Assessment and Future Directions of Nonlinear Model Predictive ...

The Potential of Interpolation 65Definition 1 (Feasibility and robust positive invariance). Given a system,stabilizing feedback and constraints (1,2,3), a set S⊂R nx is feasible iff S⊆S 0 .Moreover, the set is robust positive invariant iffx ∈S ⇒ (A − BK)x ∈S, ∀[A B] ∈ Ω. (6)Definition 2 (MAS). The largest feasible invariant set (no other feasible invariantset can contain states outside this set) is uniquely defined and is calledthe Maximal Admissible Set (MAS, [7]).Define the closed-loop predictions for a given feedback K as x(k) =Φ k x(0);u(k) =−KΦ k−1 x(0); Φ = A − BK, then, under mild conditions [7] the MASfor a controlled LTI system is given byS =n⋂{x : Φ k x ∈S 0 } = {x : Mx ≤ 1}, (7)k=0with n a finite number. In future sections, we will for the sake of brevity usethe shorthand notation λS ≡{x : Mx ≤ λ1}. The MCAS (maximum controladmissible set) is defined as the set of states stabilisable with robust constraintsatisfaction by the specific control sequence:u i = −Kx i + c i , i =0, ..., n c − 1,u i = −Kx i , i ≥ n c .(8)By computing the predictions given a model/constraints (1,3) and control law(8), it is easy to show that, for suitable M,N, the MCAS is given as ([18, 19]):S MCAS = {x : ∃C s.t. Mx+ NC ≤ 1}; C =[c T 0 ... c T n c−1] T . (9)In general the MAS/MCAS are polyhedral and hence ellipsoidal invariant sets[9], S E = {x|x T Px ≤ 1}, are suboptimal in volume [12]. Nevertheless, unlike thepolyhedral case, a maximum volume S E is relatively straightforward to computefor the LPV case. However, recent work [11, 3] has demonstrated the tractabilityof algorithms to compute MAS for LPV systems. This algorithm requires anouter estimate, e.g. S 0 , constraints at each sample (also S 0 )andthemodelΦ.2.3 Background for InterpolationDefine several stabilizing feedbacks K i ,i=1,...,n,withK 1 the preferred choice.Definition 3 (Invariant sets). For each K i , define closed-loop transfer matricesΦ ij and corresponding robust invariant sets S i andalsodefinetheconvexhull S :Φ ij = A j − B j K i , j =1, ..., m; S i = {x : x ∈S i ⇒ Φ ij x ∈S i , ∀j}, (10)S Co{S 1 ,...,S n }. (11)