Fragility of Polyhedral Partition for PWA Control Laws Designed for Constrained Discrete-Time Systems

This paper deals with the fragility margins of discrete-time Piecewise Affine (PWA) closed-loop dynamics. The chosen framework is one of the nominal linear systems in closed-loop with a PWA controller implemented using a Binary Search Tree (BST) mechanism for effective gain selection. Our objective is to preserve the properties of nominal dynamics, particularly the positive invariance under perturbations in the control law representation. The main contribution revolves around defining and constructing two distinct types of fragility margins: the Single Hyperplane Fragility Margin (sHFM) and the Multiple Hyperplane Fragility Margin (mHFM). The sHFM specifically examines the admissible perturbations for a single hyperplane defining the nominal PWA controller's partition. In contrast, the mHFM explores the admissible perturbations for multiple hyperplanes simultaneously, and the perturbed multiple hyperplanes need to adapt special operations to decouple the influence between each hyperplane. This margin exhibits the extent of freedom for perturbations across multiple hyperplanes simultaneously.