 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3392 | \begin_inset Text |
---|
| 3393 | |
---|
| 3394 | \begin_layout Plain Layout |
---|
| 3395 | \begin_inset Graphics |
---|
| 3396 | filename grafy/lqplots/jinath0r200m4.eps |
---|
| 3397 | scale 40 |
---|
| 3398 | |
---|
| 3399 | \end_inset |
---|
| 3400 | |
---|
| 3401 | |
---|
| 3402 | \end_layout |
---|
| 3403 | |
---|
| 3404 | \end_inset |
---|
| 3405 | </cell> |
---|
| 3406 | </row> |
---|
| 3407 | <row> |
---|
| 3408 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3409 | \begin_inset Text |
---|
| 3410 | |
---|
| 3411 | \begin_layout Plain Layout |
---|
| 3412 | 3 -- plný model |
---|
| 3413 | \end_layout |
---|
| 3414 | |
---|
| 3415 | \end_inset |
---|
| 3416 | </cell> |
---|
| 3417 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3418 | \begin_inset Text |
---|
| 3419 | |
---|
| 3420 | \begin_layout Plain Layout |
---|
| 3421 | 4 -- plný model s hyperstavem |
---|
| 3422 | \end_layout |
---|
| 3423 | |
---|
| 3424 | \end_inset |
---|
| 3425 | </cell> |
---|
| 3426 | </row> |
---|
| 3427 | </lyxtabular> |
---|
| 3428 | |
---|
| 3429 | \end_inset |
---|
| 3430 | |
---|
| 3431 | |
---|
| 3432 | \begin_inset Caption |
---|
| 3433 | |
---|
| 3434 | \begin_layout Plain Layout |
---|
| 3435 | Průběhy otáček |
---|
| 3436 | \begin_inset Formula $\omega$ |
---|
| 3437 | \end_inset |
---|
| 3438 | |
---|
| 3439 | a polohy |
---|
| 3440 | \begin_inset Formula $\vartheta$ |
---|
| 3441 | \end_inset |
---|
| 3442 | |
---|
| 3443 | pro simulace na jednoduchém simulátoru s profilem požadovaných otáček |
---|
| 3444 | \begin_inset Formula $\pm200$ |
---|
| 3445 | \end_inset |
---|
| 3446 | |
---|
| 3447 | a volbou počátečního odhadu |
---|
| 3448 | \begin_inset Formula $\vartheta_{0}=1,5$ |
---|
| 3449 | \end_inset |
---|
| 3450 | |
---|
| 3451 | při skutečné hodnotě |
---|
| 3452 | \begin_inset Formula $0$ |
---|
| 3453 | \end_inset |
---|
| 3454 | |
---|
| 3455 | |
---|
| 3456 | \end_layout |
---|
| 3457 | |
---|
| 3458 | \end_inset |
---|
| 3459 | |
---|
| 3460 | |
---|
| 3461 | \begin_inset CommandInset label |
---|
| 3462 | LatexCommand label |
---|
| 3463 | name "fig_grafym_pocth34" |
---|
| 3464 | |
---|
| 3465 | \end_inset |
---|
| 3466 | |
---|
| 3467 | |
---|
| 3468 | \end_layout |
---|
| 3469 | |
---|
| 3470 | \end_inset |
---|
| 3471 | |
---|
| 3472 | |
---|
| 3473 | \end_layout |
---|
| 3474 | |
---|
| 3475 | \begin_layout Subsection* |
---|
| 3476 | Simulátor PMSM |
---|
| 3477 | \end_layout |
---|
| 3478 | |
---|
| 3479 | \begin_layout Standard |
---|
| 3480 | Výsledky ze simulátoru PMSM, který více odpovídá reálnému chování stroje, |
---|
| 3481 | již dopadly hůře. |
---|
| 3482 | Je tomu především z důvodu zahrnutí složitějších efektů do simulátoru, |
---|
| 3483 | jako úbytky napětí a mrtvé časy. |
---|
| 3484 | Pro nulové požadované otáčky při užití kompenzace úbytků napětí i bez ní |
---|
| 3485 | je zdánlivě vše v pořádku, viz graf na obrázku ( |
---|
| 3486 | \begin_inset CommandInset ref |
---|
| 3487 | LatexCommand ref |
---|
| 3488 | reference "fig:nicsenedej" |
---|
| 3489 | |
---|
| 3490 | \end_inset |
---|
| 3491 | |
---|
| 3492 | a). |
---|
| 3493 | Ovšem pro profil požadovaných otáček |
---|
| 3494 | \begin_inset Formula $\pm1$ |
---|
| 3495 | \end_inset |
---|
| 3496 | |
---|
| 3497 | s i bez kompenzace je výsledný průběh stejný, viz grafy na obrázku ( |
---|
| 3498 | \begin_inset CommandInset ref |
---|
| 3499 | LatexCommand ref |
---|
| 3500 | reference "fig:nicsenedej" |
---|
| 3501 | |
---|
| 3502 | \end_inset |
---|
| 3503 | |
---|
| 3504 | b). |
---|
