Model Predictive Control

5.1 General Results for Multiparametric Nonlinear Programs 67
z 2
g1( z1, z2)= x1 g2( z1, z2)= x1 (a) Set R(¯x1) shaded in gray.
z 2
g1( z1, z2)= x3 g2( z1, z2)= x3 (c) Set R(¯x3) shaded in gray.
z 1
z 1
z 2
g1( z1, z2)= x2
g2( z1, z2)= x2
(b) Set R(¯x2) shaded in gray.
x 1
J x
*
( )
x 3
(d) Value function J ∗ (x)
Figure 5.3 Example 5.4: problem (5.10). (a)-(c) Projections of the point-toset
map R(x) for three values of the parameter x: ¯x1 < ¯x2 < ¯x3; (d) Value
function J ∗ (x).
• in example 5.2 the feasible vector space Z is unbounded (z ≥ 0),
• in examples 5.3 and 5.4 the feasible point-to-set map R(x) (defined in (5.4))
is discontinuous, as precisely defined below.
In the next sections we discuss both cases in detail.
Continuity of Point-to-Set Maps
Consider a point-to-set map R : x ∈ X ↦→ R(x) ∈ 2 Z . We give the following
definitions of open and closed map according to Hogan [137]:
z 1
x