Response Of Linear Systems Research Articles

Coherent Continuous Wave Terahertz Spectroscopy Using Hilbert Transform

Coherent continuous wave (CW) terahertz spectroscopy is an extremely valuable technique that allows for the interrogation of systems that exhibit narrow resonances in the terahertz (THz) frequency range, such as high-quality (high-Q) THz whispering-gallery mode resonators. Unfortunately, common implementations are dramatically impaired by deficiencies in the used data analysis schemes. Here, we show that the physics of the problem presents an elegant solution whose full potential has remained overlooked until now. We argue that, thanks to the causality of physical systems, Hilbert transformation can be used to analyze the frequency response of linear systems with arbitrarily narrow resonance features in coherent CW THz spectroscopy. In particular, by establishing that signals encountered in typical experiments are closely related to analytic signals, we demonstrate that Hilbert transformation is applicable even when the envelope varies rapidly compared to the oscillation period.

Laplace domain method for evaluating mean-square response of simple oscillators to nonstationary excitation

Evaluating mean-square response of linear systems to nonstationary excitation has been extensively studied, but exact closed-form solutions under various assumptions have been rarely available. This paper derives exact closed-form solutions for the mean-square response of simple oscillators subjected to nonstationary excitation which is formulated as the multiplication of a stationary excitation characterized by an arbitrary spectral density function (PSD) and an envelope function being the sum of several exponential functions. Special attention is given to the envelope function as the sum of two exponential functions because of its practical importance. While analytically evaluating the nonstationary mean square response of a SDOF (single-degree-of-freedom) system subjected to a nonstationary stochastic excitation requires carrying out a triple integral, traditional methods would conduct tedious calculus in the time, frequency or time–frequency domain. In contrast, the proposed method – much more efficient than traditional methods – primarily operates in the Laplace domain (complex plane) based on pole–residue formulations. Another advantage of the proposed pole–residue method is that meaningful physical and mathematical insights could be gained in the solution procedure. For demonstrating the effectiveness and versatility of the proposed method, two cases of the input PSD function are studied: (1) a modulated white noise process, and (2) a modulated correlated process. The correctness of the exact closed-form solutions is verified numerically by Monte Carlo simulations.

Matching the principal deformation mode method with the probability transformation method for the analysis of uncertain systems

SummaryThis paper deals with the probabilistic description of the random response of linear uncertain structural system when the uncertainties are modeled as random variables with assigned probability density function (pdf). In particular, a novel approach is presented that allows the direct evaluation of the response pdf starting from the joint pdf characterizing the structural uncertainties. It consists in matching adequately two methods, recently proposed by the authors, which are the Approximated Principal Deformation Modes method and the Probability Transformation Method. The proposed approach, obtained by this coupling, reveals to have good levels of accuracy, even for relatively high uncertainties, and of computational efficiency, as the reported numerical applications show.

The criterion for the existence of the weight of the orthogonality of equidistant elementary signals

To reduce the level of intersymbol interference and interchannel interference, it is proposed to form the transmitted signals using the basis functions obtained by shifting the impulse responses of linear systems by multiple time intervals. An algorithm for calculating the weight of the orthogonality of basis functions is proposed. A criterion for the existence of the indicated weight of orthogonality is formulated.

Novel Chebyshev wavelets algorithms for optimal control and analysis of general linear delay models

A new efficient type of Chebyshev wavelet is used to find the optimal solutions of general linear, continuous-time, multi-delay systems with quadratic performance indices and also to obtain the responses of linear time-delay systems. According to the new definition of Chebyshev wavelets, the operational matrices of integration, product, delay and inverse time and the integration matrix are derived. Furthermore, new operational matrices as the piecewise delay operational matrix and the stretch operational matrix of the desired Chebyshev wavelets are introduced to analyze systems with, in turn, piecewise constant delays and stretched arguments or proportional delays. Two novel algorithms based on newly Chebyshev wavelet method are proposed for the optimal control and the analysis of delay models. Some examples are solved to establish that the accuracy and applicability of Chebyshev wavelet method in delay systems are increased.

Correction of B0 eddy current effects in spiral MRI.

B0 eddy currents are a subtle but important source of artifacts in spiral MRI. This study illustrates the importance of addressing these artifacts and presents a system response-based eddy current correction strategy using B0 eddy current phase measurements on a phantom. B0 and linear eddy current system response measurements were estimated from phantom-based measurement and used to predict residual eddy current effects in spiral acquisitions. The measurements were evaluated across multiple systems and gradient sets. The corresponding eddy current corrections were studied in both axial spiral-in/out TSE and sagittal spiral-out MPRAGE volunteer data. Correction of B0 eddy currents using the proposed method mitigated blurriness in the axial spiral-in/out images and artifacts in the sagittal spiral-out images. The system response measurement was found to yield repeatable results over time with some variation in the B0 eddy current responses measured between different systems. The proposed eddy current correction framework was effective in mitigating the effects of residual B0 and linear eddy currents. Any spiral acquisition should take residual eddy currents into account. This is particularly important in spiral-in/out acquisitions.

