Nonlinear Drift Research Articles

Remaining useful life prediction of mechanical system based on improved adaptive fractional Lévy stable motion with statistical dependence measurement error

Prediction of remaining useful life (RUL) is a critical component of prognostics and health management (PHM) strategies for mechanical systems. Although some degradation models based on stochastic processes have played a pivotal role in the field of RUL prediction, the diffusion coefficients associated with these models are often fixed values that remain independent of the drift coefficient. Additionally, the measurement error is typically assumed to follow a Gaussian distribution that is independently and identically distributed with respect to the level of degradation. To address these constraints, an RUL prediction framework is proposed based on improved adaptive fractional Lévy stable motion (IAFLSM) with statistical dependence measurement error. The established IAFLSM model is capable of effectively characterizing the individual differences and time-varying uncertainties inherent in equipment degradation processes, with the nonlinear drift and diffusion coefficients exhibiting positive correlation in their variability. Moreover, the state space model is employed to realize the synchronous adaptive dynamic update of the drift coefficient and diffusion coefficient, thus accommodating the mechanical equipment degradation trajectory. Furthermore, a statistical dependence measurement error based on fractional Lévy stable motion is constructed, and the corresponding scale parameter incorporates a dynamic update mechanism with statistical correlation of degradation incremental behavior. The hidden variables related to performance degradation model are estimated through parameter estimation method and characteristic function. Expanding upon the proposed framework for RUL prediction, the quantification of the uncertainty intrinsic in the forecasting results is accomplished by employing the stability theorem of stable distribution and Monte Carlo technique. The RUL prediction framework is validated through the use of authentic truck rear axle full-life data and benchmark rolling bearing data. The comparative analysis results demonstrate the effectiveness and superiority of the proposed methodology.

Correcting Gain Drift in TES Detectors for Future X-Ray Satellite Missions

Changes in the operating environment of transition-edge sensor (TES) microcalorimeters can cause variations in the detector gain function over time. If not corrected, this can degrade the spectral resolution, and cause systematic errors in the knowledge of the absolute energy. The non-linear nature of the TES energy scale function and the potential for multiple, simultaneous sources of drift can make effective corrections extremely challenging. Satellite instruments typically employ an on-board calibration source to provide known reference X-ray lines. This allows real-time monitoring of the detector gain stability and provides information that can be used to correct for drifts. Here we discuss progress towards demonstrating that the energy scale requirements can be met for future instruments such as Athena X-IFU. We present measurements (from ∼ 1-12 keV) on ∼ 200 pixels in a prototype X-IFU array. We use a non-linear drift correction algorithm that uses two fiducial calibration lines (5.4 keV and 8.0 keV) to track gain and interpolate a new, corrected gain between a set of three pre-calibrated gain functions that span the anticipated range of induced drifts. We demonstrate this algorithm is effective at correcting the full gain scale in the presence of multiple sources of environmental drift.

On derivative-free extended Kalman filtering and its Matlab-oriented square-root implementations for state estimation in continuous-discrete nonlinear stochastic systems

Recent research in nonlinear filtering and signal processing has suggested an efficient derivative-free Extended Kalman filter (EKF) designed for discrete-time stochastic systems. Such approach, however, has failed to address the estimation problem for continuous-discrete models. In this paper, we develop a novel continuous-discrete derivative-free EKF methodology by deriving the related moment differential equations (MDEs) and sample point differential equations (SPDEs). Additionally, we derive their Cholesky-based square-root MDEs and SPDEs and obtain several numerically stable derivative-free EKF methods. Finally, we propose the MATLAB-oriented implementations for all continuous-discrete derivative-free EKF algorithms derived. They are easy to implement because of the built-in fashion of the MATLAB numerical integrators utilized for solving either the MDEs or SPDEs in use, which are the ordinary differential equations (ODEs). More importantly, these are accurate derivative-free EKF implementations because any built-in MATLAB ODE solver includes the discretization error control that bounds the discretization error arisen and makes the implementation methods accurate. Besides, this is done in automatic way and no extra coding is required from users. The new filters are particularly effective for working with stochastic systems with highly nonlinear and/or nondifferentiable drift and observation functions, i.e. when the calculation of Jacobian matrices are either problematical or questionable. The performance of the novel filtering methods is demonstrated on several numerical tests.

