Model Predictive Control

22 3 Polyhedra, Polytopes and Simplices
a ′ 2
x ≤ b2
a ′ 3x ≤ b3
a ′ 5x ≤ b5
a ′ 1x ≤ b1
a ′ 4x ≤ b4
Figure 3.3 H-polyhedron
for some V = [V1, . . . , Vk] ∈ Rn×k , Y = [y1, . . .,yk ′] ∈ Rn×k′ . The main theorem for
polyhedra states that any H-polyhedron is a V-polyhedron and vice-versa [100](pag.
30).
An H-polytope is a bounded H-polyhedron (in the sense that it does not contain
any ray {x + ty : t ≥ 0}). A V-polytope is a bounded V-polyhedron
P = conv(V ) (3.8)
The main theorem for polytopes states that any H-polytope is a V-polytope and
vice-versa [100](pag. 29).
The dimension of a polytope (polyhedron) P is the dimension of its affine hull
and is denoted by dim(P). We say that a polytope P ⊂ R n , P = {x ∈ R n :
P x x ≤ P c }, is full-dimensional if dim(P) = n or, equivalently, if it is possible to
fit a nonempty n-dimensional ball in P,
or, equivalently,
∃x ∈ R n , ǫ > 0 : B(x, ǫ) ⊂ P, (3.9)
∃x ∈ R n , ǫ > 0 : δ2 ≤ ǫ ⇒ P x (x + δ) ≤ P c . (3.10)
Otherwise, we say that polytope P is lower-dimensional. A polytope is referred to
as empty if
∄x ∈ R n : P x x ≤ P c . (3.11)
Furthermore, if P x
i 2 = 1, where P x
i denotes the i-th row of a matrix P x , we say
that the polytope P is normalized.