Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Aff<strong>in</strong>e functions<br />
Def<strong>in</strong>ition<br />
A function f : S → T is aff<strong>in</strong>e if<br />
f(αx1 + (1 − α)x2) = αf(x1) + (1 − α)f(x2)<br />
for all x1, x2 ∈ S and for all α ∈ R<br />
Theorem<br />
If S and T are f<strong>in</strong>ite dimensional, then f : S → T is aff<strong>in</strong>e if and only if<br />
f(x) = f0 + T (x)<br />
where f0 ∈ T and T : T → T a l<strong>in</strong>ear map (a matrix).<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 23 / 59