High-order Taylor Expansion Research Articles

Fast and direct calibration for high numerical aperture quadriwave lateral shearing interferometry using geometry model with higher-order Taylor expansion

The current research on quadriwave lateral shearing interferometry (LSI) is primarily focused on the measurements of plane wave or quasiplane wave. When directly measuring spherical waves with high numerical apertures (NA>0.35), it will generate additional systematic errors, which affect measurement accuracy. To make the quadriwave LSI applicable to the measurements of a high-NA spherical wave, this paper proposes a fast and direct calibration method by establishing the geometry model of the quadriwave LSI with high-NA spherical wave incidence. The expression for the optical path difference represented systematic errors introduced by high-NA spherical waves in the shearing wavefronts are derived, which utilize a ninth-order Taylor expansion and Zernike polynomials fitting to achieve the high accuracy required by the high-NA spherical wave incidence. Then, the systematic errors are directly calibrated in the shearing wavefronts of four directions. This paper presents the theoretical analysis and verifies the feasibility and reliability of the proposed method through simulations and experiments, achieving a good measurement accuracy.

Short-Term Collision Probability Caused by Debris Cloud

We propose a method for a fast and accurate estimation of the collision probability in a short window following the generation of a debris cloud. A multirevolution Lambert targeting problem is employed to determine all the locations where the collision probability is nonnegligible. Then, the Lambert problem and state transition matrix are used to obtain the image of the target body in the spread velocity space, where the collision probability computation is performed. To perform and control the accuracy of the integration of the collision rate needed to obtain collision probabilities, we apply high-order Taylor expansion and automatic domain splitting techniques. The method is validated using a Monte Carlo simulation and a literature-based test case. Test cases demonstrate the validity of the approach for Keplerian and perturbed dynamics.

Efficient Evaluation of PGF Based on High-Order Taylor Expansions

Efficient Evaluation of PGF Based on High-Order Taylor Expansions

Semi-Analytical Midcourse Guidance for Coordinated Interception of Multiple Exo-Atmospheric Targets

The simultaneous interception of multiple exo-atmospheric threat targets can leave no buffer time for the targets and greatly reduces targets’ penetration probability. This paper proposes centralized and decentralized two midcourse guidance methods to solve the coordinated interception of multiple exo-atmospheric targets. First, the free-time cooperative interception problem minimizing the total impulses is established, where the nonlinear dynamics of the interceptors and targets, the initial impulse constraints, and the terminal interception constraints are considered. Second, the necessary conditions for the optimal cooperative interception problem are derived and then approximated using higher-order Taylor expansions, which result in a series of analytical nonlinear equations. Then, the semi-analytical centralized guidance mode is designed to achieve simultaneous interception of targets by directly solving the analytical nonlinear equations. To improve computational efficiency, the decentralized cooperative guidance mode is further proposed by using the developed iterative freezing trajectory method, where each interceptor uses information from other interceptors in the previous iteration for prediction and distributed computation. Numerical examples verify the effectiveness and robustness of the proposed two guidance methods. Results show that the computational efficiency of the decentralized guidance mode is more efficient compared with the centralized midcourse guidance mode.

Gradient and uncertainty enhanced sequential sampling for global fit

Surrogate models based on machine learning methods have become an important part of modern engineering to replace costly computer simulations. The data used for creating a surrogate model are essential for the model accuracy and often restricted due to cost and time constraints. Adaptive sampling strategies have been shown to reduce the number of samples needed to create an accurate model. This paper proposes a new sampling strategy for global fit called Gradient and Uncertainty Enhanced Sequential Sampling (GUESS). The acquisition function uses two terms: the predictive posterior uncertainty of the surrogate model for exploration of unseen regions and a weighted approximation of the second and higher-order Taylor expansion values for exploitation. Although various sampling strategies have been proposed so far, the selection of a suitable method is not trivial. Therefore, we compared our proposed strategy to 9 adaptive sampling strategies for global surrogate modeling, based on 26 different 1 to 8-dimensional deterministic benchmarks functions. Results show that GUESS achieved on average the highest sample efficiency compared to other surrogate-based strategies on the tested examples. An ablation study considering the behavior of GUESS in higher dimensions and the importance of surrogate choice is also presented.

High-Order Steady-State Diffusion Approximations

Much like higher-order Taylor expansions allow one to approximate functions to a higher degree of accuracy, we demonstrate that, by accounting for higher-order terms in the Taylor expansion of a Markov process generator, one can derive novel diffusion approximations that achieve a higher degree of accuracy compared with the classical ones used in the literature over the last 50 years.

