Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 121exists an m>0 such that m ≤ M(x) for all x ∈ B. Since t k →∞as k →∞,for any k ∈ N one can show a j k so that t k ∈ [j k˜δ, (jk +1)˜δ). Note that for anyt ∈ [j k˜δ, (jk +1)˜δ) wehave∥∥̂x˜δ(t) − ̂x˜δ(t k ) ∥ ∥ ≤Then, by the triangle inequality∫tt k∥ ∥∥ ˙̂x˜δ(s)∥ ds ≤ K 3˜δ ≤ ε0 /2.d A (̂x˜δ(t)) ≥ d A (̂x˜δ(t k )) − ∥ ∥̂x˜δ(t) − ̂x˜δ(t k ) ∥ ∥ ≥ ε0 /2.Therefore ̂x˜δ(t) ∈ B, if t ∈ [j k˜δ, (jk +1)˜δ) ,thus∫(j k +1)˜δj k˜δM(̂x˜δ(s))ds ≥ m˜δ.This would imply that lim τ →∞∫ τ0 M(̂x δ (s))ds →∞contradicting (5).Now let ε>0 be arbitrarily given, let δ 1 := ε/(2K 1 ), and let ε 1 := ε/2. Fromthe preceding part of the proof we already know that there is a δ 2 = ̂δ(ε 1 ), andfor any 0 t ′ > 0V (t ′′ ,x ∗ (t ′′ )) − V (t ′ ,x ∗ (t ′ )) ≤− M(ˆx(s))ds. (7)t ′where M is a continuous, radially unbounded, positive definite function. TheMPC value function V is defined as∫ t′′