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

Reset elements are nonlinear filters that improve control performance beyond linear time-invariant (LTI) limits but introduce higher-order harmonics that complicate design. Although frequency-domain tools like describing functions (DFs) and higher-order sinusoidal-input describing functions (HOSIDFs) analyze reset control systems (RCS), no direct method yet quantifies the impact of higher-order harmonics on the error signal without time-domain simulations. This paper introduces a robustness factor, $\sigma_2(\omega)$, 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 $\sigma_2(\omega)$, 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.