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.

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.

In precision positioning systems, lightly damped higher-order resonance modes can induce undesirable vibrations that degrade system performance and accuracy. These resonances pose additional challenges in non-collocated dual-stage positioning systems, where they significantly limit control bandwidth. Although conventional notch filters are commonly used alongside tracking controllers to enhance bandwidth, they lack robustness when faced with system parameter uncertainties. Moreover, the effects of the delimiting resonance on disturbance rejection remain. Active damping control has been successfully used to mitigate issues related to the primary resonance mode, but its application to higher-order modes has not been explored. This paper introduces a novel control strategy, High-Pass Positive Position Feedback (HP-PPF), inspired by existing methods but designed specifically for active damping of higher-order, non-collocated modes in positioning systems. The proposed method incorporates a second-order high-pass filter within a positive feedback loop, effectively attenuating the delimiting resonance. Integrated with a PID tracking controller in a dual-loop configuration, this method enhances disturbance rejection and robustness against model uncertainties, overcoming limitations of traditional notch filter-based methods while achieving comparable bandwidth improvements. The proposed control architecture is validated through a proof-of-concept experimental setup that demonstrates the effectiveness of the underlying mathematical framework.

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.

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.

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.

Current reset elements mainly rely on the traditional zero-crossing resetting mechanism. This study introduces a reset element with a new resetting mechanism that distributes multiple resets within a single period for reset controllers with sinusoidal reference inputs. This new control element is termed "Fixed-Phase Reset Control (FPRC)". A higher-order sinusoidal input describing function is developed to analyze the frequency-domain properties of the new controller. The accuracy of this frequency-domain analytical approach is validated through simulations on three systems. Through the analysis, the new FPRC demonstrates phase lead compared to zero-crossing reset control, but it introduces nonlinearities at low frequencies.

Incorporating actively implemented resonators within elastic piezoelectric metastructures presents a unique approach for vibration attenuation, enabling the creation of tuneable low-frequency bandgaps. Through feedback control, we enhance the compactness of these metastructures by integrating resonator dynamics internally. We study the influence of varying the cross-section of the base substrate and the arrangement of transducers on bandgap generation. This influence is captured by the changes in the electromechanical coupling and stiffness of the metastructure, which appear directly in the formulas for bandgap edge frequencies in ideal conditions. This relationship is illustrated with numerical examples for realistic metastructures with a finite number of transducers. Our focus is on metastructures with sensors and actuators, employing feedback control techniques for resonator implementation as an alternative to shunt circuits. When a bandgap is generated in a finite metastructure, its edge frequencies can be calculated in closed form using the assumption of an infinite number of transducers of infinitesimal length distributed along the structure.

Bandgaps—frequency ranges with reduced vibration transmissibility in elastic structures, are an opportunity for vibration control originating from the research on elastic metamaterials. In this paper, we study the design for bandgap in slender beams with collocated piezoelectric patch transducers. While creating bandgaps using shunted transducers is a well-established research field, using structures with piezoelectric sensors, actuators, and feedback controllers for the same application has not been thoroughly explored. This paper aims to study the use of the tools originating from the active vibration control (AVC) field for bandgap generation in finite beams with collocated piezoelectric sensors and actuators. Lightly damped second-order low-pass filters are used as controllers in the same configuration as positive position feedback, widely used for active damping. To facilitate the understanding of systems behaviour, we propose a simplified model based on the Euler–Bernoulli beam theory. A modal analysis approach and an assumption of an infinite number of transducers of infinitesimal length distributed along the structure are used to predict the frequency range of the locally resonant bandgap in closed form. The experimental part of the work demonstrates the feasibility of the proposed approach for creating bandgaps in practice. Thanks to the insights from AVC, the control system can be designed purely based on experimental frequency response data without the need for a parametric model of the system. We also show that the uniform distribution of actuators is not necessary for creating bandgap, which can be achieved in a structure with a relatively small number of sparsely placed actuators and compare the obtained results with analytical predictions for ideal metastructure. Low-frequency bandgaps placed between 10 and 320 Hz are obtained in experiments.

This study delves into the nonlinear dynamics of metamaterials, exploring the dual objective of enhancing power output and achieving vibration suppression through piezoelectric energy harvesters (PEHs). Our approach is structured into a sequence of increasingly complex models that bridge mechanical resonators with their electromechanical counterparts. We initiate with (1) modeling mechanical resonators, incorporating nonlinear behaviors that are often overlooked in the linear domain. This lays the groundwork for understanding the fundamental mechanisms of vibration within metamaterials. Subsequently, we progress to (2) electromechanical resonators, where piezoelectric components are integrated, revealing a richer dynamic landscape that is influenced by the interplay of mechanical and electrical energies. The latter sections of our investigation introduce and examine (3) mechanical and (4) electromechanical internally coupled resonators. These segments unveil the role of internal couplings in steering the metamaterial’s energy harvesting capabilities and its resilience to vibrational disturbances. Through meticulous simulations and analysis, the research brings to light the significant influence of specific PEH nonlinear parameters on the system’s efficiency, offering insights for the optimization of PEHs in practical applications.

