Response Of Linear Systems Research Articles

Kurtosis Control of Amplitude-Modulated non-Gaussian Signals for Fatigue Test

The amplitude modulation method was used to generate non-Gaussian signals that acted as excitation for fatigue tests. The fatigue life of structures under non-Gaussian excitation has been proven to be closely related to the features of the amplitude modulation signal (AMS) and kurtosis of the structural response. In this study, the modelling of the AMS by Beta and Weibull distributions and the resulting kurtosis range problem is first reviewed. To solve this problem, a new method for creating an AMS based on a linear combination of Beta and Weibull distributions is proposed. To ensure that the high kurtosis of the amplitude-modulated non-Gaussian signal is correctly transferred to the structural response, the method is further developed to fulfil the specifications for the fatigue damage spectrum (FDS) by controlling the spectral content of the AMS. Herein, a Gaussian AMS with a low-pass cutoff frequency is first generated and then converted to a Weibull or Beta AMS based on the cumulative distribution function (CDF) transformation. The proposed method is verified using simulated and field-measured data. The results show that the full range of specified kurtosis is achieved with the new AMS modelling method. The high kurtosis of the non-Gaussian input signal can be transferred to the linear system response if the mean value of AMS during the period of the system impulse response is the same as AMS.

Minimal Arias intensity modification of ground motions to achieve extreme structural response

AbstractThis paper presents a methodology to minimally modify a ground motion time history to induce collapse in nonlinear single‐degree‐of‐freedom systems (SDOF). The metric used to characterize the modification is the Arias intensity. The proposed procedure is a heuristic extension of a closed‐form solution derived to achieve a target maximum response in linear systems. The methodology is presented as a potential alternative to incremental dynamic analysis (IDA) widely used in earthquake engineering.

A Causality-DeepONet for Causal Responses of Linear Dynamical Systems

A Causality-DeepONet for Causal Responses of Linear Dynamical Systems

Hybrid control for non-overshooting step responses of linear systems

This paper focuses on the problem of applying hybrid control to overcome overshoot performance limitations of linear time-invariant (LTI) systems, and puts forward a systematic method for the joint design of a hybrid controller. By modeling the whole closed-loop system in the framework of a hybrid system, an input-to-state stable (ISS) criterion for the system is established, and on the basis of this, conditions for non-overshooting step responses are obtained. In the proposed hybrid controller, apart from its input and output information, state information of the LTI plant and a basic LTI controller is used to design flow and jump sets. The advantage is that when the trigger matrix of flow and jump sets is unknown, it can be designed conveniently. Based on the conditions, the hybrid controller synthesis is formulated in linear matrix inequalities and the algorithm of solving controller matrices and the triggering matrix is given. In addition, the design method of the hybrid controller with a certain triggering matrix is derived. The effectiveness and reliability of the obtained results are demonstrated through a numerical example.

Gain-compensated metal cavity modes and a million-fold improvement of Purcell factors

Using a rigorous mode theory for gain-compensated plasmonic dimers, we demonstrate how quality factors and Purcell factors can be dramatically increased, improving the quality factors from 10 to over 26,000 and the peak Purcell factors from approximately 3000 to over 10 billion. Full three-dimensional calculations are presented for gold dimers in a finite-size gain medium, which allows one to easily surpass fundamental Purcell factor limits of lossy media. Within a regime of linear system response, we show how the Purcell factors are modified by the contributions from the projected local density of states as well as a non-local gain. Further, we show that the effective mode volume and radiative beta factors remain relatively constant, despite the significant enhancement of the Purcell factors.

A structure-preserving machine learning framework for accurate prediction of structural dynamics for systems with isolated nonlinearities

The nonlinearities present in structural systems are often found in isolated regions within the structure, such as those containing joints or interfaces. However, despite the localized nature of these nonlinearities their presence serves to couple together the modes of the underlying linear system and significantly complicate the development of appropriate reduced-order models; the localized nonlinearities have a global effect on the dynamics of the system. Further, in the presence of evolving structural health the nonlinearities can arise from accumulating damage, with dynamics distinct from those observed in the healthy state. The present work develops a data-driven formulation to identify and include the contributions of the isolated nonlinearities on the dynamics of the underlying linear structure. A novel coordinate separation is developed that decomposes those nonlinearities restricted to the isolated subdomain from the known linear system defined over the entire domain, and the influence of the isolated nonlinearities is reintroduced as an appropriately identified traction at the boundary of the isolated subdomain, referred to as the deviatoric force. In the region exterior to the nonlinear subdomain the response of the ideal linear system recovers that of the original nonlinear system. In this work, the deviatoric force component is predicted using a structure-preserving multilayer perceptron, based only on measured responses at the boundary of the isolated subdomain. Therefore introduction of the perceptron is able to bypass the direct numerical simulation of the nonlinearities within the isolated subdomain. This approach is illustrated through a simple structural system in which an interior region contains cubic nonlinearities and hysteretic damping. Once trained, the machine learning system is able to accurately predict the deviatoric force so that the ideal system recovers the response of the original system in the region outside the isolated nonlinear subdomain. Moreover, the data-driven approach is able to accurately predict the response when the system is subject to differing initial conditions and external excitation without the need for retraining, so that the proposed approach provides a robust description of the structural dynamics of the overall system.

