Order Taylor Series Expansion Research Articles

Combined Kalman and sliding innovation filtering: An adaptive estimation strategy

This paper proposes a new adaptive estimation strategy for a nonlinear system with modeling uncertainties. The extended Kalman filter (EKF) and unscented Kalman filter (UKF) are optimal estimators which have been used extensively for state estimation in literature and industry. While the EKF uses a first order Taylor series expansion to approximate nonlinearities, the UKF uses sigma points from the projected probability distribution of states. The sliding innovation filter (SIF) is a suboptimal, yet robust estimation strategy which has recently been proposed. For nonlinear systems, the extended SIF (ESIF) is formulated by using a first order Taylor series expansion like the EKF. This work proposes a novel adaptive estimation strategy which combines and balances the optimality of the EKF and UKF with the robustness of the ESIF. These new methods are referred to as the EKF-ESIF and UKF-ESIF, respectively. A time-varying sliding boundary layer is used as a means of detecting the presence of faults or uncertainties and as a criterion for switching between the EKF or UKF and the ESIF. In normal operating conditions the algorithm computes estimates using an optimal KF-based gain, and an SIF-based gain when a fault is detected. The system examined in this study consists of a magnetorheological (MR) damper with a constant current. Faults or uncertainties are introduced as unwanted behavior in the power supply in the form of undercurrent and overcurrent. The robustness of the EKF-ESIF and UKF-ESIF was validated for force estimation exerted by the MR damper and the results were compared with the standard EKF and UKF.

Finite rate of innovation sampling of Gaussian pulse streams with variable shape

The existing Finite Rate of Innovation (FRI) sampling theory can realize the efficient measurement of Gaussian pulse streams with fixed modal functions. However, it is hard to sample that signal with variable pulse width factors by a sub-Nyquist rate. To get around this challenge, we propose an extended FRI sampling method based on Taylor series expansion in this paper, where the signal is first filtered by the exponential reproducing kernel, then each pulse can be reconstructed sequentially. For a single pulse, we first prove that the pulse width factor does not interfere with the estimation process of the time delay parameter, and we also present its Cramer-Rao lower Bound (CRB). The signal Fourier coefficient can then be reformulated by the Taylor series expansion of order N−1 to realize the decoupling of the pulse width factor before the final estimation. We derive the order limit of expansion from the perspective of numerical stability, and we also give the upper bound on the estimation error of the pulse width factor for N=2. Simulation results show that, compared with the classical methods, the proposed state-of-the-art algorithm can achieve a higher precision reconstruction of the Gaussian pulse stream signal.

Hybrid Beamforming With Discrete Phase Shifts for RIS-Assisted Multiuser SWIPT System

Reconfigurable Intelligent Surfaces (RIS) provides an emerging affordable solution by leveraging massive low-cost reconfigurable reflective array elements. In this letter, by utilizing low-cost passive reflecting elements, a joint hybrid beamforming with discrete phase shifts design is studied in RIS-assisted multiuser simultaneous wireless information and power transfer (SWIPT) systems. However, the formulated sum rate maximization problem is nonconvex and very challenging to solve. To overcome this difficulty, we decouple the original maximization problem into the digital-, phase shifts and analog subproblem. Specifically, for the nonconvexity of the digital subproblem, we perform the first order Taylor series expansion on the objective function, and then an iterative strategy is developed to tackle the difficulty of the maximum transmit power constraint imposed by discrete optimization variables. Subsequently, an effective convex optimization technique was employed to optimize reflecting elements of RIS. Finally, the greedy selection method based on minimum mean squared error (MMSE) scheme is proposed to solve the analog subproblem. Numerical simulations are conducted to demonstrate the performance advantages of the proposed scheme.

High-order meshless method based on the generalized finite difference method for 2D and 3D elliptic interface problems

In this article, a high-order meshless method based on the generalized finite difference method (GFDM) is proposed to deal with the elliptic interface problem. The present method is capable of treating complex interfaces with non-homogeneous jump conditions to obtain high-order accuracy. The operation to improve the convergence order is only increasing the order of Taylor series expansion in the GFDM. Numerical examples show the L∞, L2 and H1 errors of this method can obtain 2nth order convergence by using 2nth order Taylor series expansion.

