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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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Daniele Nicolodi <nicolodi@science.unitn.it>
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1 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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2 <HTML>
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3 <HEAD>
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4 <META HTTP-EQUIV="CONTENT-TYPE" CONTENT="text/html; charset=windows-1252">
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5 <TITLE></TITLE>
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6 <META NAME="GENERATOR" CONTENT="OpenOffice.org 3.1 (Win32)">
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7 <META NAME="CREATED" CONTENT="0;0">
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8 <META NAME="CHANGEDBY" CONTENT="Adrien G">
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9 <META NAME="CHANGED" CONTENT="20090827;17351600">
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10 </HEAD>
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11 <BODY LANG="en-US" DIR="LTR">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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12 <H2>What is a ssm object?</H2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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13 <P>The ssm class is a class to use &ldquo;state space models&rdquo;
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Daniele Nicolodi <nicolodi@science.unitn.it>
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14 for simulation, identification and modeling in the toolbox. A state
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Daniele Nicolodi <nicolodi@science.unitn.it>
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15 space model is a mathematical object constituted of</P>
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16 <UL>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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17 <LI><P>A time model represented by the field &ldquo;timestep&rdquo;,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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18 either discrete or continuous (&ldquo;timestep&rdquo; is then 0)</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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19 <LI><P>A linear differential equation, which turns into a difference
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Daniele Nicolodi <nicolodi@science.unitn.it>
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20 equation it the model is time discrete. This equation is written
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Daniele Nicolodi <nicolodi@science.unitn.it>
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21 using a state space x in the form x' = Ax + Bu where A is the state
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Daniele Nicolodi <nicolodi@science.unitn.it>
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22 transition matrix and B the state input matrix, u being the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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23 exogenous input signal. In this setup, x and u are time series.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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24 <LI><P>An observation equation which allows for linear observation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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25 of the input and the state : y = Cx+Du. The variable y is the output
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Daniele Nicolodi <nicolodi@science.unitn.it>
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26 of the system.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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27 </UL>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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28 <P>Input and output variables (lines in u and y) may be named and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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29 assigned a unit. However for the state x there is more of a choice
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Daniele Nicolodi <nicolodi@science.unitn.it>
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30 since if free of choice via a basis change. The most usual choice is
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Daniele Nicolodi <nicolodi@science.unitn.it>
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31 that x contents the same units as y and it derivatives, or the same
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Daniele Nicolodi <nicolodi@science.unitn.it>
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32 units as u and its integrals. In general the size of the state space
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Daniele Nicolodi <nicolodi@science.unitn.it>
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33 is equal to the number of poles in the system.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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34 <P>In the LTPDA toolbox, lines in u,y,x are grouped in blocks (input
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Daniele Nicolodi <nicolodi@science.unitn.it>
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35 blocks, state blocks, output blocks) to ease the understanding of a
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Daniele Nicolodi <nicolodi@science.unitn.it>
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36 large system. This means the matrices A, B, C, and D are also split
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Daniele Nicolodi <nicolodi@science.unitn.it>
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37 into blocks of the corresponding size. The diagonal blocks of A may
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Daniele Nicolodi <nicolodi@science.unitn.it>
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38 represent the dynamics of a LTP subsystem (the controller, the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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39 propulsion &hellip;) or the coupling between two systems (from the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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40 propulsion to the equations of motion)... In addition the dynamic
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Daniele Nicolodi <nicolodi@science.unitn.it>
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41 equation also contains an inertia matrix M so that the parametric
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Daniele Nicolodi <nicolodi@science.unitn.it>
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42 equation is made simple. The equation becomes M x' = Ax + Bu. If not
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Daniele Nicolodi <nicolodi@science.unitn.it>
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43 user set M is supposed to be the identity.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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44 <P>Below we build a standard system to show the contents of the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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45 object.</P>
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46 <DIV CLASS="fragment"><PRE>&gt;&gt; <FONT COLOR="#000000"><FONT FACE="Courier New, monospace"><FONT SIZE=2>system = ssm(plist(</FONT></FONT></FONT><FONT COLOR="#a020f0"><FONT FACE="Courier New, monospace"><FONT SIZE=2>'built-in'</FONT></FONT></FONT><FONT COLOR="#000000"><FONT FACE="Courier New, monospace"><FONT SIZE=2>, </FONT></FONT></FONT><FONT COLOR="#a020f0"><FONT FACE="Courier New, monospace"><FONT SIZE=2>'standard_system_params'</FONT></FONT></FONT><FONT COLOR="#000000"><FONT FACE="Courier New, monospace"><FONT SIZE=2>, </FONT></FONT></FONT><FONT COLOR="#a020f0"><FONT FACE="Courier New, monospace"><FONT SIZE=2>'withparams'</FONT></FONT></FONT><FONT COLOR="#000000"><FONT FACE="Courier New, monospace"><FONT SIZE=2>, </FONT></FONT></FONT><FONT COLOR="#a020f0"><FONT FACE="Courier New, monospace"><FONT SIZE=2>'ALL'</FONT></FONT></FONT><FONT COLOR="#000000"><FONT FACE="Courier New, monospace"><FONT SIZE=2>))</FONT></FONT></FONT>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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47 M: running ssm/ssm
