Integration Interval Research Articles

A novel method for high-speed maneuvering target detection and motion parameters estimation

This paper considers the problems of high-speed maneuvering target detection and motion parameters estimation with complex motions, involving range migration (RM) and Doppler frequency migration (DFM) induced by target’s radial velocity, acceleration and jerk within the coherent integration interval. To address these problems, we propose a novel method based on the third-order keystone transform (TOKT) and first-order discrete polynomial-phase transform (FDPT) in this paper. Firstly, the TOKT is carried out to correct the third-order RM. Secondly, the matched filtering function is constructed to remove the quadratic DFM caused by radial jerk. After that, the FDPT is applied to reduce the target phase order. Then, the TOKT is employed again to eliminate the residual first-order RM related to target’s radial acceleration, and a well-focused peak is acquired after azimuth fast Fourier transform (FFT) as well as range inverse FFT. Finally, based on the position of peak, the motion parameters of maneuvering targets can be estimated efficiently. Compared with the existing algorithms, the proposed method achieves a good balance between the computational complexity and detection ability. Numerical results verify the effectiveness of the proposed method.

An Efficient Seven-Step Block Method for Numerical Solution of SIR and Growth Model

In this article, a new implicit continuous block method is developed using the interpolation and collocation techniques via Power series as the basis function. A constant step length within a seven-step interval of integration was adopted. The selected grid points were evaluated to get a continuous linear multistep method. The evaluation of the continuous method at the non-interpolation points produces the discrete schemes which form the block. The basic properties of the block method were investigated and found to be consistent, zero stable and hence convergent. The new method was tested on real life problems namely: SIR and Growth model. The results were found to compare favourably with the existing methods in terms of accuracy and efficiency. Keywords: Block method, Growth Model, implicit, power series and SIR model.

Особливості чисельного вирішення диференційних рівнянь моделей ланчестерського типу у стохастичній постановці

У статті приведений короткий огляд можливої сфери застосування модифікацій моделей ланчестерського типу, а також розглянуті деякі особливості чисельного вирішення диференційних рівнянь таких моделей у стохастичній постановці. Стохастична постановка дозволяє ураховувати випадкові фактори та елементи невизначеності, які впливатимуть на динаміку змін чисельності протиборчих угруповань, і які у певній мірі присутні у будь-яких бойових діях. На відміну від детермінованих моделей, стохастичні моделі потребують використання спеціальних методів, вибір конкретного з яких може ґрунтуватися на вимогах до ступеню їх збіжності на інтервалі інтегрування. Оцінка збіжності може слугувати також для перевірки правильності програмної реалізації обраного методу.

A Novel Integrated Fuzzy PIPRECIA–Interval Rough Saw Model: Green Supplier Selection

In this paper is presented a novel integrated fuzzy – rough Multi-Criteria Decision-Making (MCDM) model based on integration fuzzy and interval rough set theory. Model integrates Fuzzy PIvot Pairwise RElative Criteria Importance Assessment - fuzzy PIPRECIA and Interval rough Simple Additive Weighting (SAW) methods. An illustrative example for demonstration of the model is proposed that represents evaluation and supplier selection based on nine environmental criteria. Fuzzy PIPRECIA method is used for determining the significance of the following seven criteria: C1 - environmental image, C2 - recycling, C3 - pollution control, C4 - environmental management system, C5 – environmentally friendly products, C6 - resource consumption and C7 - green competencies. Iterval rough SAW method is applied for evaluation four alternatives. Results show that third criterion is most important while fourth alternative represents the best solution.

CORRECTION AND ADDITION TO "L 2 -BOUNDEDNESS OF MARCINKIEWICZ INTEGRALS ALONG SURFACES WITH VARIABLE KERNELS"

In our paper 2 -boundedness of Marcinkiewicz integrals along surfaces with variable kernels, have found that Theorems 3 and 5 are not correct. Here, reformulate Theorems 3 and 5, so that L 2 -boundedness holds as in Theorems 1 and 4. s J n +k−1(ρ) ρ n 2 Φ −1 ( ρ |ξ| )ϕ(Φ −1 ( ρ |ξ| ))dρ, where s =Φ (t)|ξ| and Φ(t) t = C2Φ � (t)ϕ(t). In our paper (8, p. 380), have taken the integration interval as (0 ,s ) in place of the correct one (s, +∞). Because of this error have led false claims in Theorems 3 and 5. So, in our paper (8), delete the sentence also show some sharp difference betweeen properties of singular integrals and the Marcinkiewicz integral with rough variable kernels. in the abstract. We delete the lines 26 through 34 in the page 371, and Theorem 5 in the page 372. We also delete the section 4. Other corrections: In the assumption of Lemma 2.3 add ν − λ> −1. In the line 9 of the page 371, exists should be exist. In the line 3 from the bottom in the page 372, we only give should be we have only to give. In the line 2 of the page 375, 1

