Model Predictive Control

244 12 Constrained Robust Optimal Control
Remark 12.3 The geometric conditions for invariance (12.34)-(12.35) hold for control
invariant sets.
The following algorithm provides a procedure for computing the maximal control
robust invariant set C∞ for system (12.2)-(12.3).
Algorithm 12.2 Computation of C∞
input: A and B, X, U and W
output: C∞
let Ω0 ← X
repeat
k = k + 1
Ωk+1 ← Pre(Ωk, W) ∩ Ωk
until Ωk+1 = Ωk
C∞ ← Ωk+1
Algorithm 12.2 generates the set sequence {Ωk} satisfying Ωk+1 ⊆ Ωk, ∀k ∈ N and
O∞ =
k≥0 Ωk. If Ωk = ∅ for some integer k then the simple conclusion is that
O∞ = ∅. Algorithm 12.2 terminates if Ωk+1 = Ωk so that Ωk is the maximal
robust control invariant set C∞ for the system (12.2)-(12.3). The same holds true
for non-autonomous systems.
Example 12.5 Consider the second order unstable system in example 12.2. The
maximal robust control invariant set of system (12.16) subject to constraints (12.17)
is depicted in Fig. 12.6 and reported below:
⎡
⎢
⎣
0 1
0 −1
0.55 −0.83
−0.55 0.83
⎤
⎡
⎤
3.72
⎥ ⎢
⎥
⎦ x ≤ ⎢ 3.72 ⎥
⎣ 2.0 ⎦
2.0
Definition 12.5 (Finitely determined set) Consider Algorithm 12.1. The set
C∞ (O∞) is finitely determined if and only if ∃ i ∈ N such that Ωi+1 = Ωi. The
smallest element i ∈ N such that Ωi+1 = Ωi is called the determinedness index.
For all states contained in the maximal robust control invariant set C∞ there
exists a control law, such that the system constraints are never violated for all
feasible disturbances. This does not imply that there exists a control law which
can drive the state into a user-specified target set. This issue is addressed in the
following by introducing the concept of robust controllable and stabilizable sets.
Definition 12.6 (N-Step Robust Controllable Set KN(O, W)) For a given target
set O ⊆ X, the N-step robust controllable set KN(O, W) of the system (12.2)
subject to the constraints (12.3) is defined as:
Kj(O, W) Pre(Kj−1(O), W), K0(O, W) = O, j ∈ {1, . . ., N}.