| 3505 | Tedy nic se neděje, i když požadavek je nenulový a tento výsledek je již |
---|
| 3506 | špatný. |
---|
| 3507 | \end_layout |
---|
| 3508 | |
---|
| 3509 | \begin_layout Standard |
---|
| 3510 | \begin_inset Float figure |
---|
| 3511 | wide false |
---|
| 3512 | sideways false |
---|
| 3513 | status collapsed |
---|
| 3514 | |
---|
| 3515 | \begin_layout Plain Layout |
---|
| 3516 | \align center |
---|
| 3517 | \begin_inset Tabular |
---|
| 3518 | <lyxtabular version="3" rows="2" columns="2"> |
---|
| 3519 | <features tabularvalignment="middle"> |
---|
| 3520 | <column alignment="center" valignment="top" width="0"> |
---|
| 3521 | <column alignment="center" valignment="top" width="0"> |
---|
| 3522 | <row> |
---|
| 3523 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3524 | \begin_inset Text |
---|
| 3525 | |
---|
| 3526 | \begin_layout Plain Layout |
---|
| 3527 | \begin_inset Graphics |
---|
| 3528 | filename grafy/lqplots/nicnedela.eps |
---|
| 3529 | scale 40 |
---|
| 3530 | |
---|
| 3531 | \end_inset |
---|
| 3532 | |
---|
| 3533 | |
---|
| 3534 | \end_layout |
---|
| 3535 | |
---|
| 3536 | \end_inset |
---|
| 3537 | </cell> |
---|
| 3538 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3539 | \begin_inset Text |
---|
| 3540 | |
---|
| 3541 | \begin_layout Plain Layout |
---|
| 3542 | \begin_inset Graphics |
---|
| 3543 | filename grafy/lqplots/nicnedelapm1.eps |
---|
| 3544 | scale 40 |
---|
| 3545 | |
---|
| 3546 | \end_inset |
---|
| 3547 | |
---|
| 3548 | |
---|
| 3549 | \end_layout |
---|
| 3550 | |
---|
| 3551 | \end_inset |
---|
| 3552 | </cell> |
---|
| 3553 | </row> |
---|
| 3554 | <row> |
---|
| 3555 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3556 | \begin_inset Text |
---|
| 3557 | |
---|
| 3558 | \begin_layout Plain Layout |
---|
| 3559 | a) požadovaná hodnota 0 |
---|
| 3560 | \end_layout |
---|
| 3561 | |
---|
| 3562 | \end_inset |
---|
| 3563 | </cell> |
---|
| 3564 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3565 | \begin_inset Text |
---|
| 3566 | |
---|
| 3567 | \begin_layout Plain Layout |
---|
| 3568 | b) požadovaná hodnota profil |
---|
| 3569 | \begin_inset Formula $\pm1$ |
---|
| 3570 | \end_inset |
---|
| 3571 | |
---|
| 3572 | |
---|
| 3573 | \end_layout |
---|
| 3574 | |
---|
| 3575 | \end_inset |
---|
| 3576 | </cell> |
---|
| 3577 | </row> |
---|
| 3578 | </lyxtabular> |
---|
| 3579 | |
---|
| 3580 | \end_inset |
---|
| 3581 | |
---|
| 3582 | |
---|
| 3583 | \end_layout |
---|
| 3584 | |
---|
| 3585 | \begin_layout Plain Layout |
---|
| 3586 | \begin_inset Caption |
---|
| 3587 | |
---|
| 3588 | \begin_layout Plain Layout |
---|
| 3589 | \begin_inset CommandInset label |
---|
| 3590 | LatexCommand label |
---|
| 3591 | name "fig:nicsenedej" |
---|
| 3592 | |
---|
| 3593 | \end_inset |
---|
| 3594 | |
---|
| 3595 | Pruběh otáček |
---|
| 3596 | \begin_inset Formula $\omega$ |
---|
| 3597 | \end_inset |
---|
| 3598 | |
---|
| 3599 | a polohy |
---|
| 3600 | \begin_inset Formula $\vartheta$ |
---|
| 3601 | \end_inset |
---|
| 3602 | |
---|
| 3603 | pro simulace na simulátoru PMSM s kompenzací úbytků napětí i bez ní pro |
---|
| 3604 | nulové požadované otáčky a pro profil |
---|
| 3605 | \begin_inset Formula $\pm1$ |
---|
| 3606 | \end_inset |
---|
| 3607 | |
---|
| 3608 | |
---|
| 3609 | \end_layout |
---|
| 3610 | |
---|
| 3611 | \end_inset |
---|
| 3612 | |
---|
| 3613 | |
---|
| 3614 | \end_layout |
---|
| 3615 | |
---|
| 3616 | \begin_layout Plain Layout |
---|
| 3617 | |
---|
| 3618 | \end_layout |
---|
| 3619 | |
---|
| 3620 | \end_inset |
---|
| 3621 | |
---|
| 3622 | |
---|
| 3623 | \end_layout |
---|
| 3624 | |
---|
