Response Of Linear Systems Research Articles

Computing Transient Response of Dynamic Systems in the Frequency Domain

AbstractFrequency-domain methods are usually more efficient computationally than time-domain methods to compute the responses of linear dynamic systems. However, a common drawback of frequency-doma...

Density-matrix renormalization group method for the conductance of one-dimensional correlated systems using the Kubo formula

We improve the density-matrix renormalization group (DMRG) evaluation of the Kubo formula for the zero-temperature linear conductance of one-dimensional correlated systems.The dynamical DMRG is used to compute the linear response of a finite system to an applied AC source-drain voltage, then the low-frequency finite-system response is extrapolated to the thermodynamic limit to obtain the DC conductance of an infinite system. The method is demonstrated on the one-dimensional spinless fermion model at half filling. Our method is able to replicate several predictions of the Luttinger liquid theory such as the renormalization of the conductance in an homogeneous conductor, the universal effects of a single barrier, and the resonant tunneling through a double barrier.

On the Output of Nonlinear Systems Excited by Discrete Prolate Spheroidal Sequences

The discrete prolate spheroidal sequences (DPSSs)—a set of optimally bandlimited sequences with unique properties—are important to applications in both science and engineering. In this work, properties of nonlinear system response due to DPSS excitation are reported. In particular, the sequences are shown to be approximately orthogonal after passing through a nonlinear, multiple-input multiple-output dynamical system under quite general conditions. This work quantifies these conditions in terms of constraints upon the higher-order generalized transfer functions characterizing the Volterra expansion of a MIMO system, the Volterra order of the system, and the DPSS bandwidth parameter $W$ and time-bandwidth parameter $NW$ . The approximate system output orthogonality allows multiple-input, multiple-output characterization of the linear narrowband response of a nonlinear system using simultaneous, DPSS excitation.

Pitfalls in the frequency response represented onto Polynomial Chaos for random SDOF mechanical systems

• The frequency response function of a stochastic single-degree-offreedom oscillator is studied. • Conditions for which the probability density function is multimodal are established. • It is shown that Polynomial Chaos then fails to provide accurate results. • Outputbased approach and multi-element generalized Polynomial Chaos are shown to return satisfying results. Uncertainties are present in the modeling of dynamical systems and they must be taken into account to improve the prediction of the models. It is very important to understand how they propagate and how random systems behave. This study aims at pointing out the somehow complex behavior of the structural response of stochastic dynamical systems and consequently the difficulty to represent this behavior using spectral approaches. The main objective is to find numerically the probability density function (PDF) of the response of a random linear mechanical systems. Since it is found that difficulties can occur even for a single-degree-of-freedom system when only the stiffness is random, this work focuses on this application to test several methods. Polynomial Chaos performance is first investigated for the propagation of uncertainties in several situations of stiffness variances for a damped single-degree-of-freedom system. For some specific conditions of damping and stiffness variances, it is found that numerical difficulties occur for the standard polynomial bases near the resonant frequency, where it is generally observed that the shape of the system response PDFs presents multimodality. Strategies to build enhanced bases are then proposed and investigated with varying degrees of success. Finally, a multi-element approach is used in order to gain robustness.

Linear and nonlinear response of the Vlasov system with nonintegrable Hamiltonian.

Linear and nonlinear response formulas taking into account all Casimir invariants are derived without use of angle-action variables of a single-particle (mean-field) Hamiltonian. This article deals mainly with the Vlasov system in a spatially inhomogeneous quasistationary state whose associating single-particle Hamiltonian is not integrable and has only one integral of the motion, the Hamiltonian itself. The basic strategy is to restrict the form of perturbation so that it keeps Casimir invariants within a linear order, and the single particle's probabilistic density function is smooth with respect to the single particle's Hamiltonian. The theory is applied for a spatially two-dimensional system and is confirmed by numerical simulations. A nonlinear response formula is also derived in a similar manner.

Comparison of Different Models of Positive Fractional Continuous-Time Linear Systems and Electrical Circuits Based on the Step Responses

The step responses of positive fractional continuous-time linear systems and electrical circuits described by different models will be investigated and comparison of the models will be performed. Solutions to the state equations of continuous-time linear systems described by Caputo and Caputo-Fabrizio derivatives and necessary and sufficient conditions of the positivity of fractional systems will be recalled. Considerations will be illustrated by numerical examples.

