Stirling's Interpolation Formula Research Articles

Outlier‐robust zonotope set‐membership filter for discrete‐time nonlinear system

AbstractTransient sensor malfunction, improper replacement measures, and large non‐Gaussian noise can cause data outliers in real‐world scenarios. In this article, an outlier‐robust zonotope set‐membership filter is proposed for uncertain nonlinear systems susceptible to outliers. In order to linearize nonlinear systems, a linearization method based on Stirling's interpolation formula is proposed. Then, an outlier detector utilizing the linearization error bounds, the prediction domain, and the output feasible domain is constructed. Based on the linearized system, a tight strip construction method is given to reduce the conservatism of the estimated enclosure. Moreover, a prediction‐correction‐backward reconstruction filter structure is designed to improve estimation accuracy and reduce conservatism. The proposed structure fuses historical data to attenuate the impact of uncertainty on the system. The robustness, guarantee, and convergence of the proposed filter are investigated. Finally, the effectiveness of the proposed approach is evaluated by two simulation examples.

Fractional central difference Kalman filter with unknown prior information

Abstract In this paper, a generalized fractional central difference Kalman filter for nonlinear discrete fractional dynamic systems is proposed. Based on the Stirling interpolation formula, the presented algorithm can be implemented as no derivatives are needed. Besides, in order to estimate the state with unknown prior information, a maximum a posteriori principle based adaptive fractional central difference Kalman filter is derived. The adaptive algorithm can estimate the noise statistics and system state simultaneously. The unbiasedness of the proposed algorithm is analyzed. Several numerical examples demonstrate the accuracy and effectiveness of the two Kalman filters.

Central Difference Kalman-probability Hypothesis Density Filter for Multiple Speakers Tracking

Aiming at nonlinear system model in multiple speakers tracking, a central difference Kalman-probability hypothesis density filter for multiple speakers tracking is proposed in this paper. Time difference of arrival for microphone array is taken as measurement, Stirling interpolation formula is utilized to derive polynomial approximations of nonlinear functions, central difference Kalman filter and Gaussian mixture probability hypothesis density filter are applied to estimate first-order statistical moment of posterior multiple speakers states, and finally multiple speakers tracking of nonlinear Gaussian system is realized while the speakers states are extracted by recursive updating. Simulation results show that the robustness of the algorithm is enhanced, and estimation accuracy of multiple speakers number and states is improved.

Satellite Formation based on SDDF Method

The technology of satellite formation flying has being a research focus in flight application. The relative position and velocity between satellites are basic parameters to achieve the control of formation flight during the satellite formation flying mission. In order to improve the navigation accuracy, a new filter different from Extended Kalman Filter (EKF) should be adopted to estimate the errors of relative position and velocity, which is based on the nonlinearity of the kinetic model for the satellite formation flying. A nonlinear Divided Difference Filter (DDF) based on Stirling interpolation formula was proposed in this paper. According to the linearity of the measurement equation for the filter, a simplified differential filter was designed by means of expanding the polynomial of the nonlinear system equation and linear approximating of the finite differential interpolation. Digital simulation experiment for the relative positioning of satellite formation flying was carried out. The result demonstrates that the filter proposed in this paper has a higher filtering accuracy, faster convergence speed and better stability. Compared with the EKF, the estimation accuracy of the relative position and velocity has improved by 77.1%and 47% respectively in the method of simplified DDF, which indicates the significance for practical applications.

A NMPC Scheme Based on Stirling's Interpolation Formula Approximation Method

For a kind of nonlinear system whose input-output function is not differentiable, we proposed a model predictive control scheme based on linearization approximation method. We linearized the object nonlinear system using Stirling's interpolation formula method, and reformulated the control performance index to a quadratic optimization problem, and then, we obtained the optimization control sequences by solving the quadratic optimization problem. In order to reduce the complexity of computation, we ignored the high-order terms associated with the linearization. The simulations show that the presented NMPC scheme can achieve a satisfactory control result, also can decrease the dissipation of control energy and control time.

