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5<meta http-equiv="Content-Language" content="en-us">
6<title>YALMIP Example : Efficient solution of KYP problems</title>
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16<div align="left">
17  <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>KYP problems</h2>
21      <hr noShade SIZE="1">
22    <p>
23    <img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
24    example requires <a href="solvers.htm#kypd">KYPD</a> and
25    an SDP-solver capable of calculating dual variables.</p>
26      <p>Many problems in control and system theory can be formulated using the celebrated
27      Kalman-Yakubovic-Popov lemma (KYP). By using this lemma, a large number of
28      problems can be formulated using LMIs. Unfortunately, many
29      practical problems leads to LMIs far too big to be efficiently solved
30      using standard semidefinite solvers.</p>
31      <p>YALMIP can be used with the dedicated solver <a href="solvers.htm#kypd">
32      KYPD</a> to efficiently solve some problems with large-scale KYP constraints.
33      In our setting, a <a href="reference.htm#kyp">KYP</a> is a matrix of the
34      form <b><font face="Tahoma">[A<sup>T</sup>P+PA PB;B<sup>T</sup>P 0] + M(x)</font></b>,
35      with <b>P</b> and the <b>x</b> being the free variables, and <b>M</b> a
36      linear operator.</p>
37      <p>The following code calculates the L<sub>2</sub>-gain of a random stable
38      system with 40 states, using the dedicated <a href="solvers.htm#kypd">KYPD</a>-solver,
39      and a standard SDP-solver.</p>
40      <table cellPadding="10" width="100%">
41        <tr>
42          <td class="xmpcode">
43          <pre>n = 40;
44A = randn(n);A = A - max(real(eig(A)))*eye(n)*1.5; % Stable dynamics
45B = randn(n,1);
46C = randn(1,n);
47
48t = sdpvar(1,1);
49P = sdpvar(n,n);
50
51F = set(kyp(A,B,P,blkdiag(C'*C,-t)) &lt; 0)
52
53sol1 = solvesdp(F,t,sdpsettings('solver','kypd'));
54sol2 = solvesdp(F,t);
55
56sol1.solvertime/sol2.solvertime % Compare solution time</pre>
57          </td>
58        </tr>
59      </table>
60      <p><img border="0" src="demoicon.gif" width="16" height="16">&nbsp; The
61      variable <b>P </b>may only enter in 1 constraint if you intend to use
62      <a href="solvers.htm#kypd">KYPD</a>, i.e. you cannot use
63      <a href="solvers.htm#kypd">KYPD</a> if you want to impose explicit
64      constraints (including&nbsp; <br>
65      positive definiteness) of <b>P</b>. However, positive definiteness of <b>P </b>
66      is in some cases implied by the KYP constraint.&nbsp; You can have multiple KYP constraints with different <b>P </b>variables.</p>
67      </td>
68    </tr>
69  </table>
70</div>
71
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