The Minimum Gain Lemma

The question of how to establish closed-loop stability for systems involving unstable plants remains a difficult problem in control engineering. Although the notion of minimum gain and the corresponding Large Gain Theorem are designed specifically to tackle this issue, they are not yet widely used. This paper aims to facilitate the practical application of minimum gain and the Large Gain Theorem. An altered definition of minimum gain broadens the applicability of the Large Gain Theorem and the novel Minimum Gain Lemma provides linear matrix inequality conditions that imply and are often equivalent to a minimum gain for unstable linear time invariant systems. Both results play crucial roles in establishing a stability result that is robust with respect to reasonable changes in parameters for a numerical example involving an unstable plant.