Electrical Response Estimation of Vibratory Energy Harvesters via Hilbert Transform Based Stochastic Averaging

Abstract Converting mechanical vibrations into electrical power with vibratory energy harvesters can ensure the portability, efficiency, and sustainability of electronic devices and batteries. Vibratory energy harvesters are typically modeled as nonlinear oscillators subject to random excitation, and their design requires a complete characterization of their probabilistic responses. However, simulation techniques such as Monte Carlo are computationally prohibitive when the accurate estimation of the response probability distribution is needed. Alternatively, approximate methods such as stochastic averaging can estimate the probabilistic response of such systems at a reduced computational cost. In this paper, the Hilbert transform based stochastic averaging is used to model the output voltage amplitude as a Markovian stochastic process with dynamics governed by a stochastic differential equation with nonlinear drift and diffusion terms. Moreover, the voltage amplitude dependent damping and stiffness terms are determined via an appropriate equivalent linearization, and the stationary probability distribution of the output voltage amplitude is obtained analytically by solving the corresponding Fokker–Plank equation. Two examples are used to demonstrate the accuracy of the obtained analytical probability distributions via comparisons with Monte Carlo simulation data.

Periodic measures for the stochastic delay modified Swift–Hohenberg lattice systems

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift–Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T-periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.

Analysis of electromechanical couplings and nonlinear carrier transport in flexoelectric semiconductors

In recent years, a linearization method has been extensively employed to investigate the electromechanical fields and carrier distribution in flexoelectric semiconductors, where the assumption of a small perturbation of carrier concentration is adopted. However, this method fails to accurately describe the realistic physical process in which a considerable variation of carrier concentration takes place. Based on fully coupled nonlinear equations, this paper presents a finite element approach to study the electromechanical couplings and nonlinear carrier transport in flexoelectric semiconductors. This method is applied to calculate the electrostatic potential in a bent piezoelectric semiconductive nanowire (NW) going beyond simple considerations and to simulate the nonlinear current–voltage (I–V) characteristics of a mechanically loaded flexoelectric p–n junction. The results indicate that the inherently nonlinear drift of carriers gives rise to the asymmetric distribution of the electric potential relative to the NW axis in the upper body. Flexoelectricity brings about a remarkable enhancement in output voltage and is responsible for the linear variation of electric potential along the length direction of the NW unless close to two ends. Furthermore, the barrier height and I–V relations of a flexoelectric p–n junction can be effectively tuned by mechanical forces due to the flexoelectric effect, the effect of which relies on the size of the p–n junction configuration. This work is a good starting point to comprehend the coupling of flexoelectricity and nonlinear carrier transport in static and dynamic cases, and offers an effective approach to numerically deal with the issues involved in flexoelectronics and piezoelectronics at the nanoscale.

On the 3D turbulence regime in a Tokamak plasma edge

We derive a reduced model for the electrostatic turbulence in a Tokamak edge, when dealing with a resistive plasma and neglecting the spatial gradient of the background density which triggers the linear drift wave response. The obtained dynamics, de facto equivalent to a Hasegawa–Wakatani model, is characterized by a constitutive relation between the electric potential Laplacian and the density, which allows to deal with a single 3D equation governing the electric potential fluctuations. We study the evolution of the model, by separating the n=0 (n indicating toroidal-like number) mode from all the other ones. Then, we linearize the dynamics of the n≠0 modes around the steady 2D spectrum, which describes the spectral features of the electrostatic 2D turbulence. We theoretically and numerically demonstrate the existence of decaying branch of the 3D turbulence, having such a 2D steady spectrum as a natural attractor. This result suggests that the basic constituent of the self-sustained non-linear drift response in the plasma edge has to be individualized in the non-linear 2D electrostatic turbulence.