Analytical Nonstationary Response of Linear Stochastic MDOF Systems Endowed with Half-Order Fractional Derivative Elements

Analytical Nonstationary Response of Linear Stochastic MDOF Systems Endowed with Half-Order Fractional Derivative Elements

The causal α-exponential and the solution of fractional linear time-invariant systems

Closed-form expressions for the impulse and step responses of commensurable linear time-invariant systems are deduced and exemplified. The algorithm is based on obtaining the solution in terms of the α-exponential monomials that generate a vector space containing the solutions of such systems. Several examples are considered, together with the numerical aspects involved in calculating the related series, which show the accuracy and effectiveness of the approach.

Wave-number domain analysis for determining the response of linear space-invariant time-varying systems

Wave-number domain analysis for determining the response of linear space-invariant time-varying systems

The linear response of stellar systems does not diverge at marginal stability

ABSTRACT The linear response of a stellar system’s gravitational potential to a perturbing mass comprises two distinct contributions. Most famously, the system will respond by forming a polarization ‘wake’ around the perturber. At the same time, the perturber may also excite one or more ‘Landau modes’, i.e. coherent oscillations of the entire stellar system which are either stable or unstable depending on the system parameters. The amplitude of the first (wake) contribution is known to diverge as a system approaches marginal stability. In this paper, we consider the linear response of a homogeneous stellar system to a point mass moving on a straight line orbit. We prove analytically that the divergence of the wake response is in fact cancelled by a corresponding divergence in the Landau mode response, rendering the total response finite. We demonstrate this cancellation explicitly for a box of stars with Maxwellian velocity distribution. Our results imply that polarization wakes may be much less efficient drivers of secular evolution than previously thought. More generally, any prior calculation that accounted for wakes but ignored modes – such as those based on the Balescu-Lenard equation – may need to be revised.

Duration and damping for ground motion characterization in frequency‐domain damage estimation of SDOF structures

AbstractIn estimating the damage to a single‐degree‐of‐freedom (SDOF) structure due to multiple earthquake events, it may be convenient to use a frequency‐based approach, in which the anticipated ground motions are characterized through power spectral density functions (PSDFs) compatible with the specified design spectra for those motions. Since a spectrum‐compatible PSDF corresponds to a fictitious stationary ground motion process of a specified duration, the duration of this process should preferably be same as the length of the stationary segment in the actual ground acceleration process for a meaningful computation of the stochastic response. On assuming the strong‐motion segment of a given ground motion to represent its stationary segment, the nonstationarity of the strong‐motion segment is evaluated for the existing strong‐motion duration definitions. It is found that the Trifunac‐Brady definition is the most reasonable strong‐motion duration definition in respect of consistently giving the longest strong‐motion segment with least nonstationarity. Further, a spectrum‐compatible PSDF can be used for estimating the stochastic response of a linear system, provided the system has the same damping ratio as that of the design spectrum from which the PSDF has been computed. This causes errors in estimating the peak response of an equivalent linear system due to its damping ratio being higher than the initial damping ratio of the nonlinear system. To address this, an empirical expression is developed for the modified damping ratio of a linear system, which would lead to the correct peak response of the system on using the PSDF obtained from the 5%‐damping design spectrum.

Linear optical response of a quantum dot ‒ metal nanoshell hybrid system

Linear optical response of a quantum dot ‒ metal nanoshell hybrid system

A Hybrid Algorithm for Adaptive Neuro-controllers

In this study, a novel hybrid algorithm consisting of the least mean square and backpropagation neural network is proposed to auto-adjust adaptive proportional integral derivative (PID) controller gains for improving the transient response of linear systems. The hybrid approach comprises the scheme of the two algorithms running in parallel and updates PID gains simultaneously. All algorithms are implemented on the same linear system and present a general framework for different scenarios such as initial PID gains, learning rates, and target functions. The results show that the presented hybrid algorithm has better accuracy, precision, F1-score, adaptability, and robustness than origin algorithms, and significantly improves the controllability in most of the system scenarios. It also exhibits better performance in periodic incremental and decremental targets compared to origin algorithms. Different hybridization levels are also simulated and are highlighted as significant features of their performance. This work can be expanded to the combination of other well-known algorithms, paving the way to significant improvements in control system applications.

