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1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
2<html>
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4<head>
5<meta http-equiv="Content-Language" content="en-us" />
6<title>YALMIP Example : Lyapunov stability</title>
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16<div align="left">
17  &nbsp;<table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
18    <tr>
19      <td width="100%" align="left" height="100%" valign="top">
20      <h2>Lyapunov stability</h2>
21      <hr noshade size="1" color="#000000" />
22      <p>Given a linear dynamic system <strong>x&#39; = Ax</strong>, our goal is to
23      prove stability by finding a symmetric matrix <b>P</b> satisfying </p>
24      <blockquote dir="ltr" style="MARGIN-RIGHT: 0px">
25        <p><span style="font-style: normal"><strong>
26        <img border="0" src="lyapun6.gif" /></strong></span></p>
27      </blockquote>
28      <p>Define a matrix <b><font face="Tahoma">A</font></b> and the symmetric matrix
29      <b><font face="Tahoma">P</font></b>.</p>
30      <table cellpadding="10" width="100%">
31        <tr>
32          <td class="xmpcode">
33          <pre>A = [-1 2;-3 -4];
34P = sdpvar(2,2);</pre>
35          </td>
36        </tr>
37      </table>
38      <p>Having <b>P</b>, we are ready to define the constraints.</p>
39      <table cellpadding="10" width="100%">
40        <tr>
41          <td class="xmpcode">
42          <pre>F = set(P &gt; 0) + set(A&#39;*P+P*A &lt; 0);</pre>
43          </td>
44        </tr>
45      </table>
46      <p>To avoid the zero solution or an unbounded solution, we constrain the trace
47      of the matrix (Of course, this is not the only way. We could have used, e.g.,
48      the constraint <b><font face="Tahoma">P&gt;I</font></b> instead)</p>
49      <table cellpadding="10" width="100%">
50        <tr>
51          <td class="xmpcode">
52          <pre>F = F + set(trace(P) == 1);</pre>
53          </td>
54        </tr>
55      </table>
56      <p>At this point, we are ready to solve our problem. But first, we display
57      the collection of constraints to see what we have defined.</p>
58      <table cellpadding="10" width="100%">
59        <tr>
60          <td class="xmpcode">
61          <pre>F
62<font color="#000000">+++++++++++++++++++++++++++++++++++++++++++++++++++
63|   ID|      Constraint|                      Type|
64+++++++++++++++++++++++++++++++++++++++++++++++++++
65|   #1|   Numeric value|     Matrix inequality 2x2|
66|   #2|   Numeric value|     Matrix inequality 2x2|
67|   #3|   Numeric value|   Equality constraint 1x1|
68+++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
69          </td>
70        </tr>
71      </table>
72      <p>We only need a feasible solution, so one argument is sufficient when we
73      call <a href="reference.htm#solvesdp">solvesdp</a> to solve the problem.</p>
74      <table cellpadding="10" width="100%">
75        <tr>
76          <td class="xmpcode">
77          <pre>solvesdp(F);
78P_feasible = double(P);</pre>
79          </td>
80        </tr>
81      </table>
82      <p>The resulting constraint satisfaction can be conveniently investigated
83      with <a href="reference.htm#checkset">checkset</a>.</p>
84      <table cellpadding="10" width="100%">
85        <tr>
86          <td class="xmpcode">
87          <pre>checkset(F)
88<font color="#000000">+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
89|   ID|      Constraint|                  Type|   Primal residual|   Dual residual|   Compl. slack|
90+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
91|   #1|   Numeric value|                   LMI|           0.32029|     2.7929e-014|     4.983e-014|
92|   #2|   Numeric value|                   LMI|            1.6706|     2.5781e-015|     1.936e-014|
93|   #3|   Numeric value|   Equality constraint|      -2.0151e-014|    -1.0991e-014|    2.2148e-028|
94+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
95          </td>
96        </tr>
97      </table>
98      <p>Minimizing, e.g., the top-left element of <b>
99      <font face="Tahoma,Arial,sans-serif">P</font></b> is done by specifying an
100      objective function.</p>
101      <table cellpadding="10" width="100%">
102        <tr>
103          <td class="xmpcode">
104          <pre>F = set(P &gt; 0) + set(A&#39;*P+P*A &lt; 0);
105solvesdp(F,P(1,1));</pre>
106          </td>
107        </tr>
108      </table>
109      </td>
110    </tr>
111  </table>
112</div>
113
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