scilab/modules/xcos/help/en_US/palettes/Continuous_pal/CLSS.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
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  24 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns5="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="CLSS">
  25     <refnamediv>
  26         <refname>CLSS</refname>
  27         <refpurpose>Continuous state-space system</refpurpose>
  28     </refnamediv>
  29     <refsection>
  30         <title>Block Screenshot</title>
  31         <para>
  32             <inlinemediaobject>
  33                 <imageobject>
  34                     <imagedata align="center" fileref="../../../../images/palettes/CLSS.png" valign="middle"/>
  35                 </imageobject>
  36             </inlinemediaobject>
  37         </para>
  38     </refsection>
  39     <refsection id="Contents_CLSS">
  40         <title>Contents</title>
  41         <itemizedlist>
  42             <listitem>
  43                 <para>
  44                     <link linkend="CLSS">Continuous state-space system</link>
  45                 </para>
  46             </listitem>
  47             <listitem>
  48                 <itemizedlist>
  49                     <listitem>
  50                         <para>
  51                             <xref linkend="Palette_CLSS">Palette</xref>
  52                         </para>
  53                     </listitem>
  54                     <listitem>
  55                         <para>
  56                             <xref linkend="Description_CLSS">Description</xref>
  57                         </para>
  58                     </listitem>
  59                     <listitem>
  60                         <para>
  61                             <xref linkend="Dialogbox_CLSS">Dialog box</xref>
  62                         </para>
  63                     </listitem>
  64                     <listitem>
  65                         <para>
  66                             <xref linkend="Defaultproperties_CLSS">Default
  67                                 properties
  68                             </xref>
  69                         </para>
  70                     </listitem>
  71                     <listitem>
  72                         <para>
  73                             <xref linkend="Interfacingfunction_CLSS">Interfacing
  74                                 function
  75                             </xref>
  76                         </para>
  77                     </listitem>
  78                     <listitem>
  79                         <para>
  80                             <xref linkend="Computationalfunction_CLSS">Computational
  81                                 function
  82                             </xref>
  83                         </para>
  84                     </listitem>
  85                     <listitem>
  86                         <para>
  87                             <xref linkend="Example_CLSS">Example</xref>
  88                         </para>
  89                     </listitem>
  90                 </itemizedlist>
  91             </listitem>
  92         </itemizedlist>
  93     </refsection>
  94     <refsection id="Palette_CLSS">
  95         <title>Palette</title>
  96         <itemizedlist>
  97             <listitem>
  98                 <para>
  99                     <link linkend="Continuous_pal">Continuous time systems
 100                         palette
 101                     </link>
 102                 </para>
 103             </listitem>
 104         </itemizedlist>
 105     </refsection>
 106     <refsection id="Description_CLSS">
 107         <title>Description</title>
 108         <para>This block realizes a continuous-time linear state-space
 109             system.
 110         </para>
 111         <para>
 112             <latex>
 113                 \begin{matrix}
 114                 \dot{x} = A \cdot x + B \cdot u \\
 115                 y = C \cdot x + D \cdot u
 116                 \end{matrix}
 117             </latex>
 118         </para>
 119         <para>
 120             where <emphasis role="bold">x</emphasis> is the vector of state
 121             variables, <emphasis role="bold">u</emphasis> is the vector of input
 122             functions and <emphasis role="bold">y</emphasis> is the vector of output
 123             variables.
 124         </para>
 125         <para>
 126             The system is defined by the <emphasis role="bold">(A, B, C, D)</emphasis> matrices and the initial state <emphasis role="bold">X0</emphasis>.
 127             The dimensions must be compatible.
 128         </para>
 129     </refsection>
 130     <refsection id="Dialogbox_CLSS">
 131         <title>Dialog box</title>
 132         <para>
 133             <inlinemediaobject>
 134                 <imageobject>
 135                     <imagedata align="center" fileref="../../../../images/gui/CLSS_gui.gif" valign="middle"/>