| 3625 | \begin_layout Standard |
---|
| 3626 | Pro profil |
---|
| 3627 | \begin_inset Formula $\pm10$ |
---|
| 3628 | \end_inset |
---|
| 3629 | |
---|
| 3630 | otáčky nezůstávají nulové a dosahují požadovaných hodnot |
---|
| 3631 | \begin_inset Formula $10$ |
---|
| 3632 | \end_inset |
---|
| 3633 | |
---|
| 3634 | respektive |
---|
| 3635 | \begin_inset Formula $-10$ |
---|
| 3636 | \end_inset |
---|
| 3637 | |
---|
| 3638 | , objevuje se zde však problém s průchody nulou, viz obrázek ( |
---|
| 3639 | \begin_inset CommandInset ref |
---|
| 3640 | LatexCommand ref |
---|
| 3641 | reference "fig:spatnypruchod0" |
---|
| 3642 | |
---|
| 3643 | \end_inset |
---|
| 3644 | |
---|
| 3645 | a). |
---|
| 3646 | V případě užití kompenzace úbytků napětí se situace ještě zhorší, viz obrázek |
---|
| 3647 | ( |
---|
| 3648 | \begin_inset CommandInset ref |
---|
| 3649 | LatexCommand ref |
---|
| 3650 | reference "fig:spatnypruchod0" |
---|
| 3651 | |
---|
| 3652 | \end_inset |
---|
| 3653 | |
---|
| 3654 | b). |
---|
| 3655 | \end_layout |
---|
| 3656 | |
---|
| 3657 | \begin_layout Standard |
---|
| 3658 | \begin_inset Float figure |
---|
| 3659 | wide false |
---|
| 3660 | sideways false |
---|
| 3661 | status collapsed |
---|
| 3662 | |
---|
| 3663 | \begin_layout Plain Layout |
---|
| 3664 | \align center |
---|
| 3665 | \begin_inset Tabular |
---|
| 3666 | <lyxtabular version="3" rows="2" columns="2"> |
---|
| 3667 | <features tabularvalignment="middle"> |
---|
| 3668 | <column alignment="center" valignment="top" width="0"> |
---|
| 3669 | <column alignment="center" valignment="top" width="0"> |
---|
| 3670 | <row> |
---|
| 3671 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3672 | \begin_inset Text |
---|
| 3673 | |
---|
| 3674 | \begin_layout Plain Layout |
---|
| 3675 | \begin_inset Graphics |
---|
| 3676 | filename grafy/lqplots/spatnypruch0.eps |
---|
| 3677 | scale 40 |
---|
| 3678 | |
---|
| 3679 | \end_inset |
---|
| 3680 | |
---|
| 3681 | |
---|
| 3682 | \end_layout |
---|
| 3683 | |
---|
| 3684 | \end_inset |
---|
| 3685 | </cell> |
---|
| 3686 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3687 | \begin_inset Text |
---|
| 3688 | |
---|
| 3689 | \begin_layout Plain Layout |
---|
| 3690 | \begin_inset Graphics |
---|
| 3691 | filename grafy/lqplots/spatnypruch0horsi.eps |
---|
| 3692 | scale 40 |
---|
| 3693 | |
---|
| 3694 | \end_inset |
---|
| 3695 | |
---|
| 3696 | |
---|
| 3697 | \end_layout |
---|
| 3698 | |
---|
| 3699 | \end_inset |
---|
| 3700 | </cell> |
---|
| 3701 | </row> |
---|
| 3702 | <row> |
---|
| 3703 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3704 | \begin_inset Text |
---|
| 3705 | |
---|
| 3706 | \begin_layout Plain Layout |
---|
| 3707 | a) bez kompenzace úbytků |
---|
| 3708 | \end_layout |
---|
| 3709 | |
---|
| 3710 | \end_inset |
---|
| 3711 | </cell> |
---|
| 3712 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3713 | \begin_inset Text |
---|
| 3714 | |
---|
| 3715 | \begin_layout Plain Layout |
---|
| 3716 | b) s kompenzací úbytků |
---|
| 3717 | \end_layout |
---|
| 3718 | |
---|
| 3719 | \end_inset |
---|
| 3720 | </cell> |
---|
| 3721 | </row> |
---|
| 3722 | </lyxtabular> |
---|
| 3723 | |
---|
| 3724 | \end_inset |
---|
| 3725 | |
---|
| 3726 | |
---|
| 3727 | \end_layout |
---|
| 3728 | |
---|
| 3729 | \begin_layout Plain Layout |
---|
| 3730 | \begin_inset Caption |
---|
| 3731 | |
---|
| 3732 | \begin_layout Plain Layout |
---|
| 3733 | \begin_inset CommandInset label |
---|
| 3734 | LatexCommand label |
---|
| 3735 | name "fig:spatnypruchod0" |
---|
| 3736 | |
---|
| 3737 | \end_inset |
---|
| 3738 | |
---|
