Non-uniform Time Research Articles

Analysis of Hyers–Ulam stability and controllability of non-linear switched impulsive systems with delays on time scales

In this article, we have used the concepts of time scales theory to discuss nonlinear switched impulsive systems with delay. Our main objective is to determine the Hyers–Ulam stability and controllability of nonlinear switched impulsive systems with delay on non-uniform time domains. To obtain the necessary and sufficient conditions for existence, Hyers–Ulam stability, and controllability, we utilize the Banach fixed-point theorem and Krasnoselskii’s fixed-point theorem. In order to demonstrate our conclusions, we have discussed some simulation-based examples along with the three tank liquid control problem and a potential practical situation related to the infectious disease with switching rules. The results of this manuscript provide all the necessary and sufficient conditions for Hyers–Ulam stability and controllability that are true for discrete, continuous as well as unified time domains simultaneously.

Low-power adaptive sampling electronic nose system with a Radon transform-based convolutional neural network for optimized gas recognition

The electronic nose (E-nose) systems provide great promise in capturing information about the identity of gaseous chemical analytes as well as their temporal variation. However, the regular and dense sampling events of the E-nose often lead to a substantial amount of redundancy in the temporal structure, which in turn limits their application in efficiency-sensitive scenarios such as advanced robotics and the Internet of Things. Herein, we introduce a novel operating mechanism for E-noses, which dynamically adjusts the time interval of operation to reduce power consumption and information content through an adaptive sampling mechanism. Three strategic schemes with different operating principles are presented to implement this mechanism: the accurate scheme targets the optimized discrimination accuracy, the vague scheme is oriented toward power-sensitive scenarios, and the balanced scheme aims to balance cost and benefit. In order to ensure that the adaptively sampled inhomogeneous data can be compatible with each other, we further propose an algorithm based on the Radon transform that can convert non-uniform time series with timestamp information into the same size tensor. Finally, we designed a convolutional neural network-based classification model for gaseous chemical analytes, providing guidance on scheme and parameter selection with an accuracy of up to 99 % and power savings of up to 96 %. Overall, this work provides a novel solution to optimize temporal redundancy in E-nose systems and a generalized approach to the corresponding deep-learning data processing.

An improved solution of the probability density evolution equation for analyzing stochastic structural responses with adaptive time steps and initial conditions

The probability density evolution method (PDEM) offers an innovative approach for analyzing the stochastic vibration responses of structures. Given the complexities inherent in real-world challenges, the probability density evolution equations (PDEE) often require numerical methods for effective resolution. Consequently, improving computational efficiency and solution accuracy is of paramount importance. Currently, the impulse function discretization method (IFDM) faces several challenges, including limited differentiability in the spatial direction for initial value conditions. Furthermore, the probability density function produced by this method frequently deviates significantly from the true value, undermining the accuracy of the PDEE. Additionally, when addressing vibration responses characterized by rapid evolutionary velocities, the use of a globally fixed uniform time step size (FUTSS) results in a marked decrease in computational efficiency. To tackle these challenges, this study introduces an improved probability density evolution method (IPDEM), which integrates the Gaussian kernel density estimation discretization method (GKDEDM) and an adaptive non-uniform time step size (ANUTSS). The advantages of the IPDEM are illustrated through two numerical simulation examples. The findings indicate that the GKDEDM not only enhances spatial differentiability but also provides a more accurate estimation of the probability density at the initial moment, thereby improving the accuracy of the PDEE. Furthermore, the ANUTSS significantly reduces the number of iterations required while preserving precision, leading to a substantial optimization of computation time.

Performance evaluation of multivariate deep-time convolution neural architectures for short-term electricity forecasting: Findings and failures

Deep learning (DL) models hold great promise in enhancing the decision-making abilities of electricity market participants and system operators in the short term, as they excel at capturing the intricate interdependencies and uncertainties in electricity consumption. Nevertheless, currently, the adoption of DL models for electricity prediction is not keeping up with the advancements in DL research. Within this framework, the aims of this study are to examine the probabilistic, multivariate, and multihorizon time series prediction of deep time convolution neural (DTCN) models. The effectiveness of DTCN in forecasting electricity consumption over daily, weekly and monthly time horizons is emperically assessed. We analyze both probabilistic and deterministic predictions. The evaluation data comprises real residential electricity consumption, average temperature and humidity with hourly granularity, collected over one year from a residential building. For evaluation, we considered accuracy and uncertainties, sensitivity to hyperparameters, the impact of exogeneous variables, as well as computational efficiency. Results show that the probabilistic models offer greater advantages when it comes to forecasting on multihorizons, especially for diverse datasets with non-uniform time series models. Unlike traditional models, the DTCN models demonstrated superior performance in capturing complex patterns and reducing forecast errors in short-term scenarios. However, their performance can be influenced by the sensitivity of hyperparameters. These findings can act as a benchmark for quantitatively assessing the performance of novel multivariate DL models in predicting electricity demand.