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Daniele Nicolodi <nicolodi@science.unitn.it>
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48 M: running ssmFromBuiltinSystem
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Daniele Nicolodi <nicolodi@science.unitn.it>
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49 M: looking for models in C:\Documents and Settings\Adrien.IFR-NB01\My Documents\MATLAB2008\LTPDA_SSM_MODELS\ltp_ssm_models
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Daniele Nicolodi <nicolodi@science.unitn.it>
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50 M: looking for models in C:\Users\Adrien.IFR-NB01\My Documents\MATLAB2008\ltpda_toolbox\ltpda\classes\@ssm\..\..\m\built_in_models
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 M: running buildParamPlist
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52 M: running ssm/ssm
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Daniele Nicolodi <nicolodi@science.unitn.it>
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53 M: running ssm/ssm
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 M: running fromStruct
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Daniele Nicolodi <nicolodi@science.unitn.it>
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55 M: running ssm/ssm
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 M: running validate
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Daniele Nicolodi <nicolodi@science.unitn.it>
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57 M: running validate
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Daniele Nicolodi <nicolodi@science.unitn.it>
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58 M: running display
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Daniele Nicolodi <nicolodi@science.unitn.it>
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59 ------ ssm/1 -------
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Daniele Nicolodi <nicolodi@science.unitn.it>
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60 amats: { [2x2] } [1x1]
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61 mmats: { [2x2] } [1x1]
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62 bmats: { [2x1] [2x2] } [1x2]
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63 cmats: { [1x2] } [1x1]
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64 dmats: { [1x1] [1x2] } [1x2]
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65 timestep: 0
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66 inputs: [1x2 ssmblock]
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67 1 : U | Fu [kg m s^(-2)]
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68 2 : N | Fn [kg m s^(-2)], On [m]
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69 states: [1x1 ssmblock]
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70 1 : standard test system | x [m], xdot [m s^(-1)]
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71 outputs: [1x1 ssmblock]
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72 1 : Y | y [m]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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73 params: (W=0.2, C=0.5, C1=0, C2=0, B=1, D=0) [1x1 plist]
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74 version: $Id: ssm_introduction_content.html,v 1.3 2009/08/28 14:20:07 adrien Exp $--&gt;$Id: ssm_introduction_content.html,v 1.3 2009/08/28 14:20:07 adrien Exp $
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75 Ninputs: 2
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76 inputsizes: [1 2]
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77 Noutputs: 1
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78 outputsizes: 1
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 Nstates: 1
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80 statesizes: 2
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81 Nparams: 6
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82 isnumerical: false
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83 hist: ssm.hist [1x1 history]
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84 procinfo: (empty-plist) [1x1 plist]
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85 plotinfo: (empty-plist) [1x1 plist]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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86 name: standard_system_params
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Daniele Nicolodi <nicolodi@science.unitn.it>
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87 description: standard spring-mass-dashpot test system
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 mdlfile:
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89 UUID: 8eafd65e-9052-4800-a694-9482e2bd7b70
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90 --------------------</PRE></DIV><P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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91 The fields &ldquo;amats&rdquo;, &ldquo;mmats&rdquo;, &ldquo;bmats&rdquo;,
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92 &ldquo;cmats&rdquo;, &ldquo;dmats&rdquo; contain the matrices of the
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93 differential and observation equation of a mass spring system with
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94 dampening.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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95 <P>The systems inputs/states/outputs are listed by blocks. There are
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Daniele Nicolodi <nicolodi@science.unitn.it>
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96 two input blocks, which correspond to the commanded signal (U) and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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97 the noise (N). There is one state block and one output block. Each
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Daniele Nicolodi <nicolodi@science.unitn.it>
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98 input has its name displayed followed with individual input variable
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99 names and units. Note that the empty ssm object does not have an
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Daniele Nicolodi <nicolodi@science.unitn.it>
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100 input block, a state block and an output block of size 0, it has no
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Daniele Nicolodi <nicolodi@science.unitn.it>
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101 such blocks, and the size fields are empty.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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102 <P>The &ldquo;params&rdquo; field is the 1x1 default plist with no
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Daniele Nicolodi <nicolodi@science.unitn.it>
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103 parameter. This is the case when the system is fully numerical,
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Daniele Nicolodi <nicolodi@science.unitn.it>
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104 parameters are deleted from the field &ldquo;params&rdquo; as they
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Daniele Nicolodi <nicolodi@science.unitn.it>
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105 get substituted .</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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106 <P>Inputs, states and output sizes are summed up in the fields N* and
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Daniele Nicolodi <nicolodi@science.unitn.it>
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107 *sizes.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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108 </BODY>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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109 </HTML>