Infinite ergodicity in generalized geometric Brownian motions with nonlinear drift.

Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g., in finance, in physics, and biology. The definition of the process depends crucially on the interpretation of the stochastic integrals which involves the discretization parameter α with 0≤α≤1, giving rise to the well-known special cases α=0 (Itô), α=1/2 (Fisk-Stratonovich), and α=1 (Hänggi-Klimontovich or anti-Itô). In this paper we study the asymptotic limits of the probability distribution functions of geometric Brownian motion and some related generalizations. We establish the conditions for the existence of normalizable asymptotic distributions depending on the discretization parameter α. Using the infinite ergodicity approach, recently applied to stochastic processes with multiplicative noise by E. Barkai and collaborators, we show how meaningful asymptotic results can be formulated in a transparent way.

Automatic drift compensation for nanoscale imaging using feature point matching

An implementation of drift compensation for imaging at the nanoscale is presented. The method is based on computer vision techniques and hence applicable to any microscope that generates images through a computer interface. The algorithm extracts and matches pairs of feature points from consecutive images to compute and compensate for probe–sample misalignments over time. The protocol also applies selection rules that enable it to withstand significant changes in image contrast. We demonstrate our fully automatic implementation by continuously imaging the same area of a Si(100) surface at the atomic scale with scanning probe microscopy over a period of 25 h at room temperature, showing that the method is robust even under the presence of non-linear drift or spontaneous changes of the probe apex. We apply our method to study the movement of pairs of tin atoms confined within a half-unit cell of the Si(111)-(7 × 7) surface and estimate the energy barrier for their diffusion at room temperature.

A novel nonlinear drift control for sharp turn of autonomous vehicles

Under the sharp turn condition, especially on low adhesion coefficient roads, drift motion can improve vehicle agility and ensure safety. During drift motion, the large sideslip angle and rear wheel slip as well as the saturation of lateral tyre force result in the complex nonlinear dynamic behaviour, such as non-unique equilibrium point and unstable vehicle dynamic. Therefore, this paper focuses on the analysis of dynamic characteristics and the design of both the equilibrium point and feedback controller for drift motion. First, a dynamics model, in which the rear wheel slip ratio is treated as a parametric variable and the influence of rear wheel lateral force on the drift motion is explicitly expressed, is established. Then the expression of steady-state yaw rate and the virtual control input is derived, and the error dynamic model is presented furthermore. To design the equilibrium point, in this paper the effect of rear tyre slip angle and slip ratio on the equilibrium point is discussed, and the condition of stable equilibrium point is proposed. In addition, the design of desired rear tyre slip angle by synthesising the cornering performance during drift motion is presented. For the drift controller design, the main idea is to design the stabilisation controller for the nonlinear dynamic system with the stable equilibrium point and to calculate static feedback gain by means of Lyapunov stability theory. To illustrate the characteristics of drift motion, the influence of the vehicle speed and rear wheel slip ratio on the vehicle path and the low damping characteristics of the dynamic system on the control performance are analysed by different simulation conditions. Moreover, the performance and effectiveness of the proposed drift control are verified by simulation.

Le Cam–Stratonovich–Boole theory for Itô diffusions

Abstract We connect the theory of local asymptotic normality (LAN) of Le Cam to Boole’s approximation of the Stratonovich stochastic integral by estimating the parameter in the nonlinear drift coefficient of an ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discretely spaced dense observations of the process. The asymptotic normality and local asymptotic minimaxity (in the Hajek–Le Cam sense) of approximate maximum likelihood estimators, approximate maximum probability estimators and approximate Bayes estimators based on Itô and Boole’s approximations of the continuous likelihood are obtained under an almost slowly increasing experimental design (ASIED) condition ( T n 6 / 7 → 0 {\frac{T}{n^{6/7}}\to 0} as T → ∞ {T\to\infty} and n → ∞ {n\to\infty} , where T is the length of the observation time and n is the number of observations) through the weak convergence of the approximate likelihood ratio random fields. Among other things, the Bernstein–von Mises type theorems concerning the convergence of suitably normalized and centered approximate posterior distributions to normal distribution under the same design condition are proved. Asymptotic normality and asymptotic efficiency of the conditional least squares estimator under the same design condition are obtained as a by-product. The log-likelihood derivatives based on Itô approximations are martingales, but the log-likelihood derivatives based on Boole’s approximations are not martingales but weighted averages of forward and backward martingales. These new approximations have faster rate of convergence than the martingale approximations. The methods would have advantages over Euler and Milstein approximations for Monte Carlo simulations.

