Uniform versions of Finsler´s lemma

Finsler´s lemma and its variants are used extensively in control and optimization literature. Although there are several variants of this lemma, it can be summarized as an equivalence of a quadratic inequality with a linear constraint to a linear matrix inequality (LMI) with possibly more extra variables than the original problem. When applying Finsler´s lemma to problems that depends of a parameter, the extra variables stemming from the use of Finsler´s lemma can also be dependent of this parameter. In this paper it is investigated when these extra variables can assume a simple structure without loss of generality and some sufficient conditions that guarantees this assumption are given. These sufficient conditions can be applied when investigating parameter dependent LMIs that emerge, for instance, when dealing with the robust stability of systems with polytopic uncertainty.