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.

Nanopositioning systems frequently encounter limitations in control bandwidth due to their lightly damped resonance behavior. This paper presents a novel Non-Minimum-Phase Resonant Controller (NRC) aimed at active damping control within dual closed-loop architectures, specifically applied to piezo-actuated nanopositioning systems. The control strategy is structured around formulated objectives for shaping sensitivity functions to meet predetermined system performance criteria. Leveraging non-minimum-phase characteristics, the proposed NRC accomplishes complete damping and the bifurcation of double resonant poles at the primary resonance peak through a constant-gain design accompanied by tunable phase variation. The NRC demonstrates robustness against frequency variations of the resonance arising from load changes and is also capable of damping higher-order flexural modes simultaneously. Furthermore, by establishing high gains at low frequencies within the inner closed-loop and integrating it with a conventional PI tracking controller, the NRC achieves substantial dual closed-loop bandwidths that can exceed the first resonance frequency. Moreover, the NRC significantly diminishes the effect of low-frequency reference signals on real feedback errors while effectively rejecting disturbances proximate to the resonance frequency. All contributions are thoroughly formulated and exemplified mathematically, with the controller’s performance confirmed through an experimental setup utilizing an industrial nanopositioning system. The experimental results indicate dual closed-loop bandwidths of 830 Hz and 755 Hz, characterized by  ± 3 dB and  ± 1 dB bounds, respectively, that surpass the resonance frequency of 710 Hz.

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 paper explores the use of reset control in systems subjected to wide-band disturbances. Such excitation may result in too rare or excessive resetting, leading to deteriorated performance. Moreover, the commonly used Describing Function (DF) approximation for the frequency-domain design of reset systems does not sufficiently represent the reset element’s behavior under such conditions as it is defined for sinusoidal excitation. To address this, we present a design approach based on analyzing the power spectral densities (PSD) of the signals in the system and using the Best Linear Approximations (BLA) of reset elements. In the first step, the dominant components in the PSD of the reset triggering signal are related to the frequency domain properties of the reset element. To benefit from resetting, it should lead to an increase in phase margins near the cross-over frequency. This is the case where the components at the cross-over frequency dominate the reset triggering signal. To ensure this, the use of a bandpass shaping filter is proposed. In the second step, the BLA of the reset element is used to represent its response to the signal with a specific PSD in the frequency domain. This information is used to tune both the reset element and the shaping filter to achieve the desired performance and minimize loss of gain at low frequencies. Closed-loop simulations show the method’s feasibility in achieving the desired behavior of the reset element, leading to improved resonance peak damping in the example studied.

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.

Reset control enhances the performance of high-precision mechatronics systems. This paper introduces a generalized reset feedback control structure that integrates a single reset-state reset controller, a shaping filter for tuning reset actions, and linear compensators arranged in series and parallel configurations with the reset controller. This structure offers greater tuning flexibility to optimize reset control performance. However, frequency-domain analysis for such systems remains underdeveloped. To address this gap, this study makes three key contributions: (1) developing Higher-Order Sinusoidal Input Describing Functions (HOSIDFs) for open-loop reset control systems; (2) deriving HOSIDFs for closed-loop reset control systems and establishing a connection with open-loop analysis; and (3) creating a MATLAB-based App to implement these methods, providing mechatronics engineers with a practical tool for reset control system design and analysis. The accuracy of the proposed methods is validated through simulations and experiments. Finally, the utility of the proposed methods is demonstrated through case studies that analyze and compare the performance of three controllers: a PID controller, a reset controller, and a shaped reset controller on a precision motion stage. Both analytical and experimental results demonstrate that the shaped reset controller provides higher tracking precision while reducing actuation forces, outperforming both the reset and PID controllers. These findings highlight the effectiveness of the proposed frequency-domain methods in analyzing and optimizing the performance of reset-controlled mechatronics systems.

Piezoelectric 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.

