To address the limitations imposed by Bode's gain-phase relationship in linear controllers, a reset-based filter called the Constant in gain- Lead in phase (CgLp) filter has been introduced. This filter consists of a reset element and a linear lead filter. However, the sequencing of these two components has been a topic of debate. Positioning the lead filter before the reset element in the loop leads to noise amplification in the reset signal, whereas placing the lead filter after the reset element results in the magnification of higher-order harmonics. This study introduces a tunable lead CgLp structure in which the lead filter is divided into two segments, enabling a balance between noise reduction and higher-order harmonics mitigation. Additionally, a filtering technique is proposed, employing a target-frequency-based approach to mitigate nonlinearity in reset control systems in the presence of noise. The effectiveness of the proposed methods in reducing nonlinearity is demonstrated through both frequency domain and time-domain analyses using a simulated precision positioning system as a case study.

We introduce a fractional-order generalization of the hybrid integrator-gain system (HIGS) with memory reset of the fractional-order operator when re-enter the integration mode. We compute the describing function for rational orders in terms of Mittag-Leffler functions. The concepts also allow for the evaluation of the higher-order harmonics. For the implementation we represent higher-order approximations by combining first-order reset elements with an integrator. The fractional-order extension without memory reset can also be approximated using the same framework. Finally we show how the approximation affects the describing function.

We consider the fractional-order version of the hybrid integrator-gain system (HIGS) including memory reset. For the implementation an explicit higher-order approximation is considered, which combines first-order reset elements with an integer-order HIGS. This framework can also be used for fractional-order extensions without memory reset. Using passivity theory we present a Circle-Criterion-like condition for the closed-loop stability based on this higher-order approximation.

In this paper, the higher-order sinusoidal-input describing function (HOSIDF) of the fractional-order hybrid integrator-gain system (HIGS) is derived analytically. The HIGS element, designed as a nonlinear component, aims to overcome limitations inherent in linear control, such as the waterbed effect. The HIGS element has been generalized by replacing the integer-order integrator with a fractional one. Here, a modified version of the fractional-order HIGS (FO-HIGS) is introduced with the aim of shaping the nonlinearity at low frequencies. Additionally, this paper demonstrates that obtaining an analytical solution for the HOSIDF of the FO-HIGS enables us to gain better insight into the tuning of the control architecture.