Stability analysis of neutral systems with mixed interval time-varying delays and nonlinear disturbances

This paper studies the problem of stability analysis for neutral systems with mixed time-varying delays and nonlinear disturbances. Improved stability criteria are obtained by applying augmented Lyapunov-Krasovskii methodology. First, a novel Lyapunov–Krasovskii functional containing the intersection relationship terms about mixed time-varying delays and their upper-lower bounds with lower bounds limited to be strictly greater than zero is fully established. Second, the upper bounds for integral terms presented are accurately estimated based on integral intervals decomposition approach. Third, a less conservative stability condition in terms of linear matrix inequality is derived by utilizing a new quadratic inequality technology. In the end, two numerical examples are given to illustrate the effectiveness and the advantages of the proposed method.

Some high order formulae for approximating Caputo fractional derivatives

In order to approximate the Caputo fractional derivative of order α, 0<α<1, we construct here a new class of formulae on the basis of B-spline interpolation. These new formulae called S1, S2 and S3 have 2−α, 3−α and 4−α order of convergence, respectively. The proposed formulae are as simple as the well-known L1 formula and the main advantage of them lies in the fact that their accuracy is fixed in the whole interval of integration while the previous formulae such as L1-2 have lower accuracy at the beginning of the interval. Hence in comparison with the previous formulae, new ones have better accuracy and their computational costs are comparable. We then modify S2 and S3 formulae for approximating the Caputo fractional derivative of order α, 1<α<2. Some numerical examples as well as two applications in solving fractional ordinary and partial differential equations (PDEs) are provided to demonstrate the applicability and accuracy of the new formulae.

Transmission and reflection characteristics of a multi-layered wall with doubly periodic interfaces

In this paper, a semi-analytical technique to derive scattering coefficients of a three-dimensional multi-layered wall with doubly periodic interfaces is presented. First, the proposed method uses the periodicity of interfaces in the surface integral equation formulation to confine the integration interval to just one period. It also simplifies the formulation by using the extended boundary condition method (EBCM) and the spectral formulation of the background Green’s function. Then, each of the scattering coefficient matrices of interfaces and layers is treated as two-port networks, and the scattering matrix of the whole structure is derived using the scattering transfer matrix method. In a limiting case, the proposed method is validated by comparing the results to a two-dimensional semi-analytical approach that is available in the literature. Then for a single layer wall, the effects of side interfaces’ roughness, layer’s thickness, and dielectric constant on reflection and transmission coefficients are analyzed.

A New In-Flight Alignment Method with an Application to the Low-Cost SINS/GPS Integrated Navigation System.

The optimization-based alignment (OBA) methods, which are implemented by the optimal attitude estimation using vector observations—also called double-vectors—have proven to be effective at solving the in-flight alignment (IFA) problem. However, the traditional OBA methods are not applicable for the low-cost strap-down inertial navigation system (SINS) since the error of double-vectors will be accumulated over time due to the substantial drift of micro-electronic- mechanical system (MEMS) gyroscope. Moreover, the existing optimal estimation method is subject to a large computation burden, which results in a low alignment speed. To address these issues, in this article we propose a new fast IFA method based on modified double-vectors construction and the gradient descent method. To be specific, the modified construction method is implemented by reducing the integration interval and identifying the gyroscope bias during the construction procedure, which improves the accuracy of double-vectors and IFA; the gradient descent scheme is adopted to estimate the optimal attitude of alignment without complex matrix operation, which results in the improvement of alignment speed. The effect of different sizes of mini-batch on the performance of the gradient descent method is also discussed. Extensive simulations and vehicle experiments demonstrate that the proposed method has better accuracy and faster alignment speed than the related traditional methods for the low-cost SINS/global positioning system (GPS) integrated navigation system

Comments on the use of block methods for solving singular boundary value problems

Singular boundary-value problems appear frequently on the modellization of many physical phenomena as in catalytic diffusion reactions, chemical kinetics, thermal-explosion theory, or electro hydrodynamics, among others. The singular Lane-Endem equation is a typical kind of equation modelling some of those problems. Unfortunately, just in few occasions the exact solutions can be obtained. In this situation the block methods have been used largely for approximating different kind of differential problems. We propose its use for solving singular boundary value problems. The proposed strategy consist in a block method combined with an appropriate set of formulas which are developed at the first subinterval to circumvent the singularity at the left end of the integration interval. Some examples are presented to validate the efficiency of the proposed strategy.

