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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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11 <title>Introduction to Statespace Models with LTPDA (LTPDA Toolbox)</title>
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15 "Presents an overview of the features, system requirements, and starting the toolbox.">
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20
21 <p style="font-size:1px;">&nbsp;</p>
22
23 <table class="nav" summary="Navigation aid" border="0" width=
24 "100%" cellpadding="0" cellspacing="0">
25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "ssm.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Statespace models"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "ssm_building.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Building Statespace models"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Introduction to Statespace Models with LTPDA</h1>
37 <hr>
38
39 <p>
40 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
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50 <BODY LANG="en-US" DIR="LTR">
51 <H2>What is a ssm object?</H2>
52 <P>The ssm class is a class to use &ldquo;state space models&rdquo;
53 for simulation, identification and modeling in the toolbox. A state
54 space model is a mathematical object constituted of</P>
55 <UL>
56 <LI><P>A time model represented by the field &ldquo;timestep&rdquo;,
57 either discrete or continuous (&ldquo;timestep&rdquo; is then 0)</P>
58 <LI><P>A linear differential equation, which turns into a difference
59 equation it the model is time discrete. This equation is written
60 using a state space x in the form x' = Ax + Bu where A is the state
61 transition matrix and B the state input matrix, u being the
62 exogenous input signal. In this setup, x and u are time series.</P>
63 <LI><P>An observation equation which allows for linear observation
64 of the input and the state : y = Cx+Du. The variable y is the output
65 of the system.</P>
66 </UL>
67 <P>Input and output variables (lines in u and y) may be named and
68 assigned a unit. However for the state x there is more of a choice
69 since if free of choice via a basis change. The most usual choice is
70 that x contents the same units as y and it derivatives, or the same
71 units as u and its integrals. In general the size of the state space
72 is equal to the number of poles in the system.</P>
73 <P>In the LTPDA toolbox, lines in u,y,x are grouped in blocks (input
74 blocks, state blocks, output blocks) to ease the understanding of a
75 large system. This means the matrices A, B, C, and D are also split
76 into blocks of the corresponding size. The diagonal blocks of A may
77 represent the dynamics of a LTP subsystem (the controller, the
78 propulsion &hellip;) or the coupling between two systems (from the
79 propulsion to the equations of motion)... In addition the dynamic
80 equation also contains an inertia matrix M so that the parametric
81 equation is made simple. The equation becomes M x' = Ax + Bu. If not
82 user set M is supposed to be the identity.</P>
83 <P>Below we build a standard system to show the contents of the
84 object.</P>
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>
86 M: running ssm/ssm
87 M: running ssmFromBuiltinSystem
88 M: looking for models in C:\Documents and Settings\Adrien.IFR-NB01\My Documents\MATLAB2008\LTPDA_SSM_MODELS\ltp_ssm_models
89 M: looking for models in C:\Users\Adrien.IFR-NB01\My Documents\MATLAB2008\ltpda_toolbox\ltpda\classes\@ssm\..\..\m\built_in_models
90 M: running buildParamPlist
91 M: running ssm/ssm
92 M: running ssm/ssm
93 M: running fromStruct
94 M: running ssm/ssm
95 M: running validate
96 M: running validate
97 M: running display
98 ------ ssm/1 -------
99 amats: { [2x2] } [1x1]
100 mmats: { [2x2] } [1x1]
101 bmats: { [2x1] [2x2] } [1x2]
102 cmats: { [1x2] } [1x1]
103 dmats: { [1x1] [1x2] } [1x2]
104 timestep: 0
105 inputs: [1x2 ssmblock]
106 1 : U | Fu [kg m s^(-2)]
107 2 : N | Fn [kg m s^(-2)], On [m]
108 states: [1x1 ssmblock]
109 1 : standard test system | x [m], xdot [m s^(-1)]
110 outputs: [1x1 ssmblock]
111 1 : Y | y [m]
112 params: (W=0.2, C=0.5, C1=0, C2=0, B=1, D=0) [1x1 plist]
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 $
114 Ninputs: 2
115 inputsizes: [1 2]
116 Noutputs: 1
117 outputsizes: 1
118 Nstates: 1
119 statesizes: 2
120 Nparams: 6
121 isnumerical: false
122 hist: ssm.hist [1x1 history]
123 procinfo: (empty-plist) [1x1 plist]
124 plotinfo: (empty-plist) [1x1 plist]
125 name: standard_system_params
126 description: standard spring-mass-dashpot test system
127 mdlfile:
128 UUID: 8eafd65e-9052-4800-a694-9482e2bd7b70
129 --------------------</PRE></DIV><P>
130 The fields &ldquo;amats&rdquo;, &ldquo;mmats&rdquo;, &ldquo;bmats&rdquo;,
131 &ldquo;cmats&rdquo;, &ldquo;dmats&rdquo; contain the matrices of the
132 differential and observation equation of a mass spring system with
133 dampening.</P>
134 <P>The systems inputs/states/outputs are listed by blocks. There are
135 two input blocks, which correspond to the commanded signal (U) and
136 the noise (N). There is one state block and one output block. Each
137 input has its name displayed followed with individual input variable
138 names and units. Note that the empty ssm object does not have an
139 input block, a state block and an output block of size 0, it has no
140 such blocks, and the size fields are empty.</P>
141 <P>The &ldquo;params&rdquo; field is the 1x1 default plist with no
142 parameter. This is the case when the system is fully numerical,
143 parameters are deleted from the field &ldquo;params&rdquo; as they
144 get substituted .</P>
145 <P>Inputs, states and output sizes are summed up in the fields N* and
146 *sizes.</P>
147 </BODY>
148 </HTML>
149 </p>
150
151 <br>
152 <br>
153 <table class="nav" summary="Navigation aid" border="0" width=
154 "100%" cellpadding="0" cellspacing="0">
155 <tr valign="top">
156 <td align="left" width="20"><a href="ssm.html"><img src=
157 "b_prev.gif" border="0" align="bottom" alt=
158 "Statespace models"></a>&nbsp;</td>
159
160 <td align="left">Statespace models</td>
161
162 <td>&nbsp;</td>
163
164 <td align="right">Building Statespace models</td>
165
166 <td align="right" width="20"><a href=
167 "ssm_building.html"><img src="b_next.gif" border="0" align=
168 "bottom" alt="Building Statespace models"></a></td>
169 </tr>
170 </table><br>
171
172 <p class="copy">&copy;LTP Team</p>
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