Toward quantitative characterization of simulated earthquake-cycle complexities

Earthquake cycle simulations based on the rate-and-state friction formulation are evolutions of nonlinear dynamical systems (NDS). The term “cycle” implies an overall stable structure that is a phase-space attractor naturally traced out by trajectories of NDS as it evolves. Quantitatively characterizing these attractors should be a basis for measuring complexities of the simulated earthquake cycles, i.e. to determine if and how regular or chaotic they are. I first revisit the textbook-standard quasi-dynamic spring-slider system from an NDS perspective, explicitly showing the attractors, their relationship with the parameters of the NDS, and how they can be characterized taken advantage of their low-dimensionality while aiming to extend the analysis to high-dimensionality. I evaluate two approaches, computing the Lyapunov exponents (LEs) and measuring correlation dimensions, with the simple spring-slider and earthquake-cycle examples whose phase-space attractors can be visually verified. I conclude LEs are too inconvenient and computationally expensive to use whereas measuring correlation dimensions is an easy and effective approach even with highly non-uniform time sampling present in all simulations. For earthquake-cycle simulations, an attractor reconstruction is performed based on Taken’s theorem to corroborate my correlation-dimension results. The current method is limited in its ability to detect chaos in a dichotomous manner, which illuminates the direction for future study. An improving ability to quantitatively characterize earthquake-cycle simulations as an overall stable structure offers new opportunities to understand exotic seismic observations such as slow-slip events and enables more informative comparison with real data, particularly from paleoseismology, which could have far-reaching implications in earthquake forecasting.

Parsing Unranked Tree Languages, Folded Once

A regular unranked tree folding consists of a regular unranked tree language and a folding operation that merges (i.e., folds) selected nodes of a tree to form a graph; the combination is a formal device for representing graph languages. If, in the process of folding, the order among edges is discarded so that the result is an unordered graph, then two applications of a fold operation are enough to make the associated parsing problem NP-complete. However, if the order is kept, then the problem is solvable in non-uniform polynomial time. In this paper, we address the remaining case, where only one fold operation is applied, but the order among the edges is discarded. We show that, under these conditions, the problem is solvable in non-uniform polynomial time.

Viscoelastic Fractional Model with a Non-Uniform Time Discretization for Laminated Glass: Experimental Validation

We discuss a novel approach, based on fractional calculus with a non-uniform time discretization, to numerically simulate interlayer viscoelastic behaviour and associated time-dependent deformation of laminated glass. Reference is made to the classic example of a simply supported laminated glass beam under long-duration loads. The fractional model is compared with some results obtained using the widely used finite element software ABAQUS 2021, which for the viscoelastic properties of the polymeric interlayer, utilizes the more traditional approach based on the Wiechert model and approximation via Prony series of the relaxation function and a uniform discretization of time for the numerical solution. The model is also validated through the comparison with experimental test. The novel approach based on fractional calculus presents two main advantages: 1) the definition of the model parameters from experimental data is simplified; and 2) the numerical implementation is easier and computationally more efficient. When a long observation time is considered, the use of a non-uniform time discretization presents the great advantage of not neglecting any part of the relaxation function. Use of traditional uniform time discretization requires the use of large time steps making it impossible to describe all the changes of the relaxation curve within the large time interval. Practical examples will be presented using viscoelastic models for Trosifol® Extra Stiff (PVB) and SentryGlas® interlayers. This methodology also shows potential to advance next generation standards for the design of structural laminated glass.

