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<title>YALMIP Example : Efficient solution of KYP problems</title>
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<h2>KYP problems</h2>
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<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
example requires <a href="solvers.htm#kypd">KYPD</a> and
an SDP-solver capable of calculating dual variables.</p>
<p>Many problems in control and system theory can be formulated using the celebrated
Kalman-Yakubovic-Popov lemma (KYP). By using this lemma, a large number of
problems can be formulated using LMIs. Unfortunately, many
practical problems leads to LMIs far too big to be efficiently solved
using standard semidefinite solvers.</p>
<p>YALMIP can be used with the dedicated solver <a href="solvers.htm#kypd">
KYPD</a> to efficiently solve some problems with large-scale KYP constraints.
In our setting, a <a href="reference.htm#kyp">KYP</a> is a matrix of the
form <b><font face="Tahoma">[A<sup>T</sup>P+PA PB;B<sup>T</sup>P 0] + M(x)</font></b>,
with <b>P</b> and the <b>x</b> being the free variables, and <b>M</b> a
linear operator.</p>
<p>The following code calculates the L<sub>2</sub>-gain of a random stable
system with 40 states, using the dedicated <a href="solvers.htm#kypd">KYPD</a>-solver,
and a standard SDP-solver.</p>
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<pre>n = 40;
A = randn(n);A = A - max(real(eig(A)))*eye(n)*1.5; % Stable dynamics
B = randn(n,1);
C = randn(1,n);
t = sdpvar(1,1);
P = sdpvar(n,n);
F = set(kyp(A,B,P,blkdiag(C'*C,-t)) < 0)
sol1 = solvesdp(F,t,sdpsettings('solver','kypd'));
sol2 = solvesdp(F,t);
sol1.solvertime/sol2.solvertime % Compare solution time</pre>
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<p><img border="0" src="demoicon.gif" width="16" height="16"> The
variable <b>P </b>may only enter in 1 constraint if you intend to use
<a href="solvers.htm#kypd">KYPD</a>, i.e. you cannot use
<a href="solvers.htm#kypd">KYPD</a> if you want to impose explicit
constraints (including <br>
positive definiteness) of <b>P</b>. However, positive definiteness of <b>P </b>
is in some cases implied by the KYP constraint. You can have multiple KYP constraints with different <b>P </b>variables.</p>
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