Harmonic analysis of the arctangent function regarding the angular error introduced by superimposed Fourier series for application in sine/cosine angle encoders

We present a rigorous analytical method for harmonic analysis of the angular error of rotary and linear encoders with sine/cosine output signals in quadrature that are distorted by superimposed Fourier series. To calculate the angle from measured sine and cosine encoder channels in quadrature, the arctangent function is commonly used. The hence non-linear relation between raw signals and calculated angle—often thought of as a black box—complicates the estimation of the angular error and its harmonic decomposition. By means of a Taylor series expansion of the harmonic amplitudes, our method allows for quantification of the impact of harmonic signal distortions on the angular error in terms of harmonic order, magnitude and phase, including an upper bound on the remaining error term—without numerical evaluation of the arctangent function. The same approximation is achieved with an intuitive geometric approximation in the complex plane, validating the results. Interaction effects between harmonics in the signals are considered by higher-order Taylor expansion. The approximations show an excellent agreement with the exact calculation in numerical examples even in case of large distortion amplitudes, leading to practicable estimates for the angular error decomposition.

High-order synchroextracting transform for characterizing signals with strong AM-FM features and its application in mechanical fault diagnosis

In this paper, a novel time–frequency analysis (TFA) technique termed High-order Synchroextracting Transform (HSET), is proposed to better characterize the changing dynamics of multi-component signals with strong AM-FM components. Synchroextracting Transform (SET) is a TFA method has been developed recently. As a post-processing technology, unlike the parameterized TFA methods that requires different demodulation operators to be designed according to different types of signals, SET can generate high-quality time–frequency representation (TFR) only on the basis of short-time Fourier transform (STFT). However, since the core of SET is based on the assumption that the signal is composed of pure harmonics, SET cannot accurately extract the time-varying characteristics of signal with strong AM-FM components. To construct a method to well characterize the time-varying laws of strong AM-FM signals, this paper uses high-order Taylor expansion to estimate the time varying laws of the signal to obtain a more accurate frequency estimation operator, which we call high-order Synchroextracting operator (HSEO), and then HSET can be constructed through Dirichlet function to generate TFR with high TF resolution. Good reconstruction quality can be achieved through the inverse transformation of HSET while signal reconstruction operation can be completed concisely. In addition, this paper applies HSET to the processing of the measured vibration signals and feature extraction of gravitational-wave (GW) signal. Compared with other advanced TFA technologies, the effectiveness and competitiveness of the proposed method can be demonstrated.

Robustly complete synthesis of sampled-data control for continuous-time nonlinear systems with reach-and-stay objectives

Formal methods are becoming favorable for control and verification of safety-critical systems because of the rigorous model-based computation. Relying on an over-approximated model of the original system behaviors, formal control synthesis algorithms are not often complete, which means that a controller cannot necessarily be synthesized even if there exists one. The main result of this paper shows that, for continuous-time nonlinear systems, a sample-and-hold control strategy for a reach-and-stay specification can be synthesized whenever such a strategy exists for the same system with its dynamics perturbed by small disturbances. Control synthesis is carried out by a fixed-point algorithm that adaptively partitions the system state space into a finite number of cells. In each iteration, the reachable set from each cell after one sampling time is over-approximated within a precision determined by the bound of the disturbances. To meet such a requirement, we integrate validated high-order Taylor expansion of the system solution over one sampling period into every fixed-point iteration and provide a criterion for choosing the Taylor order and the partition precision. Two nonlinear system examples are given to illustrate the effectiveness of the proposed method.

Design Method of High-Order Kalman Filter for Strong Nonlinear System Based on Kronecker Product Transform.

In this paper, a novel design idea of high-order Kalman filter based on Kronecker product transform is proposed for a class of strong nonlinear stochastic dynamic systems. Firstly, those augmenting systems are modeled with help of the Kronecker product without system noise. Secondly, the augmented system errors are illustratively charactered by Gaussian white noise. Thirdly, at the expanded space a creative high-order Kalman filter is delicately designed, which consists of high-order Taylor expansion, introducing magical intermediate variables, representing linear systems converted from strongly nonlinear systems, designing Kalman filter, etc. The performance of the proposed filter will be much better than one of EKF, because it uses more information than EKF. Finally, its promise is verified through commonly used digital simulation examples.

