Assessment and Future Directions of Nonlinear Model Predictive ...

A Low Dimensional Contractive NMPC Scheme 527for a ball in R np centered at the origin. Consider the following open-loop optimalcontrol problem defined for some α>0andε>0:Pα ε,∗ (x) : min J ∗ (x, q, p) =(q,p)∈{1,...,N}×P X‖F (qτ s ,x,p)‖ 2 + α q N · min{ ε 2 , ‖F q (·,x,p)‖ 2 ∞}. (8)Note that if all the functions involved in the definition of the problem (thesystem’s map f and the control parametrization) are continuous then the costfunction is continuous in p. This together with the compactness of the set P Xguarantee that the problem Pαε,∗ (x) admits a solution for all x ∈ X and henceis well posed. Therefore, let us denote the solution of (8) for some x ∈ X byˆq(x) ∈{1,...,N} and ˆp(x) ∈ P X . These solutions are then used to define thereceding horizon state feedback given by :u(kτ s + τ) =u 1 (ˆp(x(kτ s ))) ∀τ ∈ [0,τ s [. (9)The stability result associated to the resulting feedback strategy is stated in thefollowing proposition :Proposition 1. If the following conditions hold :1. The function f in (3) and the parametrization map are continuous and satisfythe strong contraction property (see definition 4). Moreover, the system(3) satisfies assumption 2.1.2. For all x ∈ X and all admissible u = U pwc (·,p), the solution of (3) is definedfor all t ∈ [0,Nτ s ] and all p ∈ P X . (No explosion in finite time shorter thanNτ s ).3. The control parametrization is translatable on P X in the sense of definition 2.Then, there exist sufficiently small ε>0 and α>0 such that the recedinghorizon state feedback (9) associated to the open-loop optimal control problem(8) is well defined and makes the origin x = 0 asymptotically stable for theresulting closed loop dynamics with a region of attraction that contains X. ♭Proof. The fact that the feedback law is well defined directly results fromthe continuity of the functions being involved together with the compactness ofP X . Let us denote by x cl (·) the closed loop trajectory under the receding horizonstate feedback law. Let us denote by V (x) the optimal value of the cost function,namely : V (x) =J ∗ (x, ˆq(x), ˆp(x)).♭ V is continuous V (x) can clearly be written as followsV (x) =inf{}V 1 (x),...,V N (x); V q (x) :=minp∈P XJ ∗ (x, q, p). (10)But for given q, J ∗ (x, q, p) is continuous in (x, p), therefore V q (·) isacontinuousfunction of x. SinceV is the sum of N continuous functions (V j ) j=1,...,N ,it is continuous itself.