| 3739 | Pruběh otáček |
---|
| 3740 | \begin_inset Formula $\omega$ |
---|
| 3741 | \end_inset |
---|
| 3742 | |
---|
| 3743 | a polohy |
---|
| 3744 | \begin_inset Formula $\vartheta$ |
---|
| 3745 | \end_inset |
---|
| 3746 | |
---|
| 3747 | pro simulace na simulátoru PMSM s kompenzací úbytků napětí i bez ní pro |
---|
| 3748 | reprezentativní model (3 -- plný model bez hyperstavu) |
---|
| 3749 | \end_layout |
---|
| 3750 | |
---|
| 3751 | \end_inset |
---|
| 3752 | |
---|
| 3753 | |
---|
| 3754 | \end_layout |
---|
| 3755 | |
---|
| 3756 | \end_inset |
---|
| 3757 | |
---|
| 3758 | |
---|
| 3759 | \end_layout |
---|
| 3760 | |
---|
| 3761 | \begin_layout Standard |
---|
| 3762 | V případě profilu požadovaných otáček |
---|
| 3763 | \begin_inset Formula $\pm200$ |
---|
| 3764 | \end_inset |
---|
| 3765 | |
---|
| 3766 | poskytuje simulátor PMSM lepší výsledky, je však důležité užít kompenzace |
---|
| 3767 | úbytků napětí. |
---|
| 3768 | Průběhy otáček |
---|
| 3769 | \begin_inset Formula $\omega$ |
---|
| 3770 | \end_inset |
---|
| 3771 | |
---|
| 3772 | a polohy |
---|
| 3773 | \begin_inset Formula $\vartheta$ |
---|
| 3774 | \end_inset |
---|
| 3775 | |
---|
| 3776 | pro simulátor PMSM bez kompenzace úbytků je zobrazen na grafech obrázek |
---|
| 3777 | ( |
---|
| 3778 | \begin_inset CommandInset ref |
---|
| 3779 | LatexCommand ref |
---|
| 3780 | reference "fig_grafy_pmsm200necp" |
---|
| 3781 | |
---|
| 3782 | \end_inset |
---|
| 3783 | |
---|
| 3784 | ). |
---|
| 3785 | Přínos kompenzace úbytků je pak patrný ze srovnání s grafy obrázek ( |
---|
| 3786 | \begin_inset CommandInset ref |
---|
| 3787 | LatexCommand ref |
---|
| 3788 | reference "fig_grafy_pmsm200scp" |
---|
| 3789 | |
---|
| 3790 | \end_inset |
---|
| 3791 | |
---|
| 3792 | ). |
---|
| 3793 | Srovnání dosažených ztrát shrnuje následující tabulka: |
---|
| 3794 | \end_layout |
---|
| 3795 | |
---|
| 3796 | \begin_layout Standard |
---|
| 3797 | \align center |
---|
| 3798 | \begin_inset Tabular |
---|
| 3799 | <lyxtabular version="3" rows="5" columns="3"> |
---|
| 3800 | <features tabularvalignment="middle"> |
---|
| 3801 | <column alignment="center" valignment="top" width="0"> |
---|
| 3802 | <column alignment="center" valignment="top" width="0"> |
---|
| 3803 | <column alignment="center" valignment="top" width="0"> |
---|
| 3804 | <row> |
---|
| 3805 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3806 | \begin_inset Text |
---|
| 3807 | |
---|
| 3808 | \begin_layout Plain Layout |
---|
| 3809 | |
---|
| 3810 | \series bold |
---|
| 3811 | použitý model |
---|
| 3812 | \backslash |
---|
| 3813 | kompenzace úbytků |
---|
| 3814 | \end_layout |
---|
| 3815 | |
---|
| 3816 | \end_inset |
---|
| 3817 | </cell> |
---|
| 3818 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3819 | \begin_inset Text |
---|
| 3820 | |
---|
| 3821 | \begin_layout Plain Layout |
---|
| 3822 | s kompenzací |
---|
| 3823 | \end_layout |
---|
| 3824 | |
---|
| 3825 | \end_inset |
---|
| 3826 | </cell> |
---|
| 3827 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3828 | \begin_inset Text |
---|
| 3829 | |
---|
| 3830 | \begin_layout Plain Layout |
---|
| 3831 | bez kompenzace |
---|
| 3832 | \end_layout |
---|
| 3833 | |
---|
| 3834 | \end_inset |
---|
| 3835 | </cell> |
---|
| 3836 | </row> |
---|
| 3837 | <row> |
---|
| 3838 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3839 | \begin_inset Text |
---|
| 3840 | |
---|
| 3841 | \begin_layout Plain Layout |
---|
| 3842 | 1 -- redukovaný model |
---|
| 3843 | \end_layout |
---|
| 3844 | |
---|
| 3845 | \end_inset |
---|
| 3846 | </cell> |
---|
