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.

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.

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.

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.

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.

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.

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.

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.

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.