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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
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2 "http://www.w3.org/TR/1999/REC-html401-19991224/loose.dtd">
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3
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4 <html lang="en">
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5 <head>
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6 <meta name="generator" content=
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7 "HTML Tidy for Mac OS X (vers 1st December 2004), see www.w3.org">
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8 <meta http-equiv="Content-Type" content=
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9 "text/html; charset=us-ascii">
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10
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11 <title>Introduction to Statespace Models with LTPDA (LTPDA Toolbox)</title>
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12 <link rel="stylesheet" href="docstyle.css" type="text/css">
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13 <meta name="generator" content="DocBook XSL Stylesheets V1.52.2">
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14 <meta name="description" content=
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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16 </head>
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17
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18 <body>
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19 <a name="top_of_page" id="top_of_page"></a>
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20
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21 <p style="font-size:1px;">&nbsp;</p>
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22
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23 <table class="nav" summary="Navigation aid" border="0" width=
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24 "100%" cellpadding="0" cellspacing="0">
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25 <tr>
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26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
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27
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28 <td valign="baseline" align="right"><a href=
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29 "ssm.html"><img src="b_prev.gif" border="0" align=
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30 "bottom" alt="Statespace models"></a>&nbsp;&nbsp;&nbsp;<a href=
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31 "ssm_building.html"><img src="b_next.gif" border="0" align=
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32 "bottom" alt="Building Statespace models"></a></td>
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33 </tr>
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34 </table>
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35
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36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Introduction to Statespace Models with LTPDA</h1>
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37 <hr>
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38
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39 <p>
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40 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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41 <HTML>
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42 <HEAD>
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43 <META HTTP-EQUIV="CONTENT-TYPE" CONTENT="text/html; charset=windows-1252">
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44 <TITLE></TITLE>
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45 <META NAME="GENERATOR" CONTENT="OpenOffice.org 3.1 (Win32)">
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46 <META NAME="CREATED" CONTENT="0;0">
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47 <META NAME="CHANGEDBY" CONTENT="Adrien G">
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48 <META NAME="CHANGED" CONTENT="20090827;17351600">
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49 </HEAD>
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50 <BODY LANG="en-US" DIR="LTR">
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Daniele Nicolodi <nicolodi@science.unitn.it>
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51 <H2>What is a ssm object?</H2>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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52 <P>The ssm class is a class to use &ldquo;state space models&rdquo;
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53 for simulation, identification and modeling in the toolbox. A state
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Daniele Nicolodi <nicolodi@science.unitn.it>
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54 space model is a mathematical object constituted of</P>
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55 <UL>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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56 <LI><P>A time model represented by the field &ldquo;timestep&rdquo;,
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57 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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58 <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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59 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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60 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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61 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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62 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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63 <LI><P>An observation equation which allows for linear observation
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Daniele Nicolodi <nicolodi@science.unitn.it>
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64 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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65 of the system.</P>
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66 </UL>
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67 <P>Input and output variables (lines in u and y) may be named and
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68 assigned a unit. However for the state x there is more of a choice
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69 since if free of choice via a basis change. The most usual choice is
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70 that x contents the same units as y and it derivatives, or the same
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71 units as u and its integrals. In general the size of the state space
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72 is equal to the number of poles in the system.</P>
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73 <P>In the LTPDA toolbox, lines in u,y,x are grouped in blocks (input
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74 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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75 large system. This means the matrices A, B, C, and D are also split
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76 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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77 represent the dynamics of a LTP subsystem (the controller, the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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78 propulsion &hellip;) or the coupling between two systems (from the
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Daniele Nicolodi <nicolodi@science.unitn.it>
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79 propulsion to the equations of motion)... In addition the dynamic
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80 equation also contains an inertia matrix M so that the parametric
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81 equation is made simple. The equation becomes M x' = Ax + Bu. If not
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82 user set M is supposed to be the identity.</P>
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83 <P>Below we build a standard system to show the contents of the
