Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Aff<strong>in</strong>ely constra<strong>in</strong>ed LMI is LMI<br />
Comb<strong>in</strong>e LMI constra<strong>in</strong>t F (x) ≺ 0 and x ∈ S with S aff<strong>in</strong>e:<br />
Write S = x0 + M with M a l<strong>in</strong>ear subspace of dimension k<br />
Let {ej} k j=1<br />
be a basis for M<br />
Write F (x) = F0 + T (x) with T l<strong>in</strong>ear<br />
Then for any x ∈ S:<br />
⎛<br />
⎞<br />
k<br />
k<br />
F (x) = F0 + T ⎝x0 + xjej ⎠ = F0 + T (x0) + xjT (ej)<br />
<br />
j=1<br />
constant<br />
j=1<br />
<br />
l<strong>in</strong>ear<br />
= G0 + x1G1 + . . . + xkGk = G(x)<br />
Result: x unconstra<strong>in</strong>ed and x has lower dimension than x !!<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 46 / 59
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