Revisiting the generalized first-order reset element with shaping filters

Reset control provides a nonlinear approach for improving closed-loop performance beyond the limitations of linear time-invariant controllers. However, the reset action inevitably introduces higher-order harmonics, which may degrade tracking performance, distort the reset signal, and reduce the reliability of frequency-domain predictions obtained via describing-function analysis. This paper revisits the generalized first-order reset element with shaping filters and develops a systematic framework for suppressing undesired reset-induced higher-order harmonics. Analytical conditions are derived for the shaping-filter coefficients to increase the asymptotic attenuation slope of the magnitudes of the higher-order sinusoidal-input describing functions (HOSIDFs), | H n ( ω )|, as the fundamental excitation frequency ω approaches zero. By modifying the asymptotic attenuation behavior of these higher-order harmonics (which are generated throughout the considered excitation-frequency range), the proposed design provides stronger harmonic suppression in frequency regions where reset action is undesired, while preserving the beneficial first-order harmonic phase advantage near the desired cross-over frequency. The reduction in nonlinear behavior is verified through HOSIDF analysis and a superposition-law test, demonstrating that higher-order shaping filters make the reset element behave more closely to a linear system at a certain range of frequencies. Experimental validation on an industrial motion stage demonstrates improved tracking performance, reduced higher-order harmonic content, and selective activation of the reset action in the intended frequency region.