Closed Form Expressions of Linear Discrete Time System Responses

Previously published closed form expressions of linear discrete time responses are generalized and presented with new proofs. They are based on backward difference formulation, shown to offer some important simplifications and closer analogy with continuous time responses. The expressions are related to basic responses, i.e., responses corresponding to unity numerator transfer functions. Those are in turn related to the fundamental solutions of the difference equations involved, adding an extra insight into the nature of these responses. These expressions apply without any restrictions on the poles or the zeros of the systems. Further, efficient evaluations of the fundamental solutions can thus be extended to the general responses, and evaluated directly at any timestep or symbolically.

Responses of standard and fractional linear systems and electrical circuits with derivatives of their inputs

Responses of standard and fractional linear systems and electrical circuits with derivatives of their inputs

Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach

A frequency domain methodology is developed for stochastic response determination of multi-degree-of-freedom (MDOF) linear and nonlinear structural systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates/DOFs is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant DOFs can facilitate the formulation of the equations of motion in a less labor intensive manner. Specifically, relying on the generalized matrix inverse theory, a Moore-Penrose (M-P) based frequency response function (FRF) is determined for a linear structural system with singular matrices. Next, relying on the M-P FRF a spectral input-output (excitation-response) relationship is derived in the frequency domain for determining the linear system response power spectrum. Further, the above methodology is extended via statistical linearization to account for nonlinear systems. This leads to an iterative determination of the system response mean vector and covariance matrix. Furthermore, to account for singular matrices, the generalization of a widely utilized formula that facilitates the application of statistical linearization is proved as well. The formula relates to the expectation of the derivatives of the system nonlinear function and is based on a Gaussian response assumption. Several linear and nonlinear MDOF structural systems with singular matrices are considered as numerical examples for demonstrating the validity and applicability of the developed frequency domain methodology.

Mechanism for Hall conductance of two-band systems against decoherence

The Kubo formula expresses a linear response of the quantum system to weak classical fields. Previous studies showed that the environment degrades the quantum Hall conductance. By studying the dynamics of dissipative two-band systems, in this paper we find that the formation of system-environment bound states is responsible for the Hall conductance immune to the effect of the environment. The bound states can form only when the system-environment couplings are below a threshold. Our results may be of both theoretical and experimental interest in exploring dissipative topological insulators in realistic situations, and may open new perspectives for designing active quantum Hall devices working in realistic environments.

An approach to quantify the influence of ground motion uncertainty on elastoplastic system acceleration in incremental dynamic analysis

Inclusion of ground motion–induced uncertainty in structural response evaluation is an essential component for performance-based earthquake engineering. In current practice, ground motion uncertainty is often represented in performance-based earthquake engineering analysis empirically through the use of one or more ground motion suites. How to quantitatively characterize ground motion–induced structural response uncertainty propagation at different seismic hazard levels has not been thoroughly studied to date. In this study, a procedure to quantify the influence of ground motion uncertainty on elastoplastic single-degree-of-freedom acceleration responses in an incremental dynamic analysis is proposed. By modeling the shape of the incremental dynamic analysis curves, the formula to calculate uncertainty in maximum acceleration responses of linear systems and elastoplastic single-degree-of-freedom systems is constructed. This closed-form calculation provided a quantitative way to establish statistical equivalency for different ground motion suites with regard to acceleration response in these simple systems. This equivalence was validated through a numerical experiment, in which an equivalent ground motion suite for an existing ground motion suite was constructed and shown to yield statistically similar acceleration responses to that of the existing ground motion suite at all intensity levels.

Linear response theory for open systems: Quantum master equation approach

A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.

Non-Markovian linear response theory for quantum open systems and its applications.

The Kubo formula is an equation that expresses the linear response of an observable due to a time-dependent perturbation. It has been extended from closed systems to open systems in recent years under the Markovian approximation, but is barely explored for open systems in non-Markovian regimes. In this paper, we derive a formula for the linear response of an open system to a time-independent external field. This response formula is available for both Markovian and non-Markovian dynamics depending on parameters in the spectral density of the environment. As an illustration of the theory, the Hall conductance of a two-band system subjected to environments is derived and discussed. With the tight-binding model, we point out the Hall conductance changes from Markovian to non-Markovian dynamics by modulating the spectral density of the environment. Our results suggest a way to the controlling of the system response, which has potential applications for quantum statistical mechanics and condensed matter physics.

