非线性时变大系统递阶控制的PGOPO法

提出了一类新型的线性、有界算子--分段广义正交多项式算子(PGOPO),建立了其 主要性质及运算规则;随后将PGOPO法用于求解非线性时变大系统最优控制问题.在这种新 型逼近运算中将PGOPO法和改进型关联预估法相结合,得出显式递阶递推算法更易于计算机 计算和推广.数值仿真实例说明了给出的算法是有效的.     

多维分段广义正交多项式算子及其在分布参数系统辨识中的应用

在一维分段广义正交多项式的基础上，提出多维分段广义正交多项式及其相应的正交多项式算子的定义，总结归纳了多维分段广义正交多项式算子的基本性质和主要运算规则，并将二维分段广义正交多项式算子法应用于一类非线性分布参数系统的辨识，数值算例表明，即使在存在量测噪声的情况下，使用较小的分段数和正交基项数也能得到较好的辨识效果。     

Galerkin approximationwith Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time (CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman (GHJB) equations. The Galerkin approximation with Legendre polynomials (GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved. Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.     

Solution of variational problems via general orthogonal polynomials

The general orthogonal polynomials approximation is employed to solve variational problems. The operational matrix of integration is applied to reduce an integral equation to an algebraic equation with expansion coefficients. A simple and straightforward algorithm is then developed to calculate the expansion coefficients of the general orthogonal polynomials. The proposed method is general and various classical orthogonal polynomial approximations of the same problem can be obtained as a special case of the derived results.     

Application of general discrete orthogonal polynomials to optimal-control systems

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.     

基于二类切比雪夫正交多项式非参数混合模型的图像分割

有参混合模型需要假设模型为某种已知的参数模型,而实际数据往往很难假设出这种参数模型的分布.为此,提出一种二类切比雪夫正交多项式的非参数图像混合模型分割方法.首先,设计出一种基于二类切比雪夫正交多项式的图像非参数混合模型,每一个模型的平滑参数根据误差方法和最小的准则进行计算.然后,利用随机期望最大(SEM)算法求解正交多项式系数和每一个模型的权重.此方法不需要对模型作任何假设,可以有效克服有参混合模型与实际数据分布不一致的问题.实验表明,该方法比高斯混合模型分割效率更高,并比其他非参数正交多项式混合模型有更好的分割效果.     

Partial Differential Equations and Bivariate Orthogonal Polynomials

In 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras.     

Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.      

一种新型的非参数模型及其在系统辨识中的应用

本文给出了一个求解广义正交多项式的微分运算矩阵的新方法，应用对连续线性系统的脉冲响应函数进行正交逼近的方法来讨论脉冲响应函数的实现问题，得到了一类新型的非参数模型，并导出了利用该模型来辨识连续线性系统的脉冲响应函数的算法，最后给出了例子证实本文所给方法的有效性。     

General orthogonal polynomials analysis of linear optimal control systems incorporating observers

The analysis of linear time-invariant optimal control systems incorporating observers is approached using general orthogonal polynomials. The operational matrix for the forward integration of general orthogonal polynomials is derived and used to simplify the system of equations into the successive solution of a set of algebraic equations. An illustrative example is included and a comparison is made of solutions obtained by different orthogonal polynomials.     

Evaluation of spline functions for digital filtering problems

Methods for the evaluation of spline functions for digital filtering in data processing systems are developed. Basis polynomials of general form and basis discrete orthogonal polynomials are considered. Computations are organized by solving constrained optimization problems. Recurrences for the system of normalized discrete orthogonal polynomials and their derivatives are obtained. The proposed spline functions on discrete orthogonal polynomials reduce the computational cost and approximation errors compared with the case of general polynomials. The results of the statistical simulation of the application of spline functions based on discrete orthogonal polynomials in digital filtering problems are presented.     

Properties and performance of orthogonal neural network in function approximation

Backpropagation neural network has been applied successfully to solving uncertain problems in many fields. However, unsolved drawbacks still exist such as the problems of local minimum, slow convergence speed, and the determination of initial weights and the number of processing elements. In this paper, we introduce a single‐layer orthogonal neural network (ONN) that is developed based on orthogonal functions. Since the processing elements are orthogonal to one another and there is no local minimum of the error function, the orthogonal neural network is able to avoid the above problems. Among the five existing orthogonal functions, Legendre polynomials and Chebyshev polynomials of the first kind have the properties of recursion and completeness. They are the most suitable to generate the neural network. Some typical examples are given to show their performance in function approximation. The results show that ONN has excellent convergence performance. Moreover, ONN is capable of approximating the mathematic model of backpropagation neural network. Therefore, it should be able to be applied to various applications that backpropagation neural network is suitable to solve. © 2001 John Wiley & Sons, Inc.     