In this article, the problem of the optimal reset control design for Lipschitz nonlinear systems is addressed. The reset controller includes a base linear controller and a reset law that enforces resets to the controller states. The reset law design is strongly dependent on the appropriate design of the base controller. For this reason, in this article, the base controller and reset law are simultaneously designed. More precisely, an optimal dynamic output feedback is considered as the base controller which minimizes the upper bound of a quadratic performance index, and a reset law is used to improve the transient response of the closed-loop system. This design is done in a full offline procedure. The problem is transformed into a set of linear matrix inequalities (LMIs), and the reset controller is obtained by solving an offline LMI optimization problem. Finally, two examples are presented to illustrate the effectiveness and validity of the proposed method.

Metamaterials have marked notable advancements in vibration damping and energy harvesting. However, the specific impact of internal coupled resonators, encompassing both linear and nonlinear types, has received limited attention from researchers. This study aims to delve into this underexplored area. Utilizing a distributed parameter model grounded in modal analysis, our research investigates the effects of these coupled resonators on metamaterial functionality. We particularly emphasize the influence of varying the position of attached masses and its consequent impact on the bandgap properties. Through developing theoretical and mathematical models for metastructural beams with internally coupled resonators, our approach facilitates future simulations and analyses. This investigation not only provides pivotal insights for the design and optimization of metamaterials but also underscores the possible potential of manipulating resonator properties to broaden their practical applications.

Reset controllers have demonstrated their efficacy in enhancing transient responses, such as the overshoot and response time in motion control systems. Designing these systems to meet specific transient requirements requires a method for analyzing transient responses. However, the inherent nonlinearity of reset control systems presents challenges in this regard, limiting their widespread application. This study introduces a novel method for analyzing the step responses of closed-loop reset control systems. By decomposing the step response of the reset system into piece-wise functions, with each piece-wise function computed based on linear systems, this analysis method offers new insights into understanding reset systems. Experimental validation conducted on eleven reset Proportional-Integral-Derivative (PID) control systems implemented on a precision motion stage confirms the effectiveness of the proposed method. The experimental results also underscore the applicability of the method as a tool for selecting optimized parameters and reset control structures to achieve enhanced transient responses.

When a linear controller is replaced by a reset controller, it is possible to keep the gain behaviour essentially the same, while improving the phase behaviour. However, because reset control is nonlinear, higher order harmonics appear, which may deteriorate the results. In this paper, reset controllers are combined with fractional derivatives, decreasing the magnitude of higher order harmonics. In this way, better results are found in simulations of a Clegg integrator, of a FORE (first order reset element), and of a CgLp (constant in gain lead in phase) controller. The reason why better results are found is explained.

The dynamics of metastructures, incorporating both conventional and internally coupled resonators, are investigated to enhance vibration suppression capabilities through a novel mathematical framework. A close-form formulation and a transfer function methodology are introduced, integrating control system theory with metastructure analysis, offering new insights into the role of internal coupling. The findings reveal that precise internal coupling, when matched exactly to the stiffness of the resonator, enables the clear formation of secondary bandgaps, significantly influencing the vibration isolation efficacy of the metastructure. Although the study primarily focuses on theoretical and numerical analyses, the implications of adjusting mass distribution on resonators are also explored. This formulation methodology enables the adjustment of bandgap characteristics, underscoring the potential for adaptive control over bandgaps in metastructures. Such capabilities are crucial for tailoring the vibration isolation and energy harvesting functionalities in mechanically resonant systems, especially when applied to demanding heavy-duty applications.

This work proposes a novel nonlinear Proportional-Integral (PI) controller, which utilizes a generalized first-order reset element. The proposed element can achieve similar magnitude-characteristics as its linear counterpart but with less phase lag at the open-loop crossover frequency (i.e. the control bandwidth), according to a sinusoidal-input describing function (SIDF) analysis. The same can be achieved with an existing reset-based integrator, the Clegg integrator (CI). However, it is known that a Proportional-CI (PCI) element can excessively generate higher-order harmonics of its input, which are neglected in the SIDF-analysis. Furthermore, a PCI can cause a limit cycle when placed in closed-loop with certain types of plants. The novel PI controller proposed in this work can prevent the limit cycle and can reduce the generation of higher-order harmonics, while retaining the beneficial phase advantage that is associated with existing reset-based PI controllers.

In non-collocated compliant positioning systems, the parasitic resonance peak induces undesirable vibrations, limiting control bandwidth. Despite conventional notch filters being employed alongside PID controllers for improving bandwidth, parasitic resonance effects persist in disturbance rejection. This paper introduces an overactuation-based solution, utilizing additional actuators for active damping control to enhance closed-loop disturbance rejection within a PID-based control architecture. Integrating distributed piezoelectric bender actuator sensor pairs in a collocated configuration further improves damping. A formulated mathematical framework substantiates the benefits, validated by an experimental setup serving as a proof of concept. The proposed solution effectively suppresses parasitic resonance, enhances end-effector disturbance rejection, and achieves higher control bandwidth in the positioning system.

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.