Assessment and Future Directions of Nonlinear Model Predictive ...

Hybrid MPC: Open-Minded but Not Easily Swayed 25Note that when W N (x, Ω) ≤ µV N (x), we have ¯κ N (x, Ω) =ω 1 and π N (x, Ω) ={ω 2 ,ω 3 , ..., ω M ,w M }. The closed loop generated by this algorithm isWe use ψ(k, x, Ω, ¯κ N ) to denote the solution to (8).x + = f(x, ¯κ N (x, Ω)) (8)Ω + = π N (x, Ω) . (9)Theorem 2. Let Assumption 3.1 hold. For each ρ ∈ (0, 1) there exist n ◦ ∈ Nand positive real numbers K and α such that for all x ∈X and ΩW N (f(x, ¯κ N (x, Ω)), π N (x, Ω)) − W N (x, Ω) ≤−ρσ(x) (10)σ(ψ(k, x, Ω, ¯κ N )) ≤ Kσ(x)exp(−αk) ∀k ∈ N (11)for all horizon N and memory horizon M satisfying N ≥ M + n ◦ .Robustness with Respect to Measurement NoiseLet us now comment on the possible extra robustness that the MPC with memoryalgorithm may bring to the stability of a closed loop. Suppose the stability ofthe closed loop obtained by standard MPC has some robustness with respect to(bounded) measurement noise characterized as (perhaps for x in some compactset)V N (f(x, κ N (x + e))) − V N (x) ≤−σ(x)/2+α v |e|where N is large enough and α v > 0. Let us choose some µ>1. Let us be givensome Ω = {ω 1 , ..., ω M }. Then it is reasonable to expect for M and N − Msufficiently large, at least for systems such as that with a disjoint attractor, thatW N (f(x, ω 1 ),π N (x + e, Ω)) − W N (x, Ω) ≤−σ(x)/2+α w |e|with α w > 0 (much) smaller than α v as long as W N (x, Ω) is not way far off fromV N (x), say W N (x, Ω) ≤ 2µV N (x). Now consider the closed loop (8)-(9) undermeasurement noise. Suppose W N (x + e, Ω) ≤ µV N (x + e). Then ¯κ N (x + e, Ω) =ω 1 .Forµ sufficiently large it is safe to assume W N (x, Ω) ≤ 2µV N (x). Thereforewe haveW N (f(x, ¯κ N (x + e, Ω)), π N (x + e, Ω)) − W N (x, Ω) ≤−σ(x)/2+α w |e| .Now consider the other case where W N (x + e, Ω) >µV N (x + e). Then define˜Ω := {v 0 , ..., v M−1 } where {v 0 ,v 1 , ...} =: v and V N (x + e) =J N (x + e, v).Note then that W N (x + e, ˜Ω) =V N (x + e) and it is safe to assume W N (x, ˜Ω) ≤2µV N (x) aswellasW N (x, ˜Ω) ≤ W N (x, Ω) forµ large enough. Note finally that¯κ N (x + e, Ω) =¯κ N (x + e, ˜Ω) =v 0 and π N (x + e, Ω) =π N (x + e, ˜Ω) inthiscase. Hence