Iterative Formula Research Articles

Principle and Control of Active Engine Mount Based on Magnetostrictive Actuator

Engine mount system affects the automobile NVH performance. Active mounts would achieve excellent vibration isolation and relative displacement control performance in a broad frequency bandwidth by outputting controlled force to the mounting system. The actuator and control method of the active mounts determine the system performance. In this paper, an active mount based on the smart material, i.e., Terfenol-D rod, is proposed, which mainly includes three parts: rubber spring, magnetostrictive actuator (MA), and hydraulic amplification mechanism (HAM). Dynamic model of the active mount is correspondingly established. A state feedback control method based on x-LMS (Least-Mean-Square) algorithm is proposed as well. Specifically, with the consideration of the unmeasurable state parameters in the active mounting system, an x-LMS state feedback controller with the system state as the reference signal is constructed by employing Sage-Husa Kalman filter to realize the state estimation of the active mounting system. Then a detailed analysis of the proposed control method is conducted, with deriving iterative formula of tap-weight vector. Sequentially, the problem of the dependence on the excitation signal in the x-LMS algorithm is addressed. The feasibility and capability of the proposed control method are verified and evaluated by simulation of a two-degree-of-freedom active mounting system.

Fast and Stable Iterative Algorithm for Searching Wheel–Rail Contact Point Based on Geometry Constraint Equations

Abstract This paper proposes a fast and stable iterative algorithm for wheel–rail contact geometry based on constraint equations, which can be implemented in dynamic wear simulations that real-time profile updating is needed. Further, critical factors that determine convergence and iteration stability are analyzed. A B-spline is adopted for wheel–rail profile modeling because it does not contribute to changes in the global shape of curves. It is found that the smoothness of the first and second derivative curves significantly affects the numerical stability of the Jacobian matrix, which determines the increments in iterations. Moreover, a damped Newton's iteration formula with a scaling factor of 0.5 is proposed considering the convergence rate and out-of-bound issues for the updated step. The influence of the initial iteration parameters on the convergence is studied using Newton fractals. The range within ±3 mm, centered on the target contact point, is found to be an unconditionally stable domain. The proposed method could achieve convergence within 10 and 30 steps under thread and flange contact conditions, respectively.

Fractal interwoven resonator based penta-band metamaterial absorbers for THz sensing and imaging

This paper presents a unique penta-band metamaterial absorber platform for terahertz imaging systems.The proposed fractal metamaterial absorber (FMMA) consists of fractal triangle section metasurfaces. By combining fractal resonators posing different operation skills in the same unit cell, the absorber shows multiband spectral response. The proposed unit cell structure operates at five near perfect absorption modes corresponding to the frequency bands of 1.1 THz, 3.4 THz, 4.9 THz, 5.9 THz, and 7.8 THz, respectively. Based on the fractal metamaterial absorber array, we also propose a sensing pixel design for bimaterial cantilever array sensing systems. The single pixel assembles 4x6 fractal resonator array and SiO_2-Al bimaterial microcantilevers. The sensing region of the FMMA pixel can bend the bimaterial cantilevers effectively at multiple modes, enhancing the imaging capacity. The effective medium theory is executed to visualize the constitute parameters during the absorption and reveal the origin of the rising modes. The absorption mechanism is also discussed based on the surface current distributions and electric field profiles. The numerical outcomes prove that the proposed fractal metamaterial unit cell is a promising candidate as an absorbing platform for THz band sensing and imaging applications. The derived iterative formula used in the fractal design procedure is explained for further investigations of microelectromechanical systems (MEMS) compatible compact absorber arrays.

