Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Convex sets<br />
Def<strong>in</strong>ition<br />
A set S <strong>in</strong> a l<strong>in</strong>ear vector space X is convex if<br />
x1, x2 ∈ S =⇒ αx1 + (1 − α)x2 ∈ S for all α ∈ (0, 1)<br />
The po<strong>in</strong>t αx1 + (1 − α)x2 with α ∈ (0, 1) is a convex comb<strong>in</strong>ation of x1<br />
and x2.<br />
Def<strong>in</strong>ition<br />
The po<strong>in</strong>t x ∈ X is a convex comb<strong>in</strong>ation of x1, . . . , xn ∈ X if<br />
x :=<br />
n<br />
αixi, αi ≥ 0,<br />
i=1<br />
n<br />
αi = 1<br />
Note: set of all convex comb<strong>in</strong>ations of x1, . . . , xn is convex.<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 15 / 59<br />
i=1