Optimal control of linear distributed-parameter systems via polynomial series

A method for finding the optimal control of linear distributed-parameter systems using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into a Taylor polynomial by the use of a suitable transformation. In this paper, the optimal control of a distributed-parameter system is simplified into the solution of a linear two-point boundary value problem, and, as a result, the optimal control is obtained via a Taylor series. It is shown that the implementation of Taylor series for this problem involves the use of an ill-conditioned matrix commonly known as the Hilbert matrix. The optimal control of linear distributed-parameter systems using other polynomial series is then calculated by transforming the properties of the Taylor series into other polynomial series. The formulation is straightforward and convenient for digital computation. An illustrative example is given.     

The operational matrix of polynomial series transformation

This paper introduces the operational matrix of polynomial series transformation T that may be applied to transform any polynomial series basis vector to the Taylor polynomials. The matrix is determined for most commonly used polynomial series expansions, such as the Chebyshev, the Laguerre, the Legendre and the Hermite. Using the polynomial series transformation matrix, the corresponding operational matrix of integration of a polynomial series, may easily be determined. Finally, it is shown that all the approximate methods using polynomial-based operational matrices of integration may be connected to the Taylor-series method.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.     

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.     

基于约束Jacobi基的多项式反函数逼近及应用

求解多项式反函数是CAGD中的一个基本问题.提出一种带端点Ck约束的反函数逼近算法.利用约束Jacobi基作为有效工具, 推导了它与Bernstein基的转换公式,采用Bernstein多项式的升阶、乘积、积分与组合运算, 给出了求解反函数系数的具体算法.该算法稳定、简易, 克服了以往计算反函数的系数时每次逼近系数需全部重新计算的缺陷.最后通过具体逼近实例验证了文中算法的正确性和有效性, 同时给出了它在PH曲线准弧长参数化中的应用.     

Solution of integral equations via generalized orthogonal polynomials

A new approximation method using a generalized orthogonal polynomial (GOP) is employed for solving integral equations. The integration operational matrix of the GOP, which can represent all kinds of individual orthogonal polynomial, is developed. The dependent variables in the integral equation are assumed to be expressed by a GOP series. A set of algebraic equations is obtained from the integral equation. The calculation of coefficients is straightforward and easy. Examples are given, and the results obtained from individual orthogonal polynomial approximations are compared with each other. It is found that nearly all individual orthogonal polynomials, except Hermite polynomials, offer excellent results.     

Analysis and parameter estimation of bilinear systems via generalized orthogonal polynomials

An effective method of using generalized orthogonal polynomials (GOP) for analysing and identifying the parameters of a process whose behaviour can be modelled by a bilinear equation is presented. The integration operational matrix and the operational matrix for the product of tⁱ with the GOP vector are derived. These two kinds of operational matrices of the GOP are related to any type of individual orthogonal polynomial. By expanding the state and control functions into a series of GOP, the bilinear equation can be converted into a set of linear algebraic equations. The expansion coefficients of state variables are solved from these linear algebraic equations. The unknown parameters are evaluated by using the least squares method in conjunction with the individual orthogonal polynomial expansion. Two examples are given to illustrate the validity of the method. Very satisfactory results are obtained.     

Solution of a scaled system via generalized orthogonal polynomials

Generalized orthogonal polynomials which include all types of orthogonal polynomial are introduced first. Using the idea of orthogonal polynomials that can be expressed by a Taylor power series and vice versa, the operational matrix for the integration of the generalized orthogonal polynomials is first derived. A stretched operational matrix of diagonal form is also derived. Both the operational matrix for the integration and the stretched operational matrix of generalized orthogonal polynomials are applied to solve functional differential equations. The characteristics of each kind of orthogonal polynomial in solving the scaled system is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The inversion of only one matrix, which has the same dimension as the state variables, is required. The expansion coefficients of the state variables are obtained by the proposed recursive formula. Much computer time is thus saved and computational results are obtained that are very accurate compared with previous methods.     

The Use of the Ritz Method and Laplace Transform for Solving 2D Fractional‐Order Optimal Control Problems Described by the Roesser Model

In this article a numerical solution is presented for a class of two‐dimensional fractional‐order optimal control problems (2D‐FOOCPs) with one input and two outputs. To implement the numerical method, the Legendre polynomial basis is used with the aid of the Ritz method and the Laplace transform. By taking the Ritz method as a basic scheme into account and applying a new constructed fractional operational matrix to estimate the fractional and integer order derivatives of the basis, the given 2D‐FOOCP is reduced to a system of algebraic equations. One of the advantages of the proposed method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, satisfactory results are obtained in just a small number of polynomials order. The convergence of the method is extensively investigated and finally two illustrative examples are included to show the validity and applicability of the novel proposed technique in the current work.     

Analysis of nonlinear circuits by using differential Taylor transform

An effective but not widely used method, differential Taylor transform, is introduced for the analysis of the nonlinear electrical circuits. To apply the method the differential transform of the mathematical model of the system is obtained first, and then the response function is evaluated by using the inverse transform of the differential spectra. The inverse transform can be written in the form of Taylor series. The method is described with two examples for nonlinear electrical circuits.     

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.     