Attitude estimation by divided difference filter in quaternion space

This article considers the application of Divided Difference Filter (DDF) to the orientation estimation, based on a quaternion-error continuous-discrete time model. DDF is a nonlinear estimator that in contrast to Taylor's expansion of extended Kalman Filter (EKF), exploit the polynomial approximations as a multivariable extension of Stirling's interpolation formula and require no derivatives. The DDF can be based on 1st and 2nd order Stirling's interpolation, which is named the divided difference filter-1st order (DDF1) and the divided difference filter-2nd order (DDF2). The orientation kinematics is defined in a quaternion vector space that unlike the Euler angle representation does not have any singularity problem. The presented nonlinear orientation model is an exact error model and is independent of the rigid body dynamics. The nonlinear process model includes six error-states in which only non-scalar elements of quaternion error vector are included in the error-state equations. The fourth element of quaternion error vector, which obeys unit norm constraint, is removed from system states to alleviate the estimated error covariance matrix divergence. The measurement system is a MARG sensor, which consists of a tri-axial rate gyro, a tri-axial accelerometer and a tri-axial magnetometer. The nonlinear measurement model is obtained based on the principals of magnetometer and accelerometer and the properties of the quaternion vector space. For the presented nonlinear orientation model, the performance of three filters namely DDF, EKF and Unscented Kalman Filter (UKF) is compared for different sampling frequencies in terms of the rms error, the captured area under the error norm curve, the estimated state variance and the computational cost. It is shown that under the same initial angle-error conditions, DDFs and UKF are more robust than EKF. The DDFs perform better than unscented Kalman filter (UKF) although the computational load for UKF is less. Among DDF1 and DDF2, DDF2's performance is slightly better but with more computation load. In the case of no initial angle-error conditions, the performance of the four filters is the same especially when the low noise level condition is considered.

Nonlinear estimation comparison for ballistic missile tracking

Nonlinear filtering for state estimation has been recently advanced to include more accurate and stable alternatives. The extended Kalman filter (EKF), the first and most widely used approach (applied as early as the late 1960s and developed into the early 1980s), uses potentially unstable derivative-based linearization of nonlinear process and/or measurement dynamics. The unscented Kalman filter (UKF), developed after around 1994, approximates a distribution about the mean using a set of calculated sigma points. The central difference filter (CDF), or divided difference filter (DDF), developed after around 1997, uses divided difference approximations of derivatives based on Stirling's interpolation formula and results in a similar mean, but a different covariance from the EKF and using techniques based on similar principles to those of the UKF. We compare the performance of these three approaches to the coordinate conversion problem within the larger problem of ballistic missile tracking under various sensor configurations, target dynamics, measurement update/sensor communication rates, and measurement noise. The coordinate conversion problem here specifically deals with conversion of spherical measurements to Cartesian estimates and vice versa. The importance of filter stability in some cases is emphasized as the EKF shows possible divergence due to linearization errors and overconfident state covariance, while the UKF shows possibly slow convergence due to overly large state covariance approximations. The CDF demonstrates relatively consistent stability, despite its similarities to the UKF. The requirement that the CDF and UKF expected state covariances be positive definite is demonstrated to be unrealistic in a case involving multisensor fusion, indicating the necessity for the reportedly more robust and efficient square-root implementation. Strategies for taking advantage of the strengths (and avoiding the weaknesses) of each filter are proposed.

Real-time human motion estimation using biomechanical models and non-linear state-space filters.

