[723] | 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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| 10 | \font_sans default |
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| 11 | \font_typewriter 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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