Energy dissipation and evolutions of the nonlocal Cahn-Hilliard model and space fractional variants using efficient variable-step BDF2 method

In this work, an energy stable BDF2 scheme with general nonuniform time steps is developed for the nonlocal Cahn-Hilliard model to capture the multi-scale behavior of evolution efficiently and accurately. The fully discrete numerical scheme is demonstrated to inherit main physical properties of the continuous model, namely, mass conservation and energy dissipation. Recently, various variants of the Cahn-Hilliard model with nonlocal operators and fractional Laplacian have attracted great attention in terms of theoretical analysis and numerical approximation, while there is less emphasis on the comparisons among the different variant models. By comparing the impact of various parameters and coarsening behaviors, numerical results suggest that the nonlocal Cahn-Hilliard model bears a strong resemblance to the space fractional Cahn-Hilliard model derived from H−1 gradient flow of the nonlocal energy. Nevertheless, the order of the fractional Laplacian appears to affect only the rate of energy decay, without altering the profile of the solution for the fractional model relating to the H−β (0<β<1) gradient flow of the classical Ginzburg-Landau energy functional in one dimensional case.

An analysis of hybrid impulsive prey-predator-mutualist system on nonuniform time domains

Abstract In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.

The semi-implicit DLN algorithm for the Navier-Stokes equations

AbstractDahlquist, Liniger, and Nevanlinna design a family of one-leg, two-step methods (the DLN method) that is second order, $$\varvec{A}-$$ A - and $$\varvec{G}-$$ G - stable for arbitrary, non-uniform time steps. Recently, the implementation of the DLN method can be simplified by the refactorization process (adding time filters on the backward Euler scheme). Due to these fine properties, the DLN method has strong potential for the numerical simulation of time-dependent fluid models. In the report, we propose a semi-implicit DLN algorithm for the Navier-Stokes equations (avoiding non-linear solver at each time step) and prove the unconditional, long-term stability, and second-order convergence with the moderate time step restriction. Moreover, the adaptive DLN algorithms by the required error or numerical dissipation criterion are presented to balance the accuracy and computational cost. Numerical tests will be given to support the main conclusions.

Asynchronous sampled-data control for connected vehicles with heterogeneous delays

This study investigates an asynchronous sampled-data control problem of vehicular platoons, with heterogeneous sampling, subjected to actuator delays. Without a synchronized clock, a completely asynchronous sampled-data controller is designed for each follower, where the state of the ith follower itself and its neighboring vehicles are sampled at their own sampling time instants. The caused closed-loop tracking error dynamics for the entire platoon considering the effect of the nonuniform sampling time intervals, heterogeneous vehicle dynamics, inter-vehicle topology and heterogeneous time delays. To simplify the stability analysis and controller design, the tracking-error dynamics of the entire platoon are decomposed into individual subsystems with reduced-order dynamics. Based on Lyapunov stability theory, the optimal conditions are explored to design an asynchronous sampled-data controller to guarantee the desired stability performance. Moreover, the exact values for the maximum allowable sampling interval and time delay are calculated for each follower using the designed feedback controller gain. The proposed asynchronous sampled-data control method is extended to a vehicular platoon using an event-based sampling scheme. Numerical examples are used to verify the effectiveness of the proposed sampled-data control method.

Dynamic Motion Transition: A Hybrid Data-driven and Model-driven Method for Human Pose Transitions.

The rapid, accurate, and robust computation of virtual human figures' "in-between" pose transitions from available and sometimes sparse inputs is of fundamental significance to 3D interactive graphics and computer animation. Various methods have been proposed to produce natural lifelike transitions of human pose automatically in recent decades. Nevertheless, conventional pure model-driven methods require heuristic knowledge (e.g., least motion guided by physics laws) and ad-hoc clues (e.g., splines with non-uniform time warp) that are difficult to obtain, learn, and infer. With the fast emergence of large-scale datasets readily available to animators in the most recent years, deep models afford a powerful alternative to tackle the aforementioned challenges. However, pure data-driven methods still suffer from the remaining challenges such as unseen data in practice and less generative power in model/domain/data transfer, and the measurement of the generative power has always been omitted in these works. In essence, data-driven methods solely rely on the qualities and quantities of training datasets. In this paper, we propose a hybrid approach built upon the seamless integration of data-driven and model-driven methods, called Dynamic Motion Transition (DMT), with the following salient modeling advantages: (1) The data augmentation capability based on the limited human locomotion data capture and the concept of force-derived directly from physical laws; (2) Force learning by which skeleton joints are driven to move, and the Conditional Temporal Transformer (CTT) being trained to learn the force change in the local range, both at the fine level; and (3) At the coarse level, the effective and flexible creation of the subsequent step motion using Dynamic Movement Primitives (DMP) until the target is reached. Our extensive experiments have confirmed that our model can outperform the state-of-the-art methods under the newly devised metric by virtue of the least action loss function. In addition, our novel method and system are of immediate benefit to many other animation tasks such as motion synthesis and control, and motion tracking and prediction in this bigdata graphics era.

