Robust Performance Analysis and Nonlinearity Shaping for Closed-Loop Reset Control Systems

Abstract

Reset elements are nonlinear filters that have shown the ability to improve control performance beyond the limits of linear time-invariant (LTI) systems, but they introduce higher order harmonics that can compromise closed-loop performance. Although frequency-domain tools like describing functions (DFs) and higher order sinusoidal-input DFs (HOSIDFs) analyze reset control systems (RCSs), no direct method yet quantifies the impact of higher order harmonics on the error signal without time-domain simulations. This article introduces a robustness factor, σ₂(ω), which quantifies the increase in the root-mean-square (rms) value of the error signal due to HOSIDFs, enabling RCS to rely solely on first-order DF characteristics while accounting for nonlinear effects. By using this robustness factor, a systematic method for designing pre- and post-filters is developed to ensure a predefined bound on σ₂(ω), thereby limiting the influence of higher order harmonics without altering first-order DF behavior. The proposed framework is validated through a case study on a planar precision positioning stage, demonstrating how the robustness factor guides the reduction of nonlinearities and improves performance predictability.