Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 129(ii) Let the pair (¯x, ū)| [ti+δ − ,t i+T p] be an optimal solution to P(x(t i + δ − ),T c −δ, T p − δ). ThenV ∗ (t i + δ − )=V ti (t i + δ − , ¯x(t i + δ − ))=∫t i+T pt i+δL(¯x(s), ū(s)) ds + W (¯x(t i + T p )).Now, extend the process (¯x, ū) totheinterval[t i + δ − ,t i + T p + δ − ]usinginthe last δ seconds the auxiliary control law k aux . This is an admissible (theset S is invariant under k aux and is contained in X), suboptimal solution toP(x(t i + δ + ),T c − δ, T p − δ). ThereforeV ∗ (t i + δ + )=V ti+1 (t i + δ + , ¯x(t i + δ + ))≤∫t i+T p+δt i+δL(¯x(s), ū(s)) ds + W (¯x(t i + T p + δ))= V ∗ (t i + δ − )+W (¯x(t i + T p + δ)) − W (¯x(t i + T p ))+∫t i+T p+δt i+T pL(¯x(s), ū(s)) ds.Using RSC5a in the interval [t i + T p ,t i + T p + δ] weobtainV ∗ (t i + δ + ) ≤ V ∗ (t i + δ − ) for all i ≥ 0as required.✷