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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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  <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Simulations</h1>
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<P>The function simulate can use ssm object to produce simulations. 
</P>
<H2>Simulation example. 
</H2>
<P>The following closed loop system is built. 
</P>
<DIV CLASS="fragment"><PRE><FONT SIZE=2>&gt;&gt; sys = ssm(plist(</FONT><FONT COLOR="#a020f0">'built-in'</FONT>, <FONT COLOR="#a020f0">'standard_system_params'</FONT>, <FONT COLOR="#a020f0">'setnames'</FONT>, {<FONT COLOR="#a020f0">'W'</FONT> <FONT COLOR="#a020f0">'C'</FONT>}, <FONT COLOR="#a020f0">'setvalues'</FONT>, [-0.2 -0.5]));
&gt;&gt; sys.modifTimeStep(0.01);
<FONT SIZE=2>&gt;&gt; sys.duplicateInput(</FONT><FONT COLOR="#a020f0">'U'</FONT>,<FONT COLOR="#a020f0">'Negative Bias'</FONT>);
&gt;&gt; controller = ssm(plist( <FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0">'amats'</FONT><FONT COLOR="#000000">,cell(0,0), </FONT><FONT COLOR="#a020f0">'bmats'</FONT><FONT COLOR="#000000">,cell(0,1), </FONT><FONT COLOR="#a020f0">'cmats'</FONT><FONT COLOR="#000000">,cell(1,0), </FONT><FONT COLOR="#a020f0">'dmats'</FONT><FONT COLOR="#000000">,{-1}, </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0">'timestep'</FONT><FONT COLOR="#000000">,0.01, </FONT><FONT COLOR="#a020f0">'name'</FONT><FONT COLOR="#000000">,</FONT><FONT COLOR="#a020f0">'controller'</FONT><FONT COLOR="#000000">, </FONT><FONT COLOR="#a020f0">'params'</FONT><FONT COLOR="#000000">,plist, </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0">'statenames'</FONT><FONT COLOR="#000000">,{}, </FONT><FONT COLOR="#a020f0">'inputnames'</FONT><FONT COLOR="#000000">,{</FONT><FONT COLOR="#a020f0">'Y'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'outputnames'</FONT><FONT COLOR="#000000">,{</FONT><FONT COLOR="#a020f0">'U'</FONT><FONT COLOR="#000000">} ));</FONT>
------ ssm/1 -------
      amats: {  [2x2]  }  [1x1]
      mmats: {  [2x2]  }  [1x1]
      bmats: {  [2x2]   [2x1]  }  [1x2]
      cmats: {  [1x2]   
                [1x2]  }  [2x1]
      dmats: {  [1x2]    []     
                [1x2]    []    }  [2x2]
   timestep: 0.01
     inputs:  [1x2 ssmblock]
         1 : N | Fn [kg m s^(-2)], On [m]
         2 : Negative Bias | Fu [kg m s^(-2)]
     states:  [1x1 ssmblock]
         1 : standard test system | x [m], xdot [m s^(-1)]
    outputs:  [1x2 ssmblock]
         1 : Y | y [m]
         2 : U | U &gt; 1 []
     params: (empty-plist) [1x1 plist]
    version: $Id: ssm_simulation_content.html,v 1.3 2009/08/28 14:20:07 adrien Exp $
    Ninputs: 2
 inputsizes: [2 1]
   Noutputs: 2
outputsizes: [1 1]
    Nstates: 1
 statesizes: 2
    Nparams: 0
isnumerical: true
       hist: ssm.hist [1x1 history]
   procinfo: (empty-plist) [1x1 plist]
   plotinfo: (empty-plist) [1x1 plist]
       name: assembled( standard_system_params + controller))
description: 
    mdlfile: 
       UUID: 163d7103-063b-4a57-af7e-b08d22fe42c1
--------------------</PRE></DIV><P>
Then we wish to use the inputs of N for a correlated force noise and
measurement noise, &ldquo;Negative Bias&rdquo; for a sinewave, and
there will be an observation DC offset.</P>
<P>We want as an output the controller output &ldquo;U&rdquo; and the
sensor output &ldquo;y&rdquo;.</P>
<DIV CLASS="fragment"><PRE>&gt;&gt; ao1 = ao(plist(<FONT COLOR="#a020f0">'FCN'</FONT>,<FONT COLOR="#a020f0">'sin(0:0.01:100)'</FONT>));