Laguerre series approximation of infinite dimensional systems

Laguerre-Fourier approximations of stable systems are shown to exhibit many desirable properties for various classes of infinite dimensional systems. Specifically, time domain supremum and L¹ norm convergence results, and frequency domain H^∞ norm convergence results, are given for Laguerre-Fourier approximations. It is also shown that the theory of Laguerre polynomials solves explicitly the problem of determining Laguerre-Fourier approximations for a large class of delay systems. Furthermore, it is believed that these results are important for the study of orthonormal series identification as a general technique for identification of infinite dimensional systems.     

Application of orthogonal collocation on finite elements to a flow problem

The method of orthogonal collocation on finite elements (OCFE) has been used to solve the flow problem of Newtonian fluid in an internally finned tube. Cylindrical coordinates were employed and Legendre shifted orthogonal polynomials were employed as basis functions. An alternating direction implicit (ADI) method was used to solve the resulting set of equations. Better accuracy was achieved by increasing the number of elements for given total interior collocation points. Although, in general, for given total interior collocation points the OCFE method was found marginally superior to the finite difference method in terms of accuracy, the computational time requirement was much higher for the method of orthogonal collocation on finite elements.     

Solutions of linear dynamic systems by generalized orthogonal polynomials

Generalized orthogonal polynomials that represent all types of orthogonal polynomial are introduced in this paper. Using the idea of orthogonal polynomial functions that can be expressed by power series, and vice versa, the operational matrix for integration of a generalized orthogonal polynomial is first derived and then applied to solve the equations of linear dynamic systems. The characteristics of each kind of orthogonal polynomial in relation to solving linear dynamic systems is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The operational matrix is simpler than those of conventional orthogonal polynomials. Hence the expansion coefficients are more easily calculated from the proposed recursive formula when compared with those obtained from conventional orthogonal polynomial approximations.     

一类系统参数辨识的按段多重一般正交多项式方法

本文介绍了按段多重一般正交多项式系及其基本性质,并把它们应用于参数可分离系统的参数辨识.由于采用了按段低阶正交多项式多重逼近技术,该方法具有计算量少、结果精度高、可递推计算及不需要被辨识参数的初始估计等优点.本文提出了两个算法,并成功地应用于发酵过程细菌生长动力学模型的参数辨识.     

多元Chebyshev正交多项式混合模型及其在医学图像分割中的应用

针对原有一元正交多项式混合模型只能根据灰度特征分割图像的问题,提出一种基于多元Chebyshev正交 多项式混合模型的多维特征的医学图像分割方法。首先,根据Fouricr分析方法与张量积理论推导出图像的多元 Chcbyshcv正交多项式,并构建多元正交多项式的非参数混合模型,用最小均方差(MISE)估计每一个模型的平滑参 数;然后,用EM算法求解正交多项式系数和模型的混合比。此方法不需要对模型作任何假设,可以有效克服“模型失 配”问题。通过实验,表明了该分割方法的有效性。     

Solutions of convolution integral and integral equations via double general orthogonal polynomials

Double general orthogonal polynomials are developed in this work to approximate the solutions of convolution integrals, Volterra integral equations, and Fredholm integral equations. The proposed method reduces the computations of integral equations to the successive solution of a set of linear algebraic equations in matrix form; thus, the computational complexity is considerably simplified. Furthermore, the solutions obtained by the general orthogonal polynomials include as special cases solutions by Chebyshev polynomials, Legendre polynomials, Laguerre polynomials, or Jacobi polynomials. A comparison of the results obtained via several different classical orthogonal polynomial approximations is also presented.     

A continuous method model for solving general variational inequality

Based on the projection operator, this paper presents a continuous method model for solving general variational inequality problems (VIPs) with bound constraints. A main feature of the proposed model is that it does not involve any form of matrix information in analysing its convergence properties. Under some reasonable assumptions, the convergence results of the proposed method model are established. Numerical results on some problems show that the proposed approach is efficient and can be applied to solve large scale VIPs with bound constraints.     

Solution of discrete scaled systems via general discrete orthogonal polynomials

General discrete orthogonal polynomials are introduced to analyse and approximate the solution of a class of discrete scaled systems. Using the general discrete shift transformation matrix, together with the general discrete scale matrix, the discrete scaled system can be reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of general discrete orthogonal polynomials can be determined simply by the derived algorithm. Examples are included to show the applicability of the general discrete orthogonal polynomial approximations.