非完整约束Hamilton动力系统保结构算法

Based on the concept of variational integrator and the Lagrange-d’Alembert principle with dual variables, a high-order structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints was proposed. Based on the variational integrator, a discretization form of the Lagrange-d’Alembert principle with dual variables was obtained by means of appropriate polynomials and quadrature rules. On the basis of this discretization form, a numerical integration method was given with displacements at both ends of the integral interval as independent variables. Meanwhile, the nonholonomic constraints were strictly met at the endpoints of the integral interval and the control points within the interval. The symmetric property of the proposed algorithm was proved. Numerical examples show that, the proposed algorithm has a high convergence order, strictly meets the nonholonomic constraints and has good long-time behaviors.

A class of approximate computation method for antenna directivity

Directivity is one of the core performance parameters for an antenna, and its accurate computation and measurement have been receiving attention in the past decades. While quite a number of computational methods for antenna directivity are developed, a comprehensive comparison among these methods, including their advantages and disadvantages, has not yet been reported. In the literature, most of these methods can be roughly classified as two categories, i.e. numerical methods and analytical methods. The numerical methods include those simplified numerical methods for accessing the radiation pattern. For example, some methods assume that the radiation pattern is only elevation-angle dependent, and an approximate truncation of the integral interval is often employed. The analytical methods are developed, where the electromagnetic radiation intensity and the total radiation power for a specific antenna are derived analytically. Nevertheless, so far there have been less efforts dedicated to developing general computational methods based on the antenna's planar near-field theory and associated computational schemes, although indeed quite a few of advanced mathematical approaches have been used to obtain the accurate directivity, including the series expansion based on the spherical wave scheme and the Fourier expansion scheme. In spite of these efforts, one can see that a general accurate numerical scheme for the antenna directivity is still urgently required. As an important supplement to these efforts, this paper adopts or proposes four numerical integration methods regarding planar near-field measurement of the antennas under test. Based on the planar near-field theory, microwave near-field data, and fast Fourier transform (FFT), this paper discusses the numerical integration-based algorithm schemes based on the simulated (or measured) near-field data and relevant numerical uncertainty estimation. Specifically, an antenna with accurate directivity is chosen for illustrating the accuracy assessment. First, a set of near-field data under the well-established probing condition is simulated and then used as an input for subsequent numerical calculation of the antenna directivity value. Four different numerical integration methods are employed and their possible numerical errors are discussed. Consequently, the antenna directivity values are obtained respectively from the four different numerical methods. It is revealed that the numerically evaluated directivity values from the near-field data are roughly at most 0.6 dB larger than the accurate value. This work represents a substantial step toward a reliable estimation of the antenna directivity from the near-field data of an antenna.

Codeless Acquisition of GNSS Signals

Theoretical characteristics of codeless GPS-signal acquisition in the L1 band are described and analyzed. The results obtained allow for a reliable choice of a coherent integration interval, calculation of signal detection thresholds, and estimation of signal detection probability. The operability and feasibility of the technique for codeless signal acquisition by frequency has been experimentally confirmed with the use of a software receiver.

Closed-Form Asymptotic Approximation of Target’s Range Information in Radar Detection Systems

The echoes of the radar systems can provide useful information about the target, including range, velocity, shape, and angular direction. Extensive studies have utilized the information of the target to improve system performance, whereas the problem of a favorable closed-form asymptotic approximation of the target's range information (RI) is seldom investigated. In this paper, we address the problem of obtaining a closed-form asymptotic approximation of the target's RI in all SNR regions for radar detection systems. The RI is formulated as the mutual information (MI) between the random range and the received signals with complex additive white Gaussian noise (CAWGN). The basis of our scheme is to employ the a posteriori probability density function (PDF) of the range to extract RI from the echoes. We show the a posteriori PDF is a function of the autocorrelation function (ACF) of signal and the cross-correlation function (CCF) of signal-noise. By dividing the integration interval of the a posteriori PDF into the signal-noise interval and noise interval, a closed-form asymptotic approximation of RI is derived based on the probability of range distinguishability and the normalized a posteriori entropies of low and high signal-to-noise ratio (SNR). The results also reveal an interesting relationship between the RI and the uncertainty of the target. As special cases, the closed-form approximations of RI in low and high SNR are obtained. Moreover, for the problem of slightly looser in medium SNR approximation, a better approximation is derived based on a linear model approach. Numerical results are presented to validate the proposed theory and verify the effectiveness of the proposed approximation.

