Linear Matrix Inequalities in Control

Convex sets
Definition
A set S in a linear vector space X is convex if
x1, x2 ∈ S =⇒ αx1 + (1 − α)x2 ∈ S for all α ∈ (0, 1)
The point αx1 + (1 − α)x2 with α ∈ (0, 1) is a convex combination of x1
and x2.
Definition
The point x ∈ X is a convex combination of x1, . . . , xn ∈ X if
x :=
n
αixi, αi ≥ 0,
i=1
n
αi = 1
Note: set of all convex combinations of x1, . . . , xn is convex.
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 15 / 59
i=1