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

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.

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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

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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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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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

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.@en

In this paper, a fractional-order extension of a negative position feedback (NPF) controller for active damping is proposed. The design of the controller is motivated by the frequency-domain loop shaping analysis, and the controller dynamics are defined to maintain the high-pass characteristics of an integer-order NPF. The proposed controller provides greater attenuation of a resonance peak of a flexible plant than the integer order equivalent with the same high-frequency gain. The stability and influence of tuning parameters on the behaviour of the proposed controller are analysed. The efficiency and feasibility of the fractional-order controller are demonstrated by implementing it on an experimental setup.@en

This study evaluates three recursive Bayesian input and state estimation algorithms, as introduced in the field of Structural Health Monitoring, for estimating modal contributions for high-tech compliant mechanisms. The aim of estimating modal contributions is the use for active vibration control. High-tech compliant motion stages allow for different sensor configurations, making new and interesting performance evaluations of these filters possible. The algorithms used, namely, the Augmented Kalman Filter (AKF), Dual Kalman Filter (DKF) and Gilijns de Moor Filter (GDF) are implemented on a compliant motion stage for guidance flexure deformation estimation. Our results show the GDF performs overall best, with good estimation performance and real-world tuning capability.

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Metamaterials are artificial structures with properties that are rare or non-existent in nature. These properties are created by the geometry and interconnection of the metamaterial unit cells. In active metamaterials, sensors and actuators are embedded in each unit cell to achieve greater design freedom and tunability of properties after the fabrication. While active metamaterials have been used in vibration control applications, the influence of applied control architectures on damping performance has not been thoroughly studied yet. This paper discusses the relationship between suitable control architectures for increased damping in finite active metamaterials and the number of damped modes. A metamaterial beam consisting of links with measured and actuated joints is considered. Optimal controllers for each of the considered scenarios are designed in the modal domain using linear-quadratic regulator (LQR). We show that, when all modes of a structure should be damped, the optimal solution can be reduced to a decentralised controller. When modes in a smaller range of frequencies are targeted, distributed controllers show better performance. The results are confirmed with experiments.

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Elastic metamaterials incorporating locally resonating unit cells can create bandgap regions with lower vibration transmissibility at longer wavelengths than the lattice size and offer a promising solution for vibration isolation and attenuation. However, when resonators are applied to a finite host structure, not only the bandgap but also additional resonance peaks in its close vicinity are created. Increasing the damping of the resonator, which is a conventional approach for removing the undesired resonance peaks, results in shallowing of the bandgap region. To alleviate this problem, we introduce an elastic metamaterial with resonators of fractional order. We study a one-dimensional structure with lumped elements, which allows us to isolate the underlying phenomena from irrelevant system complexities. Through analysis of a single unit cell, we present the working principle of the metamaterial and the benefits it provides. We then derive the dispersion characteristics of an infinite structure. For a finite metastructure, we demonstrate that the use of fractional-order elements reduces undesired resonances accompanying the bandgap, without sacrificing its depth.

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Active vibration control (AVC) is crucial for the structural integrity, precision, and speed of industrial machines. Despite advancements in nonlinear control techniques, most AVC techniques predominantly employ linear feedback control due to their simplicity and ability to be designed in the frequency domain. In this paper, we introduce a reset-based nonlinear bandpass filter that uses velocity feedback to improve transient damping of vibrating structures. The approach is motivated from an energy-based mechanistic analysis, which incentivizes the use of reset. A novel feature of our approach is that it works for non-ideal, naturally damped systems, and enables control design in the frequency domain, inline with industrial practice. We demonstrate the effectiveness of this new filter by numerical simulations and experimental validation on a single degree-of-freedom flexure stage.@en

In this paper, we introduce a new representation for open-loop reset systems. We show that at steady-state a reset integrator can be modelled as a parallel interconnection of the base-linear system and piece-wise constant nonlinearity. For sinusoidal input signals, this nonlinearity takes a form of a square wave. Subsequently, we show how the behaviour of a general open-loop reset system is related to the nonlinearity of a reset integrator. The proposed approach simplifies the analysis of reset elements in the frequency domain and provides new insights into the behaviour of reset control systems.

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