Quantum control of surface acoustic-wave phonons.

One of the hallmarks of quantum physics is the generation of non-classical quantum states and superpositions, which has been demonstrated in several quantum systems, including ions, solid-state qubits and photons. However, only indirect demonstrations of non-classical states have been achieved in mechanical systems, despite the scientific appeal and technical utility of such a capability1,2, including in quantum sensing, computation and communication applications. This is due in part to the highly linear response of most mechanical systems, which makes quantum operations difficult, as well as their characteristically low frequencies, which hinder access to the quantum ground state3-7. Here we demonstrate full quantum control of the mechanical state of a macroscale mechanical resonator. We strongly couple a surface acoustic-wave8 resonator to a superconducting qubit, using the qubit to control and measure quantum states in the mechanical resonator. We generate a non-classical superposition of the zero- and one-phonon Fock states and map this and other states using Wigner tomography9-14. Such precise, programmable quantum control is essential to a range of applications of surface acoustic waves in the quantum limit, including the coupling of disparate quantum systems15,16.

A Modified Newmark Scheme for Simulating Dynamical Behavior of MDOF Nonlinear Systems

The Newmark integration scheme, originally developed for simulating responses of linear dynamical systems, is modified in this paper to effectively model nonlinearities through introduction of the incremental displacements and the effective mass, damping and tiffness matrices. Based on the results of comparisons with the analytical solutions and the Runge-Kutta (RK) method for well-known nonlinear oscillators, the proposed scheme is found to capture accurately dynamical behaviors of nonlinear systems such as mplitude-dependent stiffening or softening natural frequencies, jump phenomena, superharmonic resonances, sub-harmonic resonances, and even chaos and bifurcations. The proposed scheme is more efficient than the RK method for large scale nonlinear ystems with some type of sparsity for which a targeted algebraic equations solver can be employed to speed up the solution for each time step.

A probability transformation method (PTM) for the dynamic stochastic response of structures with non-Gaussian excitations

PurposeThis paper aims to present an approach for the probabilistic characterization of the response of linear structural systems subjected to random time-dependent non-Gaussian actions.Design/methodology/approachIts fundamental property is working directly on the probability density functions of the actions and responses. This avoids passing through the evaluation of the response statistical moments or cumulants, reducing the computational effort in a consistent measure.FindingsIt is an efficient method, for both its computational effort and its accuracy, above all when the input and output processes are strongly non-Gaussian.Originality/valueThis approach can be considered as a dynamic generalization of the probability transformation method recently used for static applications.

Generalized Characteristic Ratios: Definition, Assignment, and Application

ABSTRACT This paper introduces the generalized characteristic ratios concept to shape the transient response of minimum-phase linear time-invariant systems. Unlike the previously defined characteristic ratios devoted to all-pole transfer functions, the generalized characteristic ratios are defined for zero-including systems. The proposed definition is based on the separation of amplitude and time specifications. This means that the generalized characteristic ratios are assigned to reach a low overshoot step response while the generalized time constant is chosen to adjust the transient response speed, independently. Attaining a monotonically decreasing frequency response is the basis of the mentioned assignment. The efficiency of the generalized characteristic ratios to design zeros including controllers like PID and lead (lag) controllers is verified through numerical simulations.

Closed-Form Impulse Responses of Linear Time-Invariant Systems: A Unifying Approach [Lecture Notes

In many signal processing applications, filtering is accomplished through linear time-invariant (LTI) systems described by linear constant-coefficient differential and difference equations since they are conveniently implemented using either analog or digital hardware [1]. An LTI system can be completely characterized in the time domain by its impulse response or in the frequency domain by its frequency response, which is the Fourier transform of the system's impulse response. Equivalently, using the Laplace transform [or the z-transform in the case of discrete-time (DT) systems] as a generalization of the Fourier transform, any continuous-time (CT) or DT LTI system can be characterized by its transfer function (or system function) in the s-domain or the z-domain, respectively. In this article, we explain how to find impulse responses of LTI systems described by differential and difference equations directly in the time domain without resorting to any transform methods or recursive procedures.