Research on multi-moving target location algorithm based on improved TDOA/FDOA

Wireless positioning and tracking technology are key technologies in applying wireless sensor networks and have vital research significance and application value. In this paper, in terms of positioning multiple non-coincident dynamic targets, aiming at the problem of low precision and large computational complexity when locating multiple dynamic targets by the two-step least squares algorithm and the constrained total least squares algorithm, an improved constrained total least squares algorithm is proposed. This algorithm fully considers the constraints, introduces the Lagrangian multiplier technology and the quasi-Newton BFGS iterative formula, avoids the calculation of the Hessian matrix, reduces the amount of calculation, and improves the positioning accuracy. Simulation experiments show that when the measurement error and sensor position error are moderate, the ICTLS positioning algorithm has a smaller RMSE than the TSWLS and CTLS positioning algorithms, showing higher positioning accuracy and stronger robustness. Secondly, aiming at the problem that the target is close to the reference node or any coordinate axis causes the positioning error of the traditional positioning algorithm to increase sharply, an optimized two-step least squares algorithm is proposed. This algorithm corrects the defects of the TSWLS algorithm by selecting the reference station and rotating the coordinate system again, and improves the positioning performance of the algorithm while reducing the amount of calculation.

Evaluating intraocular lens power formula constant robustness using bootstrap algorithms.

Bootstrapping is a modern technique mostly used in statistics to evaluate the robustness of model parameters. The purpose of this study was to develop a method for evaluation of formula constant uncertainties and the effect on the prediction error (PE) in intraocular lens power calculation with theoretical-optical formulae using bootstrap techniques. In a dataset with N=888 clinical cases treated with the monofocal aspherical intraocular lens (Vivinex, Hoya) constants for the Haigis, the Castrop and the SRKT formula were optimised for the sum of squared PE using nonlinear iterative optimisation (interior point method), and the formula predicted spherical equivalent refraction (predSEQ) and the PE were derived. The PE was bootstrapped NB=1000 times and added to predSEQ, and formula constants were derived for each bootstrap. The robustness of the constants was calculated from the NB bootstrapped models, and the predSEQ was back-calculated from the NB formula constants. With bootstrapping, the 90% confidence intervals for the a0/a1/a2 constants of the Haigis formula were -0.8317 to -0.5301/0.3203 to 0.3617/0.1954 to 0.2100, for the C/H/R constants of the Castrop formula they were 0.3113 to 0.3272/0.1237 to 0.2149/0.0980 to 0.1621, and for the A constant of the SRKT formula they were 119.2320 to 119.3028. The back-calculated PE from the NB bootstrapped formula constants standard deviation for the mean/median/mean absolute/root mean squared PE were 5.677/5.735/0.401/0.318 e-3 dpt for the Haigis formula, 5.677/5.735/0.401/0.31829 e-3 dpt for the Castrop formula and 14.748/14.790/0.561/0.370 e-3 dpt for the SRKT formula. We have been able to prove with bootstrapping that nonlinear iterative formula constant optimisation techniques for the Haigis, the Castrop and the SRKT formulae yield consistent results with low uncertainties of the formula constants and low variations in the back-calculated mean, median, mean absolute and root mean squared formula prediction error.

Accelerated convergence strategy of weighted least squares method for image reconstruction

The weighted least squares algorithm is effective in image reconstruction, in which the weight is determined by a covariance matrix of model error. To speed up the iterative convergence speed of the WLS method, we express the WLS iterative formula in a matrix form with an iterative relaxation coefficient and propose relaxation strategies based on minimizing the spectral radius of the iterative matrix. The numerical simulations demonstrate that the relaxation strategies proposed in this paper have the better reconstruction performances.

Push-Sum Based Algorithm for Constrained Convex Optimization Problem and Its Potential Application in Smart Grid

Dear Editor, This letter aims to investigate the optimization problem where the decision variable is contained in a closed convex set. By combing the gradient decent-like method with the push-sum algorithm framework, we design the distributed iterative formulas under the condition that the considered graphs sequence is time-varying and unbalanced. Under some standard assumptions on the problem setting and the decaying step-sizes, we analyze the convergence property of the generated variables sequence under the proposed iterative formulas.

