[1469] | 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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[723] | 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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[1469] | 7 | \maintain_unincluded_children false |
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[723] | 8 | \language english |
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[1469] | 9 | \language_package default |
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| 31 | \use_amsmath 1 |
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| 32 | \use_esint 1 |
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[1469] | 33 | \use_mhchem 1 |
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[723] | 35 | \cite_engine basic |
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[1469] | 37 | \use_indices false |
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[723] | 38 | \paperorientation portrait |
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[1469] | 39 | \suppress_date false |
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| 41 | \index Index |
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| 42 | \shortcut idx |
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| 43 | \color #008000 |
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| 44 | \end_index |
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[723] | 45 | \secnumdepth 3 |
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| 46 | \tocdepth 3 |
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| 47 | \paragraph_separation indent |
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[1469] | 48 | \paragraph_indentation default |
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[723] | 49 | \quotes_language english |
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| 50 | \papercolumns 1 |
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[723] | 58 | \end_header |
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| 59 | |
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| 60 | \begin_body |
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| 61 | |
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| 62 | \begin_layout Standard |
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| 63 | System |
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[1469] | 64 | \begin_inset Formula |
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| 65 | \[ |
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| 66 | y_{t}=y_{t-1}+bu_{t}+e_{t} |
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| 67 | \] |
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[723] | 68 | |
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| 69 | \end_inset |
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| 70 | |
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| 71 | is assumed to fullfill causality, i.e. |
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| 72 | |
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| 73 | \begin_inset Formula $y_{t-1}\rightarrow u_{t}\rightarrow y_{t}$ |
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| 74 | \end_inset |
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| 75 | |
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| 76 | |
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| 77 | \end_layout |
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| 78 | |
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| 79 | \begin_layout Standard |
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| 80 | with change in notation, i.e. |
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| 81 | |
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| 82 | \begin_inset Formula $y_{t-1}\rightarrow u_{t-1}\rightarrow y_{t}$ |
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| 83 | \end_inset |
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| 84 | |
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| 85 | the same system is: |
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[1469] | 86 | \begin_inset Formula |
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| 87 | \[ |
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| 88 | y_{t}=y_{t-1}+bu_{t-1}+e_{t} |
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| 89 | \] |
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[723] | 90 | |
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| 91 | \end_inset |
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| 92 | |
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| 93 | and can be transformed as |
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[1469] | 94 | \begin_inset Formula |
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| 95 | \begin{eqnarray*} |
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[723] | 96 | x_{t+1} & = & x_{t}+bu_{t}+e_{t}\\ |
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[1469] | 97 | y_{t} & = & x_{t} |
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| 98 | \end{eqnarray*} |
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[723] | 99 | |
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| 100 | \end_inset |
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| 101 | |
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| 102 | Morale: |
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| 103 | \end_layout |
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| 104 | |
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| 105 | \begin_layout Itemize |
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| 106 | for transcription from ARX to State-Space, the state is one time-step shorter |
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| 107 | than the regressor in ARX! |
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| 108 | \end_layout |
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| 109 | |
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| 110 | \begin_layout Subsection* |
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| 111 | Delays in |
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| 112 | \begin_inset Formula $u$ |
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| 113 | \end_inset |
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| 114 | |
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| 115 | : |
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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 | System |
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[1469] | 120 | \begin_inset Formula |
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| 121 | \[ |
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| 122 | y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t} |
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| 123 | \] |
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[723] | 124 | |
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| 125 | \end_inset |
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| 126 | |
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| 127 | is |
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[1469] | 128 | \begin_inset Formula |
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| 129 | \begin{eqnarray*} |
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[723] | 130 | \left[\begin{array}{c} |
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| 131 | x_{t+1}\\ |
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[1469] | 132 | u_{t} |
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| 133 | \end{array}\right] & =\left[\begin{array}{cc} |
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[723] | 134 | 1 & c\\ |
