Model Predictive Control

12.2 Robust Invariant Sets 243
subject to the state constraints
−10 10
x(t) ∈ X = x :
≤ x ≤ , ∀t ≥ 0
−10
10
−1 1
w(t) ∈ W = w : ≤ w ≤ , ∀t ≥ 0
−1 1
(12.37)
The maximal robust positive invariant set of system (12.36) subject to constraints (12.37)
is depicted in Fig. 12.5 and reported below:
⎡
0.89
⎤
−0.44
⎡ ⎤
8.04
⎢ −0.89
⎢ −1
⎢ 0
⎣ 1
0.44
0
−1
0
⎥ ⎢
⎥ ⎢ 8.04 ⎥
⎥ ⎢
⎥ x ≤ ⎢ 10 ⎥
⎢
⎥ ⎢ 10 ⎥
⎦ ⎣ 10 ⎦
0 1 10
x2
10
5
0
O∞
-5
X
-10
-10 -5 0 5 10
Figure 12.5 Maximal Robust Positive Invariant Set of system (12.36) subject
to constraints (12.37).
Robust control invariant sets are defined for systems subject to controllable
inputs. These types of sets are useful to answer questions such as: “Find the set
of states for which there exists a controller such that the system constraints are
never violated for all possible disturbances”. The following definitions introduce
the different types of robust control invariant sets.
Definition 12.3 (Robust Control Invariant Set) A set C ⊆ X is said to be a
robust control invariant set for the system in (12.2) subject to the constraints in
(12.3), if
x(t) ∈ C ⇒ ∃u(t) ∈ U such that f(x(t), u(t), w(t)) ∈ C, ∀ w(t) ∈ W, ∀ t ∈ N+
Definition 12.4 (Maximal Robust Control Invariant Set C∞) The set C∞ ⊆
X is said to be the maximal robust control invariant set for the system in (12.2)
subject to the constraints in (12.3), if it is robust control invariant and contains all
robust control invariant sets contained in X.
x1