| 3847 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3848 | \begin_inset Text |
---|
| 3849 | |
---|
| 3850 | \begin_layout Plain Layout |
---|
| 3851 | \begin_inset Formula $9,84\cdot10^{8}$ |
---|
| 3852 | \end_inset |
---|
| 3853 | |
---|
| 3854 | |
---|
| 3855 | \end_layout |
---|
| 3856 | |
---|
| 3857 | \end_inset |
---|
| 3858 | </cell> |
---|
| 3859 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3860 | \begin_inset Text |
---|
| 3861 | |
---|
| 3862 | \begin_layout Plain Layout |
---|
| 3863 | \begin_inset Formula $8,54\cdot10^{6}$ |
---|
| 3864 | \end_inset |
---|
| 3865 | |
---|
| 3866 | |
---|
| 3867 | \end_layout |
---|
| 3868 | |
---|
| 3869 | \end_inset |
---|
| 3870 | </cell> |
---|
| 3871 | </row> |
---|
| 3872 | <row> |
---|
| 3873 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3874 | \begin_inset Text |
---|
| 3875 | |
---|
| 3876 | \begin_layout Plain Layout |
---|
| 3877 | 2 -- redukovaný model s hyperstavem |
---|
| 3878 | \end_layout |
---|
| 3879 | |
---|
| 3880 | \end_inset |
---|
| 3881 | </cell> |
---|
| 3882 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3883 | \begin_inset Text |
---|
| 3884 | |
---|
| 3885 | \begin_layout Plain Layout |
---|
| 3886 | \begin_inset Formula $1,12\cdot10^{9}$ |
---|
| 3887 | \end_inset |
---|
| 3888 | |
---|
| 3889 | |
---|
| 3890 | \end_layout |
---|
| 3891 | |
---|
| 3892 | \end_inset |
---|
| 3893 | </cell> |
---|
| 3894 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3895 | \begin_inset Text |
---|
| 3896 | |
---|
| 3897 | \begin_layout Plain Layout |
---|
| 3898 | \begin_inset Formula $1,97\cdot10^{5}$ |
---|
| 3899 | \end_inset |
---|
| 3900 | |
---|
| 3901 | |
---|
| 3902 | \end_layout |
---|
| 3903 | |
---|
| 3904 | \end_inset |
---|
| 3905 | </cell> |
---|
| 3906 | </row> |
---|
| 3907 | <row> |
---|
| 3908 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3909 | \begin_inset Text |
---|
| 3910 | |
---|
| 3911 | \begin_layout Plain Layout |
---|
| 3912 | 3 -- plný model |
---|
| 3913 | \end_layout |
---|
| 3914 | |
---|
| 3915 | \end_inset |
---|
| 3916 | </cell> |
---|
| 3917 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3918 | \begin_inset Text |
---|
| 3919 | |
---|
| 3920 | \begin_layout Plain Layout |
---|
| 3921 | \begin_inset Formula $7,98\cdot10^{7}$ |
---|
| 3922 | \end_inset |
---|
| 3923 | |
---|
| 3924 | |
---|
| 3925 | \end_layout |
---|
| 3926 | |
---|
| 3927 | \end_inset |
---|
| 3928 | </cell> |
---|
| 3929 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3930 | \begin_inset Text |
---|
| 3931 | |
---|
| 3932 | \begin_layout Plain Layout |
---|
| 3933 | \begin_inset Formula $2,82\cdot10^{5}$ |
---|
| 3934 | \end_inset |
---|
| 3935 | |
---|
| 3936 | |
---|
| 3937 | \end_layout |
---|
| 3938 | |
---|
| 3939 | \end_inset |
---|
| 3940 | </cell> |
---|
| 3941 | </row> |
---|
| 3942 | <row> |
---|
| 3943 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3944 | \begin_inset Text |
---|
| 3945 | |
---|
| 3946 | \begin_layout Plain Layout |
---|
| 3947 | 4 -- plný model s hyperstavem |
---|
| 3948 | \end_layout |
---|
| 3949 | |
---|
| 3950 | \end_inset |
---|
| 3951 | </cell> |
---|
| 3952 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 3953 | \begin_inset Text |
---|
| 3954 | |
---|
| 3955 | \begin_layout Plain Layout |
---|
| 3956 | \begin_inset Formula $4,92\cdot10^{9}$ |
---|
| 3957 | \end_inset |
---|
| 3958 | |
---|
| 3959 | |
---|
| 3960 | \end_layout |
---|
| 3961 | |
---|
| 3962 | \end_inset |
---|
| 3963 | </cell> |
---|
| 3964 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 3965 | \begin_inset Text |
---|
| 3966 | |
---|
| 3967 | \begin_layout Plain Layout |