Generalized isogeometric boundary element method for uncertainty analysis of time-harmonic wave propagation in infinite domains

This paper proposes a novel generalized nth order perturbation isogeometric fast multipole boundary element method for time harmonic wave propagation in infinite domains. The non-uniform rational B-splines are employed to construct structural geometries and discretize boundary integral equations. The randomness of wave number for incident plane wave is considered as the source of system uncertainty. The generalized nth order perturbation method is employed to model the physical field depending on the input random variable. The nth order derivatives of field functions and kernel functions in the boundary integral equations are derived by generalized nth order Taylor series expansions with a small perturbation parameter. The subtraction of singularity technique is used to evaluate the singular integrals and the fast multipole method is applied to accelerate the solution. The Monte Carlo simulations are conducted in numerical examples to demonstrate the validity and correctness of the proposed algorithm.

The correlation of substitution effects across populations and generations in the presence of nonadditive functional gene action.

Allele substitution effects at quantitative trait loci (QTL) are part of the basis of quantitative genetics theory and applications such as association analysis and genomic prediction. In the presence of nonadditive functional gene action, substitution effects are not constant across populations. We develop an original approach to model the difference in substitution effects across populations as a first order Taylor series expansion from a “focal” population. This expansion involves the difference in allele frequencies and second-order statistical effects (additive by additive and dominance). The change in allele frequencies is a function of relationships (or genetic distances) across populations. As a result, it is possible to estimate the correlation of substitution effects across two populations using three elements: magnitudes of additive, dominance, and additive by additive variances; relationships (Nei’s minimum distances or Fst indexes); and assumed heterozygosities. Similarly, the theory applies as well to distinct generations in a population, in which case the distance across generations is a function of increase of inbreeding. Simulation results confirmed our derivations. Slight biases were observed, depending on the nonadditive mechanism and the reference allele. Our derivations are useful to understand and forecast the possibility of prediction across populations and the similarity of GWAS effects.

A Solution Approach for a Class of Parametric Linear Programming Problems

Depending on the nature, objectives, and constraints of the decision variables; linear programming, nonlinear programming, integer programming, mixed integer programming etc. can be classified. Extensive research has been conducted to solve all types of these problems in a parametric context. In this paper, to solve optimization problems having uncertainties represented by a single parameter on the objective function, a systematic linearization approach is developed considering the parametric expression as nonlinear. In the proposed approach, the objective function is considered as nonlinear which is converted into linear by using first order Taylor series expansion at the points making the parametric costs zero. Thus, the optimal solution is obtained from the constructed linear programming problem. In this way, by determining the intervals in which the optimal solution changes, the solution of the parametric linear programming problem is obtained. A numerical experiment is illustrated to present the effectiveness of the proposed approach.

Accurate explicit analytical solution for Colebrook-White equation

This paper presents a direct closed form analytical solution for the implicit type equation of Colebrook-White for friction factor calculation. Based on the developed expression, the friction factor can be directly calculated given the fluid flow parameters namely the relative roughness and the Reynolds number. Obtained expression is explicit and it does not require any iterative calculation as it is built reposing on the analytical solution of a cubic equation of real coefficients. The used cubic equation is obtained through a 3rd order Taylor series expansion of the original Colebrook-White equation around an expansion point reasonably defined. The reliability of the proposed expression is proved through comparison with original Moody chart values as well as others results issued from previous approximations found in literature and the exact solution of Colebrook-White equation. The accuracy and the robustness of the established expression over other explicit correlations were emphasized. The presented closed form expression was proved to be simple, robust and accurate. Additionally it has the advantages of covering all the range of applicability of the original Colebrook-White equation.

Efficient 2nd Order Taylor Series Expansion Model to Integrate Stiff and Non-Linear ODEs: Comparison to Rk45-Type Integrators

Efficient 2nd Order Taylor Series Expansion Model to Integrate Stiff and Non-Linear ODEs: Comparison to Rk45-Type Integrators

Investigation into the influence of bearings nonlinear forces in unbalance identification

