On Static O'Shea-Zames-Falb Multipliers for Idempotent Nonlinearities

In this paper, we investigate static O'Shea-ZamesFalb (OZF) multipliers for slope-restricted and idempotent nonlinearities. For the analysis of nonlinear feedback systems, the powerful framework of integral quadratic constraint has been frequently employed, where the core of this framework is capturing the behavior of nonlinearities by multipliers. Among them, OZF multipliers are known to be effective for slope-restricted nonlinearities. However, OZF multipliers only grasp the rough slope properties of the nonlinearities, and hence cannot distinguish nonlinearities with the same slope property. To address this issue, we focus on the fact that some nonlinearities that are important in control engineering satisfy idempotence. By actively using the idempotence property, we first show that we can enlarge (relax) the set of standard OZF multipliers for slope-restricted nonlinearities. In particular, for slope-restricted and idempotent nonlinearities with slope [0, 1], we can provide a clear understanding on how the set of standard OZF multipliers is enlarged. We finally illustrate the effectiveness of the newly proposed multipliers by numerical examples on stability analysis of nonlinear feedback systems. Keywords: idempotent nonlinearities, static O'Shea-ZamesFalb multipliers.