A General Solution for the Errors in Variables (EIV) Model with Equality and Inequality Constraints

Targeting the adjustment of the errors-in-variables (EIV) model with equality and inequality constraints, a general solution that is similar to the classical least square adjustment is proposed based on the penalty function and the weight in measurement. Firstly, we take the equality constraints as inequality constraints that do not satisfy the constraint conditions and construct the penalty functions of equality and inequality constraints, respectively. Thus, the inequality constrained optimization problem is transformed into an unconstrained optimization problem. Then the detailed calculation formula and approximate accuracy evaluation formula of the general solution are deduced. The iteration formula of the general solution is easy regarding comprehension and applicable in implementation. It can not only solve the EIV model with equality and inequality constraints respectively, but also address the EIV model with equality and inequality constraints simultaneously. In addition, it can promote the Gauss–Markov (G-M) model with equality and inequality constraints. Finally, three examples (i.e., equality constraints, inequality constraints and those with equality and inequality constraints) are validated, indicating that the general solution is effective and feasible. The results show that the general solution is effective and feasible.

Iterative isotropization of the upper and lower bounds for accurate prediction of elastic properties of particulate composites

The Voigt-Reuss (VR) and the Hashin-Shtrikman (HS) formulas are widely applied to estimate the upper and lower bounds of effective elastic properties of particulate composites, but they are not able to predict the specific values of the properties. The commonly used micromechanics models are inaccurate for composites that have large contrast of phase properties and high volume-fraction of inclusions. In this paper, we first demonstrate that the gap between the upper and the lower bounds is caused by the anisotropy existing in the micromechanics model. We then introduce a novel iterative isotropization procedure to remove the anisotropy and thus to eliminate the bound gap. Upon the convergence of the iteration procedure, the upper and the lower bounds become coincident, which provides a prediction of the property value. The validation and comparison against both experimental data and finite element modeling results show that, the new iterative formulas are able to predict the effective elastic properties more accurately than the commonly used micromechanics models, especially for particulate composites that have large contrast of phase properties and high volume-fraction of inclusions.

Financial Factors Contribution to SMEs’ Profitability

Micro, small, and medium enterprises play a very important role in driving the economy both nationally and internationally. The level of profitability is one indicator of the performance of SMEs. One of the factors that affect profitability is the financial factor. Therefore, this study attempts to analyze the contribution of financial factors (Own Capital, Loan Capital, Assets, Trade Credit, Production Costs, Sales, Average Wages, Taxes, and Lagged Profitability) to the profitability of SMEs. The population consists of all SMEs in the city of Malang. The sampling technique used an iteration formula and obtained a sample of 116 SMEs in Malang that received financing from Islamic financial institutions, banks, and non-banks. Questionnaires and observational interviews collected data. The data analysis technique used Multiple Linear Regression. The result of the determination test on this research model is 0.839 (84%). And based on the results of the partial test, of the nine independent variables in the study, only three variables have a significant positive effect on the profitability of SMEs, namely Own Capital, Loan Capital, and Lagged Profitability. And of the three variables that provide the largest contribution or influence on profitability is Own Capital.

Binary Newton Calculation Method of Residual Stress Based on the Indentation Energy Difference Theory

Residual stress is a key parameter to evaluate the structural reliability of energy equipment. The indentation method has the characteristics of being nondestructive and easy to operate to calculate the residual stress of test materials, which has a broad application prospect in the field testing of energy equipment. However, because of the effect of preloading and data acquisition delay, the problem of indentation data fluctuation is prominent, and the indentation energy coefficient fitted by the traditional least square method is not consistent with the theoretical law, making it difficult to carry out stable calculations. In this paper, the Newton iteration formula of a binary nonlinear formula is derived based on the univariate Newton iteration formula, which is introduced into the data processing of residual stress, which increases the weight of the data in the stability stage and reduces the influence of the fluctuation data on the fitting results, so that the indentation energy coefficient is accurately calculated. Combined with the basic principle of indentation energy difference theory, the precise and efficient measurement of residual stress is realized.