Weak mean attractors of stochastic p-Laplacian delay lattice systems driven by nonlinear noise

This paper deals with the existence and uniqueness of weak pullback mean random attractors for a class non-autonomous stochastic p-Laplacian delay lattice systems with local Lipschitz nonlinear terms driven by nonlinear noise defined on the entire integer set Z. We first establish the global well-posedness to non-autonomous stochastic p-Laplacian delay lattice system in C([τ,∞),L2(Ω,ℓ2)) when the nonlinear diffusion terms and drift terms are locally Lipschitz continuous functions, based on the uniform estimates of approximate solutions in a finite time-interval as well as an appropriate stopping time. We then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors in product Hilbert space L2(Ω,F;ℓ2)×L2(Ω,F;L2((−ρ,0),ℓ2)) under the condition that the constant λ in the system is relatively large.

An integrate-and-fire approach to Ca2+ signaling. Part I: Renewal model

In computational neuroscience integrate-and-fire models capture the spike generation by a subthreshold dynamics supplemented by a simple fire-and-reset rule; they allow for a numerically efficient and analytically tractable description of stochastic single cell as well as network dynamics. Stochastic spiking is also a prominent feature of Ca2+ signaling which suggests to adopt the integrate-and-fire approach for this fundamental biophysical process. The model introduced here consists of two components describing 1) activity of clusters of inositol-trisphosphate receptor channels and 2) dynamics of the global Ca2+ concentrations in the cytosol. The cluster dynamics is given in terms of a cyclic Markov chain, capturing the puff, i.e., the punctuated release of Ca2+ from intracellular stores. The cytosolic Ca2+ concentration is described by an integrate-and-fire dynamics driven by the puff current. For the cyclic Markov chain we derive expressions for the statistics of the interpuff interval, the single-puff strength and the puff current assuming constant cytosolic Ca2+. The latter condition is often well approximated because cytosolic Ca2+ varies much slower than the cluster activity does. Furthermore, because the detailed two-component model is numerically expensive to simulate and difficult to treat analytically, we develop an analytical framework to approximate the driving puff current of the stochastic cytosolic Ca2+ dynamics by a temporally uncorrelated Gaussian noise. This approximation reduces our two-component system to an integrate-and-fire model with a nonlinear drift function and a multiplicative Gaussian white noise, a model that is known to generate a renewal spike train, i.e., a point process with statistically independent interspike intervals. The model allows for fast numerical simulations, permits to derive analytical expressions for the rate of Ca2+ spiking and the coefficient of variation of the interspike interval, and to approximate the interspike interval density and the spike train power spectrum. Comparison of these statistics to experimental data is discussed.

Uniform large deviation principles of fractional stochastic reaction-diffusion equations on unbounded domains

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space.

High-Precision Tribometer for Studies of Adhesive Contacts

Herein, we describe the design of a laboratory setup operating as a high-precision tribometer. The whole design procedure is presented, starting with a concept, followed by the creation of an exact 3D model and final assembly of all functional parts. The functional idea of the setup is based on a previously designed device that was used to perform more simple tasks. A series of experiments revealed certain disadvantages of the initial setup, for which pertinent solutions were found and implemented. Processing and correction of the data obtained from the device are demonstrated with an example involving backlash and signal drift errors. Correction of both linear and non-linear signal drift errors is considered. We also show that, depending on the research interests, the developed equipment can be further modified by alternating its peripheral parts without changing the main frame of the device.

