Multipliers for forced Lurye systems with slope-restricted nonlinearities

TL;DR

This paper explores the use of dynamic multipliers to ensure stability and bounded power gain in Lurye systems with slope-restricted nonlinearities, extending their guarantees to include finite incremental gain and noise sensitivity.

Contribution

It demonstrates that multipliers can guarantee bounded closed-loop power gain and introduces methods to derive multipliers for stability analysis, revisiting classical tools and assumptions.

Findings

Multipliers guarantee bounded closed-loop power gain.

Dynamic multipliers can be derived using classical tools.

Phase limitations of multipliers are inherited from discrete-time counterparts.

Abstract

Dynamic multipliers can be used to guarantee the stability of Lurye systems with slope-restricted nonlinearities, but give no guarantee that the closed-loop system has finite incremental gain. We show that multipliers guarantee the closed-loop power gain to be bounded and quantifiable. Power may be measured about an appropriate steady state bias term, provided the multiplier does not require the nonlinearity to be odd. Hence dynamic multipliers can be used to guarantee such Lurye systems have low sensitivity to noise, provided other exogenous signals have constant steady state. For periodic excitation, the closed-loop response can apparently have a subharmonic or chaotic response. We revisit a class of multipliers that can guarantee a unique, attractive and period-preserving solution. We show the multipliers can be derived using classical tools and reconsider assumptions required for their application. Their phase limitations are inherited from those of discrete-time multipliers. The multipliers cannot be used at all frequencies unless the circle criterion can also be applied; this is consistent with known results about dynamic multipliers and incremental stability.