Elastic piezoelectric metastructures with actively implemented resonators offer an opportunity for novel vibration attenuation solutions, thanks to the possibility of creating bandgaps at low frequencies, their tuneability and compactness. We focus on metastructures with sensors and actuators, where the resonators are implemented using feedback control techniques, an alternative to commonly used shunt circuits. For bandgap creation in finite structures, unit-cell-based dispersion analysis is unsuitable since it lacks information on modal behaviour. As an alternative, a modal analysis approach can be used to calculate the frequency range of a locally resonant bandgap in closed form using the assumption of an infinite number of transducers of infinitesimal length distributed along the structure. The predictions obtained using this approach are accurate if a sufficiently high number of transducers is used, and the number required increases with the increasing target frequency. Despite the recent developments in the field, it remains to be seen what the sufficient number is in a specific situation. In this paper, we show that for low-frequency bandgaps in cantilevers, the minimal number of transducers is equal to the number of the dominant vibration mode at the targeted range of frequencies. Increasing the number of transducers above this value increases the vibration attenuation in the bandgap region but does not result in its widening. The result is demonstrated using numerical analysis.

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 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 note, we present an extension of the nonlinear negative imaginary (NI) systems theory to reset systems. We define the reset negative imaginary (RNI) and reset strictly negative imaginary (RSNI) systems and provide a state-space characterization of these systems in terms of linear matrix inequalities. Subsequently, we establish the conditions for the internal stability of a positive feedback interconnection of a (strictly) negative imaginary linear time-invariant plant and a reset (strictly) negative imaginary controller. The applicability of the proposed method is demonstrated in a numerical example of a reset version of a positive position feedback (PPF) controller for a plant with resonance.

This paper explores the combination of a Hybrid Reluctance Actuator (HRA) with a Hybrid Tunable Magnet Actuator (HTMA) to realize a high bandwidth actuator that can generate low-frequency forces with greater efficiently. The HTMA allows desired forces to be sustained without continuous coil heating by manipulating the remnant magnetisation of an AlNiCo magnet. This enables the actuator to exert force through two modes of operation: by magnetisation of the Tunable AlNiCo Magnet (TM) or by inducing a proportionally force-dependent field. The second mode may furthermore be used to compensate for unwanted variations in forces during magnetisation. Although FEM analyses provide an understanding of the actuator behaviour in steady states, it is inefficient to integrate transient FEM models with accurate hysteresis models. Hence, firstly, an analytical framework is presented to determine the transient behaviour and the comparative energy efficiency of the two actuator modes. Then, a control strategy is presented for the operation of the combined actuator to track a reluctance force step reference. An experimental setup is designed and tested to validate the concept and control method.

In nanopositioning systems, the control bandwidth is frequently limited due to the presence of lightly damped resonant dynamics. Active Damping Control is typically integrated with tracking control within an inner-loop configuration to mitigate dominant resonant dynamics and enable higher bandwidths. The paper discusses that, in such architectures, feedforward control based on plant dynamics inversion is insufficient to achieve the intended feedforward objectives. In response to this limitation, the study introduces a delay-based input shaping and feedforward framework combined with a dual closed-loop feedback control system that includes active damping. The feedforward filter, derived from partial inner closed-loop dynamics inversion, facilitates precise, delayed tracking of reference signals. This configuration implements a unity-gain shaping filter, effectively reducing tracking feedback errors caused by reference inputs. Furthermore, the study presents a simulated example employing a simplified dynamic model of an industrial nanopositioning system to demonstrate enhancements in closed-loop periodic tracking performance through the proposed feedforward and input-shaping methodology.

This article introduces an output prediction method for a general class of closed-loop reset control systems. The considered type of system consists of a linear time-invariant (LTI) part which is connected in feedback with a reset controller that (partially) resets (a part of) its states when its input is equal to zero. Given some practical assumptions on the reset element’s input signal, the system output can be accurately predicted when the system is subject to a sinusoidal input. One benefit of this approach is that it provides an intuitive frequency-domain representation of the system. Another benefit is that output prediction can be done based solely on a frequency-response function (FRF) of the LTI part of the system. This article also introduces an accurate and computationally efficient algorithm which can – based on the predicted output – compute a closed-loop pseudo-sensitivity. This pseudo-sensitivity represents the ratio between the maximum absolute value of the system’s output and the amplitude of its input, similar to the closed-loop sensitivity functions for LTI systems.