In the field of sports biomechanics and rehabilitation engineering, the possibility of computing, in real time, the angular displacements and derivatives of human joints, from a video of motion sequences, represents an appealing goal. In particular, applications of biofeedback protocols in rehabilitation can benefit from this capability. The focus of the investigation was concerned with the application of biomechanical models, comprising of a kinematic chain and surface envelopes, and state-space filters, to the computation, in real time and with high accuracy, of the angular data and derivatives. By minimising the distances, measured with TV cameras, between the 2D marker projections and the corresponding back-projected markers located on the mannequin, the configuration of the biomechanical model was automatically updated. The use of state-space estimation allowed the computation of smooth derivatives of the orientation data. Owing to the non-linearity of the functions involved, the derivatives of the observation model were obtained through a multidimensional extension of Stirling's interpolation formula. Proper algorithms were developed to cope with the model calibration, initialisation and data labelling. Extensive experiments on real and simulated motions proved the reliability (maximum angular error less than 1 degree, maximum point reconstruction less than 1 mm) of the developed system, which is robust to false matching caused by marker occlusions. Moreover, orientation artifacts due to skin motion can be reduced by a factor of 50%.

Easy and accurate state estimation for nonlinear systems

Two new state estimation filters for nonlinear systems are derived in this paper. The filters are based on a multi-dimensional extension of Stirling's interpolation formula for approximation of the nonlinear transformations. The filters can be shown to perform better than filters based on Taylor approximations. Yet, they can be implemented more easily as no derivatives are required. Thus, it is believed that the new filters can replace well-known estimators, such as the extended Kalman filter (EKF) and its higher order relatives.

Numerical method for the nonlinear Fokker-Planck equation

A practical method based on distributed approximating functionals ~DAFs! is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo @Phys. Rev. E 54, 931 ~1996!# to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the ‘‘flat’’ region in the kinetic potential fails for the ‘

Solution of nonlinear Fokker-Planck equations.

A finite-difference method for solving a general class of linear and nonlinear time-dependent Fokker-Planck equations, which is based on a K-point Stirling interpolation formula, is suggested. It has a fifth-order convergence in time and a 2Kth-order convergence in space and allows one to achieve a given level of accuracy with a slow (or even without) increase in the number of grid points. The most appealing features of the method are perhaps that it is norm conserved, and equilibrium preserving in the sense that every equilibrium solution of the analytic equations is also an equilibrium solution of the discretized equations. The method is applied to a nonlinear stochastic mean-field model introduced by Kometani and Shimizu [J. Stat. Phys. 13, 473 (1983)], which exhibits a phase transition. The results are compared with those obtained with other methods that rely on not too well controlled approximations. Our finite-difference scheme permits us to establish the region of validity and the limitations of those approximations. The nonlinearity of the system is found to be an obstacle for the application of Suzuki's scaling ideas, which are known to be suitable for linear problems. But what is most remarkable is that this nonlinearity allows for transient bimodality in a globally monostable case, even though there is no ``flat'' region in the potential. \textcopyright{} 1996 The American Physical Society.

Accurate solution to the time-independent Schrödinger equation using Stirling's interpolation formula

Stirling's interpolation formula is used to obtain a matrix representation of the time-independent Schrödinger equation. It is shown that an N-point Stirling interpolation formula is able to give an order of ϵ 2 N accuracy to the kinetic energy matrix. The potential matrix is diagonal and can be calculated exactly without involving any integration. Therefore, this formulation is able to yield any desired accuracy for a given discretized real space. A general formula is given explicitly for constructing the Hamiltonian matrix elements to any order of accuracy.

A Hybrid Fitting Method for Multiplet of K X-Ray

A new fitting method is presented for analysis of K X-ray multiplets including very close-lying peaks. The method is composed of two routines. First, we fitted the second derivatives of a mathematical model for peaks to the data transformed by a second differentiation filter associated with Stirling's interpolation formula, then determined the profile parameters (width and centroid) of a peak by a nonlinear least-squares method. Second, we fitted the profile to raw data to evaluate areas of component peaks by a linear least-squares method, keeping the profile parameters constant This analysis method, named the “hybrid method”, was applied to the analysis of the K X-ray spectra of As, Ag and Eu and is shown to be more reliable and effective for unravelling multiplets than conventional methods.

