K-d tree based approach for point location problem in explicit model predictive control

Explicit Model Predictive Control (EMPC) produces control laws defined over a set of polyhedral regions in the state space, and the online computation of EMPC is to find the corresponding control law according to a given state by searching in a lookup table, called point location problem. This paper presents an approach of constructing a hybrid data structure called constructed k-d tree(CKDT), which combines the k-dimensional tree (k-d tree) with the binary search tree (BST) for point location in such polyhedral sets. To maintain a 'full' and balanced constructed tree the number of affine control laws is used as the basis for choosing the candidate hyperplanes (HPs) during the main construction process of CKDT, thus increasing offline efficiency by reducing the number of candidate HPs requiring computation. This methodology can be applied to the EMPC of high dimensional problems as the k-d tree - a main part of the CKDT approach - has already been successfully used to solve high dimensional problems in the field of computer science and engineering. The method involves a trade-off between memory storage requirement and online efficiency. A complexity analysis of the approach in the runtime and storage requirements is provided. The advantages of the method are supported by two examples in the paper.