基于多尺度边缘表示的图像增强快速算法

低对比度结构广泛存在于各种数字图像之中，研究如何通过后期处理增强数字图像的对比度是很有意义的。灰度图像对比度的高低总是与图像灰度梯度幅值的大小相联系，受这种思想的启发，提出了一种基于图像多尺度边缘表示的，利用对信号小波变换模极大值的拉伸和Hermite插值多项式实现的图像增强快速算法。此算法可以实现对噪声的抑制和对图像中不同尺度特征的增强。数值实验结果表明，该算法增强效果明显，运算速度快，是一种实用性较强的图像对比度增强算法。     

Robustness analysis for linear dynamical systems with linearlycorrelated parametric uncertainties

The authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties. Linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered. A design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given. The proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space, preserving system poles in the given region. The solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space. It is proved that for stability problems for continuous-time systems with independent perturbations on polynomial coefficients, this method generates the four extreme Kharitonov polynomials     

Reply to remarks on the paper ‘A direct method for computing optimal feedback control for linear systems’

The method or generalized orthogonal polynomials (GOP) is applied to the analysis and optimal control of time-varying systems. The proposed GOP can represent all kinds of individual orthogonal-polynomial and non-orthogonal Taylor series. The operational matrix for the forward and backward integration of the generalized orthogonal polynomials, and the operational matrix of the product of r' and the generalized orthogonal-polynomial vector are derived and applied to time-varying systems. By using these three kinds of operational matrices, the computational algorithm for calculating the expansion coefficients is very simple and effective. Three satisfactory examples illustrate the usefulness of the method.     

Early Automatic Differentiation: The Ch’in-Horner Algorithm

A method for numerical solution of polynomial equations appeared in the 1247 a.d. book The Nine Sections of Mathematics, by Ch’in Kiu-shao. This procedure was rediscovered independently by W. G. Horner [1773–1827] at the beginning of the 19th century. Since their algorithm produces values of Taylor coefficients of polynomials, it can be viewed as an early example of automatic differentiation. For polynomials, their method is shown to be computationally equivalent to automatic generation of Taylor coefficients as introduced by R. E. Moore in 1962 for use on digital computers.     

Generalization of generalized orthogonal polynomial operational matrices for fractional and operational calculus

A method is given for the approximation of generalized orthogonal polynomials (GOP) to solve the problems of fractional and operational calculus. A more rigorous derivation for the generalized orthogonal polynomial operational matrices is proposed. The Riemann-Liouville fractional integral for repeated fractional (and operational) integration is integrated exactly, then expanded in generalized orthogonal polynomials to yield the generalized orthogonal polynomial operational matrices. The generalized orthogonal polynomial operational matrices perform as s^α(α ≥ αε R) in the Laplace domain and as fractional (and operational) integrators in the time domain. Using these results, inversions of the Laplace transforms of irrational and rational transfer functions are solved in a simple way. Very accurate results are obtained.     

Artificial neural networks based modeling for solving Volterra integral equations system

Properly designing an artificial neural network is very important for achieving the optimal performance. This study aims to utilize an architecture of these networks together with the Taylor polynomials, to achieve the approximate solution of second kind linear Volterra integral equations system. For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system. Then we apply the suggested neural net for adjusting the numerical coefficients of given expansions in resulting system. Consequently, the reported architecture using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. The proposed method was illustrated by several examples with computer simulations. Subsequently, performance comparisons with other developed methods was made. The comparative experimental results showed that this approach is more effective and robust.     

Embedded image coding using laplace transform for Turkish letters

In this paper, a different cryptographic method is introduced by using a Power series transform. A new algorithm for cryptography is produced. The extended Laplace transform of the exponential function is used to encode an explicit text. The key is generated by applying the modular arithmetic rules to the coefficients obtained in the transformation. Here, ASCII codes used to hide the mathematically generated keys to strengthen the encryption. Text steganography is used to make it difficult to break the password. The made encryption is reinforced by image steganography. To hide the presence of the cipher text, it is embedded in another open text with a stenography method. Later, this text is buried in an image. For decryption, it is seen that the inverse of the Power series transform can be used for decryption easily. Experimental results are obtained by making a simulation of the proposed method. As a result, it is stated that the proposed method can be used in crypto machines.

     

用多项式近似的图象逆滤波及空间移变系统图象的恢复

光(电)成像系统的特性会引起图象降质，但如果能够根据系统的传递函数确定其逆滤波函数，就可以对这种降质图象进行一定的恢复．为此，提出了一种用多项式近似的图象逆滤波的图象恢复方法，该方法就是首先将连续的逆滤波函数按泰勒级数展开，并用多项式来近似表示，通过对用多项式表达的用于图象恢复的逆滤波函数作反傅里叶变换，就可得到恢复图象在空间域中的近似运算公式，该运算是图象信号及其各阶导数的线性组合，而不是复杂的反卷积操作．同时还详细分析了方法的原理，并推导了算法公式，最后给出了空移不变和移变系统图象的恢复处理结果．实验表明，该方法特别适合于空间移变系统降质图象的恢复，如场曲恢复．     

New analytical method for gas dynamics equation arising in shock fronts

This work suggests a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving nonlinear homogeneous and nonhomogeneous time-fractional gas dynamics equations. The FHATM is an innovative adjustment in Laplace transform algorithm (LTA) and makes the calculation much simpler. The proposed technique solves the nonlinear problems without using Adomian polynomials and He’s polynomials which can be considered as a clear advantage of this new algorithm over decomposition and the homotopy perturbation transform method. In this paper, it can be observed that the auxiliary parameter ?$?$, which controls the convergence of the HATM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more qualitative difference in analysis between HATM and other methods. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. The proposed method is illustrated by solving some numerical examples.