Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.     

Application of discrete Chebyshev polynomials to the optimal control of digital systems

The shift transformation matrix for discrete Chebyshev polynomials is introduced in this study. The discrete variational principle combined with the idea of penalty function is taken to construct the modified discrete Euler-Lagrange equations. Then, the discrete Chebyshev series are applied to simplify the modified equations into a set of linear algebraic ones for the approximations of state and control variables of digital systems. It is seen that this technique is quite straightforward and simple, and computing time can be saved considerably.     

Functional approximation for inversion of Laplace transforms via polynomial series

A method for finding the inverse of Laplace transforms using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into Taylor polynomials by use of a suitable transformation. In this paper, the cross product of a polynomial series basis vector is derived in terms of Taylor polynomials, and as a result the inverse of the Laplace transform is obtained, using the most commonly used polynomial series such as Legendre, Chebyshev, and Laguerre. Properties of Taylor series are first briefly presented and the required function is given as a Taylor series with unknown coefficients. Each Laplace transform is converted into a set of simultaneous linear algebraic equations that can be solved to evaluate Taylor series coefficients. The inverse Laplace transform using other polynomial series is then obtained by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are also given,     

递阶控制最优化的新方法

本文基于Ｃｈｅｂｙｓｈｅｖ级数给出了一种大系统递阶控制新方法，主要思想是将微分方程转化成代数方程，整个算法简单，方便，文中用一个例子说明了该方法的有效性。     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

基于Chebyshev小波的分布参数系统辨识

工程实际和社会系统中广泛存在着分布参数系统,因而研究分布参数系统的辨识与控制具有重要意义.但由于其复杂性,对分布参数系统的辨识研究十分困难.借助于Chebyshev多项式的逼近性质,以及小波的时频特性,构造了Chebyshev小波,并利用其积分运算矩阵,运用于分布参数系统的辨识,从而将一类分布参数系统的辨识问题转化为一般代数问题.并且考虑了初始条件和边界条件对辨识结果的影响,因此具有较好的适用性,仿真结果证实了该方法的有效性.     

Error analysis of the Chebyshev series solution of linear time invariant systems

The first part of this paper calculates the error between a function and its m-term Chebyshev series expansion. The second part calculates the error between the integral of a function and its m-term Chebyshev series obtained using the usual operational matrix of integration. The final part derives the error between the exact solution of a state equation and the m-term Chebyshev series solution obtained by the Chebyshev polynomial method.     

Minimization of a closed-loop response to a fixed input for SISOsystems

It is shown that the problem of minimizing a regulated response of a single-input/single-output system due to a fixed bounded input can be converted, via polynomial techniques, to a linear infinite-dimensional Chebyshev data fitting problem. Approximating feasible solutions within any specified degree of accuracy can be obtained by converting the original problem into a sequence of increasingly large, finite-dimensional Chebyshev approximation problems, for which solution stable and efficient numerical methods exist. A direct formula for calculating tight upper-bounds to the approximation error is provided. The link between the present algebraic approach and the Dahleh and Pearson functional analytic one (1988) is also discussed     

Solution of linear two-point boundary-value problems via shifted Chebyshev series

In this paper, the shifted Chebyshev polynomial functions approximation is extended to solve the linear ordinary differential equation of the two-point boundary-value problem. The linear ordinary differential equation of boundary-value problems are reduced to the linear functional differential equation of the initial-value problem. A new time-domain approach to the derivation of a Chebyshev transformation matrix is presented. Using the derived Chebyshev transformation matrix together with the Chebyshev integration matrix, the solution of the linear functional ordinary differential equation of initial-value problem can be obtained via shifted Chebyshev series. Two examples are given and the satisfactory computational results are compared with those of the exact solution.     

