Assessment and Future Directions of Nonlinear Model Predictive ...

128 F.A.C.C. Fontes, L. Magni, and É. GyurkovicsV ti (t, ¯x(t)) =∫t i+T ptL(¯x(s), ū(s))ds + W (¯x(t i + T p )). (14)Suppose that the worst disturbance scenario did not occur and so, at time t, weare at state x ∗ (t) which is, in general, distinct from ¯x(t). Because such scenario ismore favorable, and by the assumption on the existence of value to the differentialgamewehavethatV ti (t, x ∗ (t)) ≤ V ti (t, ¯x(t)) for all t ∈ [t i ,t i + δ). (15)We may remove the subscript t i from the value function if we always choose thesubscript t i to be the sampling instant immediately before t, that is (recall that⌊t⌋ π =max i {t i ∈ π : t i ≤ t})For simplicity define the functionV (t, x) :=V ⌊t⌋π (t, x).V ∗ (t) =V (t, x ∗ (t)).We show that t ↦→ V ∗ (t) is decreasing in two situations:(i) on each interval [t i ,t i + δ)∫ tV ∗ (t) ≤ V ∗ (t i ) −(ii) from one interval to the othertherefore yielding the result.t iM(¯x(s)) ds for all t ∈ [t i ,t i + δ), and all i ≥ 0;V ∗ (t i + δ + ) ≤ V ∗ (t i + δ − ) for all i ≥ 0;(i) The first assertion is almost immediate from (15), (14) and (13).V ∗ (t) ≤ V ti (t, ¯x(t))= V ti (t i , ¯x(t i )) −∫ tt i∫ t= V ti (t i ,x ∗ (t i )) −∫ t≤ V ∗ (t i ) −for all t ∈ [t i ,t i + δ), and all i ≥ 0;t it iM(¯x(s)) dsL(¯x(s), ū(s)) dsL(¯x(s), ū(s)) ds