Model Predictive Control

30 3 Polyhedra, Polytopes and Simplices
3.4.7 Set-Difference
x2
10
8
6
4
2
0
-2
-4
-6
-8
-10
-10 -5 0 5 10
x1
Figure 3.7 Illustration of the Chebychev ball contained in a polytope P.
x3
5
4
3
2
1
0
1
0
x2
-1-0.5 0 0.5 1 1.5
-1
-2
Figure 3.8 Illustration of a projection of a 3-dimensional polytope P onto a
plane.
The set-difference of two polytopes Y and R0
x1
R = Y \ R0 := {x ∈ R n : x ∈ Y, x /∈ R0}, (3.22)
in general, can be a nonconvex and disconnected set and can be described as a Pcollection
R = m i=1 Ri, where Y = m i=1 Ri
(R0 Y). The P-collection R =
m i=1 Ri can be computed by consecutively inverting the half-spaces defining R0
as described in the following Theorem 3.1.
Note that here we use the term P-collection in the dual context of both Pcollection
and its underlying set (cf. Definitions 3.3 and 3.4). The precise statement
would say that R = Y \ R0, where R is underlying set of the P-collection R =
{Ri} m i=1 . However, whenever it is clear from context, we will use the former, more
compact form.
Theorem 3.1 [38] Let Y ⊆ Rn be a polyhedron, R0 {x ∈ Rn
: Ax ≤ b}, and
¯R0
m {x ∈ Y : Ax ≤ b} = R0 Y, where b ∈ R , R0 = ∅ and Ax ≤ b is a minimal