Assessment and Future Directions of Nonlinear Model Predictive ...

Coordination of Networked Dynamical Systems 585A suitable generalization of the standard CG selection logic is given, for themaster part, by{min ‖w − r(t +¯τ f )‖ 2ŵ(t, t+¯τ f )= w∈V( ˆXΨ if V( ˆX (t +¯τ f |t b )) is non-empty(t+¯τ f |t b ))ŵ(t − 1,t+¯τ f − 1), otherwise(7)where Ψ = Ψ ′ > 0 p and ‖w‖ 2 Ψ := w′ Ψw, and V( ˆX (t +¯τ f |t b )) collects all stepvirtual commands, whose corresponding constraints vector predicted evolutionsstarting at time t +¯τ f from any x ∈ ˆX (t +¯τ f |t b ) are satisfied for all future timeinstants. The rationale underlying the above strategy hinges upon the propertyof virtual command sequences ensuring that if w is an admissible command attime t from the state x, it will be as such in all future time instants if constantlyapplied. Note that V( ˆX (t +¯τ f |t b )) may be empty, this means that the actualuncertainty on x(t +¯τ f ) is so large that we cannot use (7) to compute an admissiblevirtual command. However, the previously computed virtual commandŵ is still admissible and we are authorized to send it to the slave CG unit forbeing applied at time t.The slave part of the CG logic is far simpler and reduces toŵ(·,t), if available and ‖ŵ(·,t) − r(t)‖2w(t) ={Ψw< ‖w(t − 1) − r(t)‖ 2 Ψ ww(t − 1), otherwise(8)The basic properties of the above master/slave strategy have been described indetails in [4] and are here briefly condensed.Theorem 1. Let the assumptions A1-A2 hold true. Consider the system (1)along with the CG master/slave (7)-(8) selection strategy. Then:Abnormal case (possibly unbounded time-delay):1. The cardinality of ˆX (t +¯τf |t b ) may become unbounded;2. The set V( ˆX (t +¯τ f |t b )) is finitely determined;3. V(¯τ f |0)) ≠ ∅, need not imply that V(X (t+¯τ f |t b )) will be non-empty in somefuture time-instant;4. c(t) ∈Cfor all t ∈ Z + ;5. The overall system remains asymptotically stable but tracking performancemay be lost, viz. w(t) ↛ w r as r(t) ≡ r;Normal case (bounded time-delay):1. The cardinality of ˆX (t +¯τf |t b ) is bounded;2. If the set V( ˆX (t +¯τ f |t b )) is empty at a certain time instant t, itremainsempty for a finite number of steps only;3. If the set ˆX (t +¯τ f |t b ) consists of a single vector, then V( ˆX (¯τ f |0)) non-emptyimplies V( ˆX (t +¯τ f |t − ¯τ b )) non-empty for all t ∈ Z + ;