Formal analysis of continuous-time systems using Fourier transform

To study the dynamical behavior of the engineering and physical systems, we often need to capture their continuous behavior, which is modeled using differential equations, and perform the frequency-domain analysis of these systems. Traditionally, Fourier transform methods are used to perform this frequency domain analysis using paper-and-pencil based analytical techniques or computer simulations. However, both of these methods are error prone and thus are not suitable for analyzing systems used in safety-critical domains, like medicine and transportation. In order to provide an accurate alternative, we propose to use higher-order-logic theorem proving to conduct the frequency domain analysis of these systems. For this purpose, the paper presents a higher-order-logic formalization of Fourier transform using the HOL-Light theorem prover. In particular, we use the higher-order-logic based formalizations of differential, integral, transcendental and topological theories of multivariable calculus to formally define Fourier transform and reason about the correctness of its classical properties, such as existence, linearity, time shifting, frequency shifting, modulation, time scaling, time reversal and differentiation in time domain, and its relationships with Fourier Cosine, Fourier Sine and Laplace transforms. We use our proposed formalization for the formal verification of the frequency response of a generic n-order linear system, an audio equalizer and a MEMs accelerometer, using the HOL-Light theorem prover.

A study on dynamic model of steady-state visual evoked potentials

Objective. Significant progress has been made in the past two decades to considerably improve the performance of steady-state visual evoked potential (SSVEP)-based brain–computer interface (BCI). However, there are still some unsolved problems that may help us to improve BCI performance, one of which is that our understanding of the dynamic process of SSVEP is still superficial, especially for the transient-state response. Approach. This study introduced an antiphase stimulation method (antiphase: phase ), which can simultaneously separate and extract SSVEP and event-related potential (ERP) signals from EEG, and eliminate the interference of ERP to SSVEP. Based on the SSVEP signals obtained by the antiphase stimulation method, the envelope of SSVEP was extracted by the Hilbert transform, and the dynamic model of SSVEP was quantitatively studied by mathematical modeling. The step response of a second-order linear system was used to fit the envelope of SSVEP, and its characteristics were represented by four parameters with physical and physiological meanings: one was amplitude related, one was latency related and two were frequency related. This study attempted to use pre-stimulation paradigms to modulate the dynamic model parameters, and quantitatively analyze the results by applying the dynamic model to further explore the pre-stimulation methods that had the potential to improve BCI performance. Main results. The results showed that the dynamic model had good fitting effect with SSVEP under three pre-stimulation paradigms. The test results revealed that the parameters of SSVEP dynamic models could be modulated by the pre-stimulation baseline luminance, and the gray baseline luminance pre-stimulation obtained the highest performance. Significance. This study proposed a dynamic model which was helpful to understand and utilize the transient characteristics of SSVEP. This study also found that pre-stimulation could be used to adjust the parameters of SSVEP model, and had the potential to improve the performance of SSVEP-BCI.

Astrophysical gyrokinetics: turbulence in pressure-anisotropic plasmas at ion scales and beyond

We present a theoretical framework for describing electromagnetic kinetic turbulence in a multi-species, magnetized, pressure-anisotropic plasma. The turbulent fluctuations are assumed to be small compared to the mean field, to be spatially anisotropic with respect to it and to have frequencies small compared to the ion cyclotron frequency. At scales above the ion-Larmor radius, the theory reduces to the pressure-anisotropic generalization of kinetic reduced magnetohydrodynamics (KRMHD) formulated by Kunz et al. (J. Plasma Phys., vol. 81, 2015, 325810501). At scales at and below the ion-Larmor radius, three main objectives are achieved. First, we analyse the linear response of the pressure-anisotropic gyrokinetic system, and show it to be a generalization of previously explored limits. The effects of pressure anisotropy on the stability and collisionless damping of Alfvénic and compressive fluctuations are highlighted, with attention paid to the spectral location and width of the frequency jump that occurs as Alfvén waves transition into kinetic Alfvén waves. Secondly, we derive and discuss a very general gyrokinetic free-energy conservation law, which captures both the KRMHD free-energy conservation at long wavelengths and dual cascades of kinetic Alfvén waves and ion entropy at sub-ion-Larmor scales. We show that non-Maxwellian features in the distribution function change the amount of phase mixing and the efficiency of magnetic stresses, and thus influence the partitioning of free energy amongst the cascade channels. Thirdly, a simple model is used to show that pressure anisotropy, even within the bounds imposed on it by firehose and mirror instabilities, can cause order-of-magnitude variations in the ion-to-electron heating ratio due to the dissipation of Alfvénic turbulence. Our theory provides a foundation for determining how pressure anisotropy affects turbulent fluctuation spectra, the differential heating of particle species and the ratio of parallel and perpendicular phase mixing in space and astrophysical plasmas.