Rotating machines are key components in several industrial sectors, mainly, in energy generation. In this way, a rotating component supported on hydrodynamic bearings creates typical problems, being high vibrations amplitudes due to unbalance, one of the most common. To avoid failures and ensure a safe operation, the rotor should be balanced, and influence coefficient methods are usually used. Since balancing by the traditional influence coefficients method requires trial masses, the machine can sometimes experience a long setup time, causing financial losses. Also, this method assumes a linear rotor response that can hamper the balancing procedure for nonlinear situations, which happens when the rotor experiences high vibrational motion. Thus, this work proposes a balancing identification that considers nonlinear bearings and avoids trial masses. For this, a mixed-integer gradient-based optimization is presented. The theoretical model of a rotor supported by hydrodynamic bearings is obtained using the Finite Element method and solving the Reynolds equation. In order to save computational time, the bearing forces are approximated by a fifth order Taylor series expansion. The presented results show that nonlinear bearing consideration can improve the machine diagnosis.

GNSS-Based Non-Negative Absolute Ionosphere Total Electron Content, its Spatial Gradients, Time Derivatives and Differential Code Biases: Bounded-Variable Least-Squares and Taylor Series.

Global navigation satellite systems (GNSS) allow estimating total electron content (TEC). However, it is still a problem to calculate absolute ionosphere parameters from GNSS data: negative TEC values could appear, and most of existing algorithms does not enable to estimate TEC spatial gradients and TEC time derivatives. We developed an algorithm to recover the absolute non-negative vertical and slant TEC, its derivatives and its gradients, as well as the GNSS equipment differential code biases (DCBs) by using the Taylor series expansion and bounded-variable least-squares. We termed this algorithm TuRBOTEC. Bounded-variable least-squares fitting ensures non-negative values of both slant TEC and vertical TEC. The second order Taylor series expansion could provide a relevant TEC spatial gradients and TEC time derivatives. The technique validation was performed by using independent experimental data over 2014 and the IRI-2012 and IRI-plas models. As a TEC source we used Madrigal maps, CODE (the Center for Orbit Determination in Europe) global ionosphere maps (GIM), the IONOLAB software, and the SEEMALA-TEC software developed by Dr. Seemala. For the Asian mid-latitudes TuRBOTEC results agree with the GIM and IONOLAB data (root-mean-square was < 3 TECU), but they disagree with the SEEMALA-TEC and Madrigal data (root-mean-square was >10 TECU). About 9% of vertical TECs from the TuRBOTEC estimates exceed (by more than 1 TECU) those from the same algorithm but without constraints. The analysis of TEC spatial gradients showed that as far as 10–15° on latitude, TEC estimation error exceeds 10 TECU. Longitudinal gradients produce smaller error for the same distance. Experimental GLObal Navigation Satellite System (GLONASS) DCB from TuRBOTEC and CODE peaked 15 TECU difference, while GPS DCB agrees. Slant TEC series indicate that the TuRBOTEC data for GLONASS are physically more plausible.

Large‐eddy simulation of subsonic turbulent jets using the compressible lattice Boltzmann method

SummaryThe lattice Boltzmann method (LBM) is a powerful technique for the computational modeling of a wide variety of single‐s and multiphase flows involving complex geometries. Although the LBM has been demonstrated to be effective for the solution of incompressible flow problems, there are limitations when this methodology is applied to the solution of compressible flows, especially for flows at high Mach numbers. In this article, we investigate strategies to overcome some of the limitations associated with the application of LBM to compressible flows. To this purpose, one of the key contributions of this study is the synthesis and integration of previous efforts concerning the formulation of LBM for the large‐eddy simulation (LES) of compressible turbulent flows in the subsonic flow regime. It is shown how certain limitations of applying the LBM to compressible flows can be addressed by using either a higher order Taylor series expansion of the Maxwell–Boltzmann equilibrium distribution function or using the Kataoka and Tsutahara (KT) LBM model formulation for compressible flows. The proposed LBM/LES methodology for compressible flows has been combined with the Kirchhoff integral formulation for computational aeroacoustics and used to simulate the flow and acoustic fields of compressible jet flows at high subsonic speeds with practical relevance for providing a better understanding of problems associated with jet noise. In this context, simulations of the physics associated with the jet flow and concomitant noise in the near‐ and far‐field regimes were conducted using the proposed framework of a compressible LBM/LES and Kirchhoff integral method. The results of the subsonic isothermal and nonisothermal jet flow simulations for the flow and acoustic fields have been compared with available numerical and experimental results with generally good to excellent agreement.

