On the Exactness of a Stability Test for Discrete-Time Lur'e Systems With Slope-Restricted Nonlinearities

In this note it is shown that the famous multiplier absolute stability test of R. O'Shea, G. Zames, and P. Falb is necessary and sufficient if the set of Lur'e interconnections is lifted to a Kronecker structure. An explicit method to construct the destabilizing static nonlinearity is presented.