Assessment and Future Directions of Nonlinear Model Predictive ...

286 H. Chen et al.Hence, at each sampling time t k ≥ t 1 with the actual state x(t k ), the movinghorizon control defined in (6) is obtained via (existence assumed)u ∗ k∫ ∞:= arg min max ‖z 1 (t)‖ 2 − γ 2 ‖w(t)‖ 2 dt subject to (7). (9)u∈U w∈Wt kFor the closed-loop system with (6), if we define the piecewise continuous functionV (x(t),t)aswe can state the following result:V (x(t),t):=V k (x(t)), t ∈ [t k ,t k+1 ),t k ≥ t 0 (10)Proposition 1. For any τ > t 0 , the moving horizon closed-loop system givenby (1) and (6) is dissipative in the sense of∫ τV (x 0 ,t 0 )+where γ(t) is piecewise constant and defined ast 0γ(t) 2 ‖w(t)‖ 2 −‖z 1 (t)‖ 2 dt ≥ V (x(τ),τ) (11)γ(t) :=γ i , ∀t ∈ [t i ,t i+1 ),t i ≥ t 0 . (12)Proof: We denote the sequence of the sampling times as t 0 ,t 1 ,t 2 , ··· and considerthat t k