---|
| 3968 | \begin_inset Formula $9,45\cdot10^{5}$ |
---|
| 3969 | \end_inset |
---|
| 3970 | |
---|
| 3971 | |
---|
| 3972 | \end_layout |
---|
| 3973 | |
---|
| 3974 | \end_inset |
---|
| 3975 | </cell> |
---|
| 3976 | </row> |
---|
| 3977 | </lyxtabular> |
---|
| 3978 | |
---|
| 3979 | \end_inset |
---|
| 3980 | |
---|
| 3981 | |
---|
| 3982 | \end_layout |
---|
| 3983 | |
---|
| 3984 | \begin_layout Standard |
---|
| 3985 | \begin_inset Float figure |
---|
| 3986 | wide false |
---|
| 3987 | sideways false |
---|
| 3988 | status collapsed |
---|
| 3989 | |
---|
| 3990 | \begin_layout Plain Layout |
---|
| 3991 | \align center |
---|
| 3992 | \begin_inset Tabular |
---|
| 3993 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 3994 | <features tabularvalignment="middle"> |
---|
| 3995 | <column alignment="center" valignment="top" width="0"> |
---|
| 3996 | <column alignment="center" valignment="top" width="0"> |
---|
| 3997 | <row> |
---|
| 3998 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 3999 | \begin_inset Text |
---|
| 4000 | |
---|
| 4001 | \begin_layout Plain Layout |
---|
| 4002 | \begin_inset Graphics |
---|
| 4003 | filename grafy/lqplots/sim200necp1.eps |
---|
| 4004 | scale 40 |
---|
| 4005 | |
---|
| 4006 | \end_inset |
---|
| 4007 | |
---|
| 4008 | |
---|
| 4009 | \end_layout |
---|
| 4010 | |
---|
| 4011 | \end_inset |
---|
| 4012 | </cell> |
---|
| 4013 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4014 | \begin_inset Text |
---|
| 4015 | |
---|
| 4016 | \begin_layout Plain Layout |
---|
| 4017 | \begin_inset Graphics |
---|
| 4018 | filename grafy/lqplots/sim200necp2.eps |
---|
| 4019 | scale 40 |
---|
| 4020 | |
---|
| 4021 | \end_inset |
---|
| 4022 | |
---|
| 4023 | |
---|
| 4024 | \end_layout |
---|
| 4025 | |
---|
| 4026 | \end_inset |
---|
| 4027 | </cell> |
---|
| 4028 | </row> |
---|
| 4029 | <row> |
---|
| 4030 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 4031 | \begin_inset Text |
---|
| 4032 | |
---|
| 4033 | \begin_layout Plain Layout |
---|
| 4034 | 1 -- redukovaný model |
---|
| 4035 | \end_layout |
---|
| 4036 | |
---|
| 4037 | \end_inset |
---|
| 4038 | </cell> |
---|
| 4039 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4040 | \begin_inset Text |
---|
| 4041 | |
---|
| 4042 | \begin_layout Plain Layout |
---|
| 4043 | 2 -- redukovaný model s hyperstavem |
---|
| 4044 | \end_layout |
---|
| 4045 | |
---|
| 4046 | \end_inset |
---|
| 4047 | </cell> |
---|
| 4048 | </row> |
---|
| 4049 | <row> |
---|
| 4050 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 4051 | \begin_inset Text |
---|
| 4052 | |
---|
| 4053 | \begin_layout Plain Layout |
---|
| 4054 | \begin_inset Graphics |
---|
| 4055 | filename grafy/lqplots/sim200necp3.eps |
---|
| 4056 | scale 40 |
---|
| 4057 | |
---|
| 4058 | \end_inset |
---|
| 4059 | |
---|
| 4060 | |
---|
| 4061 | \end_layout |
---|
| 4062 | |
---|
| 4063 | \end_inset |
---|
| 4064 | </cell> |
---|
| 4065 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4066 | \begin_inset Text |
---|
| 4067 | |
---|
| 4068 | \begin_layout Plain Layout |
---|
| 4069 | \begin_inset Graphics |
---|
| 4070 | filename grafy/lqplots/sim200necp4.eps |
---|
| 4071 | scale 40 |
---|
| 4072 | |
---|
| 4073 | \end_inset |
---|
| 4074 | |
---|
| 4075 | |
---|
| 4076 | \end_layout |
---|
| 4077 | |
---|
| 4078 | \end_inset |
---|
| 4079 | </cell> |
---|
| 4080 | </row> |
---|
| 4081 | <row> |
---|
| 4082 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 4083 | \begin_inset Text |
---|
| 4084 | |
---|
| 4085 | \begin_layout Plain Layout |
---|
| 4086 | 3 -- plný model |
---|