Contactless handling systems for substrates hold significant potential in enhancing chip manufacturing yields by allowing the use of thinner and larger substrates, eliminating the risks associated with physical contact. This article introduces a novel contactless force actuator, employing the active air-bearing working principle, designed with a compact structure to effectively actuate substrates. The actuator features a continuous deformable air-bearing surface composed of compliant-based actuation unit cells, ensuring ease of fabrication to meet tight air-bearing tolerances. A modular design with seven unit cells is designed and manufactured to validate the performance. The results confirm that the proposed contactless actuator can be used to levitate and actuate the substrate simultaneously, in which case the maximum actuation force in the x -axis is determined to be 90 mN and a 42.5- μ m fly height in the z -axis is achieved.

Piezoelectric nanopositioning systems, typically guided by flexure mechanisms, are limited to low closed-loop bandwidths due to their lightly damped resonance behavior. Active damping controllers (ADCs) have been widely employed to suppress the dominant first mode and enable higher bandwidths; however, their performance is severely degraded in the presence of delay, while significant higher-order modes often remain unaddressed, further restricting precision. This paper proposes a simple loop-shaping methodology that incorporates a non-minimum-phase (NMP) filter in series with a linear damping controller to restore sufficient phase margins at open-loop crossover frequencies, thereby mitigating delay-induced degradation in closed-loop damping performance. The methodology is further extended to a parallel damping control structure that enables simultaneous suppression of both the first dominant and higher-order modes. Experimental validation on a piezoelectric nanopositioner demonstrates the effectiveness of the proposed strategy, achieving up to 25.5 dB attenuation of higher-order resonances under significant delay. In conjunction with a standard proportional–integral (PI) motion controller and a non-minimum-phase resonant controller (NRC) targeting the first mode, the overall control architecture extends the closed-loop bandwidth to 435 Hz, surpassing the system’s first resonance frequency, without compromising low-frequency dynamics.

This study presents a shaped reset feedback control strategy to enhance the performance of precision motion systems. The approach utilizes a phase-lead compensator as a shaping filter to tune the phase of reset instants, thereby shaping the nonlinearity in the first-order reset control. The design achieves either an increased phase margin while maintaining gain properties or improved gain without sacrificing phase margin, compared to reset control without the shaping filter. Then, frequency-domain design procedures are provided for both Clegg Integrator (CI)-based and First-Order Reset Element (FORE)-based reset control systems. Finally, the effectiveness of the proposed strategy is demonstrated through two experimental case studies on a precision motion stage. In the first case, the shaped reset control leverages phase-lead benefits to achieve zero overshoot in the transient response. In the second case, the shaped reset control strategy enhances the gain advantages of the previous reset element, resulting in improved steady-state performance, including better tracking precision and disturbance rejection, while reducing overshoot for an improved transient response.

This article introduces output prediction methods for two types of systems containing sinusoidal-input uniformly convergent (SIUC) elements. The first method considers these elements in combination with single-input single-output linear time-invariant (LTI) systems before, after, and in parallel to them. The second method considers a multiple-input multiple-output LTI system where each input is controlled by an SIUC element. The output prediction only requires frequency-response functions of the LTI elements and is fully accurate for sinusoidal inputs.

In this paper, a robust nonlinear control scheme is designed for the motion control of a class of piezo-actuated nano-positioning systems using frequency-domain analysis. The hysteresis, the nonlinearity in the piezoelectric material, degrades the precision in tracking references with high frequency contents and different travel ranges. The hysteresis compensation by the inverse model, as the state-of-the-art solution, is not reliable alone. Therefore, a control framework with robustness against the remaining nonlinearity is needed. It is shown that there is an unavoidable limitation in robust linear control design to improve the performance. A robust control methodology based on a complex-order element is established to relax the limitation. Then, a constant-in-gain-lead-in-phase (CgLp) reset controller is utilized to realize the complex-order control. The control design is based on the sinusoidal input describing function (SIDF) and the higher-order SIDF (HOSIDF) tools. A constrained optimization problem is provided to tune the control parameters. The achieved improvements by the CgLp control is validated by the simulation.