Assessment and Future Directions of Nonlinear Model Predictive ...

120 F.A.C.C. Fontes, L. Magni, and É. GyurkovicsNow, suppose that due to disturbances we have no means of guaranteeing thatall the hypothesis of the lemma are satisfied for the trajectory x ∗ we want toanalyse. Instead some hypothesis are satisfied on a neighbouring trajectory ˆxthat coincides with the former at a sequence of instants of time. 1 Furthermore,suppose that instead of approaching the origin we would like to approach someset containing the origin. These are the conditions of the following lemma.Definition 1. Let A be a nonempty, closed subset of IR n . The function x ↦→d A (x), fromIR n to IR, denotes the distance from a point x to the set A (i.e.,d A (x) :=min y∈A ‖x − y‖).We say that a function M is positive definite with respect to the set A ifM(x) > 0 for all x ∉ A and M(x) =0for some x ∈ A.Lemma 3 (A generalization of Barbalat’s lemma). Let A be subset of IR ncontaining the origin, and M : IR n → IR be a continuous function which ispositive definite with respect to A.Let ∆>0 be given and for any δ ∈ (0,∆) consider the functions x ∗ δ and ˆx δfrom IR + to IR n satisfying the following properties:• The function x ∗ δ is absolutely continuous, the function ˆx δ is absolutely continuouson each interval [iδ, (i +1)δ), for all i ∈ IN 0 ,andˆx δ (iδ) =x ∗ δ (iδ) forall i ∈ IN 0 .• There exist positive constants K 1 , K 2 and K 3 such that for all δ ∈ (0,∆)‖ẋ ∗ δ(·)‖ L ∞ (0,∞) ε 0 for all k ∈ N.Without loss of generality, we may assume that t k+1 − t k ≥ ˜δ, k ∈ IN. LetR ≥ K 2 be such that the set B = {x ∈ R n : d A (x)) ≥ ε 0 /2and ‖x‖ ≤R} isnonempty. Since B is compact, M is continuous and M(x) > 0,ifx ∈ B, there1 In an NMPC context, ˆx would represent the concatenation of predicted trajectories;see equation (12).