- Timestamp:
- 12/02/13 13:54:59 (11 years ago)
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library/bdm/design/arx2statespace_notes.lyx (modified) (8 diffs)
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library/bdm/design/arx2statespace_notes.lyx
r723 r1469 1 #LyX 1.6.4created this file. For more info see http://www.lyx.org/2 \lyxformat 3451 #LyX 2.0 created this file. For more info see http://www.lyx.org/ 2 \lyxformat 413 3 3 \begin_document 4 4 \begin_header 5 5 \textclass article 6 6 \use_default_options true 7 \maintain_unincluded_children false 7 8 \language english 9 \language_package default 8 10 \inputencoding auto 11 \fontencoding global 9 12 \font_roman default 10 13 \font_sans default 11 14 \font_typewriter default 12 15 \font_default_family default 16 \use_non_tex_fonts false 13 17 \font_sc false 14 18 \font_osf false … … 17 21 18 22 \graphics default 23 \default_output_format default 24 \output_sync 0 25 \bibtex_command default 26 \index_command default 19 27 \paperfontsize default 20 28 \use_hyperref false … … 23 31 \use_amsmath 1 24 32 \use_esint 1 33 \use_mhchem 1 34 \use_mathdots 1 25 35 \cite_engine basic 26 36 \use_bibtopic false 37 \use_indices false 27 38 \paperorientation portrait 39 \suppress_date false 40 \use_refstyle 0 41 \index Index 42 \shortcut idx 43 \color #008000 44 \end_index 28 45 \secnumdepth 3 29 46 \tocdepth 3 30 47 \paragraph_separation indent 31 \ defskip medskip48 \paragraph_indentation default 32 49 \quotes_language english 33 50 \papercolumns 1 … … 36 53 \tracking_changes false 37 54 \output_changes false 38 \author "" 55 \html_math_output 0 56 \html_css_as_file 0 57 \html_be_strict false 39 58 \end_header 40 59 … … 43 62 \begin_layout Standard 44 63 System 45 \begin_inset Formula \[ 46 y_{t}=y_{t-1}+bu_{t}+e_{t}\] 64 \begin_inset Formula 65 \[ 66 y_{t}=y_{t-1}+bu_{t}+e_{t} 67 \] 47 68 48 69 \end_inset … … 63 84 64 85 the same system is: 65 \begin_inset Formula \[ 66 y_{t}=y_{t-1}+bu_{t-1}+e_{t}\] 86 \begin_inset Formula 87 \[ 88 y_{t}=y_{t-1}+bu_{t-1}+e_{t} 89 \] 67 90 68 91 \end_inset 69 92 70 93 and can be transformed as 71 \begin_inset Formula \begin{eqnarray*} 94 \begin_inset Formula 95 \begin{eqnarray*} 72 96 x_{t+1} & = & x_{t}+bu_{t}+e_{t}\\ 73 y_{t} & = & x_{t}\end{eqnarray*} 97 y_{t} & = & x_{t} 98 \end{eqnarray*} 74 99 75 100 \end_inset … … 93 118 \begin_layout Standard 94 119 System 95 \begin_inset Formula \[ 96 y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t}\] 120 \begin_inset Formula 121 \[ 122 y_{t}=y_{t-1}+bu_{t}+cu_{t-1}+e_{t} 123 \] 97 124 98 125 \end_inset 99 126 100 127 is 101 \begin_inset Formula \begin{eqnarray*} 128 \begin_inset Formula 129 \begin{eqnarray*} 102 130 \left[\begin{array}{c} 103 131 x_{t+1}\\ 104 u_{t}\end{array}\right] & =\left[\begin{array}{cc} 132 u_{t} 133 \end{array}\right] & =\left[\begin{array}{cc} 105 134 1 & c\\ 106 0 & 0\end{array}\right] & \left[\begin{array}{c} 107 x_{t}\\ 108 u_{t-1}\end{array}\right]+\left[\begin{array}{c} 135 0 & 0 136 \end{array}\right] & \left[\begin{array}{c} 137 x_{t}\\ 138 u_{t-1} 139 \end{array}\right]+\left[\begin{array}{c} 109 140 b\\ 110 1\end{array}\right]u_{t}+e_{t}\\ 111 y_{t} & = & x_{t}\end{eqnarray*} 112 113 \end_inset 114 141 1 142 \end{array}\right]u_{t}+e_{t}\\ 143 y_{t} & = & x_{t} 144 \end{eqnarray*} 145 146 \end_inset 147 148 149 \end_layout 150 151 \begin_layout Standard 152 System #2: 153 \begin_inset Formula 154 \[ 155 y_{t}=ay_{t-1}+a_{2}y_{t-2}+bu_{t-1}+cu_{t-2}+e_{t} 156 \] 157 158 \end_inset 159 160 is 161 \begin_inset Formula 162 \begin{eqnarray*} 163 \left[\begin{array}{c} 164 x_{t+1}\\ 165 x_{t}\\ 166 u_{t} 167 \end{array}\right] & =\left[\begin{array}{ccc} 168 a_{1} & a_{2} & c\\ 169 1 & 0 & 0\\ 170 0 & 0 & 0 171 \end{array}\right] & \left[\begin{array}{c} 172 x_{t}\\ 173 x_{t-1}\\ 174 u_{t-1} 175 \end{array}\right]+\left[\begin{array}{c} 176 b\\ 177 \\ 178 1 179 \end{array}\right]u_{t}+e_{t}\\ 180 y_{t} & = & x_{t} 181 \end{eqnarray*} 182 183 \end_inset 184 185 186 \end_layout 187 188 \begin_layout Standard 115 189 Notation in BDM: 116 190 \end_layout … … 126 200 \begin_layout Standard 127 201 System 128 \begin_inset Formula \[ 129 y_{t}=y_{t-1}+bu_{t-1}+g+e_{t}\] 202 \begin_inset Formula 203 \[ 204 y_{t}=y_{t-1}+bu_{t-1}+g+e_{t} 205 \] 130 206 131 207 \end_inset 132 208 133 209 must be modelled as 134 \begin_inset Formula \[ 210 \begin_inset Formula 211 \[ 135 212 \left[\begin{array}{c} 136 213 x_{t+1}\\ 137 one\end{array}\right]=\left[\begin{array}{cc} 214 one 215 \end{array}\right]=\left[\begin{array}{cc} 138 216 1 & c\\ 139 0 & 0\end{array}\right]\left[\begin{array}{c} 140 x_{t}\\ 141 one\end{array}\right]+\left[\begin{array}{c} 217 0 & 0 218 \end{array}\right]\left[\begin{array}{c} 219 x_{t}\\ 220 one 221 \end{array}\right]+\left[\begin{array}{c} 142 222 b\\ 143 1\end{array}\right]u_{t}+e_{t}\] 223 1 224 \end{array}\right]u_{t}+e_{t} 225 \] 144 226 145 227 \end_inset
