Stability of systems with sector nonlinearities: A comparison of various inequalities

A common method of deriving stability conditions of the "multiplier" type involves two steps." 1) the feedback equation is transformed into a new equation containing a pair of positive operators 2) an appropriate multiplier is introduced. Depending on which of these steps is performed first, two different results are obtained, and it has not been clear in the past whether either result is preferable to the other. It is shown here that, under suitable hypotheses, it is preferable to perform step 1) first, i.e., a better stability condition is obtained if the transformation into positive operators is performed before the multiplier is introduced.