 136                 </imageobject>
 137             </inlinemediaobject>
 138         </para>
 139         <itemizedlist>
 140             <listitem>
 141                 <para>
 142                     <emphasis role="bold">A matrix</emphasis>
 143                 </para>
 144                 <para>A square matrix.</para>
 145                 <para>Properties : Type 'mat' of size [-1,-1].</para>
 146             </listitem>
 147             <listitem>
 148                 <para>
 149                     <emphasis role="bold">B matrix</emphasis>
 150                 </para>
 151                 <para>
 152                     The <emphasis role="bold">B</emphasis> matrix, [] if system has no input.
 153                 </para>
 154                 <para>Properties : Type 'mat' of size ["size(%1,2)","-1"].</para>
 155             </listitem>
 156             <listitem>
 157                 <para>
 158                     <emphasis role="bold">C matrix</emphasis>
 159                 </para>
 160                 <para>
 161                     The <emphasis role="bold">C</emphasis> matrix , [] if system has no output.
 162                 </para>
 163                 <para>Properties : Type 'mat' of size ["-1","size(%1,2)"].</para>
 164             </listitem>
 165             <listitem>
 166                 <para>
 167                     <emphasis role="bold">D matrix</emphasis>
 168                 </para>
 169                 <para>
 170                     The <emphasis role="bold">D</emphasis> matrix, [] if system has no D term.
 171                 </para>
 172                 <para>Properties : Type 'mat' of size [-1,-1].</para>
 173             </listitem>
 174             <listitem>
 175                 <para>
 176                     <emphasis role="bold">Initial state</emphasis>
 177                 </para>
 178                 <para>A vector/scalar initial state of the system.</para>
 179                 <para>Properties : Type 'vec' of size "size(%1,2)".</para>
 180             </listitem>
 181         </itemizedlist>
 182     </refsection>
 183     <refsection id="Defaultproperties_CLSS">
 184         <title>Default properties</title>
 185         <itemizedlist>
 186             <listitem>
 187                 <para>
 188                     <emphasis role="bold">always active:</emphasis> yes
 189                 </para>
 190             </listitem>
 191             <listitem>
 192                 <para>
 193                     <emphasis role="bold">direct-feedthrough:</emphasis> no
 194                 </para>
 195             </listitem>
 196             <listitem>
 197                 <para>
 198                     <emphasis role="bold">zero-crossing:</emphasis> no
 199                 </para>
 200             </listitem>
 201             <listitem>
 202                 <para>
 203                     <emphasis role="bold">mode:</emphasis> no
 204                 </para>
 205             </listitem>
 206             <listitem>
 207                 <para>
 208                     <emphasis role="bold">regular inputs:</emphasis>
 209                 </para>
 210                 <para>
 211                     <emphasis role="bold">- port 1 : size [1,1] / type
 212                         1
 213                     </emphasis>
 214                 </para>
 215             </listitem>
 216             <listitem>
 217                 <para>
 218                     <emphasis role="bold">regular outputs:</emphasis>
 219                 </para>
 220                 <para>
 221                     <emphasis role="bold">- port 1 : size [1,1] / type
 222                         1
 223                     </emphasis>
 224                 </para>
 225             </listitem>
 226             <listitem>
 227                 <para>
 228                     <emphasis role="bold">number/sizes of activation
 229                         inputs:
 230                     </emphasis>
 231                     0
 232                 </para>
 233             </listitem>
 234             <listitem>
 235                 <para>
 236                     <emphasis role="bold">number/sizes of activation
 237                         outputs:
 238                     </emphasis>
 239                     0
 240                 </para>
 241             </listitem>
 242             <listitem>
 243                 <para>
 244                     <emphasis role="bold">continuous-time state:</emphasis>
 245                     yes
 246                 </para>
 247             </listitem>
 248             <listitem>
 249                 <para>
 250                     <emphasis role="bold">discrete-time state:</emphasis> no
 251                 </para>
 252             </listitem>
 253             <listitem>
 254                 <para>
 255                     <emphasis role="bold">object discrete-time state:</emphasis>
 256                     no
 257                 </para>
 258             </listitem>
 259             <listitem>
 260                 <para>
 261                     <emphasis role="bold">name of computational function:</emphasis>
 262                     <emphasis role="italic">csslti4</emphasis>
 263                 </para>
 264             </listitem>
 265         </itemizedlist>
 266     </refsection>
 267     <refsection id="Example_CLSS">
 268         <title>Example</title>
 269         <para>This sample example illustrates how to use CLSS block to simulate
 270             and display the output waveform <emphasis role="bold">y(t)=Vc(t)</emphasis> of the RLC circuit shown below.
 271         </para>
 272         <para>
 273             <inlinemediaobject>
 274                 <imageobject>
 275                     <imagedata align="center" fileref="../../../examples/clss_d1.png" valign="middle"/>
 276                 </imageobject>
 277             </inlinemediaobject>
 278         </para>
 279         <para>The equations for an RLC circuit are the following. They result from
 280             Kirchhoff's voltage law and Newton's law.
 281         </para>
 282         <para>
 283             <latex><![CDATA[
 284 \begin{matrix}
 285 i = i_L = i_C = C \frac{\mathrm{d} v_C }{\mathrm{d} t} \\
 286 Ri_L + L \frac{\mathrm{d} i_L }{\mathrm{d} t} + v_C = u_0(t)
 287 \end{matrix}
 288       ]]></latex>
 289         </para>
 290         <para>The R, L and C are the system's resistance, inductance and
 291             capacitor.