Study on Urban Proximity Prediction Based on Doppler Radar Gust Front Characteristics and Urban Microclimate Characteristics

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Controllability and Observability Criteria for Systems Described by Fractional Differential Equations

Based on the state response of fractional order singular linear systems with impulses, the sufficient and necessary conditions for complete controllability and observability of fast subsystems are studied and given, and the criteria for complete controllability and observability of fast subsystems are further established. These assumptions are too strong to synthesize the controllability of slow subsystems and fast subsystems. The method proposed in this paper does not need these assumptions, The approximate controllability of Hilfer fractional order integro differential equations is studied by using the order method. The controllability and observability criteria of the system described by fractional order differential equations are derived. When the rank of its controllability discrimination matrix M and observability discrimination matrix N is full, the fractional order system is controllable and observable.

Hardware in the loop simulation and implementation of a dragonfly-like MAV using clap and fling mechanism

The main aim of this paper is to model and simulate flight dynamics and to design a controller for a dragonfly-like micro aerial vehicle (MAV). This MAV has flapping wings using a clap and fling mechanism with a rigid abdomen. Kane’s method is applied to derive the longitudinal motion equations of the MAV as a multi-body system. Then, the aerodynamic lift and drag forces of the MAV are obtained. These equations are substituted in the derived motion equations. Furthermore, a new structure for a dragonfly-like flapping wing MAV is presented in which abdomen motion is considered in the longitudinal mode. In this work, the use of the abdomen is also formulated. A change in the motor’s speed generates differential thrust that creates a pitch moment as a control input. State space linearized equations of motion are presented by using appropriate assumptions in the aerodynamics, and dynamic nonlinear equations. To validate the linearized model, the response of the open-loop linear system is compared with the nonlinear one. In addition, the positive role of the abdomen in the open loop is discussed and analyzed. Finally, an LQR controller is designed for the pitch mode which is robust against disturbances. The validity, effectiveness, and performance of the derived equations and designed autopilot are shown by implemented hardware in the loop testbed using a fixed abdomen. Results demonstrate a good agreement between theoretical and experimental responses with accurate hovering at the desired angle.

LMI-Based Determination of the Peak of the Response of Structured Polytopic Linear Systems

This paper addresses the problem of determining the peak of the response to a linear time-invariant (LTI) signal of a linear system whose system matrices are rational functions of an uncertainty vector constrained into a convex bounded polytope. The uncertainty can be time-invariant, bounded-rate time-varying or arbitrarily time-varying. A novel approach based on linear matrix inequalities (LMIs) is proposed for obtaining upper bounds of the sought peak based on the construction of a structured polynomial Lyapunov function in the state and in the uncertainty. A priori and a posteriori conditions for establishing optimality of the obtained upper bounds are also provided. As shown by some numerical examples, which includes the model of an electric circuit, the proposed approach may have significant advantages with respect to the existing methods in terms of conservatism or computational burden.

Analysis of non-Markovian passive quantum linear systems’ response to single-photon input fields*

In this paper, we analyze the response of passive quantum linear systems in a non-Markovian environment to single photon input fields. Based on an augmented modelling method for non-Markovian quantum systems, analytic forms of output intensity and output covariance function in steady-state are derived. These results allow us to quantitatively analyze the influence of non-Markovian environment on system response under a framework of linear system theory.

Quantum speed limits on operator flows and correlation functions

Quantum speed limits (QSLs) identify fundamental time scales of physical processes by providing lower bounds on the rate of change of a quantum state or the expectation value of an observable. We introduce a generalization of QSL for unitary operator flows, which are ubiquitous in physics and relevant for applications in both the quantum and classical domains. We derive two types of QSLs and assess the existence of a crossover between them, that we illustrate with a qubit and a random matrix Hamiltonian, as canonical examples. We further apply our results to the time evolution of autocorrelation functions, obtaining computable constraints on the linear dynamical response of quantum systems out of equilibrium and the quantum Fisher information governing the precision in quantum parameter estimation.

An approach of dynamic response analysis of nonlinear structures based on least square Volterra kernel function identification

Abstract Analysis of the dynamic response of a complex nonlinear system is always a difficult problem. By using Volterra functional series to describe a nonlinear system, its response analysis can be similar to using Fourier/Laplace transform and linear transfer function method to analyse a linear system's response. In this paper, a dynamic response analysis method for nonlinear systems based on Volterra series is developed. Firstly, the recursive formula of the least square method is established to solve the Volterra kernel function vector, and the corresponding MATLAB programme is compiled. Then, the Volterra kernel vector corresponding to the nonlinear response of a structure under seismic excitation is identified, and the accuracy and applicability of using the kernel vector to predict the response of a nonlinear structure are analysed. The results show that the Volterra kernel function identified by the derived recursive formula can accurately describe the nonlinear response characteristics of a structure under an excitation. For a general nonlinear system, the first three order Volterra kernel function can relatively accurately express its nonlinear response characteristics. In addition, the obtained Volterra kernel function can be used to accurately predict the nonlinear response of a structure under the similar type of dynamic load.