A Modified Law of Heat Conduction of Thermoelasticity with Fractional Derivative and Relaxation Time

(1) In the present work, a new modified thermoelasticity theory with fractional order has been constructed based on fractional calculus and Taylor series expansion of time-fractional order. The models of fractional thermoelasticity proposed by Sherief et al. [H. H. Sherief, A. M. A. El-Sayed and A. M. Abd El-Latief, Int. J. Solids Struct. 47, 269 (2010)], Ezzat [M. A. Ezzat, Phys. B 406, 30 (2011)] and Lord and Shulman with one relaxation time [H. W. Lord and Y. H. Shulman, J. Mech. Phys. Solids 15(5), 299 (1967)] as well as coupled thermoelasticity [M. A. Biot, J. Appl. Phys. 27, 240 (1956)] follow as limiting cases. This modified model is applied to an infinitely long annular cylinder. The inner and outer surfaces of the cylinder are traction free and subjected to known surrounding temperatures. Laplace transform technique will be used to get the solutions of all physical quantities. Some comparisons are shown in figures and tables to assess the effects of the fractional-order parameters in the studied fields. Results of some earlier researchers have been deduced from the current formulation. Finally, a conclusion about the new modified model has been promoted according to the analysis and numerical results.

Estimation of Involuntary Components of Human Arm Impedance in Multi-Joint Movements via Feedback Jerk Isolation.

Stable and efficient coordination in physical human-robot interaction requires consideration of human feedback behavior. In unpredictable tasks, where voluntary cognitive feedback is too slow to guarantee desired task execution, the human must rely on involuntary intrinsic and reflexive feedback. The combined effects of these two feedback mechanisms and the inertial characteristics can be summarized in the involuntary impedance components. In this work, we present a method for the estimation of the involuntary impedance components of the human arm in multi-joint movements. We apply force perturbations to evoke feedback jerks that can be isolated using a high pass filter and limit the duration of the estimation interval to guarantee exclusion of voluntary cognitive feedback. Dynamic regressor representation of the rigid body dynamics of the arm and first order Taylor series expansion of the feedback behavior yield a model that is linear in the involuntary impedance components. The constant values of the inertial parameters are estimated in a static posture maintenance task and subsequently inserted to estimate the remaining components in a dynamic movement task. The method is validated with simulated data of a neuromechanical model of the human arm and its performance is compared to established methods from the literature. The results of the validation demonstrate superior estimation performance for moderate movement velocities, and less influence of the variability of the movements. The applicability to real data and the plausibility of the limited estimation interval are successfully demonstrated in an experiment with human participants.

A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems

Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined, either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.

Stochastic Finite Element Method for Modelling Heat Transfer in the Building Envelope

Improving the energy efficiency of the buildings is one of the most effective and fastest ways of reduction of the carbon dioxide emission. However, in the assessment of the energy demand of the buildings, numerous factors are uncertain, i.e. layer thickness, material parameters, climatic conditions, etc. In the present study, mathematical model was developed for analyzing temperature distribution and heat flux in the wall with thermal conductivity of insulation as a random parameter. The results obtained employing a stochastic perturbation technique were compared against the results of the Monte Carlo simulation. Stochastic perturbation technique has been implemented using the tenth order Taylor series expansion. The direct differential method was used to determine the values of Taylor’s coefficient. The obtained results indicate good accordance of the stochastic perturbation technique with the Monte Carlo method. Afterwards, the expected value of the heat flux and its variance were studied for the reference year for a city in the Central Europe. Two cases of the external wall were investigated, in which the thermal insulation was localized either on the internal or the external side of the wall. Performed analyses serve as a good method for assessing the reliability of results obtained using standard, deterministic approach.

