Assessment and Future Directions of Nonlinear Model Predictive ...

A Low Dimensional Contractive NMPC Scheme 525Assumption 2.1. For all finite horizon T>0, the following asymptotic propertyholds :[]lim‖x 0‖→∞mint∈[0,T ]‖F (t, x 0 , u)‖ = ∞ (5)minu∈W [0,T ]for all compact subset W ⊂ R m . (In other words, infinitely fast state excursionsneed infinite control)♭Note that assumption 2.1 is rather technical since it only excludes systems withfinite inverse escape time.2.1 Piece-Wise Constant Control ParametrizationLet some sampling period τ s > 0 be given. One way to define a low dimensionalparametrization of piece-wise constant control profiles over the time interval[0,Nτ s ] that belongs to a closed subset U ⊂ R m is to follow the following twostep procedure :1. First, define a mapC : P → R m ×···×R m p ❀ C(p) = ( u 1 (p),...,u N (p) ) ; u i (p) ∈ R m .2. Project C(p) on the admissible subset U N using the projection map P U ,namely :P U N ◦ C : P → U ×···×U p ❀ P U N ◦ C(p)= ( P U (u 1 (p)),...,P U (u N (p)) ) /3. For all t ∈ [(k − 1)τ s ,kτ s ], the control is given by u(t) = P U (u k (p)) =:U pwc (t, p).Definition 1. The map C defined above is called the parametrization mapwhile for given C and U, thefamily { U pwc (·,p) } is called a P-admissiblep∈Pparametrization of control profiles.♭Definition 2. A P-admissible parametrization is said to be translatable if andonly if for each p ∈ P, thereexistssomep + ∈ P such that u i (p + )=u i+1 (p) forall i ∈{1,...,N − 1}♭Definition 3. A P-admissible parametrization {U pwc (·,p)} p∈P is called proper if∫ t2and only if for all t 1 0 be given together with an associated P-admissible control parametrization {U pwc (·,p)} p∈P .