Zames-Falb multipliers for MIMO nonlinearities

Abstract

In their celebrated 1968 paper on nonlinear stability, Zames and Falb determined a class of multipliers that preserve positivity of monotone SISO nonlinearities. They conjectured that their results might also hold for incrementally positive, norm-bounded MIMO nonlinearities. In this note, we demonstrate that their conjec ture regarding MIMO nonlinearities holds true only if a further restriction is applied. Specifically, we show that it suffices either to restrict the nonlinearity to be the gradient of a convex real-valued function or to restrict the multiplier to be a real-valued function of frequency. Copyright © 2000 John Wiley & Sons, Ltd.