<FONT COLOR="#000000">&gt;&gt; ao_out = sysCL.simulate( plist(</FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'NOISE VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'Fn'</FONT><FONT COLOR="#000000"> </FONT><FONT COLOR="#a020f0">'On'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'COVARIANCE'</FONT><FONT COLOR="#000000">, [1 0.1 ; 0.1 2] , </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'AOS VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'Fu'</FONT><FONT COLOR="#000000">} ,</FONT><FONT COLOR="#a020f0">'AOS'</FONT><FONT COLOR="#000000">, ao1 ,</FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'CONSTANT VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'On'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'CONSTANTS'</FONT><FONT COLOR="#000000">, 35, </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'RETURN STATES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'x'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'RETURN OUTPUTS'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'y'</FONT><FONT COLOR="#000000"> </FONT><FONT COLOR="#a020f0">'U &gt; 1'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'SSINI'</FONT><FONT COLOR="#000000"> , {[100;3]}, </FONT><FONT COLOR="#a020f0">'TINI'</FONT><FONT COLOR="#000000">, 0));</FONT>
&gt;&gt; iplot(ao_out([1, 2, 3]));</PRE></DIV><P>
<IMG SRC="images/simulate.png" NAME="images1" ALIGN=LEFT WIDTH=567 HEIGHT=454 BORDER=0><BR CLEAR=LEFT>It
turns out the system output (blue) is not much like the state (red),
causing the control (green) to waste a lot of energy. The state is
not experimentally available, but might be obtained through
filtering. The kalman method is so far the only filtering method
implemented in the toolbox.</P>
<DIV CLASS="fragment"><PRE><FONT COLOR="#000000">&gt;&gt; ao_est = sysCL.kalman( plist(</FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'NOISE VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'Fn'</FONT><FONT COLOR="#000000"> </FONT><FONT COLOR="#a020f0">'On'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'COVARIANCE'</FONT><FONT COLOR="#000000">, [1 0.1 ; 0.1 2] , </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'AOS VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'Fu'</FONT><FONT COLOR="#000000">} ,</FONT><FONT COLOR="#a020f0">'AOS'</FONT><FONT COLOR="#000000">, ao1 ,</FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'CONSTANT VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'On'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'CONSTANTS'</FONT><FONT COLOR="#000000">, 35, </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'OUTPUT VARIABLE NAMES'</FONT><FONT COLOR="#000000">, {</FONT><FONT COLOR="#a020f0">'y'</FONT><FONT COLOR="#000000">}, </FONT><FONT COLOR="#a020f0">'OUTPUTS'</FONT><FONT COLOR="#000000"> , ao_out(2), </FONT><FONT COLOR="#0000ff">...</FONT>
<FONT COLOR="#a020f0"> 'RETURN STATES'</FONT><FONT COLOR="#000000">, 1, </FONT><FONT COLOR="#a020f0">'RETURN OUTPUTS'</FONT><FONT COLOR="#000000">, 1 ));</FONT>
&gt;&gt; iplot(ao_out(2), ao_est(1), ao_out(1))</PRE></DIV><P>
<IMG SRC="images/kalman.png" NAME="images2" ALIGN=LEFT WIDTH=567 HEIGHT=454 BORDER=0><BR CLEAR=LEFT>In
this example the estimate (blue) of the state (green) is
satisfactory. It leads us to think that such a filter should be used
to provide with the input of the controller. 
</P>
<P>However, the DC offset correction by the kalman filter is one
information that is not available under usual circumstances. 
</P>
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