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.

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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.@en

Reset controllers have demonstrated their effectiveness in enhancing performance in precision motion systems. To further exploiting the potential of reset controllers, this study introduces a parallel-partial reset control structure. Frequency response analysis is effective for the design and fine-tuning of controllers in industries. However, conducting frequency response analysis for reset control systems poses challenges due to their nonlinearities. We develop frequency response analysis methods for both the open-loop and closed-loop parallel-partial reset systems. Simulation results validate the accuracy of the analysis methods, showcasing precision enhancements exceeding 100% compared to the traditional describing function method. Furthermore, we design a parallel-partial reset controller within the Proportional-Integral-Derivative (PID) control structure for a mass-spring-damper system. The frequency response analysis of the designed system indicates that, while maintaining the same bandwidth and phase margin of the first-order harmonics, the new system exhibits lower magnitudes of higher-order harmonics, compared to the traditional reset system. Moreover, simulation results demonstrate that the new system achieves lower overshoot and quicker settling time compared to both the traditional reset and linear systems.

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The frequency response analysis describes the steady-state responses of a system to sinusoidal inputs at different frequencies, providing control engineers with an effective tool for designing control systems in the frequency domain. However, conducting this analysis for closed-loop reset systems is challenging due to system nonlinearity. This paper addresses this challenge through two key contributions. First, it introduces novel analysis methods for both open-loop and closed-loop reset control systems at steady states. These methods decompose the frequency responses of reset systems into base-linear and nonlinear components. Second, building upon this analysis, the paper develops closed-loop higher-order sinusoidal-input describing functions for reset control systems at steady states. These functions facilitate the analysis of frequency-domain properties, establish a connection between open-loop and closed-loop analysis. The accuracy and effectiveness of the proposed methods are successfully validated through simulations and experiments conducted on a reset Proportional–Integral–Derivative (PID) controlled precision motion system.

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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.

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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.

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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.@en

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.

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In this work, the proportional Clegg integrator (PCI), a resetting proportional-integrator (PI) element, is studied with the aim of improving the performance of an industrial motion stage currently controlled by a linear controller. A novel parallel continuous reset (CR) architecture, based on the PI, is presented, along with frequency-based tuning guidelines, similar to linear time-invariant (LTI) loopshaping techniques. Open-loop higher order sinusoidal input describing functions (HOSIDFs) and pseudo-sensitivities computed through analytically derived approximate closed-loop HOSIDFs were effectively applied to predict steady-state performance. The experimental results, obtained on a wire bonding machine, confirmed that resonance-induced vibrations of the machine's base frame can be suppressed more effectively by adopting a PCI-PID controller compared to the currently used linear controller. The novel structure does not only reduce unwanted excitation of higher order harmonics of the base frame resonance, such as the series CR architecture recently introduced in literature, but also avoids amplification of noise when implemented in practice. With the novel parallel structure, a significant (32%) decrease in the root mean square (rms) of the settling error could be achieved when compared to the linear controller currently used and the series CR reset structure.

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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.@en

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.@en

This paper proposes an effective approach to en-hance bandgap depth in metastructures with high damping ratios, utilizing piezoelectric actuators coupled with notch filters for improved vibration isolation. The active control strategy focuses on dynamically attenuating specific resonant frequencies through the application of notch filters within the actuator control loops. AI algorithm, in particular Reinforcement Learning, is employed to optimize the notch filter parameters, thereby fine-tuning the system's response. Numerical validation reveals that this approach not only maintains system stability but also significantly deepens the bandgap. The results highlight that utilizing notched piezo-actuation achieves a more pronounced bandgap depth in overdamped systems compared to traditional piezo-actuated resonators, marking a substantial advancement in vibration control technologies.

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Metastructures with internally coupled resonators promise enhanced vibration control and energy harvesting capabilities by theoretically enabling multiple bandgaps. This paper investigates the feasibility of these theoretical benefits under practical constraints, particularly the challenge of merging multiple bandgaps in continuous systems. Employing a closed-form analytical approach alongside FEM simulations and experimental validation, the study reveals that while internal coupling can modify bandgap behavior, achieving precise stiffness alignment and bandgap merging remains challenging. The findings indicate that practical applications may not fully realize the predicted advantages and also present more challenges in merging multiple bandgaps created in such metastructures, even for metastructures with advanced manufacturing precision and design optimization. The paper contributes to the understanding of the dynamic behavior of internally coupled metastructures and outlines directions for future research to bridge the gap between theory and application.

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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.@en

This paper examines the stability implications of integrating piezoelectric actuators into metamaterial beams, focusing on the compensation of structural damping and its effect on the system's dynamic performance. Metamaterials, characterized by their unique bandgap properties, offer potential in various engineering applications, including vibration control and energy harvesting. However, structural damping inherent in such systems can degrade these properties, prompting the use of piezoelectric actuators as a compensatory mechanism. Through a distributed parameter model and modal analysis, this study explores the temporal and spatial dynamics of the metamaterial beam and investigates how piezoelectric actuation influences the natural frequencies and mode shapes, with a particular emphasis on stability thresholds. Employing root locus analysis, the paper visualizes the transition of system stability across different levels of actuation voltage, highlighting the delicate balance between enhanced performance and stability. The findings delineate a clear operational voltage range, within which piezoelectric actuation improves bandgap properties without compromising system stability.

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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.

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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.

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This study explores the optimization of bandgap characteristics in locally resonant metastructures through advanced artificial intelligence (AI) and optimization algorithms, focusing on the accurate estimation of resonator damping ratios. By developing a novel mathematical framework for metastructure analysis, this research diverges from traditional methods, offering a more nuanced approach to bandgap manipulation. This research significantly improves metastructure modeling accuracy by precisely estimating resonator and structural damping ratios, enhancing model fidelity crucial for analysis, control strategies, and design optimization. Through a combination of model simulations and experimental validation, the efficacy of the Hybrid Genetic Algorithm-Particle Swarm Optimization (GA-PSO) algorithm is demonstrated, highlighting its potential for practical applications in engineering metastructures. This paper not only provides a robust method for estimating damping ratios but also opens new avenues for future research, including the application of machine learning techniques and the development of intelligent materials. The findings of this study contribute to the foundational understanding necessary for the advancement of mathematical modeling metamaterials, with broad implications for industries where precise vibration control is crucial.

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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.

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Recent studies have shown that tunable magnets (soft permanent magnets) can significantly reduce Joule heating in electromagnetic actuators. To achieve high motion accuracy and repeatability, this article proposes a novel actuator design with a linearized force-flux relation. In prior designs of variable reluctance tunable magnet actuators, the force and flux are related quadratically via a C-shaped actuator. Hybrid tunable magnet actuators based on biased fluxes are developed using lumped parameter models. Using finite element analysis, it is shown that the force-flux relation is symmetric linear around the mid position depending on the magnetic flux direction in the magnet. Within a position range of ±500 μm and a force range of ±20 N, the linear fit produces a negligible error of 0.08 N. Finally, this linear relationship is validated with a 0.03-N error in an experimental setup.

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