Assessment and Future Directions of Nonlinear Model Predictive ...

On the Computation of Robust Control Invariant Sets 137that, by definition of inner supporting constraint, S j⋂ { x : c⊤j x ≤ 1 } equalsthe empty set. Thus,Q c (˜λ ˜C k ) ⋂ ˜Ck+1 = Q c (˜λ ˜C k ) ⋂ (⋃n c=( S j ) ⋂ (j=1⋂n cm=1⋂n cm=1{ x : c ⊤ mx ≤ 1 }) =⋃n c⋂⊆ (S j { x : c⊤j x ≤ 1 }) =∅.j=1{ x : c ⊤ mx ≤ 1 })⋃n cj=1()⋂ ⋂n cS j ( { x : c ⊤ mx ≤ 1 })m=1That is, Q c (˜λ ˜C k ) ⋂ ˜Ck+1 = ∅. This implies that ˜C k+1 ⊆ Q(˜λ ˜C k ).5 Numerical ExampleIn this example, region X = { x : ‖x‖ ∞ ≤ 15 } is subdivided into the subregionsX 1 , X 2 and X 3 . These subregions are defined as follows:X 1 = { x ∈ X : x 1 − x 2 ≤ 0 }X 2 = { x ∈ X : x 1 − x 2 > 0andx 1 + x 2 ≥ 0 }X 3 = { x ∈ X : x 1 − x 2 > 0andx 1 + x 2 < 0 }Consider the following piecewise affine system:⎧ [ ] [ ] [ ]11 0 1x + u + w if x ∈ X0111 0[ ] [ ] [ ]⎪⎨ 1 1 0 1x + =x + u + w if x ∈ X0.5 1.521 0[ ] [ ] [ ]1 −0.5 1 1⎪⎩ x + u + w if x ∈ X0 1.531.5 0In this example it is assumed that U = { u ∈ R : ‖u‖ ∞ ≤ 2 } andW = { w ∈ R : ‖w‖ ∞ ≤ 0.1 }. The contracting factor for algorithm 3 has beenset equal to 0.95. In figure 1 the sequence of outer bounds Ĉk is displayed (theouter approximation of the convex hull operator presented in [12] has been used).The most inner polyhedron is used in algorithm 3 as initial guess to obtain arobust control invariant set. In figure 2 a sequence of sets ˜C k leadingtoarobustcontrol invariant set is displayed. The most inner polyhedron is a robust controlinvariant set for the piecewise affine system.