Assessment and Future Directions of Nonlinear Model Predictive ...

528 M. Alamir♭V is radially unboundedSince the control parametrization is supposed to be continuous, the set ofcontrols given by :{ }U := U pwc (t, p),(t,p)∈[0,Nτ s]×P Xis necessarily bounded. using assumption 2.1 with W = U gives the results.♭ Finally it is clear that V (0) = 0 since zero is an autonomous equilibriumstate.Decreasing properties of VTwo situations have to be distinguished :Case where ˆq(k) > 1. In this case, let us investigate candidate solutions forthe optimization problem Pαε,∗(xcl(k + )) where x cl (k + ) is the next state on theclosed loop trajectory, namely :()x cl (k + )=F τ s ,x cl (k),u 1 (ˆp(x cl (k))) .A natural candidate solution to the optimal control problem Pαε,∗(xcl(k + )) isthe one associated to the translatable character of the control parametrization,namelyp cand (k + ):=ˆp + (x cl (k)) ; q cand (k + ):=ˆq(x cl (k)) − 1 ≥ 1. (11)In the following sequel, the following short notations are usedˆp(k) =ˆp(x cl (kτ s )) ; ˆq(k) =ˆq(x cl (kτ s )) ; V (k) =V (x cl (k)).By the very definition of p + , it comes that :‖F (q cand (k + )τ s ,x cl (k + ),p cand (k + ))‖ 2 = ‖F (ˆq(k)τ s ,x cl (k), ˆp(k))‖ 2= V (x cl (k)) − α ˆq(k)N min{ε, ‖Fˆq(k)(·,x cl (k), ˆp(k))‖ 2 ∞ }, (12)and since V (x cl (k + )) satisfies by definition, one has :V (x cl (k + )) ≤‖F (q cand (k + )τ s ,x cl (k + ),p cand (k + ))‖ 2 ++ α ˆq(k) − 1 min{ε, ‖Fˆq(k)−1 (·,x cl (k + ),p cand (k + ))‖ 2N∞},This with (12) gives :V (x cl (k + )) ≤ V (x cl (k)) − α ˆq(k)N min{ε, ‖Fˆq(k)(·,x cl (k), ˆp(k))‖ 2 ∞} ++ α ˆq(k) − 1Nmin{ε, ‖Fˆq(k)−1 (·,x cl (k + ),p cand (k + ))‖ 2 ∞ }. (13)