On the lifting problems and their connections with piecewise affine control law design

Lifting as geometric operation can be defined as a pseudo-inverse of orthogonal projection. It has received attention in different fields and applications (mechanics, geometry, control, etc). Numerous studies have been dedicated to the existence conditions of a convex lifting in a higher dimension for a given cell complex. It is worth noting that this notion can be extended for a polyhedral partition for applications in control theory, and it is recently shown to be the key step in solving the inverse parametric linear/quadratic programming problems. The present paper presents in a succinct manner the main elements of this topological problem with specific attention to the case of polyhedral partitions and their liftings. Furthermore, we are interested from the practical point of view, in the use of these concepts in control system design. Practically, a construction for the Voronoi diagram class is presented. Secondly, a methodological result is presented which leads to the modification of partitions guaranteeing a theoretically liftable result. In addition, a generic constructive procedure for the partitions based on convexity, continuity and linear (or quadratic) programming is proposed. In order to bring the discussion closer to the control related formulations, the correspondence between convex liftings of a given partition in ℝ^d onto (d+1)-space and n-space with n > d+1 is provided. Finally an analysis with respect to predictive control related problems will conclude our contribution.