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[1469] | 135 | 0 & 0 |
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| 136 | \end{array}\right] & \left[\begin{array}{c} |
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[723] | 137 | x_{t}\\ |
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[1469] | 138 | u_{t-1} |
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| 139 | \end{array}\right]+\left[\begin{array}{c} |
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[723] | 140 | b\\ |
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[1469] | 141 | 1 |
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| 142 | \end{array}\right]u_{t}+e_{t}\\ |
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| 143 | y_{t} & = & x_{t} |
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| 144 | \end{eqnarray*} |
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[723] | 145 | |
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| 146 | \end_inset |
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| 147 | |
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[1469] | 148 | |
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| 149 | \end_layout |
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| 150 | |
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| 151 | \begin_layout Standard |
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| 152 | System #2: |
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| 153 | \begin_inset Formula |
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| 154 | \[ |
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| 155 | y_{t}=ay_{t-1}+a_{2}y_{t-2}+bu_{t-1}+cu_{t-2}+e_{t} |
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| 156 | \] |
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| 157 | |
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| 158 | \end_inset |
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| 159 | |
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| 160 | is |
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| 161 | \begin_inset Formula |
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| 162 | \begin{eqnarray*} |
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| 163 | \left[\begin{array}{c} |
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| 164 | x_{t+1}\\ |
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| 165 | x_{t}\\ |
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| 166 | u_{t} |
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| 167 | \end{array}\right] & =\left[\begin{array}{ccc} |
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| 168 | a_{1} & a_{2} & c\\ |
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| 169 | 1 & 0 & 0\\ |
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| 170 | 0 & 0 & 0 |
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| 171 | \end{array}\right] & \left[\begin{array}{c} |
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| 172 | x_{t}\\ |
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| 173 | x_{t-1}\\ |
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| 174 | u_{t-1} |
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| 175 | \end{array}\right]+\left[\begin{array}{c} |
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| 176 | b\\ |
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| 177 | \\ |
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| 178 | 1 |
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| 179 | \end{array}\right]u_{t}+e_{t}\\ |
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| 180 | y_{t} & = & x_{t} |
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| 181 | \end{eqnarray*} |
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| 182 | |
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| 183 | \end_inset |
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| 184 | |
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| 185 | |
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| 186 | \end_layout |
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| 187 | |
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| 188 | \begin_layout Standard |
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[723] | 189 | Notation in BDM: |
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| 190 | \end_layout |
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| 191 | |
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| 192 | \begin_layout Standard |
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| 193 | xrv = [y{0}, u{-1}] |
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| 194 | \end_layout |
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| 195 | |
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| 196 | \begin_layout Subsection* |
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| 197 | Constant term |
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| 198 | \end_layout |
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| 199 | |
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| 200 | \begin_layout Standard |
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| 201 | System |
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[1469] | 202 | \begin_inset Formula |
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| 203 | \[ |
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| 204 | y_{t}=y_{t-1}+bu_{t-1}+g+e_{t} |
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| 205 | \] |
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[723] | 206 | |
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| 207 | \end_inset |
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| 208 | |
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| 209 | must be modelled as |
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[1469] | 210 | \begin_inset Formula |
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| 211 | \[ |
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[723] | 212 | \left[\begin{array}{c} |
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| 213 | x_{t+1}\\ |
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[1469] | 214 | one |
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| 215 | \end{array}\right]=\left[\begin{array}{cc} |
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[723] | 216 | 1 & c\\ |
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[1469] | 217 | 0 & 0 |
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| 218 | \end{array}\right]\left[\begin{array}{c} |
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[723] | 219 | x_{t}\\ |
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[1469] | 220 | one |
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| 221 | \end{array}\right]+\left[\begin{array}{c} |
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[723] | 222 | b\\ |
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[1469] | 223 | 1 |
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| 224 | \end{array}\right]u_{t}+e_{t} |
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| 225 | \] |
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[723] | 226 | |
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| 227 | \end_inset |
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| 228 | |
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| 229 | where one is an extra state variable. |
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| 230 | \end_layout |
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| 231 | |
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| 232 | \end_body |
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| 233 | \end_document |
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