Assessment and Future Directions of Nonlinear Model Predictive ...

136 T. Alamo et al.1+f ⊤ λ1, ∀x ∈ S (9)r⋃c ⊤ 1x ≤ 1, ∀x ∈ρ T l (10)l=14.3 Robust Control Invariant Set: Proposed AlgorithmThe following algorithm serves to compute a robust control invariant set for apiecewise affine system:Algorithm 3(i) Set k = 0 and choose a contracting factor ˜λ ∈ (0, 1).(ii) Make ˜C 0 equal to the outer approximation of C ∞ obtained by means ofalgorithm 2.(iii) Given ˜C k = { x : Hx ≤ h }, obtainT i = Q i (˜λ ˜C k ), i =1,...,r.(iv) Obtain Q c (˜λ ˜C ⋃k )= r Q c i (˜λ ˜C k )= nc ⋃S j by means of property 4.i=1j=1(v) For every j =1,...,n c obtain { x : c ⊤ j x ≤ 1 }, the inner supportingconstraint of S j over Q(˜λ ˜C ⋃k )=r T i . This can be achieved by means ofi=1property 5.(vi) Make ˜C k+1 = nc ⋂{ x : c ⊤ j x ≤ 1 }.j=1(vii) If ˜C k+1 ⊆ Q( ˜C k+1 )then ˜C k+1 is a robust control invariant set. Stop. Else,set k = k + 1 and go to step (iii).Bearing in mind the λ contractive procedure of [14], a contracting factor˜λ ∈ (0, 1) has been included in the algorithm. Note that algorithm 3 finishes onlyif ˜C k+1 ⊆ Q( ˜C k+1 ). In virtue of the geometrical condition of robust invariance[14], it is inferred that ˜C k is a robust control invariant set. That is, if algorithm 3finishes then a robust control invariant set is obtained. This set serves as an innerapproximation of C ∞ . Due to the approximate nature of the algorithm, it is notguaranteed that algorithm 3 converges to a robust control invariant set. Note,however, that it can be shown that each one of the obtained sets ˜C k constitutesan inner approximation of C k . The proof of this statement is based on the fact