SCALAR: Self-Calibrated Acoustic Ranging for Distributed Mobile Devices

Acoustic ranging has been viewed as a promising Human-Computer Interaction (HCI) technology in many scenarios, such as Augmented Reality (AR)/Virtual Reality (VR) and smart appliances. Most ranging systems with distributed devices undergo an extra calibration process to remove the timing errors. However, the calibration process needs user intervention. Furthermore, it should assume that the clock drifts are linear and stable, which is disabled within tens of minutes. In this paper, we introduce a self-calibrated acoustic ranging system that achieves sub-millimeter accuracy on distributed asynchronous devices. Based on our theoretical timing model, we precisely cancel both the system delay and the nonlinear clock drift with carefully designed Orthogonal Frequency-Division Multiplexing (OFDM) ranging signals. Our synchronization scheme achieves a timing accuracy of 1.9 microseconds, which allows us to build large-scale virtual acoustic arrays. Based on such a calibration scheme, our localization system achieves a ranging error of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rm{0.39}~mm$</tex-math></inline-formula> within three meters in real-world experiments.

Approximation properties of residual neural networks for Kolmogorov PDEs

In recent years residual neural networks (ResNets) as introduced by He et al. [17] have become very popular in a large number of applications, including in image classification and segmentation. They provide a new perspective in training very deep neural networks without suffering the vanishing gradient problem. In this article we show that ResNets are able to approximate solutions of Kolmogorov partial differential equations (PDEs) with constant diffusion and possibly nonlinear drift coefficients without suffering the curse of dimensionality, which is to say the number of parameters of the approximating ResNets grows at most polynomially in the reciprocal of the approximation accuracy $ \varepsilon &gt; 0 $ and the dimension of the considered PDE $ d\in \mathbb{N} $. We adapt a proof in Jentzen et al. [20] - who showed a similar result for feedforward neural networks (FNNs) - to ResNets. In contrast to FNNs, the Euler-Maruyama approximation structure of ResNets simplifies the construction of the approximating ResNets substantially. Moreover, contrary to [20], in our proof using ResNets does not require the existence of an FNN representing the identity map, which enlarges the set of applicable activation functions.

A positivity-preserving numerical algorithm for stochastic age-dependent population system with Lévy noise in a polluted environment

This paper develops a model for describing a stochastic age-dependent population system (SADPS) with Lévy noise in a polluted environment. As the model includes a nonlinear drift term and Lévy noise, it is difficult, if not impossible, to derive an analytical solution of the model. This study thus aims at developing a numerical algorithm that is based on a modified truncated Euler-Maruyama (EM) method and positivity preserving. We first study global positivity for the solution of the SADPS model in a polluted environment. Subsequently, we present a semi-discrete Galerkin finite element method in an age variable, and construct a full-discrete numerical approximation using modified truncated EM scheme on time to preserve positivity. Under certain suitable regularity conditions, estimates of convergence error of the full-discrete and a semi-discrete scheme are presented. Finally, a numerical example is given to illustrate the theoretical results.

Dynamical stability of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise

In this paper, two problems related to FitzHugh–Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh–Nagumo lattice systems driven by nonlinear Wong–Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong–Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli–Arzelà theorem and uniform tail-estimates. We then show the upper semicontinuity of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model and study the convergence of attractors when the delay approaches zero. That is, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved.

Statistical Learning of Nonlinear Stochastic Differential Equations from Nonstationary Time Series using Variational Clustering

Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary non-linear drift and non-linear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based clustering approach includes a quadratic programming (QP) problem with equality and inequality constraints. We couple the QP problem to a closed-form likelihood function approach based on suitable Hermite-expansions to approximate the parameter values of the SDE model. The classification problem provides a smooth indicator function, which enables us to recover the underlying temporal parameter modulation of the one-dimensional SDE. As shown by the numerical examples, the clustering approach recovers a hidden functional relationship between the SDE model parameters and an additional auxiliary process. The study builds upon this functional relationship to develop closed-form, non-stationary, data-driven stochastic models for multiscale dynamical systems in real-world applications.