Analysis on the Establishment and Management of Library Resource Base Based on Modern Information Technology

The current rapid development of modern network information technology, along with the arrival of this trend, has profoundly changed people’s living and learning methods. In the past, the management of book resources in the traditional period has gradually exposed many drawbacks, and it is difficult to meet the technological development of the information age. Combined with the complexity of book management parameters, high variable dimensions, and the characteristics of multiattribute data points including not only numerical attributes but also category attributes and mixed attributes, the fuzzy clustering algorithm is combined with attribute weighted optimization, and then, the optimization is derived. Iterative formula and form show a weighted clustering algorithm to perform cluster analysis on relevant influencing factors in book data management. Digital library is not a library entity: it corresponds to various real social activities of public information management and dissemination and is manifested in various new information resource organizations and information dissemination services. As a fuzzy association rule mining algorithm, FP-growth algorithm is obviously better than Apriori algorithm in execution efficiency. However, due to the lack of fuzzy attributes and the large space complexity, the FP-growth algorithm cannot achieve effective multilayer association rule mining when dealing with large transaction databases, such as library databases. First, the algorithm divides the large-scale book transaction database into several sub-databases according to the transaction of the first item and constructs the corresponding sub-FP-tree structure; then, the parent items that are not frequent items in the hierarchical tree are filtered out in real time to reduce the scanning space.

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.

Robust Adaptive Beamforming Based on Sparse Bayesian Learning and Covariance Matrix Reconstruction

Sparse Bayesian learning (SBL) exploits sparse information from a Bayesian perspective by making sparse prior assumptions about the signal. It has good flexibility in modeling sparse signals and can also achieve good estimation results when the number of samples is low. In this letter, we propose a robust adaptive beamforming (RAB) method based on interference-plus-noise covariance matrix (INCM) reconstruction using SBL. This method derives the fixed point iteration formula of the signal under the Laplace prior distribution by maximizing the marginal likelihood function of the data received by the array. Then the sampling position of the coarse grid in the SBL model is regarded as an adjustable parameter, and it is refined through iteration. By using the superiority of SBL in modeling sparse signals, the steering vectors and power of the source signals are precisely estimated, and an accurate INCM is reconstructed to obtain a high output SINR. The only prior knowledge required is imprecise angular sector knowledge of the desired signal (DS). Simulation results demonstrate that our proposed method achieves near-optimal output SINR over a wide range of input SNR and a limited number of snapshots and even achieves good performance in the single snapshot situation.

A New Secant Type Method with Quadratic-Order Convergence

Using the development of the second approximation, a variation of the standard secant technique for nonlinear problems has been developed. The iterative formula is created using Taylor series expansion, which includes an estimate of the second derivative of Θ(μ). It is demonstrated that the new approaches have quadratic convergence. In comparison with the Newton technique employing seven test functions in the practical application, it turns out that the performance of the method is efficient.

Boundary element method with particle swarm optimization for solving potential problems

In this paper, the boundary element method (BEM) based on the particle swarm optimization (PSO) algorithm is proposed for solving potential problems. At present, the traditional Gaussian elimination method still needs to improve for computing results accuracy. The PSO algorithms with substantial accuracy used to improve the accuracy of potential problems. So it is introduced the PSO to solve the system of matrix equations derived from the boundary integral equation (BIE). For the presented method, unknown solutions to the system of matrix equations are considered particles, and the optimal solution is sought through group collaboration. Some parameters in this model play a vital role, so confirming them and initializing the positions and velocities of the particles are needed. Then, the positions of the particles are constantly updated with the iteration formula until obtained the optimum solutions. Furthermore, the optimum solution are arrived at in which the cycle is jumped when meeting the conditions (i. e. given the maximum number of iterations or the fitness value). There are examples with constant element and linear element and quadratic element types of boundary, in turn. The three numerical models of 2D potential problems are analyzed and discussed. They certified that the PSO algorithm has feasibility and good convergence. Most importantly, it is proved that the PSO applied in BEM has higher accuracy compared with the Gaussian elimination method for solving potential problems.

