Assessment and Future Directions of Nonlinear Model Predictive ...

Conditions for MPC Based Stabilization of Sampled-Data Nonlinear Systems 43xEmx (T ) =y (1)Emx (T ) =y (2)x =y (0)0Eξ 0ξ 1Eζ 1Aξ 2Eζ 0Aζ 2Al 1 T l 2 Tl1Tl 2 Tl 1 T l 2cu = u(p,0)Tm2TT * l TT*= T*+ lTT * = T * + 2 lT0 =u (1)u (p,1)1 0mu (p,2)u (2)T2 0tFig. 1. Sketch to the Algorithm(iii) Find the solution u ∗ = {u ∗ 0 ,...,u∗ N−1 } to the problem P T,h A (N,ζA ju (j+1) = {u ∗ 0 ,...,u∗ l−1 } and u(p,j+1) = {u ∗ l−l ,...,u∗ l−1 }.(iv) j = j +1.A schematic illustration of the Algorithm is sketched in Figure 1.), letTheorem 2. Suppose that the conditions of Theorem 1 hold true. Then thereexists a β ∈KL, and for any r>0 there exists a h ∗ > 0 such that for any fixedN ≥ N ∗ , h ∈ (0,h ∗ ] and x 0 ∈ Γ , the trajectory of the l-step exact discrete-timesystemξk+1 E = F l E (ξE k , v l,h(ζk A )), ξE 0 = φE(x l 0, u c ) (15)with the l-step receding horizon controller v l,h obtained by the predictionζ A k+1 = F A l(yk+1 , v p ( ζkA )), ζA0 = φ A (x l 0, u c ) (16)satisfies that ξ E k∈ Γ max(h) and∥ ξEk∥ ∥ ≤ max{β(∥ ∥ξ E0∥ ∥ ,kTm ) , r }for all k ≥ 0. Moreover, ζkA ∈ Γ max(h), aswell,and∥ { (∥ ∥∥ ζA k ≤ max β ∥ζ A 0 ,kTm ) + δ 1 , r }where δ 1 can be made arbitrarily small by suitable choice of h.Proof. The proof is given in the Appendix.