Assessment and Future Directions of Nonlinear Model Predictive ...

Computational Aspects of Approximate Explicit NMPC 187The following rule is applied when no feasible solution to the NLP problem(10)–(11) was found at some of the points w i ∈ W , w i ≠ w 0 ,wherethesetW = {w 0 ,w 1 ,w 2 , ..., w N1 } is defined in Procedure 1.Heuristic splitting rule 1. (handling infeasibility)Consider the following two cases:1. The set of the feasible points in X 0 includes the center point w 0 and someof the points w i ∈ W , w i ≠ w 0 (the set W = {w 0 ,w 1 ,w 2 , ..., w N1 } is definedin Procedure 1). Then, split X 0 into two types of hyper-rectangles byhyperplanes containing some of the feasible points w i ∈ W :i. Hyper-rectangles X f 1 ,Xf 2 , ..., Xf N fcontaining only feasible points.ii. Hyper-rectangles X nf1 ,Xnf 2 , ..., XnfN nfcontaining some infeasible points.Denote the number of the new hyper-rectangles N s = N f +N nf . The optimalchoice of dividing hyperplanes is the one which minimizes the number N s ofthe new hyper-rectangles.2. The center point w 0 of X 0 is the only feasible point. Then, split X 0 on allstate space axes by hyperplanes through w 0 .□The following rule is applied when there is no feasible solution to the NLPproblem (10)–(11) at the center point w 0 of the hyper-rectangle X 0 .Heuristic splitting rule 2. (handling infeasibility)If there is no feasible solution of the NLP (10)–(11) at the center point w 0 ofX 0 , split the hyper-rectangle X 0 by a hyperplane through w 0 and orthogonal toan arbitrary axis.□The following rule is used when the NLP problem (14)–(15) in Procedure 2has no feasible solution.Heuristic splitting rule 3. (handling infeasibility)If the NLP problem (14)–(15) in Procedure 2 is infeasible, split the hyperrectangleX 0 by a hyperplane through its center and orthogonal to an arbitraryaxis.□3.5 Approximate Algorithm for Explicit Solution of Mp-NLPsAssume the tolerance ε>0 of the cost function approximation error is given.The following algorithm is proposed to design explicit NMPC controller for constrainednonlinear systems:Algorithm 1. (approximate explicit mp-NLP)1. Initialize the partition to the whole hyper-rectangle, i.e. P = {X}. Markthehyper-rectangle X as unexplored.2. Select any unexplored hyper-rectangle X 0 ∈ P . If no such hyper-rectangleexists, the algorithm terminates successfully.