New Explicit Integration Algorithms with Controllable Numerical Dissipation for Structural Dynamics

This paper presents a new family of explicit time integration algorithms with controllable numerical dissipation for structural dynamic problems by utilizing the discrete control theory. Firstly, the equilibrium equation of the implicit Yu-[Formula: see text] algorithm is adopted, and the recursive formulas of velocity and displacement for the explicit CR algorithm are used in the algorithms. Then, the transfer function and characteristic equation of the algorithms with integration coefficients are obtained by the [Formula: see text] transformation. Furthermore, their integration coefficients are derived according to the poles condition. It was indicated that the proposed algorithms possess the advantages of second-order accuracy, self-starting, and unconditional stability for linear systems and nonlinear systems with softening stiffness. The numerical dissipation of the algorithms is controlled by the spectral radius at infinity [Formula: see text]. It was also shown that the proposed algorithms have the same poles as the Yu-[Formula: see text] algorithm, and thus the same numerical properties. Compared with the implicit Yu-[Formula: see text] algorithm, the proposed algorithms are explicit in terms of both the displacement and velocity formulas. Finally, the effectiveness of the proposed algorithms in reducing the undesired participation of higher modes for solving the dynamic responses of linear and nonlinear systems has been demonstrated.

Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the little-studied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.

Fast Regulation Control of a Class of Input-delayed Linear Systems with Pre-feedback

A control problem of a class of input-delayed linear systems is considered in this paper. Due to the delay τ in the input, any designed feedback controller can only be engaged after t ≥ τ. Then, this can become the cause of slow regulation since any feedback information cannot be available during the delay. So, the initial function defined for -τ ≤ t ≤ 0 is engaged as an ‘initial non-feedback input’ for 0 ≤ t ≤ τ, which governs the system behavior during this initial time period. There have been numerous research results on the control of input-delayed linear systems by far. Yet, there have been no results on the examination and design of this initial function. Utilizing a time optimal control in the existing results, we show that if some pre-feedback as the initial function is engaged, the system response of the input-delayed linear system can be much improved, and a bang-bang input function is a good candidate as a pre-feedback which can provide better starting state values for the state feedback controller in order to perform the fast regulation. Two examples are given for the illustration of our results.

Closed Form Expressions of Linear Continuous Time System Responses

General closed form expressions of linear continuous time system responses of an arbitrary order are derived, by first relating them to basic responses, i.e., responses corresponding to unity numerator transfer functions. Those are then related to the fundamental solutions of the underlying differential equations. These expressions apply to all systems without any restrictions on the poles or the zeros, further the systems may be noncausal. We derive responses for all regular types of inputs, impulse, step, ramp, parabola, etc., in addition for all generalized derivatives of the impulse. All the presented results have a direct counterpart in results presented in Sigurðsson et al. (2017) on discrete time systems based on the fundamental solution of the associated difference equation. Efficient evaluations of the fundamental solutions along with their derivatives and integrals can thus be extended to the responses and are readily implemented, e.g., as Matlab functions. Such results may be presented symbolically as functions of time or evaluated numerically at any sequence of times without time stepping.

Analysis of Effects due to Right Half Plane Zeros in PI Controller based Hydro Turbine

In this paper, some detrimental time-domain characteristics (zero-crossings, overshoot(due to zeros) and initial undershoot) are analyzed using the system transfer function model, for identifying the presence of these effects in step response of non-minimum phase linear systems. Moreover, regarding initial undershoot, a theorem for detection using the state-space model, without the need to obtain transfer function, is proposed. In addition to detection, a theorem for estimating the percentage of initial undershoots is also proposed. Applications of these theorems are carried out on load frequency control of hydro-electric power plant which employs hydro-turbine, a non-minimum phase system. Moreover, the effect of a PI controller on initial undershoot is discussed for a general non-minimum phase linear system and illustrated for load frequency control of hydro-electric power plant.

求解周期性分段线性系统动态响应的高效数值方法

An efficient method based on the parametric variational principle (PVP) was proposed for computing the dynamic responses of periodic piecewise linear systems with multiple gap-activated springs. Through description of gap-activated springs with the PVP, the complex nonlinear dynamic problem was transformed to a standard linear complementary problem. This method can avoid iterations and updating the stiffness matrix in the computing process and can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method (PIM), an efficient numerical time-integration method was developed to obtain the dynamic responses of the system. This method indicates that there are a large number of identical elements and zero elements in the matrix exponents of a periodic structure, and saves computation load and computer storage by avoiding repeated calculation and storage of these elements. Numerical results validate the proposed method. The dynamic behaviors of a 5-DOF piecewise linear system under harmonic excitations were analyzed, including the stable periodic motion, the quasi-periodic motion and the chaotic motion. In comparison with the Runge-Kutta method, the proposed method has satisfactory correctness and efficiency.