Mode excitation hysteresis of a flexible cylinder undergoing vortex-induced vibrations

A series of experiments was performed to investigate the modal excitation of a tensioned flexible cylinder in a uniform flow due to flow-induced vibrations. Experiments were conducted in a recirculating flow channel using a flexible cylinder with relatively low aspect ratio such that low structural mode numbers were excited. The spanwise motion of the cylinder was measured using standard motion tracking techniques with high-speed cameras. Hysteresis is observed in the response of the cylinder dependent on whether the flow speed is increased or decreased from the previous experiment. This observation differs from predictions in the literature regarding hysteresis and is attributed to hysteresis in the transition between excited structural modes coupled with the cylinder wake. It is also found that the flexible cylinder is unable to sustain excitation of asymmetric modes in the in-line direction unless the fluid-structure interaction excites a 1:1 frequency response between the in-line (IL) and cross-flow (CF) directions, resulting in a pedaling mode response. The inability to excite asymmetric modes is consistent with the response of linear systems undergoing a symmetric drag load and is consistent with the cylinder undergoing a preferred figure eight shape motion when excited. Distributed fluid forces are derived from the structural characteristics and body motions illustrating the transition of the distribution of added mass and excitation forces on the body.

A semi-parametric polynomial frequency response estimator for identification of weakly non-linear systems using multi-amplitude MLS measurements - part II: Extension and application of the Hp estimator

In this article the family of polynomial estimators for the estimation of the linear system response of weakly non-linear systems is examined regarding estimation errors and applicability. The Hp3 estimator is introduced as an extension for measurement setups having both input and output noise and it is shown how the results obtained with the polynomial estimators are related to the parameters of Volterra models. The bias-variance tradeoff of the estimation error is derived and methods for obtaining the minimum error are presented.

Evolutionary frequency response function of linear systems subjected to earthquake accelerograms using the adaptive chirplet decomposition

In seismic engineering, in order to reproduce the typical characteristics of real earthquakes ground-motion time history, several approaches has been proposed in literature. In this study the adaptive chirplet signal decomposition is adopted to analyze recorded accelerograms in order of defining appropriately evolutionary power spectra [1].The present study focuses on a method to evaluate in closed-form the evolutionary frequency response function, that is required to evaluate the statistics of the response of linear structural systems [2], once the adaptive chirplet signal decomposition is adopted.

Berry Fermi liquid theory

We develop an extension of the Landau Fermi liquid theory to systems of interacting fermions with non-trivial Berry curvature. We propose a kinetic equation and a constitutive relation for the electromagnetic current that together encode the linear response of such systems to external electromagnetic perturbations, to leading and next-to-leading orders in the expansion over the frequency and wave number of the perturbations. We analyze the Feynman diagrams in a large class of interacting quantum field theories and show that, after summing up all orders in perturbation theory, the current–current correlator exactly matches with the result obtained from the kinetic theory.

Influence of rough surface on the ranging distribution of constant fraction discriminator

Aiming effect of the rough surface on the pulse laser ranging distribution of Constant fraction discriminator (CFD), based on the traditional laser radar equation and rough surface characteristic, incoherent pulse laser detection equation of plane target is derived considering launch process as the shock response of linear system. Based on principle of CFD, the probability density function of ranging distribution of CFD is derived. The pulse laser CFD distribution with rough plane is deduced with the echo equation and probability density function. The influence of different plane angles, reflectivity, noise characteristics and pulse width on the distance distribution is analyzed by simulation. The simulation results show that CFD can effectively suppress walk error caused by different angles of rough plane.

An edge determination algorithm for exact computation of the frequency response of linear interval systems

A novel algorithm, called the edge determination algorithm, for exact computation of the frequency response of a linear interval system is proposed. The algorithm formulates candidate curves for the frequency response boundaries as cubic Bezier curves. The edge determination algorithm operates on the cubic Bezier control points of these curves to obtain those, or their parts, that are on the frequency response boundaries. It presents the frequency response boundaries as an array whose entries are the cubic Bezier control points of the curves on the boundaries. Examples for two different cases are presented to illustrate the mechanics and validity of the algorithm.

Control of fractional periodic discrete-time linear systems by parametric state feedback matrices

In this article stabilization of fractional-order periodic discrete-time linear system by a complete nonlinear parametric approach for pole assignment with respect to stability and reachability of system via state feedback is proposed. To modify the dynamic response of linear system we should change the poles of state feedback matrices. Nonlinear parametric feedback matrix makes some freedoms in conditions such as minimum norm of feedback matrix. But it makes large monodromy matrix and changing all of eigenvalues makes some problems. Reassigning a part of bad spectrums, leaving the rest of the spectrums invariant, we have lower-ordered matrix to modify the dynamic response of linear system and we can make nonlinear parametric for this new lower-ordered matrix by less expenses and better stability conditions. Numerical examples illustrate the effectiveness of the proposed approaches.