A high-efficient splitting step reduced-dimension pure meshless method for transient 2D/3D Maxwell’s equations in complex irregular domain

In this work, a high-efficient and accurate pure meshless method is designated as splitting step reduced-dimension finite pointset method, which is first proposed to solve the time-dependent 2D/3D Maxwell equations in lossy medium with periodic or perfect conducting boundary condition. The proposed method is mainly motivated by following three points. (a) The coupled 2D/3D Maxwell equations are decomposed into several mutually uncoupled 1D partial differential equations through a splitting step method. (b) The space-derivatives of the above those 1D systems are discretized by the 1D finite pointset method based on the Taylor expansion and the weighted least squares method. Meanwhile, the time-derivatives are approximated by second-order scheme. (c) The local refinement technique is adopted at the selected region to improve the numerical accuracy, and the multi-CPUs parallel computing is used to enhance the computational efficiency. All the numerical results show the proposed scheme has second-order convergence rate, and which has higher computational efficiency than the traditional finite pointset method for solving the 2D/3D Maxwell equations, especially for the case of 3D problems or higher-order Taylor expansion in finite pointset method. Subsequently, the advantage of the proposed scheme applied in complex irregular domain is illustrated.

Variational Optimization of Continuous Matrix Product States.

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

High-Order Taylor Expansion for Wind Field Retrieval Based on Ground-Based Scanning Lidar

The uniform and linear wind models have been commonly used for wind field retrieval in meteorological community. However, the accuracy and robustness of the retrieval results can be quite unsatisfactory due to the mismatch between these models and the real wind distribution, especially under complex wind conditions. In this article, a nonlinear model based on high-order Taylor expansion is proposed to deal with this limitation, and the combination of ridge regression and decomposition-iteration process (denoted as Ridge-DI method) is further introduced to solve the model with high accuracy and robustness. A case study on simulation and field experiment shows that the proposed method with the third-order Taylor expansion can reduce the mean root-mean-square errors (RMSEs) of the retrieved velocities by more than 16.84% in comparison with traditional methods.

High-Order Synchroextracting Chirplet Transform for Accurate Instantaneous Frequency Estimation and Its Application in Fault Diagnosis of Rotary Machinery

Time-frequency analysis (TFA) technology is an effective tool for extracting time-varying characteristics of a signal, but for strong AM-FM signals, time-frequency representation (TFR) with high time-frequency (TF) resolution is often difficult to generate. The measured vibration signals of mechanical equipment often contain a large number of non-stationary features and are polluted by noise, which hinders the use of TFA technologies. In this paper, we propose a TFA method termed high-order synchroextracting chirplet transform (HSECT) to effectively process strong AM-FM signals. On the basis of Chirplet transform (CT), we first determine the best linear demodulation operator at each moment through the local maximum amplitude of the generated TFR. Subsequently, the amplitude and phase of the signal are subjected to high-order Taylor expansion, and a new frequency estimation operator termed high-order synchroextracting chirplet operator (HSECO) is obtained by the high-order approximation of the time-varying laws of the signal. HSECO can accurately extract the instantaneous frequency of strong AM-FM signal and the construction of HSECT is completed by Dirichlet function at last. HSECT can generate TFR with high energy concentration from which the time-varying law of strong AM-FM signal can be extracted accurately. Meanwhile, the signal reconstruction operation can be completed concisely. In addition, the proposed method is applied to the fault diagnosis of the rotor rub impact and variable speed equipment and compared with other TFA methods. Simulation and experiments can verify the effectiveness and competitiveness of the proposed method.

A bivariate Chebyshev polynomials method for nonlinear dynamic systems with interval uncertainties

A bivariate Chebyshev polynomials approach is proposed to estimate the dynamic response bounds of nonlinear systems with interval uncertainties. The existing collocation method directly searches the maximum and minimum values of the surrogate model in the entire interval space by the scanning method (SM). The presence of too many uncertain parameters will lead to expansive computational cost. To overcome this shortcoming, the dynamic response is decomposed by a bivariate function decomposition (BFD), established based on high-order Taylor expansion, into the sum of multiple univariate and bivariate response functions. The above univariate and bivariate functions are fitted using Chebyshev polynomials, and polynomial coefficients are obtained through one-dimensional (1D) and two-dimensional (2D) interpolation points. Thus, the solution of the nonlinear dynamic systems with uncertain parameters can be transformed into that of univariate and bivariate Chebyshev interval functions. The extremum values of the low-dimensional Chebyshev interval functions can be found by SM, and then the bounds of dynamic response are acquired by interval arithmetic. Since SM searches for extreme values only in 1D and 2D uncertain domains, the amount of calculation is reduced compared to searching the whole uncertain space. The efficiency, practicability and effectiveness of the proposed interval uncertainty analysis method are proved by three dynamic examples.

