1 | #LyX 1.6.4 created this file. For more info see http://www.lyx.org/ |
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2 | \lyxformat 345 |
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3 | \begin_document |
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4 | \begin_header |
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5 | \textclass article |
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6 | \use_default_options true |
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7 | \language english |
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8 | \inputencoding auto |
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9 | \font_roman default |
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15 | \font_sf_scale 100 |
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16 | \font_tt_scale 100 |
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17 | |
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18 | \graphics default |
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19 | \paperfontsize default |
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20 | \use_hyperref false |
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21 | \papersize default |
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22 | \use_geometry false |
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23 | \use_amsmath 1 |
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24 | \use_esint 1 |
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25 | \cite_engine basic |
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26 | \use_bibtopic false |
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27 | \paperorientation portrait |
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28 | \secnumdepth 3 |
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29 | \tocdepth 3 |
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30 | \paragraph_separation indent |
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31 | \defskip medskip |
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32 | \quotes_language english |
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33 | \papercolumns 1 |
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34 | \papersides 1 |
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35 | \paperpagestyle default |
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36 | \tracking_changes false |
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37 | \output_changes false |
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38 | \author "" |
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39 | \end_header |
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40 | |
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41 | \begin_body |
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42 | |
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43 | \begin_layout Standard |
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44 | System |
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45 | \begin_inset Formula \[ |
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46 | y_{t}=y_{t-1}+bu_{t}+e_{t}\] |
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47 | |
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48 | \end_inset |
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49 | |
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50 | is assumed to fullfill causality, i.e. |
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51 | |
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52 | \begin_inset Formula $y_{t-1}\rightarrow u_{t}\rightarrow y_{t}$ |
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53 | \end_inset |
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54 | |
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55 | |
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56 | \end_layout |
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57 | |
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58 | \begin_layout Standard |
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59 | with change in notation, i.e. |
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60 | |
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61 | \begin_inset Formula $y_{t-1}\rightarrow u_{t-1}\rightarrow y_{t}$ |
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62 | \end_inset |
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63 | |
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64 | the same system is: |
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65 | \begin_inset Formula \[ |
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66 | y_{t}=y_{t-1}+bu_{t-1}+e_{t}\] |
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67 | |
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68 | \end_inset |
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69 | |
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70 | and can be transformed as |
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71 | \begin_inset Formula \begin{eqnarray*} |
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72 | x_{t+1} & = & x_{t}+bu_{t}+e_{t}\\ |
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73 | y_{t} & = & x_{t}\end{eqnarray*} |
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74 | |
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75 | \end_inset |
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76 | |
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77 | Morale: |
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78 | \end_layout |
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79 | |
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80 | \begin_layout Itemize |
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81 | for transcription from ARX to State-Space, the state is one time-step shorter |
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82 | than the regressor in ARX! |
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83 | \end_layout |
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84 | |
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85 | \begin_layout Subsection* |
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86 | Delays in |
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87 | \begin_inset Formula $u$ |
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88 | \end_inset |
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89 | |
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90 | : |
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91 | \end_layout |
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92 | |
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93 | \begin_layout Standard |
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94 | System |
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95 | \begin_inset Formula \[ |
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96 | y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t}\] |
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97 | |
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98 | \end_inset |
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99 | |
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100 | is |
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101 | \begin_inset Formula \begin{eqnarray*} |
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102 | \left[\begin{array}{c} |
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103 | x_{t+1}\\ |
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104 | u_{t}\end{array}\right] & =\left[\begin{array}{cc} |
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105 | 1 & c\\ |
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106 | 0 & 0\end{array}\right] & \left[\begin{array}{c} |
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107 | x_{t}\\ |
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108 | u_{t-1}\end{array}\right]+\left[\begin{array}{c} |
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109 | b\\ |
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110 | 1\end{array}\right]u_{t}+e_{t}\\ |
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111 | y_{t} & = & x_{t}\end{eqnarray*} |
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112 | |
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113 | \end_inset |
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114 | |
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115 | Notation in BDM: |
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116 | \end_layout |
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117 | |
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118 | \begin_layout Standard |
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119 | xrv = [y{0}, u{-1}] |
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120 | \end_layout |
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121 | |
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122 | \begin_layout Subsection* |
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123 | Constant term |
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124 | \end_layout |
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125 | |
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126 | \begin_layout Standard |
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127 | System |
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128 | \begin_inset Formula \[ |
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129 | y_{t}=y_{t-1}+bu_{t-1}+g+e_{t}\] |
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130 | |
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131 | \end_inset |
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132 | |
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133 | must be modelled as |
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134 | \begin_inset Formula \[ |
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135 | \left[\begin{array}{c} |
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136 | x_{t+1}\\ |
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137 | one\end{array}\right]=\left[\begin{array}{cc} |
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138 | 1 & c\\ |
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139 | 0 & 0\end{array}\right]\left[\begin{array}{c} |
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140 | x_{t}\\ |
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141 | one\end{array}\right]+\left[\begin{array}{c} |
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142 | b\\ |
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143 | 1\end{array}\right]u_{t}+e_{t}\] |
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144 | |
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145 | \end_inset |
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146 | |
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147 | where one is an extra state variable. |
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148 | \end_layout |
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149 | |
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150 | \end_body |
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151 | \end_document |
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