Solutions of time-varying TS-fuzzy-model-based dynamic equations using a shifted Chebyshev series approach

In this paper, the approach of a shifted Chebyshev series is proposed to solve the time-varying Takagi–Sugeno (TS) fuzzy-model-based dynamic equations. The new method simplifies the procedure of solving the time-varying TS-fuzzy-model-based dynamic equations into the successive solution of a system of recursive formulae only involving the matrix algebra. Based on the presented recursive formulae, an algorithm involving only straightforward algebraic computation is also proposed in this paper. The new proposed approach is non-iterative, non-differential, non-integral, straightforward, and well adapted to the computer implementation. The computational complexity can therefore be reduced remarkably. The first illustrated numerical example shows that the proposed method based on the shifted Chebyshev series can yield better and more satisfactory results than the Runge–Kutta method. The second illustrated example for the pendulum system with the vibration in the vertical direction on the pivot point is given to demonstrate the application of the proposed approach.     

Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions

A numerical technique is presented for the solution of fourth-order integro-differential equations. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

Predictive control for a class of distributed delay systems using Chebyshev polynomials

In this paper, the model predictive control (MPC) is developed for linear time-varying systems with distributed time delay in state. The Chebyshev operational matrices of product, integration and delay are utilized to transform the solution of distributed delay differential equation to the solution of algebraic equations. The Chebyshev functions are also applied to derive approximate solution of finite horizon optimal control problem involved in MPC. The proposed method is simple and computationally advantageous. Illustrative example demonstrates the validity and applicability of the technique.     

基于Chebyshev神经网络的图像复原算法

退化图像的点扩散函数难以准确确定,为此,提出一种基于Chebyshev正交基函数的前向神经网络图像复原算法。该算法以一组Chebyshev正交基为隐层神经元的激励函数,采用BP算法对权值进行修正,达到收敛目标。给出2类Chebyshev神经网络的实现步骤及其相应衍生算法的图像恢复实现步骤。实验结果表明,该算法能较好地实现图像复原。     

Analysis of linear time-varying systems and bilinear systems via shifted Chebyshev polynomials of the second kind

Linear time-varying systems and bilinear systems are analysed via shifted Chebyshev polynomials of the second kind. Using the operational matrix for integration and the product operational matrix, the dynamical equation of a linear time-varying system (or bilinear system) is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of shifted Chebyshev polynomials of the second kind can be determined by using the least-squares method. Illustrative examples show that shifted Chebyshev polynomials of the second kind having a finite number of terms are more accurate than either the Legendre or Laguerre methods.     

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.     

Uncertain Method for Optimal Control Problems With Uncertainties Using Chebyshev Inclusion Functions

In this paper, a new uncertain analysis method is developed for optimal control problems, including interval variables (uncertainties) based on truncated Chebyshev polynomials. The interval arithmetic in this research is employed for analyzing the uncertainties in optimal control problems comprising uncertain‐but‐bounded parameters with only lower and upper bounds of uncertain parameters. In this research, the Chebyshev method is utilized because it generates sharper bounds for meaningful solutions of interval functions, rather than the Taylor inclusion function, which is efficient in handling the overestimation derived from the wrapping effect due to interval computations. For utilizing the proposed interval method on the optimal control problems with uncertainties, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, by using some algebraic manipulations, they are converted into the ordinary differential equation. Afterwards, the Chebyshev inclusion method is employed to achieve the solution of the system. The final results of the Chebyshev inclusion method are compared with the interval Taylor method. The results show that the proposed Chebyshev inclusion function based method better handle the wrapping effect than the interval Taylor method.     

Eliciting dependability requirements: a control cases based approach

At present,great demands are posed on software dependability.But how to elicit the dependability requirements is still a challenging task.This paper proposes a novel approach to address this issue.The essential idea is to model a dependable software system as a feedforward-feedback control system,and presents the use cases+control cases model to express the requirements of the dependable software systems.In this model,while the use cases are adopted to model the functional requirements,two kinds of control cases(namely the feedforward control cases and the feedback control cases)are designed to model the dependability requirements.The use cases+control cases model provides a unified framework to integrate the modeling of the functional requirements and the dependability requirements at a high abstract level.To guide the elicitation of the dependability requirements,a HAZOP based process is also designed.A case study is conducted to illustrate the feasibility of the proposed approach.