Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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<strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong><br />
Def<strong>in</strong>ition<br />
(More general:) A l<strong>in</strong>ear matrix <strong>in</strong>equality is an <strong>in</strong>equality<br />
F (X) ≺ 0<br />
where F is an aff<strong>in</strong>e function mapp<strong>in</strong>g a f<strong>in</strong>ite dimensional vector space<br />
X to the set H of Hermitian matrices.<br />
Allows def<strong>in</strong><strong>in</strong>g matrix valued LMI’s.<br />
F aff<strong>in</strong>e means F (X) = F0 + T (X) with T a l<strong>in</strong>ear map (a matrix).<br />
With Xj basis of X , any X ∈ X can be expanded as<br />
X = n<br />
j=1 xjXj so that<br />
F (X) = F0 + T (X) = F0 +<br />
which is the standard form.<br />
n<br />
j=1<br />
xjFj<br />
with Fj = T (Xj)<br />
Siep Weiland and Carsten Scherer (DISC) <strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> <strong>in</strong> <strong>Control</strong> Class 1 44 / 59
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