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

The aim of this article is to establish a numerical scheme to achieve the theoretical accuracy near the weak singularity at t=0 in solving the two-dimensional time-fractional nonlinear mixed diffusion-wave equation (TFNMDWE). The governing problem involves diffusion term with time-fractional Caputo derivative (TFCD) of order ν1(0<ν1<1), and wave term with TFCD of order ν2(1<ν2<2). To handle the singularity at t=0, we use linearized L1 method to discretize both the TFCDs on nonuniform time meshes. By using the nonuniform L1 method to approximate TFCD and central difference operator for the space derivative approximation, the considered problem is converted to an equivalent system of equations. Then, we use the Alternating Direction Implicit (ADI) approach to develop a numerical scheme for solving the resulting system of coupled equations. Further, we prove stability analysis of the scheme. Numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFNMDWEs with smooth and non-smooth exact solutions to describe the accuracy of numerical scheme. The illustrated examples confirm that the scheme has second-order accuracy in space, and order of convergence (OC) in time direction is min(2−ν1,3−ν2,γν1,γ(ν2−1)), where γ is the mesh grading parameter used in construction of the nonuniform meshes. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Existence, controllability and Hyers–Ulam stability of a hybrid neutral switched system with impulsive effects

In this study, our main focus is on the analysis of the existence, controllability, and Hyers–Ulam stability of a neutral switched system with impulsive effects on non-uniform time domains. First, we use the Banach fixed point theorem and the time scales calculus to explore the existence, Ulam's type stability, and controllability of a considered switched system. In the end, we examine the controllability results through numerical simulations. This article aims to identify the essential requirements for achieving controllability and stability on different time scales.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Quantifying Coincidence in Non-Uniform Time Series with Mutual Graph Approximation: Speech and ECG Examples

Compressive sensing and arbitrary sampling are techniques of data volume reduction challenging the Shannon sampling theorem and expected to provide efficient storage while preserving original information. Irregularity of sampling is either a result of intentional optimization of a sampling grid or stems from sporadic occurrence or intermittent observability of a phenomenon. Quantitative comparison of irregular patterns similarity is usually preceded by a projection to a regular sampling space. In this paper, we study methods for direct comparison of time series in their original non-uniform grids. We also propose a linear graph to be a representation of the non-uniform signal and apply the Mutual Graph Approximation (MGA) method as a metric to infer the degree of similarity of the considered patterns. The MGA was implemented together with four state-of-the-art methods and tested with example speech signals and electrocardiograms projected to bandwidth-related and random sampling grids. Our results show that the performance of the proposed MGA method is comparable to most accurate (correlation of 0.964 vs. Frechet: 0.962 and Kleinberg: 0.934 for speech signals) and to less computationally expensive state-of-the-art distance metrics (both MGA and Hausdorf: O(L1+L2)). Moreover, direct comparison of non-uniform signals can be equivalent to cross-correlation of resampled signals (correlation of 0.964 vs. resampled: 0.960 for speech signals, and 0.956 vs. 0.966 for electrocardiograms) in applications as signal classification in both accuracy and computational complexity. Finally, the bandwidth-based resampling model plays a substantial role; usage of random grid is the primary cause of inaccuracy (correlation of 0.960 vs. for random sampling grid: 0.900 for speech signals, and 0.966 vs. 0.878, respectively, for electrocardiograms). These figures indicate that the proposed MGA method can be used as a simple yet effective tool for scoring similarity of signals directly in non-uniform sampling grids.

