Efficient point location via subdivision walking with application to explicit MPC

An explicit (or closed-form) solution to Model Predictive Control (MPC) results in a polyhedral subdivision of the state-space when the system and constraints are linear, and the cost is linear or quadratic. Within each region the optimal control law is an affine function of the current state, so the online evaluation is reduced to determining the region containing the current state measurement, known as a point-location or set membership problem. In this paper we present the subdivision walking method, which is based on the idea of travelling from a seed point in a known seeded region, in the direction of the state measurement, by walking from one region to the next until the region of interest is found. The algorithm requires minimal pre-computation, and achieves significant computational savings for many control problems.