ハイブリッドあてはめ法 二重線解析のための新しい方法

An analysis method, named the Hybrid Fit Method, for very close-lying peaks is investigated. The method is composed of two steps of analysis. Firstly we fit the 2nd derivatives of profiles to the data deformed by a double differentiation filter associated with Stirling's interpolation formula in order to derive the widths and the centroids of cornponent peaks, and then fit the profiles to the raw data with these values fixed in order to determine the areas of component peaks. The fitting routine is the Marquardt algorithm with a modified Neal loop. The method is compared with a conventional fit method using doublets synthesized of 14.4 keV γ-rays with various mixing ratios and separations in respect of the following view-points: the effect of tail, using four different modified Robinson tails, and the reliability of fit, using four independent doublets. Also a comparison is made between the two method in fitting to the well-known 81 keV doublet of 133Ba with the exp·Erfc type tail having various fixed decay constants. The hybrid fit method has an advantage over the conventional fit method with the less affections of the tail function and the values of shape parameters, especially in the case of very close-lying peaks or doublets containing a small peak lying on the lower energy shoulder of the dominant peak, and so it improves the reliability of fit several times or more.

A new algorithm for rapid and precise centering of reflections in computer-controlled neutron diffractometers

A rapid and unique method for centering reflections on a computer-controlled, four-circle neutron diffractometer is presented. The method is based on numeric differentiation using the three-point form of Stirling's interpolation formula and its most outstanding features are the speed of operation and the high precision of the derived diffractometer coordinates.

A refined algorithm for finite-difference synthesis of cam profiles

The present paper deals with the synthesis of a cam profile when the prescribed acceleration pattern is given in numerical form. It is primarily an extension of previous work by the author with a view to upgrading the numerical accuracy of the synthesized cam profile, hence broadening the scope of application to a general class of high-speed cams. The technique is to differentiate the Stirling interpolation formula or the Gregory-Newton interpolation formula, thereby refining the prescribed initial acceleration data so that it will be correct to differences of high order. The effectiveness and the accuracy of this technique is demonstrated by a numerical example.

A Method for the Calculation of the Normal-Mode Frequencies in Irregular Rooms

This paper will present a method to solve for the normal-mode frequencies (standing-wave frequencies) in rooms of irregular shape. Since, in general, a direct solution of this problem is not possible, a numerical approximation procedure is presented. The theoretical derivation and solution for the normal-mode frequencies in an irregularly shaped room is done by using forward and backward difference formulas in approximating the second partial space derivatives in the acoustic wave equation. An alternative derivation, using Stirling's interpolation formula, will be presented. Boundary conditions (walls) are applied with an approximation of the image technique. A computer program will be shown that was written to calculate the lowest-mode wavelength constant k in a specified irregularly shaped room. Results of computer calculations for a cubic and pentagonal room will also be shown.

Periodic Solutions of van der Pol's Equations with Large Damping Coefficientlambda = 0 sim 10

A method of computing a periodic solution of van der Pol's equation is devised reducing the problem to the solution of a certain equation by means of Newton's method. For computing the value of the derivative necessary to apply Newton's method, the properties of variation of the orbit in the phase plane are used and, for step-by-step numerical integration of differential equations, a somewhat new method based on Stirling's interpolation formula combined with an ordinary Adams' extrapolating integration formula is used. The periodic solutions are actually computed for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda = 0 \sim 10</tex> and the minute but important change of the amplitude described by van der Pol's equation is found.

Elements of Finite Differences, also Solutions to Questions set for Part I of the Examinations of the Institute of Actuaries

THE first edition of this work, which is intended for students preparing for the first examination of the Institute of Actuaries, appeared in 1902. The present edition only differs from its predecessor in the addition to part i. of an alternative demonstration of Lubbock's formula for approximate summation, and of three chapters dealing with Stirling's Interpolation Formula, Interpolation for functions of two variables, and Interpolation of inverse functions.

An Application of Stirling's Interpolation Formula

"An Application of Stirling's Interpolation Formula." The American Mathematical Monthly, 12(2), pp. 43–44