| 4087 | \end_layout |
---|
| 4088 | |
---|
| 4089 | \end_inset |
---|
| 4090 | </cell> |
---|
| 4091 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4092 | \begin_inset Text |
---|
| 4093 | |
---|
| 4094 | \begin_layout Plain Layout |
---|
| 4095 | 4 -- plný model s hyperstavem |
---|
| 4096 | \end_layout |
---|
| 4097 | |
---|
| 4098 | \end_inset |
---|
| 4099 | </cell> |
---|
| 4100 | </row> |
---|
| 4101 | </lyxtabular> |
---|
| 4102 | |
---|
| 4103 | \end_inset |
---|
| 4104 | |
---|
| 4105 | |
---|
| 4106 | \begin_inset Caption |
---|
| 4107 | |
---|
| 4108 | \begin_layout Plain Layout |
---|
| 4109 | Průběhy otáček |
---|
| 4110 | \begin_inset Formula $\omega$ |
---|
| 4111 | \end_inset |
---|
| 4112 | |
---|
| 4113 | a polohy |
---|
| 4114 | \begin_inset Formula $\vartheta$ |
---|
| 4115 | \end_inset |
---|
| 4116 | |
---|
| 4117 | při užití simulátoru PMSM bez užití kompenzace úbytků napětí a s profilem |
---|
| 4118 | požadovaných otáček |
---|
| 4119 | \begin_inset Formula $\pm200$ |
---|
| 4120 | \end_inset |
---|
| 4121 | |
---|
| 4122 | |
---|
| 4123 | \end_layout |
---|
| 4124 | |
---|
| 4125 | \end_inset |
---|
| 4126 | |
---|
| 4127 | |
---|
| 4128 | \begin_inset CommandInset label |
---|
| 4129 | LatexCommand label |
---|
| 4130 | name "fig_grafy_pmsm200necp" |
---|
| 4131 | |
---|
| 4132 | \end_inset |
---|
| 4133 | |
---|
| 4134 | |
---|
| 4135 | \end_layout |
---|
| 4136 | |
---|
| 4137 | \end_inset |
---|
| 4138 | |
---|
| 4139 | |
---|
| 4140 | \end_layout |
---|
| 4141 | |
---|
| 4142 | \begin_layout Standard |
---|
| 4143 | \begin_inset Float figure |
---|
| 4144 | wide false |
---|
| 4145 | sideways false |
---|
| 4146 | status collapsed |
---|
| 4147 | |
---|
| 4148 | \begin_layout Plain Layout |
---|
| 4149 | \align center |
---|
| 4150 | \begin_inset Tabular |
---|
| 4151 | <lyxtabular version="3" rows="4" columns="2"> |
---|
| 4152 | <features tabularvalignment="middle"> |
---|
| 4153 | <column alignment="center" valignment="top" width="0"> |
---|
| 4154 | <column alignment="center" valignment="top" width="0"> |
---|
| 4155 | <row> |
---|
| 4156 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 4157 | \begin_inset Text |
---|
| 4158 | |
---|
| 4159 | \begin_layout Plain Layout |
---|
| 4160 | \begin_inset Graphics |
---|
| 4161 | filename grafy/lqplots/sim200cp1.eps |
---|
| 4162 | scale 40 |
---|
| 4163 | |
---|
| 4164 | \end_inset |
---|
| 4165 | |
---|
| 4166 | |
---|
| 4167 | \end_layout |
---|
| 4168 | |
---|
| 4169 | \end_inset |
---|
| 4170 | </cell> |
---|
| 4171 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4172 | \begin_inset Text |
---|
| 4173 | |
---|
| 4174 | \begin_layout Plain Layout |
---|
| 4175 | \begin_inset Graphics |
---|
| 4176 | filename grafy/lqplots/sim200cp2.eps |
---|
| 4177 | scale 40 |
---|
| 4178 | |
---|
| 4179 | \end_inset |
---|
| 4180 | |
---|
| 4181 | |
---|
| 4182 | \end_layout |
---|
| 4183 | |
---|
| 4184 | \end_inset |
---|
| 4185 | </cell> |
---|
| 4186 | </row> |
---|
| 4187 | <row> |
---|
| 4188 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 4189 | \begin_inset Text |
---|
| 4190 | |
---|
| 4191 | \begin_layout Plain Layout |
---|
| 4192 | 1 -- redukovaný model |
---|
| 4193 | \end_layout |
---|
| 4194 | |
---|
| 4195 | \end_inset |
---|
| 4196 | </cell> |
---|
| 4197 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4198 | \begin_inset Text |
---|
| 4199 | |
---|
| 4200 | \begin_layout Plain Layout |
---|
| 4201 | 2 -- redukovaný model s hyperstavem |
---|
| 4202 | \end_layout |
---|
| 4203 | |
---|
| 4204 | \end_inset |
---|
| 4205 | </cell> |
---|
| 4206 | </row> |
---|
| 4207 | <row> |
