Assessment and Future Directions of Nonlinear Model Predictive ...

Sampled-Data MPC for Nonlinear Time-Varying Systems 123The target set Θ is a closed set, contains the origin and is robustly invariantunder no control. That is, x(t; t 0 ,x 0 ,u,d) ∈ Θ for all t ≥ t 0 ,allx 0 ∈ Θ, andalld ∈D([t 0 ,t)) when u ≡ 0. We further assume that f is a continuous functionand locally Lipschitz continuous with respect to x.Consider a sequence of sampling instants π := {t i } i≥0 with constant intersamplingtimes δ > 0 such that t i+1 = t i + δ for all i ≥ 0. Let the controlhorizon T c and prediction horizon T p ,withT c ≤ T p , be multiples of δ (T c = N c δand T p = N p δ with N c ,N p ∈ IN). Consider also a terminal set S (⊂ IR n ), aterminal cost function W :IR n → IR and a running cost function L :IR n ×IR m → IR. The optimization problem is a finite horizon differential game wherethe disturbance d acts as the maximizing player and the control u acts as theminimizing player. We shall assume that the minimizing player uses a sampleddatainformation structure. The space of the corresponding strategies over [t 1 ,t 2 ]we denote by K([t 1 ,t 2 ]). For any t ∈ π, let ktaux ∈K([t + T c ,t+ T p ]) be an apriori given auxiliary sampled-data strategy. The quantities time horizons T c andT p , objective functions L and W , terminal constraint set S, the inter-samplingtime δ, and auxiliary strategy kt aux are the quantities we are able to tune —the so-called design parameters — and should be chosen to satisfy the robuststability condition described below.At a certain instant t ∈ π, we select for the prediction model the controlstrategy for the intervals [t, t + T c )and[t + T c ,t+ T p ) in the following way. Inthe interval [t, t + T c ), we should select, by solving an optimization problem, thestrategy k t in the interval [t, t + T c ]. The strategy ktaux , known a priori, is usedin the interval [t + T c ,t+ T p ].The robust feedback MPC strategy is obtained by repeatedly solving on-line,at each sampling instant t i , a min-max optimization problem P, to select thefeedback k ti , every time using the current measure of the state of the plant x ti .P(t, x t ,T c ,T p ): Min k∈K([t,t+Tc])Max d∈D([t,Tp])subject to:∫t+T ptL(x(s),u(s))ds + W (x(t + T p )) (9)x(t) =x tẋ(s) =f(s, x(s),u(s),d(s)) a.e. s∈ [t, t + T p ] (10)x(s) ∈ X for all s ∈ [t, t + T p ]u(s) ∈ U a.e. s∈ [t, t + T p ]x(t + T p ) ∈ S, (11)whereu(s) =k t (s, x(⌊s⌋ π )) for s ∈ [t, t + T c )u(s) =k auxt (s, x(⌊s⌋ π )) for s ∈ [t + T c ,t+ T p ).