Parameterised ellipsoid and its applications to S-lemma and control of uncertain, time-varying linear systems

The celebrated S-lemma was originally proposed to ensure the inclusion of a quadratic inequality within the intersection of multiple quadratic inequalities. This paper addresses a natural extension: can the S-lemma be applied to relate a quadratic inequality to the convex hull of quadratic inequalities, which is considered as the dual to their intersection. We provide an affirmative answer by introducing a novel S-lemma that extends the classical result to the convex hull setting. This extension is achieved through a new parameterisation of the convex hull, leading to conditions that are both necessary and sufficient. We further demonstrate the applicability of the new result to key problems in control theory, including stability analysis, domain of attraction estimation, and more importantly, control design for uncertain, time-varying linear discrete-time systems, with or without constraints. We show that our approach yields less conservative results compared to those based on quadratic Lyapunov functions.