Recurrent neural network-induced Gaussian process

In this study, we develop a recurrent neural network-induced Gaussian process (RNNGP) to model sequence data. We derive the equivalence between infinitely wide neural networks and Gaussian processes (GPs) for a relaxed recurrent neural network (RNN) with untied weights. We compute the covariance function of the RNNGP using an analytical iteration formula derived through the RNN procedure with an error-function-based activation function. To simplify our discussion, we use the RNNGP to perform Bayesian inference on vanilla RNNs for various problems, such as Modified National Institute of Standards and Technology digit identification, Mackey–Glass time-series forecasting, and lithium-ion battery state-of-health estimation. The results demonstrate the flexibility of the RNNGP in modeling sequence data. Furthermore, the RNNGP predictions typically outperform those of the original RNNs and GPs, demonstrating the efficiency of the RNNGP as a data-driven model. Moreover, the RNNGP can quantify the uncertainty in the predictions, which implies the significant potential of the RNNGP in uncertainty quantification analyses.

Feature Nonlinear Transformation Non-Negative Matrix Factorization with Kullback-Leibler Divergence

This paper introduces a Feature Nonlinear Transformation Non-Negative Matrix Factorization with Kullback-Leibler Divergence (FNTNMF-KLD) for extracting the nonlinear features of a matrix in standard NMF. This method uses a nonlinear transformation to act on the feature matrix for constructing a NMF model based on the objective function of Kullback-Leibler Divergence, and the Taylor series expansion and the Newton iteration formula of solving root are used to obtain the iterative update rules of the basis matrix and the feature matrix. Experimental results show that the proposed method obtains the nonlinear features of data matrix in a more efficient way. In object recognition and clustering tasks, better accuracy can be achieved over some typical NMF methods.

Resilient Filtering of Nonlinear Complex Dynamical Networks Under Randomly Occurring Faults and Hybrid Cyber-Attacks

This paper investigates the problem of recursive resilient filtering for discrete-time nonlinear dynamical networks with randomly switching topologies under randomly occurring faults and hybrid cyber-attacks. The attacks considered are composed of DoS attacks describing by binary random sequences and deception attacks, the amplitude of which obeys the Gaussian mixing distribution. A recursive resilient filter dependent on DoS attack sequences is designed to simultaneously estimate system states and occurred faults, where the desired gain is parameterized via the matrix iterative formula of filtering error covariance. Furthermore, the maximum posterior probability estimation is calculated to identify DoS attack sequences with the help of the classical Bayesian rule. In particular, a modified recursive filtering algorithm is established by replacing the corresponding parameters in the previously derived formulas about filter gains and covariance matrices. Finally, the effectiveness of the proposed recursive filtering algorithm is demonstrated by a simulation example.

Study on the Electromagnetic Scattering Characteristics of High-Speed Target With Non-Uniform Plasma via the FCC-FDTD Method

The face-centered cube (FCC) grid has more sampling points than the traditional Yee cell (as it contains a total of 14 grid points, of which six are center points and eight are vertices, and each grid point has three field components). Thus, FCC is better than the former in dealing with the dispersion error of anisotropic media. On the basis of this feature, this article first derived the FCC-finite-difference time-domain (FDTD) iterative formula for nonuniform plasma on the basis of FCC grid and verifies the accuracy of the algorithm. This algorithm is combined with the nonuniform plasma flow field model of hypersonic vehicles, and the backward single-station radar cross sections (RCSs) of vertical polarization (VV) and horizontal polarization (HH) are calculated under different flight altitudes, flight velocities, and external conditions. Results showed that as flying height increased or flying speed decreased, it results in an increase in RCS. Different external magnetic fields and incident angles also had an obvious impact on the electromagnetic scattering characteristics. From the perspective of electromagnetic wave polarization, both VV polarization and HH polarization have a great impact on RCS. These results provide a theoretical basis for studying the electromagnetic scattering characteristics of nonuniform plasma targets.