Chebyshev Approximation Theory Research Articles

A generalisation of de la Vallée-Poussin procedure to multivariate approximations

The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vallée-Poussin procedure. In this paper we demonstrate that under certain assumptions the classical de la Vallée-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation. There are two main advantages of our approach. First of all, it provides an elegant geometrical interpretation of the procedure. Second, the corresponding basis functions are not restricted to be monomials and therefore can be extended to a larger family of functions.

Membership-Function-Dependent Stability Analysis for Polynomial-Fuzzy-Model-Based Control Systems via Chebyshev Membership Functions

In this article, we investigate the stability analysis of a polynomial-fuzzy-model-based control system by employing a new form of approximate membership functions called Chebyshev membership functions (CMFs) based on the Lyapunov stability theory. The membership-function-dependent method shows that the approximate functions can introduce useful information to relax the conservativeness of stability analysis. Therefore, in the case of the same order and operating domain, how to design the approximate functions to introduce more information to relax stability is a concern. In this article, given the Chebyshev norm relationship between the approximate membership functions and the original ones, CMFs can be obtained by the Chebyshev approximation theory and the Remez/Remez-type iterative algorithm, which own the minimum of the maximum absolute approximation error. Together with CMFs, the boundary information of premise variables on the overall operating domain is considered to reduce the conservativeness of stability analysis. Furthermore, to further relax the stability analysis without increasing the order of CMFs, the operating domain is divided into subdomains, and piecewise Chebyshev membership functions (PCMFs) are proposed to facilitate the stability analysis. Together with PCMFs, the subdomain boundary information of premise variables is introduced into stability conditions. A simulation example is given to demonstrate the effectiveness of the proposed approach.

Adaptive Kinematic Mapping Based on Chebyshev Interpolation: Application to Flexure-Jointed Micromanipulator Control

For flexure-jointed mechanisms and manipulators, accurate kinematic models typically involve complex nonlinear descriptions of elastic behavior and are ill-suited for real-time control computations. An alternative approach is to use multidimensional function approximation methods, with optimization via online kinematic identification and calibration. In this article, a function interpolation approach is proposed based on Chebyshev approximation theory for near-optimal minimization of maximum positioning errors. The method allows fast calibration procedures using a small number of data points. The implementation involves a correction mapping that operates on command input variables before an approximate inverse kinematic model is applied. An adaptation algorithm is further proposed that can be used to update and refine mappings: 1) in a localized space for improved precision for the current task, or 2) globally by using calibration points chosen to match the Chebyshev nodes of the overall workspace. Results are shown for simulation of a flexure-jointed X–Y motion stage and for experiments on a X–Y–Z micromanipulation platform with Delta-type parallel kinematics. For the experiments, direct measurement of platform position was achieved using a microscope vision system. The proposed method gave order-of-magnitude improvements in positioning accuracy compared with the pseudo-rigid-body modeling approach and was found to out-perform direct visual servoing when operating with similar image capture rates.

Chebyshev Approximation and the Global Geometry of Model Predictions.

Complex nonlinear models are typically ill conditioned or sloppy; their predictions are significantly affected by only a small subset of parameter combinations, and parameters are difficult to reconstruct from model behavior. Despite forming an important universality class and arising frequently in practice when performing a nonlinear fit to data, formal and systematic explanations of sloppiness are lacking. By unifying geometric interpretations of sloppiness with Chebyshev approximation theory, we rigorously explain sloppiness as a consequence of model smoothness. Our approach results in universal bounds on model predictions for classes of smooth models, capturing global geometric features that are intrinsic to their model manifolds, and characterizing a universality class of models. We illustrate this universality using three models from disparate fields (physics, chemistry, biology): exponential curves, reaction rates from an enzyme-catalyzed chemical reaction, and an epidemiology model of an infected population.

Polar Affine Arithmetic

Uncertainties practically arise from numerous factors, such as ambiguous information, inaccurate model, and environment disturbance. Interval arithmetic has emerged to solve problems with uncertain parameters, especially in the computational process where only the upper and lower bounds of parameters can be ascertained. In rectangular coordinate systems, the basic interval operations and improved interval algorithms have been developed in the numerical analysis. However, in polar coordinate systems, interval arithmetic still suffers from issues of complex computation and overestimation. This article defines a polar affine variable and develops a polar affine arithmetic (PAA) that extends affine arithmetic to the polar coordinate systems, which performs better in many aspects than the corresponding polar interval arithmetic (PIA). Basic arithmetic operations are developed based on the complex affine arithmetic. The Chebyshev approximation theory and the min-range approximation theory are used to identify the best affine approximation. PAA can accurately keep track of the interdependency among multiple variables throughout the calculation procedure, which prominently reduces the solution conservativeness. Numerical examples implemented in MATLAB programs show that, compared with benchmark results from the Monte Carlo method, the proposed PAA ensures completeness of the exact solution and presents a more compact solution region than PIA when dependency exists in the calculation process. Meanwhile, a comparison of affine arithmetic in polar and rectangular coordinates is presented. An application of PAA in circuit analysis is quantitatively presented and potential applications in other research fields involving complex variables in polar form will be gradually developed.

Nonlinear response analysis of a rotor system with a transverse breathing crack under interval uncertainties

Parametric uncertainties are present in complex mechanical systems due to various reasons such as material dispersion and wear. This paper investigates the effects of interval uncertain parameters on the dynamic behaviors of a rotor system with a transverse breathing crack in the shaft. The uncertainties are modeled as uncertain-but-bounded interval inputs on the basis that no sufficient prior information is available to define their precise probabilistic distributions. A finite element rotor model is formulated and the harmonic balance method (HBM) is employed to solve the deterministic dynamic problem. Based on the Chebyshev approximation theory, a surrogate model for the uncertain problem is established and then used to determine the bounds of the nonlinear dynamic responses. The accuracy verification is performed by comparing with the scanning method. Simulations are carried out considering different uncertainties with several uncertain degrees. It will provide some references for early crack fault detection and condition monitoring in rotor systems with uncertainties included.

