Model Predictive Control

242 12 Constrained Robust Optimal Control
Definition 12.1 (Robust Positive Invariant Set) A set O ⊆ X is said to be
a robust positive invariant set for the autonomous system (12.1) subject to the
constraints in (12.3), if
x(0) ∈ O ⇒ x(t) ∈ O, ∀w(t) ∈ W, t ∈ N+
Definition 12.2 (Maximal Robust Positive Invariant Set O∞) The set O∞ ⊆
X is the maximal robust invariant set of the autonomous system (12.1) subject to
the constraints in (12.3) if O∞ is a robust invariant set and O∞ contains all the
robust positive invariant sets contained in X that contain the origin.
Theorem 12.1 (Geometric condition for invariance) A set O ⊆ X is a robust
positive invariant set for the autonomous system (12.1) subject to the constraints
in (12.3), if and only if
O ⊆ Pre(O, W) (12.34)
The proof of Theorem 12.1 follows the same lines of the proof of Theorem 10.1. ✷
It is immediate to prove that condition (12.34) of Theorem 12.1 is equivalent
to the following condition
Pre(O, W) ∩ O = O (12.35)
Based on condition (12.35), the following algorithm provides a procedure for computing
the maximal robust positive invariant subset O∞ for system (12.1)-(12.3)
(for reference to proofs and literature see Chapter 10.3).
Algorithm 12.1 Computation of O∞
input fa , X, W
output O∞
let Ω0 ← X
repeat
k = k + 1
Ωk+1 ← Pre(Ωk, W) ∩ Ωk
until Ωk+1 = Ωk
O∞ ← Ωk
Algorithm 12.1 generates the set sequence {Ωk} satisfying Ωk+1 ⊆ Ωk, ∀k ∈ N and
it terminates if Ωk+1 = Ωk so that Ωk is the maximal robust positive invariant set
O∞ for system (12.1)-(12.3).
Example 12.4 Consider the second order stable system in Example 12.1
x(t + 1) = Ax(t) + w(t) =
0.5 0
1 −0.5
x(t) + w(t) (12.36)