Model Predictive Control

266 12 Constrained Robust Optimal Control
Then, for all x ∈ X0, limk→∞ d(x(k), O) = 0. X0 is called the region of attraction.
Remark 12.10 Theorem 12.7 holds also for the CROC-OL (12.41)–(12.45). The
region of attraction will be X OL
0 .
Remark 12.11 Assumptions A4 and A5 in Theorem 12.7 imply that stage cost q(x,u)
and terminal cost p(x) in (12.41) need to be zero in O. Note that this cannot be
obtained with p = 2 in (12.41).
Remark 12.12 Robust control invariance of the set Xf in Assumption (A2) of Theorem
12.7 implies that problem (12.107) in Assumption (A3) is always feasible.
Remark 12.13 Robust control invariance of the set O in Assumption (A2) of Theorem
12.7 is also implicitly guaranteed by Assumption (A3) as shown next.
From Remark 12.11, Assumption (A3) in O becomes minu maxw a ,w p(p(A(wp )x +
B(w p )u + Ew a ) ≤ 0 for all x ∈ O. From Assumption (A4), this can be verified only
if it is equal to zero, i.e. if it exists a u ∈ U such that A(w p )x + B(w p )u + Ew a ∈ O
for all w a ∈ W a w p ∈ W p (since O is the only place where p(x) can be zero). This
implies the robust control invariance of O.
12.9 Literature Review
An extensive treatment of robust invariant sets can be found in [50, 51, 48, 52].
The proof to Theorem 12.1 can be fond in [152, 85] For the derivation of the algorithms
12.1, 12.2 for computing robust invariant sets (and their finite termination)
see [10, 44, 152, 111, 147].
Min-max robust constrained optimal control was originally proposed by Witsenhausen
[255]. In the context of robust MPC, the problem was tackled by Campo
and Morari [67], and further developed in [5] for SISO FIR plants. Kothare et
al. [158] optimize robust performance for polytopic/multi-model and linear fractional
uncertainty, Scokaert and Mayne [224] for additive disturbances, and Lee and
Yu [165] for linear time-varying and time-invariant state-space models depending
on a vector of parameters θ ∈ Θ, where Θ is either an ellipsoid or a polyhedron.
Other suboptimal CROC-Cl strategies have been proposed in [158, 25, 159]. For
stability and feasibility of the robust RHC (12.38), (12.104) we refer the reader
to [37, 173, 181, 21].