An interval precise integration method for transient unbalance response analysis of rotor system with uncertainty

Abstract A non-intrusive interval precise integration method (IPIM) is proposed in this paper to analyze the transient unbalance response of uncertain rotor systems. The transfer matrix method (TMM) is used to derive the deterministic equations of motion of a hollow-shaft overhung rotor. The uncertain transient dynamic problem is solved by combing the Chebyshev approximation theory with the modified precise integration method (PIM). Transient response bounds are calculated by interval arithmetic of the expansion coefficients. Theoretical error analysis of the proposed method is provided briefly, and its accuracy is further validated by comparing with the scanning method in simulations. Numerical results show that the IPIM can keep good accuracy in vibration prediction of the start-up transient process. Furthermore, the proposed method can also provide theoretical guidance to other transient dynamic mechanical systems with uncertainties.

Comparative study of four-bar hyperbolic function generation mechanism with four and five accuracy points

Kinematic mechanisms are synthesized for a task. Function generation provides precise displacement at output links that obeys a given functional relations. This article describes the synthesis of four-bar mechanism for the hyperbolic function generation with four and five accuracy point, which is further optimized using least square method. This research is concerned with development of mathematical formulation based on Freudenstein–Chebyshev approximation theory, extended to four- and five-point synthesis function generation problem. The objective function is analyzed for the structural error between the generated function and the desired function. Resulting nonlinear equations are converted into set of linear equations applying the compatibility conditions and are solved using Gauss elimination method. The formulation is proposed for five position synthesis for algebraic and trigonometric function generation problem. Associated structural error is estimated. Comparison of estimated error through the formulation is carried out with the reported errors through graphical method. The error for hyperbolic function is estimated. Attempt is made to minimize the error through simple of least square technique. The results obtained are compared with Freudenstein–Chebyshev approximation method. Three hyperbolic functions, namely sinh(x), cosh(x) and tanh(x), are used to demonstrate the effectiveness of the proposed synthesis method.

Chebyshev approximation problem of multidimensional IIR digital filters in the frequency domain

This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. Chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the Chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.

Chebyshev Approximation by Linear Combinations of Fixed Knot Polynomial Splines with Weighting Functions

In this paper, we derive conditions for best uniform approximation by fixed knots polynomial splines with weighting functions. The theory of Chebyshev approximation for fixed knots polynomial functions is very elegant and complete. Necessary and sufficient optimality conditions have been developed leading to efficient algorithms for constructing optimal spline approximations. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). In this paper, we extend these results to the case when the model function is a product of fixed knots polynomial splines (whose parameters are subject to optimization) and other functions (whose parameters are predefined). This problem is nonsmooth, and therefore, we make use of convex and nonsmooth analysis to solve it.

Design and Implementation of the Stone Elliptical Arc Contour Fitting Algorithm

Elliptical arcs have been widely used in stone contours, but the circular saw of the existing stone bridge cutting machine can only cut stone contours which compose of long lines and large radius arcs. The elliptical arc is often fitted by small line segments or small arc segments, and the circular saw cannot be used to process the stone elliptical arc contour. Therefore, a better choice of fitting the elliptical arc in the stone contour is the four-arc fitting algorithm. The traditional four-arc fitting ellipse algorithm can only fit the whole ellipse, in order to fit elliptical arcs, a new fitting algorithm of four-arc fitting elliptical arc is designed. Firstly, identify the stone contour and analyze the elliptical entities section from the DXF file, and divide the elliptical arc curve into several arcs according to the angular relationship. Then adopting the Chebyshev approximation theory of four-arc fitting ellipse, the preliminary fitting of the elliptical arc is performed, and re-fit the start section and end section of the elliptical arc curve, the elliptical arc fitting algorithm is designed. In addition, the maximum value of fitting error has been calculated. Finally, by using the Visual C++ as the developing software, the stone elliptical arc contour fitting algorithm and its processing trajectory simulation are realized. Testing results show that the fitting algorithm can meet the requirements of the circular saw cutting the stone elliptical arc contour.

Design of FIR Lowpass Differentiator and it's Applications in Airborne Gravimetry

AbstractBased on Chebyshev approximation theory, a finite impulse response (FIR) lowpass differentiator, which suits for airborne gravimetry application, is designed. Then, we use a mathematical model to test the performance of this filter. Finally, the vertical acceleration is determined using the filter based on the actual data from GPS stationary test and kinematic test. The results indicate that the accuracy of the acceleration derived from GPS height is about ±1‐2 mGal when using a filtering period of 100 s.

Generalized simultaneous Chebyshev approximation

In this note, we develop, without assuming the Haar condition, a generalized simulaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and contains it as a special case. Our results also contain those in [1] and [3] as a special case, and the two conjectures proposed by C. B. Dunham in [2] are proved to be true in the case of simultaneous approximation.

A constructive development of Chebyshev approximation theory

A constructive development of Chebyshev approximation theory

Chebyshev approximation with non-continuous weight functions

The theory of Chebyshev approximation with respect to a positive continuous (ordinary) weight function is well understood and the Remez algorithm can be used in most cases of practical interest. The problem of Chebyshev approximation with respect to a non-negative, non-continuous weight function is now under investigation by the author, who would like to hear from any reader having a problem of this type. In return for such information the author will supply the theory and methods developed to date.

Unisolvence on Multidimensional Spaces

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.