PWA Control Functions: From the Projection of mpQP Solution and Back to the Convexification by Lifting

This paper focuses on the geometric properties of the Piece-Wise Affine (PWA) feedback function as they appear from the optimal solution of the multi-parameter quadratic programming (mpQP) problem. Such optimization problems are popular formulations, for example, in the design of model-based predictive controllers (MPC) for discrete linear systems subject to input and state constraints. The paper considers such a PWA function as input data and provides a method for reconstructing a feasible convex set and a PWA curve within it, which retrieves the identical structure of the solution in the original parametric feasible set. The proposed method involves establishing and decomposing the topology structure of the polyhedral critical regions, which form the domain of the PWA function by means of a graph of interconnections. The regions are split into the boundary and interior collections using convex-concave lifting. The explicit solution is then merged based on the convex-concave liftings to reconstruct the feasible domain and the PWA curves.