Kinetic derivation of Cahn-Hilliard fluid models

A compressible Cahn-Hilliard fluid model is derived from the kinetic theory of dense gas mixtures. The fluid model involves a van der Waals and Cahn-Hilliard gradient energy, a generalized Korteweg's tensor, a generalized Dunn and Serrin heat flux, and Cahn-Hilliard-type diffusive fluxes. Starting from the BBGKY hierarchy for gas mixtures, a Chapman-Enskog method is used-with a proper scaling of the generalized Boltzmann equations-as well as higher-order Taylor expansions of pair distribution functions. A Euler and van der Waals model is obtained at zeroth order, while the Cahn-Hilliard fluid model is obtained at first order, involving viscous, heat, and diffusive fluxes. The Cahn-Hilliard extra terms are associated with intermolecular forces and pair interaction potentials.

A univariate Chebyshev polynomials method for structural systems with interval uncertainty

This paper presents an effective univariate Chebyshev polynomials method (UCM) for interval bounds estimation of uncertain structures with unknown-but-bounded parameters. The interpolation points required by the conventional collocation methods to generate the surrogate model are the tensor product of each one-dimensional (1D) interpolating point. Therefore, the computational cost is expensive for uncertain structures containing more interval parameters. To deal with this issue, the univariate decomposition is derived through the higher-order Taylor expansion. The structural system is decomposed into a sum of several univariate subsystems, where each subsystem only involves one uncertain parameter and replaces the other parameters with their midpoint value. Then the Chebyshev polynomials are utilized to fit the subsystems, in which the coefficients of these subsystems are confirmed only using the linear combination of 1D interpolation points. Next, a surrogate model of the actual structural system composed of explicit univariate Chebyshev functions is established. Finally, the extremum of each univariate function that is obtained by the scanning method is substituted into the surrogate model to determine the interval ranges of the uncertain structures. Numerical analysis is conducted to validate the accuracy and effectiveness of the proposed method.

Residual high-order wavefront curvature error compensation for the polar format algorithm in widefield spotlight SAR imaging

ABSTRACT Polar format algorithm (PFA) is an efficient image formation algorithm for spotlight-mode synthetic aperture radar (SAR). However, the derivation of PFA relies upon the approximation of planar wavefront, and the resulting wavefront curvature error (WCE) causes severe SAR image degradation, e.g., distortion and defocus. The distortion stems from the first-order WCE and the defocus mainly comes from the quadratic WCE. To eliminate the defocusing effects, the existing space-variant post-filtering algorithms (SVPFAs) mainly compensate the quadratic term of WCE and ignore the residual high-order terms. Actually, the residual odd-order WCE will also broaden the main-lobe of the target and the residual even-order WCE will lead to the problem of sidelobe asymmetry. To tackle these issues, this letter proposes a modified SVPFA to correct both the quadratic and the residual high-order WCE. First, based on the derivation of PFA, the accurate expression of WCE is obtained. Second, the quadratic and residual high-order WCE are separated without complicated high-order two-dimensional Taylor expansion. Finally, the WCE is corrected by phase compensation and geometric distortion correction, respectively. The effectiveness of the proposed algorithm is verified by simulation experiments.

Robust Pencil Beam Pattern Synthesis With Array Position Uncertainty

In many practical applications, the antenna arrays are not accurately implemented as desired, resulting in the degradation of sidelobe level. This letter proposes a robust pencil beam pattern synthesis (PBPS) algorithm that is adaptive to the uncertainties of array position. The proposed algorithm, first, sets up the position-variation-related PBPS optimization problem, and then rewrites the position-related parameter, i.e., the array factor, into a high-order Taylor expansion with respect to the array position variation. Thereafter, an iterative processes are designed to estimate the position variation. Finally, the optimal array excitation is calculated by solving the related PBPS problem, in which the estimated array position is used. Simulations with both isotropic element pattern and active element pattern are carried out and the results are compared with the results achieved from the existing work to validate the superiority of the proposed algorithm.

Uncertainty propagation on a nonlinear measurement model based on Taylor expansion

In this paper, the propagation of uncertainty on a nonlinear measurement model is presented using a higher-order Taylor series. As the derived formula is based on a Taylor series, it is necessary to compute the partial derivatives of the nonlinear measurement model and the correlation among the various products of the input variables. To simplify the approximation of this formula, most previous studies assumed that the input variables follow independent Gaussian distributions. However, in this study, we generate multivariate random variables based on copulas and obtain the covariances among the products of various input variables. By applying the derived formula to various cases regardless of the error distribution, we obtained the results that coincide with those of a Monte-Carlo simulation. To apply high-order Taylor expansion, the nonlinear measurement model should be continuous within the range of the input variables to allow for differentiation, and be an analytic function in order to be represented by a power series. This approach may replace some time-consuming Monte-Carlo simulations by choosing the appropriate order of the Taylor series, and can be used to check the linearity of the uncertainty.