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84 object.</P>
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85 <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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86 M: running ssm/ssm
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87 M: running ssmFromBuiltinSystem
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Daniele Nicolodi <nicolodi@science.unitn.it>
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88 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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89 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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90 M: running buildParamPlist
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91 M: running ssm/ssm
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92 M: running ssm/ssm
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93 M: running fromStruct
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94 M: running ssm/ssm
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95 M: running validate
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96 M: running validate
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97 M: running display
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98 ------ ssm/1 -------
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99 amats: { [2x2] } [1x1]
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100 mmats: { [2x2] } [1x1]
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101 bmats: { [2x1] [2x2] } [1x2]
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102 cmats: { [1x2] } [1x1]
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103 dmats: { [1x1] [1x2] } [1x2]
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104 timestep: 0
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105 inputs: [1x2 ssmblock]
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106 1 : U | Fu [kg m s^(-2)]
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107 2 : N | Fn [kg m s^(-2)], On [m]
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108 states: [1x1 ssmblock]
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109 1 : standard test system | x [m], xdot [m s^(-1)]
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110 outputs: [1x1 ssmblock]
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111 1 : Y | y [m]
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112 params: (W=0.2, C=0.5, C1=0, C2=0, B=1, D=0) [1x1 plist]
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113 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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114 Ninputs: 2
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115 inputsizes: [1 2]
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116 Noutputs: 1
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117 outputsizes: 1
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Daniele Nicolodi <nicolodi@science.unitn.it>
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118 Nstates: 1
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119 statesizes: 2
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120 Nparams: 6
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121 isnumerical: false
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122 hist: ssm.hist [1x1 history]
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123 procinfo: (empty-plist) [1x1 plist]
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124 plotinfo: (empty-plist) [1x1 plist]
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Daniele Nicolodi <nicolodi@science.unitn.it>
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125 name: standard_system_params
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Daniele Nicolodi <nicolodi@science.unitn.it>
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126 description: standard spring-mass-dashpot test system
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Daniele Nicolodi <nicolodi@science.unitn.it>
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127 mdlfile:
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128 UUID: 8eafd65e-9052-4800-a694-9482e2bd7b70
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129 --------------------</PRE></DIV><P>
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130 The fields &ldquo;amats&rdquo;, &ldquo;mmats&rdquo;, &ldquo;bmats&rdquo;,
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131 &ldquo;cmats&rdquo;, &ldquo;dmats&rdquo; contain the matrices of the
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132 differential and observation equation of a mass spring system with
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133 dampening.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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134 <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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135 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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136 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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137 input has its name displayed followed with individual input variable
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Daniele Nicolodi <nicolodi@science.unitn.it>
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138 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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139 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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140 such blocks, and the size fields are empty.</P>
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Daniele Nicolodi <nicolodi@science.unitn.it>
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141 <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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142 parameter. This is the case when the system is fully numerical,
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143 parameters are deleted from the field &ldquo;params&rdquo; as they
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144 get substituted .</P>
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145 <P>Inputs, states and output sizes are summed up in the fields N* and
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146 *sizes.</P>
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147 </BODY>
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148 </HTML>
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149 </p>
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150
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151 <br>
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152 <br>
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153 <table class="nav" summary="Navigation aid" border="0" width=
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154 "100%" cellpadding="0" cellspacing="0">
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155 <tr valign="top">
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156 <td align="left" width="20"><a href="ssm.html"><img src=
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157 "b_prev.gif" border="0" align="bottom" alt=
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158 "Statespace models"></a>&nbsp;</td>
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159
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160 <td align="left">Statespace models</td>
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161
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162 <td>&nbsp;</td>
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163
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164 <td align="right">Building Statespace models</td>
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165
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166 <td align="right" width="20"><a href=
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167 "ssm_building.html"><img src="b_next.gif" border="0" align=
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168 "bottom" alt="Building Statespace models"></a></td>
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169 </tr>
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170 </table><br>
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171
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172 <p class="copy">&copy;LTP Team</p>
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173 </body>
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174 </html>