Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 125Theorem 1. Assume condition RSC is satisfied and that the differential gamesP(t, x t ,T c ,T p ) have a value for all x t ∈ X and t ≥ t 0 . Then, the robust MPCstrategy robustly stabilizes the system to the target set Θ.ProofThe proof starts by establishing a monotone decreasing property of the MPCValue function. Then the application of the generalized Barbalat’s Lemma yieldsthe robust stability result.Let V (t, x) be the value function of P(t, x, T c − (t − t i ),T p − (t − t i )) witht i = ⌊t⌋ π .Letalsoˆx be the concatenation of predicted trajectories ¯x for eachoptimization problem. That is for i ≥ 0ˆx(t) =¯x i (t) for all t ∈ [t i ,t i + δ) (12)where ¯x i is the trajectory of a solution to problem P(t i ,x ∗ (t i ),T c ,T p ). Note thatˆx coincides with x ∗ at all sampling instants t i ∈ π, but they are typically notidentical on [t i ,t i + δ), since they correspond to different disturbances.The following lemma establishes a monotone decreasing property of V .Lemma 4. ([9], Lemma 4.4) There exists an inter-sample time δ > 0 smallenough such that for any t ′