Evaluation of piecewise affine control via binary search tree

Abstract

We present an algorithm for generating a binary search tree that allows efficient evaluation of piecewise affine (PWA) functions defined on a polyhedral partitioning. This is useful for PWA control approaches, such as explicit model predictive control (MPC), as it allows the controller to be implemented online with small computational effort. The computation time is logarithmic in the number of regions in the PWA function.

Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra

“…An alternative approach, which will be used here for comparison, was proposed by Tøndel et al in [30], where a binary search tree is constructed on the basis of the geometric structure of the polyhedral partition 2 by utilizing the facets of the regions as separating hyperplanes to divide the polyhedral partition at each tree level. This however, can lead to a worst case combinatorial number of subdivisions of existing regions and therefore to an additional increase in the number of regions to be considered during the search procedure.…”

“…In comparison, the binary search tree of [30] for this case consists of 815 unique hyperplanes. For each such hyperplane 2N P LPs must be solved in the preprocessing phase to compute the index set which corresponds to 4 185 840 linear programs.…”

“…Consider the following piecewise affine system from [25] 7.9 · 10 6 linear programs which are needed to construct the binary search tree of [30] (this does not correspond to the worst case scenario). Since the cost function of a given CFTOC solution is not necessarily convex, one cannot use the method of [6] to perform the local search, and therefore one must perform a sequential search as outlined in Section VI-B on possible candidates.…”

“…Due to the high number of regions, the binary search tree of [30] was not applicable to this example (denoted by in Table III), since it would require the solution of 3.5·10 9 LPs already in the preprocessing stage to determine the index set before building the binary search tree. In contrast, in the preprocessing stage, the here proposed algorithm has to solve 228 860 LPs to obtain the bounding boxes for all regions.…”

“…This enables the algorithm to work for controller partitions with a large number of regions, which is demonstrated on numerical examples. In order to show its efficiency, the algorithm is compared with the procedure proposed in [30] where a binary search tree is pre-computed over the controller state space partition.…”