Real-time estimation of state-of-charge in lithium-ion batteries using improved central difference transform method

Abstract The accurate and real-time estimation of state-of-charge (SOC) in lithium-ion batteries (LIBs) are crucial for battery management system (BMS) in electric vehicles. The SOC estimation is affected by several factors like temperature, aging and many other battery characteristics, making it challenging. In this study, improved central difference transform Kalman filter method based on square root second-order central difference transform (SRCDKF) was utilized for real-time estimation of SOC in LIBs. The hybrid pulse power characterization (HPPC) tests were combined with recursive least squares (RLS) method to identify the second-order equivalent circuit model parameters. To avoid high order Taylor series expansion and complicated multi-parameter adjustment in other Kalman filters, the CDKF with square root second-order difference transform is developed to generate Sigma point. The effectiveness of the proposed SRCDKF was then verified by pulse discharge and urban dynamometer driving schedule (UDDS) testing, and the results were compared with those obtained from extended Kalman filter (EKF) and unscented Kalman filter (UKF). In particular, the proposed algorithm provided small error within 2% under UDDS testing, and the convergence time was earlier than those obtained with the other two algorithms. The proposed SRCDKF can also guarantee the non-negative covariance and reduce the computational complexity. Overall, these findings look promising for future BMS SOC estimation in practice.

Measurement Extraction for Two Closely-Spaced Objects Using an Imaging Sensor

This paper considers measurement extraction for two closely-spaced objects with unknown equal intensities in an imaging sensor's focal plane array (FPA). Given a screen of FPA data, the first part of the measurement extractor, target location estimator, can extract the location estimates for two targets or one, with the corresponding accuracy given by the Cramer Rao lower bound (CRLB). The second part of the measurement extractor, target detector, selects among the hypotheses of two resolved targets and a single one using information-theoretic criteria and hypothesis tests. Simulation results have been conducted to evaluate the measurement extraction performance including the probability of resolving the two hypotheses, and the efficiency and unbiasedness of the target location estimates for the selected hypothesis using different hypothesis detection schemes. The generalized likelihood ratio test (GLRT) based on linearized observation model using second order Taylor series expansion is most appealing as it provides an explicit expression of the probability of detecting two targets as a function of the target separations, the signal-to-noise ratio at a given false resolution probability. It is shown that the simulation-based resolution performance for the GLRT using the estimated center location of the two targets matches well with the analytic performance assuming known center.

Iterative technique for optimizing two-channel: Quadrature mirror, bi-orthogonal and graph filter bank

An iterative technique is proposed to optimally design the two-channel linear phase (LP) FIR quadrature mirror filter bank (2c-LP-FIR-QMFB). In each iteration, first, the frequency-domain reconstruction error function of the 2c-LP-FIR-QMFB is approximately converted into a convex function, by approximating all its sub-functions using first order Taylor series expansion. Then, the approximated reconstruction error function is minimized as the optimization objective function while the frequency selectivity performance metric function of the sub-filter is taken as the constraint function. Further, the optimal step-size optimization problem for each iteration is solved using a similar iterative strategy. Additionally, the proposed technique can also be utilized to optimize the coefficients of a class of structurally perfect reconstruction two-channel LP FIR bi-orthogonal filter bank, as well as a class of two-channel graph filter bank. Simulation results reveal that the proposed technique outperforms the previously existing techniques in terms of the frequency-domain reconstruction error and/or the frequency selectivity performance metric of the sub-filter. Particularly, for the graph filter bank, the reconstruction error of the proposed technique is only (3.1e-11)% of its previous counterpart. The proposed technique might also be useful to optimize the coefficients of other classes of two-channel traditional signal or graph signal filter banks.

Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives

In the last few years, the theory of fractional calculus has been successfully used in thermoelasticity theories and many models of thermoelasticity with fractional order are established by several authors. In the present article, a new model of three-phase-lag thermoelastic heat conduction of higher-order time-fractional derivatives has been derived based on fractional calculus. Using the approach of the Taylor series expansion of time-fractional order developed by Jumarie (Comput Math Appl 59:1142, 2010), an alternative construction model is established extending Ezzat and others (Arch Appl Mech 82:557, 2012) and Roychoudhuri (J Therm Stress 30:231, 2007) models. This new model includes high-order time-fractional derivative approximations of three-phase-lags in the heat flux vector, the temperature gradient and in the thermal displacement gradient. We applied the resulting formulation to an infinite non-homogeneous orthotropic thermoelastic functionally graded medium having a spherical cavity with a power-law distribution of material properties along the radial direction. The effects of high-order time-fractional derivative parameters and non-homogeneity index on various distributions are discussed in detail and represented graphically and tabular forms. Finally, to illustrate the validity and accuracy of the proposed model, a comparison was made with various previous models, which are considered as special cases of our model.