Classifier-Based Nonuniform Time Slicing Method for Local Community Evolution Analysis

With the rapid expansion of the scale of a dynamic network, local community evolution analysis attracts much attention because of its efficiency and accuracy. It concentrates on a particularly interested community rather than considering all communities together. A fundamental problem is how to divide time into slices so that a dynamic network is represented as a sequence of snapshots which accurately capture the evolutionary events of the interested community. Existing time slicing methods lead to inaccurate evolution analysis results. The reason is that they usually rely on a linear strategy while the community evolution is a nonlinear process. This paper investigates the problem and proposes a classifier-based time slicing method for local community evolution analysis. First, a classifier is trained for judging whether there is a community in the given network snapshot is identified as the continuing of the community defined by the given node subset. The features for classification include internal cohesion degree and external coupling degree. Second, a time slicing method is proposed based on the trained classifier. As the network evolves, the method continuously uses the classifier to predict whether there is a community in the newest network identified as the continuing of the interested community. Whenever the answer is negative, an evolutionary event is presumed to have occurred and a new time slice is generated. Experimental results show that compared with existing time slicing methods, our proposed method achieves higher recognition rate for given redundancy ratio.

Parameter identification method of nonuniform and under-sampled blade tip timing based on extended DFT and compressed sensing

Vibration amplitude and frequency are the two most important indicators that characterize the health status of high-speed rotating blades, but the signal obtained by blade tip timing (BTT) technology, one of the best rotating blade vibration monitoring methods, is seriously nonuniform and under-sampled, which makes these two indicators difficult to identify. In view of this problem, the paper proposes a parameter identification method for the nonuniform and under-sampled BTT signal based on extended Discrete Fourier transform and compressed sensing (CS), with the Fourier integral transformation as the goal. It realizes the frequency analysis of nonuniform under-sampled signals by constructing and optimizing the transformation basis function instead of the exponential basis in the traditional FFT transformation in the extended frequency range, and then constructs a CS model through the obtained blade vibration frequency. The complete waveform of the blade vibration is restored by using a small number of under-sampled signals, thus obtaining the blade vibration amplitude and vibration frequency. On the one hand, the method proposed in this paper breaks through the limitation of Nyquist’s sampling theorem, and the number of analytical spectral lines is no longer limited to the number of sampling points, which improves the frequency resolution. On the other hand, only a small number of measurement signals can be reconstructed to achieve a complete vibration signal. The feasibility and reliability of the proposed method are verified by mathematical modeling, simulation analysis, and experimental testing. The results indicate that when the number of sensors is greater than or equal to four, the time domain and frequency domain signals of blade vibration can be accurately analyzed based on the proposed method, the vibration amplitude error is less than 0.01 mm, the frequency error is less than 0.1 Hz, and it has good anti-interference performance.

A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black–Scholes model

The Black–Scholes (B–S) equation has been recently extended as a kind of tempered time-fractional B–S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B–S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.

Anomaly prediction of CT equipment based on IoMT data

BackgroundLarge-scale medical equipment, which is extensively implemented in medical services, is of vital importance for diagnosis but vulnerable to various anomalies and failures. Most hospitals that conduct regular maintenance have been suffering from medical equipment-related incidents for years. Currently, the Internet of Medical Things (IoMT) has emerged as a crucial tool in monitoring the real-time status of the medical equipment. In this paper, we develop an IoMT system of Computed Tomography (CT) equipment in the West China Hospital, Sichuan University and collected the system status time-series data. Novel multivariate time-series classification models and frameworks are proposed to predict the anomalies of CT equipment. The important features that are closely related to the equipment anomalies are identified with the model.MethodsWe extracted the real-time CT equipment status time-series data of 11 equipment between May 19, 2020 and May 19, 2021 from the IoMT, which includes the equipment oil temperature, anode voltage, etc. The arcs are identified as labels of anomalies due to their relationship with decreased imaging quality and CT equipment failures. To improve prediction accuracy, the statistics and transformations of the raw historical time-series data segment in the sliding time window are used to construct new features. Due to the particularity of time-series data, two frameworks are proposed for splitting the training and test sets. Then the Decision Tree, Support Vector Machine, Logistic Regression, Naive Bayesian, and K-Nearest Neighbor classification models are used to classify the system status. We also compare our model to state-of-the-art models.ResultsThe results show that the anomaly prediction accuracy and recall of our method are 79% and 77%, respectively. The oil temperature and anode voltage are identified as the decisive features that may lead to anomalies. The proposed model outperforms the others when predicting the anomalies of the CT equipment based on our dataset.ConclusionsThe proposed method could predict the state of CT equipment and be used as a reference for practical maintenance, where unexpected anomalies of medical equipment could be reduced. It also brings new insights into how to handle non-uniform and imbalanced time series data in practical cases.