This paper proposes a novel discrete-time (DT) implementation of the generalized Clegg integrator (GCI), which is an integrator that resets its state to a fraction of the original state when its input is equal to zero. The implementation is derived by discretizing a continuous-time (CT) GCI using the Tustin discretization method. By means of a numerical validation it is shown that the state of the DT GCI is identical to its CT counterpart when both are subject to an input which is linearly interpolated between samples, as expected when using this discretization method. For a general CT input which is not linearly interpolated between samples, a numerical comparison is made between the state of the novel DT GCI and the CT GCI. At samples with linear behaviour, the state mismatch is equivalent to the one observed between their linear counterparts. At samples with resetting behaviour, the mismatch even reduces compared to previous samples, as a consequence of (partially) resetting the state mismatch.

Loop-shaping is widely used in precision motion control, but conventional approaches — focused on phase margin and open-loop gain — are inadequate for piezo positioning systems where open-loop phase critically affects performance. This paper proposes generalized loop-shaping guidelines tailored for nonlinear piezo-actuated stages. A constant-in-gain lead-in-phase reset controller is developed to implement the guidelines by overcoming waterbed effect in linear control. An intuitive methodology for shaping filter design is presented to ensure reliable reset control implementation. Using (higher-order) sinusoidal input describing functions, nonlinear motion control is designed. Experiments demonstrate closed-loop bandwidth flatness (±[jls-end-space/]1 dB) and enhanced sensitivity function.

Reset control systems (RCSs) can achieve performance beyond that of conventional linear time-invariant (LTI) controllers, while also allowing analysis directly in the frequency domain using measured frequency response functions (FRFs). Despite this potential, existing frequency-domain stability approaches are typically restricted to specific RCS architectures and commonly depend on parametric plant models, which limits their applicability in practice. In this paper, a generalized H_β framework is developed for the most comprehensive class of RCS structures, incorporating pre-, post-, and parallel LTI filters, as well as nonzero after-reset values. Based on this formulation, an FRF-based representation corresponding to the H_β transfer function is derived, and frequency-domain sufficient conditions are established to certify the H_β-based quadratic stability criterion. As a result, the proposed framework enables direct FRF-based assessment of quadratic stability and convergence for the considered class of reset control systems, using the measured plant FRF together with the known controller and filter transfer functions, without requiring an explicit parametric plant model. The effectiveness and practical relevance of the method are demonstrated through an illustrative industrial case study.

AbstractPiezoelectric nanopositioning systems exhibit low damping and resonance modes that are highly sensitive to loading conditions, resulting in performance degradation under payload variations. Conventional damping and robust control methods typically address these challenges separately, overlooking the coupling between damping and tracking dynamics as well as the influence of higher-order resonant modes. This paper proposes a dual-loop control framework that integrates active damping with mixed-sensitivity H∞ synthesis to achieve robust reference tracking and disturbance rejection under large resonance frequency variations. A Non-Minimum-Phase Resonant Controller (NRC) is implemented in the inner loop to suppress the dominant resonance and reduce system uncertainty. Generalized plant formulation and systematic weighting design guidelines of arbitrary order are developed to explicitly incorporate higher-order modes in the outer loop H∞ synthesis. The proposed approach is validated through simulations and experiments on an industrial piezoelectric nanopositioning system, demonstrating improved robustness and precision across the full payload range.

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.

The sinusoidal input describing function (SIDF) is a powerful tool for control system analysis and design, with its reliability directly impacting the performance of the designed control systems. This study improves both the accuracy of SIDF analysis and the performance of closed-loop reset feedback systems through two main contributions. First, it introduces a method to identify frequency ranges where SIDF analysis becomes inaccurate. Second, these identified ranges correlate with dominated high-order harmonics that can degrade system performance. To address this, a shaped reset control strategy is proposed, incorporating a shaping filter that tunes reset actions to suppress these harmonics. A frequency-domain design procedure for the shaped reset control system is then demonstrated in a case study, where a proportional–integral–derivative (PID)-based shaping filter effectively reduces high-order harmonics and eliminates limit cycles issues under step inputs. Finally, simulations and experiments on a precision motion stage validate the shaped reset control, confirming improved SIDF analysis accuracy, enhanced steady-state performance over linear and reset controllers, and the elimination of limit cycles under step inputs.

This study introduces a modified version of the constant-in-gain, lead-in-phase (CgLp) filter, which incorporates a feedthrough term in the first-order reset element (FORE) to reduce the undesirable nonlinearities and achieve an almost constant gain across all frequencies. A backward calculation approach is proposed to derive the additional parameter introduced by the feedthrough term, enabling designers to easily tune the filter to generate the required phase. This article also presents an add-on filter structure that can enhance the performance of an existing LTI controller without altering its robustness margins. A sensitivity improvement indicator is proposed to guide the tuning process, enabling designers to visualize the improvements in closed-loop performance. The proposed methodology is demonstrated through a case study of an industrial wire bonder machine, showcasing its effectiveness in addressing low-frequency vibrations and improving overall control performance.

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.