Parameter Estimation of a Horizontally Multilayered Soil With a Fast Evaluation of the Apparent Resistivity and its Derivatives

The information of earth structure is curial for designing the grounding system and evaluating the transmission line parameters. In order to obtain the earth structure using the measured data, an efficient algorithm for the parameter estimation of a horizontally multilayered soil by using a novel integration method is proposed. In the proposed algorithm, an adaptive iteration process will terminate till the predefined error reaches the requirement, where the earth parameters, such as the layer depth, layer resistivity, and layer number, are adjusted accordingly with the help of the Levenberg-Marquardt algorithm. In each iteration, the key is to evaluate the apparent resistivity and its partial derivatives with respect to soil parameters, which are expressed in the form of general Sommerfeld integral (GSI). In this paper, the GSI is divided into two parts. The first part with an infinite upper limit is integrated along a novel complex integration path instead of the conventional real axis path, which makes the evaluation of the GSI highly efficient due to the fast damping property of the transformed kernel. The second part with the finite integral interval is calculated using the cubic spline technique to have a semi-analytic expression, which can avoid evaluating the oscillatory kernel. In comparison with the results in the published literatures, numerical examples show that the proposed soil parameters estimation approach can reach good accuracy, and is robust and efficient. The reason is simple as the GSIs are calculated by using the well-designed and sophisticated approach. With extensive numerical experiments, the proposed algorithm is believed to be a capable and applicable candidate in engineering to invert the information of the earth structure.

Definite quadrature formulae of 5-th order with equidistant nodes

We construct sequences of definite quadrature formulae of order five which use equidistant nodes. The error constants of these quadratures are evaluated and simple a posteriori error estimates derived under the assumption that the integrand's fifth derivative does not change its sign in the integration interval.

On construction of Lyapunov functions for scalar linear time-varying systems

It is known that the construction of Lyapunov functions for scalar linear time-varying systems is related with solutions to the scalar Lyapunov differential equation, whose solution involves both improper integrals and double integrals, and thus are not easy to compute in general. This paper establishes a systematic method for constructing Lyapunov functions for scalar linear time-varying systems. The constructed Lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval. Explicit conditions are imposed on the weighting function and the integral interval such that the Lyapunov function is both positive definite and uniformly bounded, and its time-derivative is negative definite. As a result, constructive solutions to the associated scalar Lyapunov differential equations are also obtained. The established method includes some existing ones as special cases. Examples demonstrate the effectiveness of the proposed methods.

A new third order convergent numerical solver for continuous dynamical systems

Continuous dynamical systems are attempted to solve via proposed third order convergent numerical solver. The solver is shown to possess third order convergence on the basis of containing O(h4) term in the leading coefficient of the local truncation error which has further been employed for the local and global error bounds whereas its increment function is found to be Lipchitz continuous. Consistency of the solver has extensively been discussed using Taylor’s series expansion. The necessary and sufficient conditions for the solver to be stable have been proved accompanying the linear stability function obtained via Dahlquist test problem. In order to illustrate the performance of the solver, few scalar and vector-valued dynamical systems of both linear and nonlinear nature have numerically been solved in comparison to two well-known numerical solvers having same order of convergence as that of the proposed solver. Whereupon, the proposed solver is found to contain the smallest errors. The analysis of comparison is based upon three kinds of errors; namely, maximum absolute global relative error, absolute relative error determined at the final nodal point along the integration interval and the ℓ2-error norm. MATLAB software having version 9.3.0.713579 (R2017b), using Intel(R) Core(TM) i3-4500U processor running on 1.70 GHz has been used.

Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates

The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.

Minimizing Integration Interval in the Calculation of the Center of Gravity for an Elution Curve

For the case of nonanalytical chromatography, elution curves have been simulated for the adsorbent layers different in relative lengths. By adjusting integration interval, the calculation errors are obtained for the center of gravity of an elution curve depending on the value of this interval, normalized with respect to its ratio to the time of the elution curve maximum. The obtained dependences make it possible to control the minimization of the integration interval and thereby to exclude from the calculated values the parts of the experimental curve that lead to additional errors in the calculation of moments owing to the drift and fluctuations of the baseline signal.