---|
| 4208 | <cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none"> |
---|
| 4209 | \begin_inset Text |
---|
| 4210 | |
---|
| 4211 | \begin_layout Plain Layout |
---|
| 4212 | \begin_inset Graphics |
---|
| 4213 | filename grafy/lqplots/sim200cp3.eps |
---|
| 4214 | scale 40 |
---|
| 4215 | |
---|
| 4216 | \end_inset |
---|
| 4217 | |
---|
| 4218 | |
---|
| 4219 | \end_layout |
---|
| 4220 | |
---|
| 4221 | \end_inset |
---|
| 4222 | </cell> |
---|
| 4223 | <cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4224 | \begin_inset Text |
---|
| 4225 | |
---|
| 4226 | \begin_layout Plain Layout |
---|
| 4227 | \begin_inset Graphics |
---|
| 4228 | filename grafy/lqplots/sim200cp4.eps |
---|
| 4229 | scale 40 |
---|
| 4230 | |
---|
| 4231 | \end_inset |
---|
| 4232 | |
---|
| 4233 | |
---|
| 4234 | \end_layout |
---|
| 4235 | |
---|
| 4236 | \end_inset |
---|
| 4237 | </cell> |
---|
| 4238 | </row> |
---|
| 4239 | <row> |
---|
| 4240 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none"> |
---|
| 4241 | \begin_inset Text |
---|
| 4242 | |
---|
| 4243 | \begin_layout Plain Layout |
---|
| 4244 | 3 -- plný model |
---|
| 4245 | \end_layout |
---|
| 4246 | |
---|
| 4247 | \end_inset |
---|
| 4248 | </cell> |
---|
| 4249 | <cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none"> |
---|
| 4250 | \begin_inset Text |
---|
| 4251 | |
---|
| 4252 | \begin_layout Plain Layout |
---|
| 4253 | 4 -- plný model s hyperstavem |
---|
| 4254 | \end_layout |
---|
| 4255 | |
---|
| 4256 | \end_inset |
---|
| 4257 | </cell> |
---|
| 4258 | </row> |
---|
| 4259 | </lyxtabular> |
---|
| 4260 | |
---|
| 4261 | \end_inset |
---|
| 4262 | |
---|
| 4263 | |
---|
| 4264 | \begin_inset Caption |
---|
| 4265 | |
---|
| 4266 | \begin_layout Plain Layout |
---|
| 4267 | Průběhy otáček |
---|
| 4268 | \begin_inset Formula $\omega$ |
---|
| 4269 | \end_inset |
---|
| 4270 | |
---|
| 4271 | a polohy |
---|
| 4272 | \begin_inset Formula $\vartheta$ |
---|
| 4273 | \end_inset |
---|
| 4274 | |
---|
| 4275 | při užití simulátoru PMSM s užitím kompenzace úbytků napětí a s profilem |
---|
| 4276 | požadovaných otáček |
---|
| 4277 | \begin_inset Formula $\pm200$ |
---|
| 4278 | \end_inset |
---|
| 4279 | |
---|
| 4280 | |
---|
| 4281 | \end_layout |
---|
| 4282 | |
---|
| 4283 | \end_inset |
---|
| 4284 | |
---|
| 4285 | |
---|
| 4286 | \begin_inset CommandInset label |
---|
| 4287 | LatexCommand label |
---|
| 4288 | name "fig_grafy_pmsm200scp" |
---|
| 4289 | |
---|
| 4290 | \end_inset |
---|
| 4291 | |
---|
| 4292 | |
---|
| 4293 | \end_layout |
---|
| 4294 | |
---|
| 4295 | \end_inset |
---|
| 4296 | |
---|
| 4297 | |
---|
| 4298 | \end_layout |
---|
| 4299 | |
---|
| 4300 | \begin_layout Subsection* |
---|
| 4301 | Shrnutí |
---|
| 4302 | \end_layout |
---|
| 4303 | |
---|
| 4304 | \begin_layout Itemize |
---|
| 4305 | kompenzace úbytků napětí se daří jen při vyšších otáčkách, při nižším moc |
---|
| 4306 | nefunguje |
---|
| 4307 | \end_layout |
---|
| 4308 | |
---|
| 4309 | \begin_layout Itemize |
---|
| 4310 | výhoda využití hyperstavu je především v přesnějším řízení pro redukovaný |
---|
| 4311 | model a dále v lepším zvládnutí špatného počátečního odhadu |
---|
| 4312 | \begin_inset Formula $\vartheta_{0}$ |
---|
| 4313 | \end_inset |
---|
| 4314 | |
---|
| 4315 | |
---|
| 4316 | \end_layout |
---|
| 4317 | |
---|
| 4318 | \begin_layout Itemize |
---|
| 4319 | proti špatným průchodům nulou při nízkých otáčkách zatím nic nepomáhá -- |
---|
| 4320 | ani hyperstav ani kompenzace |
---|
| 4321 | \end_layout |
---|
| 4322 | |
---|
| 4323 | \end_body |
---|
| 4324 | \end_document |
---|