Model Predictive Control

26 3 Polyhedra, Polytopes and Simplices
3.4 Basic Operations on Polytopes
We will now define some basic operations and functions on polytopes. Note that
although we focus on polytopes and polytopic objects most of the operations described
here are directly (or with minor modifications) applicable to polyhedral
objects. Additional details on polytope computation can be found in [261, 123, 97].
All operations and functions described in this chapter are contained in the MPT
toolbox [161, 160].
3.4.1 Convex Hull
The convex hull of a set of points V = {V i } NV
i=1 , with V i ∈ R n , is a polytope defined
as
conv(V ) = {x ∈ R n NV
: x = α i V i , 0 ≤ α i ≤ 1,
i=1
NV
α i = 1}. (3.15)
The convex hull operation is used to switch from a V-representation of a polytope
to an H-representation. The convex hull of a union of polytopes Ri ⊂ R n , i =
1, . . .,NR, is a polytope
conv
NR
i=1
Ri
i=1
:= {x ∈ R n NR
: x = α i x i , x i ∈ Ri, 0 ≤ α i ≤ 1,
i=1
NR
α i = 1}.
(3.16)
An illustration of the convex hull operation is given in Figure 3.5. Construction
of the convex hull of a set of polytopes is an expensive operation which is exponential
in the number of facets of the original polytopes. An efficient software
implementation is available in [96].
3.4.2 Envelope
The envelope of two H-polyhedra P = {x ∈ R n : P x x ≤ P c } and Q = {x ∈ R n :
Q x x ≤ Q c } is an H-polyhedron
env(P, Q) = {x ∈ R n : ¯
P x x ≤ ¯
P c , ¯
Q x x ≤ ¯ Q c }, (3.17)
where ¯ P xx ≤ ¯ P c is the subsystem of P xx ≤ P c obtained by removing all the
inequalities not valid for the polyhedron Q, and ¯ Qxx ≤ ¯ Qc is defined in a similar
way with respect to Qxx ≤ Qc and P [32]. In a similar fashion, the definition can
be extended to the case of the envelope of a P-collection. An illustration of the
envelope operation is depicted in Figure 3.6. The computation of the envelope is
relatively cheap since it only requires the solution of one LP for each facet of P and
Q. Note that the envelope of two (or more) polytopes is not necessarily a bounded
set (e.g. when P ∪ Q is shaped like a star).
i=1