 292         </para>
 293         <para>
 294             We define the capacitor voltage <literal>Vc</literal> and the
 295             inductance current <literal>iL</literal> as the state variables
 296             <literal>X1</literal> and <literal>X2.</literal>
 297         </para>
 298         <para>
 299             <latex><![CDATA[
 300 \begin{array}{c}
 301 \mbox{$x_1 = i_L$}\\
 302 \mbox{$x_2 = v_C$}\\
 303 \end{array}
 304
 305 then
 306
 307 \begin{array}{c}
 308 \mbox{$\dot{x_1} = \frac{\mathrm{d} i_L }{\mathrm{d} t}$}\\
 309 \mbox{$\dot{x_2} = \frac{\mathrm{d} v_C }{\mathrm{d} t}$}
 310 \end{array}
 311       ]]></latex>
 312         </para>
 313         <para>thus</para>
 314         <para>
 315             <latex>x_1 = i_L = C\frac{\mathrm{d} v_C }{\mathrm{d} t} = C\dot{x_2}</latex>
 316         </para>
 317         <para>Rearranging these equations we get:</para>
 318         <para>
 319             <latex><![CDATA[
 320 \begin{array}{c}
 321 \dot{x_1} = -\frac{R}{L} x_1 - \frac{1}{L} x_2 + \frac{1}{L} u_0(t) \\
 322 \dot{x_2} = \frac{1}{C} x_2
 323 \end{array}
 324       ]]></latex>
 325         </para>
 326         <para>These equations can be put into matrix form as follows,</para>
 327         <para>
 328             <latex><![CDATA[
 329 \begin{bmatrix}\dot{x_1}\\\dot{x_2}\end{bmatrix}
 330 =
 331 \begin{bmatrix}
 332 -\frac{R}{L} & -\frac{1}{L}\\
 333 -\frac{1}{C} & 0
 334 \end{bmatrix}
 335 \begin{bmatrix}x_1\\x_2\end{bmatrix}
 336 +
 337 \begin{bmatrix}\frac{1}{L}\\0\end{bmatrix}
 338 u_0(t)
 339       ]]></latex>
 340         </para>
 341         <para>The required output equation is</para>
 342         <para>
 343             <latex><![CDATA[
 344         y(t) = \begin{bmatrix} 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} + \begin{bmatrix} 0 \end{bmatrix} \cdot u
 345       ]]></latex>
 346         </para>
 347         <para>The following diagram shows these equations modeled in Xcos where
 348             R=10Ω, L=5 mΗ and C=0.1µF; the initial states are x1=0 and x2=0.5.
 349         </para>
 350         <para>To obtain the output Vc(t) we use CLSS block from Continuous time
 351             systems Palette.
 352         </para>
 353         <para>
 354             <link type="scilab" linkend="scilab.zcos/xcos/examples/continuous_pal/CLSS_Example.zcos">
 355                 <inlinemediaobject>
 356                     <imageobject>
 357                         <imagedata align="center" fileref="../../../../examples/continuous_pal/CLSS_Example.zcos" valign="middle"/>
 358                     </imageobject>
 359                 </inlinemediaobject>
 360             </link>
 361             <scilab:image><![CDATA[
 362 importXcosDiagram(SCI + "/modules/xcos/examples/continuous_pal/CLSS_Example.zcos");
 363 xcos_simulate(scs_m, 4);
 364 ]]></scilab:image>
 365         </para>
 366     </refsection>
 367     <refsection id="Interfacingfunction_CLSS">
 368         <title>Interfacing function</title>
 369         <itemizedlist>
 370             <listitem>
 371                 <para>SCI/modules/scicos_blocks/macros/Linear/CLSS.sci</para>
 372             </listitem>
 373         </itemizedlist>
 374     </refsection>
 375     <refsection id="Computationalfunction_CLSS">
 376         <title>Computational function</title>
 377         <itemizedlist>
 378             <listitem>
 379                 <para>SCI/modules/scicos_blocks/src/c/csslti4.c (Type 4)</para>
 380             </listitem>
 381         </itemizedlist>
 382     </refsection>
 383 </refentry>